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Full text of "Radio meteorology"



f^-^.^1 






UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary 

NATIONAL BUREAU OK STANDARDS • A. V. Astin, Director 



Radio Meteorology 



B. R. Bean and E. J. Dutton 

Central Radio Propagation Laboratory* 

National Bureau of Standards 

Boulder, Colorado 



*The CRPL was transferred to the 

Enviromental Science Services Administration 

while this Monograph was in press. 




National Bureau of Standards Monograph 92 
Issued March 1, 1966 



Yor sale by the Superintendent of Documents, U. S. Government Printing Office 
Washington, D. C. 20402 Price ?2.75 



Library of Congress Catalog Card No. 65-60033 



Foreword 

This volume brings together the work done in radio meteorology 
over the past ten years at the National Bureau of Standards' Central 
Radio Propagation Laboratory (CRPL). This decade has seen the 
development, on an international scale, of great emphasis upon the effects 
of the lower atmosphere on the propagation of radio waves. The CRPL 
group has concentrated upon the refraction of radio waves as well as the 
refractive index structure of the lower atmosphere on both synoptic and 
climatic scales. These are the areas of radio meteorology that are treated 
in this volume, with additional chapters on radio-meteorological param- 
eters and the absorption of radio waves by the various constituents of the 
lower atmosphere. An effort has been made to include results obtained 
in other laboratories both in the United States and abroad. 

A. V. AsTiN, Director. 



Preface 

The authors express their gratitude and appreciation to the many 
members of the Central Radio Propagation Laboratory staff and others 
with whom they have had the pleasure of working. This volume would 
not have been possible without their assistance, encouragement, and 
stimulation. Particular acknowledgements are due : R. L. Abbott, Mrs. 
L. Bradley, Miss B. A. Gaboon, C. B. Emmanuel, L. Fehlhaber,* V. R. 
Frank, Miss S. Gerkin, Mrs. B. J. Gibson, J. Grosskopf,* J. W. Herbstreit, 
J. D. Horn, J. A. Lane,t J. R. Lebsack, R. E. McGavin, Miss F. M. 
Meaney, C. M. Miller, K. A. Norton, A. M. Ozanich, Jr., the late Mrs. 
G .E. Richmond, L. P. Riggs, P. L. Rice, C. A. Samson, W. B. Sweezy, 
G. D. Thayer, Mrs. B. J. Weddle, E. R. Westwater, Mrs. P. C. Whittaker 
and W. A. Williams. 

B. R. Bean 
E. J. Button 



*Fernmeldetechnisches, Darmstadt, Germany. 
tRadio Research Station, Slough, England. 



Contents 

Foreword iii 

Preface iv 

Chapter 1. The radio refractive index of air 1 

1.1. Introduction 1 

1.2. Dielectric constant of moist air 2 

1.3. Constants in the equation for A'^ 4 

1.4. Errors in the practical use of the equation for A^ 9 

1.5. Presentation of A^ data 13 

1.6. Conclusions 20 

1.7. References 21 

Chapter 2. Measuring the radio refractive index 23 

2.1. The measurement of the radio refractive index 23 

2.2. Indirect measurement of the radio refractive index 23 

2.3. Direct measurement of the refractive index 30 

2.4. Comparison between the direct and indirect methods of measure- 

ment 38 

2.5. Radiosonde lag constants 38 

2.5.1. Introduction 38 

2.5.2. Theory of sensor time lags 29 

2.5.3. Radiosonde profile analysis 41 

2.5.4. Conclusion 43 

2.6. References 45 

Chapter 3. Tropospheric refraction 49 

3.1. Introduction 49 

3.2. Limitations to radio ray tracing 52 

3.3. An approximation for high initial elevation angles 53 

3.4. The statistical method 54 

3.5. Schulkin's method 54 

3.6. Linear or effective earth's radius model 56 

3.7. Modified effective earth's radius model 59 

3.8. The exponential model 65 

3.9. The initial gradient correction method 77 

3.10. The departures-from-normal method 77 

3.11. A graphical method 80 

3.12. Derivations 1 82 

3.13. References 87 

Chapter 4, N climatology 89 

4.1. Introduction 89 

4.2. Radio-refractive-index climate near the ground 89 

4.2.1. Introduction 89 

4.2.2. Presentation of basic data 89 

4.2.3. Worldwide values of A^o 98 

4.2.4. Climatic classification by A^, 102 

4.2.5. Applications 105 

4.2.6. Critical appraisal of results 107 

4.2.7. Conclusions 108 



4.3. On the average atmospheric radio refractive index structure over 

North America 109 

4.3.1. Introduction 109 

4.3.2. Meteorological data and reduction techniques 109 

4.3.3. Average A'^o structure 111 

4.3.4. Continental cross sections 122 

4.3.5. Delineation of climatic characteristics 129 

4.4. The climatology of ground-based radio ducts 132 

4.4.1. Introduction 132 

4.4.2. Meteorological conditions associated with radio 
refractive index profiles 132 

4.4.3. Refractive conditions due to local meteorological 
phenomena 134 

4.4.4. Background 135 

4.4.5. Description of observed ground-based atmospheric 

ducts 140 

4.5. A study of fading regions within the horizon caused by a surface 

duct below a transmitter 146 

4.5.1. Introduction 146 

4.5.2. Regions and extent of fading within the horizon in 

the presence of superref raction 147 

4.5.3. Theory and results 149 

4.5.4. Sample computations 157 

4.6. Air mass refractive properties 163 

4.6.1. Introduction 163 

4.6.2. Refraction of radio rays 164 

4.6.3. Conclusions 170 

4.7. References 170 

Chapter 5. Synoptic radio meteorology 173 

5.1. Introduction 173 

5.2. Background 173 

5.3. Refractive index parameters 185 

5.4. A synoptic illustration 195 

5.5. Surface analysis in terms of A''o 197 

5.6. Constant pressure chart analysis 205 

5.7. Vertical distribution of the refractive index using A units 211 

5.8. Summary 223 

5.9. References 224 

Chapter 6. Transhorizon radio-meteorological parameters 229 

6.1. Existing radio-meteorological parameters 229 

6.1.1. Introduction 229 

6.1.2. Parameters derived from the A'^ profile 233 

6.1.3. Parameters involving thermal stability 238 

6.1.4. Composite parameter 238 

6.1.5. Potential refractive index (or modulus), K 239 

6.1.6. Vertical motion of the atmosphere 240 

6.1.7. Discussion of the parameters 241 

6.1.8. Comparison of some parameters 243 

6.1.9. Some exceptions and anomalies 243 

6.1.10. Conclusions 246 



6.2. An analysis of VHF field strength variations and refractive index 
profiles 247 

6.2.1. Introduction 247 

6.2.2. Radio and meteorological data 247 

6.2.3. Classification of radio field strengths by profile types. 248 

6.2.4. Prediction of field strength for unstratified conditions. 250 

6.2.5. The effect of elevated layers on the Illinois paths 251 

6.2.6. Elevated layers at temperature inversions 252 

6.2.7. The influence of small layers 255 

6.2.7. Conclusions 257 

6.3. A new turbulence parameter 258 

6.3.1. Introduction 258 

6.3.2. The concept of thermal stability 258 

6.3.3. The adiabatic lapses of TV 261 

6.3.4. The turbulence parameter, II 262 

6.3.5. Comparison of radio-meteorological parameters 263 

6.3.6. Conclusion 266 

6.4. References 266 

Chapter 7. Attenuation of radio waves 269 

7.1. Introduction 269 

7.2. Background 269 

7.3. Attenuation by atmospheric gases 270 

7.4. Estimates of the range of total gaseous absorption 280 

7.5. Total radio path absorption 283 

7.6. Derivation of absorption estimate for other areas 286 

7.7. Attenuation in clouds 291 

7.8. Attenuation by rain 292 

7.9. Rainfall attenuation climatology 297 

7.10. Rain attenuation effects on radio systems engineering 298 

7.11. Attenuation by hail 302 

7.12. Attenuation by fog 303 

7.13. Thermal noise emitted by the atmosphere 304 

7.14. References 308 

Chapter 8. Applications of tropospheric refraction and refractive index 

models 311 

8.1. Concerning the bi-exponential nature of the tropospheric radio 
refractive index 311 

8.1.1. Introduction and background 311 

8.1.2. TV structure in the I.e. A. O. atmosphere 312 

8.1.3. Properties of the dry and wet term scale heights 314 

8.1.4. Refraction in the bi-exponential model 32 1 

8.1.5. Extension to other regions 321 

8.2. Effect of atmospheric horizontal inhomogeneity upon ray tracing. _ 322 

8.2.1. Introduction and background 322 

8.2.2. Canterbury 323 

8.2.3. Cape Kennedy 324 

8.2.4. Ray bending 326 

8.2.5. Comparisons 328 

8.2.6. Extension to other regions 331 

8.2.7. Conclusions 333 



8.3. 



8.4. 



8.5. 
Chapter 9. 

9.1. 
9.2. 

9.3. 
9.4. 
9.5. 
9.6. 



Comparison of observed atmospheric radio refraction effects with 
values predicted through the use of surface weather observations __ 

8.3.1. Introduction 

8.3.2. Theory 

8.3.3. The CRPL standard atmospheric radio refractive 

index profile sample 

Comparison with independent data 

Comparison with experimental results 

Discussion of results 

of atmospheric refraction errors in radio height finding. 

Introduction 

Refractive index 

Ray theory 

Use of the effective earth's radius 

Meteorological parameters 

Calculation and correlation of height errors 

Estimation of the average gradient 

Regression analysis 

Height error equations 

Conclusions 



8.3.4 

8.3.5 

8.3.6 
Correction 

8.4.1 

8.4.2 

8.4.3 

8.4.4 

8.4.5 

8.4.6 

8.4.7 

8.4.8 

8.4.9 

8.4.10. 

References 

Radio-meteorological charts, graphs, tables, and sample com- 
putations 

Sample computations of atmospheric refraction 

Tables of refraction variables for the exponential reference 

atmosphere 

Ref ractivity tables 

Climatological data of the refractive index for the United States _ . 

Statistical prediction of elevation angle error 

References 



333 
333 
334 

340 
342 
344 
353 
356 
356 
357 
357 
360 
362 
363 
363 
364 
372 
373 
373 

375 
375 

393 
396 
404 
404 
423 



Chapter 1. The Radio Refractive 
Index of Air 

1.1. Introduction 

The last few decades have seen a remarkable increase in the practical 
utilization of the radio spectrum above 30 Mc/s. This, in turn, has 
focussed attention upon the mechanisms by which these radio waves are 
propagated. Since radio energy at these frequencies is not normally 
reflected by the ionosphere, variability in the characteristics of the re- 
ceived fields is attributed to variations in the lower atmosphere and, in 
particular, to the radio refractive index. 

The radio refractive index is central to all theories of radio propagation 
through the lower atmosphere. The atmosphere causes a downward 
curvature of horizontally launched radio waves which is normally about 
one quarter of that of the earth. Under unusual meteorological con- 
ditions, however, the radio energy may be confined to thin layers near 
the earth's surface with resultant abnormally high field strengths being 
observed beyond the normal radio horizon. At other times a transition 
layer between differing air masses will give rise to the reflection of radio 
energy. In addition to these gross profile effects, the atmosphere is 
always more or less turbulent, with the result that radio energy is scattered 
out of the normal antenna pattern. 

It is not the purpose of the present discussion to emphasize the inter- 
play of various propagation mechanisms, as has been done, for example, 
in a classic paper by Saxton [1]', but rather to emphasize that the refrac- 
tive index of the troposphere is of central concern in the propagation of 
radio waves at frequencies above 30 Mc/s. In what follows, then, the 
classical equation for the radio refractive index will be considered, a sum- 
mary of the recent determinations of the constants in this equation given, 
the errors in the practical use of the equation will be analyzed, and, finally, 
the normal gross features of atmospheric refractive index structure will 
be described. 

1.2. Dielectric Constant of Moist Air 

Debye [2] has considered the effect of an impressed electric field upon 
the dielectric constant of both non-polar and polar molecules, a polar 



1 Figures in brackets indicate the literature references on p. 21. 



2 RADIO REFRACTIVE INDEX OF AIR 

molecule being one with a strong permanent dipole moment. He derives 
the same result from both classical and quantum mechanical methods; 
i.e., the polarizability is composed of two effects: one due to the distortion 
of all of the molecules by the impressed field and the second arising from 
an orientation effect exerted upon polar molecules. The polarization, P, 
of a polar liquid under the influence of a high-frequency radio field is 
given by 



P{o:) 



6 - 1 M 

£ + 2 p 



47riV 



ao + 



1 



3/cr 1 + iojrj 



(1.1) 



where : e is the dielectric constant, 

M is the molecular weight, 

p is the density of the liquid, 

A^ is Avogadro's number, 

ao is the average polarizability of the molecules in the liquid, 
assuming no interaction between molecules, 

M is the permanent dipole moment, 

k is Boltzmann's constant, 

T is the absolute temperature, 

T is the relaxation time required for external field-induced orien- 
tations of the molecules to return to random distribution 
after the field is removed, 

CO = 2'wf where / is the frequency of the external field. 

One concludes from Debye's analysis that for external fields with fre- 
quencies less than 100 Gc/s, wr « 1 with the result that (1.1) is written^ 



€ - 1 M 


A^ttN 


2 


e + 2 p " 


3 


L""" ^ ^kTA 



(1.2) 



The dispersive effect of the 22.5 Gc/s water-vapor absorption line is not 
expected to be important below 30 Gc/s although, as shall be seen, experi- 
mental evidence indicates some dispersion due to the 60 Gc/s absorption 



^ The effect of the relaxation time upon interpretation of measurements of dielectric 
constant has been discussed by Saxton [1]; he has also given a description of measure- 
ment techniques involved in the determination of dielectric constant during the war 
years [3]. 



DIELECTRIC CONSTANT OF MOIST AIR 3 

line of oxygen. For non-polar gases (n = 0), this equation becomes 

e - 1 M _ 4TNao 

6 + 2 p " 3 ^^-"^^ 

which is well approximated by 

e - 1 ^ ^ 4:irNao (1.4) 

for gases at low pressures. 

This equation may be rewritten, by assuming the perfect gas law, as 

e-l =K[^ (1.5) 

where K[ is a constant. The result for polar gases, (1.2), may be written 

(1.6) 
which, assuming (1.5), can be rewritten as 

e- 1 ::=K^|(a +|) (1.7) 

where K'^, A, and B are constants. For a mixture of gases, Dalton's 
law of partial pressures is assumed to hold with the result that we can 
sum the effects of polar and non-polar gases and hence obtain 



e - 1 =« 1^ 47riV 
M 



2 

"" + SkTA 



— 1 = 2] K'li -^ + 22 ^20 i^[Ag-{--~]- (1 



•8) 



For the troposphere, however, we need only consider the effects of 
CO2, dry air (non-polar gases), and water vapor (a polar gas) such that 

e - 1 = K,\ :^ + AV, I ( A + I ) + K{, ^ (1.9) 

where Pd is the pressure of dry air, c is the partial pressure of water vapor, 
and Pc the partial pressure of CO2. The equation for refractive index, n, is 
obtained using the expression n = a/m^, where ^i, the permeability, may 



4 RADIO REFRACTIVE INDEX OF AIR 

be assumed to be approximately unity for air. Since 



n = \/l + (/xe - 1) , 
one may employ the approximation 

and obtain the familiar form of practical application^ 

N = (n - 1) 10*' = K, ^ + K, -- + K,jr^-\-K,^ (1.10) 
where A'l, A'o, A'3, and A'4 are constants. 

1.3. Constants in the Equation for N 

A survey of the various deterniinations of Ai, A2, and A3 was recently 
made by Smith and Weintraub [4] to arrive at a set of "best values" to 
represent the mean of previous independent determinations. In radio 
work one is interested in propagation through the troposphere, therefore, 
the composition of air should be taken to include an average amount of 
carbon dioxide. However, laboratory measurements usually are made 
on C02-free air due to variable concentrations of CO2 in the laboratory. 
Hence Smith and Weintraub have adjusted the the values of e — 1 
originally published for C02-free air by raising them to 0.02 percent to 
correspond to a 0.03 percent CO2 content. It should be noted that this 
correction of 0.02 percent is essentially of academic interest, since as 
shall be seen, the final equation for A'^ will be considered accurate to 0.5 
percent. The value of 0.02 percent was obtained by noting that the 
value of dielectric constant for C02-free air, e', in the expression 

e - 1 = — 2^ 

could be utilized for applications in the real atmosphere by expressing 
total atmospheric pressure, P t, as 

Pt = Pd-{- Pc 



5 Henceforth, A^ will be used to denote {n — 1)10^ and will not be used again for 
Avogadro's number. This slight inconsistency was adopted to maintain notational 
agreement with both Debye's early work and the later work in radio meteorology. 



CONSTANTS IN EQUATION FOR A^ 5 

By assuming a constant COi content of 0.03 percent one obtains, from 
Dalton's law of partial pressures, 



Pd = (1 - 3 X 10"') Pt 



and 



P. = 3 X lO-'P, 



with the result that the dielectric constant for the real, dry atmosphere, 
e, may be written 

. K,{\ - 3 X 10~^)P, ^ A' 4(3 X 10~') Pt 

€ - 1 = ~ \ . 



One may then adjust measurements of C02-free air upwards by 
\rTz\ ~ 1 10' = 3 X 10"'(^ - l) percent ^ 0.02 percent, 



since Ki/Ki ^^ 5/3. Such values are given in table 1.1 on a real, rather 
than an ideal, gas basis. The first determination shown, that of Barrell 
[5], is an average of the constant term (n for X == oo ) of the optical Cauchy 
dispersion equations for standard air used in three of the principal 
measurement laboratories of the world. Theoretical considerations indicate 
that the dielectric constant for dry air will be the same for optical and 
radio frequencies. Barrell's value is converted to dielectric constant 
from the relationship \/m« with ju, the permeability, taken as unity at 
optical frequencies. Unless otherwise noted, the standard error is used 
here and throughout the remainder of this discussion. 



Table 1.1. Dry air refractive index and dielectric constant at 0°C and 1 atm 



Reference 


Frequency of 
measurement 


Measured 

N 


(« - 1) 10« 


Year 


Barrel! [5] 


Optical'' 


287. 7 ± 0. 


575. 6 ± 0. 1 


1951 


Birnbaum,Kryder,and Lyons[40] 


9,000 Mc/s 




575.8 ±0.3 


1951 










Essen and Froome (11] _. 


24,000 Mc/s 


288. 2 ± 0. 1 


576. 1 ± 0. 2 


1951 






Mean value of {« — 1) 10^ 
Mean value of N^' 




288. 0. ± 0. 


575. 7 ± 0. 1 





» Maryott and Buckley [6] have determined a mean value for the dielectric constant from the simple 
average of eight different determinations that, when adjusted to the pressure, temperature, and value of/x 
assumed in table 1.1, yield a value of N = 287. 7 ± 0. 15, which is in agreement with Barren's value. 



•> Derived from ;* 
meability. 



Vm* where m — 1 = 0. 4 X 10"'' is taken for radio froquencies to account for the per- 



6 RADIO REFRACTIVE INDEX OF AIR 

The statistical mean of the dielectric constant is then converted to 
refractive index. The constant Ki is evaluated from 



N = Kv 



T ' 



(1.11) 



which stems from (1.10) when e = 0. Setting N = 288.04, Pa = 1013.25 
mbar, T = 273 °K and solving for K^: then^ 



Ki = 77.607 ± 0.13 



'K 



mbar ' 



(1.12) 



The constants K2 and K^ have been evaluated from a survey of deter- 
minations of the dielectric constant of water vapor in the microwave 
region by Birnbaum and Chatterjee [7]. These values, determined from 
the data of table 1.2, are 



and 



K2 = 71.6 ± 8.5 °K/mbar, 
K, = (3.747 ± 0.031) 10'(°K)Vmbar 



(1.13) 
(1.14) 



and were obtained as the weighted means of the various determinations, 
the weights being taken inversely proportional to the square of the prob- 
able errors. Although Birnbaum and Chatterjee considered only those 
data that would allow a determination of Ko and Kz over a wide range 
of temperature, they conclude their results are in satisfactory agreement 
with those of Stranathan [8] at 1 Mc/s, Phillips [9] at 3000 Mc/s, Grain 
[10] at 9000 Mc/s, and Essen and Froome [11] at 24,000 Mc/s. 



Table 1.2. Values of the constants K2 and Kz used by Birnbaum and Chatterjee 



Observer 


A'2 


Ki X 10"5 Temperature range 


Birnbaum and Chatterjee [7] 


69. 43 ± 13. 02 
77. 75 ± 21. 70 
61. 47 ± 21. 70 
72. 86 ± 7. 05 


°C 
3.774 ± 0.043 25-103 


Groves and Sugden [12] 

Hurdis and Smyth [13] 


3.742 ± 0.097 110-211 
3.765 ± 0.096 111-249 


Stranathan [8]_ . ._- 


3. 736 ± 0. 025 21-189 







* The development here follows that of Smith and Weintraub. Grain's [10] de- 
termination of the dielectric constant of dry air (e = 1.000572) indicates that Ki may 
be somewhat lower than the "best value" of Smith and Weintraub, since his value, 
Ki = 77.10, is about 40 standard deviations lower than that of Smith and Weintraub. 



CONSTANTS IN EQUATION FOR A^ 
The full equation for the refractive index is now 



^ + 72 I + 3.75 X 10^ jr. 



N = 11. Q ^ + 72 ^ + 3.75 X 10' 7^ , (1.15) 



after reducing the values to three figures where significant. These are 
the constants recommended by Smith and Weintraub to yield an overall 
accuracy of ± 0.5 percent in A^.^ Equation (1.15) may be simplified by 
assuming Pt = P = Pd + e and obtaining 

N = 11.6 I - 5.6 1^ + 3.75 X lO' |^ . (1.16) 

For practical work in radio-meteorological studies, (1.16) may be sim- 
plified to the two-term expression 



I + 3.73 X 10' f. 



N = 77.6 ~ + 3.73 X lO' ^ , (1.17) 



which yields values of A^^ within 0.02 percent of those obtained by (1.16) 
for the temperature range of —50 to 40 °C; i.e., a maximum error of 0.1 
A'' unit, and a standard error of 0.5 percent of about 4.5 A'^ units. 

This is accomplished by dividing the second and third terms of (1.16) 
by e/T and solving for the composite constant, K^, in relation 

2:^^-5.6 = 1, (1.18) 

which, for T = 273 °K, results in: 

^5 = 3.73 X 10'(°KVmbar). (1.19) 

Equation (1.17) is commonly Avritten as 

iV = ^(p + ^). (1.20) 

Although this last equation is in widespread use by radio scientists 
throughout the world, it is by no means adopted on an international scale. 



^ The values of K2 and Kz given in (1.13) and (1.14) are those of Birnbaum and 
Chatterjee as reported by Smith and Weintraub. A consideration of the ratio of 
Ko/Ki shows the Smith-Weintraub value to be about 0.2 percent greater than that 
obtained from the original data, indicating an arithmetic conversion error. This 
error is, however, totally absorbed in the rounding to three significant figures in 
(1.15). 



8 



RADIO REFRACTIVE INDEX OF AIR 



Another set of values in common use is that of Essen and Froome (11) 
which is listed in table 1.3 along with values determined by several other 
authors. Subsequent microwave refractometer determinations of these 
constants by Saito [14] in Japan yielded a value of Ki lower than that of 
either Smith and Weintraub or of Essen and Froome and nearer to Grain's 
determination. Saito's values of Ko and K^ lie between those of Smith 
and Weintraub, and Essen and Froome. A recent French microwave 
determination of these constants by Battaglie, Boudouris, and Gozzini 
[15], has found the Smith and Weintraub values of Ki, Ko, and K^ given 
by (1.16) fit their experimental data within measurement error. The use 
of the Essen and Froome values in radio geodesy appears to be based 
upon Wadley's [16] comparison of optical and radio surveys and his con- 
clusion that the best (radio) consistency is obtained when using Essen 
and Froome's constants in conjunction with their determination of the 
velocity of light. 



Table 1.3. Table of constants used by different authors 



Pd e e 

N = {n - I) 10« = A'l — + K.- + Kz — 



Reference 



Schelleng, Burrows, and Ferrell [20] 

Englund, Crawford, and Mumford [41] 

Waynick [35] 

Smith- Rose and Stickland [36] 

Burrows and Attwood NRDC [37] 

Meteorol. Factors in Prop. [40] 

Grain [10] 

Craig etal. [32] 

Essen and Froome [11] 

Smith and Weintraub [4] 

Essen [38] 

Saito [14] 

Battaglia, Boudouris, and Gozzini [15]. 
Magee and Grain [42] 



Date 



1933 
1935 
1940 
1943 
1946 
1946 
1948 
1951 
1951 
1952 
1953 
1955 
1957 
1958 



Ki 



79.0 

79.1 

79.0 

79.0 

79.0 

79.0 

77.10 

79.0 

77.64 

77.6 



77.26 
77.6 

77.5 



Ki 



67.5 
68.3 
68.5 
68.0 
68.0 
68.0 



(79) 
64.68 
72.0 
75.0 
67.5 
72.0 
65.0 



1.35 

3.81 

3.72 

3.8 

3.8 

3.8 



3.8 

3.718 

3.75 

3.68 

3.77 

3.75 

3.70 



The particular constants in the equation for A^ given by Smith and 
Weintraub are considered to be good to 0.5 percent in N for frequencies 
up to 30,000 Mc/s and normally encountered ranges of pressure, tempera- 
ture, and humidity. Experimental determinations of the variability of 
the radio refractivity, {n — 1)10^, with frequency have been carried out 
by Essen and Froome [11] and are summarized in table 1.4. 

Remembering that the dispersion of refractive index would be expected 
to be greatest at frequencies slightly off the water vapor resonance at 
22 Gc/s and the oxygen resonance at 60 Gc/s [17] one concludes that for 
the normally used frequencies, / < 30 Gc/s, the various gases give a 
frequency variation of A^ well within the limits of accuracy given by Smith 



ERRORS IN USE OF EQUATION FOR A^ 

Table 1.4. Refractivities of water vapor and dry gases 
(Dry gases at °C, 760 mm Hg; water vapor at 20 °C, 10 mm Hg) 



Gas 


9 Gc/s 


24 Gc/s 


72 Qc/s 


Water vapor 

Dry air 

Oxygen 


60. 7 ± 0. 2 
288. 10 ± 0. 10 
266. 2 ± 0. 2 


60. 7 ± 0. 2 
288. 15 ± 0. 10 
266. 4 ± 0. 2 


61.0 ±0.2 
287.66 ±0.11 
263. 9 ± 0. 2 



and Weintraub. With the exception of oxygen the same is true up to 
72 Gc/s. This conclusion is somewhat strengthened by the resuUs of 
Hughs and Armstrong [18] at 3000 Mc/s which agree within 1.5 percent 
of those of Essen and Froome. It is important to note that the differ- 
ences in formulas are greater than the frequency variation and, further, 
as shall be seen in the next section, the errors of determining P, T, and e 
are sufficiently large to mask even the differences in formulas. 



1.4. Errors in the Practical Use of the Equation for N 

A high degree of accuracy of temperature, pressure, and water vapor 
pressure measurements is necessary for precise determinations of the 
refractive index from (1.20). If one assumes that the formula for A^ is 
exact, then a relation between small changes in N and small changes in 
temperature, pressure, and vapor pressure may be evaluated from 



,,, dN -.^ . dN . . dN .j^ 



(1.21) 



assuming that the errors in P, T, and e are unrelated. 

The partial derivatives may be evaluated by reference to some standard 
atmosphere to yield the approximate expression 



AN' = aAT + bAe + cAP. 



(1.22) 



The root-mean-square (rms) error is then 



AN = {(aATY + (bAeY + (cAPy\ 



2i 1/2 



(1.22) 



Typical values of the constants a, b, and c, based upon the International 
Civil Aviation Organization (ICAO) standard atmosphere and assuming 
60 percent relative humidity, are given in table 1.5 for various altitudes. 



10 RADIO REFRACTIVE INDEX OF AIR 

It was mentioned earlier that the differences in refractive index ob- 
tained by the two most commonly used formulas, those of Essen and 
Froome and of Smith and Weintraub, are small compared to the error 
made in observing P, T, and e. For example, by assuming errors of ±2 
mbar in P, ±1 °C in T, and ±5 percent in relative humidity (RH) com- 
mon in radiosonde measurements with sea-level values of P = 1013 
mbar, T = 15 °C and 60 percent RH, a standard error of 4.1 A^ units may 
be obtained compared to the difference of about 0.5 A'^ units between the 
values obtained from the two formulas. It is seen from table 1.6 that 
(over the normal range of sea-level temperatures, —50 °C to +40 °C) 
the differences in the formulas are comparable to those associated with 
surface meteorological measurements but are significantly less than the 
errors that may arise in radiosonde observations. 

In addition to these errors, there is an additional source of error in the 
uncertainty of the constants in the equation for N. Equation (1.22) 
may be written to include this additional source of error as 

AA^' = a^T + 6Ae -f cAP + dAA^ + /AK2 + g^Kz, 

and, again assuming that the errors are uncorrected, the rms error may 
be evaluated as before. Table 1.7 lists the percentage error arising from 
errors in the various surface meteorological observations as well as those 
of the constants. It is seen that the errors in constants constitute no 
more than 30 percent of the total error. All sources of error combined 
will yield errors no larger than 1.5 percent in N. Thus, although the 
Smith and Weintraub equation has a stated accuracy of 0.5 percent due to 
the constants alone, total errors of nearly twice that figure may occur 
under conditions of extremely high humidity. 

Currently one must use radiosonde data for estimation of A'^ gradients, 
with the result that the overall accuracy is determined much more by the 
errors in the meteorological sensors than by errors between constants in 
the equation for A''. Until such time as better measurement methods for 
T and e are developed, there appears to be little or no need for more 
accurate determinations of the constants in (1.10). The different deter- 
minations of the constants now available are in essential agreement. For 
example, the use of the Essen and Froome expression will give values of 
A^ that generally lie within the standard error of ±0.5 percent of the 
Smith-Weintraub expression. It a])pears desirable for our pur})oses to use 
the Smith-Weintraub constants since they represent the weighted mean 
of several independent determinations, noting also that the value of K2 
given by Essen and Froome was obtained by extrapolation from optical 
measurements rather than direct measurements at radio frequencies. 



ERRORS IX USE OF EQUATION FOR A^ 



11 



Table 1.5. Values of the constants a, b, and c in the expression AN = aAT + bAe 
+ cAP, for the ICAO Standard Atmosphere and 60 percent relative humidity 



Altitude 


A^ 


T 


P 


e 


a 


b 


c 


km 




°C 


mbar 


mbar 


°K-i 


mbar~' 


mbar-^ 





319 


15.0 


1013 


10.2 


-1.27 


4.50 


0.27 


1 


277 


8.2 


893 


6.5 


-1.09 


4.72 


0.28 


3 


216 


-4.5 


701 


2.6 


-0.86 


5.17 


0.29 


10 


92 


-50.3 


262 


0.04 


-0.50 


7.52 


0.30 


20 


20 


-56.5 


55 





-0.09 


7.96 


0.35 


50 


0.2 


9.5 


0.8 





-0. 0008 


4.67 


0.27 



Table 1.6. Comparison of errors in determining ^ from meteorological measurements 

assuming no error in the equation for N 

[p = 1013 mbar and RH = 60 percent] 



Source of error 


-50 °C 


0°C 


+ 15 °C 


+40 °C 


Difference in formula.s: 


Smith-Weintraub [4] (three term).__ 
Essen and Froome [11] (three term).. 


352. 61 
352. 79 


306. 18 
306. 07 


318. 79 
318.28 


419.55 
417.19 




Difference from Smith-Weintraub 


+0.18 


-0.11 


-0.51 


-2.34 








Surface weather observations (±1 mbar of pressure, 
±0. 1 °C of T, and ±1 percent RH), (all errors in rnis 
A^ units). error 


±0.38 


±0.43 


±0.82 


±2.83 


Radiosonde observations 
of T, and ±5 percent R 


(±2 mbar of pressure, ±1 °C 
H) (all errors in A" units). rms 
error 


±1.73 


±2.02 


±4.07 


±14.19 



Table 1.7. Percentage contribution of errors in surface meteorological measurements 
and constants in the equation for N 



Temp (°C) 


Percent of total variance due to: 


Total rms 




AT 


Ae 


AP 


AA'i 


AA'2 


AK} 


error (A'-units) 


-50 



15 

40 


16.7 
6.2 
1.8 
0.3 


0.0 
41.3 
64.1 
69.3 


81.0 

35.6 

7.9 

0.5 


2.3 
1.0 
0,2 
0.0 


0.0 
5.7 
10.0 
12.7 


0.0 
10.2 
16.0 
17.2 


0.387 
.476 
.955 

3.378 



-50 


0.158 


0.005 


0.348 


0.059 


0.001 


0.002 


0.387 





.119 


.306 


.284 


.048 


.114 


.152 


.476 


15 


.127 


.765 


.269 


.045 


.302 


.382 


.955 


40 


.188 


2.811 


.248 


.040 


1.202 


.399 


3.378 



This opinion is particularly reinforced by Essen's subsequent determina- 
tion of K2 and A'3 at 9200 Mc/s, where closer agreement was obtained 
with the Smith-Weintraub values (note items 9, 10, 11 in table 1.3). The 
difference in K2 and /C3 accounts for most of the final difference in A'^ ob- 
tained from the two formulas. Although the great precision of Essen and 
Froome's work is reflected in their standard deviation of experimental 



12 RADIO REFRACTIVE INDEX OF AIR 

values being about one-tenth that of other workers, one must recognize 
that systematic errors undoubtedly contribute to the disagreement in 
values obtained by different workers. It appears that there exists a real 
need for a new determination carried out with the greatest possible care 
and over as large a pressure and temperature range as possible. 

The need for further measurements is particularly evident by the 
several Essen and Froome determinations. Their values of the dielectric 
constant have been determined with impressive precision and absolute 
accuracy over a wide range of frequencies, but their concomitant deter- 
minations of the constants in the equation for A'^ differ from one another 
as much as the Smith- Weintraub and the 1951 Essen and Froome values. 
It is clear that the use of either set of Essen and Froome values permits 
the calculation of either relative or absolute values of A'' with equal pre- 
cision. However, the question of comparative absolute accuracy with 
the results of other workers remains. 

As a further example, drawn from radio geodesy, consider the electrical 
path length 

R = / ndS, (1.23) 

Jo 

where »S indicates the radio path. If we assume that the measurements 
are made in a stratum of constant n, then (1.23) becomes 

R = n ds = nS = S{1 -\- N X 10"'), 

Jo 

and 

S = R/{1 -{-NX 10-') c^R{l - N X 10-'). (1.24) 

An error in ^V thus directly produces an error in radio distance determina- 
tion. In typical radio geodesic field work (see, for example, Wadley, 
[16]) A^ would be determined from pressure, dry-bulb temperature, and 
wet-bulb temperature readings. The partial pressure of water vapor may 
be obtained from Sprung's [19] (psychrometric) formula 

e ^ e's - 0.00067 (T - T') P (1.25) 

where e', denotes the saturation vapor pressure at the wet-bulb tempera- 
ture, r'. For sea-level conditions of P = 1013 mbar, T = 15 °C and 
T — T' = 4.1 °C (60 percent RH) one finds, neglecting errors in the 
constants, that 

AA^' ^ 0.28AP - 4.32A7^ - 6.96Ar. (1.26) 



PRESENTATION OF A^ DATA 13 

Assuming errors of it 0.1 mbar for P and ±0.25 °C for T and T' (de- 
rived from considering Wadley's field data as being accurate to ±5 units 
of the next significant figure beyond the tabulated whole degrees Fahren- 
heit for wet and dry bulb temperatures and hundredths of an inch of 
mercury for total air pressure) one finds that errors in A^ may be as large 
as 2.75 N units, which is much greater than the 0.5 N unit difference given 
by the two formulas for N. Again it is seen that the difference in the 
constants used in the two most widely accepted expressions for N yields 
an error of the same size or smaller than that produced by errors of meas- 
urement of the necessary meteorological parameters. 



1.5. Presentation of N Data 

There is now, in the literature, an almost bewildering choice of modi- 
fications to basic N data when presented as a function of height. The 
underlying principle is always to remove the syste?natic decrease of N with 
height, h, in an assuincd standard atmosphere. This arises from the fact that, 
the curvature of a radio ray, C, is given by ^ 



C = - "^rcos0 (1.27) 

n dh 



where 6 is the local elevation angle of the ray. Since the curvature of a 
radio ray is proportional to the gradient of the refractive index, specifica- 
tion of a model for dn/dh specifies the curvature of radio rays in that model 
atmosphere. For example, it is customary for radio engineers to think in 
terms of effective earth radius factors [20]. This convenient fiction, 
discussed in chapter 3, makes straight the actual curved path of a radio 
ray in the atmosphere by presenting it relative to an imaginary earth 
larger in radius by a factor k than the radius of the real earth, a, thus 
maintaining the relative curvature between earth and radio ray. 

This is expressed as 

1 I dn 1 

+ - -,7 cos d = y- -\- , 

a n dh ka 



curvature of curvature of curvature of curvature of 

earth radio ray effective earth straight ray 



^Equation (1.27) as well as general ray curvature considerations are elegantly 
derived by Millington [21] for the effective earth's radius model. The same equation 
is also derived in chapter 3. 



14 RADIO REFRACTIVE INDEX OF AIR 

Thus k is defined by 



1 + iT" COS 6 

n an 



which, for rays tangential to the earth (60 = 0) and assuming n in unity, 
is approximated by 

^ - 1 + a dn/dh ' ^^-^^^ 



It is customary to set 



dn/dh = - 7- , (1-30) 



and thus obtain k = 4/3. Since l/4a is close to the observed gradients 
of n, this represents a "standard atmosphere" gradient and one may re- 
move the "standard" decrease of N with height by adding a quantity, 8N 

m = {h/4a) 10' (1.31) 

to N{h) thus obtaining the "B unit" or 

B(h) = [(n - 1) + h/ia] lO' = N(h) + (/i/4a) lO'. (1.32) 

This method of presenting refractive index profiles has been shown to 
be very useful in emphasizing the departure from standard of the atmos- 
phere over southern California [22]. The principal advantage of the 
B unit is that any departure from a vertical line, and thus constant n 
gradient of dn/dh = — l/4a, equals a departure in refractive bending from 
that in the effective earth's radius atmosphere with k = 4/3. 

A very similar approach is used to aid in the study of extended ranges 
of radio waves. The most pronounced case of this phenomena occurs 
when dn/dh = — (1/a) which, from (1.29), gives k = 00 or an effective 
earth of infinite radius. This then implies, for radio purposes, the earth is 
flat and comnmnication is between "radio-visible" terminals. For this 
special study the modified index of refraction, M, is defined by 

M{h) = [{n - 1) + h/a] lO' = N(h) + (h/a) lO'. (1.33) 

A gradient of dM/dh = implies that A = oc and also implies a theo- 
retically infinite range of radio signals. M units were used to present the 



PRESENTATION OF N DATA 15 

meteorological data gathered by the Canterbury Project [23] in their 
intensive study of ranges of over-water radar signals. Both B and M 
units assume a standard atmosphere with a linear decrease of A'^ with 
height and thus introduce a correction which increases linearly with 
height. If the actual atmosphere had in fact the assumed A^ distribution 
then, for example, B{h) = B{0). Recent studies [24, 25, 26] have shown 
than an exponential decrease of A'^ with height is a more realistic model of 
the true atmosphere. 

For example, the data given on figure 3.3 were chosen to represent the 
extremes of average N profile conditions over the United States. The 
Miami, Fla., profile is typical of warm, humid, sea-level stations that 
tend to have maximum refraction effects while the Portland, Me., profile 
is associated with nearly minimum sea-level refraction conditions. Al- 
though Ely, Nev., has a much smaller surface A^ value than either Miami 
or Portland, its A^ profile falls within the limits of the maximum and 
minimum sea-level profiles. The A'^ distribution for the 4/3 effective 
earth's radius atmosphere is also shown on figure 3.3. 

It is quite evident that the 4/3 earth distribution has about the correct 
slope in the first kilometer above the earth's surface but decreases much 
too rapidly above that height. It is also seen that the observed refrac- 
tivity distribution is more nearly an exponential function of height than a 
linear function as assumed by the effective earth's radius model. The 
exponential decrease of N with height is sufficiently regular as to permit a 
first approximation of average vV structure from surface conditions alone. 
Consider that 



N{h) = Nsexp i-h/H), (1.34) 



where H iasi scale height appropriate to the value of A'' at zero height, Ns- 
Average values of N^ and H for the United States are approximately 313 
and 7 km respectively. As used here, scale height is simply that height 
at which N{h) is l/e of A''^ under the assumption of (1.34). 
One would exj)ect the gradient 



^^ = - (Ns/H) exp i-h/H) (1.35) 



also to be sufficiently well-behaved as to allow prediction of at least its 
general features. In fact a high (;orrelation between AN, the simple 
difference between A" at the earth's surface and at 1 km above the earth's 
surface, and A^,, has been observed in a number of regions. This is 



16 RADIO REFRACTIVE INDEX OF AIR 

particularly true of climatological data such as 5-year averages for given 
months. In the United States the expression [26] 

-AN = 7.32 exp (0.005577iV.), (1.36) 

TlnAN . T's = 0.93, 

where r denotes the standard correlation coefficient, has been found, while 
in Germany [27] the expression 

-AN = 9.30 exp (0.004:5Q5Ns), (1.37) 

^InAiV • Ns — 0.6, 

has been reported. A recent article by Lane [28] indicates that a similar 
correlation, rin^" 'Ns= 0.85, is found over the British Isles. Although 
the slopes and intercepts of this relation between AA'^ and Ng obviously 
vary with climate, one concludes, for the purposes of the present discus- 
sion, that the observational evidence corroborates the assumption of an 
average exponential decrease of A^ with height. The general application 
of the exponential model nmst await considerable work upon the part of 
the radio climatologists. 

This has naturally led to the definition of a new unit that adds a correc- 
tion factor to account for this exponential decrease [29, 30]. This new 
unit. A, is given by 

A(h) = Nih) + A^.[l - exp {-h/H)] (1.38) 

It is i)ossible to calculate a theoretical value of the scale of N by 
as.suming a distribution of water vapor. This is, however, quite a 
complex procedure. Futhermore, the value obtained depends upon the 
model of the water vapor distribution, and no definite conclusion can be 
justified considering the extreme variation of water vapor concentration 
with season, geographic location, and height above the earth's surface. A 
convenient and simple alternative is to adopt a value for H from the 
average (n — 1) variation with height in the free atmosphere. Several 
such values of H were determined by reference to the National Advisory 
Committee for Aeronautics (NACA) standard atmosphere and recent 
climatological studies of atmospheric refractive index structure [26]. It 
is seen in table 1.8 that H varies from 6.56 to 7.63 km in the NACA 
standard atmosphere, depending on the value of the relative humidity 
assumed. The value of // = 7.01 km for SO jiercent relative humidity is 



PRESENTATION OF N DATA 



17 



in close agreement with H = 6.95 km obtained from climatological 
studies of (n — 1). 

Table 1.8. Determination of the effective scale height, H, for the radio refractive index 



Source 


Humidity // 




percent 
100 

80 

60 
As observed 


6.56 


Climatological data 
adopted for this discussion. 


7.01 
7.63 
6.95 
7.0' 



*The value of H = 7. km was arbitrarily adopted for use in the present discus- 
sion. Note that this value is smaller than the usual scale height near sea level of 
8. km for the density distribution of the air. Although the value H is quite arbi- 
trary, it is evident from table 1. 8 that H should lie between 6. 5 and 7. 5 km. 



A comparison of B, M, N, and .4 units is shown on figure 1.1 for arctic, 
temperate, and tropical climates. It is quite evident that both M and B 
over-correct the profile to produce increasing values with height while 
the A unit tends to yield a constant value at heights in excess of 2 km. 
It is noted, however, that important departures of the Nik) profile from 
normal within the first few kilometers are emphasized by both the A and 
B profiles. 

In radio meteorology, as in other branches of meteorology, it is often 
convenient to express the potential value of the refractive index referred 
to some standard pressure level. In practice this is found by adding to 
the values at any height the product of the iV lapse rate and the height. 
This has already been done in the case of the B unit, while the A unit 
simply adds the total decrease of N in an exponential atmosphere from 
the surface to the height under consideration. Yet another approach is 
to replace the values of pressure, temperature, and vapor pressure in the 
expression for A^ with their values at some desired pressure level. This 
unit, called the potential refractive modulus, (/>, is then defined by 



77.(5 



Po + 4810 -} 



(1.39) 



where d is the potential temperature and eo the potential partial pressure 
of water vapor, both referred to the reference pressure Po. This formula 
is that of Katz [31] with the Smith-Weintraub (constants. 
The potential temperature is defined as 



d = T{Po/p)' 



(1.40) 



18 



RADIO REFRACTIVE INDEX OF AIR 



\ 


ISACHS 


ON, N.V\ 


/.T, FEBRUARY 
1 








b/ 


y 


\ 


s. 








V 




/ 


( 




N( 


\ 












/ 


/ 










k 








> 


/ 








A=N(h)H 

-B=N(h)i 

M=N(h)-( 


3i3.o[i-exp.(-^)] 


k^ 




K 








M 








^ 


i. 


_^ 






' 



\ 


WA 


SHINGT 


DN, DC, 


MAY 


1 








B, 


y 


> 


\ 










A 




^ 






N 


{h)\ 












/ 


/ 










\ 








, 


/ 












> 


\ 




/ 








-jh-^^ 












^U 


_^_ 


-^ 





\ 


c/ 


\NTON 1 


SLAND, 


AUGUST 










B-' 


X 


\ 


\ 










A 




^ 


/ 






N(h\ 












/ 


y' 










\ 








/ 












N 






/ 








M 










^^^ 


5>^ 


__- 







50 100 150 200 250 300 350 400 450 500 550 

N 



Figure 1.1. Variation of N, A, B, and M profiles with climate. 



PRESENTATION OF A^ DATA 19 

and eo as 

eo = e(Po/p), (1.41) 

where R is the universal gas constant, w the molecular weight of air, and 
Cjc the specific heat of air at constant pressure. By setting Po = 1000 
mbar and noting that R/mCp = 0.286, one obtains 

• . = ^(^r + 3.73X.O-|,(l<^r. (1.42) 

Thus, it is seen that <^ is obtained by correcting the dry air term of A'^, 
77.6p/T, by the factor (lOOO/p)"-^" and the water vapor term by the 
factor (1000/p)°-^-'^. In well-mixed air, 6 and Co are independent of height 
and conseciuently is independent of height. If the temperature lapse 
rate is less than the dry adiabatic rate, the potential temperature increases 
with height and, conseciuently, the first term of (/> decreases with height. 
If the lapse of the partial pressure of water vapor is less than than given 
by (1.41) then eo and the second term of </> increase with height, conversely, 
if e decreases more rapidly with height than the decrease given by (1.41), 
eo and the second term of decrease with height. Abundant examples of 
the vertical distribution of </> under various meteorological conditions are 
given throughout volume 13 of the Radiation Laboratory Series [32]. 
The potential refraction modulus has recently been used by Jehn [33] to 
illustrate the variability of A^ about a classic polar-front wave. 

A conceptually similar approach to that above is to arbitrarily adopt a 
reduced-to-sea-level value of refract ivity 

A^o = A^. exp (-H/i/7.0) (1.43) 

where h is in kilometers, that effectively removes the station elevation 
dependence of N s [34] and allows the emphasis of air-mass differences. 

Since the effects of the atmosphere upon the propagation of radio waves 
are dependent upon either the absolute value of n or its gradient, we may 
use this as a criterion for the choice of appropriate units to represent the 
A^ profile. One may recover both n and its gradient from a knowledge of 
the height structure alone. The corrections are additive in the case of 
B, M , and A and nmltiplicative for A^'o. The values B, M, and A are in 
convenient form for refraction calculations since it is the gradient above 
the surface of the earth that is needed. The value, A'^o, removes the effect 
of station elevation and t hus is convenient for mapping of surface condi- 
tions in a fashion similar to sea-level pressure rather than station pressure. 
One cannot obtain the gradient and absolute value of n from a 0(/O distri- 
bution, although <l>(h) may be graphically calculated from the familiar 



20 



RADIO REFRACTIVE INDEX OF AIR 



characteristic diagram of the meteorologist; even for that process one 
must first calculate the potential values of both temperature and water 
vapor pressure. The relative merits of the various units are summarized 
in table 1.9. 

1.6. Conclusions 

The above discussion has emphasized the following points: 

(a) The differences in constants in the expression for A'' are small com- 
pared with the error inherent in the formula. 

(b) The refractive index is effectively nondispersive for frequencies 
below, say, 50 Gc/s. 

(c) The choice of atmospheric model for A'^, and concomitant units, 
depends upon the application at hand. 

(d) The atmosphere, on the average, yields an exponential d stribution 
of A^" wath height. 

(e) The use of an exponential model facilitates the preparation of 
climatic maps of A'' and makes clear the effect of non-normal A^ structure 
upon the bending of radio rays. 



Table 1.9. Cotnparison of various units iised in radio meteorology 



Unit 



B(ft) = .V(/i) + (/i/4a)10i', 



M(h)=N(h) + (h/a)W, 



N(,{h) = N(h) c\p(h/-.0). 



.■nh)=N(h) 
+313|l-exp(-/!/7.0)], 



.(A)^^X 



6/,ooo+^-^» 



Oradient 



dBjh) _dN{h) 
dh " dh 



+ (l/4a)106, 



dM(h)_dNih) 
dh " dh 



+ (l/a)10«, 



dA'o(ft) dN(h) ( h 



dh 






dA(h) _ dN(h) 313 
dh ~ dh ^ 1 
exp(-ft/7.0), 

d0(fe) _ d<i>(h) de 
dh de dh 

d<j>{h) deo. 



+ 



deo dh 



Referred 
to: 



Surface. 



Surface. 



Sea-level. 



Surface. 



1000 mbar 



Advantages 



Vertical distribution 
identifies "4/3 earth" 
conditions; absolute 
value and gradient of n 
easily recovered. 

Trapping or ducting 
layers identified by 
{dM/dh)^0, absolute 
value and gradient of 
n easily recovered. 

Removes effects of both 
height and station ele- 
vation; facilitates map- 
ping since N{h) is 
easily recovered. 

Removes effects of height 
n and gradient easily 
recovered. 



Removes the effects of 
height and station ele- 
vation, may be de- 
rived from character- 
istic meteorological 
diagram. 



Disadvantages 



Over-corrects 
N(h ) above 
the first few 
kilometers. 



Grossly over- 
corrects N{h) 
above the 
trapping 
layer. 

Gradient on n 
not easily re- 
covered. 



Does not re- 
move the ef- 
fect of station 
elevation. 

Neither n or 
dn/dh may 
be recovered 
from </){ft); 
must calcu- 
late both 
from basic p, 
T, and e data 



REFERENCES 21 

1.7. References 

Saxton, J. A. (Sept. 1951), Propagation of metre radio waves beyond the normal 

horizon, Proc. lEE 98, 360-369. 
Debye, P. (1957), Book, Polar Molecules, pp. 89-90 (Dover Publ. Co., New 

York, N.Y.). 
Saxton, J. A. (1947), The anomalous dispersion of water vapour at very high 

frequencies. Parts I-IV, Book, Meteorological Factors in Radio Wave Propa- 
gation, pp. 278-316 (The Physical Society, London, England). 
Smith, E. K., and S. Weintraub (Aug. 1953), The constants in the equation for 

atmospheric refractive index at radio frequencies, Proc. IRE 41, 1035-1037. 
Barrel!, H. (May 1951), The dispersion of air between 2500 A and 6500 A, J. Opt. 

Soc. Am. 41, 295-299. 
Maryott, A. A., and F. Buckley (1953), Table of dielectric constants and electric 

dipole moments of substances in the gaseous state, NBS Circular 537. 
Birnbaum, G., and S. K. Chatterjee (Feb. 1952), The dielectric constant of water 

vapor in the microwave region, J. App. Phys. 32, 220-223. 
Stranathan, J. J. (Sept. 1935), Dielectric constant of water vapor, Phys. Rev. 48, 

538-544. 
Phillips, W. C. (July 1950), The permittivity of air at a wave-length of 10 centi- 
meters, Proc. IRE 38, 786-790. 
Crain, C. M. (Sept. 1948), The dielectric constant of several gases at a wave- 
length of 3.2 centimeters, Phys. Rev. 74, No. 6, 691-693. 
Essen, L., and K. D. Froome (Oct. 1951), The refractive indices and dielectric 

constants of air and its principal constituents at 24,000 Mc/s, Proc. Phys. 

Soc. (London, England) 64, 862-875. 
Groves, L. G., and S. Sugden (1935), Dipole moments of vapors II, J. Chem. 

Soc. England 30, 971-974. 
Hurdis, E. C, and C. P. Smyth (1942), Dipole moment, induction and resonance 

in nitroethane and some chloronitroparaffins, J. Am. Chem. Soc. 64, 28-29. 
Saito, S. (Aug. 1955), Measurement at 9,000 Mc of the dielectric constant of air 

containing various quantities of water vapor, Proc. IRE 43, No. 8, 1009. 
Battaglia, A., G. Boudouris, et A. Gozzini (May 1957), Sur I'indice de refraction 

de I'air humide en microondes, Ann. de Telecomm. 12, 181-184. 
Wadley, T. L. (1957), The tellurometer system of distance measurement. Empire 

Survey Review 14, No. 105, 100-111, and 14, No. 106, 1232-1239. 
Born, M., and E. Wolf (1959), Book, Principles of Optics, p. 89 (Pergamon Press, 

New York, N.Y., and London, England). 
Hughs, J. v., and H. L. Armstrong (May 1952), The dielectric constant of dry 

air, J. Appl. Phys. 23, 501-504. 
Sprung, A. (1888), Bestimm. d. Luftfeuchtigk. mit Htilfe d. Assmann'schen 

Asperationspsychrometers, Das Wetter 5, 105-109. 
Shelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave 

propagation, Proc. IRE 21, 427-463. 
Millington, G. (Jan. 1957), The concept of the equivalent radius of the earth in 

tropospheric propagation, Marconi Rev. 20, No. 126, 79-93. 
Smyth, J. B., and L. G. Trolese (Nov. 1947), Propagation of radio waves in the 

lower atmosphere, Proc. IRE 35, 1198-1202. 
Report of P'actual Data from the Canterbury Project (1951), Vols. I-III (Dept. 

Sci. Indus. Research, Wellington, New Zealand). 
Anderson, L. J. (Apr. 1958), Tropospherit; bending of radio waves. Trans. Am. 

Geophys. Union 39, 208-212. 



22 RADIO REFRACTIVE INDEX OF AIR 

[25] Bauer, J. R., W. C. Mason, and F. A. Wilson (27 Aug. 1958), Radio refraction in 
a cool exponential atmosphere. Tech. Rept. No. 186, Lincoln Laboratory, 
Massachusetts Institute of Technology, Cambridge, Mass. 

[26] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric re- 
fractive index, Proc. IRE 47, No. 5, 740-755. 

[27] Bean, B. R., L. Fehlhaber, and J. Grosskopf (Jan. 1962), Die Radiometeorologie 
und ihre Bedeutung fiir die Ausbreitung der m-, dm,- und cm-Wellen auf 
grosse Entfernungen, 15, 9-16, Nachrichtentechnische Zeitschrift. 

[28] Lane, J. A. (1961), The radio refractive index gradient over the British Isles, 
J. Atmospheric Terrest. Phys. 21, Nos. 2/3, 157-166. 

[29] Bean, B. R., and L. P. Riggs (July-Aug. 1959), Synoptic variation of the radio 
refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97. 

[30] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the 
vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio 
Prop.), No. 2, 249-258. 

[31] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient 
of refractive modulus in homogeneous air, potential modulus, Book, Propaga- 
tion of Short Radio Waves, ed. D. E. Kerr, pp. 198-199 (McGraw-Hill Book 
Co., Inc., New York, N.Y.). 

[32] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient 
of refractive modulus in homogeneous air, potential modulus. Book, Propaga- 
tion of Short Radio Waves, ed. D. E. Kerr, p. 189 (McGraw-Hill Book Co., 
Inc., New York, N.Y.). 

[33] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic scale 
radio meteorology, J. Meteorol. 17, 264. 

[34] Bean, B. R., and R. M. Gallet (Oct. 1959), Applications of the molecular re- 
fractivity in radio meteorology, J. Geophys. Res. 64, No. 10, 1439-1444. 

[35] Waynick, A. H. (Oct. 1940), Experiments on the propagation of ultra-short-wave 
Proc. IRE, 28, 468-475. 

[36] Smith-Rose, R. L., and A. C. Stickland (Mar. 1943), A study of propagation 
over the ultra-short-wave radio link between Guernsey and England on wave- 
lengths of 5 and 8 meters, J. lEE 90, No. Ill, 12-24. 

[37] Burrows, C. R., and S. S. Atwood (1949), Radio wave propagation, Consolidated 
Summary Technical Report of the Committee on Propagation, NDRC, p. 219 
(Academic Press, Inc., New York, N.Y.). 

[38] Essen, L. (Mar. 1953), The refractive indices of water vapour, air, oxygen, 
nitrogen, hydrogen, deuterium, and helium, Proc. Phys. Soc. B 66, 189-193. 

[39] Birnbaum, G., S. J. Kryder, and H. Lyons, (Aug. 1951), Microwave measure- 
ments of the dielectric properties of gases, J. Appl. Phys. 22, 95-102. 

[40] Meteorological Factors in Radio Wave Propagation (1946), (The Physical So- 
ciety, London, England) Foreword. 

[41] Englund, C. R., A. B. Crawford, and W. W. Mumford (July 1935), Further 
results of a study of ultra-short-wave transmission phenomena. Bell Sys. 
Tech. J. 13, 369-387. 

[42] Magee, J. B., and C. M. Grain (Jan. 1958), Recording microwave hygrometer, 
Rev. Sci. Instr. 29, 51-54. 



Chapter 2. Measuring the Radio 
Refractive Index 

2.1. The Measurement of the Radio 
Refractive Index 

The radio refractive index is defined as the ratio of the speed of propa- 
gation of radio energy in a vacuum to the speed in a specified medium. 
The radio refractive index may be measured directly if the measuring 
instrument is sensitive to the speed of propagation. Refractive index is 
measured indirectly by measuring temperature, pressure, and humidity 
with subsequent conversion to the refractive index as indicated in the 
previous chapter. The direct method employs radio frequency refrac- 
tometers to determine the refractive index; the indirect method, standard 
weather observations. The direct method appears preferable because 
accuracy is dependent upon a single sensor, rather than three and, of 
course, because A'^ values are obtained directly. However, refractometers 
are relatively complex and expensive devices requiring no small degree of 
skill to maintain. Hence, as yet, refractometers are not in general use, 
or even available in sufficient quantity to permit large-scale mapping of 
refractive index structures. The bulk of the synoptic and climatological 
mapping of refractive index is still based on the indirect method of 
measurement. The laborious task of converting the measured parame- 
ters to refractive index has been somewhat alleviated by the development 
of special analog computers [1]' and digital computers. 



2.2. Indirect Measurement of the Radio 
Refractive Index 

The accuracy of the determination of the refractive index from standard 
weather observations has been discussed in chapter 1. Figure 2.1 illus- 
trates the degree of accuracy to be expected in the measurement of the 
refractive index as a function of the accuracies of the sensors for sea-level 
conditions. It is assumed that the errors are additive. Planes of equal 



' Figures in brackets indicate the literature references on p. 45. 

23 



24 



MEASURING THE RADIO REFRACTIVE INDEX 




Figure 2.1. Accuracy of (he delermination of refractivity as a function of the accuracies 

of the meteorological sensors. 



accuracy are shown for accuracies of ±0.1, ±0.5, ±1.0, and ±2.0 A'^. 
It can be seen that a measurement error of ±1.0 A'^ can be contributed 
by each sensor if the sensor errors exceed ±0.8 °C for temperature, ±3.7 
mbar for total pressure, and ±0.22 mbar for vapor pressure measure- 
ments. It is apparent that extreme accuracy is required in the measure- 
ment of the water pressure. The temperature can be measured easily to 
within several tenths of a degree, the total pressure to within several 
millibars, but the humidity is considerably less responsive to the same 
degree of relative accuracy. If the wet and dry bulb hygrometer tech- 
nique is used to measure the humidity with subsequent conversion using 
either the psychrometric formula of Sprung- [2] or List^ [3], the degree of 
accuracy is seen to be a function of temperature. For an error of no 
greater that ±1.0 A^, the relative humidity must be accurate to within 
0.33 percent at 35 °C, while at °C an accuracy of 3.0 percent is necessary. 



e = Cs — c , c IS a constant. 

566 

e = Cs — [fi(l + Co/u.)] p(AT'), Ci and c^ are constants. 



INDIRECT MEASUREMENT 25 

Experimenters at the National Physical Laboratories [4] have shown that 
when an Assnian hygrometer is used with thermometers accurate to 
within 0.1 °C, the optimum expected accuracy in the determination of the 
water vapor pressure is 0.2 mbar. Errors three times as great can be 
expected at extreme conditions. Hence, it would appear that measure- 
ment of water vapor limits the accuracy of the determination of the 
refractive index to approximately ±1.0 A'^ units. 

Temperature, pressure, and humidity are standard measurements of 
the world's weather services. Methods of measuring these parameters 
are fairly standard. Usually the temperature is read from mercurial or 
alcohol thermometers, the pressure from mercurial barometers, and the 
huniidity from a conversion of wet- and dry-bulb thermometers. The 
degree of accuracy of these measurements is usually a function of the care 
exercised by the observer. Thermometers protected by radiation shields 
are usually accurate to within ±0.1 °C, barometers to within ±1.0 mbar. 
Reading errors can easily be in excess of the instrument error, especially 
in the determination of humidity where the wet-bulb depression is subject 
to many sources of error. Contamination of the wick or water, insuffi- 
cient wetting, and inadequate aspiration are con^mon sources of error. 
The wet-bulb determination can be used below freezing, if proper pre- 
cautions are observed. 

Automatic-recording systems have been devised for measurement of 
temperature, pressure, and humidity. The simplest are the hygrother- 
mograph and the microbarograph, in which the sensors are connected by 
mechanical linkages to pens or chart recorders. The pressure is recorded 
by means of an aneroid capsule, the temperature by means of a bimetal 
strip or a curved Bourdon tube, and the humidity by means of a hair 
hygrometer. The accuracy of such devices limits the determination of 
N to within 2 or 3 A^ units. 

Sensors producing electrical outputs are used to measure meteorological 
parameters in a variety of automatic-recording systems, such as strip- 
chart recorders, punched-paper tape, and magnetic-tape recorders. The 
more common temperature sensors include resistance thermometers, 
thermocouples, and thermistors. The platinum resistance thermometer, 
an international standard, is capable of measuring temperature in still 
air to within ±0.05 °C if used in a well-compensated bridge circuit. 
However, the platinum resistance thermometer is velocity-sensitive in a 
moving air stream. Thermocouples avoid this problem and yield short- 
term accuracies of approximately ±0.1 °C with time constants measured 
in milliseconds. 

The total atmospheric pressure can be measured by a variety of elec- 
trical sensors. The simplest device is the pressure potentiometer where 
an aneroid capsule is mechanically linked to a potentiometer. One or 
two millibars can be considered the limit of accuracy in a differential 



26 MEASURING THE RADIO REFRACTIVE INDEX 

device operating over a range of ±100 nibar with reference to an average 
value with time constants of perhaps 5 msec. The capacitive micro- 
phone is capable of accuracies of ±0.01 mbar and lag constants of several 
milliseconds over a limited range (±2 mbar). The resolution diminishes 
to about ±0.1 mbar over the normal range of variations of atmospheric 
pressure. The capacitive microphone requires correction for significant 
temperature variations. Strain-gage pressure transducers have approxi- 
mately the same characteristics as capacitive microphones with some 
degradation in lag constants. Differential strain gauge transducers oper- 
ating over a range of 4 to 5 mbar may yield accuracies of several hundred- 
ths of a millibar and are relatively free of temperature effects. 

Relative humidity reciuires the greatest care in measurement. The 
common lithium chloride strip is accurate to within ±5 percent in relative 
humidity. In general, the lag constant is of the order 8 to 10 sec for 
temperatures in excess of °C (see sec. 2.5). Other sensors under de- 
velopment show promise. The barium fluoride strip [5] yields accuracies, 
hysteresis effects, and response times far superior to lithium chloride but 
has the disadvantage of rapid aging. Phosphorous pentoxide sensors [6] 
and aluminum oxide sensors [7] appear to be quite promising. The most 
accurate method that is available for measuring water vapor pressure is 
the wet-dry bulb technique using electrical thermometers (±0.02 mbar). 
The lag constant will be relatively large, as it is a function of the wetting 
properties of the wick, the rate of aspiration, and even the relative 
humidity, and is thus recommended only for temperatures above —24 °C. 

Measurements up to several thousand feet can be made from towers or 
with tethered balloons or a wiresonde. Only the sensors need be sent 
aloft. All auxiliary devices are on the ground, and long cables transmit 
the information from the sensors to the recording equipment [8, 9]. 

The radiosonde is in almost universal use for high-altitude measure- 
ment of the meteorological parameters influencing the refractivity. There 
are many models of the radiosonde, differing from country to country. 
The principle of operation for present American radiosondes is illustrated 
in figure 2.2. An aneroid capsule is used as the active element of a 
baroswitch. The temperature sensor, humidity sensor, and a reference 
resistance are alternately switched into the grid circuit of a blocking 
oscillator as the baroswitch wiper moves under the action of decreasing 
pressure. The blocking oscillator controls the pulse rate of the rf trans- 
mitter; hence, the pulse rate of the transmission is indicative of the value 
of the sensor being sampled. The number of the contact energized is a 
measure of the pressure; the switching sequence permits identification 
of temperature and humidity values. A constant rate of ascent of the 
balloon is assumed so that the values of the parameters can be identified 
with the proper altitude. 



INDIRECT MEASUREMENT 



27 



RF 
TRANSMITTER 



BLOCKING 
OSCILLATOR 











.' 


— < 


* 






















( 


















_, 




T 




RH 




































•s 










< 

< 
< 

- 


:r 


EF 


EF 


?EN 


E 
CE 




ANEROID 
CAPSULE 



BAROSWITCH 



Figure 2.2. Block diagram of the radiosonde. 



There is no uniformity of sensors used in the world's radiosondes. 
Hence, it is difficult to compare measurements from different countries. 
At one international comparison at Payerne, Switzerland, in 1956, 14 
nations participated [10]. The results indicated significant differences. 
Temperature measurements corresponded to within ±1.5 °C for night 
flights but corresponded only to withm ±3.5 °C for daytnne flights. 
Pressure measurements agreed well at low altitudes but indicated a dis- 
persion of ±1.5 mbar above 9,000 m (29,000 ft), and ±2.5 mbar above 
16,000 m (50,000 ft). Humidity comparisons were poor, indicating that 
15 percent would be the most optimistic estnnate of the standard devia- 
tion from the mean for all flights. On the average, it is estimated that 
the standard American temperature sensor indicated a value approxi- 
mately 1.5 °C below the mean for all sondes used in the test. The pres- 
sure determination was below the mean by 0.5 mbar, whereas the humid- 
ity sensor could not be quantitatively evaluated due to the erratic be- 
havior of all sensors. 

No attempt was made at these comparison trials to determine the 
absolute accuracy of any radiosonde. Although the absolute accuracy 
of the American sonde has not been determined, satistical evaluation of 



28 



MEASURING THE RADIO REFRACTIVE INDEX 



the uniformity of American radiosondes has been conducted by both the 
U. S. Air Force and the U. S. Army Signal Corps. ResuUs indicate a 
standard deviation for the temperature sensor of 0.8 °C to 6,000 m and 
1.0 °C above that ahitude. The standard deviation for the pressure 
determination was 2.2 mbar below 9,000 m and 1.1 mbar above 9,000 m. 
Under ideal conditions the humidity sensor (lithium chloride) exhibited 
a standard deviation of 5 percent. This accuracy is possible only if the 
element is not subjected to high humidity (95 to 100 percent), or satura- 
tion by liquid water, and if the temperature is above °C. The response 
of the element is especially poor where both temperature and humidity 
are low. The measurement means little if the relative humidity is below 
15 percent at a temperature of 20 °C, 20 percent at a temperature of 
°C, or 30 percent at a temperature of —30 °C. 

Hence, "under ideal conditions" at sea level, the standard deviation 
in the determination of the refractivity from radiosonde data is approxi- 
mately 3 A^ units; at 1 km, the standard deviation would be approxi- 



16.7% 




33.3% 















10 


0- 


66.7% 








; 






/ 






.^ 




83. 
5( 


3*. 
D% 


"i 


\y 






M"66.7^33.3 


S% 



50% 



66.7% 



DECREASING 



CHANGE IN RH 



100— 50 % - 



83.3-33.3% 



100-33.3% 



< 400 
if) 

g 300 

< 200 



16.7% 





83 

6 
"8 


5.3- 

S.7- 
3.3 


-100% 
















/ 








^ 


il 


H 


■^* 


i 


9> 


V 


66. 


7% 


^ 


"33.3-50% 





33.3% 














— 


66 


7- 


00% 

p 






/ 










/ 






/ 


\ 




y 


/ 


50- 
83.3% 


y^ 




Y 


«^ 


'33.3-667% 



50% 



66.7% 



INCREASING 



CHANGE IN RH 

50-100% 




:/t33.3-IOO°/ 



% 



-20 -40 -20 -40 -20 -40 -20 -40 
TEMPERATURE (°C) 



Figure 2.3. Factors affecting the lag constant of the lithium chloride humidity sensor. 

(.\fter Wexler.) 



INDIRECT MEASUREMENT 



29 



TEMPERATURE 
SENSOR 




TEMPERATURE 
HUMIDITY 
SENSOR 



PRESSURE f 
SENSOR 




BLOCKING OSCILLATOR 
[an /AMT-6(xa-l)l 



Figure 2.4. CUnger-Slraiton radiosonde. 
(Transmission in N units.) 



mately 2 A^ units. This then would appear to be the uhiniate precision 
with which present conventional radiosonde sensors can yield the refrac- 
tivity. 

The lag constants of the radiosonde sensors are also of importance. 
Since the sonde is rising at a relatively rapid rate, it passes into regions 
of changing refractivity before the sensors are aware of it. The lag 
coefficient associated with the radiosonde introduces an error in the esti- 
mation of the true gradient. Lag constants of sensors have been ana- 
lyzed by Wexler [11] and by Bean and Button [12]. Some correction can 
be made to the radiosonde data. Wexler (fig. 2.3) shows that the lag 
coefficient of the lithium chloride strip is a function not only of the 
temperature, but also of the absolute value of the relative humidity, as 
well as of the size and direction of the gradient. 

The radiosonde samples temperature and humidity in sequence rather 
than simultaneously. Several experimenters have devised means to cor- 
rect this deficiency. Misme [13] decreased the cycling time in one radio- 



30 



MEASURING THE RADIO REFRACTIVE INDEX 



L 




\ 300 GRAM 

RADIOSONDE 
/ BALLOON 






FREQ a Tyv 


Tyv 395 MC L 




RECEIVER 
395 MC 




FREQUENCY 
METER 
UNIT 


_K>.K. 


T 
























RECORDER 




ANALOG 

COMPUTER 

UNIT 
























T 405 MC 1 


T 


RECEIVER 
405 MC 




FREQUENCY 

METER 

UNIT 




|i_i 




.KKN^ 












FREQ. a T 



Figure 2.5. Schematic diagram of the Navy Electronics Laboratory (Thiesen) system. 

sonde so that many more samples of each parameter were produced per 
unit time. CHnger and Straiton [14] developed a radiosonde that com- 
bines the parameters so that the output signal is in terms of the refractive 
index (see fig. 2.4). Since the wet and dry terms are additive, a parallel 
combination of independent conductances can be used. The dry term 
sensor has a conductance proportional to temperature and inversely pro- 
portional to pressure; the wet term sensor is such that it not only yields 
a value proportional to the relative humidity, but also adjusts the relative 
magnitude of the two terms. Thiesen [15] devised a similar radiosonde 
utilizing two separate sondes on the same balloon (see fig. 2.5). The 
temperature and humidity information are combined at the ground sta- 
tion by means of an analog computer. For captive balloon application, 
Hirao and Akita [8] and Crozier [9] developed similar systems using wet 
and dry bulb thermistors to produce direct output in refractive index. 



2.3. Direct IVleasurement of the Refractive Index 

The resonant frequency of a microwave cavity is a function of its 
dimensions and the refractive index of its contents. Hence, if a cavity is 
open to the atmosphere, the resonant frequency changes as the refractive 
index of the air passing through it changes. If a sealed reference cavity is 



DIRECT MEASUREMENT 31 

used for comparison, the difference between resonant frequencies becomes 
a convenient measure of the refractive index variations in the sampling 
cavity. This method of measurement was used in three different instru- 
ments: the Grain type, the Birnbaum type, and the Vetter type. The 
Grain refractometer utiHzes the cavities as frequency determining ele- 
ments in ultra-stable oscillators. The difference in frequency is then the 
unit of measure. The Birnbaum refractometer utilizes the cavities as 
passive resonance frequencies of the cavities. The time difference between 
resonances is the unit of measure. The Vetter instrument, an improve- 
ment on Sargent's [16] modification of the Birnbaum's refractometer, 
utilizes servo techniques to achieve a null system, thus eliminating the 
necessity for extreme electronic stability. 

The resonant cavities are significant components in any of these instru- 
ments. Of prime importance is the temperature coefficient of the cavity. 
Most cavities today are made of invar having a temperature coefficient 
of approximately one part per million per degree centigrade [17, 18]. 
This is equivalent to 1 N unit per degree centigrade. Further tempera- 
ture compensation of the cavity has produced temperature coefficients of 
0.2 and 0.1 A'^ unit per degree centigrade [19]. Most recently, improved 
invar cavities, when compensated, yielded temperature coefficients of 
0.03 N units per degree centigrade [20]. The possibility of improving 
cavity performance by use of special ceramics [21, 22], with temperature 
coefficients (without compensation) of 0.1 N units per degree centigrade 
has been investigated. 

The response time of the refractometer is a function of how much of 
the end plates of the cavity can be opened to the air without appreciable 
loss of resonant characteristics. It has been found [23, 24], that as much 
as 92 percent of the end plate area could be eliminated without serious 
resonant degradation. Such end plates do not impede the flow of air, 
and the response time of the instrument is essentially instantaneous. 

In general, the accuracy of the refractometer is at least one order of 
magnitude smaller than can be achieved by indirect measurement. With 
proper care, these instruments are capable of discerning changes of the 
refractive index that are less than a tenth of an A'^ unit. As a relative 
instrument, i.e., used to measure variations about an undetermined mean, 
the time constant is such as to easily allow detection of rates of up to 
100 c/s. 

The development of microwave refractometers led to an immediate in- 
creased interest in the fine structure of refractive index variations and its 
application to radio wave propagation. A summary of this development 
has been given by Herbstreit [25]. Detailed investigations of the inhomo- 
geneities of refractive index structure using spaced cavities yielded signi- 
ficant results pertinent to the spectrum of turbulence [26, 27]. Refrac- 
tometers have been widely used in aircraft [18, 28, 29]. The Grain 



32 



MEASURING THE RADIO REFRACTIVE INDEX 



refractonieter [29, 30], as illustrated in figure 2.6, utilizes one sealed cavity 
as a reference cavity and a ventilated cavity as the sampling element. 
Each of these cavities serves as a frequency-determining element of a 
Pound [31] oscillator. Very good cavity resonance is required. The 
Grain refractonieter is so designed that the cavity resonators operating 
in the 9400-Mc/s range are slightly different in resonant frequency. The 
resultant difference frequency is the center frequency from which depar- 
tures due to changing refractive index are measured. Recent models 
utilize a 43 Mc/s center frequency. As the refractive index of the con- 
tents of the sampling cavity changes, the difference frequency at the 
output of the mixer changes according to 



f — 



Then, since 



AN = An 10^ 
a change of 1 A^ unit corresponds to a change in the resonant frequency 




SAMPLING 
CAVITY 



REFERENCE 
CAVITY 














OSCILLATOR 




OSCILLATOR 




















MIXER 


























AMP 








DISCRIW 


INATOR 




RECO 


f\UtR 



Figure 2.6. The Crain refractonieter. 



DIRECT MEASUREMENT 



33 



SAMPLING 
CAVITY 




J — • — 


' 










PHASE 
METER 




RECORDER 






T 




1 







REFERENCE 
CAVITY 

Figure 2.7. The Bimbaum refradomeler. 



of the sampling cavity of approximately 9.4 kc/s if the nominal operating 
frequency is 9400 Mc/s. 

For the measurement of small scale variations of refractive index, a 
modified Grain refractometer utilizing a 10.7-Mc/s center frequency with 
a 200-kc/s linear range has been used [29]. Scales of 1 to 20 A^ units for 
full scale deflections of a 1-mA chart recorder have been used. 

The Birnbaum refractometer [32], illustrated in figure 2.7, applies the 
resonance principle in a somewhat different fashion. Both the reference, 
or sealed, cavity and the sampling, or open, cavity are passive elements. 
The cavities are slightly different in frequency and are of the transmission 
type having crystal detectors at the output. 

A single klystron, frequency modulated by a sawtooth voltage and 
having an output frequency linear with time, excites each cavity in se- 
quence. During the frequency excursion of the klystron, the respective 
cavity resonances are excited, and a pulse formed at each crystal detector. 
Since the resonant frequencies of the cavities differ, the two output pulses 
will be displaced in time. If the modulation on the klystron is periodic, 
then the output pulses will be also displaced in phase, the displacement 
being a function of the difference in resonant frequencies. Any change in 
the refractive index of the contents of the sampling cavity will alter the 
phase difference between the two pulse trains. The relative phase be- 
tween the two outputs can then be measured by an electronic phase meter. 



34 MEASURING THE RADIO REFRACTIVE INDEX 

The phase meter in use is merely a multivibrator having a constant ampli- 
tude output. The pulse from each cavity alternately switches the multi- 
vibrator "on" and "off." The width of the constant amplitude output 
pulse from the multivibrator is a measure of the time difference between 
pulses. The resulting train of constant amplitude variable width pulses 
is then applied to appropriate recording circuits. The Birnbaum refrac- 
tometer is adaptable to multi-cavity operation. A single klystron may 
be used to sweep simultaneously a number of spaced sampling cavities. 
A single reference cavity permits simultaneous comparison between the 
several sampling cavities, and important consideration when the scale and 
form of refractive index variations are desired ([27]. 

The accuracy of the Birnbaum instrument is dependent upon maintain- 
ing a linear frequency sweep. This problem can be circumvented in the 
manner of Sargent [16], who modified the Birnbaum refractometer to 
operate as a microwave hygrometer. A servomechanism is used to tune 
the sampling cavity to the resonant frequency of the reference cavity. 
The servomechanism positions a tuning probe in the sampling cavity. 
The depth of penetration is the measure of the refractive index of the 
contents of the sampling cavity. This technique minimizes the depend- 
ence on the sweep characteristics of the klystron. 

The Vetter refractometer [20], figure 2.8, virtually eliminates the de- 
pendence of the refractometer on electronic characteristics and shifts the 
limitations to the cavities themselves. In addition, while previous in- 
struments are primarily relative refractive index indicators, the Vetter 
refractometer was developed as an absolute refractive index device by 
using klystron stabilization techniques [33, 34, 35, 36]. 

Figure 2.8 is a simplified diagram of the Vetter refractometer illustrat- 
ing the basic principle of operation. The reference cavity is excited by a 
klystron which is modulated by a small sine voltage on the repeller. Any 
output at the fundamental modulating frequency at the detector of the 
reference cavity is compared in phase to the original modulating signal, 
and an error signal is applied to the repeller of the klystron to lock the 
center frequency of the klystron to the reference cavity. At coincidence, 
the fundamental disappears in the output of the reference cavity. The 
same klystron excites the sampling cavity. Any fundamental appearing 
at the output after phase comparison to the modulating signal develops 
another error signal. This error signal is used to drive a mechanical 
servomechanism which tunes the reference cavity to the resonant fre- 
quency of the sampling cavity with concomitant shifting of the klystron 
center frequency. This then is a double loop system; the reference cavity 
controls the klystron; the sampling cavity controls the reference cavity. 

Tuning of the reference cavity is accomplished by a motor-driven 
probe that penetrates the reference cavity. The design of the probe is 



DIRECT MEASUREMENT 



35 



6> 



MODULATING 
SINE WAVE 



PHASE SENSITIVE 

AMPLIFIER 
? 



ERROR SIGNAL 




SAMPLING 
CAVITY 



SERVO 
AMPLIFIER 




SERVO 



ERROR SIGNAL 



READOUT 



PHASE SENSITIVE 
AMPLIFIER 



Figure 2.8. Simplified diagram of the Vetter ref Tactometer. 



such as to permit very nearly a linear function of AA'^ versus probe pene- 
tration over the range of operation. Once the probe is calibrated with 
the cavity, the relative calibration is static and practically independent 
of the contents of the cavity. Although the reference cavity is sealed, 
there is no necessity for a vacuum. Absolute calibration is accomplished 
by determining the zero point, or the intercept, where A'' = 0, by evacuat- 
ing the sampling cavity. 

Since the Vetter refractometer is mechanically tuned by a servosystem 
having considerable inertia, the high-frequency response is relatively poor. 
The upper frequency response is 10 c/s. Accuracy and simplicity of 
operation are its distinct advantages. 

Refractometers have been used to measure both the surface value and 
the gradient of the refractive index. Although the units so far described 
were designed originally as ground-based equipment, modifications have 



36 MEASURING THE RADIO REFRACTIVE INDEX 

been made to convert them to airborne use. However, where vertical 
gradients to high ahitude are desired, practical aspects of aircraft opera- 
tion such as the turning radius and the angle and rate of climb or glide 
are factors influencing how well the vertical gradients are estimated. 
Considerable horizontal variation in N is often observed. In addition, 
the time involved in ascent or descent is relatively long, and the measure- 
ments are valid only under static conditions for the total time of measure- 
ment. Due to expense and weight, conventional refractometers are not 
economical or practical for use in balloon ascent, whether free or tethered. 
This led to the development of several lightweight refractometers to be 
used either in wiresonde, radiosonde, or dropsonde applications. 

Figure 2.9 is a block diagram of a lightweight (6 lb), expendable refrac- 
tometer developed by Deam [37] and Deam and Cole [38] for use as a 
balloon-borne unit or a dropsonde. The device is a Pound oscillator [31], 
operating at nominal frequency of 403 Mc/s; the instantaneous frequency 
is determined by coaxial cavity. The refractive index is sampled by the 
cavity, and is reflected as a capacitance in the tuned circuit of the oscil- 
lator. 

Operational tests indicate the electronics are sufficiently stable to pro- 
duce the desired accuracy although the cavity size and mass appear to 
introduce a sizable time constant. Present accuracy is estimated to be 
better than zb5 A'^ units for a complete profile. 

The Hay refractometer [39] was developed as a compromise between 
the microwave refractometer and the conventional radiosonde (see fig. 
2.10). It lacks the accuracy of the microwave refractometer by an order 
of magnitude but, weighing only 7 lb, it incorporates the lightweight 
feature of the radiosonde. As a balloon-borne instrument it fills a need 
for more accurate, faster probing of the refractive index at the higher 
elevations. 

This refractometer is a 10-Mc/s oscillator whose frequency is deter- 
mined by an air-sensing capacitor. A change in the refractive index of 
the air passing through the capacitor is reflected as a frequency change 
in the oscillator. The sensing capacitor is alternately switched with a 
reference capacitor of identical design but which is protected from the 
free atmosphere. The use of a reference capacitor diminishes the problem 
of temperature compensation since both capacitors will be at very nearly 
the same temperature. To further improve the temperature charac- 
teristics, the 12 plates of each capacitor are made of invar and are sepa- 
rated by 0.25-in. quartz spacers. The output of the 10-Mc/s oscillator 
is doubled in frequency and transmitted to the ground station. 



DIRECT MEASUREMENT 



37 



CORRECTION 
VOLTAGE. 



DETECTOR 



HF 
OSCILLATOR 



AMP 



PHASE 
DETECTOR 



4.5 MC 
OSCILLATOR 




CRYSTAL 
MODULATOR 



CAVITY 



Figure 2.9. The Deani expendable refradonieter {modified Pound oscillator). 



REFERENCE 
CAPACITOR 



r 



- _r 



. ^ 



SWITCH 



OSCILLATOR 



DOUBLER 
AMPLIFIER 



^ 



SENSING 
CAPACITOR 



Figure 2.10. The Hay refradometer. 



38 MEASURING THE RADIO REFRACTIVE INDEX 

2.4. Comparison Between the Direct and 
Indirect Methods of Measurement 

Absolute accuracy may not be the only consideration in the comparison 
between the direct and indirect method of measurement. Although 
refract ometers may be capable of superior accuracy, the factors of require- 
ments, economics, and availability of competent technical personnel may 
outweigh this advantage. Refractometers are relatively expensive, some- 
what complex, and require competent technical personnel to maintain, 
calibrate, and operate them. 

In many cases where average values or long-term statistics are adequate, 
the use of refractometers may not be indicated. The data on refractive 
index structure derived from weather service data has long been used 
successfully for the determination of average conditions. 

Where extreme accuracy is required, the use of refractometers is indi- 
cated. Radar and radio navigation are examples where accurate esti- 
mates of both surface values and gradient are necessary to determine the 
refraction through the atmosphere. The necessity for true vertical 
gradients would demand the use of a balloon-borne or dropsonde type of 
refractometer. In many applications the indirect method may be suffi- 
cient; however, determination of the fine structure of the refractive index 
appears to be presently limited to some type of radio-frequency refrac- 
tometer. 

2.5. Radiosonde Lag Constants 

2.5.1. Introduction 

The determination of N from radiosonde data is subject to all of the 
errors inherent in the radiosonde observation. Recently Wagner [40] 
has analyzed the errors in A^ arising from time lag of the sensing elements, 
data transmission techniques, and significant level-selection criteria. Of 
these sources of error, Wagner concludes that the time lag of the sensing 
elements is the most serious source of error. Further, for the southern 
California coastal inversions, Wagner concludes that only the time lag of 
the humidity strip need be considered. A similar conclusion has been 
reached by Clarke [41] for practical applications involving ship-borne 
radar and over-water air-to-air communications. Although there is a 
significant correction associated with the time lag of the U.S. radiosonde's 
lithium chloride humidity-sensing element, there is also a time lag in the 
temperature element, which, as will be shown, must also be taken into 
consideration. The correction for the temperature element yields a two- 
fold correction to A^ due to the actual error in temperature and the ancil- 
lary correction in vapor pressure resulting from the more correct estimate 
of the true saturation vapor pressure. This arises from the fact that 



RADIOSONDE LAG CONSTANTS 39 

when the lithium chloride element measures relative humidity it must be 
used with the saturation vapor pressure. Since the saturation vapor 
pressure of water is a function of temperature, an error in temperature 
produces an error in the estimated water vapor pressure. 

The lag constant of the lithium chloride (LiCl) humidity element be- 
comes significantly larger for temperatures below °C [42]. Wexler [11] 
has made detailed studies of the lag constants of the LiCl elements at low 
temperatures under laboratory conditions. Bunker [43], however, has 
found quite different lag constants in the free atmosphere. He attributes 
this discrepancy in lag constants to the laboratory-determined values 
which were obtained for isothermal conditions; whereas, in rising through 
the free atmosphere, the radiosonde normally passes from warm to cooler 
air. Bunker has raised a serious question, namely that the temperature 
lag of the lithium chloride element is possibly as important as the iso- 
thermal humidity lags studied by Wexler. It is quite possible that the 
interplay of these two lags could produce a total effect either greater or 
less than the humidity lag alone. We now have a quandary since there is 
not currently in the literature a complete analysis of the interplay of the 
temperature and humidity lags of the LiCl humidity element. Although 
Bunker considers some aspects of this problem, he does not consider the 
case of decreasing humidity and increasing temperature, a typical condi- 
tion giving rise to the superrefraction of radio waves. The differences 
between Wexler's and Bunker's estimates of the LiCl lag constants are so 
great as to make one wonder at the validity of applying any correction for 
this effect. Since neither Wexler's nor Bunker's tabulation of lag con- 
stants is complete, the choice of lag constant appears to be arbitrary. 
Wexler's lag constants will be adopted for the present discussion. Both 
the temperature and humidity elements are corrected (as much as is 
possible) for their respective time lags in this chai)ter for the purpose of 
preparing refractive index profiles. 

In what follows the theory of sensor time lag will be examined. Data 
from several climatically diverse locations will then be examined to 
illustrate the relative importance of the various lag constant corrections 
under conditions of superrefraction of radio waves. 

2.5.2. Theory of Sensor Time Lags 

Middleton and Spilhaus [44] give 

f = - -X (^' - "•) (2.1) 

as the basic differential equation of the time lag of a meteorological 
sensor, measuring the variable 6, where t is time, X the appropriate lag 



40 MEASURING THE RADIO REFRACTIVE INDEX 

constant, and the subscripts i and e stand for the indicated and environ- 
mental values, respectively. The solution to (2.1) depends upon the 
manner in which the environmental value of 6 varies. For example, if it 
is assumed that de varies linearly with time, 

Be -^ e, + ^t, (2.2) 

one obtains for the solution to (2.2) 

Be - 9, = +^X [1 - exp (-t/\)], (2.3) 

as compared to 

^ [(/3X - Dexp (t/X)]d, 
^ /3X exp m - exp {t/\) ' ^ ^ 

under the assumption that de varies exponentially with time, 

de = 00 exp i-^t). (2.5) 

For a column of air, one normally knows the initial reading of the 
sensing strij), 0o, and for an assumed linear decrease of 6 with height the 
coefficient (8 becomes 

rj _ Bj Pq /„ n\ 

^ ~ t-\[\ - exp (-«/X)]' ^ ^ 

with the result that (2.2) is written 

6e — Qo -{- ', 7T\ 1 . /x M ■ (2-7) 

i — X[l — exp ( — V^)J 

Once the value of X is determined, estimates of the true properties of the 
air can be found at all heights up to the point where the gradient changes. 
One may proceed by a different course by noting in (2.1) that 

dt ~ dh ' dt ~ ^ dh' ^ ^ 

where R is ascension rate of the radiosonde (assumed a constant 300 
m/min). 

If it is further assumed that di varies linearly between reported values 
{an assumption compatible with radiosonde reporting procedure), then 

R^ = R ^-'"-' ~['\ (2.9) 

dh hk+i — hk 

where the /cth and the (A- + l)st layers are the boundaries considered. 



THEORY OF SENSOR TIME LAGS 41 

Thus the environmental value of 6 can be determined from 

de,k+i = d.,k+i + RX ^ '^' ~ I'-' , (2.10) 

which involves, in a simple fashion, only the indicated or actually meas- 
sured values of the parameter d. 

One assumes that 6, and Oc are identically the same at /?. = and that 
successive application of (2.10) will yield a more realistic estimate of the 
distribution of 6 with height. When one applies a correction procedure 
of the form of (2.10) to temperature, where the temperature lag constant, 
\t, is always 3 sec [45], one obtains immediately the corrected temperature 
profile. The same is true for humidity (provided the temperature is 
greater than °C), when X/ is assumed to be always 10 sec [40]. Although 
there is some indication [46] that, for room temperatures, the humidity 
lag constant may be nearer 5 sec. For temperatures less than °C, how- 
ever, X/ is a function orf true temperature, true value of relative humidity, 
fe, and change of the true value of fe (see fig. 2.3). This means that one 
must use an iterative solution for fe since X/ will change as one's estimate 
of the true value of /^ and A/^, changes. Since our knowledge of X/ is essen- 
tially empirical, the correction procedure is limited to the temperatures 
and values of fe and Afe reported by Wexler [11]. 

In applying the above equations one generally assumes that the time 
lags are always known and that the environmental and indicated values 
are identical at the base of each layer [47]. These two conditions are 
approximately satisfied only for the ground layer since the total lag con- 
stant of the humidity strip is not known. It is not clear that any correc- 
tion for sensor lag may be made above the initial layer since, for subse- 
quent layers, the initial indicated and environmental values are not 
identical and, further, lag constants have not been determined for this 
case. 

2.5.3. Radiosonde Profile Analysis 

The utility of the above lag constant corrections is illustrated by 
analyzing past radiosonde data for the occurrence of ground-based radio 
ducts. A ground-based radio duct is one in which the gradient of N is 
sufficient to refract a radio ray to the same curvature as that of the earth. 
Thus, for ducting, since ray curvature is given by the gradient of the 
refractive index, 

4t < ~ = -157 A^ units/km (2.11) 

where ro is the earth's radius. The data analyzed were from the months 
of expected maximum duct occurrence at Fairbanks, Alaska (Feb.), 



42 MEASURING THE RADIO REFRACTIVE INDEX 

Washington, D.C. (May), and Swan Island, W.I. (Aug.). All data were 
for the year 1953. 

As an example of past work, Wagner [40] has assumed that the refrae- 
tivity lagged its environmental value according to (2.3) with the result 
that 

A^, - N, = /3.V X^ [1 - exp {-I/Xm)]. (2.12) 

After an analysis of the meteorological conditions of his area of applica- 
tion, he set \n = Xf = 10 sec; i.e., the time lag of A^ derived from pres- 
sure, temperature, and humidity was identically that of the humidity 
sensor for temperature <0 °C. Comparing this method of A'-lag correc- 
tion with the uncorrected data (first and last columns of table 2.1), one 
notes an increase in both intensity and incidence of ducts in all climates. 
This is particularly marked for Fairbanks, where a sixfold incidence in- 
crease is obtained. However, when one makes individual time-lag correc- 
tions for both temperature and humidity by means of (2.10) a quite 
different picture of duct statistics is obtained. For example, the Fair- 
banks data indicate a twofold increase rather than a sixfold increase. On 
the other hand, Washington shows an incidence of nine rather than six 
ducts with a marked increase in N gradient when one corrects for both 
temperature and humidity. 

The change of ducting statistics at all three locations obtained by the 
two methods of time-lag correction yields paradoxical results. The near 
contradiction of the two correction procedures is easily explained and 
serves as an illustration of the necessity of correcting both the tempera- 
ture and humidity elements for general application to ducting statistics. 

Consider typical temperature and humidity conditions associated with 
ground-based ducts within each climate. Such cases are shown in figure 
2.11. It is sufficient to note that the temperate ducts arise from typical 
radiation inversion conditions of increasing temperature and decreasing 
relative humidity with height; the arctic ducts are associated with the 
intense arctic radiation inversion with ground temperature near —25 °C 
and nearly constant relative humidity with respect to height; the tropical 
ducts, however, appear to be due to slight decreases of both temperature 
and humidity with height at temperatures near +25 °C. The effect of 
sensor lag upon these different gradients is always to make the indicated 
gradient appear less than the true or environmental gradient. Thus 
correcting for sensor lags makes the temperature and humidity gradients 
more intense. This, in turn, effects the resultant N gradient. One may 
write 

dN = f~dT + -^ de + jp dP (2.13) 



RADIOSONDE PROFILE ANALYSIS 



43 



Table 2.L Comparison of ground-based ducting statistics derived from various sensor 

time lag corrections 



Station 


No. 
profiles 


Reported data 


Corrected for T and / 
sensor lag 


Assuming X.v=X/=10 
sec and eq (12) 




No. 
ducts 


Average 
gradient 


No. 
ducts 


Average 
gradient 


No. 
ducts 


Average 
gradient 


Washington, D.C., 
(May), temperate 
climate 

Swan Island, (Aug.), 
tropical climate. __ 

Fairbanks, Alaska, 
(Feb.), arctic climate.. 


123 
62 
51 


1 

15 
2 


A'" units/km 

-163 
-186 
-211 


9 

20 

4 


N units/km 

-202 
-206 
-212 


6 
23 
12 


-181 
-238 
-250 



which, for normal conditions of 15 °C, 1013 mbar, and 60 percent RH, is 
approximated by 



AAT ^ -1.27 AT + 4.50 Ae + 0.27 AP. 



(2.14) 



Since, for ducting, a large negative gradient of A^ is required, the effect 
of temperature sensor lag correction is to make temperate and arctic 
temperature increases wath height more pronounced with a resultant 
larger contribution to a negative A^ gradient. Humidity lag corrections 
under all conditions lead to a larger decrease of RH with height than 
indicated and concomitant more rapid decrease of N with height. Such 
an explanation does not seem so evident for tropical conditions, however, 
where both temperature and relative humidity decrease with height. 
This apparent paradox is explained by the relatively great change in 
saturation vapor pressure of water associated with a small change of 
temperature near 25 °C which then produces the required large decrease 
of e and A'' with height. It is seen, then, that the temperature sensor lag 
correction produces an added humidity correction. Thus one must cor- 
rect both sensing elements for the purpose of preparing A^ profiles. Even 
in the case of the arctic inversion the interplay of the temperature and 
humidity lag corrections is very important, as has been noted by Yerg 
[48]. 

Thus, the correction for humidity sensor lag alone tends to overestimate 
ducting incidence since the corrected relative humidity decrease, coupled 
with the indicated temperature increase, produces a greater decrease in 
water vapor pressure than is actually present. 

2.5.4. Conclusion 

It appears from the present study that if studies of refractive index 
profile characteristics are to include sensor lag correction, then allowance 
should be made for both temperature humidity sensor time lags, regard- 
less of climate. Any systematic application of these conclusions to large 



44 



MEASURING THE RADIO REFRACTIVE INDEX 



bodies of data, however, must await the appearance in the literature of 
effective lag coefficients that combine the effects of both temperature and 
humidity lags upon the LiCl element. 



auw 












200 






















100 






















r\ 













300 



200 



100 



SWAN 


SLAND 












8-2C 


)-53 


1500 GMT 


































































I 


















1 






1 





































\ 












\ 






\ 










\ 


\ 


Due 



WASHINGTON, 


D.C. 












8-19 


-53, 


0900 


GMT 





























































'i 


- 




t 










Thickness 

i 1 










1 








25 50 75 100 -40 -30 -20 -10 10 20 30 40 

Relative Humidity Temperature in Degrees Centigrade 

Figure 2.11. Temperalure and humidity profiles. 



REFERENCES 45 

2.6. References 

Johnson, W. E. (Dec. 1953), An analogue computer for the solution of the radio 

refractive index equation, J. Res. NBS 51, No. 6, 335-342. 
Sprung, A. (1888), Bestimm. d. Luftfeuchtigk. mit Hiilfe d. Assmann'schen 

Asperationspsychrometers, Das Wetter 5, 105-109. 
List, R. J. (1958), Smithsonian Meteorological Tables, Washington, D.C. (Pub- 
lished by the Smithsonian Institution). 
National Physical Laboratory, Dept. Science Research (1960), The refractive 

index of air for radio waves and microwaves (Teddington, Middlesex, England). 
Jones, F. E., and A. Wexler (July 1960), A barium fluoride film hygrometer 

element, J. Geophys. Res. 65, No. 7, 2087-2095. 
Macready, P. B., Jr. (1960), Field applications of MRI Model 901 water vapor 

meter, Internal Memo Report, Meteorol. Research Inc. of Altadena, Calif. 
Stover, C. M. (Apr. 1961), Preliminary report on a new aluminum humidity 

element. Internal Tech. Memo, Sandia Corporation, Albuquerque, N. Mex. 
Hirao, K., and K. I. Akita (Oct. 1957), A new type refractive index variometer, 

J. Radio Res. Lab. 4, No. 18, 423-437. 
Crozier, A. L. (Apr. 1958), Captive balloon refractovariometer. Rev. Sci. Instr. 

29, No. 4, 276-279. 

Cline, D. E. (1957), International radiosonde comparison tests. Tech. Memo 
NRM-1907, U.S. Army Signal Eng. Lab. Task NR 3-36-11-402. 

Wexler, A. (July 1949), Low temperature performance of radiosonde electric 
hygrometer elements, J. Res. NBS 43, 49-56. 

Bean, B. R., and E. J. Dutton (Nov. 1961), Concerning radiosondes, lag con- 
stants, and refractive index profiles, J. Geophys. Res. 66, No. 11, 3717-3722. 

Misme, P. (Jan. 1956), Methode de mesure thermodynamique de I'indioe de 
refraction de I'air — description de la radiosonde MDI, Ann. Telecommun. 11, 
No. 1, 81-84. 

Clinger, A. H., and A. W. Straiton (May 1960), Adaptation of the radiosonde for 
direct measurement of radio refractive index. Bull. Am. Meteorol. Soc. 41, 
No. 5, 250-252. 

Thiesen, J. F. (Apr. 1961), Direct measurement of the refractive index by radio- 
sonde, Bull. Am. Meteorol. Soc. 42, No. 4, 282. 

Sargent, J. A. (May 1959), Recording microwave hygrometer, Rev. Sci. Instr. 

30, 345-355. 

Lement, B. S., C. S. Roberts, and B. L. Averbach (Mar. 1951), Determination of 

small thermal expansion coefficients by a micrometric dilatometer method. 

Rev. Sci. Instr. 22, No. 3, 194-196. 
Bussey, H. E., and G. Birnbaum (Sept.-Oct. 1953), Measurement of variation in 

atmospheric refractive index with an airborne microwave refractometer, J. Res. 

NBS 51, No. 4, 171-178. 
Grain, C. M., and C. E. Williams (Aug. 1957), Method of obtaining pressure and 

temperature in sensitive microwave cavity resonators. Rev. Sci. Instr. 28, 

No. 8, 620-623. 
Vetter, M. J., and M. C. Thompson, Jr. (June 1962), An absolute microwave 

refractometer, Rev. Sci. Instr. 33, 65()-660. 
Thompson, M. C, Jr., F. E. Freethey, and D. M. Waters (Oct. 1958), Fabrication 

techniques for ceramic X-band cavity resonators, Rev. Sci. Instr. 29, No. 10, 

865-868. 
Eraser, D. W., and E. G. Holmes (1953), Precision frequency control techniques, 

Unal Report Proj., 229-298 (Georgia List. Tech., State Eng. Exp. Sta.). 



46 MEASURING THE RADIO REFRACTIVE INDEX 

Adey, A. W. (1957), Microwave refractometer cavity design, Can. J. Tech. 34, 
519-521. 

Thompson, M. C, Jr., F. E. Freethey, and D. M. Waters (July 1959), End plate 
modification of A'-band TE cavity resonators, IRE Trans. Microwave Theory 
Tech. MTT-7, No. 3, 388-389. 

Herbstreit, J. W. (July 1960), Radio refractometry, NBS Tech. Note 66. 

Birnbaum, G. (Apr. 1951), Fluctuations in the refractive index of the atmosphere 
at microwave frequencies, Phys. Rev. 82, 110-111. 

Birnbaum, G., and H. E. Bussey (Oct. 1955), Amplitude, scale, and spectrum of 
refractive index inhomogeneities in the first 125 meters of the atmosphere, 
Proc. IRE 43, 1412-1418. 

Grain, C. M., and A. P. Deam (Aug. 1951), Measurement of Tropospheric Index 
of Refraction Profiles with an Airplane-Carried Direct Reading Refractometer, 
Electrical Engineering Research Laboratory, University of Texas, Austin, Tex. 

Grain, C. M. (Oct. 1955), Survey of airborne microwave refractometer measure- 
ments, Proc. IRE 43, No. 10, 1405-1411. 

Grain, G. M. (May 1950), Apparatus for recording fluctuations in the refractive 
index of the atmosphere at 3.2 centimeter wavelength. Rev. Sci. Instr. 21, 
No. 5, 456-457. 

Pound, R. V. (Nov. 1946), Electronic frequency stabilization of microwave 
oscillators, Rev. Sci. Instr. 17, No. 11, 490-505. 

Birnbaum, G. (Feb. 1950), A recording microwave refractometer, Rev. Sci. 
Instr. 21, No. 2, 164-176. 

Murray, G., and T. B. Watkins (Apr. 1957), Automatic frequency control, U.S. 
Patent 2,788,445. Filed Feb. 5, 1952, granted Apr. 9, 1957. 

Silsbee, R. H. (Jan. 1956), A high sensitivity paramagnetic resonance spectrom- 
eter, Gruft Lab. Tech. Report 221, Harvard Univ., Cambridge, Mass. 

Jung, P. (Oct. 1960), Transistorized frequency stabilization for reflex klystrons 
used in magnetic resonance, J. Sci. Instr. 37, 372-374. 

Smith, M. J. A. (Oct. 1960), Frequency stabihzation of klystrons, J. Sci. Instr. 37, 
398-399. 

Deam, A. P. (May 1959), An expendable atmospheric radio refractometer, EERL 
Report 108, Univ. of Texas, Austin, Tex. 

Deam, A. P., and C. F. Cole, Jr. (Jan. 1960), Development of a lightweight 
expendable microwave refractometer, EERL Report 5-46, Univ. of Texas, 
Austin, Tex. 

Hay, D. R., H. C. Martin, and H. E. Turner (June 1961), Lightweight refrac- 
tometer. Rev. Sci. Instr. 32, No. 11, 693-697. 

Wagner, N. K. (July 1960), An analysis of radiosonde effects on measured fre- 
quency of occurrence of ducting layers, J. Geophys. Res. 65, 2077-2085. 

Clarke, L. G. (1960), Theory of atmospheric refraction. Part II of Meteorological 
Aspects of Radio-Radar Propagation, pp. 31-82, NWRF3 1-0660-035 (U.S. 
Navy Weather Research Facility, Norfolk, Va.). 

Dunmore, F. W. (1938), An electric hygrometer and its application to radio 
meteorography, J. Res. NBS 20, 723-744. 

Bunker, A. F. (1953), On the determination of moisture gradients from radiosonde 
records, Bull. Am. Meteorol. Soc. 34, 406-409. 

Middleton, W. E. K., and A. F. Spilhaus (1953), Book, Meteorological Instru- 
ments, p. 63 (Univ. of Toronto Press, Toronto, Canada). 

Sion, E. E. (Jan. 1955), Time constants of radiosonde thermistors. Bull. Am. 
Meteorol. Soc. 36, 16-21. 

Wexler, A., S. Garfinkel, F. Jones, S. Hasegawa, and A. Krinsky (Aug. 1955), 
A fast responding electric hygrometer, J. Res. NBS 55, 71-78. 



REFERENCES 47 

[47] Wagner, N. K. (May 1961), The effect of time constant of radiosonde sensors on 
the measurement of temperature and humidity discontinuities in the atmos- 
phere, Bull. Am. Meteorol. Soc. 42, 317-321. 

[48] Yerg, D. G. (May 1950), The importance of water vapor in microwave propaga- 
tion at temperatures below freezing, Bull. Am. Meteorol. Soc. 31, 175-177. 



Chapter 3. Tropospheric Refraction 

3.1. Introduction 

If a radio ray is propagated in free space, where there is no atmosphere, 
the path followed by the ray is a straight line. However, a ray that is 
propagated through the earth's atmosphere encounters variations in 
atmospheric refractive index along its trajectory that cause the ray path 
to become curved. The geometry of this situation is shown in figure 3.1, 
which defines the variables of interest. The total angular refraction of 
the ray path between two points is designated by the Greek letter r, and 
is commonly called the "bending" of the ray. The atmospheric radio 
refractive index, n, always has values slightly greater than unity near the 
earth's surface (e.g., 1.0003), and approaches unity with increasing height. 
Thus ray paths usually have a curvature that is concave downward, as 
shown in figure 3.1. For this reason, downward bending is usually defined 
as being positive. 

If it is assumed that the refractive index is a function only of height 
above the surface of a smooth, spherical earth (i.e., it is assumed that the 
refractive index structure is horizontally homogeneous), then the path of 
a radio ray will obey Snell's law for polar coordinates: 

riir^ cos 6-2 = niVi cos 9i, (3.1) 

the geometry and variables used with this equation are shown in figures 
3.1 and 3.19. With this assumption r may be obtained from the following 
integral, 

Ti 2 = - / cot 5 — , (3.2) 

which can be derived as shown later in the chapter or as derived by Smart 

The elevation angle error, e, is an important quantity to the radar 
engineer since it is a measure of the difference between the apparent 



Figures in V:)rackets indicate the literature references on p. 87. 

49 



50 



TROPOSPHERIC REFRACTION 




Figure 3.1. Geometry of the refraction of radio waves. 



elevation angle, ^o, to a target, as indicated by radar, and the true eleva- 
tion angle. Under the same assumptions made previously, e is given as a 
function of r, n, and by 



e = Arctan 



cos T — sin T(tan 6) — 



tan ^0 — sin t — cos r tan 



(3.3) 



INTRODUCTION 



51 




Figure 3.2. Differential geometry used in the derivation of the effective-earth' s-radius- 

model atmosphere. 



The apparent range to a target, Re, as indicated by a radar, is defined as 
an integrated function of n along the ray path, 



r 

Jo 



Re = I ndR = 



ndh 
sin d 



(3.4) 



However, the maximum range error {R^ minus the true range) likely to 
be encountered is only about 200 m, hence the evaluation of (3.4) is not 
of great importance unless one is dealing with an interferometer or phase- 
measuring system. 



62 TROPOSPHERIC REFRACTION 

The integral for r, (3.2), cannot be evaluated directly without a knowl- 
edge of the behavior of n as a function of height. Consequently, the 
approach of the many workers in this field has been along two distinct 
lines: the use of numerical integration technicjues and approximation 
methods to evaluate r without full knowledge of 7i as a function of height, 
and the construction of mode) ?i-atmospheres in order to evaluate average 
atmospheric refraction. The following sections are devoted to a discus- 
sion of these methods. 



3.2. Limitations to Radio Ray Tracing 

The user should keep in mind that the equations given in the preceding 
section are subject to the following restrictions of ray tracing: 

(1) The refractive index should not change appreciably in a wavelength. 

(2) The fractional change in the spacing between neighboring rays 
(initially parallel) must be small in a wavelength. 

Condition (1) will be violated if there is a discontinuity in the refractive 
index (which will not occur in nature), or if the gradient of refractive 
index, dn/dr, is very large, in which case condition (2) will also be violated. 
Condition (1) should be satisfied if 



(dn /dh) per km 

T^ < (J.VVZjkc, 



where refract ivity, A^, is defined as .V = (n — 1) X 10^ and/kc is the carrier 
frequency in kilocycles per second [2]. Condition (2) is a basic require- 
ment resulting from Fermat's principle for geometrical optics. An atmos- 
pheric condition for which both conditions (1) and (2) are violated is 
known as "trapping" of a ray, and it can occur whenever a layer exists 
with a vertical decrease of A'' greater than 157 A'^-units per kilometer. A 
layer of this type is called a "duct," and the mode of propagation through 
such a layer is similar to that of a waveguide [3]. Taking into account 
refractive index gradients, a cutoff frequency may be derived for wave- 
guidelike propagation through a ducting layer [4]. 

In addition to the above limitations, it should be remembered that the 
postulate of horizontal homogeneity, made in order to use (3.1), is not 
realized under actual atmospheric conditions; some degree of horizontal 
inhomogeneity is always ])resent (see chapter 8). 



HIGH INTIAL ELEVATION ANGLES 



53 



3.3. An Approximation for High Initial 
Elevation Angles 

A method may be derived for determining ray bending from a knowl- 
edge only of n at the end points of the ray path, if it is assumed that the 
initial elevation angle is large. Equation (3.2) in terms of refractivity, 
N, is equal to 



Ti,2 = - / cot d dN ■ 10" 



(3.5) 



assuming w = 1 in the denominator. Integration by parts yields: 



Tl,2 = 



Ni.e., 



cot 8 dN ■ 10" 



A^,,e, 



N coid ■ 10" 



JA^i.fli 



i.A^j 



Bx.Ni 



N 
sin^0 



dd ■ 10' 



(3.6) 



Note that the ratio, N/s,m^d, becomes smaller with increasing 6 for values 
of 6 close to 90°. If point 1 is taken at the surface, then di = ^o and 
A^i = A^,. Then for ^o = 10°, No = and do = 7r/2, the last term of 
(3.6) amounts to only 3.5 percent of the entire equation, and for the same 
values of ^"2 and 62, but with ^0 = 87 mrad (-^5°), the second term of (3.6) 
is still relatively small (--^10 percent). Thus it would seem reasonable 
to assume that for 

do > 87 mrad (-^5°), 



the bending, n ,2, between the surface and any point, r, is given sufficiently 
well by 



or 



Tl,2 = — 



A cot X 10" 



^r.»r 



-I AT ,,.9(1 
Tl,2 = Ns cot do X 10-"^ - Nr cot Or X IQ-^ 



(3.8) 



The term —Nr cot dr X 10~^ is practically constant and small with re- 
spect to the first term, for a given value of do and r, in the range ^0 > 87 
mrad. Thus ti,2 is seen to be essentially a linear function of N s in the 
range do > 87 mrad. For bending through the entire atmosphere (to a 
point where Nr = 0), and for do < 87 mrad, (3.8) reduces to 



T = N, cot ^0 X 10- 



(3.9) 



54 TROPOSPHERIC REFRACTION 

For initial elevation angles less than about 5°, the errors inherent in this 
method exceed 10 percent (except near the surface) and rise quite rapidly 
with decreasing ^o- 



3.4. The Statistical Method 

Another method for determining high-angle bending is the statistical 
linear regression technique developed by Bean, Gaboon, and Thayer [5]. 
It has been found that for normal conditions and all heights the righthand 
integral of (3.6) is approximately a linear function of A''^ (^o, r constant) 
for 00 < 17 mrad ('~1°) and that the second term of (3.8) tends to be 
constant. Thus (3.6) reduces to a linear equation, 

ri,2 = hNs + a, (3.10) 

where h and a are constants (as in tables 9.1 to 9.9) and N s is the surface 
refractivity. 

The form of (3.10) is very attractive, since it implies two things: 

(1) The value of n ,2 may be predicted with some accuracy as a function 
only of N s (surface height and ^o constant), a parameter which may be 
observed from simple surface measurements of the common meteorological 
elements of temperature, pressure, and humidity. 

(2) The simple linear form of the equation indicates that, given a 
large number of observed ri,2 versus N s values for many values of h and 
da, the expected (or best estimate) values of h and a can be obtained by 
the standard method of statistical linear regression. 

This is what was done to obtain values listed in tables 9.1 to 9.9. 

Tables 9.1 to 9.9 also show the values of the standard error of estimate, 
SE, to be expected in predicting the bending, and the correlation coeffi- 
cients, r, for the data used in predicting the lines. Linear interpolation 
can be used between the heights given to obtain a particular height that 
is not listed in the tables. For more accurate results, plot the values of 
r from the tables (for desired N s) against height, and then plot the values 
of the standard error of estimate on the same graph. Then connect these 
points with a smooth curve. This will permit one to read the r value and 
the SE value directly for a given height. 

3.5. Schulkin's Method 

Schulkin has presented a relatively simple, numerical integration 
method of calculating bending for A-pro files obtained from ordinary 
significant-level radiosonde (or "RAOB") data [6]. The A profile ob- 
tained from the RAOB data consists of a series of values of A for different 



SCHULKIN'S METHOD 55 

heights; one then assigns to N(h) a Hnear variation with height in between 
the tabulated profile points, so that the resulting A^ versus height profile 
is that of a series of interconnected linear segments. Under this assump- 
tion, (3.2) is integrable over each separate linear A'^-segment of the profile 
(after dropping the n term in the denominator, which can result in an 
error of no more than 0.04 percent in the result), yielding the following 
result : 

An.2(rad) ^ - f"'"' cot d dn ^ '^^'^' ~ ""'^ 



tan di + tan 62 ' 
or 

ATi.2(mrad) ^ — „ . . — . (3.11) 

tan 0] + tan d-i 

For the conditions stated above, this result is accurate to within 0.04 
percent or better of the true value of An, 2, an accuracy that is usually 
better than necessary. Thus it is possible to simplify (3.11) further by 
substituting 6 for tan 6; this introduces an additional error that is less than 
1 percent if 6 is under 10° (^^175 mrad). Now (3.11) becomes 



2 (A/", — N'>) 
Ari,2(mrad) ^ - q^ j^ q^ , (3-12) 

(mrad) (mrad) 



where 6 may be determined from (3.58). 

The bending for the whole profile can now be obtained by summing 
up the An, 2 for each pair of profile levels: 

r.(mrad) S t ^ ^f "f"^'' . (3.13) 

(mrad) (mrad) 



This is Schulkin's result. The degi'ee of approximation of (3.13) is 
quite high, and thus most recent "improved" methods of calculating r will 
reduce to Schulkin's result for the accuracy obtainable from RAOB or 
other similar data. Thus, provided that the A'^-profile is known, (3.13) is 
the most useful form for computing bending (for all practical purposes) 
that should concern the communications or radar engineer. Some other 
methods have been published which are actually the same as Schulkin's, 
but have some additional desirable features; e.g., the method of Anderson 
[7] employs a graphical approach to avoid the extraction of s(iuare roots 
to obtain 0^. 



56 TROPOSPHERIC REFRACTION 

3.6. Linear or Effective Earth's Radius Model 

The classical method of accounting for the effects of atmospheric 
refraction of radio waves is to assume an effective earth's radius, a^ = ka, 
where a is the true radius of the earth and k is the effective earth's radius 
factor. This method, advanced by Schelleng, Burrows, and Ferrell [8], 
assumes an earth suitably larger than the actual earth so that the curva- 
ture of the radio ray may be absorbed in the curvature of the effective 
earth so that the relative curvature of the two remains the same, thus 
allowing that radio rays be drawn as straight lines over this earth rather 
than curved rays over the true earth. This method of accounting for 
atmospheric refraction permits a tremendous simplification in the many 
practical problems of radio propagation engineering although the height 
distribution of refractive index implied by this method is not a very 
realistic representation of the average refractive index structure of the 
atmosphere. This section will consider the refractive index structure 
assumed by the effective earth's radius model and how this differs from 
the observed refractive index structure of the atmosphere. Further, the 
limits of applicability of the effective earth's radius approach will be ex- 
plored and a physically more realistic model, the exponential, will be 
described for those conditions where the effective earth's radius model is 
most in error. 

It is instructive to give a derivation of the expression relating the curva- 
ture of radio rays to the gradient of refractive index. In figure 3.2 a 
wave front moves from AB to A'B' along the ray path. If the phase 
velocity along AA'isv and v + dv along BB', then, from considering the 
angular velocity, 

'- = '-^ (3.14) 

p p -j- dp 

or 

^ = ^ (3.15) 

V p 

where p is the radius of curvature of the arc A A'. Now, since the phase 
velocity, v, is 

V = - (3.16) 

n 

where c is the velocity of light in vacuo, one obtains 

dv^_dn (3^^) 

V n 



EFFECTIVE EARTH'S RADIUS 57 

combining (3.15) and (3.17), the familiar equation, 

i=-^f^, (3.18) 

p n dp 

is obtained. If the ray path makes an angle 6 with the surface of constant 
refractive index 

dr = dp cos d (3.19) 

and 

1 = _ i ^ cos e. (3.20) 

p n dr 

If the curvature of the effective earth is defined as 



tte a p 
then 



(3.21) 



and 



Ue = ka ^ .. ^ .. (3.22) 

1/a — 1/p 



k = ^ . (3.23) 

a dn 

1 + — 77 cos d 

n dn 



For the small values of 6 normally used in tropospheric propagation, cos 6 
may be set equal to unity. Further, by setting 

I - - fa ^'-'^^ 

one obtains the familiar value of k = 4/3 for the effective earth's radius 
factor. By assuming that the gradient of 7i is constant, a linear model 
of N versus height has been adopted. 
For this model, the bending 



ri.2 = — / cot d dn (3.25) 

is written 

-'=£'1^'* (3.26) 



58 



TROPOSPHERIC REFRACTION 



since 



and 



N = No - ^ 10 
4a 



dn = dN X 10-' = - ? 
4a 



(3.27) 



(3.28) 



Further, for the case hi = ho ^ 0, and 

< 00 < 10°, 

where do is the initial elevation angle of a ray, (3.26) may be approximated 
by 



TO,h — 



dli 
4ad ' 



The angle 6 may be determined from 



7. - )dl-h 2{N - No) +l(h- ho) • 10' 



1/2 



u-iir 



For the case when ^o = 0, (3.29) becomes 

1 I" dh 



TO.h 



2\/6a '^'^ Vh VQ 
1 



To.h — 



Vq 



1 Ih 

Vh/a. 



Now, from the geometrical relationship, 



To.h = ' h (do ~- Oh), 



(3.29) 



(3.30) 
(3.31) 



(3.32) 



(3.33) 



(3.34) 



one finds, for do = 0, 



do.h ~ a(ro,/i + dh), 



(3.35) 



MODIFIED EFFECTIVE EARTH'S RADIUS 59 

which upon substitution from (3.33) and (3.31) gives 



do.ft = V2/i(4/3) a, (3.36) 



or, more familiarly, 



do,h = V2kah . (3.37) 



A very convenient working formula is derived from (3.37) hy k = 4/3, 
a = 3960 miles and using units of miles for the ground distance to the radio 
horizon, do,h, and feet for the antenna height, h: 

do.h = V2h miles (3.38) 

This is the familiar expression often used in radio propagation engineer- 
ing for the distance to the radio horizon. 

3.7. Modified Effective Earth's Radius Model 

The effective earth's radius model, although very useful for engineering 
practice, is not a very good representation of actual atmospheric A^ struc- 
ture. For example, the data on figure 3.3 represent the average of 
individual radiosonde observations over a 5-yr period at several locations 
chosen to represent the extremes of refractive index profile conditions 
within the United States. The Miami, Fla., profile is typical of warm, 
humid sea-level stations that tend to have maximum refraction effects 
while the Portland, Me., profile is associated with nearly minimum sea- 
level refraction conditions. Although Ely, Nev., has a much smaller 
surface A'^ value than either Miami or Portland, it is significant that when 
its A'^ profile is plotted in terms of altitude above sea level, it falls within 
the limits of the maximum and minimum sea level profiles. It is to take 
advantage of this simplification, that altitude above sea level rather than 
height above ground is frequently used throughout this monograph. The 
A'' distribution for the 4/3 earth atmosphere is also shown on figure 3.3. 
It is quite evident that the 4/3 earth distribution has about the correct 
slope in the first kilometer above the earth's surface but decreases much 
too rapidly above that height. It is also seen, by noting that figure 3.3 is 
plotted on semi-logarithmic paper, that the observed refractivity distri- 
bution is more nearly an exponential function of height than a linear 
function of height as assumed by the 4/3 earth atmosphere. One might 
expect the refractivity to decrease exponentially with height since the 
first term of the refractivity equation (1.20) involving P/T, comprises at 
least 70 percent of the total and is proportional to air density, a well- 
known exponential function of height. 



60 



TROPOSPHERIC REFRACTION 



N 



40 



















































N 


^ 


Miami, August, 15:00 G.M.T. 
^Portland, Me., February, 15.00 G.M.T 






\ 


^ 






y, i^sL 


bruary 


, 15:00 GM.T 
















\ 


\ 


rApproximate data limits for 
\ individual radiosonde profiles 

Y^^^^ r Maximum 
\ ^^^^ rMean 
\ ^^^>/ yMlnimum 












^ 


* 


\ 


■ 


/3 ea 


rth pre 


ifiles 



2 4 6 8 10 

Altitude Above Mean Sea Level in Kilometers 

Figure 3.3. Typical N versus height distrihulions. 



One might wonder, in the light of the data of figure 3.3, why the effective 
earth's radius approach has served so well for so many years. It appears 
that this success is due to the 4/3 earth model being in essential agree- 
ment with the average A^ structure near the earth's surface which largely 
controls the refraction of radio rays at the small values of 6^ common in 
tropospheric communications systems. 

It would seem that the deficiency of the effective earth's radius ap- 
proach could be lessened by modifying that theory in the light of the 
average A^ structure of the atmosphere. An indication of the average A^ 
structure was obtained by examining a variety of N profiles which were 
carefully selected from 39 station-years of individual radiosonde obser- 
vations to represent the range of A'^ profile conditions during summer and 
winter at 13 climatically diverse locations. The results of this examina- 
tion are given in table 3.1. 



MODIFIED EFFECTIVE EARTH'S RADIUS 



61 



Table 3.1. 



Refradivity statistics as a function of altitude above sea level as derived 
from individual radiosonde observations 

















Altitude 


N 


Maximum A^ 


Minimum A' 


Range* 




{km) 










4. 




197.1 


209.5 


186. 5 


23.0 


5. 




172.3 
151.4 


184.0 
161.0 


165.0 
146.0 


19.0 


6. 




15.0 


7. 




134.0 


139.5 


129.5 


10.0 


8 




118.4 
104.8 


121.5 
108.0 


113.3 
100.0 


8.2 


9_ 




8.0 


10. 




92.4 


97.0 


86.0 


11.0 


11. 




81.2 


86.0 


70.0 


16. 


12. 




70.7 


76.0 


60.5 


15,5 


14. 




53.2 


60.0 


44.5 


15.5 



•Range = maximum A^ — minimum A^. 

It is interesting to note that the range of A^ values has a minimum at 
8 to 9 km above sea level but is systematically greater above and below 
that altitude. The average value of 104.8 at 9 km corresponds to a 
similar value reported by Stickland [9] as typical of the United Kingdom. 
Further the altitude of 8 km corresponds to the altitude reported by 
Humphreys [10] where the atmospheric density is nearly constant regard- 
less of season or geographical location. Since the first term in the expres- 
sion for refractivity is proportional to air density, and the water-vapor 
term is negligible at an altitude of 9 km, the refractivity also tends to be 
constant at this altitude. It seems quite reasonable, then, to adopt a 
constant value of A^ = 105 for 9 km, thus further facilitating the specifica- 
tion of model atmospheres. Further, as also noted in chai)ter 1, when 
the values of table 3.1 are plotted such as on figure 3.3, it is seen that the 
data strongly suggest that N may be represented by an exiwnential 
function of height of the form: 

N{h) = iVoexp {-bh}, 



in the altitude range of 1 to 9 km above .sea level. 

The following recommendation is made when dealing with problems 
involving ground-to-ground communications systems or other types of 
low-altitude radio propagation problems where the ray paths involved do 
not exceed 1, or at most 2, km above the earth's surface: use the effective 
earth's radius method to solve the associated refraction problems. The 
u.ser should refer to the tables in chapter 9, where effective earth's radius 
factors are tabulated along with other refractivity variables. Table 9.27 
may be entered with N ^ and table 9.28 may be entered with AN{N s sub- 
tracted from the N value at 1 km above the surface). In both these 
tables linear interpolation will suffice for any practical problem. The 
variables listed in these tables are for the exponential model of N{h) that 
is covered in the following .subsection. 



62 TROPOSPHERIC REFRACTION 

When the effective earth's radius treatment is used, height is 
calculated as a function of distance, for a ray with 60 = 0, with the 
equation h = d^/2ka, where d is the distance, k is the effective earth's 
radius factor, and a is the true radius of the earth ('^6373 km). The 
errors likely to be incurred when using this equation, assuming as a true 
atmosphere an exponential N{h) profile as given in the following sub- 
section will not exceed 5 percent for heights of 1 km or less. 

The preceding background discussion has presented the material neces- 
sary for the consideration of the suitability of various models of refrac- 
tivity to describe atmospheric refraction of radio waves. As a guide to 
what follows, let us ask what a logical sequence of models (or assumptions) 
would be to describe the effects of atmospheric refraction. 

One such sequence might be: 

(1) Assume an invariant model that is near to the actual average 
conditions and facilitates the calculation of radio field strengths. This 
has been done by the 4/3 earth model. 

(2) Assume a variable effective earth's radius factor for the calculation 
of radio field strengths in various climatic regions. This approach has 
been followed by Norton, Rice, and Vogler [11]. When it has become 
apparent that the effective earth's radius approach is inadequate, one 
might proceed by : 

(3) Correctmg the effective earth's radius model by assuming a more 
realistic A^ structure in the region where that model is most in error. This 
"modified effective earth's radius" model would then maintain, for some 
applications, the advantages of the original model. 

(4) Assume an entirely new model of A'^ structure guided by the 
average N structure of the atmosphere. 

It is assumed that models (3) and (4) would allow for seasonal and 
climatic changes of the average refractive index structure of the atmos- 
phere. 

In the following sections, models (3) and (4) will be developed and 
tested by their relative agreement with the ray bendings obtained from 
actual long-term average A^^ profiles. 

The first model of atmospheric refractivity that will be considered is 
based upon the effective earth's radius concept in the first kilometer. In 
this atmosphere N is assumed to decay linearly with height from the sur- 
face hs to 1 km above the surface hs + 1. This linear decay is given by 

N{K) = N, -\- {h - hs) AN,hs < h < hs + I, (3.39) 

where 

-AA^ = 7.32 exp (0.005577 A^.). (3.40) 



MODIFIED EFFECTIVE EARTH'S RADIUS 



63 



This last relationship comes from the observed relationship between 
6 to 8 year averages of daily observations of A'^s and AA^, the difference 
between N s and the value of A'' at 1 km above the earth's surface: 

-AA^ = A^,, - A^(l km). 

It is evident from figure 3.4 that for average conditions, a relationship 
exists between AA'' and A^'^. The least squares determination given by 
(3.40) was based upon 888 sets of monthly mean values of AA'^ and N s 
from 45 United States weather stations representing many diverse cli- 
mates. This relationship between AA'^ and N s is expected to represent 
the best estimate of a majority of individual profiles and certainly will 
closely agree with average conditions for the United States with one 
notable exception, southern California in the summer. These data, al- 
though shown on figure 3.4, were excluded from the least squares deter- 
mination due to their singular large range of AA'' compared to their small 
range of A''^ which resulted in a marked "finger" of points rising from the 
main body of the data. This obvious departure of data points (24 points 
out of a total of 912) plus the well-known unique nature of the southern 
California summer climate were taken as sufficient justification for ignor- 
ing these points, although, as shall be seen, the ray bendings based upon 



-AN 



90 




































80 
70 




































































^^_— 


60 






























■ . . ■ - 


^ 


























■;, .i 


^ 


^^ 


7^ 


























'■><' 


■ffirf^' 






























*ir«^ 


'^■'; 


\ 


~^^ 




rr^r^r, 






30 






'.v'JJj:, 


■^ 


'i0 


M 


M 


si:^ 










888 sets of data 




































20 
15 









































































































230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 



Figure 3.4. Eighl-year average N, versus 6-year average AN at 0300 and 1500 GMT 



64 TROPOSPHERIC REFRACTION 

the AA^^ obtained from (3.40) are in rather good agreement with the values 
calculated from actual A'' profiles, even those observed in southern 
California. 

Equation 3.40 offers a convenient method of specifying various models 
of the refractivity structure of the atmosphere, since it allows an estima- 
tion of the value of A^ at 1 km in addition to the two values already 
known; i.e., A''^ and A'' = at /i = oo. 

It may be further assumed that A'^ decreases exponentially from hs + 1 
to a constant value of 105 at 9 km above sea level. In this altitude 
range N is defined by: 

A^ = A^i exp \-c{h - h, - l)\,hs + I < h < 9 km, (3.41) 

where 

^ S - h, 105 ' 

and A^^i is the value of A'' at 1 km above the surface. 

Above the altitude of 9 km, where less than 10 percent of the total 
bending occurs, a single exponential decrease of A'^ may be assumed. The 
coefficients in the exponential expression: 

A^ = 105 exp { -0.1424 (h - 9)},h> 9 km, (3.42) 

were determined by a least squares analysis of The Rocket Panel data 
[12]. This expression is also in agreement with the ARDC Model Atmos- 
phere 1956 [13] and Dubin's [14] conclusion that a standard density- 
distribution may be used to determine the refractivity distribution at 
altitudes in excess of 20,000 ft. 

The three-part model of the atmosphere expressed by (3.39)-(3.42) has 
the advantage of the effective earth's radius approach, particularly for 
such applications as point-to-point radio relaying over distances up to, 
say, 100 mi, where the radio energy is generally confined to the first 
kilometer, plus being in reasonably good agreement with the average A^ 
structure of the atmosphere. The reader is cautioned, however, that 
application of this model to mode-theory calculations would be mislead- 
ing, since the resultant diffraction region fields would be enhanced by the 
addition of strong reflections from the r?-gradient discontinuities at /is + 1 
and at 9 km. The specific combinations of A^s, hs, and AA^, that define 
the CRPL Reference Atmosphere — 1958 are given in table 3.2. 

The station elevations corresponding to given combinations of N s and 
AA'^ were chosen to correspond with an average decay of A^ with station 
elevation. Although the error in neglecting this height dependence has 



EXPONENTIAL MODEL 65 

Table 3.2. Constants for the CRPL Reference Atmosphere — 1958 



N, 


/!, 


a' 


-AN 


k 


Oe 


c 




ft 


mi 






mi 


per km 








3960. 0000 





1.00000 


3960. 00 





200 


10,000 


3961.8939 


22. 3318 


1. 16599 


4619. 53 


0. 106211 


250 _-.. 


5,000 


3960. 9470 


29. 5124 


1. 23165 


4878. 50 


0.114559 


301 


1,000 


3960. 1894 


39. 2320 


1.33327 


5280. 00 


0. 118710 


313.. _. 


700 


3960. 1324 


41.9388 


1.36479 


5403. 88 


0. 121796 


350 





3960. 0000 


51.5530 


1. 48905 


5896. 66 


0. 130579 


400-. 





3960. 0000 


68. 1295 


1. 76684 


6996. 67 


0. 143848 


450 





3960. 0000 


90. 0406 


2. 34506 


9286. 44 


0. 154004 



Note: Oe is the effective earth's radius and is equal to a'k, a' = a + hs, where hs is the altitude of the earth's 
surface above sea level, a = 3960 miles and c = 1/8— /is In Ni/105. 



been estimated to be no more than a few percent, it could be important 
in such high precision applications as radar tracking of earth satellites. 
It should be remembered in subsequent applications that a unique feature 
of these reference atmospheres is the dependence of N s on the altitude of 
surface above sea level. This feature was built in so that the reference 
atmospheres would be completely specified by the single parameter N s. 

3.8. The Exponential Model 

The next model of the atmosphere to be considered may be specified 
by assuming a single exponential distribution of A^: 



where 



N ^ Ns exp {-Ce {h - hs)], 



Ce = In 



Ns 



N{1 km) 



In 



Ns 



Ns + AN 



(3.43) 
(3.44) 



and these equations are used to determine A'^ at all heights. This model 
of atmospheric refractivity is a close representation of the average re- 
fractivity structure within the first 3 km. Further, the single exponential 
model has the advantage of being an entire function, and therefore is easily 
used in theoretical studies. This model of the atmosphere has been 
adopted for use within the National Bureau of Standards with specific 
values of the constants in (3.43) and (3.44). These constants are given in 
table 3.3 and specify the CRPL Exponential Reference Atmosphere — 
1958. 

Figure 3.5 compares the A'^ structure of the above two models plus the 
4/3 earth model. It can be seen that the assumption agrees with the 
reference atmosphere in the first kilometer, which is to be expected since 
A'^g = 301 is the value required to yield the 4/3 gradient from figure 3.5. 
Figure 3.5 illustrates the essential agreement of the reference atmosphere 



66 TROPOSPHERIC REFRACTION 

Table 3.3. Table of the constant Ce for the CRPL exponential radio refractivity 

atmospheres 



N=1S 


^exp [— c,(ft 


-h,)] 


AN 


Ns 


Ce 






(per km) 











22. 3318 


200.0 


0. 118400 


29.5124 


250.0 


. 125625 


30. 0000 


252.9 


. 126255 


39. 2320 


301.0 


. 139632 


41.9388 


313.0 


. 143859 


50. 0000 


344.5 


. 156805 


51. 5530 


350.0 


. 159336 


60. 0000 


377.2 


. 173233 


68. 1295 


400.0 


. 186720 


70. 0000 


404.9 


. 189829 


90. 0406 


450.0 


. 223256 



with the Rocket Panel and ARDC data. The exponential reference at- 
mosphere is also shown on figure 3.5 for N s = 313, the average value of 
the United States. The exponential reference atmosphere appears to be 
a reasonable single line representation of A'^ throughout the height interval 
shown. The differences between the various models becomes more 
apparent by examining their agreement with observed N profiles over the 
first 10 km as in figure 3.6. 

The reference and exponential reference atmospheres are given for the 
N profiles corresponding to near-maximum N s (Lake Charles, La.) and 
near minimum-at-sea-level Ns conditions (Caribou, Me.). The two 
reference atmospheres were determined solely from the N s values of each 
profile. Several observations can be made of these data. First, the 4/3 
earth model closely represents the slope of the minimal N s profile over 
the first kilometer, but then decreases too rapidly with height. Note, 
however, that the 4/3 earth model with its constant decay of 39.2 N units 
per kilometer would be a very poor representation of the maximum pro- 
file which decreases over 66 N units in the first kilometer. The exponen- 
tial reference atmosphere is in good agreement with the initial N distribu- 
tion but tends to give values systematically low above approximately 
3 km. At first glance, the exponential reference atmosphere does not 
appear to be as good a representation of the two observed profiles as the 
reference atmosphere, particularly above approximately 5 km. Subse- 
quent analysis of the refraction obtained from the two model atmospheres 
will show that this systematic disagreement of the exponential reference 
atmosphere in the 5- to 20-km interval is a minor defect of the model 
compared to its closer agreement with observed N distributions over the 
first 1 to 3 km. This is particularly true for the higher values of N s such 
as that for Lake Charles. 

The above models are more in agreement with long-term mean N pro- 
files than is the 4/3 earth model. The application at hand would aid in 
deciding which of the reference atmospheres would be most useful. To 



EXPONENTIAL MODEL 



67 



o 

Q 
< 



500 

300 
200 

100 
70 
50 

30 
20 

10 
7 
5 

3 
2 



0.7 
0.5 

0.3 
02 

0.1 



1 ' 


1 


1 I.I 1 

RFFFRFMr.F 


1 1 1 ! ' ' 1 1 1 r- ■ 
RFFRflr.TIVITY ATMnc;PHFRF 


1 


V „ f-, f-,, f 


1 ~ " — 1 1 1 


\ ICRPL reference 




Height Interval 


Expression for N |J 


^|«/ 1 refractivity 
^^v atmosDhere -1958 


hjS h< Pj+I 


N=N5-{h-hg) AN 










hs+l<h< 9km 






^^! 


' N^^ 301 
350 


N=N,exp{-(g.',^i'n ^^ )(h-h3-l)} 




Nr^ 


'^\ 


HUU 




h>9km 


N = I05 exp {-0.1424 {h-9)} 






"T 


S> 








1 






— 






















— 










\ 


S' 


V 








1 






































\ 




sHi. 






et Panel 






































\ 




N'nV 
















































V 


^ 


1 'II 










































6 


^N= 313 exp (-0.1439 h) 








































^\ 
























































\ 


\ 


\ 






















































\ 


\ 


\ 


~ 






-^ 


Least Squares fit to Rocket Panel data 


























^ 








forh > 9 km- adjusted to pass through 
























^ 






























' 








N= 105 at 9 km 




























\ 


















1 
















- — 


- N UistriOution tor a 
4/3 Earth Amosphere 




^\ 


^ 




































\ 


k\ 






rCRPL reference refractivity 
Atmosphere -1958 










_! 1 1 1 1 1 1 






\ 


K 












n 






















\ 


^ 


s. 








































Ati 


Ah 

■no. 


sph 


7 n 
ere 


wd 


el- 
95 


s 




^ 




V 


~' 


Vr. 


j/j 


exf. 


iH 


1.14 


59 


h) 






































r 


\V 






















































^s, v 






















































N^vN 
























































\, 
































CRPL exponential reference " 
atmosphere - 1958 






O 
































, 




'^ 


X 




































^ 


^ 






















































\ 


\ 


\ 






















































\ 


% 


\ 


\ 



4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 
ALTITUDE IN KILOMETERS ABOVE SEA LEVEL 

Figure 3.5. CRPL reference refractivity atmospheres — 1958. 



aid in distinguishing between the various models, the following sections 
will be concerned with comparing the ray bending between the models. 
A comparison of the ray paths in the reference atmospheres in the 4/3 
earth model will illustrate the systematic differences between these 
models. Such a comparison is given in figure 3.7 for a distance of 200 mi 
and a height of 14,000 ft and in figure 3.8 for a distance of 800 mi and a 
height of 240,000 ft. The particular graphical presentation used in 
figures 3.7 and 3.8 shows the 4/3 earth rays as straight lines. It is noted 
that the 4/3 earth ray at 0o = is in relatively good agreement with the 
values from the reference atmosphere for distances out to 200 mi and 
heights up to 14,000 ft, but systematically departs from tlu^ n^ference 



68 



TROPOSPHERIC REFRACTION 



m 
















! 


\. 










1 




1 1 


360 

340 




Reference atmosphere 

Exponential reference 

atmosphere 




\ 






















^ 


320 
300 
280 
260 
240 
??0 














\ 


V 


















\ 




















\ 


NX 


\ 






































^\ 


\ 


















\ \ 


\ \\ 


X/ 














200 


//Vy = 384.1 Lake Charles, La. 
July, 03:00 G.M.T 








^ 


\\\ 

\ "^ 
\ \ 
\ \ 


\\> 




1 1 








N^^ 302.8 Caribou, Me. 
March, 03:00 CM.T. 




180 
160 

140 




















"1 


\ 






















\ 
















A 


\ 


^ 










\4/J Earth\ 


120 

100 

90 














^ 


\ 


\ 
\ 

\ 


\ 
\ 

\ 


\ 


\ 












\ 




\ 

\ 
\ 

\ 


\ 
\ 

\ 

\ \ 


V 



2 3 4 5 6 7 

HEIGHT IN KILOMETERS 



Figure 3.6. Comparison of reference atmospheres with observed N profiles. 



EXPONENTIAL MODEL 



69 



atmosphere for greater distances and heights. For a range of 600 mi, 
where the ray reaches heights of about 200,000 ft, the 4/3 earth ray is 
some 9000 ft lower than the N s = 400 reference atmosphere and 36,000 ft 
lower than the N s = 250 reference atmosphere. This height discrepancy 
is due to the 4/3 earth model's unrealistically large A'^ gradient at great 
heights with resultant increased bending. 



,"^3e^.^ 




Figure 3.7. Comparison of rays in the CRPL reference refradivity atmospheres — 1968 
and the 4/3 earth atmosphere. 



t^=S' 




3.8. Comparison of rays in the CRPL reference refradivity atmospheres — 1968 
and the 4/3 earth atmosphere. 



70 



TROPOSPHERIC REFRACTION 



en 

< 

< 
CC 




T at 70 km 
Mean profiles: 15.12 mrad 
Mean exp. ref.; 15.23 mrad 
4/3 earth ; 30.25 mrad 



6 8 10 12 14 

h-hs IN KILOMETERS 

Figure 3.9. Bending versus height. 



16 



20 



Further, the bending in the 4/3 earth atmosphere is compared with 
that in the exponential reference atmosphere in figure 3.9. The bending 
in an "average" atmosphere is also given. This average atmosphere is 
a composite of the 5-year mean profiles for both summer and winter at 
the 11 U.S. radiosonde stations enumerated in the following paragraph, 
and was used as a readily available measure of average conditions. The 
important point made by figure 3.9 is that the 4/3 earth model is sys- 
tematically in disagreement with average bending; at low heights it gives 
too little bending, while at high altitudes it gives too much bending. The 
exponential reference atmosphere does not appear to be systematically 
biased, and deviates less than 5 percent from the average atmosphere. 
It is significant that the exponential reference and the average atmosphere 
are in essential agreement as to the shape of the r-height curve. 

It would now be instructive to compare the bendings obtained from 
the various models with values obtained from each of the 5-year mean 
A^ profiles from different climatic regions. The 5-year mean A'^ profiles 
were obtained for both summer and winter for a variety of climates as 
represented by the states of Florida, Texas, Maine, Illinois, Nevada, 
California, North Dakota, Washington, Nebraska, Wyoming, and by the 
District of Columbia. 

Comparisons of the bending obtained from the 4/3 earth model and 
the bendings obtained from the 5-year mean A^ profiles with the reference 
atmospheres are shown in figures 3.10 and 3.11. These figures were 
selected to illustrate the range of agreement between the models and the 
expected long-term average bendings. Figure 3.10 gives a comparison 



EXPONENTIAL MODEL 



71 



(I) 



24 



22 



20 



18 



< 16 
Q 
< 
DC 

-J 14 



^ 12 



10 



I Washington, D.C. 
August, 1500 

2 Omaha, Nebr. 
August, 1500 

3 Santa Maria, Calif 
August, 1500 




200 



300 



400 



500 



N. 



Figure 3.10. Comparison of j versus Nb as obtained from CRPL reference atmos- 
pheres — 1958 and 5-year mean radiosonde data for ^o = 0, h — he = 3 km. 



for a small initial elevation angle, do = 0, and a small height increment, 
h — hs = 3 km, and shows that both reference atmospheres tend to set a 
lower limit to the bendings. In this case, the exponential reference atmos- 
phere appears to be in better agreement with the expected long-term 
mean bendings than does the reference atmosphere. The numbered data 
points for Washington, D.C, Omaha, Nebr., and Santa Maria, Calif., are 
of special interest. Washington and Omaha have the only long-term 



72 



TROPOSPHERIC REFRACTION 



h 



1.6 



1.5 



1.4 



(f) 

-z. 

< 1.3 

Q 
< 

-J 1.2 



1.0 



0.9 



08 



0.7 



1 


/ 


4/3 Earth Bending = 5.1 nnrad 




0o = l5°, h-hs = 70km 


/ 


Exponential ~~~~-.,^_^^^^^ / 
Reference ^^~^~W^ 




Atr 


nosphere / ! 

/^Reference 










/ 





























200 



300 



400 



500 



Nc 



Figure 3.11. Comparison of t versus Ne as obtained from CRPL reference atmos- 
pheres — 1958 and 5-year mean radiosonde data for do = 15°, h — ha = 70 km. 



mean A^ profiles with initial N gradients (i.e., — 112/km and — 106/km, 
respectively) that are significantly greater than would be expected from 
the AA^ versus A^^ relationship. Both of these stations have an unusually 
large humidity decrease near the ground. The third point, Santa Maria, 
Calif., is of interest since it is in relatively good agreement with the refer- 
ence atmospheres, even though it represents the southern California 
summer climate which was excluded from the original AN versus Ns 



EXPONENTIAL MODEL 73 

relationship. This agreement is attributed to the fact that the reference 
atmosphere is a good representation of the A^ distribution below the 
California elevated inversion and to the fact that a majority of the bend- 
ing is accomplished below the elevated inversion height of about 500 m. 
Further, it can be easily shown that the bending integral is increasingly 
insensitive to strong A'' discontinuities as the height increases. 

Figure 3.11 shows a similar comparison for a high initial elevation angle, 
00 = 15° and a large height increment, h — hs = 70 km. This compari- 
son shows that both of the reference atmospheres are in closer agreement 
with the long-term mean bendings than are the 4/3 earth bendings. Note 
that, whether r is predicted from A^^ or AN, the 4/3 earth model gives but 
a single value of bending that is outside the limits of the values of r ob- 
tained from the long-term mean profiles. 

In considering the comparisons of figures 3.10 and 3.11, one might ask 
if they reflected the form of the basic equation for bending; namely, at 
low angles is r determined by the A^ gradient throughout the A'" profile, 
and at high angles is r essentially a function of the value of A^ at both ends 
of the A^ profile (i.e., the limits of integration). Thus one might expect 
the deviations to be smaller if the comparisons were made on the basis of 
a function of the A^ gradient such as AA^, particularly for small values of 
do. Such a comparison is given by figures 3.12 and 3.13 for the same 
initial elevation angles and height increment as before. It is seen that 
the AA^-specified reference atmospheres improve the agreement for the 
low-angle case, but decidedly decrease the agreement for the high-angle 
case. 

A numerical evaluation of the root mean scjuare (rms) deviation of the 
long-term mean bendings from both the reference atmospheres deter- 
mined as a function of both AA^ and A''^ was made for a variety of initial 
elevation angles for the height increments 3 and 70 km. Root mean 
square deviations were not calculated for the 4/3 earth model since it was 
felt that this model was obviously in marked disagreement with the long- 
term mean bendings under these conditions. Figure 3.14 summarizes 
the rms deviations for the h — hs = 3-km case. It is seen that for 
do < 10 mrad (about 0.5°), the AA^-specified reference atmospheres have 
the smaller rms deviations. Also, the exponential reference atmospheres, 
whether specified by AA'^ or A^^, have smaller rms deviations than the 
reference atmosphere. 

It is seen for the 70-km case, figure 3.15, that the A^s-specified reference 
atmospheres have a significantly smaller rms deviation than the AA^- 
specified atmospheres for do > 5 mrad. Again it is seen that the exponen- 
tial i-eference atmosphere generally has the smaller rms deviation for 
values of do less than 10 mrad. Howev(M', the slightly smaller rms devia- 
tions associated with the reference atmosphere for ^o > 10 mrad reflect 
that model's closer agreement with the actual A structure of the atmos- 
phere at high heights. 



74 



TROPOSPHERIC REFRACTION 



C/0 

< 

< 

or 



h- 



24 



22 



20 



18 



14 



12 



10 



1 Washington, D.C. 
August, 1500 








/ 


2 Omaha, Nebr. 
August, 1500 

3 Santa Maria, Calif. 








// 


August, 1500 


.' 


2 




/ 


'fc 


^, M Mg 
















Exponeniiui — 

Reference 

Atmosphere 


/ 

/ / 
/ / 










• 


* / / 


,3 








• 

4 


.•/ 




^Reference 
Atmosphere 






/ 


^ 












/ 


x^ 


73 Eai 


'th Bending 




// 










/ 











20 30 40 50 60 70 80 90 

-AN 



Figure 3.12. Comparison of t versus AN as obtained from CRPL reference atmos- 
pheres — 1958 and 6-year mean radiosonde data for 0o = 0, h. — he = S km. 



EXPONENTIAL MODEL 



75 



C/) 
< 
< 



1.5 



1.4 



1.2 



^ 1.0 



0.9 



08 



0.7 



0.6 



1 [ ] 


/ 


V 


4/3 Earth Bending = 5.1 mrad 






E XP'~"^^i^'^i'^l 








Reft 
Atn 


"irence ^*y/^ 
losphere ^ 


7^ 

Reference 
<\tmospht 




^0=15° 


h-hs = ' 


rakm 




3re 






• y 


• 


2 
.2 








•• 


» 












/ *• 














/ • 












1 






1 


Washing 


fon,D.C. 




1 






2 


August, IbUU 
Onnaha,Nebr. 
August, 1500 








3 


Santa IV 
August, 


aria,Ca 
1500 


if. 



20 30 40 50 60 70 80 90 

-AN 



Figure 3.13. Comparison of t versus AN as obtained from CRPL reference atmos- 
pheres — 1958 and 6-year mean radiosonde data for do = 15° h — hs = 70 km. 



76 



TROPOSPHERIC REFRACTION 




q: I 



0.1 



Mean bending is that for 
reference atnnosphere Ns=330 



•Reference Atmosphere 
■Exponential Reference Atnnosphere 



0.01 



0.1 



I 10 

IN MILLIRADIANS 



100 



1000 



Figure 3.14. Covi-parison of percent RMS deviations of 5-year mean profile bendings 

about CRPL reference atmospheres using two parameters, Ng and AN for 

h — hs = S km. 



100 



Eh 




Mean bending is that for 
reference atnnosphere Ns = 330 



•Reference Atmosphere 
-Exponential Reference Atmosphere 



1.0 10 

IN MILLIRADIANS 



1000 



Figure 3.15. Comparison of percent RMS deviations of 5-year mean profile bendings 

about CRPL reference atmospheres using two parameters, Ns and AN for 

h - hs = 70 km. 



DEPARTURES-FROM-NORMAL METHOD 77 

3.9. The Initial Gradient Correction Method 

The importance of the initial gradient in radio propagation, where the 
initial elevation angle of a ray path is near zero, has long been recognized. 
For example if dn/dh = — 1/a (the reciprocal of the earth's radius), then 
the equation for r is indeterminate, an expression of the fact that the 
ray path remains at a constant height above the earth's surface. This is 
called ducting, or trapping of the radio ray. The effect of anomalous 
initial A^-gradients on ray propagation at elevation angles near zero, and 
for gradients less than ducting ( | dN/dh \ < 157 N units/km, or dN/dh > 
— 157 A^ units/km) may also be quite large. A method has been devel- 
oped for correcting the predicted refraction (from the exponential refer- 
ence atmosphere) to account for anomalous initial A'^ gradients, assuming 
that the actual value of the initial gradient is known [2]. 
The result is 

Th = Th (Ns, So) + [riooiN*, do) - T100{N„ do)], (3.45) 

where th (N s,do) = r at height h, for the exponential reference atmosphere 
corresponding to A''^, and N s* is the A''s for the exponential reference 
atmosphere that has the same initial gradient as the observed intiial 
gradient; t^qq is r at a height of 100m. 

This procedure has the effect of correcting the predicted bending by 
assuming that the observed initial gradient exists throughout a surface 
layer 100 m thick, calculating the bending at the top of the 100-m-thick 
layer, then assuming that the atmosphere behaves according to the ex- 
ponential reference profile corresponding to the observed value of A^^ for 
all heights above 100 m. This approach has proved quite successful in 
predicting T for initial elevation angles under 10 mrad, and will, of course, 
predict trapping when it occurs. 

3.10. The Departures-From-Normal Method 

A method of calculating bending by the use of the exponential model 
of N{h) together with an observed N{h) profile can sometimes be advan- 
tageously employed [15]. This method is primarily intended to point out 
the difference between actual ray bending and the average bending that is 
predicted by the exponential N{h) profile and is a powerful method of 
identifying air mass refraction effects. 

The exponential model described in section 3.8 can be expected to 
represent average refractivity profile characteristics at any given location, 
but it cannot be expected to depict accurately any single refractivity pro- 
file selected at random, even though it may occasionally do so. In order 



78 TROPOSPHERIC REFRACTION 

to study the differences between individual observed N(h) profiles and the 
mean profiles predicted by the exponential model, a variable called the 
A unit has been developed; it is defined simply as the sum of the observed 
A'' at any height, h, and the refractivity drop from the surface to the 
height, h, which is predicted by the exponential profile for a given value 
of A^.. 

Thus 

A{Ns,h) = N(h) + Ns (l-exp{ -c,/i}). (3.46) 

Thus (3.46) adds to N{h) the average decrease of N with height, so that 
if a particular profile should happen, by coincidence, to be the same as 
the corresponding exponential profile, the value of A {N s,h) for this pro- 
file would be equal to N s for all heights. The above analysis shows that 
the difference between A{Ns,h) from N s, 8A{Ns,h), is a measure of the 
departure of N(h) from the normal, exponential profile: 

5.4(A^,,^) = AiNs,h) - Ns = A^(/i)— iV. exp{ -c./i}. (3.47) 

It seems logical that the application of the A unit to bending would indi- 
cate the departures of bending from normal, in some way, just as it indi- 
cates departures of refractivity, A'^, from normal. This is indeed the case, 
as can be seen in figure 3.16, where for an A'^s = 313.0 exponential atmos- 
phere, .4(313.0, h) is plotted on one set of graphs for various typical air 
masses, and the corresponding bending departures from normal are shown 
in the second set of graphs corresponding to the same air masses. Ob- 
viously, the bending departures between layers are highly analogous to 
the A unit variation. It can be seen from figure 3.16 that the similarity 
exists, although it is less, for higher initial elevation angles. The simi- 
larity also decreases with increasing height, owing to the fact that the 
bending departures from normal are an integrated effect, and at low initial 
elevation angles are more sensitive to A^ variations at the lower heights. 
This causes an apparent damping of the bending departures from normal 
at greater heights. However, the A-unit variation is not similarly in- 
fluenced; hence, a loss of similarity arises at large heights above the earth's 
surface. 

If (3.46) is differentiated and substituted into (3.2), the following equa- 
tion results: 



k = Uk -T Uk+1 

(rad) [rad) (.rad) (rad) 



^A{N. 



'k+\ 

X 10-', (3.48) 



1 



DEPARTURES-FROM-NORMAL METHOD 



79 



s 

WINTER 




CP MP MT cT 

WINTER WINTER SUMMER SUMMER 




\ ;; : 



320 340 



1 




1 

/ 

"1 


- 


-) 

1 


- 


-<» 


_ 


r>A 



3.h), 


N U 


I 1 


1 1 


- 


I 
\ 


1 


1 - 

} ' 

1 




-2 0-2 2-4 -2 



T^l^-Tdnrad) 

Figure 3.16. A-unit profiles for typical air ynasses and refraction deviation from normal. 



where 



A.4(A^,) = ANih) + A[iV,|l-exp(-c,/i)!] 
= AN{h) + N sCeexp{-Ceh)Ah, 



and TNs{h) is the value of r tabulated for various atmospheres in tables 
9.10 to 9.17, dk and dk+i are in milliradians and must be from the N s ex- 
ponential atmosphere used. The value of AA{N s) is obtained from sub- 
traction of the A value at layer level, k, from the value of A at layer, A' + l. 
The A value may be obtained by adding any given N{h) value, obtained 
from RAOB or other similar data, to a value of A^s[l ~exp{ —ch]] for the 
same height which may be obtained from figure 3.17. Since TNs{h) has 
been calculated only for a few of the exponential atmospheres, these being 
the A^, - 200.0, 252.9, 289.0, 313.0, 344.5, 377.2, 404.9, and 450.0 atmos- 
pheres, one of these atmospheres must be used in the calculation of bend- 
ing by the departures method. The selection of the particular atmos- 
phere to be used is based on the value of the gradient of N, dN/dh, be- 
tween the surface of the earth and the first layer considered. In table 3.2 
are shown the ranges of the gradient for the choice of a particular ex- 
ponential atmosphere. 



80 



TROPOSPHERIC REFRACTION 








50 



300 



350 



100 150 200 250 

Ns[l-exp(ch)] IN N UNITS 

Figure 3.17. Graphic representation of Ns[l — exp(— ch)] in N iinits versus height. 



3.11. A Graphical Method 

Weisbrod and Anderson [16] present a handy graphical method for 
computing refraction in the troposphere. Rewriting and enlarging (3.11), 
one obtains 



N,- iV,+i 



T(mrad) = 2^ tttttt: z — n: — o — \ 

k^o 500 (tan dk + tan dk+i) 



(3.49) 



where r will be the total bending through n layers. Terms for the de- 
nominator can be determined from figure 3.18. Equation 3.49 is essen- 
tially Schulkin's result with only the approximation, tan dk ~ Ok, for 
small angles, omitted. 

The procedure for using figure 3.18 follows. Enter on the left margin 
at the appropriate Ns — N{h). Proceed horizontally to the proper 
height, h, interpolating between curves if necessary. Use the solid height 
curves when N s — N{h) is positive and the dashed curves when N s — 
N{h) is negative. Then proceed vertically to the assumed ^o and read 
500 tan 6 along the right margin. 



GRAPHICAL METHOD 



81 




1000 



J 1000 



Figure 3.18. Graphic representation of Snelis Law for finding 500 tan d. 



82 TROPOSPHERIC REFRACTION 

3.12. Derivations 

The approximate relation between ^i and 62 is derived here. This rela- 
tion holds for small increments of height and small ^'s. The relationship 
was used in making all sample computations in preceding sections. 
Since for small ^'s 

cos ^1^1- ~ and cos 02 = 1 - - , (3.50) 

and knowing 

r, = ri + ^h (fig. 3.19) 

then substituting in (3.1) yields 

n,{n + A/i)( 1 - f ) = n:n( 1 - f ) , (3.51) 

or 

i2 



'2 . dl 



UiVi + UiAh — riiri -- — n^Ah -- ^ UiTi — UiVi — . (3.52) 



Dividing by Vi 



. UiAh niQ\ Ah dl ^ e\ ,_ __. 

^2 + -^ - -^ - n^ - 2 ^ ni - n. -- . (3.53) 



Since the term — n2 (A/i/rO {61/2) is small with respect to the other 
terms of (3.53) it may be neglected, and thus: 

n2Ah b\ ^ d\ 



or 



n2 + -^— - n2 77 = ni - ni -^ (3.54) 

T\ Z Zi 



^2 ~ ^1 ^2A/l . , , . 

-n2 7,- = -ni — + (ni - n2). (3.55) 



If one now divides both sides of (3.55) by ni and assumes 
(ni — ri2)/r?2 = Wi — 712 and nxln-i = 1, (3.55) may be arranged to yield 



72 ^ x/^i + -— - 2(ni - 71,). (3.56) 



DERIVATIONS 83 

Writing (3.56) in terms of A^ units, 



02(mrad) ^ ^0i + — X lO' - 2iNr - N2) (3.57) 

if 01 is in milliradians. 

Generalizing (3.57) for the ^'th and the (A: + l)st layers, 



0,+i(mrad) ^ xJeUmrsid) + ^"^ ^ X lO' - 2{N, - N,+r). (3.58) 

Also from the geometry shown in figure 3.19, a useful relationship for 
Ti,2 can be obtained. Tangent lines drawn at A and B will be respec- 
tively perpendicular to ri and r-z, since ri and r2 describe spheres of refrac- 
tive indices Ui and n2 concentric with 0. Therefore, 

angle AEC ^ angle AOB ^ <}> 

also, in triangle A EC 

angle ACE = 180° - angle CAE - angle A EC 

= 180° - di - <t>. (3.59) 

But from triangle DCB 

angle ACE = angle DCB = 180° - ti,2 - ^2. (3.60) 

Thus 

180° - ri.2 - 02 = 180° - di - <t>, 
or 

T1.2 - (^ + (01 - 02). (3.61) 

Now since <f> in radians = d/a, where d is distance along the earth's 
surface : 

T1.2 = - + (01 - 02), (3.62) 

a 

or the bending of a ray between any two layers is given in terms of the 
distance, d, along the earth's surface from the transmitter (or receiver), 
the earth's radius, a, and the elevation angles 0i and 02 (in radians) at the 
beginning and end of the layer. 



84 



TROPOSPHERIC REFRACTION 




Figure 3.19. Bending geometry on a spherical earth with concentric layers. 



If one considers figure 3.20, Snell's law in polar coordinates (3.1) and 
the refraction formula (3.2) may be obtained from the more familiar 
form of Snell's law. 

Assume that the earth is spherical and that the atmosphere is arranged 
in spherical layers. In figure 3.20 let C be the center of the earth, the 
observer and COZ the direction of his zenith. Let n and n+dn be the 
indices of refraction in two adjacent thin layers M and M' . Let LP be 
the section of a ray in M' which finally reaches the observer at 0. At 
P it is refracted along PQ. Similarly, it is refracted at the surfaces be- 
tween successive layers and the final infinitesimal element of its path is 
TO. 

Draw the radii CP and CQ. Let angle PQF = 6, angle LPS = 6 -\- dd, 
and angle QPF = \p. Then, since the radius CP is perpendicular at P to 



DERIVATIONS 



85 




-EARTH'S 
SURFACE 



Figure 3.20. Geometry for the derivation of Snell's Law in spherical coordinates. 



the bounding surface between layers M and M', by Snell's law we have 

(n+rfn) sin [90° - id+dd)] = n sin i/^. (3.63) 

Now from the triangle CQP-, in which CQ = r and CP = r -\- dr, and 



2The assumption involved in this triangle is that the path of the ray in M' is a 
straight line, which, of course, can only be true in an isotropic medium. Hence, it 
can only be true for an infinitesimal layer in the troposphere. Thus only a differential 
form of Snell's law, (3.65), in polar coordinates, can be obtained by the use of the 
geometry of figure 3.20; not the finite form, (3.68), which has the same appearance. 



86 TROPOSPHERIC REFRACTION 

angle CQP = 90° + 6, we have, from the law of sines, 

r sin (90° + 6) = r sin (90° - 0) = (r + dr) sin xj^. (3.64) 

Eliminating sin i/^ from (3.63) and (3.64) we then have 

(n + dn)(r + dr) sin [90° - (^ + dd)] = nr sin (90° - 6), 
or 

(n + dn)ir + rfr) cos (0 + dd) = nr cos 6. (3.65) 

Multiplication of the (n + (in) and (r + dr) terms, ignoring differential 
products, yields 

{nr + ndr + rdn) cos (0 + dd) = nr cos 0, 
or 

{nr + n(/r + rdn) [cos ^ cos ((/0) — sin 6 sin (c?0)] 

= nr cos 0. (3.66) 

Since cos {dd) = 1, and sin {dd) ^ c?^, another multiplication, again ignor- 
ing products of differentials, yields: 

ndr cos d + rdn cos d — nr sin d dd = 0, 

or, dividing all terms by nrcos 0, 



- + — -ta^nddd = 0. (3.67) 

r n 



Now if (3.67) is integrated between any two thin layers of refractive 
indices ni and n2, whose radial distances from the earth's center are ri and 
r2, and the initial elevation angles of a radio ray entering the layers are di 
and 02: 



dr . dn / . «7^ . ^2 , , n2 , , cos ^2 ^ 

[- / — — / tan ddd = (n j- ^n — + ^n = 

, r Jn, n Je, ri ni cos ^i 



or, taking antilogs of both sides, 

r2n2 cos 02 
rini cos di 



= 1, 



REFERENCES 87 

whence 

riiri co.s di = ti2r2 cos do, (3.68) 

which is Snell's law for polar coordinates, (3.1). 
In figure 3.20, it can be seen that 

PF dr 
'^"'=QF = 7d-^- (3.69) 

where </> is the angle at the earth's center between r and COZ. Substitut- 
ing (3.69) in (3.67) 

dn 
tan dd(f) -\ tan Odd = 0, 

n 



or 



dn 
id4> - dd) tan = - — . (3.70) 



Since, by considering (3.61) for infinitesimal angles, 

dr = d<t) - de, 



or, in (3.70) 



or 



dr tan 6 = — — 

n 



dti 

dr = -cote — . (3.71) 



n 



Integration of (3.71) yields (3.2). 



3.13. References 

[1] Smart, W. M. (1931), Book, Spherical Astronomy, Ch. 3 (Cambridge Univ. 
Press, London, England). 

[2] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric re- 
fractive index, Proc. IRE 47. No. 5, 740-755. 

[3] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric 
refraction and its relation to wave guides and diffraction, Book, Meteorological 
Factors in Radio-Wave Propagation, pp. 80-127 (The Physical Society, 
London, England). 

[4] Freehafer, John E. (1951), Tropospheric refraction. Book, Propagation of Short 
Radio Waves, pp. 9-22 (McGraw-Hill Book Co., Inc. New York, N.Y.). 

[5] Bean, B. R., and B. A. Cahoon (Nov. 1957), The use of surface weather observa- 
tions to predict the total atmospheric bending of radio waves at small elevation 
angles, Proc. IRE, 45, 1545-1546. 



88 TROPOSPHERIC REFRACTION 

[6] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere, 

Proc. IRE 40, 554-561. 
[7] Anderson, L. J. (Apr. 1958), Tropospheric bending of radio waves, Trans. Am. 

Geophys. Union 39, 208-212. 
[8] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave 

propagation, Proc. IRE 21, 427-463. 
[9] Stickland, A. C. (1947), Refraction in the lower atmosphere and its application to 

the propagation of radio waves. Book, Meteorological Factors in Radio Wave 

Propagation, pp. 253-267 (The Physical Society, London, England). 
[10] Humphreys, W. J. (1940), Book, Physics of the Air, p. 82 (McGraw-Hill Book 

Co., Inc., New York, N.Y.). 
[11] Norton, K. A., P. L. Rice, and L. E. Vogler (Oct. 1955), Use of angular distance 

in estimating transmission loss and fading range for propagation through a 

turbulent atmosphere over irregular terrain, Proc. IRE 43, 1488-1526. 
[12] The Rocket Panel (1952), Pressures, densities, and temperatures in the upper 

atmosphere, Phys. Rev. 88, 1027-1032. 
[13] Handbook of Geophysics for Air Force Designers (1957), Geophysics Research 

Directorate (Air Force Cambridge Research Center, ARDC, USAF). 
[14] Dubin, M. (Sept. 1954), Index of refraction above 20,000 feet, J. Geophys. Res. 

59. 339-344. 
[15] Bean, B. R., and E. J. Dutton (May-June 1960), On the calculation of departures 

of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3, 

259-263. 
[16] Weisbrod, S., and L. J. Anderson (Oct. 1959), Simple methods for computing 

tropospheric and ionospheric refractive effects on radio waves, Proc. IRE 47, 

1770-1777. 



Chapter 4. N Climatology 

4.1. Introduction 

The contents of this chapter include a study of the surface variation of 
the radio refractive index on a worldwide scale in terms of a reduced-to- 
sea-level form of the index that gives a significantly more accurate descrip- 
tion of refractive index variations than the nonreduced form. 

The mean vertical structure of the refractive index parameter in the 
troposphere over central North America is presented, again in terms of 
a reduced-to-sea-level form of A^. 

A climatological treatment of the phenomenon of the atmospheric duct, 
or waveguide, and associated fading regions is also presented. The 
chapter is concluded with a discussion of refraction of radio waves in 
various air masses. It is demonstrated that refraction differences within 
air masses arise from departures of refractive index structure from normal. 



4.2. Radio-Refractive-Index Climate Near the Ground 

4.2.1. Introduction 

The radio refractive index of air, w, is a function of atmospheric pres- 
sure, temperature, and humidity, thus combining in one parameter three 
of the normal meteorological elements used to specify climate. In the 
following sections we will examine the variability of n during different 
seasons of the year and in differing climatic regions. The systematic 
dependence of n upon station elevation will make it necessary to consider 
a method of expressing n in terms of an equivalent sea-level value in 
order to see more clearly the actual climatic differences of the various 
parts of the world. After a consideration of the n climate of the world, 
the application of this information to such practical problems as the 
prediction of radio field strength and the refraction of radio waves will be 
discussed. 



4.2.2. Presentation of Basic Data 

Near the surface of the earth, for VHF and UHF frequencies, n is a 
number of the order of 1.0003. Since, for air, n never exceeds unity by 

89 



90 



N CLIMATOLOGY 



more than a few parts in 10^. it is convenient to consider the chmatic 
variation of n in terms of 



N = {n - 1)10«, 



(4.1) 



as defined in chapter I. The notation N s is used to indicate that (4.1) 
has been evaluated from standard surface weather observation. 

To obtain long-term average values of A^, one should properly average 
individual observations over many years. This is difficult to do since, 
in general, only summaries of weather observations are readily available. 
However, long-term average values of temperature, pressure, and 
humidity are available and may be converted into an "average" value 
of A''. This "average" N differs from the true average since the inter- 
correlation of pressure, temperature, and humidity is neglected. This 
difference was examined by an analysis of 2 years of weather records of 
the months of February and August at an arctic location (Fairbanks, 
Alaska), a temperate zone location (Washington, D.C.), and a tropical 
location (Swan Island, W.I.). These data, given in table 4.1, indicate 
that the difference between the two methods was never more than 1.5 A'' 
units and that the average difference was less than 1 A'' unit, which is small 
compared to commonly observed seasonal and geographic variations of 
20 to 100 A^ units. 



Table 4.L Two-year average value of Ns versus the value of Ns calculated from average 
temperature, pressure, and humidity 



Fairbanks; 

February. 

August 

Washington: 

February. 

August 

Swan Island: 

February. 

August 





N, 


(P, 


T, 


RH) 


N,-N, (P, T, EH) 


314.0 
320.5 








313.0 
320.0 


1.0 
0.5 


305.5 
356.0 








340.5 
354.5 


1.0 
1.5 


362.0 
387.5 








362.5 
388.0 


0.5 
0.5 



Average. 



0.83 



On this basis it was decided to use the long-term means given in the 
United Nations' monthly publication, Climatic Data for the World. This 
publication is particularly advantageous for our present study since it re- 
ports the fictitious value of the relative humidity needed to obtain the 
actual average vapor pressure from the saturated vapor pressure of the 
reported mean temperature [1].^ 



^Figures in brackets indicate the literature references on p. 170. 



PRESENTATION OF BASIC DATA 91 

Data from 306 weather stations were obtained in order to give reason- 
able geographical coverage. In general, 5 years of records were obtained 
for each station from the period 1949 to 1958, preference being given to 
the years 1954 through 1958. A noticeable exception, however, was 
Russia, for which only 1 year of data (IGY) is reported in Climatic Data 
for the World; thus all charts are drawn with dashed contours for Russia. 
There are vast expanses of ocean for which there are no meteorological 
observing stations. Climatic atlases were utilized in order to present 
estimates of world climate in these locales. A reasonable coverage of 
the sparse data areas of the world was made by estimating temperature 
from sea surface isotherms [2] and humidity from charts of seasonal 
average depression of the wet bulb temperature [3]. Pressure was esti- 
mated for these locations from average winter and summer pressure 
charts. 

When these data were converted to A^ [4] and charts prepared, a pro- 
nounced altitude dependence could be seen, as in figure 4.1. Figure 4.1 
and the following charts of N variations across the United States are from 
an extensive N climatology now being prepared at the Central Radio 
Propragation Laboratory. Although the present study is primarily 
aimed toward worldwide variations, it is felt that the U.S. data better 
illustrate the height dependence of Ng and the subsequent reduction 
process employed. It is noted that the coastal areas display high values of 
N s, while the inland areas have lower values. There are low values of 
N s corresponding to the Appalachian and Adirondack Mountains and a 
decrease with increasing elevation of the Great Plains until the lowest 
values are observed in the Rocky Mountain region and the high plateau 
area of Nevada. A corresponding gradient is observed from the west 
coast eastward. Crosshatching encloses areas where the terrain changes 
so rapidly that it was felt the data were inadequate to obtain realistic 
contours of N s. 

The altitude dependence of N can be studied in terms of the "dry" and 
"wet" components of N. These components are those of the two-term 
expression in (1.17). The dry term, D, 



i) = ^^ , (4.2) 



is proportional to air density and normally constitutes at least 60 percent 
of A. 

The average variation of density with altitude in the atmosphere may 
be expressed in the first approximation as 



p = pocxp {-z/H\ (4.3) 



92 



N CLIMATOLOGY 




Figure 4.L Mean Ns, August 0200 local time. 



where z is the akitude, po the average sea level density of moist air and 
H the average scale height between zero and z. It is useful to introduce 
the concept of an effective scale height, H*, for the average variation of 
refractive index in the atmosphere. Many studies have shown that the 
average refractive index variation with height is quite well represented, 
to a first approximation, by a formula similar to (4.3) [5, 6, 7]. It is 
possible to calculate a theoretical value of this effective scale height using 
a distribution of water vapor. This is, however, quite a complex pro- 
cedure. Furthermore, the value obtained depends upon the model of the 
water vapor distribution, and no definite conclusion can be justified 
considering the extreme variation of water vapor concentration with 
season, geographic location, and height above the earth's surface. A 
convenient and simple alternative is to adopt a value for H* from the 
average (n — 1) variation with height in the free atmosphere. Several 
such values of H* were determined by reference to the NACA standard 
atmosphere [8] and recent climatological studies of atmospheric refractive 
index structure [9]. 

It is seen from table 1.8, chapter 1, that H* varies from 6.56 to 7.63 
km in the NACA standard atmosphere, depending on the value of rela- 
tive humidity assumed. The value of H* = 6.95 km obtained from 
chmatological studies of (n - 1) variations over the first kilometer above 



PRESENTATION OF BASIC DATA 93 

the earth's surface from nearly 2 million radiosonde observations from 
many diverse climates. 

At the time of analysis of the map series presented here, a value of 
effective scale height, H* = 9.46 km, was in use. This form was 



Do = D, exp j^l 



(4.4) 



in which the value H* = 9.46 km was determined from the NACA dry 
standard atmosphere. A Do chart is shown on figure 4.2, presenting a 
gradient that is remarkably free of detail as compared to the N ^ chart of 
figure 4.1, and is easily drawn for all areas of the country. 

An investigation of the elevation dependence of the surface wet term 

W. = ^^A^^ (4.5) 



revealed low correlations of log W s and height, indicating that W s is not 
a marked exponential function of elevation. Contours of Ws for all 
sections of the country are shown on figure 4.3. 

The maps that follow are completed in terms of a single reduced form 

A^o ^ {Ds + Ws) exp j^^l =^ Ns exp j^[, (4.6) 

where N s is reduced by the dry term effective scale height, H* = 9.46 km. 
Figure 4.4 gives the A^o contours for the same time as the previous maps 
of Do and W s- The A^o maps are no more difficult to prepare than the W s 
maps and have effectively removed the station height dependence of N s. 
One might wonder at the advisability of arbitrarily reducing the wet term 
by the dry term correction. For the coastal areas of the country, where 
the exponential height correction factor is nearly unity, this amounts 
simply to adding the Do and W s maps for the mountain areas. Where the 
height correction factor is large, the W s values are small with the result 
that the gradient of the N isopleths obtained from the Do and W s maps is 
essentially maintained on the Ao maps. As an example, for the series of 
maps under discussion, the {Do -\- W s) difference between Reno, Nev. 
(1,340 m elevation), and Oakland, Calif. (5.5 m elevation), is 21 N units, 
while the Ao difference is 19 A^ units. 

The effects of this correction on the worldwide values can be seen from 
figure 4.5, where As is plotted versus station elevation in kilometers. A 
sample line illustrates the decay of As with height for Ao = 348. The 
value of As for any other value of A^o would be obtained from a line parallel 
\o the Ao = 348 line but having a zero intercept equal to the new value of 



94 



A^ CLIMATOLOGY 




Figure 4.2. Mean Do, Aug^^st 0200 local time. 




Figure 4.3. Mean We, August 0200 local time. 



PRESENTATION OF BASIC DATA 



95 




Figure 4.4. Mean No, August 0200 local time. 



No. The advantage of adopting No is illustrated by the reduction in 
range from 190 A^ units for iV^ to 115 A^ units for No, thus diminishing 
the number of contours of the resulting maps. 

It would appear that by removing the influence of station elevation it 
would be more efficient to estimate A''^ from A^^o charts rather than from 
charts of A''^. As a test of this hypothesis, N s and A^'o contour charts were 
prepared for both summer and winter from only 42 of the 62 U.S. Weather 
Bureau stations for which 8-year means of N s are available. The remain- 
ing 20 stations, distributed at random about the country, were used as a 
test sample by estimating their 8-year mean value of N s from the A'^o and 
N s contours. Summertime examples of these charts are given by figures 
4.6 and 4.7. Note that due to the reduced range of N, the A^'o charts are 
drawn every 5 A'^ units as compared to the 10-A^-unit contours of the N s 
charts. The individual deviations of the values obtained from the con- 
tour maps with the actual 8-year means are listed in table 4.2. By com- 
paring the root mean square (rms) deviations of 10.7 A'^ units in winter 
and 13.0 N units in summer obtained by estimating N s from the N s con- 
tours with the 2.7-A^-unit rms of estimating N s from A'^o contours, one 
concludes that it is at least 4 times more accurate to estimate N s from A''o 
contours than from those of A^^. An inspection of the individual devia- 
tions in table 4.2 indicates that the A^o contour method is particularly 
efficient at elevations in excess of 1,200 m or where the terrain is changing 



96 



A^ CLIMATOLOGY 




0.5 1.0 1.5 2.0 2.5 3.0 

STATION ELEVATION IN KILOMETERS 



Figure 4.5. Worldwide values of Ns versus height for August. 



PRESENTATION OF BASIC DATA 



97 




Figure 4.6. Test chart of mean Ns August 0200 local time. 




Figure 4.7. Test chart of mean No August 0200 local time. 



98 



N CLIMATOLOGY 



rapidly with respect to horizontal distance. As a further practical con- 
sequence, one notes the remarkable similarity between the A^'o contours 
of figures 4.4 and 4.7, even though the latter contours were derived from 
only two-thirds of the original data. This indicates that any desired level 
of accuracy may be maintained with fewer stations (and less expense) by 
the use of No. 

Table 4.2. Deviations of estimated 8-year means of Ns {calculated from contour charts) 
from actual 8-year means selected at random from 20 Weather Bureau stations 



Test station 



Sacramento, Calif.. 

Portland, Oreg 

San Diego, Calif... 

Mobile, Ala 

Fresno, Calif 



Boston, Mass 

Grand Rapids, Mich. 

Columbia, Mo 

Minneapolis, Minn... 
Cincinnati, Ohio 



Des Moines, Iowa 

Pendleton, Oreg 

Billings, Mont 

Burns, Oreg 

Salt Lake City, Utah. 



Reno, Nev 

Pocatello, Idaho 

Denver, Colo 

Colorado Springs, Colo. 
Flagstaff, Ariz 



Root mean square deviation. 



Height 



meters 
7 
8 
11 
66 
86 



210 
239 
255 
271 

294 

455 

1,088 

1,262 

1,288 

1.340 
1,355 
1,625 
1.882 
2,131 



February 1400 



Actual 

8-yr 
mean 
value 

of JV, 



N units 
315.6 
316.2 
314.2 
326.6 
310.6 

308.6 
304.4 
300.8 
301.1 
302.5 

300.9 
295.9 
269.3 
268.1 
266.3 

259.6 
264.7 
244.9 
237.1 
237.8 



Deviation* 



map 



N tinits 
7.6 
4.2 
-5.8 
6.6 
9.6 

-6.4 

0.4 

-.2 

.1 

-.5 

3.9 

1.9 

-2.3 

-23.9 

1.3 

-20.4 
-2.3 
-8.1 
-15.9 
-26.2 

10.7 



No 
map 



A'^ units 
0.8 
-.5 
-2.4 
4.8 
3.4 

-0.5 

-5.9 

2.4 

0.6 

.7 

2.3 

0.4 

.1 

-3.2 

-0.8 

-6.8 

0.4 

.7 

-.6 

2.3 

2.7 



August 0200 



Actual 
8-yr 
mean 
value 
of iV, 



A'' units 
329.6 
337.7 
348.1 
376.0 
326.2 

347.5 
340.5 
348.7 
338.5 
344.1 

343.1 
300.9 
285.6 
271.3 
279.5 

277.6 
269.7 
276.6 
272.4 
261.4 



Deviation 



Ns 
map 



A'' units 
1.0 
19.7 
16.1 
6.0 
5.2 

-7.5 
-1.5 
-2.3 
-0.5 
-2.9 

-1.9 

2.9 

5.6 

-15.7 

8.5 

-29.4 
-3.3 
-1.4 
-6.6 

-36.6 

13.0 



ATo 
map 



A^ units 
1.8 
6.0 
3.5 
0.6 
4.2 

-0.4 

.1 

-2.5 

2.7 

-2.8 

-0.1 
-3.1 

1.2 
-4.4 

4.4 

1.6 

0.0 

.3 

1.1 

-2.2 

2.7 



♦Deviation equals the actual long-term mean minus the value obtained from map contours. 



4.2.3. Worldwide Values of N„ 

Mean values of No were calculated at each of the 306 selected stations 
and charts were prepared for each month of the year. The charts for 
February and August are given on figures 4.8 and 4.9. It is seen that 
the values of No for sea-level stations vary from 390 in the maritime 
tropical areas to 290 in the deserts and high plateaus. The interiors of 
continents and mountain chains in middle latitudes are reflected by low 
values as compared to those of coastal areas. Further, such pronounced 
climatic details as the Indian monsoon and the effects of coastal mountain 
ranges blocking prevailing winds and producing rain shadows are indi- 
cated by these No contours. Regional climatological data of No for the 
United States are given in chapter 9. 



WORLDWIDE VALUES OF A^o 



99 




Figure 4.8. Worldwide values of mean No /or February. 




Figure 4.9. Worldwide values of mean 'No for August. 



100 



A^ CLIMATOLOGY 




Figure 4.10. Annual range of monthly mean Ns. 




Figure 4.n. Minimum monthly mean value o/No. 



WORLDWIDE VALUES OF A^o 



101 




Figure 4.12. Year-to-year range of monthly mean Ng/or February. 




P'iGURE 4.13. Year-to-year range of monthly mean Na/or August. 



102 N CLIMATOLOGY 

The annual variation of N s is indicated on figure 4.10 by contours of 
the difference between the maximum and minimum monthly means ob- 
served throughout the year. It is quite remarkable how clearly climatic 
differences are evidenced by the yearly range of N s. The prevailing 
transport of moist maritime air inland over the west coasts of North 
America and Europe is indicated by relatively small annual ranges (20 to 
30 A'^ units), while, for example, the east coast of the United States with a 
range of 40 to 50 A^ units or more reflects the invasion of that area from 
time to time by such diverse air masses as arctic continental and tropical 
maritime. The largest annual ranges of N ^ (90 N units) are observed in 
the Sudan of Africa and in connection with the Indian monsoon. 

An additional A'^o map (fig. 4.11) was prepared from the minimum 
monthly mean value of A^^ observed throughout the year to supplement 
the range map in order that an estimation might be made of both the 
minimum and maximum monthly mean N s expected durmg the year. 

A measure of the variability of the February and August mean values 
of A'"^ is given by monthly range maps (fig. 4.12 and 4.13) determined 
from monthly averages from 5 years of data. Ranges are given by the 
maximum difference of the five individual monthly mean values. In con- 
touring the two variability maps only those terrestrial regions having 
reasonable data coverage are included. Dashed contours are shown for 
areas of sparse or unreliable data. The general picture of the worldwide 
distribution of A''^ variability is that of a number of continentally located 
cells of moderate range accompanied by somewhat random small-scale 
variation over ocean areas. Regions of large range, from 40 to as much 
as 70 A^ units, are present, however, in Australia and on islands of the 
adjoining oceans, on the African equatorial plateau near the Cameroons, 
and in the Great Basin of the southwestern United States. Common to 
all these areas of large year-to-year variability, at least during the summer 
season, are high mean temperatures ranging from about 25 to 30 °C, the 
variability being due to relatively small variations of humidity. It is felt 
that when a more dense network of stations is available for a longer period 
of record, say 10 years, areas of high monthly variability are likely to be 
more extensive in tropical and desert areas than indicated on our present 
maps. 

4.2.4. Climatic Classification by N^ 

The annual cycle of A^« at each station was examined for the purpose 
of deriving similarities of climatic pattern. As one form of climatic 
classification, the annual mean value of A'^s at each station was plotted 
versus the annual range at the station. When this was done, several 
distinct groupings of data seemed evident. These groupings, described 



CLIMATIC CLASSIFICATION BY No 



103 



in table 4.3, are intended to give a general idea of the geographic and 
climatic character of the majority of the stations found within given 
values of range and yearly mean of N s. 



Table 4.3. Characteristics of climatic types 



Type 


Location 


Annual 
mean N, 


Annual 
range of A'', 


Characteristics 


I. Midlatitude- 
coastal. 


Near the sea or in lowlands on 
lakes and rivers, in latitude 
belts between 20° and 50°. 


N units 
300 to 350 


N units 
30 to 60 


Generally subtropical 
with marine or modi- 
fied marine climate. 


n. Subtropical- 
Savanna. 


Lowland stations between 30°N 
and 25°S, rarely far from the 
ocean. 


350 to 400 


30 to 60 


Definite rainy and dry 
seasons, typical of 
Savanna climate. 


III. Monsoon- 
Sudan. 


Monsoon — generally between 
20° and 40°N, Sudan— across 
central Africa from 10° to 20°N _ 


280 to 400 


60 to 100 


Seasonal extremes of 
rainfall and tem- 
perature. 


IV. Semiarid- 

mountain. 


In desert and high steppe regions 
as well as mountainous re- 
gions above 3,000 ft. 


240 to 300 


to 60 


Year-round dry climate 


V. Continental- 
Polar. 


In middle latitudes and polar 
regions. (Mediterranean cli- 
mates are included because of 
the low range resulting from 
characteristic dry summers.) 


300 to 340 


to 30 


Moderate or low annual 
mean temperatures. 


VI. Isothermal- 
equatorial. 


Tropical stations at low eleva- 
tions between 20°N and 20°S, 
almost exclusively along sea- 
coasts or on islands. 


340 to 400 


to30 


Monotonous rainy 
climates. 



For a given classification of refractive-index climate, diverse meteor- 
ological climates and geographical regions may be represented. Note, 
for example, that type V of table 4.3 includes stations from Mediterranean 
and marine as well as polar climates. Mediterranean stations in this 
category fail to attain a high range because of the characteristic dryness 
of the subtropical high-pressure pattern that is generally found in this 
area during the summer months. Polar and marine climates in this 
group maintain a low range due to suppressed humidity effects as a result 
of low to moderate year-around average temperatures. 

Annual trends of N s for stations typical of each climatic division are 
shown by figure 4.14. 

Yet another facet of the climate is the year-to-year variation of the 
monthly mean value of N s. Five consecutive years of monthly means 
were prepared for each of the six typical stations whose annual cycles are 
shown in figure 4.14. Then, for each month, the absolute value of the 
difference between consecutive years was obtained. These values were 
then averaged for all months and are listed in the second column of table 
4.4. 



104 



N CLIMATOLOGY 



TYPE I 

MID-LATITUDE COASTAL 

WASHINGTON, D.C. 



TYPE II 
SUBTROPICAL - SAVANNA 
MIAMI, FLA. 



TYPE m 

MONSOON -SUDAN 
JODPHUR, INDIA 




TYPE EZ: 
SEMI-ARID MOUNTAIN 
DENVER, COLO. 



TYPE Y TYPE 21 

CONTINENTAL -POLAR ISOTHERMAL EQUATORIAL 
OSLO, NORWAY CANTON ISL., S. PACIFIC 0. 




JFMAMJJASONOJ JFMAMJJASONDJ JFMAMJJASONOJ 
Figure 4.14. Representative annual cycles of Ns /or the major climatic types. 



Another measure of the variation of monthly mean values of N ^ is 
obtained by differencing the maximum and minimum values occurring for 
a given month during the 5-year period. These differences are also given 
in table 4.9 for the months of February, May, August, and November. 



Table 4.4. Year-to-year differences of monthly mean Ns 







Maximum difTerences between 


monthly 




Differences between monthly means 


means over a 5-yr period 


Climatic* 


in successive years for the same 
month, averaged for all seasons over 






type 












a 5-yr period 


Feb. 


May 


Aug. 


Nov. 


I 


5.7 


6.0 


16.5 


17.0 


7.0 


n 


5.4 


8.5 


6.5 


8.0 


11.0 


III 


8.9 


16.5 


14.5 


20.0 


6.5 


IV_... 


5.4 


10.5 


11.5 


13.5 


6.5 


V 


4.7 


5.5 


11.5 


10.0 


5.0 


VI. 


7.1 


9.5 


25.5 


8.5 


8.5 



*Climatic types are the same as those in table 4.3. 



APPLICATIONS 



105 



4.2.5. Applications 

The communications engineer usually has available a small amount of 
measured field-strength data from limited tests of a particular system. 
He must then estimate the expected signal level or practical range of that 
system, or other systems, for other tunes of the year, other years, and in 
other areas. The variation of signal level from month to month and 
climate to climate can be explained, in part, by its observed correlation 
with Ns. 

Pickard and Stetson [10, 11] were among the first to note the correlation 
of Ns and received field strengths. The correlation of Ns and field 
strength over a particular path has been studied quantitatively [12, 13] 
and found to be highest (correlation coefficients of 0.8 to 0.95) when the 
variables are averaged over periods of a week to a month. This latter 
study has shown that the regression coefficient (decibel change in field 
strength per unit change in N s) varies diurnally from 0.14 dB in the after- 
noon hours to 0.24 dB per unit change of N s iu the early morning hours. 
This correlation is so sufficiently consistent that Gray [14] and Norton [15] 
have utilized it in their recent prediction methods of transmission loss in 
a band from 100 to 50,000 Mc/s. In addition, the coefficient 0.2 dB per 
unit change in A''^ has been tentatively adopted by CCIR Study Group V 
in their revision of the 30- to 300-Mc/s tropospheric-wave propagation 
curves to account for the geographic and seasonal variations of field 
strengths. The estimates of field strength variations attributed to N s 
given in table 4.5 are based upon the CCIR coefficient. 

If one assumes, for comparison only, that the worldwide average value 
of A^s is 330 and that one is able to estimate the field strength level of a 
particular communications system at a given distance and for N s = 330, 
then the above correlations would indicate that the climatic variations of 
fields given in table 4.5 might be expected. 



Table 4.5. Climatic variation of hypothetical communications system relative to pre- 
dicted value for Ns = 330, assuming a 0.2-dB variation per unit change in Ns 



Climatic type* 


Expected yearly 

mean field strength 

level relative to 

AT. =330 


Expected annual 

range on the above 

assumption 


I 

II 

Ill 

IV 

V... . . 


dB 

-6 to +4 
+4 to +14 
-10 to +14 
-18 to -6 
-6 to +2 
+2 to +14 


dB 

6 to 12 
6 to 12 
12 to 20 
Oto 12 
to 6 


VI 


Oto 6 







•Climatic types are the same as those in table 4.3. 



106 N CLIMATOLOGY 

The data of table 4.5 indicate, for example, that identically equipped 
tropospheric communications systems could display as much as a 32-dB 
difference in mean signal-strength level due to the climatic difference of 
say, Denver, Colo., and the tropics. Further, one might expect the 
monthly mean field strength of this hypothetical system to vary through- 
out the year from less than 12 dB in the high plains near Denver to as 
much as 20 dB in the African Sudan. 

Under this same assumption, figures 4.10 and 4.11 allow the communi- 
cations engineer to estimate the expected maximum and minimum 
monthly mean field strength expected throughout the year. 

The year-to-year variations of the monthly mean A''^ listed in table 4.4 
indicate that the monthly mean of field strength for a particular month 
may differ in successive years by as little as 1.0 dB for chmatic category 
V in November or as much as 5.1 dB for category VI in May. 

Another application of these worldwide charts is to aid in estimating the 
refraction of radio waves. The most convenient method of accounting 
for the effects of atmospheric refraction is by means of the effective- 
earth's-radius concept (see chapter 3) of Schelleng, Burrows, and Ferrell 
[16]. The effective earth's radius, Ug, is determined from 

tte = I ;^V' (4-7) 

I' + Ilj 

where a is the true radius of the earth, n is the refractive index, and 
dn/dh is the initial n gradient with respect to height. A great simplifica- 
tion of propagation calculations is accomplished by assuming that dn/dh 
is a constant, thus allowing radio rays to be drawn as straight rays over a 
fictitious earth of radius a^ rather than curved rays over the true earth of 
radius a. This simplification allows, for example, the distance to the 
radio horizon, d, of a radio ray leaving an antenna of height, h, to be 
calculated from d = ■\/2ae h. 

One notes, however, that the determination of a^ involves dn/dh as 
well as n and that our A^o charts allow only an estimation of n. This dis- 
parity may be resolved by utilizing the observation that N s is highly 
correlated with the value of A^ at 1 km above the surface. The difference 
between A^^ and N at 1 km is denoted AN. It has been noted [9] that the 
correlation coefficient between ^nJAA^j and Ns is 0.926 for 888 sets of 
data from 45 U.S. weather stations representing many diverse climates. 

The regression equation 

-AiV = 7.32 exp {0.005577 N s} (4.8) 

results when both variables are averaged over 6 to 8 years of record. 



CRITICAL APPRAISAL OF RESULTS 107 

Approximating dn/dh in (4.7) by AN, we may determine that the radio 
horizon distance of an antenna located 150 m above the earth would vary 
from 48 km when A''^ = 200 to 59 km when Ns = 400. Yet another ap- 
plication of the N s charts is to the exponential models of the decrease of 
refractive index with height which have been proposed to date [9, 17]. 
These models are completely specified by N s and may be used to account 
for seasonal and geographic variations of such refraction effects as radar 
range and elevation angle errors. 

4.2.6. Critical Appraisal of Results 

The world maps presented above were based upon data from 306 
weather stations. This number of stations appears to be consistent with 
the scale of map used. The map scale is so small, however, that only 
large climatic differences can be expected to be discerned. For the 
climate of any given area one should refer to detailed studies of N such 
as those currently in preparation for the United States at the Central 
Radio Propagation Laboratory. 

The accuracy of the present charts may be assessed from the charts of 
maximum range, R, of monthly means as given by figures 4.12 and 4.13. 
The standard deviation of the individual monthly means may be esti- 
mated from [18] 0.43 R, where the coefficient 0.43 is appropriate for five 
individual observations. Since, in general, R < 20 N units, then 
0.43 i? < 9N units, although this standard deviation may be as large as 26 
N units for the month of February in Australia and 17 N units in the 
southwest of the United States during August, or in the African Sudan 
during February. 

Further, the standard error of estimating a 5-year mean from five 
individual monthly values is determined from 

0A3R 

_ ) 
\/n 

where n for our case is 5 and thus the error of the 5-year mean would be 
0.192 R. Remembering that R < 20 N units and assuming perfect skill 
in drawing the contours, one would expect the standard error of estimate 
to be less than 4 A^ units. This standard error can be as large as 12 A" 
units in Australia where R = GO N units. 

The value of A^s at each of the 20 test stations of table 4.2 was estimated 
from the A^o contours with an rms error oi 5 N units which is consistent 
with the standard error of estimate obtained from the range charts. In 
the large areas of sparse data, such as the oceans and Russia, this un- 
certainty rises to about 10 N units and thus the contours in these regions 
are dashed. 



108 N CLIMATOLOGY 

At the time the present study was initiated it was felt that Ns should 
be reduced to sea level by at least the dry term correction factor as in 
(4.6). The absence of published work on models of A^ structure in the 
free atmosphere encouraged the decision to rest on prudence and adopt 
this dry term correction factor. Since that time several effective ex- 
ponential models of the free atmosphere have been demonstrated [17, 9]. 
In future work a smaller value of H* on the order of 7.0, which corresponds 
to the N decay in the free atmosphere, will be adopted. The adoption 
of any value of H* between 6.5 and 7.5 km would have reduced the range 
of A^'o values on figure 4.4 by no more than 2 A'' units. Since this reduction 
in range is more than an order of magnitude less than the reduction of 
(4.6) used to obtain No, it appears that the basic advantage of the method 
has been realized. 

A map of A^'o, such as that of figure 4.4, which represents a large con- 
tinental area, may easily be compared and merged with maps for ocean 
areas. This would be more difficult with N s, since, for example, the 
strong gradient over California mainly represents the rapid altitude 
variations of the Sierra Nevada Mountains. It has also been demon- 
strated that A^'o is a better indicator of tropospheric storms and air 
masses than N s when considered on a synoptic or "daily weather map" 
basis [19]. 

Also (4.6) aids in comparing air properties as a function of altitude at 
the same position. The variations of No will represent the local depar- 
tures between this quantity and the value in a standard atmosphere and 
will show the perturbations in the structure of the atmosphere produced 
by fronts, air masses, and other synoptic features. Although any value of 
H* between 6.5 and 7.5 km will remove the gross altitude dependence of 
the refractive index, the choice of value of H* within this range could de- 
pend on the application. The synoptic application, which is discussed in 
detail in chapter 5, would be best served by a scale height near 8.0 km, 
whereas the objectives of the climatic chart usage would best be met by 
a scale height near 6.5 km. This seeming paradox is easily understood in 
terms of the physical interpretation of the various scale heights. For 
example, H* = 8.0 km is typical of a perfectly dry atmosphere and its use 
results in emphasis of humidity differences between air masses, whereas 
H* = 6.5 km corresponds to a saturated atmosphere and minimizes 
moisture differences. Thus it would appear that eventually one might 
use a value of H* as indicated by the application at hand. As a practical 
matter, however, H* = 7.0 km appears to reach a desirable compromise 
between the objectives of the two preceding examples. 

4.2.7. Conclusions 

With the above critical appraisal in mind, the salient conclusions of the 
present study are : 



DATA AND REDUCTION TECHNIQUES 109 

(a) The radio refractive index varies in a systematic fashion with 
climate and different climates may be identified by the range and mean 
values of the refractive index. 

(b) It is 4 or 5 times more accurate to estimate the station value of the 
index from charts of the reduced-to-sea-level index than from charts of 
the station value. This improved accuracy results from using a method 
that allows height dependence to be accurately taken into account. 

(c) Identically equipped tropospheric communications systems might 
be expected to vary as much as 30 dB in monthly mean signal level in 
different climatic regions, and the annual range of monthly mean field 
strength could be as high as 20 dB in the Sudan of Africa and as low as 
to 6 dB in the high plains of the western United States. 



4.3. On the Average Atmospheric Radio Refractive 
Index Structure Over North America 

4.3.1. Introduction 

As has been already noted, the radio refractive index of the atmosphere 
combines three of the meteorological elements normally used to specify the 
state of the atmosphere on either a synoptic or a climatological basis. 
This fact has led to its being used as a synoptic tool [19, 20, 21, 22, 23] and 
as a measure of climatic characteristics [7, 24, 25, 26, 27]. 

The present treatment is concerned with the degree to which the average 
A^ structure in the vertical direction reflects the gross differences in climate 
over the North American continent. Diurnal and seasonal range graphs 
of N at the earth's surface also shed light upon climatic characteristics. 

4.3.2. Meteorological Data and Reduction Techniques 

The basic data used in this study are the significant level data of the 
radiosonde observations from the 18 weather stations shown on figure 4.15 
for the 5-year period 1952-57. These observations were converted by 
means of (1.20) to radio refractivity, N. 

The significant level data were collected for the values of N at the 
earth's surface and within height increments centered upon 0.25, 0.5, 0.75, 
1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 km. Each value of N was referenced to the 
center of its height increment by use of the average atmosphere, 

N{h) = Aoexp [-h/1.0\, (4.9) 

which has been shown to be a reasonable model for the decrease of the 
refractive index with height for the United States. By use of this ex- 
ponential model, the gross height dependence of N within each height 



no 



N CLIMATOLOGY 




Figure 4.15. Radiosonde stations used in this study. 



increment was effectively removed and there remains a more reliable 
estimate of the mean N at the center of the height increment. The final 
value of A^'o, for example, at 1 km over Washington, D.C., would be 300 
with a standard deviation of 15.7 N units for 152 observations. The 
error in the mean is then determined to be 1.3 from 



s(N) = 



s(N) 
Vk 



(4.10) 



with k = 152. An examination of similar data for all stations and levels 
used shows that s{N) is generally less than 1.0 A" unit. 

In addition, when considering atmospheric cross sections, the mean 
values of A were referred to sea-level by means of (4.9) thus further em- 
phasizing climatic differences [28]. 



AVERAGE A^o STRUCTURE 111 

4.3.3. Average N„ Structure 

The first series of charts presented in this analysis are those showing 
the time variation of A^'o both vertically above each station as well as at 
the standard ground-observing level. A slight dichotomy in data sample 
exists since the vertical data are monthly averages of radiosonde observa- 
tions taken twice a day, corresponding in time to the hours 0300 and 1500 
GMT, while the surface data are 8-year means for the even hours of the 
day local time [29]. This climatic variation at each station is represented 
by a two-part chart prepared for each station, the first part depicting 
seasonal changes in the mean value of No throughout the first 3.5 km 
above the station and the second showing the seasonal and diurnal changes 
of No at station elevation. These two presentations give a more complete 
climatic picture of a location than do the usual unidimensional annual 
cycle graphs. 

Tatoosh Island (TTI), figure 4.16, off the coast of Washington State, 
illustrates typical features of a marine west coast climate. The seasonal 
profile changes indicate moderate gradients the year around with a small 
summer maximum in the No gradients in keeping with the cool tempera- 
ture regime and small diurnal temperature range of this maritime- 
dominated climate. This consistency of weather conditions is further 
emphasized by the almost complete lack of diurnal pattern in the surface 
data throughout the year and small seasonal change in No for any hour 
of the day. 

The seasonal profile chart of Oakland (OAK), figure 4.17, indicates less 
maritime effect than at Tatoosh Island and shows clearly the influence of 
the summer subsidence inversion on the northeastern edge of the Pacific 
high-pressure area by both the low values of No and increased gradients 
at about 1.0 km. The surface data show the moderate seasonal and 
diural cycles for all months of the year expected in a Mediterranean 
climate. 

The effects of the Pacific High are more pronounced further south along 
the California coast. The vertical profile data for San Diego (SAN), 
figure 4.18, show clearly that during the winter months the strongest 
gradients of A^o are near the surface as one would expect in a maritime 
dominated climate while, by contrast, in the summer the well-known 
Southern California elevated inversion produces strong gradients aloft. 
The subsiding air above the inversion is reflected by a characteristic in- 
crease of No with height. The maritime effect is also evident at the 
surface in the low diurnal range of about 10 N units. It is significant 
that the presence of the elevated layer produces a smaller diurnal range 
in the summer than in the winter. 

The semitropical nature of the humid periphery of the Gulf of Mexico 
is reflected by the data for Brownsville, Tex. (BRO), figure 4.19, which 
show a strong seasonal cycle and pronounced diurnal range in the summer 



112 



N CLIMATOLOGY 




Figure 4.16. Diurnal, seasonal, and vertical variation of No for Tatoosh Island, 

Wash. (TTI). 



due in large measure to the very high summer temperatures in this locale 
with resultant high water-vapor capacity. It is also quite striking that 
the general level of A^o is 25 at 30 iV" units above that of the west coast. 
The high A^o values of the Gulf regions are also found at Miami (MIA), 
figure 4.20, although the moderating effect of continual onshore winds 
produces less pronounced vertical gradients and smaller diurnal ranges 
than Brownsville, particularly during the summer months. 



AVERAGE iVo STRUCTURE 



113 




Figure 4.17. Diurnal, seasonal, and vertical variation of No/or Oakland, Calif. (OAK). 



The interplay of polar and maritime air masses along the middle east 
coast of the United States is reflected in the strong seasonal range of 50 A^^ 
units at Washington, D.C. (DCA), figure 4.21. The summertime diurnal 
range of 15 A" units reflects the moderating influence of maritime air so 
common along the central east coast in the summer. 

The latitudinally controlled lower mean temperatures of the northeast 
coastal regions are reflected in the generally lower values of A'^o plus 



114 



N CLIMATOLOGY 




Figure 4.18. Diurnal, seasonal, and vertical variation 0/ No/or San Diego, Calif. (SAN). 



smaller seasonal, diurnal, and vertical ranges as illustrated by Portland, 
Me. (PWM), figure 4.22. The curious low in late autumn appears to be 
the result of advection of air from the continental interior and is perhaps 
indicative of the Indian summer of New England. The long New Eng- 
land winter appears in the surface data as a large area of nearly constant 
No. 

The above data reflect the south-to-north change from humid sub- 
tropical to marine-modified continental climates typical of the lee coasts 



AVERAGE No STRUCTURE 



115 




Figure 4.19. Diurnal, seasonal, and vertical variation 0/ No/or Brownsville, Tex. (BRO). 



of continents. By contrast, the island-like station of Cape Hatteras 
(HAT), figure 4.23, reflects both the characteristic strong seasonal range 
of a lee coast station that is dominated by dry, cold continental air in the 
winter and warm, moist maritime air in the summer and the small diurnal 
ranges for all months of the year of a maritime modified climate. 

The data for Denver, Colo. (DEN), figure 4.24, strongly reflect the 
influence of the climatic controls of altitude and continentality. The level 
of A''o is intermediate to that of Oakland and Washington. The vertical 



116 



A^ CLIMATOLOGY 




Figure 4.20. Diurnal, seasonal, and vertical variation of No /or Miami, Fla. {Ml A). 



profile chart for this station indicates the mild influx of tropical air in the 
summer at the surface and the convective mixing of this air to great heights 
by the strong thunderstorm activity common to the area. The high No 
values between 2 and 3 km are perhaps due to superior air subsiding on 
the lee slopes of the Rocky Mountains. Relatively intense diurnal ranges 
are apparent in both summer and winter, as the day to night variations of 
this high plains climate are controlled to a large extent by radiational heat- 
ing and cooling through the thin atmospheric blanket at high elevations. 



AVERAGE No STRUCTURE 



117 




Figure 4.21. Diurnal, seasonal, and vertical variation 0/ No for Washington, D.C. 

(DCA). 



The strong warm-season influx of tropical maritime air from the Gulf 
of Mexico up the Mississippi Valley is spectacularly in evidence at 
Columbia, Mo. (CBI), figure 4.25. The most significant feature of these 
charts is the weak year-round diurnal range of 10 A^ units or less accom- 
panied by a strong seasonal range of nearly 50 N units. 

In summary, it is seen that the west-coast stations display a rather 
uniform low average value of A^o accompanied by small seasonal and 
diurnal ranges. This is due to the continual onshore advection of cool 



118 



N CLIMATOLOGY 




Figure 4.22. Diurnal, seasojial, and vertical variation ofNofor Portland, Me. (PWM). 



maritime air that keeps the mean temperatures both low and uniform. 
By comparison, the continental stations show relatively large diurnal and 
seasonal ranges controlled in large measure by the radiative heating and 
cooling, summer to winter and day to night. The east-coast stations 
display a general increase of iVo from north to south arising from the 



AVERAGE No STRUCTURE 



119 




Figure 4.23. Diurnal, seasonal, and vertical variation of No /or Cape Hatieras, N.C. 

{HAT). 



general increase of mean temperature with resultant increase in water- 
vapor capacity. These same stations have a marked seasonal range due 
to the interplay of continental and maritime effects. The relatively 
small summertime diurnal ranges along the east coast reflect the strong 
influence of maritime air upon this region during that season. 



120 



N CLIMATOLOGY 




Figure 4.24. Diurnal, seasonal, and vertical variation of No /or Denver, Colo. (DEN). 



AVERAGE No STRUCTURE 



121 




Figure 4.25. Diurnal, seasonal, and vertical variation of No for Columbia, Mo. (CBI). 



122 



N CLIMATOLOGY 



4.3.4. Continental Cross Sections 

The vertical profile data obtained for the stations of figure 4.15 provide 
a means to construct A'^o cross sections traversing varied climatic zones 
and geographic regions. 

The first cross section of this series, figure 4.26, is taken along the 
Pacific coast of central North America from Canada to Mexico. Intru- 
sion of polar air at the northern end of the wintertime cross section is 
responsible for the relatively flat N gradient over Tatoosh. The 320 
A^'o-isopleth covering most of the coast southward from Tatoosh shows the 
uniform modifying influence of the ocean which, at this time of year, is 
considerably warmer than the continent. Over southern California the 
low at 500 m results from the drying-out effects of the Pacific high inver- 
sion. On the summer cross section moderate A'"o-values typify the coast. 
The minimum at Oakland stems perhaps from the upwelling of relatively 
cool ocean waters off the California coast. Striking evidence of the 
Pacific inversion is apparent from the low at 1 km between Oakland and 
San Diego that results from significant decreases in the vapor pressure 
term contribution to N within the dry subsiding air aloft. 




SAN TTI 
2 

DISTANCE (KmxIO') 



Figure 4.26. North-south No cross section along the western coast of the United States. 



CONTINENTAL CROSS SECTIONS 



123 



FEBRUARY 




^ >25 



335 






I AUGUST I 



PWM 




OCA HAT 



COF PWM 
2 

DISTANCE (KmxIO') 




Figure 4.27. North-soxdh No cross section along the eastern coast of the United States. 



The cross section for the eastern seaboard of the United States, figure 
4.27, presents a considerably different picture of refractive index chmate 
than the Pacific coast. On the winter cross section the area between 
Portland and Washington falls within a low produced by frequent occur- 
rence of polar air masses during this season of the year. Southward to- 
ward Cocoa, Fla. (COF) there is a significant A'' increase that corresponds 
to the considerable latitudinal temperature gradient that exists on the 
east coast of the continent during the winter months. On the summer- 
time cross section the temperature gradient difference between the two 
coasts is even more pronounced and is reflected in the strong east-coast A^'o 
difference of about 40 N units as compared to the Tatoosh-San Diego 
difference of about 10 A'' units. The vertical gradients are also seen to 
increase in a systematic fashion from north to south, reflecting the in- 
creased vapor pressure gradients in the warmer southern regions. 

The extensive continental cross section from Isachsen (IC) to Balboa 
(BLB), figure 4.28, discloses several interesting features. On the winter- 
time chart the relatively high A^o values between Isachsen and Churchill 
(YQ) indicate the presence of very cold, dense, dry air. To the south over 
Bismarck, N.Dak. (BIS), and the Great Plains, relatively warm, dry air of 



124 



A^ CLIMATOLOGY 




YQ BIS CBI BRO 

234567 

DISTANCE (KmxIO 



Figure 4.28. North-south No cross section from Isachsen, North West Territories (IC). 
through Churchill (YQ) to Balboa, Canal Zone (BLB). 



CONTINENTAL CROSS SECTIONS 



125 



FEBRUARY I 



325 \ V J ' 

\ 

/ •'530\ \ 

/ / ^ ^ 




AUGUST 



TTI 
3 4 

DISTANCE (KmxIO' 




Figure 4.29. Wesi-eoM No cross section for the northern United States. 



lower density creates a refractive index low. Southward from the 
Great Plains a considerable A^'o gradient is encountered as the Gulf source 
region for tropical maritime air is approached. Latitudinal temperature 
gradients produce a continued increase in A^'o to the southern extreme of 
the chart at Balboa in the tropics. The A^o gradients of the summer cross 
section are largely thermally controlled, ranging from cool temperatures 
and low A'^o in the polar regions to a maximum between Brownsville and 
Balboa near the warm-season heat equator. 

Three zonal cross sections have been prepared extending across the 
northern, central, and southern portions of the United States. The 
February chart for the northern cross section, Tatoosh to Portland, figure 
4.29, exhibits lows at both coasts, with slightly higher values of A^o iu the 
cold interior of the continent. The summertime chart clearly shows the 
intrusion of tropical maritime air from the Gulf of Mexico pushing north- 
ward over the Great Plains. The central U.S. cross section, figure 4.30, 
shows, on the wintertime chart, slightly lower values of A^o on the west 
coast at Oakland than at Hatteras. East of the Rockies, the chinook 
winds of the prevailing westerlies assist in the formation of a relatively 
warm, dry air mass of characteristically low A^o- On the summer map, 



126 



N CLIMATOLOGY 



I FEBRUARY 
330 




I AUGUST 



DEN CBI HAT OAK 

2 3 4 

DISTANCE (KmxIO') 



Figure 4.30. West-east No cross section for the central United States. 



the influence of the Pacific high is in evidence over Oakland, while in the 
interior of the continent the influence of tropical maritime air is noticeable 
up the Mississippi Valley. The zonal cross section across the southern 
United States, figure 4.31, shows a transition from the Mediterranean 
climate of the Pacific coast to the humid subtropical type from Browns- 
ville eastward. Here again the east-coast A^o values, from a higher 
temperature regime, are larger than those in the west. The summer- 
time chart is similar to the winter one with the A^-gradient intensified all 
along the route. 

Figure 4.32 is for the longest cross section, extending approximately 
22,000 km from Canton Island (CAN) to Wiesbaden (WSB) in northern 
Europe. The winter chart shows a gradual decrease in N from Canton 
Island to Brownsville and Cocoa, with very uniform A^ structure over the 
Atlantic to Ship "J," and onwards to Wiesbaden. The summer chart 
shows a double maximum, one at Canton and the second over the warm 
North American continent between Brownsville and Cocoa. The gradual 
decrease in mean temperature northward across the Atlantic Ocean is 
evidenced by the rather uniform decrease of A^o between Cocoa and 
Wiesbaden. 



CONTINENTAL CROSS SECTIONS 



127 




Figure 4.31. West-east No cross section for the southern United States. 



128 



N CLIMATOLOGY 



3.0 



'325 



CAN 




BRO COF .r 

10 '^3 

DISTANCE (KmxIO ) 



FEBRUARY 




I AUGUST I 




"J" WSB 
20 25 



Figure 4.32. West-east No cross section from Canton Island, South Pacific (CTN) 
through Wiesbaden, Germamj (WSB). 



CLIMATIC CHARACTERICS 129 

4.3.5. Delineation of Climatic Characteristics 

Since climate is really a synthesis of weather elements taken over a 
period of time, it is apparent that no fixed climatic boundaries exist in 
nature. These boundaries shift from year to year with changing weather 
and with the addition of new data into the climatic averages. Climatic 
borders, then, represent transition zones between so-called "core climates" 
of one type or another. A "core climate" presumably maintains its 
climatic personality consistently over a period of time. The coniferous 
forest regions of the far north of Canada, for example, maintain a con- 
tinuously frigid climate under the arctic inversion during the heart of the 
winter season. Since a simple system is required for effective classifica- 
tion, the congeries of minute climatic areas arising from extensive climatic 
division are, for practical purposes, coordinated into areas having broad 
similarity of climatic character. Thus we shall only attempt here to look 
for general patterns in our limited data sample. 

The A^'o data for the station elevation of figures 4.16 to 4.25 contain 
information about the radio refractive index climate of the respective 
stations that may be converted to a pair of indices, one seasonal, one 
diurnal, useful in classifying climate. The diurnal index is the ratio 
obtained by dividing the difference between the highest and lowest hourly 
means for August by a similar difference for February. This ratio is then 
plotted versus a ratio of the maximum difference for the 12 mean A'^o 
values at 0200 divided by the maxmium differences for the 1400 means. 

The results of this analysis can be seen on figure 4.33, where the largest 
pair of ratios are observed for Denver and Bismarck, a consequence of 
the strong continentality effects at these inland stations. Denver, it is 
to be noted, displays a large seasonal ratio whereas Bismarck represents 
the extreme in diurnal ratios. Brownsville, Washington, and Columbia 
show sizable seasonal ratios coupled with slightly greater than normal 
diurnal ratios. Summer-to-winter climate changes are considerable at 
these three locations. 

The remainder of the stations display the near-unity value of both 
ratios that would be expected of maritime-dominated climates. 

A comprehensive, though simple, index of climatic variation of the 
vertical distribution of A'' is particularly difficult to envisage. One such 
index is the ratio, R{AN), of the summer to winter value of AA^ versus the 
mean value of AA^ (the absolute difference in A^ at the earth's surface and 
at 1 km above the earth's surface). The gradient, AA'', has received wide 
engineering application and is currently being mapped on a worldwide 
basis by the International Consultative Committee for Radio [30]. When 
this ratio is plotted versus the yearly mean value of AA^, as on figure 4.34, 
one again obtains about the same climatic demarcation as above. For 
example, the maritime-dominated climates generally have values of 
RiAN) of 1.2 or less. 



130 



N CLIMATOLOGY 





2.0^ 


® 










DEN 








1.8- 








C=3 










fee 










Cd 


1.6- 








«a: 










•r^ 










CO 


1.4- 


® 






-3Z 
U-J 




BRO 






OO 


1.2- 


®DCA 






^^ 










-a: 


1.0_ 
( 


CAM HAT PU/U® ^^' 

SAN® (^MiA PWM^ 
COF® ® 
'" OAK 




®BIS 


1 1 1 1 1 
) 1,0 2,0 


3,0 


1 1 
4,0 




MEAN DIURNAL RATIO 



Figure 4.33. Comparison of mean seasonal to mean diurnal ratio of No at the earth's 

surface. 



The more southerly and humid stations have relatively high mean AA^ 
values in excess of 55 A'^ units/km. The single European station, Wies- 
baden, shows both low average value of AA'^ arising from the low average 
temperatures and small seasonal variation associated with the continuous 
onshore advection of cool, moist air from the Atlantic Ocean. The values 
R{AN) < 1.0 for the Canadian stations of Churchill and Isachsen reflect 
the steep gradients associated with the intense stratification during the 
long winter night. The high, dry climate of Denver produces low 
mean values of AA^ but a high seasonal range. The interplay of conti- 
nental and maritime effects along the east coast produces relatively high 
values of R{AN) while the presence of the Pacific high inversion layer 
below 1 km accounts for similar values along the west coast. The ex- 
tremely high value of R{AN) of 1.8 for Tripoli (TPI) arises principally 
from the intense summertime A^ gradients that exist near the surface in 



CLIMATIC CHARACTERISTICS 



131 





2.0j 




■^ 






^ 


1 


® 


ce: 


1.8. 


TPI 


<S 






^ 


1.6. 




CXL 






^ 


1.4. 
I.2_ 


DCA ®^^' ^ ®HAT 


U-] 

en 

-a: 
en 

C_3 


^BIS ® OAK® 

T '" CTN 


*£ 




MIA® ^®^'^° 
®WSB ""^ OOF 


cz> 


l.0_ 


CO 




®YQ n^n 


U_J 




BLB 


OO 


.8_ 
3 


®IC 


40 50 60 TO 




YEARLY MEAN AN 



Figure 4.34. Yearly mean and range ratio of AN. 



this locale. In the lowest layers of the atmosphere warm, moist air from 
the Mediterranean Sea is advected inland underneath superheated air 
from the Sahara Desert. Along the zone of contact between these air 
masses of widely differing moisture content, spectacular drops in vapor 
pressure and, consequently, in the wet term component of A'^ occur. 
During the winter, moderate temperatures and prevailing offshore flow 
combine to produce near-normal gradients of N. Consequently the 
summer-to-winter AiV ratio turns out to be quite high for this borderland 
station. 

The analysis above represents a rough and crude first attempt to under- 
stand the climatic variations of N . It is expected that such classifica- 
tions will improve with the addition of data from more diverse locations 
and, in particular, as our knowledge of the role of water vapor in climatic 
variability increases. 



132 A^ CLIMATOLOGY 

4.4. The Climatology of Ground-Based Radio Ducts 

4.4.1. Introduction 

The occurrence of atmospheric ducts places limitations upon ray tracing 
of VHF-UHF radio waves. Ducting is defined as occurring when a radio 
ray originating at the earth's surface is sufficiently refracted so that it 
either is bent back towards the earth's surface or travels in a path parallel 
to the earth's surface. Although the rigorous treatment of ducting in- 
volves consideration of the full wave eciuation solution [31] rather than a 
simple ray treatment, the present study will be based upon a geometrical 
optics definition of the limiting case in which ray tracing techniques may 
be used. This simple criterion is then applied to several years of radio- 
sonde observations from stations typical of arctic, temperate, and tropical 
climates to derive estimates of the variation of the occurrence of radio 
ducts with climatic conditions. 

4.4.2. Meteorological Conditions Associated With Radio 
Refractive Index Profiles 

The path followed by a radio ray in the atmosphere is dependent upon 
the gradient of the refractive index along that path. Of the vertical and 
horizontal gradient components that compose the path gradient, the 
horizontal gradient is normally negligibly small. Thus, the atmosphere 
is considered horizontally homogeneous and only the vertical gradient of 
the refractive index is utilized. The numerical value of the vertical 
gradient of the index of refraction depends on the vertical distribution 
of atmospheric temperature, humidity, and pressure. 

Normally, temperature and humidity decrease with height in the atmos- 
phere, since turbulence prevents any great changes in structure. How- 
ever, there are periods of time in which the air becomes fairly calm, 
whereupon temperature inversions and humidity lapses can be built up 
and maintained. Temperature inversions have a twofold importance in 
that (a) they can be widespread in area and persist over a relatively long 
period of time, and (b) they exercise a stabilizing influence on air motion 
such that turbulence is suppressed and strong humidity gradients may 
develop. Layers in which there is intense superrefraction to the point of 
duct formation may be formed as a result of these gradients and trapping 
of radio waves may follow. The inversions may start at ground level or 
at some greater height. The thickness of the layer can show great 
variability. Three processes that form temperature inversions are: 

Advection: Advection is the horizontal flow of air having different heat 
properties. Such a process is of importance in microwave propagation 
since it may lead to a different rate of exchange of heat and moisture be- 
tween the air and the underlying ground or ocean surface, thus affecting 



RADIO REFRACTIVE INDEX PROFILES 133 

the physical structure of the lowest layers of the atmosphere. This proc- 
ess results in air with different refractive index characteristics being 
brought into the area. The most common and important case of advec- 
tion is that of dry air above a warm land surface flowing out over a cold 
sea. This type of advection frequently appears in the English Channel 
during the summer when the weather has been fine for several days. 

Advective duct formation depends on two quantities: (a) the excess of 
the unmodified air temperature above that of the water surface, and (b) 
the humidity deficit (the difference between the water vapor pressures of 
the modified and unmodified air). If these quantities are large, especially 
the humidity deficit, an intensive duct may form. 

Important advective processes can also occur over land, but the condi- 
tions required for duct formation occur less frequently. However, a duct 
may be formed when dry, warm air flows over cold, wet ground with 
resultant temperature and humidity structure previously discussed. 

Radiation: Differences in daytime and nighttime radiation are the 
causes of diurnal variation in refractive conditions. A subref ractive layer 
may be present during the day, especially at the time of maximum surface 
heating. Clear skies and light surface winds at night result in consider- 
able cooling of the earth, thus causing the formation of temperature in- 
versions. The surface heat loss produced by nocturnal radiation is a 
prime factor in the formation of temperature inversions. Atmospheric 
stratification formed by such a combination of meteorological parameters 
may jjroduce a trapping layer. A temperature inversion is seldom strong 
enough to produce a duct in the middle and low latitudes, but it is of 
major importance in the formation of ducts in the northern latitudes. 
Low stratus clouds or extreme amounts of moisture (as in the troj^ics) 
tend to prevent loss by radiation which lessens the possibility of duct 
formation. 

If the ground temperature in a nocturnal inversion falls below the 
dewpoint temperature, the water vapor in the lowest layers of the air 
condenses and the heat of condensation is released directly to the air. 
Under conditions of radiative fog formation, the humidity lapse tends to 
counteract any temperature inversion present and may cause substandard 
refraction if the humidity "inversion" is sufficiently strong. However, 
the temperature inversion may be strong enough to keep the layer 
standard or superrefractive. 

Subsidence: Subsidence is the slow settling of air from a high-pressure 
system. The air is heated by adiabatic compression as it descends and 
spreads out in a layer well above the earth's surface. This process pro- 
duces stable layers and inversions of temperature with an accompanying 
decrease in relative humidity. Since the air has come from a high level 
in the atmosphere, it is dry and may overlay a cooler, moist air mass. 



134 A^ CLIMATOLOGY 

This type of inversion may cause the formation of an elevated super- 
refractive layer where the air temperature usually decreases immediately 
above the ground, rises through the inversion layer and decreases above 
the layer. This is a common occurrence which may be observed at any 
time. Subsidence has a tendency to destroy subrefractive layers and to 
intensify superrefractive layers. Although the effects of subsidence are 
generally observed at high levels, they are occasionally observed at lower 
levels, especially in the subtropics. Since subsidence frequently occurs 
in the lee of mountains and in the southeastern regions of northern 
hemisphere highs, elevated ducts may also be observed. 

Conditions inimical to ducting are those which mduce mixing of the 
lower atmosphere. Small scale atmospheric motions (turbulence) and 
consequent mixing and mass exchange result from differential surface 
heating and surface roughness effects. Both of these processes work to 
destroy stratification. They ultimately result in uniform vertical distri- 
bution of moisture through considerable depths of the lower atmosphere 
and the establishment of neutral temperature lapse rates. Accordingly, 
where the process of mechanical and convectively induced mixing are at 
work, the probability of the occurrence of ducting is vanishingly small. 
Thus, few, if any, ducts are observed over snow-free, low albedo land 
areas from midmorning to late afternoon when the skies are clear, or in 
areas of moderate to great surface roughness when the surface winds are 
more than a few meters per second, irrespective of cloud conditions or the 
time of the day. 



4.4.3. Refractive Conditions Due to Local 
Meteorological Phenomena 

Land and sea breezes may produce ducts along the coastal regions since 
the winds are of thermal origin, resulting from temperature differences 
between land and sea surfaces. During the day, when the land gets 
warmer than the sea, the air above the land rises and is replaced by air 
from the sea, thus creating a circulation from the sea to the land, called a 
sea breeze. During the night, the land becomes colder than the sea and a 
circulation, called a land breeze, is set up in the opposite direction. This 
type of circulation is generally shallow and does not extend higher than 
a few hundred meters above the land or sea surface. 

A land or sea breeze may modify the refractive conditions in different 
ways depending upon the distribution of moisture in the lower layers. 
Since these breezes are of a local nature and generally extend only a few 
miles, only coastal locations are usually affected. The very nature of sea 
and land breezes results in a marked refractive index pattern. With a 



BACKGROUND 135 

sea breeze a duct may be formed over the water due to subsidence. The 
land breeze is accompanied by subsiding air over the land with resultant 
duct formation. 

The formation of fog results in a decrease of superrefractive or ducting 
possibilities. When a fog forms by nocturnal radiation, the water content 
of the air remains practically the same; however, part of the water 
changes from the gaseous to the liquid phase thus reducing the vapor 
pressure. The resulting humidity lapse rate tends to counteract the 
temperature inversion and cause above standard refraction. However, 
the temperature inversion may be strong enough to keep the layer stand- 
ard or superrefractive. This process may also occur with advection fogs. 

The nighttime temperature profile is a result of the interaction between 
nocturnal radiation, turbulence, and heat conduction. The associated 
refractive index profiles are such that a radar duct begins to form about 
the time of sunset, developing quickly during the early evening, more 
slowly after midnight, and dissipating rapidly after sunrise. This is 
mainly an inland effect resulting from large diurnal temperature varia- 
tions observed in the interiors of large continents. However, a shallow 
body of water may have an appreciable diurnal temperature variation, as 
compared to the open ocean so that superrefraction may occur over such 
a location from time to time. 

It is generally recognized that radiosonde observations (RAOB's) do 
not have a sufficiently high degree of accuracy to be completely acceptable 
for use in observing changes in the degree of stratification of the very 
lowest layers of the atmosphere; however, until more accurate methods 
such as meteorological towers and refractometer measurements are more 
commonly used, the RAOB will continue to be used as a basis for fore- 
casting the occurrence of superrefractive conditions. 



4.4.4. Background 

The property of the atmosphere basic to radio ray tracing is the radio 
refractive index of the atmosphere, n, which for VHF-UHF frequencies 
at standard conditions near the surface, is a number of the order of 1.0003. 
Although the refractive index is used in ray tracing theory, it is more 
convenient when evaluating refraction effects from common meteoro- 
logical observations to use the refractivity A'', which, for the frequency 
range to 30,000 Mc/s, is given by (1.20). 

When evaluating the meteorological conditions that give rise to refrac- 
tive phenomena, it is frequently instructive to examine separately the 
behavior of the dry component, D, and wet component, W, of N . 



136 



A^ CLIMATOLOGY 



The gradient of the refractivity, AA'', with respect to height may then 
be expressed : 



AN = AD -\- AW. 



(4.ii: 



Average values of AA^, AD, and AW are given in table 4.6 for two incre- 
ments between the earth's surface and 1 km above sea level for Fairbanks, 
Alaska; Washington, D.C.; and Swan Island, W.I. 



Table 4.6. Gradient of N, D, and W in N units per kilometer 



Station 


Height 
increment 


February 


August 




-AZST 


-^D 


-hW 


-^N 


-AZ) 


-^W 


Fairbanks, Alaska 

Washington, D.C 

Swan Island, W.I 


surface— 0. 5 km 
0. 5km-1.0km 

surface— 0. 5 km 
0. 5km-1.0km 

surface— 0.5 km 
0. 5km-1.0km 


37 
35 

41 
30 

39 

58 


41 
35 

34 
26 

24 
24 


-4 


7 
4 

15 
34 


31 

36 

60 
46 

47 
66 


27 
24 

28 
24 

26 
24 


4 
12 

32 
22 

21 
42 



Several general observations may be made of the data of table 4.6: 
The gradient of the dry term is relatively less variable than that of the 
wet term when considered as a function of season or height; the increase 
of AA^ from winter to summer at a particular location or from arctic to 
tropical climate at a given time is most strongly reflected in AW rather 
than in AD. The marked increase of gradient with height for Swan Island 
reflects the drop of refractivity across the interface of the trade wind 
inversion where dry subsiding air overlies the moist oceanic surface layer. 

A fundamental equation used in radio ray tracing is Snell's law, which, 
for polar coordinates, is given in chapter 3 as 



nr cos 6 = rioVo cos do, 



(4.12) 



where n is the radio refractive index of the atmosphere, r is the radial 
distance from the center of the earth to the point under consideration, and 
6 is the elevation angle made by the ray at the point under consideration 
with the tangent to the circle of radius r passing through that point. The 
radius to any point, r, is equal to (a + h), where a is the radius of the 
earth and h is the height of the point above sea level. The zero subscript 
refers to the value of n, r, or d at the earth's surface. 

For the present study, geometrical optics techniques, similar to those 
considered by Bremmer [32], are used to indicate when refraction is 



BACKGROUND 137 

sufficiently great to direct a ray either back to earth or in a circular path 
at a constant height above the earth; i.e.: 

noVo cos do ^ , 

nr ~ ' 

This condition then allows one to obtain the value of do which divides the 
rays into two groups: those that penetrate the duct and those that are 
trapped within the duct. This particular value of ^o, called the angle of 
penetration and designated dp, is obtained by noting those instances where 

Mo > 1 (414) 

nr 

and solving for the value of ^o such that (4.13) is equal to unity. 

It is instructive to consider the order of magnitude of refractive index 
gradient needed for trapping for several commonly observed refractive 
index profiles. If we rewrite Snell's law, 

ritrt cos dt = HaTa cos dd, (4.15) 

where the subscripts t and A refer to the values of the variables at the 
transmitter height and the top of the trapping layer, then trapping 
occurs when 



rit Tt 



riA Va 



> 1, (4.16) 



and the angle of penetration at the transmitter, dp, is given by setting 

cos dp = 1. (4.17) 



nt Vt 



riA Ta 



The maximum permissible n gradient for a given value of dp is then given 
by: 

^ ^ ru^^UA ^ 

t^r ta — Tt 

where ua must satisfy (4.17); i.e.: 

_ Villi cos d^. (4.19) 

Ta 



138 N CLniATOLOGY 

By designating Va = rt -\- /1.4, (4.18) becomes 



An 

Ar 






1 - 



rt + hA 



cos Or, 



(4.20) 



By rewriting (4.20) 



An 


rit 
hA 


1 - 


1 


Ar 


1+^ 
rt 



cos dr 



(4.21) 



and expanding (1 + hA/rt)~'^ and cos Op, one obtains the expression: 



Lr, 



An _ 1^1 

Ar - ~ ''' Irt + 2hAA 



(4.22) 



by neglecting terms 0p/4! and QiA/rtY, and terms beyond these. For 
the case of Gp = 0, (4.22) reduces to : 



An _ n 1 
Ar a a 



157 N units/km. 



(4.23) 



It is seen from (4.22) that the n gradient necessary to trap a radio ray 
at a given value of dp is practically independent of transmitting antenna 
height above the earth. For example, a, 6p = ray will be trapped by 
an n gradient of —157.0 A'' units/km at sea level, where Ut ~ 1.0003, 
while the necessary n gradient at 3 km above sea level will be —156.9 
A^ units/km for a,n Ut = 1.0002. This indicates, for all practical appli- 
cations, that the necessary n gradient for trapping is independent of 
altitude. Further, by considering the temperature and humidity gra- 
dients encountered in the troposphere, one is led to the conclusion that 
ducting gradients would not be expected to occur at altitudes greater 
than 3 km. In fact, Cowan's [33] investigation indicates that trapping 
gradients are nearly always confined to the first kilometer above the 
surface. 

A consideration of (4.22) indicates that the magnitude of the negative 
gradient necessary for ducting is 1/a for dp = but increases in propor- 
tion to dp'/2h as dp increases. The gradients necessary for atmospheric 
ducts as a function of 60 are given for several different profiles in figure 
4.35 An analysis of radiosonde data indicates that gradients in excess of 
0.5 N units per meter are seldom exceeded within atmospheric layers. 
It is interesting to note how rapidly the necessary gradients increase to 



BACKGROUND 



139 



1300 



1200 



1100 



1000 



0^ 900 



800 



2 X 



700 



600 



500 



400 



300 



200 



100 



















/ 


/ 


















/ 


/" 
















K 


^ 








Ele 
O.Skr 
and 


voted Lo) 
7 above g 
/km tfiic 
320, Nas 


/er~~-__^ 
round 

Ir 


^^^ 


yj 












No = 


= 300 


^ 


1 



















, 


' 


















/ 








/ 


1' 


'pproximc 
adio sond 


ite upper 
e-observ 


limit of 
edgrodlet 


Its. / 


' 






/ 


^ 


















^ 




Grour 
0^ 


d Based 
03 km th 
Nq = 320 


.oyer — — 


/ 




^ 


x; 


Ground Be 


ised Loye 






^ 








^0- 


320 




_- 




— -^ 




limum Gr 




/a 



















4 5 6 

Br, in Milliradians 



10 



Figure 4.35. Refradiviiy gradients needed for radio ducts. 



the approximate upper limit of radiosonde-observed gradients: a ground- 
based layer 100 m thick attains this gradient at 8.3 mrad while the 
maximum observed gradient is intercepted by the 30-m layer curve at 
^0 = 4.5 mrad. A third example was calculated for an elevated layer 0.5 
km above the ground and 100 m thick by assuming normal refraction 
between the ground and the base of the layer and solving for the necessary 
ducting gradient within the layer. The large values of n gradient neces- 
sary for this case explain why elevated ducts were not observed. Al- 
though the preceding examples were calculated for a ground transmitter, 
the combinations of dp, An/Ar, and Ar are very nearly the same as would 
be obtained for any other transmitter height within the first 3 km above 
the surface. 



140 N CLIMATOLOGY 

4.4.5. Description of Observed Ground- Based 
Atmospheric Ducts 

Approximately three years of radiosonde data typical of an arctic 
climate (Fairbanks, Alaska), a temperate climate (Washington, B.C.), 
and a tropical maritime climate (Swan Island, W.I.) were examined by 
means of a digital computer for the occurrence of ducts during the months 
of February, May, August, and November. The percentage occurrence 
of ducts is shown on figure 4.36. The maximum occurrences of 13.8 
percent for August at Swan Island and 9.2 percent for Fairbanks in 
February are significantly greater than the values observed at other times 
of the year. The Washington data display a summertime maximum of 
4.6 percent. These data indicate that the temperate zone maximum 
incidence is about one-half the wintertime maximum incidence in the 
arctic, and about one-third of the summertime tropical maximum. 

The range of observed values of 0p is shown in figure 4.37. The mean 
value calculated for each month as well as the maximum and minimum 
values of dp observed for the limiting cases are given for each month and 
location. The mean value of the angle of penetration under these condi- 
tions is between 2 and 3 mrad and appears to be independent of climate. 
The maximum value of dp observed during ducting is 5.8 mrad. 

The refractivity gradients observed during ducting are given on figure 
4.38. The maximum gradient of 420 N units per kilometer was observed 
during February at Fairbanks, Alaska. The mean values of N gradient 
appear to follow a slight climatic trend from a high value of 230 N units 
per kilometer at Fairbanks to a value of 190 A^ units per kilometer at 
Swan Island. 

Another property of radio ducts is their thickness, which is given in 
figure 4.39. Again there is observed a slight climatic trend as the median 
thickness increases from 66 m at Fairbanks to 106 m at Swan Island. 
These values of thickness correspond to the gradients given in figure 4.38. 
One can then obtain, by linear extrapolation, the thickness at which the 
gradient is equal to — 1/a; i.e., the height corresponding to the gradient 
just sufficient to trap the ray at do = 0. These values, shown on figure 
4.40, display an increase in the median thickness of about 25 percent for 
Swan Island, 100 percent for Washington, and 200 percent for Fairbanks, 
which results in a reversal of the climatic trend of the observed thickness 
between Fairbanks and Swan Island. This increase in height emphasized 
the preceding conclusion. Fairbanks is characterized by shallow layers 
with relatively intense gradients. 

These maximum duct widths may be used to estimate the maximum 
radio wavelengths trapped. Kerr [34] gives the maximum wavelength, 
X, trapped by given thickness, d: 

X„.ax = cy'" d''\ (4.24) 



GROUND-BASED ATMOSPHERIC DUCTS 



141 





16 




15 




14 


en 








u 


1.^ 


=3 




Q 




"D 


12 


CD 




cn 




O 




CD 


1 1 


"O 




c 




13 


lU 


O 




V 




CD 


9 



O 



c 

CD 
U 

\_ 

cu 
CL 





















































Nun 
num 


bers 
ber of 


/7 curvi 
profile 


s analy 


'sed. 


1 


.218 




















/ 


\ 




















/ 




\ 


















/ 




\ 


^ 






226 








1 








\ 






\ 








/ 


5wi 


7/7 /5/i7 


'70''^ 


A 


166 




\ 








/ 














\ 


\ 






/ 
















\ 




1 


1 
















\ 


'7 


1 








y496 












\ 


( 








S 


k 






151 




^ 


\ 




H 


'ashinc, 


/■/c/?, z; 


..-^ 


\. 


^472 
247 




y 


^-v 


\ 


'175 


^o 


1 
ir banks, Alaska -—, 
1 ^1 




447 


/ 




247^ 








^^^ 












247 





M A 



M 



N D 



Months of the Year 

Figure 4.36. Frequency of occurrence of ground-based ducts. 



by assuming a linear decrease of n within the duct. The coefficient c 
is a constant and 7 is a function of the n gradient excess over the minimum 
value of An/A/i = 1/a. Expressing X^ai in centimeters, and d in meters, 
then 



{Kl^ _ 0.157) 10;' 



(4.25) 



142 

and 



N CLIMATOLOGY 



c = 2.514 X 102. 



The maximum wavelengths trapped during ducting conditions were 
estimated by (4.24) for the maximum duct thicknesses of figure 4.40. 
These values, given in table 4.7, were determined for the month with the 
maximum occurrence of ducts, thus allowing an estimate of the radio 
frequencies likely to be affected by ducting conditions. Note, for ex- 
ample, that the data of table 4.8 indicate that 1000 Mc/s rays (X = 30 cm) 
will be trapped by 50 percent of the ducts regardless of location. 



FAIRBANKS, ALASKA 


Numbers on curves are the 
total number of observed ducts. 

\ 




, 


{ 






\\ 






1 


V 


Max 


18 \ \ 




\ j^Meon 


1 


iMin 



FEB. MAY AUG NOV 



WASHINGTON, D.C 

t 






\ 




r\ 






^\ 




'/ 


-21 


V 


Max 


3 


^ 


\ 

9 


Mean 




^ 




Min 



FEB MAY AUG NOV 





SWAN ISLAND 








^^ 


Max 


/ 








/ 




14 


Mean 


Z5, 


30 


^ 




4 




/ 


Min 


\ 


\ 




/ 







FEB MAY AUG NOV 



Months of the Year 
Figure 4.37. Angle of penetration of ground-based ducts. 



GROUND-BASED ATMOSPHERIC DUCTS 



143 



< 260 
I 



1 1 1 

FAIRBANKS, ALASKA 

\ ' ' ' 










\ Numbers on curves are the 
\ total r>umber of observed ducts 


\ 








\ 








\ 








\ 




1 




\ 


/ 


v 




18 \ 




\\ 








\ 


^Max 






^ 








^ 


\ 




^ 




\\Mean 




1 







1 1 
WASHINGTON, D.C 

1 1 








i 




J 




\ 




/ 




\ 




/ 




\ 




/ 




\ 




/ 




\ 




/ 




\ 




/ 


.3 \ 




/ ^ 




\ 




1 / 




\ 




7, 




\ 


Mox 






9 


Mean 

Min 











1 

SWAN ISLAND 

1 
















































Mox 


^ 


^ 






/ 








1 








1 




> Mean 




y 








4 


1 
/Min 






\ 



FEB MAY AUG NOV 



FEB MAY AUG NOV FEB MAY AUG NOV 



Months of the Year 
Figure 4.38. Refractivity gradients of ground-based ducts. 



Table 4.7. Estimated maximum wavelength trapped at do = 



Station 


Percentage of duets that trap wavelengths in centimeters equal to 
or shorter than tabulated value 




95 


90 


75 


50 


25 


10 


5 


Fairbanks, Alaska (February). 
Washington, D.C. (August)__. 
Swan Island (August). . . 


20.5 
7 
12 


22.6 

10 

23 


25.5 

27 

41.5 


34.0 

50 

60 


43.5 
112 
82.3 


61.5 
164 
92 


69.0 
200 
132 







Table 4.8. 


Ranges of 


surface ducting gradients 


Duct gradients 
at Washington, D.C. 


N units/km 
February 


N units/km 
May 


N units/km 
August 


TV units/km 
November 


Max 


-200 
-185 
-175 


-372 
-230 
-162 


-395 
-240 
-158 


—195 


Mean 

Min 


-170 
— 159 







144 



N CLIMATOLOGY 



o 



300 

280 

260 

240 

220 

200 

180 

160 

140 

120 

100 

80 

60 

40 

20 





















































-Washington, D.C. 
















































































































\ 










































\ 








































N 


\, 








































N 


\ 




































"^ 


"--^ 


\ 




































"^ 


^ 




^ 


^ 


































^ 


\ 






\ 


N^ 


'^ 


^ 


-^ 


---Swan Island 




















X 








^^ 






""^ 


« 




















F 


'airb 


ank. 


?, Alaska — 


>-v 




- Washington 


D.C 


































' 



















































0! 0,2 0,5 I 2 5 10 20 30 40 50 60 70 80 90 95 98 99 995 99S999 

Percentage of Observations That Equal or Exceed ttie Ordinate Value 
Figure 4.39. Observed ground-based duct thickness. 



The reader is cautioned that an atmospheric duct does not have the 
sharp boundaries of a metallic waveguide. Thus the maximum wave- 
lengths obtained by (4.24) do not represent cutoff frequencies but, as 
Kerr is so careful to emphasize, merely suggest lower limits under the 
assumptions of this rudimentary theory. 

The results given above were derived from a consideration of radio- 
sonde data. Although the radiosonde is not an extremely sensitive in- 
strument, it is readily available and is the only source of climatic informa- 
tion involving the temperature and humidity structure of the atmosphere. 
It is believed that the radiosonde data will at least yield the climatic trend 
of radio ducts as well as their probable temperature and humidity distri- 
butions. Further, it is evident that the choice of stations will definitely 
affect the percentage of ducts observed. For example, it is almost certain 
that a greater percentage of ducts would be observed over water in the 
subtropics than over Swan Island. 



GROUND-BASED ATMOSPHERIC DUCTS 



145 



With these reservations in mind, the present study has shown: 

(a) Ducts occur no more than 15 percent of the time. 

(b) The annual cycle of the incidence of ducts is reversed for the arctic 
and tropical stations studied. The arctic station has a wintertime maxi- 
mum and the tropic station a summertime maximum. The temperate 
station has a summertime maximum incidence of less than 5 percent. 

(c) The maximum initial elevation angle is limited to 5.8 mrad with 
a mean value of about 3 mrad. 

(d) The steepest gradient of A^ observed is —420 A'^ units per kilometer. 

(e) The maximum thickness of observed ducts is such as to trap radio 
waves of 1000 Mc/s and above at all locations for at least 5 percent of the 
observed ducts. 

(f) Ducts in the arctic appear to be associated with temperature in- 
versions at ground temperatures of —25 °C or less; temperate zone, with 
the common radiation inversion and accompanying humidity lapse; 
tropics, with a moderate temperature and humidity lapse for temperatures 
of about 30 °C. 



450 



400 



350 



300 



250 



Q 200 
E 

•i 150 



100 



50 







> 


^ 










































N 


u-^ 


Vashin 


gton 


D.C 


-. 
































\ 








































N 




s 
































X 








\ 


\ 


\ 






























^ 












^ 


^ 


\ 




^-^ 


'airba 


nks, 


Ale 


iska 






































—Washington, D.C. 

TS 1 1 

^^Swan Island 
































^<> 











0.1 0.2 0.5 I 2 5 10 20 30 40 50 60 70 80 90 95 98 99 995 99S 999 
Percentage of Observations That Equal or Exceed the Ordinate Value 
Figure 4.40. Maximum ground-based duct thickness. 



146 A^ CLIMATOLOGY 

4.5. A Study of Fading Regions Within the Horizon 
Caused by a Surface Duct Below a Transmitter 

4.5.1. Introduction 

Among the factors influencing the choice of an antenna site for a micro- 
wave receiver operating on a line-of-sight path is the desirability of 
locating in a region relatively free from space-wave fadeouts. Although 
line-of-sight propagation is normally characterized by steady, high, de- 
pendable signals, deep, prolonged space-wave fadeouts are observed from 
time to time. Since serious disruptions occur during such fadeouts, a 
systematic discussion of fading phenomena is of considerable interest to 
the radio circuit engineer. 

There are [35] results in detail of a year's study of signal fadeouts 
occurring at certain points within the horizon caused by ground-based 
superrefractive layers. The path studied in this case [35] has its trans- 
mitter at Cheyenne Mountain near Colorado Springs, Colo., at an eleva- 
tion of 8800 ft. Receivers are at Kendrick, Colo., 49.3 mi from the 
transmitter, and at an elevation of 5260 ft; Karval, Colo., 70.2 mi from 
the transmitter, and at an elevation of 5260 ft; and Haswell, Colo., 96.6 
mi from the transmitter, and at an elevation of 4315 ft. A study of the 
profile of this path reveals that Kendrick is well within the radio horizon, 
Karval is near, but still witnin the radio horizon, and Haswell is beyond 
the radio horizon. 

The Cheyenne Mountain study found that throughout the period of 
observation, fadeouts (of 5 dB or more) occurred regularly at Karval on a 
frequency of 1046 Mc/s, in conjunction with superrefractive A'^-profile 
conditions, and coincided with enhanced field strengths at Haswell: A 
particular instance of this, the night of June 21-22, 1952, is shown in 
figures 4.53 and 4.54. Figure 4.53 shows the field strengths recorded 
simultaneously in a 12-hr period at Haswell, Karval, and Kendrick. Here 
it is to be noted that with the sudden intense rise in field strength at 
Haswell, a progressively deepening fadeout appears in the Karval data 
relative to the monthly median, while only insignificant changes are noted 
at Kendrick. Meanwhile, figure 4.54 indicates the shift in the refrac- 
tivity profile at Haswell (presumably the same for Kendrick and Karval) 
towards conditions of superrefraction throughout the evening of 21-22 
June, 1952. 

The great number of fadeouts at Karval as compared to Kendrick 
strongly suggested a dependence of fading on distance from the trans- 
mitter within the horizon. The following sections describe the conditions 
under which fading, variations in fading, and locations that favor the 
occurrence of fading within the radio horizon (in the presence of a ground- 
based duct) occur. 



FADING REGIONS WITHIN THE HORIZON 147 

Fading, or fadeout, of a radio wave may be defined as a drop in power 
or field strength below a specified level of intensity. For a given site, 
criteria may be set up by the communications engineer to establish the 
magnitude of drop that defines the onset of fading conditions. This 
magnitude usually ranges from about 5 to 30 dB below the specified level 
of intensity. The specified level is derived from some ideal propagation 
condition. 

Fading is also a time-dependent occurrence. It may be classified as 
(a) prolonged fading, which is fading of sufficient interval to cause con- 
tinued communication disruptions, and (b) short-term Rayleigh fading, 
which is only instantaneously observed by a receiver. 

One of the main causes of deep fading of field strength within the 
horizon as compared with the free space field strength value is the A^ 
structure of the atmosphere. Theoretically, the atmosphere can be con- 
sidered horizontally homogeneous and in spherical stratifications con- 
centric with the earth, and N can be considered to decrease exponentially 
with increasing height above the earth's surface. However, in reality, 
this picture of the atmosphere is rarely, if ever, realized because of the 
synoptic meteorological conditions that are perpetually present. Strati- 
fications caused by the synpotic meteorological pattern give rise to field 
strength fading within the horizon by defocusing the lobe pattern of the 
transmitter along a given path. Whenever the rate of change of refrac- 
tivity from the surface value with height (called the gradient of A^ with 
height) is less than —157 A^ units per kilometer, a "ducting" condition is 
said to exist at the surface, and, as shown in figure 4.35, certain rays will 
tend to be "trapped" or guided within the surface duct. It is this atmos- 
pheric condition of surface ducting which will be further explored herein 
with respect to fading within the radio horizon. 

It should be realized that not all within-the-horizon fading has been 
attributed to refractivity gradient discontinuities in the lower atmos- 
phere. Misme [36] shows the influence of frontal effects and frontal 
passage on signal fading within the horizon, the frontal passage even 
occurring at a time when one would most expect a fadeout caused by a 
surface duct. 

4.5.2. Regions and Extent of Fading Within the Horizon 
in the Presence of Superrefraction 

Serious disruptions in reception from a transmitter above a duct to 
a receiver within a duct can occur at particular points within the horizon. 
It is these disruptions, and these locations, which are of interest to the 
communications engineer establishing a given transmitting-receiving 
path. Whenever these disrui)tioiis occur between a transmitter and 
receiver within the horizon, a corresponding increase in the field strength 
characterized by steady, high, dei)endable signals, is usually expected 



148 N CLIMATOLOGY 

beyond the horizon. The high signal strength is in keeping with the 
properties of a surface duct. Nevertheless, deep, prolonged fadeouts 
can occur in regions beyond the horizon as well as within the horizon. 

Price [37] has theoretically determined regions of deep radio fading 
associated with surface ducts, which he has termed "shadow zones." In 
the case of a transmitter above a surface duct, a representation of what 
occurs is shown in figure 4.43. It shows the location of a shadow zone 
above the radio horizon line in the normal interference region. In the 
interference region, a fadeout of signal strength due to the presence of 
superrefraction must be compared with the value of the field when only 
the interference pattern is present. Papers such as those by Norton [38] 
and Kirby, Herbstreit, and Norton [39] give methods for calculation of 
the normal interference field. Ikcgami [40] gives a more general method 
for the calculation of received power in the presence of ground based ducts. 

Ikegami's procedure is based on a simple geometrical optics ray-tracing 
technique of determining the power relative to free space transmission 
that is received at different locations from the transmitter. A more 
refined, yet more complex, procedure is the field-strength calculations 
along the lines followed by Doherty [41], in which a strict mode-theory 
treatment of the problem reveals that geometrical optics is not sufficient 
in the presence of refraction anomalies. Doherty expands techniques 
originally considered by Airy [42] to determine relative field strengths in 
the neighborhood of caustics (apparent ray intersections) that result 
vicinal to refraction anomalies such as ducts. 

In the procedure that follows, a model for the determination of the 
location and extent of shadow zones, rather than the determinations of the 
actual received field strengths and powers at any particular point, will be 
given for conditions typical of the temperate climate to aid the radio cir- 
cuit engineer in avoiding these troublesome interference areas. 

Washington, D.C., is taken to represent an average temperate zone 
climate. Fairly extensive work has been done in determination of various 
ducting conditions at this station as well as with the average exponential 
refractivity above this station. Therefore, Washington, D.C., will be 
used as a model for all following calculations. 

The gradient of A^ with height determines whether or not a surface 
duct will exist. Therefore, a surface ducting atmosphere corresponding 
to conditions at Washington, D.C., consisting of a ducting gradient up to 
any desired height, and the N s = 313.0 exponential atmosphere [9] above 
this height, have been chosen for this study. The ducting gradients 
chosen are maximum, mean, and minimum values for February, May, 
August, and November (representing the middle of winter, spring, sum- 
mer, and fall, respectively) taken from figure 4.38 and given in table 4.8. 
Also in this particular model the duct is assumed to be of uniform height 
throughout the region considered in the calculations. 



THEORY AND RESULTS 



149 



4.5.3. Theory and Results 

In figure 4.41, although it is known that ray theory does not explain 
what happens to the radio ray at the point A when the ray is tangent to 
the top of the duct, because of the apparent ray intersection, it is sufficient 
to define the first shadow zone as that zone between the two possible paths 
of the ray tangent to the duct at a point A (this is the ray emitted from the 
transmitter with an initial angle of ^o with the horizontal). This is possi- 
ble because it can be readily seen that no radiation will be able to enter the 
shadow zone, and thus a receiver located in the shadow zone will theoreti- 
cally receive no signal from the transmitter. The portion of the "split" 
ray that enters the duct at A will be reflected at point B to point C, where 
it "splits" again. The distance, d from point A to C projected along the 
earth's surface will be designated the "length" of the first shadow zone. 
If the surface duct is homogeneous throughout, the ray will be symmetri- 
cal about point B and, therefore, 



d, = d. = d/2 = d, 



(4.26) 



where di and dz are the "half-lengths," dhi, of the shadow zone. 




Figure 4.41. Shadow zone occurrence of a curved earth with the transmitter above 

the duct. 



150 A^ CLIMATOLOGY 

Chapter 3 shows that the bending, ti,o, of the radio wave between any 
two points in the atmosphere is given by 



ri.2 = ^ + (01 - e,), (4.27) 



where di,2 is the distance between the two points 1 and 2 (see fig. 4.42) 
projected along the earth's surface, a is the radius of the earth, and 6i 
and 62 are the angles in radians the ray makes with the horizontal at points 
1 and 2 respectively. As described in chapter 3, Schulkin [43] obtained 
Ti,2 in radians as 

r,,, = ^^^ 7 ^^^ for < 00 < 10° (4.28) 



where Arii = rii — 1, A/io = no — 1, d^ = {Si + 02)/2. In terms of refrac- 
tivity, (4.28) becomes 



{N, - N2) X 10-^ AN X 10-^ ., ^Q. 

Tl.2 = a = a • (4.29) 



Considering now the case of the half-length, dht, (4.27) becomes 



ri.2 = ^ + e„ (4.30) 

since 02 = at the top of the duct, and di = 6p. Now substituting (4.29) 
in (4.30), 

^^q^i^ = ^-' + 0„ (4.31) 



since here dm = 0p/2. Rearranging (4.31) 



dht = a 
Since 



'2AN X 10~' 
_ dp (rad) 



+ dp (rad) . (4.32) 



dp = cos-^ ^^^ ^ J2[aN X 10-^ - ^'^ ~ ^h . (4.33) 
(rad) 



THEORY AND RESULTS 



151 



RADIO RAY 




Figure 4.42. Geometry of radio ray refraction. 



where ua is the index of refraction at the top of the duct, ta is the distance 
from the center of the earth to the top of the duct, and no is the index of 
refraction at the earth's surface, it is seen that dht is uniquely determined 
from the surface gradient of A'^ with height inside the duct, provided the 
height of the duct is known. 

If one desires to know the distance, dh, for a particular height, h, from 
the reflection point of the ray in the duct (point B in fig. 4.41), where dh 
is such that 



(4.34) 



152 



A^ CLIMATOLOGY 



R^ 1 ST SHADOW ZONE 

ZONE BEYOND THE HORIZON 



DIRECT AND REFLECTED RAY 
INTERFERENCE REGION 




^GROUND BASED DUCT 
FiGUEE 4.43. Regions of fading from different effects of the radio rays. 



then (4.37) becomes 

dh = a 



2AN' X 10" 



+ (e. 



I 



dp + 02 

since 02 is no longer just equal to zero, but is such that 



< 02 < 0p, 



(4.35) 



(4.36) 



and AN' will no longer be the same as the AA'^ for the duct width. The 
value of 02 may be determined from the formula: 



2 2h 

I 

a 



2AN' 



(4.37) 



In figures 4.44 to 4.46 the shadow zone half-lengths are given as a func- 
tion of duct height for the various seasons of the year. As shown in 
figure 4.39, the 50 percent level, or median of observed ground-based duct 
thickness at Washington, D.C., is about 100 m. Figure 4.40 shows that 
the percent level of observed maximum ground-based duct thickness is 
about 200 for Washington, D.C. In table 4.9, the values oidht during the 
four seasons are shown for the 100- and 200-m surface duct heights. 

As previously mentioned, the atmosphere above the assumed duct was 
taken to coincide with the N ^ = 313.0 exponential atmosphere [9] because 
this value most nearly corresponds to the average annual N s value at 



THEORY AND RESULTS 



153 





1,000 




700 




500 


E 


300 


-^ 


200 


z 




X3 


100 


U 


70 


u 


50 


^ 




^ 


30 


c/^ 


20 


u 






10 




7 




5, 



20 



—- t-+-++i~"rit4:±±:=ti4=^ ^±::^-~-~~-^—^^z~-dpz-:±—'^'^ 


! ■ ■ ■ ; i ' i ■ ■ .; 


: 1 : ; 1 1 ■" ! ■ : • ■ ■ : 


J . ■ ; . i ' : vj i : i ■..!■';'': 1 






' ' ■ ' ■ '■ ■ ' 1 1, 


: 1 , : 1 ' 1 I ; I j ' . ; 1 ^ 


i ' j : i i i i : 1 i ! i ; ; 


r -i - i MiP^'^' <no\embe|| j I ,_. 


?===S====5:f-^--=-fi^-^^J.S=^— ^^^^^UA^f^ 




1 .i- -ds: cs B3 — '^ "n 1 . n^^inziL-'d- -t— ■--^-/riA'Ci^ ■ 


11"^^^*-^'= i" ■ i^gSPmNG^AY^^-:^^ 


■■-iiM*TT r _^^4a4-l J^M'''n=='=*^-^-L^fhcp(a GU'^'' )1 "i 


.^^ \ ...,***=*«== ='*T'^ ' ' rt^SUMMERJAUbu^^^^ 


'/-:■: \. .-*Ea = *'°°'* 1 J : : i | i ' i~"r^~n ; ! 1 M M ! 


" l"^-i==='^ [ ■ ■ ■ i .';■'■ !. i ' i 1 .; . 1 


Jy^\- : ' ^ i ^ : , M i ^ ^ 1 ^ !■ 1 MM" ! 1: 


"--+--] — [ — i — i — r~~ — "^^ — i 1 ' — r--4---~ 


1 11 . : j 


-.<ik^. -\ ."--"^"^^^--11! 1 ■ ■ 



40 



60 80 100 120 140 

DUCT HEIGHT ,h, IN m 



160 



180 200 



Figure 4.44. Shadow zone half-lengths for maximum ducting conditions at 

Washington, D.C. 



o 



1,000 
700 
500 

300 
200 

100 
70 
50 

30 
20 

10 
7 



z^im4Hi^^fW--t-+-^^r^=i:=1==^+^--tt::litnt±tb 


i- 










! . ^ : . ■ ■ . ■ . 1 [ . j 1 ! , ^ i ; 1 1 1 ; 




' j i ■ ■ : ' i i ! ' 




' ■ ■ 1 i i i i ' 1 ■ 




4 i . . . ., ....__4___4---^--Y----|^ ^44--' ■ ,,■■;,,; ; 


; ■ ; 1 


i ' ' ■ ' i 1 ■ 


JvEMBERi: 


_ 


1 ■ ■ ' . 1 ! . 1 1 ,_..... 


"'"Tl "■■ 1 , , ■ ,■ : : 1 : i : J, i^xl^ 


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'^ V^f-'C' J_ -o'»' = *-'^**'''''^ ■ • 1 • 1 ! ' ■- 




■ T^ 


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J, , i . . 


:iEiiiii--|-ii:=iiiiiqEi==iiiiiEEi^|iiiii;. 



20 40 60 80 100 120 140 

DUCT HEIGHT . h , IN m 



160 



180 200 



Figure 4.45. Shadow zone half-lengths for mean ducting conditions at 
Washington, D.C. 



154 



N CLIMATOLOGY 






Q 




60 80 100 120 140 

DUCT HEIGHT , h, IN m 



180 200 



Figure 4.46. Shadow zone half-lengths for minimum ducting conditions at 
Washington, D.C. 



Table 4.9. Half-lengths at Washington, D.C. for various duct heights and surface 

gradient conditions 





Cond. 


Duct height 




100 m 


200 m 


February 


Max 

Mean 

Min _. 


68.21 
84.50 
105. 35 


96.46 
119. 50 
148. 98 


May- 


Max 

Mean 

Min 


30.53 
52.37 
199. 42 


43.17 

74.05 

281. 95 


August 


Max 

Mean 

Min 


29.02 
49.12 
440. 73 


41.04 

69.46 

622. 53 


November 


Max 

Mean 

Min 


72.56 
123. 93 
313.61 


102. 60 
175. 24 
443. 23 



Washington, D.C. Figures 4.47 and 4.48 show the distance measured 
along the earth's surface that a ray with an initial elevation angle of zero 
will travel in the A^^^ = 313.0 exponential atmosphere. The value of 
zero for an initial elevation angle is used because the ray can be thought 
of as leaving the position A, tangent to the duct in figure 4.41 and 
"arriving" at the transmitter, even though the reverse process is actually 
what is occurring. Thus, from figures 4.47 and 4.48, knowing the height 
of the transmitter, ht, and the height of the ground-based duct, Ka, the 



THEORY AND RESULTS 



155 



400 




1,000 



2,000 3,000 4,000 5,000 

HEIGHT ,h , IN m 



6,000 7,000 



Figure 4.47. Distance versus height above duct representing average Washington, 
D.C., and Fairbanks, Alaska, atmosphere. 



distance, do, that the radio ray travels from the transmitter to the top of 
the duct is given by 



d(ht - 



hA): 



(4.38) 



i.e., the distance obtained by using the difference of ht and Ha as the value 
of height on the abscissa in figures 4.47 and 4.48 gives the distance do. 
The distance, dho, to the standard radio horizon from the transmitter is 



dho = \^2ht 



(4.39) 



if ht, the transmitter height, is in feet and rf^o is in miles. Using this fact 
and figures 4.47 and 4.48, table 4.10, which shows the ratio of do to dho, 

Table 4.10. Ratio of onset point of fading zones from transmitter to distance of radio 
horizon from the transmitter, do/dho, for various transmitter heights at Washington, D.C. 



Height of 




Duct height, meters 




transmitter 


10 


20 


50 


100 


200 


200 

300 

500 

800 

1000 

1500 

2000 

3000 - 

5000 

7000 


0.795 
.837 
.878 
.907 
.918 
.935 
.944 
.953 
.959 
.959 


0.699 
.759 
.818 
.860 
.876 
.900 
.914 
.928 
.940 
.943 


0.512 
.605 
699 
.765 
.792 
.831 
.854 
.880 
.902 
.911 


0.300 
.432 
.565 
.659 
.697 
.754 
.787 
.825 
.859 
.875 



0.187 
.375 
.509 
.563 
.644 
.692 
.747 
.799 
.825 



156 



A^ CLIMATOLOGY 



100 



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40 80 120 160 200 240 280 

HEIGHT, h, IN METERS 

Figure 4.48. Distance versus height above dud representing average Washington, D.C., 
and Fairbanks, Alaska, atmosphere. 



is derived for various transmitter heights and various duct heights. This 
table is useful in determining the portion of the distance (measured from 
the transmitter) within the horizon where fading due to the presence of 
superrefraction has no effect on transmission. 



SAMPLE COMPUTATIONS 157 

4.5.4. Sample Computations 

It can be seen from figure 4.52 that 

hi < loss region < Ha + ho, (4.40) 

where hi can be determined from figures like 4.49 to 4.51, which depend 
upon the station location and the season of the year; hA is the height of 
the duct; and /i2 is determined from figures 4.47 and 4.48. The following 
example will illustrate the determination of the shadow zone. 

As an illustrative problem, assume a 500-m transmitting antenna in the 
spring of the year at 40°N latitude, and that one desires to place the re- 
ceiving antenna at (a) 40 km and (b) 90 km from the transmitter. How 
high should the receiving antenna be to be free from fading loss caused 
by a ground-based superrefractive layer? 

Since both of the distances required are within the standard horizon 
distance, dho, which, for a 500-m transmitter is 

dho = 92.181 km, 

this is a case of within-the-horizon propagation. 

Since the 40°N latitude position is within the temperate zone, use will 
be made of the Washington, D.C., model ducting atmosphere. As pre- 
viously mentioned, 100 m is the median mean duct thickness at Washing- 
ton, D.C., and 200 m is the median maximum duct thickness observed at 
Washington. These are the two duct thicknesses to use in this calcula- 
tion. Also, since it is the spring of the year, figures 4.49 to 4.51 must be 
used to determine hi. 

For the 40-km distance, case (a), the subscript of (4.38) becomes 

ht - hA = 500 - 100 = 400 m, 
which is then entered on the abscissa of figure 4.47, where, 

do = r/400 = 83 km. 
For a 200-m-thick duct, 



300 m. 



158 



N CLIMATOLOGY 



1,000 

700 
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'0 20 40 60 80 100 120 140 160 ISO 200 

HEIGHT ,h , IN m 

Figure 4.49. Height versus distance at Washington, D.C., for maximum midspring 

(May) ducting conditions. 




20 



40 



60 80 100 120 140 

HEIGHT , h , IN m 



160 



180 200 



Figure 4.50. Height versus distance at Washington, D.C., for mean midspring 
(May) ducting conditions. 



SAMPLE COMPUTATIONS 



159 



1,000 
700 
500 

300 
200 

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^ 100 



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5 



20 



40 



60 80 100 120 140 

HEIGHT ,h , IN m 



160 



180 200 



Figure 4.51. Height versus distance at Washington, D.C., for minimum midspring 

{May) dxicting conditions. 



Therefore, in this case, 



do = dzQQ = 72 km. 



In either case, as can be seen from figure 4.41, do is greater than 40 km. 
Therefore, any height of the receiving antenna will be satisfactory at this 
point, since the first shadow zone has not yet been reached. 

For the 90-km distance, case (b), it can be seen that this distance is 
beyond the onset point of the first shadow zone, and figures 4.49 to 4.51 
must be used here. 

Figures 4.49 to 4.51 are used differently depending upon which side of 
the reflection point B of figure 4.52 one desires to locate an antenna. If 
on the lefthand side of point B, one must subtract the desired distance 
from the distance from B to the top of the duct traversed by the ray and 
use this new distance to determine height, h, in figures 4.49 to 4.51. This 
procedure must be followed every time the ray heads downward. If on 
the righthand side of point B of figure 4.52 (or any other reflection point), 
one simply uses the distance from the reflection point directly. One must 
keep in mind that any previous half-lengths must be subtracted from the 
desired distance from the transmitter, as well as subtracting do, in order 
to obtain the distance used in figures 4.49 to 4.51. 



160 



N CLIMATOLOGY 




'0 I ^h£ n 

Figure 4.52. Geometry of the shadow zone loss region. 



For the maximum midspring ducting conditions (fig. 4.49) and the 
100-m-thick duct, recalling do = 83 km, the antenna is to be located 



90 



83 = 7 km 



beyond the onset point of the shadow zones. For the 200-m duct, the 
antenna is located 

90 - 72 = 18 km 

beyond point A of figure 4.52. From figure 4.44, the half-length traveled 
by a ray in a 100-m duct under maximum midspring ducting conditions 
is 30.5 km. Thus, as described above, the distance used in figure 4.49 
to find h is 

30.5 - 7.0 = 23.5 km 



because at 25 km the ray is still sloping dov/nward and hence is on the 
lefthand side of point B of figure 4.52. From figure 4.44, dht = 43 km 
for a 200-m duct; therefore, the distance used in figure 4.49 is 



43 



18 = 25 km. 



Similarly, utilizing figures 4.45 and 4.46 for mean ducting and minimum 
ducting conditions for a 100-m and 200-m duct: 

Figure 4.50 distance to be used for the 100-m duct = 45 km, 
Figure 4.50 distance to be used for the 200-m duct = 56 km, 
Figure 4.51 distance to be used for the 100-m duct ^ 191 km. 
Figure 4.51 distance to be used for the 200-m duct = 262 km. 



SAMPLE COMPUTATIONS 



161 






Figure 4.53. 1 ,046-Mc/s fields for the night of June 21-22, 1952, at Kendrick, Karval, 

and Haswell, Colo. 

and hence 

hi for a 100-m duct under maximum spring ducting conditions = 92.3 m, 
hi for a 200-m duct under maximum spring ducting conditions = 166.0 m, 
hi for a 100-m duct under mean spring ducting conditions = 96.0 m, 
hi for a 200-m duct under mean spring ducting conditions = 187.0 m, 
hi for a 100-m duct under minimum spring ducting conditions = 99.0 m, 

and 

hi for a 200-m duct under minimum spring ducting conditions = 199.0 m. 
Figures 4.47 and 4.48 will be used to determine /i2 from the total 
distance, d. The distance to be used is obtained by subtracting the dis- 
tance, do, from the desired distance. For the 100-m duct, do = 83 km and 

d = 90 - 83 = 7 km. 

Likewise for the 200-m duct, do = 72 km and 

rf = 90 - 72 = 18 km. 

Thus it is found that for the model assumed: 

(a) /i2 for the 100-m duct height case will always be 3 m, 

(b) /i2 for the 200-m duct height case will always be 18 m. 
Since from (4.40) 



hi < loss region < /i^ + /12. 



162 



N CLIMATOLOGY 




300 



280 



260 



240 



220 



200 



180 



160 



< 140 



120 



100 



tTS tao tat IM tn >oo >ae 



ITS tlO tM »0 



Refractive Index in Units of N = (n-l)xlO® 
Figure 4.54. Vertical distributions of refradivity at Haswell, Colo., for June 21-22, 1962 



Therefore, in answer to part (b) of the problem, for maximum ducting 
conditions with a 100-m-thick duct: 

92.3 m < loss region < 3 + 100 m, or 

92.3 m < loss region < 103 m. 



INTRODUCTION 163 

With a 200-m-thick duct : 

166 m < loss region < 18 + 200 m, or 
166 m < loss region < 218 m. 
For mean ducting conditions with a 100-m-thick duct: 

96 m < loss region < 103 m. 
With a 200-m-thick duct: 

187 m < loss region < 218 m. 
For minimum ducting conditions with a 100-m-thick duct: 

99 m < loss region < 103 m, 
with a 200-m-thick duct : 

199.3 m < loss region < 218 m. 

4.6. Air Mass Refractive Properties 

4.6.1. Introduction 

The foregoing material on refractive index climatology concerned the 
geographical variation of mean A'' (or A^'o) over the surface of the earth 
and the three-dimensional distribution of mean A'^ over North America. 
This section develops another aspect of the climatological variation of the 
refractive index: the mean profile of A^ in various air masses. The 
meteorologist defines air masses as bodies of air in the troposphere having 
approximately homogeneous character at the surface. An air mass is de- 
scribed in terms of its origin, as, for example, polar continental or tropical 
maritime. 

Profiles of A'^ and values of bending of radio rays are given in terms of 
departures from normal for various air masses. 

Recent studies have led to the evaluation of radar elevation angle 
errors in different climates and air masses [43, 44]. In general, these 
studies did not emphasize the relation between air mass refractive index 
structure and the refraction of radio waves within the air mass. For 
example, mean angular bendings for radio rays passing completely through 



164 



N CLIMATOLOGY 




20 40 60 80 100 120 140 160 180 200 
ANGLE OF ARRIVAL OR DEPARTURE, ^o.lmrad) 

Figure 4.55. Comparison of total angular bending of radio rays within air masses. 

After Schulkin [1952]. 



the earth's atmosphere are given for extremes of air mass types and a 
range of initial elevation angles, do, on figure 4.55 [43]. It is seen that 
maritime tropical air produces 30 percent more bending than continental 
polar air at initial angles or arrival or departures of about 10° (175 mrad) 
and that this difference increases to about 70 percent at zero initial 
elevation angle. Further, the magnitude of the total bending increases 
rapidly with decreasing values of ^o- 

Figure 4.55 does not make it clear that differences in radio ray refrac- 
tion arise from differences in the refractive index profiles of the two air 
masses. It is the purpose of this section to show that various air masses 
have characteristic refractive index profiles and that the radio ray refrac- 
tion within each air mass is mirrored by the difference of the actual re- 
fractive index structure from a standard atmosphere of exponential form. 



4.6.2. Refraction of Radio Rays 

The angular bending of a radio ray, ti,2 (see fig. 4.56) between two 
points in the atmosphere of refractive indices rii and n2 is given by the 



REFRACTION OF RADIO RAYS 



165 



RADIO RAY 




Figure 4.56. Geometry of radio ray refraction. 



geometrical optics formula [45, 43], 



Ti,2 = — f cot 9 dn 



(4.41) 



under the assumption that dn/n = dn and 6 is the local elevation angle. 
It is also assumed that n is spherically stratified and concentric with the 
earth. The value of cot 6 is determined from Snell's law, and ti,2 is com- 
pletely determined from a knowledge of the variation of n with height. 
An empirical formula to describe the long-term average variation of n 



166 A^ CLIMATOLOGY 

with height is 

WW) = RA) - 1] 10« = 313 exp {-V6.95) (4.42) 

where h is the height above the earth's surface in kilometers and 313 is the 
long-term average value of (n — 1) 10^ at the earth's surface for the United 
States. In practice it is convenient to utilize this average N{h) function 
to refer the observed N{h) data to the common level oi h = by means of 

A{h, 313) = N{h) + 313 [1 - exp {-h/Q.95}]. (4.43) 

Chapter 1 shows that A{h,31S) is analogous in concept to potential tem- 
perature but utilizes the normal A'^ gradient rather than the adiabatic 
gradient of the potential refractive modulus of Lukes [46] and Katz [20]. 
The notation A{h, 313) is used to indicate that A is determined from the 
refractive index at the height h and the 313 exponential atmosphere. It 
has been found to be advantageous to use several atmospheres of exponen- 
tial form in applications involving different climatic regions [47]. The 
particular form of (4.43) emphasizes departures of A^ structure from 
normal as shown by recent studies of synoptic variations of N(h) about 
frontal systems [21] and, in addition, permits direct calculation of radio 
ray bending for any observed refractive index structure. When A{h 313) 
is introduced into (4.41) for ray bending, one obtains (see chapter 3, 
sec. 3.10). 

r,, = - L ' IQ "^^^^ dAih, 313) + Tih, 313) (4.44) 

JA^ n 

which indicates that the bending can be regarded as the resultant of the 
bending in the normal atmosphere, t(/i, 313), and a perturbation compo- 
nent representing departures of refractive index structure from that of the 
313 exponential atmosphere. The value of r(/i, 313) can be obtained from 
refraction tables [9], and the perturbation term can be evaluated by 
graphical methods to yield an overall accuracy of a few percent in esti- 
mating ri,2 [47]. 

Some A(h, 313) profiles were prepared from typical temperature, pres- 
sure, and humidity distributions within a range of air masses as published 
in the literature [48, 49, 50] and climatic summaries of upper air data [31]. 
Two of these profiles, one for maritime tropical air and the other repre- 
senting continental tropical air, are plotted on figures 4.57 and 4.58 to 
represent the extremes of A{h, 313) profiles. These profiles clearly show 
that the two air masses have quite different refractive index structures. 
The difference is most pronounced near the ground. At heights of 20 
km, however, A{h,313) rapidly approaches the asymptotic value of 313, 



REFRACTION OF RADIO RAYS 



167 



40 

36 

32 

28 

E 24 

- 20 

D 

^ 16 



12 
81- 



4 - 



A(h,3l3) 




SAN JUAN, W.I. 
(July Average) 



'A(h,3l3) 
= 3/3 



''313-^ 

Bn=0 




310 320 330 340 350 360 370 380 390 
A(h, 313), N UNITS 

I \ \ L \ \ \ I 



-8 



■6 -5 -4 -3 -2 -I 

T313-T (mrad) 







Figure 4.57. Comparison of A(h,313) profiles with departures from normal hendirig 

of radio waves for maritime tropical air. 

After Byers [1944]. 



168 



N CLIMATOLOGY 



40 



36 



32 



28 



EL PASO, TEXAS 
I (June Average) 



-A (h, 3/3 J 







y 



/ 



':^ 



310 320 330 340 350 360 370 380 
A(h,3l3),N UNITS 



-2 



12 3 4 

T3|3-T(mrad) 



Figure 4.68. Comparison of A(h,313) profiles with departures from normal bending 

of radio waves for continental tropical air. 

After Ratner (1945). 



REFRACTION OF RADIO RAYS 



169 



regardless of air mass type. The radio ray bending is also plotted on 
these figures relative to the value expected in the Nih) = 313 exp 
{ — /i/6.95} atmosphere. Near the ground, bending departures are seen 
to be mirrored by the A unit profiles. That is, a negative gradient of 
A'^ or yl produces a positive increase of bending. Above a few kilometers, 
however, the bending departures approach a fixed, usually nonzero, value. 
It is apparent that the asymptotic value of T313 — r at large heights is 
determined by the bending in the first few kilometers, where 60 to 75 
percent of the low-angle bending normally occurs. The marked differ- 
ences in air mass n structure and bending are confined, therefore, to the 
first few kilometers above the earth, as is illustrated by figure 3.16. The 
distribution of the meteorological elements within each air mass is re- 
flected in the A{h, 313) profiles. For example, the steep humidity 
gradient characteristic of maritime tropical air is reflected by the rapid 
decrease of A{h, 313) with height. Comparatively, the high ground-level 
temperatures and rapid decrease of temperature with height in continental 
tropical air are reflected by the increase of A{h, 313) with height wdthin 
that air mass. These, and .4-profile characteristics of other air masses, 
are listed in table 4.11 and are evident from the form of the equation for 
the refractive index at radio frecjuencies [see chapter 1, (1.20)]. 



Table 4.11. Refractive characteristics of air 7nasses. 



Ref- 
erence 



Superior (S) S/mT, [48] 
typical of Oulf coast 
(Lake Charles, La.). 



Meteorological characteristics 



Formed from subsidence of high- 
level air with resulting dry adiabatic 
temperature lapse rate and low hu- 
midity. Often found overlying other, 
more humid air masses that show a re- 
sultant characteristic drying with 
height in the lower levels. 



Refractive characteristics 



Rapid decrease of N and A 
with height in the lowermost 
layer produces a superrefrac- 
tive* layer. The overlying 
superior air is nearly normal in 
bending characteristics. 



Continental Polar (cP) 
source region. 



Characterized by low temperature 
and humidity with a pronounced tem- 
perature inversion at the surface 
created by progressive nocturnal cool- 
ing during the arctic night. 



Ground-based superrefrac- 
tive layer arising from the tem- 
perature inversion. 



Maritome Polar (mP), [48] 
typical low-level 
ground modification 
(Seattle, Wash.). 



Cool air, nearly saturated to a height 
of several kilometers. Example shows 
typical drying in lower levels when 
this air mass moves over land. Over- 
lying mP has increasing humidity and 
decreasing temperature with height. 



Continental Tropical 
(cT) source region 
(El Paso, Tex.). 



[51] 



Characterized by superheated lower 
layers, rapid decrease of temperature 
with height, and very low humidity. 



Surface layer produces near- 
normal bending. Increasing 
N in the overlying mP air pro- 
duces an elevated subrefrac- 
tive layer. 



Strong temperature lapse 
produces a subrefractive layer 
reaching to several kilometers. 



Maritime Tropical 
(mT) source region 
(Pensacola, Fla.). 



[48] 



Relatively warm air, high water- 
vapor content in lowest layer, which 
decreases rapidly with height. Mod- 
erate changes in temperature and hu- 
midity structure produce large re- 
fractive gradient changes. 



Strong superrefraction to 
great heights arising chiefly 
from rapid decrease of hu- 
midity with height. 



'Normal refraction is taken to mean t(/i, 313), superrefraction to mean j-(ft)>7(h, 313), and subrefraction 
t(/i)< r(ft, 313). 



170 N CLIMATOLOGY 

The average decrease of pressure, temperature, and humidity with 
height produces a normal decrease of A'^. If, however, the temperature 
increases with height, as in a temperature inversion, A^^ decreases more 
rapidly and radio rays are superrefracted, dependent, of course, upon the 
vapor pressure. Conversely, an unusually rapid decrease of temperature, 
or an mcrease of humidity, with height produces a subnormal decrease of 
A^ with height, or subrefraction of radio waves. In any event it is evident 
from these figures that x4-profile effects on ray bending are most pro- 
nounced at do = 0, are significantly less pronounced at do = 52 mrad 
('~3°), and continue to diminish with increasing do until, at ^o = 7r/2, 
there is no bending at all and consequently all departures are zero. 

4.6.3. Conclusions 

The work of Schulkin and others has shown that characteristic total 
bending differences in radio ray refraction exist between various air 
masses. The present study extends Schulkin's conclusion by identifying 
abnormal bending of radio rays with departures of refractive index struc- 
ture from average in the lowermost layers of the air masses. Considera- 
tion of departures of both ray bending and refractive index structure from 
their value in a standard exponential atmosphere results in a suitable 
method of cataloging air masses in terms of either refractive index struc- 
ture or bending characteristics. 

4.7. References 

Resolution No. 71 (1948), Conf. of Directors, Internal. Meteorol. Organization 

(Lausanne, Switzerland). 
World Atlas of Sea Surface Temperature Charts (1944), Hydrographic Office 

Publ. 225, chap. 4, No. 2. 
Atlas of climatic charts of the oceans (1938), U.S. Weather Bureau Publ. 1247, 

Washington, D.C. 
Johnson, W. E. (Nov.-Dec. 1953), An analogue computer for the solution of the 

radio refractive index equation, J. Res. NBS 51, No. 6, 335-342. 
Gerson, N. C. (1948), Variations in the index of refraction of the atmosphere, 

Geofis. Pura Appl. 13, 3-4. 
Bean, B. R. (Apr. 1953), The geographical and height distribution of the gradient 

of refractive index, Proc. IRE 41, No. 4, 549-550. 
Misme, P. (Nov.-Dec. 1958), Essai de radioclimatologie d'altitude dans le nord 

de la France, Anna. Telecommun. 13, No. 11-12, 303-310. 
Smithsonian Meteorological Tables (1951), Table 63, Sixth Revised Ed., Book 

(Washington, D.C). 
Bean, B. R., and G. D. Thayer (1959), CRPL Exponential Reference Atmosphere, 

NBS Mono. 4. 
Pickard, G. W., and H. T. Stetson (1959), Comparison of tropospheric reception, 

J. Atmos. Terrest. Phys. 1, 32-36. 
Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception 

at 44.1 Mc with 92.1 Mc over the 167-mile path of Alpine, New Jersey to 

Needham, Mass., Proc. IRE 38, No. 12, 1450. 



REFERENCES 171 

Bean, B. R. (1956), Some meteorological effects on scattered radio waves, IRE 

Trans. Commun. Syst. CS4, 32-38. 
Onoe, M., M. Hirai, and S. Niwa (Apr. 1958), Results of experiment of long- 
distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79. 
Gray, R. E. (Jan., Feb., Mar. 1959), The refractive index of the atmosphere as a 

factor in tropospheric propagation far beyond the horizon, IRE Natl. Conv. 

Record, Pt. 1, 3-11 (1957), Elec. Commun. 36, No. 1, 60. 
Norton, K. A. (Mar. 1956), Point-to-point radio relaying via the scatter mode of 

tropospheric propagation, IRE Trans. Commun. Syst. CS-4, 39-49. 
Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave 

propagation, Proc. IRE 21, 427-463. 
Anderson, L. J. (1958), Tropospheric bending of radio waves. Trans. Am. Geo- 

phys. Union 39, 208-212. 
Snedecor, G. W. (1946), Statistical Methods, Book, 4th Ed., pp. 97-98 (Iowa 

State College Press, Ames, Iowa). 
Bean, B. R., and L. P. Riggs (July-Aug. 1959), Synoptic variation of the radio 

refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97. 
Katz, I. (1951), Gradient of refractive modulus in homogeneous air, potential 

modulus, Propagation of Short Radio Waves by D. E. Kerr, Book, pp. 198-199 

(McGraw-Hill Book Co., Inc., New York, N.Y.). 
Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the 

vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio 

Prop.), No. 2, 249-258. 
Jehn, K. H. (June 1960), The use of potential refractive index in synoptic-scale 

radio meteorology, J. Meteorol. 17, 264. 
Bean, B. R., and J. D. Horn (Nov.-Dec. 1959), The radio refractive index near 

the ground, J. Res. NBS 63D (Radio Prop.), No. 3, 259-273. 
duCastel and P. Misme (Nov. 1957), Elements de radio cUmatologie, L'Onde 

Electrique 37, 1049-1052. 
Hay, D. R. (Dec. 1958), Air mass refractivity in central Canada, J. Phys. 36, 

1678-1683. 
Tao, K., and K. Hirao (Mar. 1960), Vertical distribution of radio refractive index 

in the medium height of atmosphere, J. Radio Res. Labs. 7, No. 30, 85-93. 
Fehlhaber, L., and J. F. Grosskopf (1959), Beitrage zur Radioklimatologie 

Westdeutschlands, technische Bericht 5546, FTZ. 
Bean, B. R., and R. M. Gallet (Oct. 1959), Applications of the molecular re- 
fractivity in radio meteorology, J. Geophys. Res. 64, No. 10, 1439-1444. 
Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic charts and data 

of the radio refractive index for the United States and the world, NBS Mono. 22. 
Report 147 (1959), IX Plenary Assembly, International Radio Consultative 

Committee, pp. 299-337 (Los Angeles, Calif.). 
Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric 

refraction and its relation to wave guides and diffraction. Book, Meteorological 

Factors in Radio Wave Propagation, pp. 80-127 (The Physical Society and 

Roy. Meteorol. Soc, London, England). 
Bremmer, H. (1949), Terrestrial Radio Waves, Book, pp. 131-138 (Elsevier 

Publ. Co., New York, N.Y.). 
Cowan, L. W. (1953), A radio climatological survey of the U.S., Proc. Conf. Radio 

Meteorol. (Univ. of Texas, Austin, Tex.). 
Kerr, D. E. (1951), Propagation of Short Radio Waves, Book, pp. 9-22 (McGraw- 
Hill Book Co., Inc., New York, N.Y.). 
Bean, B. R. (May 1954), Prolonged space wave fadeouts at 1046 Mc observed in 

Cheyenne Mountain propagation program, Proc. IRE 42, 848-853. 



172 N CLIMATOLOGY 

[36 



[37 
[38 
[39 

[40 
[41 

[42 
[43 
[44 
[45 
[46 

[47 

[48 
[49 
[50 
[51 



Misme, P. (Apr. 1956), Methode de mesure thermodynamique de I'indice de 

refraction de I'air-description de la radiosonde MDI, Ann. Telecommun. 11, 

No. 4, 81-84. 
Price, W. L. (July 1948), Radio shadow effects produced in the atmosphere by 

inversions, Proc. Phys. Soc. 61, 59-73. 
Norton, K. A. (Dec. 1941), The calculation of ground-wave field intensity over a 

finitely conducting spherical earth, Proc. IRE 29, 623-639. 
Kirby, R. S., J. W. Herbstreit, and K. A. Norton (May 1952), Service range for 

air-to-ground and air-to-air communications at frequencies above 50 Mc, 

Proc. IRE 40, No. 5, 525-536. 
Ikegami, F. (Aug. 1959), Influence of an atmospheric duct on microwave fading, 

IRE Trans. Ant. Prop. AP-7, 252-257. 
Doherty, L. K. (1952), Geometrical optics and the field at a caustic with applica- 
tions to radio wave propagation between aircraft, Res. Rept. EE138 (Cornell 

Univ., Ithaca, N.Y.). 
Airy, G. B. (1938), On the intensity of light in the neighbourhood of a caustic, 

Cambridge Phil. Trans. 6, 379-402. 
Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere, 

Proc. IRE 40, No. 5, 554-561. 
Fannin, B. M., and K. H. Jehn (May 1957), A studj' of radar elevation angle 

error due to atmospheric refraction, IRE Trans. Ant. Prop. AP-5, 71-77. 
Smart, W. M. (1931), Spherical Astronomy, ch. 3, Book (Cambridge Univ. Press, 

London, England). 
Lukes, G. D. (1944), Radio meteorological forecasting by means of the thermo- 
dynamics of the modified refractive index. Third Conf. on Prop., NRDC, 

pp. 107-113 (Committee on Propagation, Washington, D.C.). 
Bean, B. R., and E. J. Button (May- June 1960), On the calculation of departures 

of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3, 

259-263. 
Byers, H. R. (1944), General Meteorology, Book, pp. 255-277 (McGraw-Hill 

Book Co., Inc., New York, N.Y^). 
Willett, H. C. (1944), Descriptive Meteorology, Book, pp. 190-220 (Academic 

Press, Inc., New York, N.Y.). 
Trewartha, G. T. (1943), An Introduction to Weather and Climate, Book, pp. 

206-215 (McGraw-Hill Book Co., Inc., New York, N.Y.). 
Ratner, B. (1945), Upper air average values of temperature, pressure, and relative 

humidity over the United States and Alaska (U.S. Weather Bureau). 



Chapter 5. Synoptic Radio Meteorology 

5.1. Introduction 

This chapter treats the variation of refractive index structure in the 
troposphere with synoptic tropospheric disturbances. Within the scope 
of the synoptic field are timewise and spacewise variations in the atmos- 
phere from microscale fluctuations to broad-scale systems of weather map 
dimensions. 

The microscale fluctuations of the refractive index are those that one 
would expect to observe at a particular point along a radio path that 
reflect detailed terrain and weather conditions in the immediate vicinity 
of, say, the transmitter or receiver site. 

Mesoscale variations, by way of contrast, are those which cover tens 
of kilometers and thus encompass a substantial portion of a radio path. 
Examples of this type of variation are land-sea breeze effects and con- 
vection cells of thunderstorm activity. 

Large-scale weather systems, affecting vast areas, perhaps even on a 
continental scale, fall under the classification of macroscale variation. 
Examples of this type of activity in the atmosphere are sweeping air mass 
changes and frontal systems traversing thousands of kilometers on the 
earth's surface. A detailed analysis of such a system is given as an illus- 
tration later in this chapter. 

5.2. Background 

The problem of determining the vertical and horizontal distribution of 
the radio refractive index has engaged the attention of radio meteorologists 
on an international scale for the better part of two decades [1, 2, 3, 4, 5, 

By analysis of current synoptic conditions from standard weather 
charts, one may ascertain the air mass type appearing over a given region 
and, likewise, may predict with reasonable accuracy the air mass type 
expected over a particular locale in, say, 24 hours. Then, from the air 
mass profile characteristics table of section 4.4, one is able to estimate 
what the departures of refractive index (and radio ray bending) from 
normal will be over a certain region. The bending predictions permit an 



Figures in brackets indicate the literature references on p. 224. 

173 



174 SYNOPTIC RADIO METEOROLOGY 

estimate of refraction errors and the introduction of appropriate correc- 
tions for radio range and elevation angle errors for radio navigational 
equipment. 

VHF-UHF radio field strengths beyond the normal radio horizon will 
also differ from air mass to air mass. It has been known for many years 
that the seasonal cycle of VHF radio field strengths received far beyond 
the normal radio horizon were correlated with the refractive index [7, 8, 
9, 10] and that significant changes in field strength level are observed from 
air mass to air mass [11, 12, 13.] Speaking about signal level on a 60 
Mc/s beyond the horizon radio path near Boston, Mass., Hull [11] states 
that during the winter low signal levels prevail during the presence over 
the path of fresh polar air. Periods of high signal occur when a cold, dry 
polar air mass is overrun by warm, moist air of tropical maritime origin. 
Hull's analysis represents early recognition of refraction and reflection 
phenomena on a synoptic scale. Later work on seasonal changes of fields 
and A'^ represents, in a way, a summary of synoptic conditions over a 
period of time. 

Gerson [2] was one of the first to consider the variation of the radio 
refractive index, n, in terms of seasonal and air mass changes. Gerson 
divided n into two parts, one density-sensitive and the other moisture- 
sensitive. This division is equivalent to the wet- and dry-term separation 
of A'^ = (n — 1) 10^ in chapter 4, section 1. Gerson was able to measure 
seasonal thermal changes by variation in the dry term and seasonal 
moisture changes by observed variation in the wet term. Gerson pre- 
pared graphs showing a sinusoidal variation of the dry term with a warm 
season trough and a cool season crest, indicating density changes in inverse 
proportion to the temperature. The wet-term component of n, on the 
other hand, was observed to attain its maximum during the warm season 
when the dry term was at its minimum. In arctic and antarctic locales, 
the surface variation of the wet term was found to be quite small while in 
temperate and tropical climates there was a sizable annual variation of 
the moisture component. As a pertinent aside, Yerg [14] showed that 
even during the long, cold, arctic night, vertical variations in moisture 
made significant contributions to the A'' profile. Apparent ducting 
gradients, obtained by neglecting the wet term at low temperatures may, 
in actuality, be only slightly more refractive than standard. 

Continuing his investigation, Gerson next turned to the analysis of 
refractive index changes within various air masses. Using air mass data 
available in the meteorological literature of the day, he charted mean 
refractive index profiles for different air mass types. The largest initial 
values and also the largest vertical gradients of n occurred with tropical 
maritime air. The air mass with the weakest gradients and, therefore, 
the poorest refraction properties was found to be the polar continental 
type. Common to all of Gerson's air mass refractive index graphs is an 



BACKGROUND 175 

approximately exponential decrease of n with respect to height. Schel- 
leng, Burrows, and Ferrell [15] attempted to remove this systematic n 
decrease with height by utilizing a correction factor of linear form, 
although, as has been seen in chapter 3, this leads to serious overestima- 
tion of refraction effects in the more modern problem of satellite tele- 
communications. 

The work of Hay [6] confirmed and extended the observations Gerson 
had made concerning air mass profiles. Measuring air mass character- 
istics at Maniwaki, Quebec, Hay concluded that each large-scale air mass 
type in central Canada has a distinctive refractive index profile. Hay 
determined the height distribution of A'' for four basic air masses by fitting 
a second-degree polynominal to each of the four sets of air mass data, 
indicating that an average A'' profile cannot be approximated effectively 
by a linear curve except over small height increments. Hay obtained 
further discrimination by constructing a "dry-term" curve for each air 
mass. The dry-term curves display a smaller standard deviation than 
the total A'^ curves for all air masses except the continental arctic which 
is, of course, itself very dry. The largest variations in total A'' are due 
to fluctuations in the wet term. The saturation vapor pressure is approx- 
imately an exponential function of the temperature, so that during the 
warm season normal temperature changes cause the saturation vapor 
pressure and, therefore, the wet term is vary sharply. 

Contained also in Hay's paper [6] is a discussion of the use of A^ pro- 
files to estimate corrections for radio ray refraction by use of a table of 
mean effective earth's radius factors for each of four air masses within 
95 percent probability limits. 

Misme [5, 16] has had an interest in synoptic radio meteorology in con- 
nection with telecommunications networks in France and North Africa. 
Interests of this French radio engineer include vertical gradients of the 
refractive index and atmospheric reflection of radio waves. 

A reasonable question at this point is, to what extent are model ex- 
ponential atmospheres applicable to various climatic zones around the 
earth? While it is readily apparent that individual profiles vary widely 
from any sort of exponential norm, there is an increasing backlog of 
experimental evidence [17] that shows that long-term averages of synoptic 
refractive index variations do tend toward an exponential form with 
respect to height. Indeed, the 5-year mean A'^ profiles of figures 5.1 and 
5.2, representing arctic and tropical climates respectively, bear out this 
contention by showing close agreement with models developed from mid- 
latitudinal data. Indeed, these examples verify the results obtained by 
Gerson [2] for various air masses. 

Randall [4] investigated the relationship of surface meteorological data 
to surface A'^, N s, and field strength in the FM frequency band. The 
results described were drawn from a very limited data sample that covered 



176 



SYNOPTIC RADIO METEOROLOGY 



400 



300 



N 200 



100 - 





78°-50'N., 103°- 50' W. 


i 


FEBRUARY 1953-1957 


w 

v\ 

v\ 


234 PROFILES, Ns=332.5 _ 




.^N(h) 


— 


^^ ^UPPER AND LOWER 


— 


VV STANDARD DEVIATION 
^^SU-IMITS 


EXPONENTIAL REFERENCE^ ^^^^^ 


ATMOSPHERE, N^ =332.5 




2 4 6 8 10 12 14 

h(knn) 

Figure 5.1. Mean N distribution for Isachson, N.W.T., Canada. 



less than a month during the summer of 1947. Within the framework of 
this speciahzed study, however, Randall found that polar continental air 
masses were associated with low field strengths and low A^^ while tropical 
maritime air masses were associated with high field strengths and high 
N s, as shown on figure 5.3. Randall advanced the hypothesis that the 
observed field strength changes were due to the existence of characteristic 
A^ profiles typical of each air mass type. Randall was curious also as to 
the behavior of A'' and radio fields during the passage of fronts and squall 
lines. Figure 5.4 shows the results of this investigation and indicates that 



BACKGROUND 



177 



definite field strength changes may occur during frontal and squall line 
passage. Caution should again follow in the interpretation of these 
results since they represent but a single example of frontal and squall line 
passage. 

Gray [18] made measurements of correlation of A^^ and the gradient on 
N with path losses at UHF frequencies for observing points representative 
of various climatic areas around the world and concluded that changes in 
transhorizon telecommunications are strongly dependent on atmospheric 
changes. 



400 



300 



2''-46'S., I7r-43'W 

FEBRUARY 1953-1957 

274 PROFILES, Ns=37l.3 



N 200 



100 








'UPPE^ AND LOWER 
STANDARD DEVIATION 
LIMITS 



EXPONENTIAL REEERENCE- 
ATMOSPHERE, Ns=37l3 



2 4 6 8 10 12 14 

h (km) 

Figure 5.2. Mean N distribution for Canton Island, Pacific Ocean. 



178 



SYNOPTIC RADIO METEOROLOGY 






;| i 
is M 

It 

is 



15 



13 



09 



0.5 







PLOTTING KEY 




















+ 


Averoge wmd speed equal 
to or more thon lOmph 




















cP 

mP 


Polar conllnen'ol oir mass 
Polar maritime air mass 






































mT 


Tropicol morifime air moss 














mT 

+ 
























+ mP 






ml 




♦mT 


^^B 




— 




cPamP 


cPar 


nP '' 
cPsmT^ . 


nT 


+■ 


mT 






cP 


cP 


_^ 


^ 


cPamT 


cPamT 
1 


mT 








mP 


^^--^^^ 


cPamT 




rnT mT 
+ + 


*mT 






A-^ 


-- 


1- + f 
cPcPcP 


+ 
cPamT 




















1- 
cP 


f 
cPSmr 




1 

1 

























320 



330 



340 



580 



39C 



Hour// Mean Surface Refracfivity 



Figure 5.3. Field strength versus refractivity. 
Scatter diagram of select hourly median field strength vs. hourly mean refractivity July 17 to Aug. 8, 1947. 



In another article Gray [19] considered radio propagation and related 
meteorological conditions over the Caribbean Sea. Utilizing the effective 
earth's radius factor as a representative index, Gray presented an em- 
pirical curve of annual median scatter loss versus effective propagation 
distance designed to fit both Caribbean and temperate regions. Effective 
distance as defined by Gray is the angular distance in radians multiplied 
by the radius of the earth modified for normal refraction. Gray reports 
that the refractive gradient in the first 100 m is in general the determining 
factor in median scatter loss for transhorizon telecommunications, as one 
would expect from earlier refraction studies [20]. 

Other studies on refraction problems during the 1950's have led to 
systematic computation of refraction effects and to significant applica- 
tions such as the evaluation of radar elevation angle errors in differing air 
masses and climates [21 , 23]. Schulkin [21] advanced a practical and very 
fundamental method for numerical calculation of atmospheric refraction 
(radio ray bending) from radiosonde data. Figure 4.55 [after Schulkin] 
gives mean angular bendings for radio rays passing completely through 
the earth's atmosphere for two extremes of air mass type. Fannin and 
Jehn concluded that a particular refractive index profile depends on air 



BACKGROUND 



179 



mass type and the climatic controls of season and latitude, as is substan- 
tiated by mean A^ profiles for 34 weather observing sites located in or near 
four distinct air mass source regions about the world. These data show 
that a definite difference does exist in the profiles of various air masses. 
Refraction effects were found to be largest in tropical maritime air, inter- 
mediate in polar continental and polar maritime, and least in tropical 
continental. Fannin and Jehn also published graphs showing day-to-day 
variations in profiles representing the effects of air mass changes over a 
given observing station and graphs of diurnal profile variations. 

In other studies along these lines, Bean, Horn, and Riggs [23], demon- 
strated that radio ray refraction within the lower layers of an air mass is 
mirrored by the difference between the observed refractive index structure 
and that of a standard atmosphere. Figure 5.5 shows a graph of bending, 
T, plotted together with a modified refractive index profile for an example 
of summertime tropical maritime air. It is apparent that near the ground 
bending departures reflect refractive index profile departures from stand- 
ard. Figure 3.16 presents a series of graphs of departures of refractive 









^ 












3.1 



1.9 



1.7 



1.5 



1.3 



0.9 



0.7 



0.5 




.Cold front 

4 posses Rtcttfnond 



I Squoll line 

I crosses pothlCold front posses Wash. 
I230 1650 20SO 0030 

July 19, 1947 



8. 
10- 



12 Cold front passes 

y*tos^lngtool830 

— July 19- 



+7 Squ oil lino crosses 
potti 1330 JolyO 



-3 

Cold fron t posses RKJimond 
15" _4 2130 July 19 
-2 



Points 6, 1 3, a 14 missinij 



320 330 340 350 360 370 380 390 

Hourly Mean Surface Refracfivify 



Figure 5.4. Field strength and N changes during frontal passage. 

Scatter diagram showing passage of a cold front system over the Washington-Richmond path and graph 
indicating the fluctuation of hourly median field strength with time. 



180 SYNOPTIC RADIO METEOROLOGY 

index and bending from normal for each air mass and emphasizes the 
close relation between the two. This affords the synoptic radio meteor- 
ologist a set of standard reference profiles for the study of a given air mass 
or the confluence of contrasting air masses at a frontal zone. 

Arvola [24] discussed the changes in refractive index profiles caused by 
migratory weather systems. Examining a series of synoptic situations in 
the midwestern portion of the United States during November 1951 that 
gave rise to greater-than-normal refraction, Arvola found that ridges and 
accompanying subsidence effects generally gave rise to strong A^ gradients 
and enhanced signal strength over a 200-km link broadcasting at 71.75 
Mc/s. Refractive gradients were stronger in the warmer air masses and 
at times when moist air was present below the inversion created by the 
subsidence mechanism. Strong gradients which appeared behind a squall 
line later weakened with the approach of a cold front. After the passage 
of this front stratification in the cold air again increased the gradient. 

Subsequent investigations of polar continental air across central North 
America make use of reduced-to-sea-level forms of the radio refractive 
index as synpotic parameters. The reduced forms are sensitive indica- 
tors of synoptic changes and afford a clearer picture of storm structure 
than that obtained using analyses in terms of unreduced N or B units 
(defined in chapter 1). Later portions of this chapter consider in detail 
these two synoptic parameters. 

Jehn [25], at the University of Texas, used a form of potential refractive 
modulus, K, developed by Lukes [26] and Katz [27] to account for the 
height dependence feature of the refractive index. Articles by Jehn 
[28, 29] on synoptic climatology use composite analysis techniques to 
study the synoptic properties of the Texas-Gulf cyclone and the central 
United States type of cold outbreak. 

In another application of the potential refractive modulus, Flavell and 
Lane [30] have utilized Katz's K unit (see chapter 6) to study tropospheric 
wave propagation. Field strength measurements on VHF-UHF trans- 
horizon radio links over the British Isles are analyzed in terms of cross 
sections in terms of K and regional charts of AK, the difference between 
K at the surface and /^gsombar- These charts show features similar to 
those obtained by use of No or A (see chapter 1). 

The authors cite a series of measurements on a 500-km path at 877 
Mc/s on which the received signal was ordinarily below the noise level. 
The singular occasions on which the received signal could be measured at 
the normal times of radiosonde ascent all exhibited a symmetric variation 
of AK over the transmission path. These results lend credence to the 
hypothesis that synoptic disturbances play an important role in trans- 
horizon telecommunicat ions. 

Moler and Arvola [31] advanced the hypothesis that the vertical 
gradient of the refractive index is affected by broad-scale vertical motion 



BACKGROUND 



181 



SAN JUAN, W.I. 
(July Average) 




310 320 330 340 350 360 370 380 390 

A(h,3l3), N UNITS 

I I I I i \ I \ I 

-8 -7 -6 -5 -4 -3 -2 -l 

T313-T (mrad) 



Figure 5.5. Departures of t and N from normal for maritime tropical air. 



182 SYNOPTIC RADIO METEOROLOGY 

in the troposphere and suggest that moisture and temperature stratifica- 
tion is modified principally by changes in vertical velocity. This latter 
work was extended [32] by a study of mesoscale centers of horizontal air 
mass convergence and divergence in the troposphere. Horizontal con- 
vergence takes place in a low-pressure area, where the winds around the 
low have a predominant component toward a local vortex at the center. 
Upward vertical motion results from the pile-up of air in the vortex region. 
In a high-pressure area, winds have a predominant component away from 
the center of a high and subsiding air descending from higher levels takes 
the place of air transported outward from the center of the high. Local 
convergence, then, implies upward vertical motion in the lower levels of 
the troposphere, while local divergence implies downward vertical motion 
in the lower levels. It follows that local convergence centers (small- 
scale low-pressure cells) produce updrafts in the atmosphere that result in 
considerable mixing and the destruction of atmospheric layers. Centers 
of divergence (small-scale high-pressure cells) create strong temperature 
and humidity inversions by the motion of subsiding air. Such inversions 
produce large vertical refractive index gradients that partially reflect 
microwaves traveling through this meteorological environment [33]. In 
a well-mixed atmosphere, on the other hand, the primary propagation 
mechanism is believed to be scattering by turbulent fluctuations of the 
refractive index [34, 35]. 

Sea-level measurements showed N s to be essentially invariant during 
the experiment, yet signal levels ranged over more than 60 dB, a power 
factor on the order of 10^ Since scattering theory would account for a 
rise in signal level of only about 13 dB, Moler and Holden [32] conclude 
that refractive layering and thermal stability over the oceans are princi- 
pally functions of vertical wind velocities. These conclusions bear out 
earlier ones of Saxton [36] and later ones of Flavell and Lane [30] stating 
that high signal levels from a distant transmitter may well be the con- 
sequence of refractive layering and subsequent reflection of radio waves. 
In the same article Saxton also considered both the scattering of radio 
energy by turbulent eddies and the effects of superrefraction of radio 
waves produced by departures from normal of the height variation of the 
tropospheric refractive index. 

A knowledge of the vertical motion of the atmosphere becomes at this 
point central to the problem of refractive layering. A brief resume of the 
Moler-Holden method for estimating the relative magnitude and direction 
of the vertical component of the wind velocity follows. 

Moler and Holden postulate a model atmosphere bound by the follow- 
ing conditions: 



BACKGROUND 183 

(1) A barotropic level (level of nondivergence) exists at some pressure 
level greater than 500 mbar. 

(2) The horizontal wind velocity divergence changes sign at the level 
of nondivergence (LND). 

(3) The vertical velocity vector in the troposphere is proportional to 
that at the LND. 

(4) The vertical velocity vector (a) vanishes at the surface of the earth 
and (b) approaches zero at the outer reaches of the atmosphere. 

Commencing with the equation of continuity, 



l^+F-Vp + pV-F + l (pW) = 0, (5.1) 



where p is the density of air, V is the horizontal del operator, V Is the 
horizontal wind velocity vector, W is the vertical wind speed, and z is the 
vertical coordinate; and employing horizontal velocity divergence in the 
natural coordinate system of the form 



F = f^ + . f . (5.2) 

ds dn 



Moler and Holden derive for the magnitude of the vertical vector at the 
LND, upon employing the vorticity equation. 



(TF)lnd = 

p 



^ (I) (11) 

1 "" 



_7LND ^ dt J LND S dS 

(III) 



LND i OZ J 



(5.3) 



where f is the vertical component of the absolute vorticity. Moler and 
Holden reason that of the labeled integrals (I) decreases in proportion 
to the height above the LND and is « (II) at 300 mbar. Since W « V, 
(III) makes only a small contribution. The expression for (IF)lnd be- 
comes 

W.No = - i f" Vf^(f-f)dz, (5.4) 

P 7 LND f ds \dx dy/ 



184 



SYNOPTIC RADIO METEOROLOGY 


















Figure 5.6. Isobaric and streamline maps. 



which simplifies to the approximate relation 

(W) = relative magnitude of — — { T~ — ^:~ ) 

f ds \dx dy/s 

for purposes of calculation. 



(5.5) 



REFRACTIVE INDEX PARAMETERS 



185 



— ou 
-60 
-70 






/ 


\ 










/ 


A 


\ 






^ 






^ 




—80 





09 



10 



11 



12 
PST 



13 



14 



15 



Figure 5.7. X-band signal level versus time. 
Santa Barbara X-band signal level for 26 Mar. 1958. 



Moler and Holdeii then continue with a description of the large signal 
enhancement and deep fading that occur as the propagation mechanism 
varies between partial reflection and scattering on a transhorizon radio 
path along the California coastline. Reflection occurs when strong re- 
fracting layers are present within a kilometer of the surface, typifying 
meteorological conditions generally associated with the subsidence inver- 
sion frequently found along the California coast. 

Figure 5.6 shows the sea level pressure chart and a series of streamline 
analyses by Moler and Holden depicting mesoscale centers of convergence 
and divergence for a day in March along the southern California coast. 
Figure 5.7 shows the X-band signal level received at Point Loma, San 
Diego (SD) from Santa Barbara (SB) during the same day. As the 
centers of convergence along the radio path weaken, refractive layers are 
formed and the signal level rises sharply during the middle of the day. 
Later the signal level lowers again with the regeneration of convergence 
centers and the destruction of stratified layers during the afternoon hours. 



5.3. Refractive Index Parameters 

In later sections, the analysis of a synoptic disturbance in the tropo- 
sphere will be described in detail. Certain reduced forms of the refrac- 
tive index that will be useful in the ensuing discussion will be developed 
here. These forms, already discussed in chapter 1, are revisited here for 
the purj)ose of com])arison in synoptic example. 



186 



SYNOPTIC RADIO METEOROLOGY 



Figure 4.1 shows contours of the mean value of A'^ at the surface, N s, 
determined from eight years of data for August, 0200 local time. Minia- 
ture circles indicate the 62 observing stations used to analyze this chart. 
It is evident that coastal areas display high values of A'^s as compared with 
inland locations. Low values of N s are apparent along the Appalachian 
mountain chain and in the great mountain systems and inter-mountain 
plateaus of the western United States. There is a marked similarity be- 
tween the N s contours on figure 4.1 and the elevation contours of figure 
5.8. As a sensitive indicator of changes in atmospheric density, A''^ dis- 
plays a strong elevation dependence. To remove this effect the reduced- 
to-sea-level expression, A^'o, was introduced in chapter 4 as: 



No = Ns exp 



H* 



(5.6) 



where z is height in kilometers and H* = 7.0 km is the scale height. 
Scale height is the height at which the mean value of A^ has decreased to a 
fraction 1/e of its initial value. A scale height of 7.0 km is in close agree- 
ment with H* = 7.01 km for the NACA standard atmosphere with 80 
percent relative humidity and H* = 6.95 km obtained from climatic 
studies utilizing over two million observations of the variation in A^ over 
the first kilometer above the surface of the earth [20]. The A^'o contours 



'00&500 




7000 a ovERjasa 



Figure 5.8. Ground elevation above sea level. 



REFRACTIVE INDEX PARAMETERS 



187 




Figure 5.9. Idealized diagram of a fast-moving cold front. 



of figure 4.4 utilize the same data as figure 4.1. The use of A^o produces a 
simpler map with a smaller range of variation. Additionally, N ^ may 
easily be estimated from the smooth and slowly varying contours of A^'o 
providing only that station elevation is known. It was shown in chapter 
4 that N s may be more accurately estimated from charts of A^o than 
from charts of N ^ itself by a factor of 4 or 5 to 1 [37]. 

The attempt to find a workable method to compensate for the decrease 
of A'' with height has brought about the development of various model 
atmospheres discussed in chapter .3. The paragraphs that follow will 
outline briefly steps in this development that are relevant to synoptic 
studies. 

Vertical refractive index cross sections are standard working charts for 
synoptic studies. Such charts constructed from observed values of N 
suffer from a serious shortcoming in that the natural decrease of A'' with 
respect to height effectively masks contrasts between air masses in the 
lower troposphere. An idealized synoptic example depictmg the con- 
fluence of contrastmg air masses is presented on figures 5.9 and 5.10. 
When these idealized systems are analyzed in terms of A'^ as on figures 5.11 
and 5.12 the most prominent feature is the laminar structure of the A'^ 
field. 



188 



SYNOPTIC RADIO METEOROLOGY 




LEGEND 

WARM FRONT — -^^ 

ISOTHERM 

RH ISOPLETH 

CLOUD PATTERN 

PRECIP PATTERN ^^^§ 
AIR FLOW 



100 200 300 400 500 600 700 800 900 

DISTANCE, km 



Figure 5.10. Idealized diagram of a warm front. 




-500 



-400 -300 -200 -100 100 

DISTANCE IN KILOMETERS 



200 



300 



Figure 5.11. Idealized cold front in N units. 



REFRACTIVE INDEX PARAMETERS 



189 



LEGEND 
WARM FRONT 
N ISOPLETH 




300 400 500 

DISTANCE, km 

Figure 5.12. Idealized warm front in N units. 



Early attempts to compensate for the decrease of A^ with height used 
the constant gradient of the effective earth's radius theory, l/4a, where a 
is the radius of the earth. As an illustration, the strong elevated layer 
found during the summer in southern California was studied in terms of a 
form of (1.32) for B, given by 



B = N{z) + (39.2)2, 



(5.7) 



where N{z) is the value of A^ at height z in kilometers [38]. Since A^ tends 
to be an exponential function of height rather than the linear function 
assumed by the effective earth's radius theory, the B unit approach over- 
corrects when z is greater than about 1 km. 

This point is illustrated by figures 5.13 and 5.14 where the A'^ data of 
figures 5.11 and 5.12 are plotted in terms of B units. Note that the over- 
correction produces a cross section in which A'^ increases with height from 
a value of 310 at the surface to 360 at 5 km. A function of exponential 
form was designed to account for the systematic decay of density with 
height that characterizes the terrestrial atmosphere, as given by 



A = A^(z)+313[l - jexp {- f^\'] 



(5.8) 



190 



SYNOPTIC RADIO METEOROLOGY 




-500 -400 -300 



-200 -100 100 

DISTANCE IN KILOMETERS 



200 300 



Figure 5.13. Idealized cold front in B units. 



The quantity, A, enables one to discern departures of A'^ structure from 
the model atmosphere 



N = 313 exp \-f^\ 
Further, the radio-ray bending, 



(5.9) 



Tl,2 = 



iV, 



N. 



cot e dN ' 10~', 



(5.10) 



where 6 is the local elevation angle of the radio ray to spherically stratified 
surfaces of constant A'^, may be approximated by 



Tl.2 = — 



"^ 'eot d (IQ-^) 

Ai n 



dA{z, 313) + t(z, 313). (5.11) 



The term t{z, 313) is the bending in the average atmosphere given by (5.9), 
while the integral term represents the departures in bending produced by 



REFRACTIVE INDEX PARAMETERS 



191 




300 400 500 600 

DISTANCE IN KILOMETERS 



Figure 5.14. Idealized warm front in B units. 



various synoptic and air mass effects. The values of bending in the 
average atmosphere are tabulated [39] and approximate methods of calcu- 
lating the integral term in (5.11) to within a few percent have been given 
[40]. 

The next logical step is to plot the frontal cross sections, previously 
analyzed in terms of A^ and B, in A units. This has been done on figures 
5.15 and 5.16. The range of refractivity values on the new charts is re- 
duced from more than 60 to about 25 units and a pattern emerges that 
displays sharp contrasts for air mass differences associated with the 
frontal zone. Note, for the warm front case (fig. 5.16), that the A values 
increase with height until they reach a maximum associated with the up- 
gliding warm moist air overriding the frontal surface. The region of 
precipitation in advance of the front is shown as an area of high surface 
A^. In the cold-front case (fig. 5.15), the classic push of warm air aloft 
by the encroaching cold air is evidenced by the "dome" of high A values 
just before the front. Stratification in the cold air due to inversion 
effects, although impossible to detect in the A'^ charts, is clearly seen by 
the use of A unit charts. 



192 



SYNOPTIC RADIO METEOROLOGY 




-200 -100 100 

DISTANCE IN KILOMETERS 



Figure 5.15. Idealized cold front in A units. 



A units were used in subsequent cross-section analyses in order to 
throw frontal discontinuities and air mass differences into sharp relief. 
The Potential Refractivity Chart of figure 5.17 facilitates the rapid con- 
version of A^ to A. This simplification eliminates the necessity of using 
exponential tables for each individual calculation of A and thus lends con- 
siderable ease to the preparation of charts of the new parameter. 

The reader probably has already observed that the A^'o and A correc- 
tions do substantially the same thing. Their primary distinction is that 
^ is a nonlinear "add-on" correction while A^o is a multiplicative one. 
The disparity between A^o and A is tabulated in table 5.1. 

These figures are obtained by taking the zero values of A^ (for example, 
300), subtracting the add-on correction for, say, 3 km (300 — 109 = 181), 
and reducing this number, A'' = 181, to zero elevation by the A^o reduction. 
No = N exp (3/7). The 313 exponential atmosphere is adopted for a 
single reference atmosphere. The large discrepancies of table 5.1 may be 
avoided for practical applications by choosing a model near to the mean 
value of A^ of the site under study. 

The A unit has the additional advantage that, while it is a convenient 
method for height reduction, the ray bending is also readily recoverable 



REFRACTIVE INDEX PARAMETERS 



193 



8- 



LEGEND 

WARM FRONT 
A=N + NoCl-exp(-I)j 
WHERE No=3l3 
H=70 km 




300 400 500 

DISTANCE, km 



600 



Figure 5.16. Idealized warm front in A units. 



from it, requiring only a knowledge of the altitude of the observed refrac- 
tive index measurement and the local elevation angle of the radio ray. 
The potential refractive modulus of I. Katz [27], 



<P 



Po + b 



eo 



(5.12) 



where 6 is the potential temperature and eo the potential vapor pressure, 
is also in current use. The constants b and c of (5.12) are given in the 
development of modification to A'' data in chapter 1 (1.39). The potential 
refractive modulus has been employed by Jehn [25] to study polar waves 
over North America. Refractivity, A'', cannot accurately be recovered 
from (p for bending calculations unless additional information is available; 
namely, observed temperature and vapor pressure. The concept of the 
potential refractive modulus arose out of the earlier refractive modulus, 
M (see ch. 1), which may be defined by 



M 



4 



1 + 



a_ 



X 10' = N(z) + 



z 
Laj 



X 10'= 



(5.13) 



where z = height above the earth's surface and a — earth's radius. 



194 



SYNOPTIC RADIO METEOROLOGY 



A=N+Nol[-exp(chl] 
WHERE No=3l3 

c =-.143859 
h=ht. in km 




170 190 210 230 250 270 290 310 330 350 370 390 410 430 

\6 



N=(n-l)IO' 

Figure 5.17. Potential refractivity chart. 



Table 5.1. Differences between No and A at various elevations 
Value of N at z=0 



Elevation 


250 


300 


313 


350 


400 


(km) 
1 


-10 
-21 
-49 

-48 


-2 

-5 

-22 

-29 








5 

7 

5 

30 


13 


2 

3 

4 


29 
31 

68 



A SYNOPTIC ILLUSTRATION 195 

The M unit came into being out of an approach similar to that which 
led to the development of the B unit. The condition dn/dz = — 1/a (a 
radio duct) implies an effective earth of infinite radius (effective earth's 
radius factor, /v = oo, see ch. 1). The M unit is designed so that dM /dz 
= when k — oo. M units are employed from time to time in radio 
meteorological analysis. The Canterbury Project [41], for example, used 
M unit analyses in the study of ranges of over-water radar signals. 

5.4. A Synoptic Illustration 

The specialized field of synoptic radio meteorology attempts a descrip- 
tion of the variations in atmospheric refraction that arise from large 
scale weather changes such as the passage of a polar front or the move- 
ment of an air mass over a particular geographic region. The term air 
mass is used to describe a portion of the troposphere that has at the surface 
generally homogeneous properties. Although no air mass is in fact 
homogeneous, the advantages of the air mass concept as a convenient 
fiction are evident in the cataloging of meteorological observations for 
climatic or synoptic purposes. 

The region of interaction between the cold air of the poles and the 
warm air of the tropics is referred to as the polar front and is generally 
located between 30 and 60°N. From time to time a section of the polar 
front is displaced northward by a flow of warm tropical air while an adja- 
cent section is simultaneously displaced southward by a flow of polar 
air. The interaction of the flow of polar and tropical air results in the 
formation of a "wave" that moves along the polar front, often for thou- 
sands of kilometers. An example of a fully developed polar front wave is 
shown on figure 5.18(a), in the same manner that it would appear on a 
daily weather map. Across the Great Plains and eastern seaboard of the 
United States the polar front wave normally moves along the line AB in 
figure 5.18(a). An idealized space cross section along the line AB is 
shown in figure 5.18(b). The warm tropical air that flows into the warm 
sector of the wave overrides the cool air before the wave to form the 
transition zone denoted as a warm front. The cold front represents the 
transition between the generally humid air of the warm sector that has 
been forced upwards and the advancing cold polar air. Squall lines are 
drawn to represent belts of vigorous vertical convection, intense thunder 
showers, and sharp wind shifts that frequently precede fast-moving cold 
fronts. The fronts shown on a daily weather map represent the ground 
intersection of the transition zones between various air masses.^ 



2 The reader who wishes a critical appraisal of recent meteorological thinking on 
fronts, air masses, squall lines, etc., is referred to Dynamic Meteorology and Weather 
Forecasting, by Godske, Bergeron, Bjerknes, and Bundgaard, American Meteoro- 
logical Society and Carnegie Institute of Washington, D.C., 1957. 



196 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.18. The Polar Front wave. 



SURFACE ANALYSIS IN TERMS OF No 197 

As a representative example of the application of these newly developed 
units to a synoptic situation, a large-scale outbreak of continental polar 
air which took place over the United States during February 1952 is 
analyzed in terms of A^o and A [42, 43]. 

For this synoptic illustration the reduced expression, No, is used in 
preparing constant level charts of the storm at the same levels and times 
as those used in the daily weather map series of the U.S. Weather Bureau. 
The A unit, on the other hand, is employed to construct vertical cross- 
sections through the frontal system to give a three-dimensional picture of 
the synoptic changes taking place. 

5.5. Surface Analysis in Terms of N^ 

A pronounced cold front developed and moved rapidly across the 
United States during the period 18 to 21 February 1952. 

Prior to February 18, a polar maritime air mass had been moving slowly 
eastward across the Great Basin and Rocky Mountain regions. This 
system included a slow-moving cold front extending from northern Utah 
southwards into Arizona and a quasi-stationary front extending north- 
eastwards into Wyoming. With the outbreak of polar continental air 
east of the Rocky Mountains, the maritime front became more active 
and, as it moved ahead of the fast-moving polar continental front sweep- 
ing across the Great Plains, was reported as a squall line by the time it 
crossed the Mississippi River early on the morning of February 20. 
During the latter stages of the storm system, the polar maritime cold 
front-squall line was located in the developing warm sector of the polar 
front wave. The entire ensemble of cold front, polar front wave, and 
squall line then moved rapidly to the east coast by the morning of the 
21st, thus completing the sequence. 

Charts of Ao were prepared from Weather Bureau surface observations 
taken at 12-hour intervals from 0130 EST, February 18 until 0130 EST, 
February 21, 1952, or, in other words, the period of time that it took the 
polar front wave to develop and move across the country. The synoptic 
sequence is seen on figures 5.19 through 5.25., where contours of A^o are 
derived for various stages of the storm and compared to the superimposed 
Weather Bureau frontal analysis. The same procedure of comparing 
derived contours with the existing frontal pattern was followed throughout 
the present example. Observations from 62 weather stations were used 
in preparing the surface weather maps. Figure 5.19 indicates that the 
cold front extending from Utah southward displays weak A^o changes 
across the frontal interface. In the early stages of the sequence (figs. 
5.19 and 5.20) this lack of air mass contrast is evidenced in another way 
by the slight change of the position of the Ao = 290 contour encircling 
west Texas and New Mexico as the frontal system moves through that 
area. 



198 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.19. No chart for storm system OlSOE 18 Feb. 1952. 




Figure 5.20. No chxirl for storm system 1S30E 18 Feb. 1952. 



SURFACE ANALYSIS IN TERMS OF A^o 



199 




Figure 5.21. No chart for storm system 0130E 19 Feb. 1952. 




Figure 5.22. No chart for storm system 1S30E 19 Feb. 1952. 



200 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.24. No chart for storm system 1330E 20 Feb. 1962. 



SURFACE ANALYSIS IN TERMS OF No 



201 




Figure 5.25. No chart for storm systems 0130E 21 Feb. 1952. 



By comparison, the cold front sweeping down across the Great Plains 
(figs. 5.20 to 5.22) has a rather marked A^o gradient across the front, due 
in large measure to the northward flow of warm, moist air that forms a 
definite warm sector by 1330 EST on the 19th. It is perhaps significant 
that the A^o contours indicate that the various frontal systems are transi- 
tion zones rather than the sharply defined discontinuities of textbook 
examples, a point that has been enlarged upon by Palmer [44]. 

Figures 5.22 to 5.25 trace the trajectory of this vigorous push of cold 
air across the Gulf Coastal Plain and the southeastern states. The most 
spectacular gradients on this map series are in the eastern half of Texas 
where the marked contrast of cold, dry polar air of low A'^o and warm 
moist Gulf air of very high A^o occurs. A prominent feature of meteor- 
ological significance on figures 5.20 to 5.23 is the northward advection of 
tropical maritime air in the warm sector ahead of the cold front. The 
advection is evidenced by the northward bulge of high A^o over the 
Mississippi Valley on both charts. Figure 5.25 shows the synoptic situa- 
tion as the front moves off into the Atlantic Ocean and refractivity 
gradients across the continent gradually weaken. 

The variation of A'^o due to the passage of the frontal system can be 
seen on figures 5.26 to 5.30 where the 24-hour changes of A^o have been 
contoured. The 24-hour change, designated AVo, is obtained by sub- 
tracting from the current value the value of A''o observed 24 hours ago. 



202 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.26. 24-hour ANo chart, OlSOE 19 Feb. 1952. 




Figure 5.27. 24-hour ANo chart, 1330E 19 Feb. 1952. 



SURFACE ANALYSIS IN TERMS OF No 



203 




Figure 5.28. S4-hour ANo chart, 0130E SO Feb. 1962. 




Figure 5.29. 24-hour ANo chart, 1330E 20 Feb. 1962. 



204 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.30. U-hour ANo chart, 0130E 21 Feb. 1952. 



The change is determined on a 24-hour basis in order to remove effects of 
the diurnal cycle of A^'o. The AA^o charts show a general rise of Nq in the 
warm sector and a drop in A^^o behind the front amounting, in the warm 
sector, to 35 to 40 A^ units by 1330 EST on February 20 (fig. 5.29) accom- 
panied by a 40 to 50 A^ drop behind the front. 

The relative sensitivity of A^o to humidity changes is emphasized by 
the AA^'o charts. The N drop behind the cold front occurs in a region of 
increasing pressure and decreasing temperature — a combination that 
increases the dry term and depresses the wet term. The decrease in the 
wet term from rapidly dropping dewpoint more than compensates for the 
increased dry term. As an example, in the 24-hour period ending 0130 
EST on the 19th, the station pressure at Oklahoma City increased 13 mbars. 
The dry term increased 12 N units while the wet term dropped 42 A'^ units, 
giving a net change of minus 30 A'' units. This A^'o rise in the warm sector 
and the drop behind the cold front is consistent throughout the develop- 
ment of the polar front wave and appears to be what one would expect 
for this type of weather system. The present system had about a 35 A^ 
unit rise and a 40 A?^ unit drop. This general pattern might be expected 
to occur in all fast-moving cold fronts with varying intensity, depending 
upon the individual synoptic pattern. In any case, it appears that the 
Nq pattern is a sufficiently stable and conservative property of the atmos- 
phere so that it should be possible to develop forecasting rules for A^^o, 
but not, of course, without the analysis of many more N patterns. 



CONSTANT PRESSURE CHART ANALYSIS 205 

5.6. Constant Pressure Chart Analysis 

The same frontal system as above was analyzed for selected constant 
pressure levels. The 850 mbar charts (about 5,000 ft above mean sea 
level) and the 700 mbar charts (about 10,000 ft above mean sea level) 
were prepared for the times of radiosonde ascent (10 A.M. and 10 P.M. 
EST) throughout the synoptic sequence from the radiosonde reports of 
43 U.S. sounding stations. It is not necessary to reduce the 850 mbar 
or 700 mbar level data, since it is already referenced to the indicated 
constant pressure level. Contours for the charts aloft are shown on 
figures 5.31 to 5.36 while their respective 24-hour changes, Nsbo and A^too, 
are given on figures 5.37 to 5.40. 

The Nsbo charts show that the northerly flow of warm humid air within 
the w^arm sector that was so prominent on the A^o maps is also clearly in 
evidence at the 850-mbar level. Further, a change pattern similar to 
that on the A'o maps is also observed at the 850-mbar level, particularly 
on figure 5.34. That is, a rise in Agso values in the w^arm sector and a 
decrease behind the cold front is apparent. Surprisingly enough, by the 
time the frontal system is well-developed, at 1000 EST on the 20th, the 
Agso values are nearly as large as those on the surface. 

The A'voo charts are more difficult to interpret than those of A^o or 
A'sso- It appears that at this altitude the wet term is usually negligible 
and N will normally vary inversely as temperature since the pressure is, 
of course, constant at the 700-mbar level. By 1000 EST on the 19th (fig. 
5.34) an intrusion of low A" values is observed in the 700-mbar warm sector 
due to the advection of warm, low-density air northwards. The chart 
for 24 hours later (fig. 5.42) displays two prominent highs in which 
A700 = 225. One of these highs lies between the squall line and the cold 
front and the other just south of the apex of the 700-mbar wave. Inter- 
estingly enough, these two highs are due to quite different causes. The 
high centered over Atlanta appears to have arisen from the unusually high 
transport of moisture to the 10,000-ft level, since the 700-mbar wet term 
at Atlanta increases from 4.5 to 25 N units in the 24-hour period ending 
with 1000 EST on February 20. The second high, centered over Omaha, 
appears to be due to an intense dome of cold air, as indicated by the drop 
of the 700-mbar temperature from -7.3 °C to -21.4 °C in 12 hours 
preceding map time. When temperatures are below °C, the wet term 
contribution to A^ is quite small and density changes become significant 
in producing changes in A. Falling temperatures produce higher density 
air and, consequently, a region of high A^ values around Omaha as de- 
picted on the A^voo chart of figure 5.31 and AA700 chart of figure 5.40 which 
shows this change more clearly. 



206 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.31. Nsso chart, lOOOE 18 Feb. 1952. 




Figure 5.32. N700 chart, lOOOE 18 Feb. 1962. 



CONSTANT PRESSURE CHART ANALYSIS 



207 




Figure 5.33. Ngso chart, lOOOE 19 Feb. 1952. 




Figure 5.34. N700 cMrt, lOOOE 19 Feb. 1962. 



208 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.35. Nsso chart, lOOOE 20 Feb. 1952. 




Figure 5.36. N700 chart, lOOOE 20 Feb. 1952. 



CONSTANT PRESSURE CHART ANALYSIS 



209 




Figure 5.37. 24-hour ANgso chart, lOOOE 19 Feb. 1952. 




Figure 5.38. 24-hour AN700 chart, lOOOE 19 Feb. 1962. 



210 



SYNOPTIC RADIO METEOROLOGY 




Figure 5.39. 24-hour ANsbo chart, lOOOE 20 Feb. 1952. 




Figure 5.40. 24-hour AN 700 chart, lOOOE 20 Feb. 1962. 



VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 



211 



5.7. Vertical Distribution of the Refractive 
Index Using A Units 

The synoptic study of the vertical distribution of the radio refractive 
index extends the foregoing constant-level analyses by considering the 
problem of whether the air mass properties associated with this typical 
wintertime outbreak of polar air are reflected in the vertical refractive 
index structure. Charts showing the structure of the storm have been 
prepared using radiosonde measurements from stations located along a 
line normal to the frontal zone between Glasgow, Mont., and Lake 
Charles, La. (fig. 5.41). Plots of A'^ versus height along this cross section 
line were obtained at 12-hour intervals during a 4-day period and con- 
verted to A units. 

Figure 5.42 is an example cross section along the Glasgow-Lake Charles 
line analyzed in terms of unmodified A'' as in the idealized cases of figures 
5.11 and 5.12. Compare this figure with the A unit analysis of figure 
5.48 for the highlighting of air mass differences refractive-index-wise. 
Examples of the distribution of A'^ components, temperature and humidity, 
around the front are charted on figures 5.43 and 5.44. Various stages 
in the advance of this intense storm system across the continent are 
represented by figures 5.45 to 5.50 in terms of modified A'^ (A units). At 
the outset of the period of observation (fig. 5.45), the polar front was 
located over the northern Great Plains, between Rapid City, S. Dak. 




Figure 5.41. Station identificalion chart. 



212 



SYNOPTIC RADIO METEOROLOGY 




1000 1500 

DISTANCE IN KILOMETERS 

GGW RAP LBF DDC OKC LIT 

Figure 5.42. Space cross section in N units, 1600Z, 19 Feb. 1962. 



2300 



LCH 




DISTANCE IN KILOMETERS 
GGW RAP LBF DDC OKC LIT 

Figure 5.43. Temperature cross section in °C, 1500Z, 19 Feb. 1952. 



LCH 



VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 213 

and North Platte, Nebr. At this time the entire cross section is charac- 
terized by weak to moderate gradients of refractivity. On figure 5.46 
12 hours later, this front had moved some 300 km southward. The 
contrast of the southward push of polar air and the northerly advection 
of tropical maritime air from the Gulf of Mexico into the developing warm 
sector of the polar front wave is evidenced by the relatively large gradients 
in the neighborhood of Dodge City. The core of tropical maritime air has 
evidently not progressed far enough northward to displace the warm but 
dry air that had been over the Great Plains prior to the outbreak, with 
the result that a region of low A values is found between the front and the 
tropical maritime air. On figure 5.47 the effects of the Pacific front are 
apparent in the buildup of a secondary region of high A some 400 km 
ahead of the polar front, located at this time over Dodge City. Twelve 
hours later (fig. 5.48) the core of tropical maritime air has become more 
extensive and now reaches to a height of 3 km. The second (Pacific) 
front is picked up now on the cross section and the area of low A values is 
confined between the two fronts. By 0300 UT on the 20th, figure 5.49, 
the polar front is approaching Lake Charles and the Pacific front is re- 
ported on the daily weather map as a squall line. Finally, by the morning 
of February 20 (fig. 5.50), both fronts have passed to the south of Lake 
Charles, and the polar air just behind the front is characterized by rela- 
tively low A values. 

The use of space cross sections does not always yield measurements 
at the most desirable points along a frontal zone. Another method of 
arriving at the probable refractive index structure about the frontal 
interface is to plot radiosonde observations for a single station arranged 
according to observation times as on figures 5.51 to 5.57. 

The time cross section for Glasgow, Mont. (fig. 5.51), which is in the 
cold air behind the polar front for the entire period of observation of the 
storm, displays gradients of A values that are generally weak. An excep- 
tion occurs on the evening of the 19th (20/0300 UT) apparently as a 
result of subsidence effects. Rapid City, S. Dak. (fig. 5.52), similar to 
Glasgow, shows generally weak gradients throughout in the cold air be- 
hind the front. North Platte, Nebr. (fig. 5.53), exhibits moderate 
gradients with increasing A values in the post-frontal, higher-density polar 
air. This station is too far north to record much in the way of moisture 
effects at this time of year. Dodge City, Kans. (fig. 5.54), represents 
dry low ahead of the polar front and increasing A values in the cold air 
just behind the front. Oklahoma City, Okla. (fig. 5.55), represents a 
classic synoptic situation in which advection of tropical maritmie air 
from the Gulf of Mexico produces a strong high ahead of the front and a 
low within the cool, dry, polar continental air. Again in the region 
around Little Rock, Ark. (fig. 5.5G), there is warm, moist air of extremely 
high refractivity ahead of the front being replaced by polar continental 



214 



SYNOPTIC RADIO METEOROLOGY 




1000 I 1500 

DISTANCE IN KILOMETERS 



2300 



GGW RAP LBF DDC OKC LIT LCH 

Figure 5.44. Relative humidity cross section (percent), 1500Z, 19 Feb. 1952. 




DISTANCE IN KILOMETERS 

I I 

GGW RAP LBF DDC OKC LIT 

Figure 5.45. Space cross section in A units, 0300Z, 18 Feb. 1962. 



LCH 



VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 215 




2000 2500 



DISTANCE IN KILOMETERS 
DODGE CITY 



GLASGOW DODGE CITY LAKE CHARLES 

Figure 5.46. Space cross section in A units, 1500Z, 18 Feb. 1962. 

3.0 




1000 I 1500 

DISTANCE IN KILOMETERS 

GGW RAP LBF DDC OKC LIT LCH 

Figure 5.47. S-pace cross section in A units, 0300Z, 19 Feb. 1962. 



216 



SYNOPTIC RADIO METEOROLOGY 




2000 2300 

DISTANCE IN KILOMETERS 

GLASGOW DODGE CITY LAKE CHARLES 

Figure 5.48. Space cross section in A units, 1500Z, 19 Feb. 1962. 




1000 I 1500 

DISTANCE IN KILOMETERS 

GGW RAP LBF DDC OKC LIT 

Figure 5.49. Space cross section in A units, 0300Z, 20 Feb. 1962. 



2300 



LCH 



VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 217 




DISTANCE IN KILOMETERS 
I 
GLASGOW DODGE CITY LAKE CHARLES 

Figure 5.50. Space cross section in A units, 1500Z, 20 Feb. 1962. 




I80300Z I8I500Z I90300Z 



I9I500Z 200300Z 

DAY AND HOUR 



20I500Z 2I0300Z 2II500Z 



Figure 5.51. Time cross section, Glasgow, Mont., in A units. 



218 



SYNOPTIC RADIO METEOROLOGY 



3.0 



2.5 



2.0 — 



1.5 — 



0.5 



I80500Z I8I500Z 





/ \ 


1 / 


' X 




y \ 


/ / 


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r"\ 









U ^ 


^ 


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\ 


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//3?0^ 


— — 



I90500Z 



I9I500Z 200300Z 

DAY AND HOUR 



20I500Z 2I0300Z 



2II500Z 



Figure 5.52. Time cross section, Rapid City, S. Dak., in A units. 



3,000 




I80300Z 



20I500Z 



I8I500Z I90300Z I9I500Z 200300Z 

DAY AND HOUR 
Figure 5.53. Time cross section, North Platte, Nebr., in A units 



2I0300Z 2II500Z 



VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 219 




180300Z iSISOOZ I90500Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z 

DAY AND HOUR 
Figure 5.54. Time cross section, Dodge City, Kans., in A units. 



3,000 



2,500 



O 




I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z 

DAY AND HOUR 
Figure 5.55. Time cross section, Oklahoma City, Okla., in A units. 



220 



SYNOPTIC RADIO METEOROLOGY 




I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z 

DAY AND HOUR 
Figure 5.56. Time cross section, Little Rock, Ark., in A units. 



3,000 





I80300Z 



I8I500Z 



I90300Z 



I9I500Z 200300Z 

DAY AND HOUR 



20I500Z 



2I0300Z 2II500Z 



Figure 5.57. Time cross section, Lake Charles, La., in A units. 



VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 



221 




-2700 -2500 



DISTANCE FROM FRONT IN KILOMETERS 
Figure 5.58. Epoch chart, Glasgow, Mont., in A units. 



air of characteristically low A value. The time cross section for Lake 
Charles, La. (fig. 5.57), is complex but represents again the same general 
features: high A values ahead of the front and low ones behind. 

The exponential correction to the refractive index height distribution 
used in this storm series allows air mass properties to be clearly seen. By 
use of such an exponential correction, one may construct an idealized 
refractive index field about a frontal transition zone that shows the tem- 
perature and humidity contrasts of the different air masses. Further, 
when this technique is applied to the analysis of a synoptic tropospheric 
disturbance, it does indeed highlight air mass differences. 

The time cross section presentation is referred to as an epoch chart 
when the observations are presented as plus or minus time deviations with 
respect to the frontal passage. Thus, as a frontal system advances and 
passes over a station, one obtains yet another perspective of the space 
cross section. Such a presentation is given on figures 5.58 to 5.64. 
Figure 5.59 represents a typical continental station located in the polar 



222 



SYNOPTIC RADIO METEOROLOGY 



continental air mass throughout the occurrence of the storm. The essen- 
tial feature here, as in figure 5.52, is the absence of detail of A structure 
due to the presence of a uniform air mass over this station. Compare 
this figure with the epoch chart for Oklahoma City (fig. 5.62), where the 
structure of the idealized model is clearly reflected by the prefrontal A 
unit high, strong gradient across the frontal zone, and the A unit low 
behind the front. This rather fortuitous agreement is felt to be due to 
the strategic location of Oklahoma City with respect to the motion of 
contrasting air masses about the polar front. That is, this epoch chart 
represents a point of confluence of virtually unmodified polar continental 
and tropical maritime air. 




//f //f //f //f I If //f I If ijf ijf ijf ijf ijf ijf ijf ijfijf ///- // 



a5 



LEGEND 
A ISOPLETH 
COLD FRONT 



J \ L 



J \ L 



-2000 



-1500 



-500 



-200 



DISTANCE FROM FRONT , km 
Figure 5.59. Epoch chart, Rapid City, S. Dak., in A units. 



SUMMARY 



223 




■1600-1500 



-500 

DISTANCE FROM FRONT IN KILOMETERS 

Figure 5.60. Epoch chart, North Platte, Nebr., in A units. 



300 



5.8. Summary 

A survey such as this is designed to indicate the direction of radio 
meteorological research. Among the brighter prospects is the work of 
Moler et al., concerning the interrelation of refractive index structure and 
mesoscale weather changes. Other synoptic features likely to have refrac- 
tive index significance are the migratory high and low pressure cells of the 
middle latitudes, since certain surface patterns and vertical profiles occur 
frequently with a particular type of synoptic system. 

A knowledge of the meso-macroscale behavior of A'^ in a synoptic sense 
enables the propagation engineer to anticipate the occurrence of super- 
refraction or ducting conditions. Thus, meteorological conditions that 
give rise to such phenomena as prolonged space-wave fadeouts and inter- 
ference effects from elevated layers in the troposphere can to some extent 
be planned for in advance. 



224 



SYNOPTIC RADIO METEOROLOGY 




DISTANCE FROM FRONT IN KILOMETERS 
Figure 5.61. Epoch chart, Dodge City, Kans., in A units. 



5.9. References 

1] Sheppard, P. A. (1947), The structure and refractive index of the atmosphere, 

Book, Meteorological Factors in Radio Wave Propagation, (Phys. Soc. and 

Roy. Meteorol. Soc, London, England). 
2] Gerson, N. C. (1948), Variations in the index of refraction of the atmosphere, 

Geofis. Pura Appl. 13, 88. 
3] Perlat, A. (1948), Meteorology and radioelectricity, L'Onde Elec. 28, 44. 
4] Randall, D. L. (1954), A study of the meteorological effects on radio propagation 

at 96.3 Mc between Richmond, Va., and Washington, D.C., Bull. Am. Meteorol. 

Soc. 35, 56-59. 
5] Misme, P. (1957), Influence des discontinuites frontales sur le propagation des 

ondes decimetriques et centimetriques, Ann. Telecommun. 12, 189-194. 
6] Hay, D. R. (1958), Air-mass refractivity in central Canada, Can. J. Phys. 36, 

1678-1683. 
7] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception, 

J. Atmos. Terrest. Phys. 1, 32. 
8] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception 

at 44.1 Mc with 92.1 Mc over the 167-mile path of Alpine, N.J., to Needham, 

Mass., Proc. IRE 38, 1450. 
9] Bean, B. R. (1956), Some meteorological effects on scattered radio waves, IRE 

Trans. Commun. Syst. CS-4, No. 1, 32. 



REFERENCES 



225 




///-" IJF IJF IJF //F//F IJF IJF I If ijh l/h ///- /, 



LEGEND 

A ISOPLETHS 
COLD FRONT 



// 



-500 500 

DISTANCE FROM FRONT , km 

Figure 5.62. Epoch chart, Oklahoma City, Okla., in A units. 



[10] Onoe, M., M. Hirai, and S. Niwa (1958), Results of experiment of long-distance 

overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79. 
[11] Hull, R. A. (1935), Air-mass conditions and the bending of ultra-high-frequency 

waves, QST 19, 13-18. 
[12] Hull, R. A. (1937), Air-wave bending of ultra-high-frequency waves, QST 21, 

16-18. 
[13] Englund, C. R., A. B. Crawford, and W. W. Mumford (1938), Ultra-short-wave 

radio transmission through the non-homogeneous troposphere, Bull. Am. 

Meteorol. Soc. 19, 356-360. 
[14] Yerg, D. G. (1950), The importance of water vapor in microwave propagation at 

temperatures below freezing, Bull. Am. Meteorol. Soc. 31, 175-177. 
[15] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (1933), Ultra-short-wave 

propagation, Proc. IRE 21, 427-463. 
[16] Misme, P. (1960), L'influence du gradient equivalent et de la stabiHt6 atmos- 

ph6 rique dans les Haisons transhorizon au Sahara et au Congo, Ann. Telecom- 

mun. 16, 110. 
[17] Misme, P., B. R. Bean, and G. D. Thayer,(1960), Comments on "Models of the 

atmospheric radio refractive index," Proc. IRE 48, 1498-1501. 
[18] Gray, R. E. (1957), The refractive index of the atmosphere as a factor in tropo- 

spheric propagation far beyond the horizon, IRE Nat. Convention Record, 

Pt. 1, 3. 



226 



SYNOPTIC RADIO METEOROLOGY 




-500 500 

DISTANCE FROM FRONT IN KILOMETERS 

Figure 5.63. Epoch chart, Little Rock, Ark., in A units. 



1000 1700 



[19] Gray, R. E. (1961), Tropospheric scatter propagation and meteorological condi- 
tions in the Caribbean, IRE Trans. Ant. Prop. AP-9, No. 5, 492-496. 

[20] Bean, B. R., and G. D. Thayer (1959), On models of the atmospheric radio 
refractive index, Proc. IRE 47, No. 5, 740-755. 

[21] Schulkin, M. (1952), Average radio-ray refraction in the lower atmosphere, Proc. 
IRE 40, No. 5, 554-561. 

[22] Fannin, B. M., and K. H. Jehn (1957), A study of radio elevation angle error due 
to atmospheric refraction, IRE Trans. Ant. Prop. AP-2, No. 1, 71-77. 

[23] Bean, B. R., J. D. Horn, and L. P. Riggs (1960), Refraction of radio waves at 
low angles within various air masses, J. Geophys. Res. 65, 1183. 

[24] Arvola, W. A. (1957), Refractive index profiles and associated synoptic patterns, 
Bull. Am. Meteorol. Soc. 38, No. 4, 212-220. 

[25] Jehn, K. H. (1960), The use of potential refractive index in synoptic-scale radio 
meteorology, J. Meteorol. 17, 264. 

[26] Lukes, G. D. (1944), Radio meteorological forecasting by means of the thermo- 
dynamics of the modified refractive index, Third Conf. Prop., NDRC, pp. 
107-113 (Committee on Propagation, Washington, D.C.). 

[27] Katz, I. (1951), Gradient of refractive modulus in homogeneous air, potential 
modulus, Book, Propagation of Short Radio Waves, pp. 198-199 (McGraw- 
Hill Book Co., Inc., New York, N.Y.). 

[28] Jehn, K. H. (1960), Microwave refractive index distributions associated with 
the Texas-Gulf cyclone, Bull. Am. Meteorol. Soc. 41, 304-312. 



REFERENCES 



227 




-300 



500 

DISTANCE FROM FRONT IN KILOMETERS 

Figure 5.64. Epoch chart, Lake Charles, La., in A units. 



[29] Jehn, K. H. (1961), Microwave refractive-index distributions associated with the 

central United States cold outbreak, Bull. Am. Meteorol. Soc. 42, 77-84. 
[30] Flavell, R. G., and J. A. Lane (1962), The appUcation of potential refractive 

index in tropospheric wave propagation, J. Atmospheric Terrest. Phys. 24, 

47-56. 
[31] Moler, W. F., and W. A. Arvola (1956), Vertical motion in the atmosphere and its 

effects on VHF radio signal strength, Trans. Am. Geophys. Union 37. 
[32] Moler, W. F., and D. B. Holden (1960), Tropospheric scatter propagation and 

atmospheric circulations, J. Res. NBS 64D (Radio Prop.), No. 1, 81-93. 
[33] Gossard, E. E., and L. J. Anderson (1956), The effect of super-refractive layers on 

50-5,000 Mc nonoptical fields, IRE Trans. Ant. Prop. AP-4, 175-178. 
[34] Megaw, E. C. S. (1950), Scattering of electromagnetic waves by atmospheric 

turbulence. Nature 166, 1100-1104. 
[35] Booker, H. G., and W. E. Gordon (1950), A theory of radio scattering in the 

troposphere, Proc. IRE 38, 401-412. 
[36] Saxton, J. A. (1951), Propagation of metre radio waves beyond the normal 

horizon, Proc. lEE 98, 360-369. 
[37] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic charts and data 

of the radio refractive index for the United States and the world, NBS Mono. 22. 
[38] Smyth, J. B., and L. G. Trolese (1947), Propagation of radio waves in the tropo- 
sphere, Proc. IRE 35, 1198. 



228 SYNOPTIC RADIO METEOROLOGY 

[39] Bean, B. R., and G. D. Thayer (1959). CRPL exponential reference atmosphere, 

NBS Mono. 4. 
[40] Bean, B. R., and E. J. Button (1960), On the calculation of the departures of 

radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3, 

259-263. 
[41] Canterbury Project (1951), Vols. I-III (Department of Scientific and Industrial 

Research, Wellington, New Zealand). 
[42] Bean, B. R., and L. P. Riggs (1959), Synoptic variations of the radio refractive 

index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97. 
[43] Bean, B. R., L. P. Riggs, and J. D. Horn (1959), Synoptic study of the vertical 

distribution of the radio refractive index, J. Res. NBS 63D (Radio Prop.), 

No. 2, 249-258. 
[44] Palmer, C. E. (1957), Some kinem.".!;ic aspects of frontal zones, J. Meteorol. 14, 

No. 5, 403-409. 



Chapter 6. Transhorizon Radio- 
Meteorological Parameters 

6.1. Existing Radio-Meteorological Parameters 

6.1.1. Introduction 

A method of predicting the statistical distribution of field strength on 
transhorizon paths is an important requirement m tropospheric wave 
propagation. Consequently, considerable attention has been given in 
recent years (see figs. 6.1 to 6.6) to studies of the correlation between the 
measured signal level (e.g., the monthly median value) and some quantity 
derived from surface or upper air meteorological data. Figures 6.1 to 6.6 
show relationships between field strength and various meteorological 
parameters. It can be seen that there are some quite marked similarities 
(high correlation coefficients) between the two. It thus appears that if a 
reliable "radio-meteorological parameter" could be developed, then 
generally available meteorological data would replace expensive radio 
measurements in deriving the required distribution. 

Progress has already been made in this difficult problem, [1, 2]^ In 
these investigations, special attention has been given to two parameters: 
(a) the surface value of refractivity, N s, and (b) the difference, AN, 
between A''^ and N at a height of 1 km, Ni. Other groups have studied 
different parameters [3, 4, 5, 6, 7, 8], either as possible alternatives to 
N s and AN in the prediction process or as quantities which clarify the 
effect of meteorological features, such as anticyclonic subsidence, on 
signal strength. It is evident from the literature that some difference of 
emphasis exists regarding the relative merits of the parameters proposed 
to date, and particularly on the value of studies of N s- 

This chapter provides a critical survey of the present position in this 
field of radio meteorology, and indicates a new approach which incor- 
porates some aspects of all existing treatments. Section 6.1 contains a 
study of previous work and attempts to put the various views in proper 
perspective; section 6.2 discusses some selected radio data from VHF 
paths and its classification in terms of refractive index profiles, while 
section 6.3 introduces a parameter combining the concepts of refraction 
and atmospheric stability, and compares its properties with those of 
existing parameters. 



1 Figures in V)rackets indicate the literature references on p. 2()t). 

22!) 



230 



TRANSHORIZON PARAMETERS 




A S 6 6 H J F M A M J J A S S H 
6 3 I 29 26 24 21 18 17 14 12 9 7 4 I 29 27 24 22 
1947 1946 



0.0 



Figure 6.1. Weekly means of measured field intensities of W2WMN and W2XEA at 
Needham, Mass., compared with corresponding atmospheric surface refraction at 
Boston, Mass. 

(After Pickard and Stetson, 1950). 









1 






















m 








A 


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r - 0,76 
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20- 
























/ 














R 


= U,»i 


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IX — 



























oofnAMOLASO 



Figure 6.2. Correlation of the received monthly mean field strength at Campu Sa Spina 
and the gradient AN {between and 1000 m.) of the monthly median index of refraction 
at Elmas. 
(The values of field strength are those obtained after omission of superrefraction) . (After Bonavoglia, 1958) . 



INTRODUCTION 



231 











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(a) 





























168 



176 



58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 
52 

50 

48 

46 
44 
42 
40 
38 

36 

ONDJFMAMJJ ASONOJFMAMJJASOND 
1950 1951 1952 

Figure 6.3. Comparison of the monthly median basic transmission loss and refradivity 
gradient for KIXL-FM, Dallas, Tex., recorded at Austin, Tex. 

























I 1 

9 inn 


I I 

f^ CT 








































^ 






_ rr^nn n.KAT 








































/ 


1 












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SI 
























^, 




























I 


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-AN— 


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*i 






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f 


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7 














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y\ 




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1 


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i-b- 






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(b) 





























178 




180 




182 
184 


X) 
Q 


186 


c 


172 


_l 


174 




176 




178 




180 




182 




184 




186 




188 





The development of prediction techniques is especially difficult in the 
case of: 

(a) any path with terminals just v/ithin or just beyond the normal 
horizon, and 

(b) signal enhancements which occur for small percentages of the time. 
Section 6.2 therefore includes some discussion of propagation charac- 
teristics on VHF paths for which considerable radio and meteorological 
data are already available. The results obtained are particularly relevant 
to an understanding of the large differences observed between median 



232 



TRANSHORIZON PARAMETERS 




1945 



1946 



Figure 6.4. Field reception of W2XMN on 42.8 Mc/s at Needham, Mass., 1945-1946. 

(After Pickard and Stetson, 1947). 



signal levels on VHF paths of comparable length, frequency and angular 
distance. 

The primary purpose of a radio-meteorological parameter is to provide 
the best estimate of the statistical distribution of field strength (in terms of 
hourly, daily, weekly, or monthly median values as required on a specific 
path). The reliability of the parameter must be judged solely in terms of 
this requirement, and care must be exercised in assessing the value of any 
given parameter in t^rms of data obtained over limited intervals of time 
or from restricted geographical areas. Discussions later in the chapter 
will consider to what extent it is possible to develop a parameter which, 
in addition to being statistically reliable, is also characteristic of the 
physical structure of the atmosphere. 

The present practice in applying radio-meteorological parameters con- 
sists in determining an average signal level for a given distance and then 
adjusting this average level for climatic and seasonal differences by refer- 
ence to the changes in some function of the refractive index of the atmos- 
phere. We may express this procedure mathematically, for a linear 
regression model, thus: 



E = b ■ f{n) + a 



(6.1) 



where E is the field strength, and b is the regression coefficient expressing 



INTRODUCTION 



233 





-1.0 












1 1 


1 1 1 1 1 






-0.8 
-0.6 


— 


1 * ° 


° ° °^ 




J^-^^"^^ y - 








A""^ 


° < o" o ESTIMATED AVERAGE 






-0.4 


— 


^ \ 


' ° I r ^. CORRELATION COEFFICIENT 






-0.2 


" 


\, 


° 




Ll_ 
CI3 





— 


" 1 






Ll_l 


0.2 











1_1_J 


0.4 


— 










0,6 


— 


ESTIMATED I 


° US. DATA 
« GERMAN DATA 






0.8 




MEASUREMENT 
ERROR 1 








1.0 




1 1 


1 1 1 1 1 


3 


3 


2 


4 6 




sib), dB 1 



Figure 6.5. Correlation coefficients of seasonal cycles o/ Ng and Lb: night 
{2000, 2200, 2400) versus the standard deviation of Lb- 



the sensitivity of £" to a unit change in f{n). The intercept, a, is a func- 
tion of path length, antenna heights, and terrain characteristics, and can 
be derived from existing prediction procedures [9]. Comments on some 
results obtained are given in the remainder of this section. 

6.1.2. Parameters Derived From the N-Profile 

It has long been recognized that the variations in field strength on 
transhorizon paths are intimately connected with changes in the vertical 
gradient of refractive index over the path. Although our knowledge of 
the detailed fluctuations in refractive index in the troposphere is still in- 
adequate for many requirements, it is nevertheless possible to relate, 



234 



TRANSHORIZON PARAMETERS 



165 



— I — I — I — I — I — I I I r 

L(,- -0.230/^^*245.87 
= 0.868 X 




WJAS FM -PATH 12 

TRANSMITTER: PITTSBURGH, PA. 

RECEIVER: STATE COLLEGE, PA. 

DISTANCE: 188.5 KILOMETERS 

fi: 30.44 MILLIRADIANS 



295 300 305 310 315 320 325 330 335 340 350 

Ne 




35 40 45 50 55 30 35 40 45 50 

ANn, 



165 






AIMq.i 










1 


1 1 




1 






^h 


= 0.529 


/iNo 3*196.24^ 




X 


^ 


170 




'l^-an' 


0.871 -^ 


^ 


X 


-x8 


175 


^ 




1 1 









Lfy=0.595A Nqs * 198.80 



60 30 




165 




ai 


^0.3 








1 




1 






^b 


-0.760 aNqj* 204.54^ 


X 


^ 


170 


- 


rLi,-AN-0.866 


"X 


>^ 


x_ 






\ 


l^ 


XX 




175 


' 


XX ^ 

X ^,jy^ 


y"^ X 

1 


1 







ANo.5 




1 

L^=0.668aNio 


1 
*20l.32 


X 


'■li,-an= 0.873 ^ 


l^y"^- 


X 


8> 


-^r X 


XX X 




- 


^y»* 


1 


1 




Figure 6.6. Monthly median basic transmission loss versus Ns or ANh /or a path from 
Pittsburgh, Pa., to Stale College, Pa. 



PARAMETERS DERIVED FROM THE iV-PROFILE 235 

statistically, changes in signal level and functions of a parameter derived 
from routine surface and upper-air measurements of pressure, tempera- 
ture, and humidity. 

Following the work of Pickard and Stetson [10], N , and AA'' have been 
the subject of detailed studies by several workers [1, 2, 11, 12, 13, 14]. 
Only the more important conclusions are summarized here. Values of the 
correlation coefficient, r, relating monthly median values of either A^^ or 
AA'' and field strength, derived from a number of paths in diverse climatic 
conditions [2], range from 0.4 to 0.95 with a median value of about 0.7. 
An analysis has also been made of the results obtained by using (a) values 
of AA'' obtained from the surface readings and at heights other than 1 km, 
and (b) values of AN between different levels on the profile up to a height 
of 3 km. A comparison of measured field strength at frequencies near 
100 Mc/s on 20 paths, 130 to 446 km long, located in various part of the 
United States, yields the following result: the use of N s gives as good a 
correlation as any of the AA^ values, due to the high correlation between 
the surface value and these differences. The values of r relating monthly 
median values of A''^ and AA'' (the decrease in the first kilometer) have 
been obtained for the United States [15], France, [16], Germany, [17] and 
the British Isles [18]. They range from 0.60 to 0.93. Moreover, the data 
are consistent with the assumption of a reference atmosphere in which A'^ 
decreases exponentially with height [15]. These results lead at once to 
a consideration of the value of N s in predicting the geographical variation 
of monthly median field strength. This question is considered in more 
detail later in this section. 

It has been shown that, where only past radio or meteorological results 
are available, one obtains at least as good a prediction of the diurnal and 
seasonal variations of field strength from long-term meteorological data as 
from relatively short-term (say 1 year's) radio data. The annual cycle 
may be represented by a single regression coefficient of 0.18 dB/A^'-unit 
for either night or day; however, the regression coefficients for the diurnal 
cycles lie between 0.2 and 1.1 dB/N-umt and vary with distance and 
season, being greatest for paths between 175 and 200 km long and for the 
winter months. The possibility of predicting the variation of hourly 
median values of field strength within any given month by combining the 
.seasonal and diurnal correlations has also been discussed [14]. It is recog- 
nized that the development of a prediction procedure based on this ap- 
proach must account for the complex sensitivity of field strength, E, to 
changes in A'^s; this requirement in turn leads to considerations of season, 
climatic region, distance and frequency. The results for the distance 
dependence of the E-N s regression coefficient are, of course, intimately 
connected with the propagation mechanism. In particular, in the case 
of VHF paths about 200 km long there will usually be components in the 
received field due to diffraction around the earth's .surface and scattering 



236 TRANSHORIZON PARAMETERS 

from randomly dispersed eddies; on occasions there will also be a semi- 
coherent field arising from partial reflection at elevated stable layers. 
Some aspects of this complex situation are discussed in section 6.2. 

To sum up other important conclusions reached by the authors men- 
tioned above, the correlation between E and N s 

(1) increases with increasing variation of E or N s, 

(2) is greater for seasonal cycles of night-time values of the variables 
than for the midday values, and 

(3) is greater for summer diurnal cycles than for winter ones. 
Conclusions (1) and (2) are particularly important if we try to assess the 
utility of A^s in prediction work in terms of signal data from areas in which 
the variation of E is small (e.g., Western Europe), or in terms of data for 
afternoon periods only. 

Another parameter, closely related to AN, is the "equivalent gradient," 
Qe, proposed by Misme [19]. This is defined as that linear decrease of n 
with height which produces the same amount of bending as the actual 
inhomogeneous atmosphere over a given transmission path. The prob- 
lem is illustrated in figure 6.7, where the dotted line represents the actual 
ray path between T and P above an earth of radius a. It is required to 
determine the curvature, p, of the circular, full-line path which is tangen- 
tial to the real path at P and which corresponds to a constant value of 
dn/dh in a fictitious atmosphere. Boithias and Misme [20] have described 
a graphical method for calculating Qe ( = l/p), for example, when P is 
located in the center of the common volume of the antenna beams. 
Monthly median values of ^^ are obtained from the corresponding monthly 
median values of dn/dh, and Misme [19] has given tables of Qe for various 
path lengths for different months. 

It is evident that Qe, like AA^", expresses the amount of refraction pro- 
duced by the atmosphere and one might expect a high correlation between 
the two parameters. This is known to be the case in some results quoted 
by Misme [19] and which are illustrated in figure 6.8. Here the monthly 
median values of AN and Qe (for a 300 km path) are compared, together 
with the variation in N s, for Leopoldville in the Congo area. The corre- 
lation between Qe and AA^ is high, but the maximum values in Ns in 
December and January are accompanied by a local minimum in the 
values of Qe and AA^. These results, and similar ones from Dakar (W. 
Africa), have been quoted in support of arguments that N s is of limited 
value in predicting seasonal and geographical variations in field strength 
[19]. It is essential, Misme claims, to investigate the nature of the 
N ,-AN correlation in separate climatic areas; the correlation seems poor 
for some equatorial climates, probably because of the presence of semi- 
permanent elevated layers. In these conditions, Alisme feels that an 
exponential reference atmosphere and a correlation between N s and AA^ 
are not to be expected. 



PARAMETERS DERIVED FROM THE A^-PROFILE 



237 



E 

.3^ 




Figure 6.7. Geometry of the equivalent gradient. 

60 . 1 370 



50 



g 40 



30 






AN 

ge 



350 



330 



J A S N D J F M A M J 
MONTH (JULY-JUNE) 

Figure 6.8. Comparison of monthly medians of ge, AN, and Na 
(Leopoldville, Congo) 



310 



238 TRANSHORIZON PARAMETERS 

While admitting the importance of an explanation of the results dis- 
cussed above, it is difficult at present to accept the argument that Qe 
should be used instead of AA^. Indeed, it has been shown [19] that for 
a 471-Mc/s link, 160 km long, located in the Sahara region, there is a 
difference of 20 dB between the signal levels exceeded for 99 percent of 
the time during daylight hours in January and June, whereas the January 
to June variation in Qe is less than 2A^/km. This result indicates the in- 
fluence of atmospheric stability on signal strength. We defer, therefore, 
further consideration of Qe until later in the chapter merely noting at this 
stage that its derivation requires more detailed calculation (i.e., ray 
tracing) than in the case of AA^, without any appreciable benefit. 

6.1.3. Parameters Involving Thermal Stability 

The role of stable elevated layers in tropospheric propagation has been 
discussed by a number of scientists; for example, by Saxton [21], Friis, 
Crawford, and Hogg [22], and in work done by the French [23] in an 
important series of papers. These contributions have stimulated interest 
in the development of radio-meteorological parameters which depend, in 
part at least, on thermal stability, and this section considers some of the 
characteristics of these parameters. 

6.1.4. Composite Parameter 

Misme [19] has discussed the Sahara results mentioned above in rela- 
tion to the theoretical work of Voge [23]. In considering the effect of 
elevated layers, Voge introduces a parameter, t], defined by: 



area of one or several layers at a given height 

V — : : r~. 

the horizontal area, at the same height, visible from 

receiver and transmitter 



The value rj is an increasing function of the atmospheric stability measured 
between two levels in the atmosphere. Stability may be defined as the 
work, ATF, required to raise a unit mass of air from one level to another. 
We have, therefore: 

7] tends to as A IF tends to 0. 
■q tends to 1 when AW is large. 

The values of AW (in joules per gram) for Aoulef (Sahara) are given 
by Misme for the January and June months in which measurements were 
made over the 160 km path at 471 Mc/s. The relevant data are sum- 
marized in table 6.1. 



COMPOSITE PARAMETER 239 

Table 6.1. Influence of thermal stability on signal level 

(d = 160 km: f = 471 Mc/s: North Sahara Region) 





Jan. 


June 


(/. (-N/km) 


34.0 


32.5 


AW for 0. 7 to 2 km - 


0.17 


0. 0024 






Relative signal level, 99 percent value, (dB) 


20.0 


0.0 



The dominant propagation mechanisms in January and June are 
thought to be "diffuse reflection" and "scattering," respectively, and on 
this basis Misme uses the equations given by Voge to calculate the ex- 
pected difference in the January and June signal levels. With various 
assumptions regarding the properties of stable and turbulent layers, a 
value of 19 (±5) dB is obtained for the ratio of predicted field strengths in 
January and June, in good agreement with the measured value exceeded 
for 99 percent of the time. This analysis is facilitated by the fact that the 
two selected months are characteristic of well-defined climatic situations 
in the Sahara region. Nevertheless, the results indicate that a parameter 
which combines the concepts of the equivalent gradient and thermal 
stability may be of general application. Misme [19] has therefore sug- 
gested a parameter M of the form: 

M = a ige - iO -\- b [AW]") (6.2) 

where a, h, and n are constants, Qe is the equivalent gradient, and AW is 
the thermal stability defined above for a 1 km height interval. With 
a = 0.5 dB/N/km, M provides an estimate of the variation in field 
strength, E, in decibels, caused by changes in equivalent gradient and 
stability. It remains to define E in a "standard" atmosphere by selecting 
a mean value of stability and a suitable value for the term h. 

This composite parameter is thought by Misme to be more representa- 
tive of the real atmosphere (especially in tropical areas) than A^^ and AA^, 
and it is certainly of great potential value. However, only fragmentary 
radio data are available for purposes of comparison in the references 
quoted, and here again the precise value of the parameter in prediction 
work can only be determined by a more comprehensive study. 

6.1.5. Potential Refractive Index (or Modulus), K 

This parameter is defined as the value of A'^ = (n— 1)10*^ which an air 
mass would have if brought adiabatically to a standard pressure, say 
lOOOmbar, assuming a constant humidity mixing ratio. Examples of the 
use of K in radio meteorology have been given by Katz [24], and by Jehn 



240 TRANSHORIZON PARAMETERS 

[25]. It has also been applied by Flavell and Lane [6] in studies of the 
effect of anticyelonic subsidence on tropospheric propagation. Values of 
K can be derived very rapidly from upper-air data and, since plots of K 
against height or pressure do not exhibit the large systematic decrease of 
A^ with height in the conventional N{h) profile, the structure and motion 
of meteorological features are clarified considerably. (Below the conden- 
sation level, a lapse rate of — 20A''/km = dK/dh = 0.) In this respect, 
K is superficially similar to the A unit [26] derived from the exponential 
reference atmosphere. However, no quantitative results are yet avail- 
able with this parameter and, furthermore, the method of deriving K 
assumes a dry adiabatic lapse rate and a constant humidity mixing ratio. 
The presence of a condensation level in the actual atmosphere is therefore 
neglected. In addition, accurate values of A'' can only be derived directly 
from the K profile in certain restricted conditions. Pending further 
studies in this direction therefore, K and dK/dh remain more suitable for 
qualitative synoptic studies than for quantitative predictions of field 
strength variations. However, it is of interest to note that a close con- 
nection exists between K and the composite parameter M discussed above. 
Misme [27] has shown that the change of K in a given height interval, 
AK/Ah, is of the form: 

AK/Ah = ki AN/ Ah - k2AW/Ah (6.3) 

where ki and ko are constants, and AW is a measure of thermal stability 
as defined earlier. 



6.1.6. Vertical Motion of the Atmosphere 

The influence of stability has also been discussed by Moler and Arvola 
[7], and Moler and Holden [8]. These authors assume that the average 
lapse rate, dn/dh and the magnitude of local irregularities on the profile, 
are primarily determined by changes in vertical velocity. Local centers 
of convergence (low pressure cells) produce updrafts which result in con- 
siderable mixing and the dissipation of any stable layer structure. Hori- 
zontal divergence from local high-pressure centers create temperature 
inversions and associated layer-type discontinuities in the n-profile. 
These latter features are most pronounced in conditions of anticyelonic 
subsidence. 

The direction and relative magnitude of the vertical component of 
wind velocity can be estimated by techniques outlined by ]\Ioler and 
Holden, and a correlation between hourly median field strength and calcu- 
lated values of vertical velocity has been noted by these authors. These 
fundamental studies represent an important attempt to explain signal 
variations in terms of atmospheric motion, and a survey of available ex- 
perimental evidence supports the basic assumptions in this approach. It 



VERTICAL MOTION OF ATMOSPHERE 241 

is particularly interesting to consider whether local centers of convergence 
and divergence, ill-defined on Daily Weather Reports, can be distin- 
guished by observations with microbarographs and whether the results 
are correlated with signal characteristics. Some results of such an investi- 
gation, forming part of a more general study of VHF propagation in the 
United Kingdom, are shown in figure 6.9. The signal records obtained 
at a frequency of 186 Mc/s on a 140 km path are shown for 12 September 
1959 together with the pressure variations recorded at transmitting and 
receiving sites. The steady high signal at 0900 hours begins to fall as the 
pressure at the transmitter begins to decrease, and at about 1030 hours 
the median level and fading rate change abruptly as the pressure at the 
midpoint falls. Between 1200 and 1700 hours the pressure over the whole 
path is falling, and the signal characteristics remain essentially uniform. 
However, a significant increase in median level and a reduction in fading 
rate are evident at 1700 hours as the pressure begins to increase at the 
transmitter. By about 2300 hours the pressure at the center of the path 
has reached a steady value and the signal level is again high with negligi- 
ble fading. The total barometric variation between 0900 and 2200 hours 
is only 2 mbar. 

There is cbviously a fruitful field of radio-meteorological study sug- 
gested by the work outlined above, but we have to conclude that the 
results obtained, while extremely valuable in clarifying a qualitative 
relationship between signal strength and vertical motion (or stability), 
are not immediately applicable in the problem of predicting field strength 
variation. 



6.1.7. Discussion of the Parameters 

The above review emphasizes the need for a critical inspection of 
representative data, radio and meteorological, with the objective of com- 
paring the merits of the several parameters and explaining, if possible, 
some of their relative merits and limitations. Some relevant points have 
been mentioned already, and these issues are developed m more detail in 
the following discussion. 

It is important to bear in mind that in the particular problem of predict- 
ing field strength changes we are concerned with a statistical relationship 
between two quantities, the median signal level and a radiometeoro- 
logical parameter. It may be advantageous, of course, to investigate 
the merits of special parameters developed for particular climatic 
areas; for example, the value of the gradient, dn/dh, in a semipermanent 
elevated layer in the trade winds area. Nevertheless, such a parameter 
must also be statistically "reliable" if it is to be applied in prediction 
analysis. 



242 



TRANSHORIZON PARAMETERS 



IS SEPTEMBER I9S9 




2ioC e.M.T. 



1019 



e 1018 ^^- — -^^ 



D 
-Q 

E 

UJ 
(T 

Z) 
V) 

en 

UJ 

(r 
a. 



lOIT 



1016 



1015 









t\ 


\ 
\ 


/^ 




s 


/ / 
/ / 
/ / 
/ 

/ 


CLEAR SKY 

1 


CLOUD 

i 
1 1 


/ 

CLEAR SKY 

1 

1 1 



0600 



1200 
TIME (GMT) 



1800 
12 SEPT 1959 



2400 



Figure 6.9. Signal records and pressure variations at transmitter, T, and receiver, R, 
on a 140 km path; f = 186 Mc/s. 



COMPARISON OF SOME PARAMETERS 243 

6.1.8. Comparison of Some Parameters 

It is probably fair to say that, at present, the greatest interest is con- 
centrated on the relative merits of N s, AN, and Qe, for these quantities 
have received more detailed attention than any others. (It might be 
argued that, since the initial gradient of refractive index with height has a 
strong influence on refraction, this quantity should be highly correlated 
with signal level. However, the errors of measuring the initial gradient 
by radiosonde techniques effectively mask the correlation with the signal.) 
In a comparison of N s, AN, and Qe, the monthly median values of trans- 
mission loss, L, were determined for the hours of radiosonde ascents for 
20 radio paths in various parts of the United States. 

The equivalent gradient, Qe, was then calculated using standard ray- 
tracing techniques [15] assuming: 

(a) the actual antenna heights, 

(b) a smooth earth with the sea-level radius, 

(c) horizontal stratification of A'^ with a vertical distribution the same 
as that of the monthly mean. 

Standard statistical methods were then employed to determine the cor- 
relation between monthly median values of (a) E and Qe, (b) E and N s, 
and (c) E and AA'^ (surface to 1 km height). The average values of these 
correlation coefficients for the 20 paths are given in table 6.2. The paths 
studied in this comparison were the same as those listed by Bean and 
Cahoon [2] in their analysis of N ^ —AN correlation. The data used were 
representative of climatic conditions ranging from those of New England 
and the Great Lakes area to Texas and the Pacific coast. This investi- 
gation, at least, would seem to justify existing prediction procedures 
based on A^^ and AA^. It is valuable, however, to consider some of the 
results for specific paths in more detail, since they illustrate some features 
of interest directly relevant to the arguments concerning the value of 
N s and AA'^. 

Table 6.2. Average correlations, r(z), of refraction variables with field strength 
(20 paths, 130 to 446 km long,/=92 to 106 Mc/s) 



Variable 


ge 


N, 


AiV 


r(z) 


0.59 


0.70 


0.71 



6.1.9. Some Exceptions and Anomalies 

In so complex a matter as field-strength prediction, it would be un- 
reasonable to expect any radio-meteorological parameter to be of world- 
wide application to a uniform degree of reliability, and some conditions 
have already been quoted in which N s may not be a reliable parameter. 
The significance of these examples is really the issue on which views 



244 TRANSHORIZON PARAMETERS 

diverge most at the present time. Unfortunately, adequate radio data 
are not available for several areas of interest (e.g., equatorial Africa) and 
in these cases we can only propose tentative explanations based on a 
critical examination of existing results, for similar but not identical con- 
ditions. 

An examination of available data, in published and unpublished reports, 
has shown two particular examples which deserve further study; these 
are (1) climatic conditions in which stable elevated layers are persistent 
during certain seasons of the year, and (2) equable climates in which the 
annual range of N s, AN, and field strength is relatively low. Attention 
has already been drawn to an example in the first category; namely, the 
path from San Diego to Santa Ana, Calif. [2]. The well-defined coastal 
inversion in this area occurs at a height of about 0.7 km, and the asso- 
ciated stable layer has a strong influence on radio field strength. Here 
the correlation between field strength and A^ gradient for heights up to 
0.7 km above the surface is small and negative, (i.e., opposite to the general 
trend). The correlations with N s, however, and with N differences to 
heights above the base of the inversion, are about 0.8. This result sug- 
gests an explanation of some of the results, already mentioned, for Dakar 
and Leopoldville. Consider the profile of figure 6.10 typical of ascents 
made through daytime inversions in equatorial and Mediterranean areas. 
(For comparison purposes, an exponential reference atmosphere is also 
shown.) 

At Dakar, for example, an elevated layer is observed in 40 to 50 percent 
of the daytime soundings during August, frequently with the base of the 
inversion above 1 km height. In these conditions, at least, the high N s 
values measured in August are not accompanied by high values of AN 
(0 to 1 km), for the lapse rate, dN/dh, below the inversion is generally less 
steep than would be expected for the given value of N s- It would be 
valuable, therefore, to examine in more detail the distribution of the 
height of the layer, particularly in the summer months. If the data given 
by Misme contain an appreciable fraction of such profiles, then the poor 
Ns — AN correlation he discusses is not surprising. The correlation be- 
tween N s and signal level, however, could still be significant, as in the 
San Diego-Santa Ana link [2]. Further radio data are obviously required 
for areas in which elevated inversions are persistent. 

In many temperate regions, a somewhat different situation exists. In 
western Europe, for example, the annual range of monthly median values 
of A" s is 10 to 20 N units; similarly, the variation in the monthly median 
values of field strength is also small and frequently lies within the esti- 
mated measurement error. Figure 6.11 shows some results which illus- 
trate these features. Monthly median values of relative field strength 
are shown for the months January-November, 1959, for a 300-km path at 
a frequency of 174 Mc/s; the terminals being located at Lille (northern 



SOME EXCEPTIONS AND ANOMALIES 



245 



£ 



X 

o 

LiJ 

X 




Figure. 6.10. Typical profile through an inversion layer {equatorial and Mediterranean 

areas.) _^ 

20 y^ 




\— X. 



CiTX: 



\ 



X Ns 

o RELATIVE FIELD STRENGTH 






J L 



J L 



15 



X 

I- 
o 

UJ 

or 



CO 
10 Q 

_i 

LU 

5 ^ 

< 

_j 

n UJ 



JFHAMJJASOND 
MONTH OF YEAR, 1959 

Figure 6.1 1 . Variation of monthly median values of Ng and field strength for a 300 km 
path from Lille, France, to Reading, England. 



246 TRANSHORIZON PARAMETERS 

France) and Reading (England). The variations in N s obtained from the 
Crawley radiosonde station (close to the midpoint of the path) are also 
shown. Here N s has a range of ± 10 A^ units, and the signal level a range 
of no more than ±4.5 dB. In fact, from March to September the total 
variation in observed monthly median field strength is only ±1.5 dB. 
The measurements of field strength in these experiments were estimated 
to be subject to possible errors of about ±3 dB. Consequently, one 
would not be justified in assessing the value of A^^s as a radio-meteorological 
parameter on the basis of the correlation coefficient (about 0.25) calculated 
from the two curves in figure 6.11. During the period studied in this 
work (0900-2300 UT daily, January-November 1959) the highest signals 
were observed during anticyclonic winter weather, with elevated inversion 
layers at a height of about 0.6 km. Extended stable layers at this height 
were rarely seen on the sonde ascents during the summer months. This 
result may partly explain the fact that the seasonal variation in Ns 
(highest values in the summer months) is not accompanied on the Lille- 
Reading path by a corresponding variation in monthly median field 
strength. 

6.1.10. Conclusions 

A review of available data shows that no radio-meteorological param- 
eter has yet been proved to be superior to A^s or AA^ for general application 
in the prediction of field strength distributions. The value of these 
parameters has been established by studies of many paths in diverse 
climatic conditions. However, it should be noted that much of the 
radio data has been obtained at frequencies near 100 Mc/s, and there is 
a clear need for further analysis of the several parameters in conjunction 
with field strength measurements at higher frequencies. 

Pending more detailed results, it does not seem likely that the equiva- 
lent gradient, Qe, affords any significant advantage over AA'', especially 
in view of the many calculations required in its derivation. Some data 
from selected areas (Congo, Sahara, west Africa) suggest that A''^ and AA'' 
may have limited value in these regions; however, the present lack of 
adequate radio data precludes any definite conclusions. These examples, 
and allied work on vertical motion and thermal stability, emphasize the 
importance of a parameter (such as that suggested by Misme) which takes 
account of the influence of elevated layers. This approach would prob- 
ably prove fruitful not only in equatorial areas but also in temperate 
regions where the annual range of N ^ and AN is small. 

Analysis of some results obtained at a frequency of 100 Mc/s has shown 
that it has proved feasible to provide at least as good a prediction of the 
annual cycle of field strength variations from long-term meteorological 
data as from relatively short-term radio data. 



INTRODUCTION 247 

6.2. An Analysis of VHF Field Strength Variations 
and Refractive Index Profiles 

6.2.1. Introduction 

It is evident from the discussion in section 6.1 that the further develop- 
ment of radio-meteorological parameters would be assisted by a better 
understanding of the propagation mechanism on non-optical paths. In 
particular, the influence of thermal stability on signal level, fading rate 
and wavelength dependence is an important topic requiring further study. 

The effect of varying meteorological conditions on signal characteristics 
is especially marked in the case of paths of "intermediate" length, with 
terminals just beyond the radio horizon. As noted in section 6.1 on such 
a path the residual scattered field in the absence of stable layers or surface 
ducts will often be comparable with the weak diffracted field. Further- 
more, at frequencies up to, say 300 Mc/s (X > Im) relatively strong 
fields will frequently be observed in conditions favorable to the production 
of temperature inversions in the first 2 km or so above the earth's surface. 
It is the purpose of this section to discuss some aspects of radiometeor- 
ology relevant to this situation, especially the field, strength distribution 
observed on a 200 km path at frequencies between 72 and 180 Mc/s. 
The signal characteristics are analyzed in terms of a classification of 
refractive index profiles, with the objective of clarifying the relative im- 
portance of different propagation mechanisms and their influence on the 
measured field strength distribution. 

Table 6.3 lists characteristic profile types, the assumed mechanism 
associated with each type, and typical meteorological conditions. Se- 
lected references are given for each category, and special mention may be 
made here of recent work [28] in France which, to a large extent, unifies 
and extends earlier analyses based on the separate concepts of "reflection" 
and "scattering." 

6.2.2. Radio and Meteorological Data 

The analysis to follow is limited to paths between Chicago and Urbana, 
111. (fig. 6.12), since several years of radio data were available for four 
separate wavelengths between 1.67 and 4.18 m. Moreover, two radio- 
sonde stations are located on or near the path, a unique situation in radio- 
meteorological investigations. Details of the radio paths are given in 
table 6.4 (in which 6 is the total angle between the horizon rays from trans- 
mitter and receiver on a 4/3-earth profile). 

The meteorological data were obtained from the simultaneous radio- 
sonde observations made by the Weather Bureau at Joliet and at 
Chanute Air Force Base, near Rantoul, 111. The results used were those 
from the significant levels reported whenever the temperature or humidity 
departed by ±10 percent from predetermined values. 



248 



TRANSHORIZON PARAMETERS 



Table 6.3. Refractive index profile classification, probable propagation mechanisms 
and meteorological conditions 



Profile 



Assumed propagation 
mechanism 



Ref- 



Meteorological 
conditions 



Unstratified (U), monotonic decrease 
with height, gradient nowhere ex- 
ceeds twice normal for that height. 



Elevated Layer (EL), monotonic de- 
crease with height with one or more 
distinct layers with gradients at least 
twice normal for that height. 

Super-refractive (SR), same as EL but 
the layer is ground-based. 



Ducting (D), same as SR but the gra- 
dient exceeds the earth's curvature, 
1/a. 



Scattering plus diffraction 



Scattering plus diffraction 
plus reflection. 



Extended radio horizon pro- 
ducing enhanced dif- 
fracted and scattered 
components. 



Extension of radio horizon 
to include the receiver. 



[29] 

[30] 
[31] 

[32] 
[21] 
[28] 



[31] 



[33] 



Well-mixed atmosphere 
due to thermal convec- 
tion and, or wind shear 



Layer formed by subsi- 
dence inversion or lift- 
ing of radiation inver- 
sion. 

Radiation inversion 

formed during the 
night or rapid evapo- 
ration from soil after 
rain. 

Same as SR. 



Table 6.4. Chicago-to-Urbana radio path characteristics 



Station 


Distance 


d 


X 


/ 


Period of record 


WBKB-TV _... 

WNBQ-TV 

WMBI-FM 

WENR-TV 


km 
203.1 
202.9 
202.7 
202.9 


mrad 
16.3 
16.7 
16.1 
16.4 


COT 

4.18 
3.67 
3.15 
1.67 


Mc/s 
71.75 
81.75 
95.50 
179. 75 


5/51- 5/53 
10/50-10/51 
7/50- 6/52 
7/51- 6/53 



6.2.3. Classification of Radio Field Strengths 
by Profile Types 

The RAOB significant level data were converted to refractive index by 
use of (1.20). The gradient of A^ was then determined between the re- 
ported significant levels of each profile and examined as to whether the 
gradients fell into the category of linear, subrefractive, or superrefractive 
depending upon the criteria set down in table 6.5, wherein superrefractive 
is approximately twice normal and subrefractive has a positive gradient. 
Simultaneous observations of similar profile types at Rantoul and Joliet 
were necessary for entry as a distinct profile occurrence. If a super- 
refractive layer occurred above the crossover heights of the radio horizon 
tangent rays from both transmitter and receiver then it was classified as 
an elevated layer provided the reported layer heights were within 1 km 
of one another at both radiosonde stations. 

Table 6.5. N gradient classification of profile types in N-units/km 



p 


h 


Subrefractive 


Unstratified 


Superrefractive 


mb 
1000-850 
850-700 
700-600 
600-500 
500-400 


km 
-1.46 
1.46-3.01 

3. 01-4. 20 

4. 20-5. 57 

5. 57-7. 18 


-(dn/dhXQ 
-{dn/dhXO 
-(.dn/dhXO 
-(dn/dhXO 
-(dn/dhXO 


20<-(dH/dft)<60 
20<-(dn/d/i)<50 
15<-(d;i/dft)<40 
10<-(dn/d/i)<30 
10<-(dn/d/i)<25 


100<- (d«/dft) 
80<- (dn/dft) 
■JQ<-(dn/dh) 
50<- (dn/dh) 
40< -(dn/dft) 



RADIO FIELD STRENGTHS BY PROFILE TYPES 



249 



Elevated layers below the crossover height were classified as ground- 
based superrefractive layers. Elevated layers below the crossover height 
at one weather station and above that height at the other were classified 
as tilted elevated layers. 

After these characteristic profiles were isolated, the median field 
strengths for the 3-hourly periods centered upon the radiosonde observa- 
tion time were arranged into cumulative probability distributions for 
each profile type. The results are shown in figure 6.13. (There were 
relatively few examples of subrefractive profiles and no distributions for 
this category are given.) 




J.. OHIO , 



1 \. 






<^^-J. 



MISSOURI V 



KENTUCKY 






.r^ 






Figure 6.12. Location of radio path and radiosonde stations used in this study. 



250 TRANSHORIZON PARAMETERS 

Generally, the unst ratified -samples have the lowest overall field 
strengths throughout the entire distribution range. The presence of any 
layer (elevated or ground-based) tends to increase the field strength by 
10-25 dB at any percentage level of the distribution. (The exception to 
this observation, WXBQ-TV, is probably explained by the fact that the 
observations were limited to six winter months, rather than the 2-year 
period of the other stations.) Tilted elevated layers appear to produce 
the greatest enchancement of signal strength, probably as a result of 
focusing effects due to the layer tilt. 



6.2.4. Prediction of Field Strength for 
Unstratified Conditions 

The field strengths recorded during the times when the radiosonde 
ascents at both Joliet and Rantoul indicated nonstratified conditions were 
compared with the values predicted by Norton, Rice, and others [31] for 
the case of diffracted plus scattered radio waves. This particular pre- 
diction process is adjustable for the average refractive conditions over the 
path in that it adjusts the effective earth's radius factor to the initial 
gradient of A^ for the calculation of diffracted field strengths. One also 
needs the angular separation of the radio horizon rays at their intersection 
near midpath. The average initial gradient of A^ was obtained for each 
instance of unstratified profile by .simply averaging the initial gradients 
from Rantoul and Joliet, while the angular separation was obtained by 
determining the amount of radio ray refraction expected over each particu- 
lar path in atmospheres of exponential decrease with height that closely 
match the observed A^ conditions. 

Figure 6.14 illustrated a comparison of the predicted and observed field 
strengths. For WNBQ and WENR there is approximate agreement be- 
tween the two sets of data. However, the predicted values for WBKB 
and WMBI are approximately 10 to 12 dB higher than the observed 
values. 

This tendency to predict field in excess of the measured values suggests 
that the empirical data, on which the predictions are based may include 
meteorological conditions with some degree of stratification in the first 
2 km or so, even though most of the empirical data refer to afternoon 
hours in winter. It will be shown that elevated layers of moderate size 
(say a few kilometers in horizontal dimensions), that may exist undetected 
by the radiosonde, could produce field strengths on the Illinois paths com- 
parable with the median values for "unstratified" conditions shown in 
figure 6.13. Furthermore, the limits placed on the profile gradients 
specifying unstratified conditions in table 6.5 are such that some layer 
type profiles may be included in the unstratified category. Consequently, 
it is important to study in more detail the properties of the elevated layer, 



ELEVATED LAYERS ON THE ILLINOIS PATH 



251 















WBKB -TV 


PATH 206 




-- 


\ 


\ 









IINSTRATIFIEO 1188) 
GROUND BASED (53) _ 


- 




t 




N 





- GB +EL (23) 





- ELEVATED LAYER (339) 


■. 














_N 


■^^ 


^^ 




~^N 




X 










- 


- 








\ 




^Ot 






- 


- 












\, 


\ 




■•■■X_^, 




\- 


- 
















\ 


\ 
\ 


N 


- 




























0.5 I 5 10 30 50 70 90 95 99 99.5 99.9 

PERCENT OF TIME VALUE EXCEEDS ORDINATE 















WNBQ -TV 




H 219 




1 




^ 


■~N 






UNSTRATIFIED (259) 

GROUND BASED (31) 






^\ 






GB+ EL (10) 


- 




1 


\ 


\ 


ELEVATED LAYER (73) 




■■•-... 


N 








~ 


V 


- 


'"\, 


V. 












- 


~ 










N 








- 


- 












\ 


^"^-^ 


- 
















'^^ 


■~- 


— . 


- 


























^ 


- 






















- 




0.1 0.5 I 5 10 30 50 70 90 95 99 915 99.9 

PERCENT OF TIME VALUE EXCEEDS ORDINATE 



Figure 6.13. Distribution of hourly median field strengths with different radiosonde 

profile conditions. 



not only as a feature occasionally producing high field strengths, but also 
as a mechanism which, in less intense form, partly determines the strength 
of the weaker fields observed for large percentages of the time. 



6.2.5. The Effect of Elevated Layers on the Illinois Path 

The influence of elevated layers of VHF transmission beyond the 
horizon has been studied by several workers [21, 22, 28, 34, 35, 36, 37]. 
However, few investigations have contained any detailed comparisons of 
theory and experimental results. The following analysis presents such 
a comparison, using simple models of the elevated layer for the four 
Illinois paths. 



252 



TRANSHORIZON PARAMETERS 



6.2.6. Elevated Layers at Temperature Inversions 

Recent radar and refractometer investigations of tropospheric structure 
have shown that elevated layers in the refractive index distribution are 
frequently observed in the stable air of temperature inversions [38]. 

A typical value of layer thickness is 100 m, with horizontal dimensions 
of tens of kilometers. On occasions, extended layers no more than 10 m 
in thickness have been detected by refractometer soundings. In the 
present discussion we attempt to evaluate the reflection coefficient of 
these elevated layers. We may express the modulus of the reflection 
coefficient \p\, for a wave incident at a glancing angle a on a layer of 
thickness h, and horizontal extent, x, in the form: 



An 



f(a, h, x) 



(6.4) 



where f{a,h,x) is the ratio of the reflection coefficient of the model to 
that of the infinitely sharp case (i.e., the Fresnel discontinuity value, 
An/2a2). This function has been evaluated for several layer profiles [28], 



- 


' i 

WBKB- 


PATH 


206 












- 


- 


L 


NSTRi 


TIFIED 






FIELC 


STRE 


NGTHS 


■ 


- 




















/ 










1 . 


• f 


•- 






/ 




_• 




1 


t 




,..s! 




i •' 


Yy. . 




- 






: 


I . I 




r ~* 


.;^.::: 


^. 


- 


- 






1 


[ 




/^ 


-PREDICTED VALUE - 

1 1 










, 


^ 





24 
o 

u. 

H 
o 

5 12 













' 


' 


/ 




' 


- 


WNBO - PA 

1 


TH 21= 






/ 






- 












/ 








- 






^ 




/ 




. 








, . 


- 


• 


/ 


).-. 






- 












- 




**h: 


* 








- 


/ 


i , i 






' 






- 





' M ■ i 






' 


! ' 


- 


WUE 


l-PA 


TH 57 














■ 








, 










/ 


- - 


• 


■ . 


".. 


. • • 








/ 




•• 




I._V 


r A 


V* 






/ 




- 




■V. t 




-/ 






- 


_• 


• 


"if- .4.« / 














, /^ 











-16 -12 -8 -4 4 8 12 16 "0 4 8 12 16 20 24 28 32 36 40 

OBSERVED FIELD OBSERVED FIELD 

28 r 
24 

_i 

LlJ 

il 16- 

Q 
UJ 

h 12- 



-12 -8 -4 4 8 12 16 20 24 16 -12 -8 -4 4 8 12 16 20 24 

OBSERVED FIELD OBSERVED FIELD 



- 


' 1 ' ! ' i ' 

WENR-PATH 210 










1 


- 


















/ 


- 
















/ 




- 








• • 


J- 




/ 






- 








! 1 




/ 




... 




- 


• 


• 


!" 


ry- 


;•^ 








- 








■ / 










- 








/ 




• • 


• 









Figure 6.14. Com-parison of observed radio field strengths and values predicted for 

wintertime afternoon hours. 



ELEVATED LAYERS AT TEMPERATURE INVERSIONS 253 

and preliminary calculations based on this work were made to determine 
the most suitable model in the present application. It was evident from 
these calculations that a simple linear profile would yield the best agree- 
ment with the measured data, and this model was therefore adopted in 
the subsequent analysis. 

Consider the layer profile shown in figure 6.15, i.e., a linear decrease of 
An over a height interval h, with transition regions of height d. This 
model and others have been discussed by several authors, but the most 
detailed treatment is that of Brekhovskikh [39]. His analysis shows that 
for this linear model: 

I p I = An • X/Sirh sin^ a 

~ An • X/Sirha^. (6.5) 

This equation is valid if: 

(a) An • \ <K rha^ 
and (b) 4ad « X. 

In the present problem, with values of X of 1.7 to 4.2 m, An '^ 10"^, 
a ^ 0.02 condition (a) is satisfied for layer thicknesses greater than about 
20 m. In addition, condition (b) is fulfilled for the stated conditions if the 
thickness of the transition region is less than a few meters. These condi- 
tions do not seem inconsistent with available refractometer data on 
elevated layers, but a rigorous justification of the model is impossible at 
the present time. In any case, there is almost certainly no unique profile 
representative of all elevated layers. We assume here, therefore, the 
linear profile of figure 6.15 merely as a simple analytical model. It may 
be noted here that the value of | p | given by (6.5) agrees with that quoted 
by du Castel [40] but is half the value obtained in an earlier analysis [28]. 

Equation 6.5 was used to calculate reflection coefficient of the layers 
on each occasion on which these were observed in the sonde ascents. The 
results, expressed in terms of a reflection loss are compared with the 
measured values of field strength in figure 6.16. The general agreement 
is satisfactory for the assumed model. As might be expected, there is 
a considerable scatter in the data and two considerations are important in 
assessing the significance of these results. These concern sonde response 
and layer structure. The work of Wagner [41] on the response of radio- 
sondes shows that, for an elevated inversion layer with An = 3 X 10~*, 
h = 100 m, a sonde with a 10 sec time constant in the sensing elements, 
and rising at 5 m/sec will give an indicated value for An of approximately 
half the true value. The above procedure, using sonde data, therefore 
underestimates the value of |p| for an idealized infinite layer. 



254 



TRANSHORIZON PARAMETERS 



X 
(3 
LJ 

X 




REFRACTIVE INDEX 
Figure 6.15. Linear profile model of the n decrease across an elevated layer. 



On the other hand, the analysis assumes a smooth layer extending 
horizontally at least over a distance x equal to the first Fresnel zone. 
We have: 



X = ■\/2a\/a 



(6.6) 



where 2a = path length = 203 km for the Illinois paths. Hence x is of 
the order of a few tens of kilometers. In addition, we have assumed that 
the layer is horizontal and smooth over a distance x, thus neglecting con- 
vergence. If we adopt the Rayleigh criterion, the height of the surface 



INFLUENCE OF SMALL LAYERS 255 

irregularities on the layer, A/i say, must not exceed ±X/8a for the layer 
to be considered smooth; i.e., 

Ah < ±7 m (X = L67 m) 
Ah < ±17 m (X = 4.18 m). 

These values apply for a = 0.03 rad, corresponding to a layer height of 
2.5 km; for lower layers Ah will be greater due to the decrease in a. 

These conditions are not likely to be satisfied in all the examples studied 
and the analysis therefore overestimates the value of lp| in this respect. 
(Some discussion of this point has been given by Bauer and Meyer [42].) 
The limitations of sonde soundings, and the effects of layer tilt and surface 
irregularities therefore provide a partial explanation of the scatter of 
points in figure 6.16. Further detailed measurements of layer structure 
are obviously desirable. 



6.2.7. The Influence of Small Layers 

The above discussion has dealt with the particular case of extended 
elevated layers such as are often associated with temperature inversions. 
However, it seems quite possible that these layers are merely the more 
extreme examples of anisotropic irregularities which are thought to be 
prevalent in the troposphere. There is already some preliminary evidence 
supporting this concept in the results of refractometer and radar soundings 
and recent theoretical work [22, 28, 40] has developed this approach in 
detail. The relationship of this work to earlier analyses in terms of a 
"scattering" model is discussed in the references quoted and need not 
concern us here. For our purpose it is sufficient to utilize the essential 
features of the argument as the basis for a simple calculation. 

It seems reasonable to assume that even in an atmosphere which sonde 
data would lead us to classify as "well-mixed" there are often layer-type 
irregularities. Detailed evidence on the spatial form and stability of this 
type of layer or "feuillet" is so far lacking, but an inspection of some 
refractometer results suggests that horizontal dimensions of a few kilo- 
meters represent a realistic assumption. Such a layer might exist as a 
separate entity for, say, several minutes (as compared with a period of 
several hours for the extended layer in a stable inversion). 

For the following analysis, let us consider two layers of horizontal 
dimensions, x, of 2 and 10 km respectively with the following character- 
istics: 

An = 10-5 
/i = 100 m 

a = 0.01-0.03 rad (i.e., layer height of 0.4 to 2.5 km on the Illinois 
paths). 



256 



TRANSHORIZON PARAMETERS 



35 
30 


1 'WB 


B-TV 
















/ 




L 


NEA 


R PR 


OFIL 
















A 


■' 






















1 


/ 






20 
15 
10 
















\ 




• 


/ 






. 


















/ 


y 


•, 
















• 






/ 




y 












'• 






'• 


», 


/* 


. 


•• J 


• 




: 


















/ 


^ 


-f 


RED 


CTED 


VAL 


UE 
















/ 




. 
























A 





















-Lr REFLECTION LOSS IN dB 





WMBI - FM 




















/ 




























/ 


























/ 
























5 


/ 




















• 




.V 


•• 


-/• 






















• 




y 


• 
















• 






\ 


'•/ 


-•.' 






















. 


••• 


'/- 


. 


. 










. 












/ 


• 
























/ 


/ 


• 










50 


1 





1 





9 





T 





5 





3 





10 



-Lr REFLECTION LOSS IN dB 





WNBQ-TV 












































































/ 


























/ 
























■ / 


/ 




















• 




/ 


/• 




• 
















• 




/ 
























/ 


y 




. t 




















/ 


/ 






• 


















/ 


r 

















-Lr REFLECTION LOSS IN dB 





WE^ 


R-T 


V 


— 


— 












— 


~ 


— 




— 


1 


— 


~ 


" 


4 


— 


# 




/ 


^ 


r 


y 












































.. 




/ 




















, 


», 




r^ 




















/•"• 




/ 


y 




















.. 




y- 


/* 


• 


















• 


/^ 


M 





















-Lr REFLECTION LOSS IN dB 



Figure 6.16. Observed hourly median field strength versus reflection loss assuming the 

linear model of figure 6.15. 



For these conditions, the layers correspond to those of "intermediate" 
size in the analysis of Friis, Crawford, and Hogg [22]. They are defined 
by the inequality: 



\/2a \/a > X > \/2a\ 



(6.7) 



where 2a is the path length. In this case, the power received, Pr, from 
an antenna of effective aperture, Ar, with a transmitter radiating a 
power Pt from a antenna of effective aperture ^r is given by: 



Pr/Pt = {AtAr X2aV)/(2XV). 



(6.8) 



INFLUENCE OF SMALL LAYERS 257 

We can use this equation to calculate the corresponding field strength, 
for the Illinois paths, in terms of lA'/m for 1 kW radiated from a half- 
wave dipole. We have the following relations: 

A (X/2 dipole) = 0.127 X^ (6.9) 

p, (X/2 dipole) = ^2X2/300^2 (6.10) 

where E is the field strength in volts/meter if Pr is in watts. From 
(6.8), (6.9), and (6.10), we can calculate E for the two layers specified 
above. The results obtained are shown in figure 6.17 for various layer 
heights and the following models of reflection coefficient: 

(a) \p\ = An • \/8ira^h 

(b) IpI = An/2a\ 

Model (b) is the Fresnel discontinuity equation which gives the limiting 
value of \p\ towards which all models tend as the layer thickness de- 
creases. The curves in figure 6.17 show that the calculated field strength 
depends considerably on the assumed n-profile. If | p | = An • X/Sira^h, 
values of field strength comparable with the long-term median value may 
be produced by layers of about 10 km in lateral dimensions in the height 
range 0.5 to 1 km. If |p| = An/2a^, similar field strength may be pro- 
duced by layers in this height range if the lateral dimension is of the order 
of 2 km. The effect of the layer decreases with increasing height, but 
even with layer heights of 3 km, the field strength is still 1 /LtV/m or 
greater at both wavelengths for a 10 km layer with \p\ — An/2a'^. How- 
ever, it should be pointed out that the assumed value of An = 10~* is 
probably somewhat large for layers as high as 3 km. The results also 
show that model (b) (i.e., p = An/2a:-) gives field strength values which 
are higher at X = 1.67 m (/ = 179.75 Mc/s) than at X = 4.18 m (/ =71.75 
Mc/s). 

6.2.8. Conclusions 

Any departure of refractive index structure from a smooth monotonic 
decrease with height produces an increase in field strength on a 200-km 
path in the frequency band 70 to 180 Mc/s. In the particular case 
studied, elevated tilted layers result in signal enhancement of 10 to 25 dB 
over the values for unstratified conditions, at all percentage levels. (The 
importance of the tilted layer is possibly a consequence of the asymmetry 
of the path, the transmitting antenna being 200 m above ground and the 
receiving antenna 30 m.) 

The predicted field strengths, for conditions classified as unstratified 
in terms of sonde data, are in approximate agreement with observed 
results, although the scatter of the points (plus the tendency to predict 



258 TRANSHORIZON PARAMETERS 

values in excess of the measured ones) point to the influence of anisotropic 
layers or eddies of varying size and degrees of stability. This interpre- 
tation is consistent with numerical calculations based on the properties of 
"intermediate" size layers, suggested in the analysis of Friis, Crawford, 
and Hogg [22]. 

Calculations of the field strength produced by extended stable layers, 
using sonde data and a model profile with a linear lapse of n with height, 
are in reasonable agreement with the experimental results. However, 
there is probably no unique profile characteristic of elevated layers. 



6.3. A New Turbulence Parameter 

6.3.1. Introduction 

We shall here attempt to unify past work utilizing the surface value 
of the refractive index and the concept of atmospheric stability. To do 
so we shall first review the concepts of atmospheric stability from the 
viewpoint of temperature structure. In the process we shall show that 
past efforts have neglected the important role of the conditions at the 
earth's surface; then we shall extend this work to derive an analogous 
expression for the radio refractive index. 

6.3.2. The Concept of Thermal Stability 

The stratification of the earth's atmosphere by and large reflects the 
control of the earth's gravitational field. This average structure is, 
however, constantly disturbed by the intrusion of thermal plumes of 
heated air rising from the differentially heated surface of the earth, as 
well as by the mechanical mixing produced by the passage of air currents. 
It is customary to assume that parcels of air are forced upwards through 
the normal stratified atmosphere without mixing with the environmental 
air mass. Since the parcel is forced to rise, be it by convection or by 
mechanical mixing, it is also assumed that it expands and cools without 
exchange of heat, i.e., adiabatically. In such a process it can be shown 
that the pressure, P, is given by 

where T is the temperature (°K); g, the acceleration of gravity; R, the 
gas constant for air; and, where a, the constant lapse of temperature with 
height is 

(6.12) 



CONCEPT OF THERMAL STABILITY 



259 



en 

2 




1.0 1.5 2.0 2.5 3.0 

LAYER HEIGHT (km) 




1.5 2.0 

LAYER HEIGHT (km) 



2.5 



3.0 



Figure 6.17. Field strength produced by layers of horizontal dimensions 2 km and 10 
km on the Chicago-Urbana paths for two model profiles. 



260 TRANSHORIZON PARAMETERS 

The zero subscripts in (6.11) refer to conditions at the earth's surface. 
Equations (6.11) and (6.12) are taken to hold as long as saturation does 
not occur. It is further assumed that the mixing ratio 

e^eo-^ (6.13) 

-to 

is constant throughout the process. In (6.13) e is the partial pressure of 
water vapor. 

A parcel of air which is caused to rise will cool to the dew point if forced 
aloft far enough. Further rising will lead to condensation. Conversely, 
a parcel of air which sinks down will have work done upon it adiabatically 
and will become warmer than it was in its original elevated position. A 
parcel of air following a condensation curve has a value of a, of — 6°C/km 
denoted at a*. In addition, as water vapor condenses out of the parcel, 
the vapor pressure will decrease. We here assume that the vapor pressure 
will follow that of the saturation vapor pressure curve, Csz, as given by: 

esz = est exp [-^a*(z-l)], z > i, (6.14) 

where i is the height of the lifting condensation level and ^ = 0.064 (°C)~^. 
The value of ( has been found to be given [44] with sufficient accuracy for 
practical applications by 

^^0.125 (T-Trf)o (km) (6.15) 

where the zero subscript indicates that the difference between the tem- 
perature and the dewpoint, Td, need be evaluated only at the earth's 
surface. 

Consider now a parcel rising through some environmental distribution 
of temperature. The dry adiabatic lapse of temperature with height is 
greater than that observed on the average in the atmosphere (6 °C/km) 
and thus the parcel becomes cooler, and more dense, than the environ- 
mental air and will sink to its initial conditions with the removal of the 
lifting force. If, on the contrary, anywhere in its trajectory it becomes 
warmer, and thus less dense, it will become unstable and rise of its own 
buoyancy through the environmental air. Past radio-meteorological 
studies have taken the area between the adiabatic curves and the environ- 
mental temperature distribution on a pressure ('^ height) versus tempera- 
ture chart as a measure of atmospheric stability. On the general argu- 
ment that the effects of atmospheric stability, or turbulence, influence 
radio waves only through the radio refractive index, we shall extend the 
above concepts to the radio refractive index. 



CONCEPT OF THERMAL STABILITY 261 

6.3.3. The Adiabatic Lapses of N 

Introducing the dry adiabatic variations of P, T, and e into (1.20), 
chapter 1, one obtains the dry adiabatic decrease of A'^, N d- 

N, =N.[1- y-j |1 - -^^i^l (6.16) 

which is vaHd for 2 < (. The distribution of A" d with z may be evaluated 
by taking the logarithms of both sides of (6.16) 




(6.17) 



and expanding the logarithms on the righthand side of (6.17) with the 
omission of second-order terms where appropriate for our applications 
of 2 < 3 km: 

In Ad~ln Ao - r^^ (6.18) 

or 

Arf~A^oexp i-Tdz) (6.19) 

where 

When evaluated for the standard conditions of To = 288 °K (15 °C) and 
Po = 1013.2 mbar, r^ becomes 

Yd = 0.08 (km)-i. (6.21) 

It is sometimes more convenient to use the dry adiabatic scale height, 
Hd-. 

Hd = — = 12.5 km. (6.22) 

1 d 

Above the local condensation level, the expression for A becomes that of 
the wet adiabatic, A^^,: 



»..(..#•(»,(..«)-- 



+ I^, exp (I3a*z)j (6.23) 



262 TRANSHORIZON PARAMETERS 

where, for convenience, 

n 77 A ^' H w 77.6 (4810) e., 
D( = 77.6 — and Wt = y~2 

Applying the same approximations as in the derivation of (6.18), one 
obtains 

In iV^ = In A^, - r^z (6.24) 

where 



!„* fe+0«*^' 



2c 

For the standard conditions assumed above and eo corresponding to 
60 percent relative humidity, H^ '^ 7.5 km. Again, the form of the wet 
adiabatic lapse of A^ is given by 

A^^ = A^, exp i-T^z), z> i. (6.26) 

We shall now apply these results to the derivation of a refractive index 
turbulence parameter. 



6.3.4. The Turbulence Parameter, n 

Analogously to the concept of thermal stability, we define n as the area 
between the environmental N{z) curve and the appropriate adiabatic 
decrease of A^. That is 



n = 



/ \N observed ~ N adiabatic j dz (6.27) 



where the integration is arbitrarily taken from the surface to 3 km. This 
then extends the integration well above the crossover height of the radio 
horizon rays from transmitter and receiver normally encountered in 
tropospheric propagation. The integral for n may be written 



n = / Ndz - j No exp (-Tdz) dz 



l>- 



exp[-V^{z-i)]dz. (6.28) 



TURBULENCE PARAMETER, II 263 

Noting that 

(6.28) then becomes, upon integration, 



No 



n = Ndz-^ir [exp (-Tdl)-l] 

Jo Ad 

+ ^»exp ^-r^^^) {exp [-r.(3-0]-l}. (6.29) 

1 w 

n is thus dependent upon both A'^ profile characteristics and initial condi- 
tions. In fact, all the terms on the righthand side of (6.29) save 

3 

Ndz 

are determined from the conditions at the earth's surface. One might ex- 
pect, then, that 11 would constitute a useful radio-meteorological param- 
eter since it includes both the time-proven parameter A^o, the concept of 
atmospheric stability, and the integrated A''-profile characteristics. We 
shall now apply this new parameter to the radio data presented in detail 
in section 6.2. The values oi H a and Hy, utilized are those for standard 
sea-level conditions; i.e., 12.5 km and 7.5 km, respectively. 

6.3.5. Comparison of Radio-Meteorological Parameters 

The various radio-meteorological parameters discussed in section 6.1 
plus n as derived from the Rantoul and Joliet radiosondes were tested 
for their relative utility by comparison with the radio data of WBKB-TV 
and WEXR-TV (see table 6.4). Correlation coefficients between the 
radio data (3-hourly medians) of each path and the various radio- 
meteorological parameters were determined for each of the profile cate- 
gories given in table 6.5 as well as for all available data. It was not 
possible to fit all of the data into the various categories since these were 
deliberately made very strict in order to explore the utility of each 
parameter within each of the propagation mechanisms assumed for the 
profile categories (table 6.3). Thus it is observed that the sum of the 
individual categories does not equal all available data. The results of 
these correlations are given in table 6.6. Before discussing these results 
it is well to note that the crossover height of the tangent rays from each 
end of the path were calculated by detailed ray tracing [15] for each radio- 
sonde ascent. The values as determined from the Rantoul and Joliet 
soundings were averaged. This average crossover height was then used 
to determine the contribution to the values of 11 below the crossover 



264 TRANSHORIZON PARAMETERS 

height (n"), above the crossover height (n'), and Qe- As hsted in table 
6.6, n represents the value of (6.27) for the height increment zero to 3 km. 
Before proceeding, the method of calculating ge will be described. The 
concept of the equivalent gradient is closely related to that of the effective 
earth's radius. The curvature of the effective earth is given by: 

1 1 , 1 dn 

— = - -\- - -,T cos d 

re a n ah 

where r^ is the effective earth's radius; a is the radius of the earth; and 
(1/n) idn/dh) cos d is the curvature of the radio ray. 
The value of r^ is determined from the geometrical relationship 

h ^-'^ 

where c?,, the ground distance to the crossover height, h^, has already been 
determined by the ray-tracing procedures discussed in chapter 3. Thus 
one obtains 

dn \ _ 2hc 1 
dhj ~ d;' 

for 6 = and setting (cos d)/n c^ 1. In order to obtain the equivalent 
gradient as a positive quantity in the more convenient refractivity units, 
the following expression is used: 



(i _ 2^Y 



0- = Ka - t^r- 

It is evident from table 6.6 that the results obtained with IT are com- 
parable with those using 11', due supposedly to the crossover height being 
on the order of 100 m for these paths. The value 11" appears to contribute 
little to the final value of 11 and is poorly correlated with the radio data. 
An interesting exception to this observation is the subrefractive case 
where 11" yields a larger correlation coefficient than either n or U'. The 
overall conclusion that one reaches from the data of table 6.6 is that the 
most promising of the parameters considered is U, closely followed by 
AA^ and N s- In fact, the difference between the results obtained with 
n and AN gives one pause to consider if the added complexity of deter- 
mining n is worth the relatively small gain in correlation with the radio 
data. The results obtained with Qe and go (the N gradient of the initial 
layer as reported by the radiosonde) are disappointingly small and are of 
no practical significance when compared with those of H, AN, and even 
A.,. 



COMPARISON OF RADIO-METEOROLOGICAL PARAMETERS 
Table 6.6. Summary of correlations 



265 













Radio-meteorological variable 








Cate- 


Sample 














gory* 


size 






















n 


n' 


n" 


ge 


9o 


N, 


AN 


WENR.TV 




















(179.75 Mc/s) 


1 


68 


-0. 3794 


-0. 3795 


-0. 1859 


0. 1889 


-0. 1884 


0. 1414 


-0. 3465 




2 


21 


-. 2064 


-. 2683 


.2003 


.0035 


.1346 


.0531 


-.2653 




3 


101 


-.1748 


-. 1754 


.0087 


.0291 


-. 0303 


-. 0940 


-.4553 




4 


6 


-.3632 


-.3503 


-.4315 


.4990 


-.4149 


.0905 


.0873 




5 


196 


-. 5665 


-. 5732 


-.0312 


.0555 


-.1772 


.3870 


-. 5640 




6 


518 


-.4624 


-. 4602 


-.0169 


.0722 


-. 1762 


.2889 


-. 4544 


WBKB-TV 




















(71.75 Mc/s) 


1 


67 


-.2952 


-.2976 


-.2231 


-.0926 


.0791 


.0107 


-. 1687 




2 


27 


-. 2071 


-. 1843 


-. 1378 


-. 1037 


.0939 


.1678 


-. 1608 




3 


113 


-. 2691 


-.2709 


.0763 


.1652 


-.0810 


.0703 


-. 3486 




4 


6 


-. 2445 


-. 2202 


-.6849 


.3084 


-.3686 


.2573 


.3485 




5 


213 


-.5519 


-. 5562 


-. 0576 


.0624 


-.1152 


.4168 


-. 4635 




6 


564 


-. 4384 


-.4398 


.0087 


.0430 


-. 1027 


.2128 


-. 3505 



♦Category 

l = Unstratifled. 

2= Ground-based duct. 

3 = Elevated layer. 

4 = Subrefractive layer. 
5=1 through 4 combined. 

6= All data for period of record. 



The above data also allow one to evaluate a parameter of the form 



H, = age-^ bW, 

which is analogous to Misme's recently suggested M parameter [19]. The 
parameter Hi given above is physically more realistic for radio purposes 
since it incorporates a mea.sure of the refractive index turbulence rather 
than thermal stability. The correlations obtained between field strength 
and Hi are 



WENR-TV: 
WBKB-TV : 



0.46 
0.44 



which represents, for the significant figures carried, no improvement over 
the use of 11 alone. This last analysis suggests that it might be desirable 
to consider other combinations such as 

H. = a.AN + 62 n. 

The correlations for this case are: 

WENR-TV: 0.49 
WBKB-TV: 0.44 



which represents a slight improvement over IT alone. 



266 TRANSHORIZON PARAMETERS 

One might well wonder why the correlations of Qe and field strengths 
yield a higher correlation relative to that with AA'^ or A^s on a monthly 
median basis than on a daily basis. This is apparently due to the non- 
linear nature of the averaging process of g e which is dependent upon the 
nonlinear weighting of the N{h) profile inherent in the refraction process. 
On the other hand, the average AA'^ or A^^^ obtained from the average N{h) 
curve is the same as that obtained from averaging AA'' or N s. Correla- 
tions on n and field strength on a monthly median basis are now under 
study. 

6.3.6. Conclusions 

In conclusion, we reach the opinion that the results of previous experi- 
ence with the dependence of radio refraction upon N s and the concept 
of stability are incorporated into IT. The resultant correlation with 
radio data on a 3-hourly median basis is perhaps the most encouraging 
obtained to date, but still, appears to be a marginal improvement over 
the time-honored, and much simpler, parameter AA'^. This conclusion 
could well change with broader experience on other paths and radio 
frequencies. 

6.4. References 

[1] Bean, B. R., and F. M. Meaney (Oct. 1955), Some applications of the monthly 

median refractivity gradient in tropospheric propagation, Proc. IRE 43, No. 

10, 1419-1431. 
[2] Bean, B. R., and B. A. Cahoon (Jan.-Feb. 1961), Correlation of monthly median 

transmission loss and refractive index profile characteristics, J. Res. NBS 65D 

(Radio Prop.), No. 1, 67-74. 
[3] Misme, P. (Mar .-Apr. 1960), Le gradient equivalent mesure direct et calcul 

theorique, Ann. Telecommun. 15, No. 3-4, 93. 
[4] Misme, P. (Nov.-Dec. 1960), Quelques aspects de la radio-climatologie, Ann. 

Telecommun. 15, No. 11-12, 266. 
[5] Misme, P. (Jan.-Feb. 1961), Essais de radio climatologie dans le bassin du 

Congo, Ann. Telecommun. 16, No. 1-2, 29. 
[6] Flavell, R. G., and J. A. Lane (Jan. 1963), The application of potential refractive 

index in tropospheric wave propagation, J. Atmos. Terrest. Phys. 24, 47-50. 
[7] Moler, W. F., and W. A. Arvola (Aug. 1956), Vertical motion in the atmosphere 

and its effect on VHF radio signal strength, Trans. Am. Geophys. Union 37, 

399-409. 
[8] Moler, W. F., and D. B. Holden (Jan.-Feb. 1960), Tropospheric scatter propaga- 
tion and atmospheric circulations, J. Res. NBS 64D (Radio Prop.), No. 1, 81-93. 
[9] Rice, P. L., A. Longley, and K. A. Norton (1959), Prediction of the cumulative 

distribution with time of ground wave and tropospheric wave transmission 

loss. Part I, NBS Tech. Note 15. 
[10] Pickard, G. W., and H. T. Stetson (Jan. 1950), Comparison of tropospheric 

reception, J. Atmos. Terrest. Phys. 1, 32-36. 
[11] Bonavoglia, L. (Dec. 1958), Correlazione fra fenomeni meteorologici e propa- 

gazione altre I'orizzonte sul Mediterraneo, Alta Frequenza 27, 815. 



REFERENCES 267 

[12] Gray, R. E. (Sept. 1961), Tropospheric scatter propagation and meteorological 

conditions in the Caribbean, IRE Trans. Ant. Prop. AP-9, 492-496. 
[13] Onoe, M., M. Hirai, and S. Niwa (Apr. 1958), Results of experiment of long- 
distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79. 
[14] Bean, B. R., L. Fehlhaber, and J. Grosskopf (Jan. 1962), Die Radiometeorologie 

und ihre Bedeutung fiir die Ausbreitung der m-, dm-, and cm-Wellen auf 

grossen Enternungen, Nachrichtentechnische Zeit. 15, 9-16. 
[15] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric 

refractive index, Proc. IRE 47, No. 5, 740-755. 
[16] Misme, P. (Nov.-Dec. 1958), Essai de radiocHmatologie d'altitude dans le nord 

de la France, Ann. Telecommun. 13, No. 11-12, 303-310. 
[17] Bean, B. R., (Mar. 1962), The radio refractive index of air, Proc. IRE 50, No. 3, 

260-273. 
[18] Lane, J. A. (Feb.-Mar. 1961), The radio refractive index gradient over the 

British Isles, J. Atmos. Terrest. Phys. 21, No. 2-3, 157-166. 
[19] Misme, P. (May-June 1961), L'influence du gradient equivalent et de la stabilite 

atmospherique dans les liaisons transhorizon au Sahara et au Congo, Ann. 

Telecommun. 16, No. 5-6, 110. 
[20] Boithias, L., and P. Misme (May-June 1962), Le gradient equivalent; nouvelle 

determination et calculgraphique, Ann. Telecommun. 17, No. 5-6, 133. 
[21] Saxton, J. A. (Sept. 1951), Propagation of metre radio waves beyond the normal 

horizon, Proc. lEE 98, 360-369. 
[22] Friis, H. T., A. B. Crawford, and D. C. Hogg (May 1957), A reflection theory for 

propagation beyond the horizon, Bell Syst. Tech. J. 36, 627. 
[23] duCastel, F., P. Misme, A. Spizzichino, and J. Voge (1958-1960), Resultats 

experimentaux en propagation tropospherique transhorizon (a series of 10 

papers), Ann. Telecommun. 13, 14, and 15. 
[24] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Meteor- 
ology of the refraction problem. Book, Propagation of Short Radio Waves by 

D. E. Kerr, pp. 198-199 (McGraW-Hill Book Co., Inc., New York, N.Y.). 
[25] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic scale 

radio meteorology, J. Meteorol. 17, 264. 
[26] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the 

vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio 

Prop.), No. 2, 249-258. 
[27] Misme, P. (1962), Private communication. 
[28] duCastel, F., P. Misme, and J. Voge (1960), Sur le role des phenomenes de 

reflexion dans la propagation louitaine des ondes ultracourtes, Electromagnetic 

Wave Propagation, Book, p. 671 (Academic Press, London and New York). 
[29] Booker, H. G., and W. E. Gordon (Apr. 1950), A theory of radio scattering in the 

troposphere, Proc. IRE 38, No. 4, 401-412. 
[30] Villars, F., and V. F. Weisskopf (Oct. 1955), On the scattering of radio waves by 

turbulent fluctuations in the atmosphere, Proc. IRE 43, No. 10, 1232-1239. 
[31] Norton, K. A., P. L. Rice, H. B. Janes, and A. P. Barsis (Oct. 1955), The rate of 

fading in propagation through a turbulent atmosphere, Proc. IRE 43, No. 10, 

1341-1353. 
[32] Smyth, J. B., and L. G. Trolese (Nov. 1947), Propagation of radio waves in the 

lower atmosphere, Proc. IRE 35, No. 11, 1198-1202. 
[33] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric 
refraction and its relation to wave guides and diffraction. Book, Meteorological 
Factors in Radio Wave Propagation, pp. 80-127 (The Physical Society, 

London, England). 



268 TRANSHORIZON PARAMETERS 

[34] Gossard, E. E., and L. J. Anderson (Apr. 1956), The effect of super refractive 
layer on 50-5000 Mc nonoptical fields, IRE Trans. Ant. Prop. AP-4, No. 2, 
175-178. 

[35] Starkey, B. J., W. R. Turner, S. R. Badcoe, and G. F. Kitchen (Jan. 1958), The 
effects of atmospheric discontinuity layers up to the tropopause height on 
beyond-the-horizon propagation phenomena, Proc. lEE, Pt. B, 105, Suppl. 
8, 97. 

[36] Abild, V. B., H. Wensien, E. Arnold, and W. Schilkorski (1952), tJber die Aus- 
breitung ultrakurzer Wellen jenseits des Horizontes unter besonderer Beruck- 
sichtigung der meteorologischen Einwiskungen, Tech. Hausmitteilungen des 
Nordwestdeutschen Rundfunks, p. 85. 

[37] Northover, F. H. (Feb. 1952), The anomalous propagation of radio waves in the 
1-10 metre band, J. Atmospheric Terrest. Phys. 2, 106-129. 

[38] Lane, J. A., and R. W. Meadows (Jan. 1963), Simultaneous radar and refrac- 
tometer soundings of the troposphere. Nature 197, 35. 

[39] Brekhovskikh, L. M. (1960), Waves in layered media. Book (Academic Press, 
New York and London, England). 

[40] duCastel, F. (1961), Propagation tropospherique et faisceaux hertziens trans- 
horizon. Book, p. 90 (Editions Chiron, Paris). 

[41] Wagner, N. K. (1960), An analysis of radiosonde effects on measured frequency 
of occurrence of ducting layers, J. Geophys. Res. 65, 2077-2085. 

[42] Bauer, J. R., and J. H. Meyer (Aug. 1958), Microvariations of water vapor in the 
lower troposphere with applications to long-range radio communications, 
Trans. Am. Geophys. Union 39, 624. 

[43] Saxton, J. A. (July- Aug. 1961), Quelques reflexions sur la propagation des ondes 
radioelectrique a travers la troposphere, L'Onde Elect. 40, 505. 

[44] Hewson, E. W., and R. W. Longley (1944), Meteorology, Theoretical and Ap- 
plied, Book, p. 352 (John Wiley & Sons, New York). 



Chapter 7. Attenuation of Radio Waves 

7.1. Introduction 

The advent of tropospheric forward scatter techniques has made possi- 
ble communications over longer distances with higher frequencies than 
has been heretofore thought practicable. The limitations imposed by 
gaseous absorption, and by scattering by raindrops, upon the power re- 
quirements of a conmiunications system for this application become more 
important with increasing distance and frequency. It has been common 
in the past to evaluate propagation path attenuation due to absorption by 
multiplying the ground separation of the terminals by the value of the 
absorption calculated for surface meteorological conditions [1]^ or avoid 
the problem by restricting the communications system to frequencies 
that are essentially free of absorption [2]. This is in contrast to another 
approach [3] which actually used the absorption along the ray path. 

The following sections of this chapter will be devoted to a descriptive 
treatment of absorption of radio waves by raindrops and gaseous oxygen 
and water vapor in the atmosphere. 

Unless otherwise specified, the following conditions will be assumed in 
this chapter: (1) All attenuations will be expressed in terms of decibel 
loss per unit length of the propagation path (dB/km). The attenuations 
due to different causes are simply added to give the total attenuation in 
decibels. (2) In this treatment average conditions of temperature, drop- 
let size, and droplet distribution are assumed for the radio path in order 
to approximate conditions met in practice. 

7.2. Background 

The attenuation experienced by radio waves is the result of two effects: 
(1) absorption and (2) scattering. At wavelengths greater than a few 
centimeters, absorption by atmospheric gases is generally thought to be 
negligibly small except where very long distances are concerned. How- 
ever, cloud and rain attenuation have to be considered at wavelengths 
less than 10 cm, and are particularly pronounced in the vicinity of 1 and 
3 cm. 



' Figures in brackets indicate the literature references on p. 308. 

2G9 



270 ATTENUATION OF RADIO WAVES 

It is helpful to recall that when an incident electromagnetic wave passes 
over an object whose dielectric properties differ from those of the sur- 
rounding medium, some of the energy from the wave is (a) absorbed by 
the object and heats the absorbing material (this is called true absorp- 
tion), and (b) some of the energy is scattered, the scattering being gener- 
ally smaller and more isotropic in direction the smaller the scatterer is 
with respect to the wavelength of the incident energy. 

In the case of point-to-point radio communications we are interested 
in the total attenuation of the scattering energy caused by losses resulting 
from both the true absorption and the scattering. 



7.3. Attenuation by Atmospheric Gases 

The major atmospheric gases that need to be considered as absorbers 
in the frequency range of 100 to 50,000 Mc/s are water vapor and oxygen. 
For these frequencies the gaseous absorption arises principally in the 
1.35 cm line (22,235 Mc/s) of water vapor and the series of lines centered 
around 0.5 cm (60,000 Mc/s) of oxygen [4]. The variations of these 
absorptions with pressure, frequency, temperature, and humidity are 
described by the Van Vleck [4, 5] theory of absorption. The frequency 
dependence of these absorptions is shown in figure 7.1 [4]. 

In connection with figure 7.1, the water vapor absorption values have 
been adjusted to correspond to the mean absolute humidity, p, (grams of 
water vapor per cubic meter) for Washington, D.C., 7.75 g/m^ The 
reason for this adjustment is that water vapor absorption is directly pro- 
portional to the absolute humidity [6] and thus variations in signal in- 
tensity due to water vapor absorption may be specified directly in terms 
of the variations in the absolute humidity of the atmosphere. 

It can be seen from figure 7.1 that the water vapor absorption exceeds 
the oxygen absorption in the frequency range 13,000 to 32,000 Mc/s, 
indicating that in this frequency range, the total absorption will be the 
most sensitive to changes in the water vapor content of the air, while 
outside this frequency range the absorption will be more sensitive to 
changes in oxygen density. Only around the resonant frequency corres- 
ponding to X = 1.35 cm is the water vapor absorption greater than the 
oxygen absorption. The absorption equations and the conditions under 
which they are applicable have been discussed by Van Vleck [4], and the 
best values to use for this section of the report have been taken from 
Bean and Abbott [3]. 

The Van Vleck theory describes these absorptions from 100 Mc/s to 
50,000 Mc/s in the following manner. The oxygen absorption at 



ATTENUATION BY ATMOSPHERIC GASES 



271 




100 200 500 1,000 2,000 5,000 10,000 20,000 50,000 

Frequency in Mc/s 

Figure 7.1. Atmospheric absorption by the 1.35 cm line of water vapor and the 0.5 cm 

line of oxygen. 



272 



ATTENUATION OF RADIO WAVES 



T = 293 °K and standard atmospheric pressure in decibels per kilometer, 
7i, is given by the expression: 



Ti 



0.34 


Ai'i 1 Aj^2 




X2 


+ 

ji + AA (2 + iy + A.l 

1 


Al^2 




(- 


- 0' + ^'i 



(7.1) 



where X is the wavelength for which the absorption is to be determined 
and where Ai'i and Ai'2 are line-width factors with dimensions of em~^ 
This formula is based on the approximations of collision broadening theory. 
This theory postulates that, although the electromagnetic energy is 
freely exchanged between the incident field and the molecules, some of 
the electromagnetic energy is converted into thermal energy during 
molecular collisions and thus a part of the incident electromagnetic energy 
is absorbed. The term in (7.1 )involving Aj'i gives the nonresonant 
absorption arising from the zero freciuency line of oxygen molecules while 
the terms involving Az^2 describe the effects of the several natural resonant 
absorptions of the oxygen molecule which are in the vicinity of 0.5 cm 
wavelength. The (2 ± 1/X) (cm~^) terms are the portion of the shape 
factors that describe the decay of the absorption at frequencies away from 
the resonant frequency (the number 2 is the reciprocal of the centroid 
resonant wavelength 0.5 cm). 

The water vapor absorption at 293 °K arising from the 1.35 cm line, 
72, is given by: 



72 

P 



3.5 X 10" 

X2 



AV3 



AX 1.35/ 



+ AV3 

+ 



Avs 



\X ^ 1.35/ 



+ AV3 



(7.2) 



where p is the absolute humidily and Afg is the line width factor of the 
1.35 cm water vapor absorption line. The additional absorption arising 
from absorption bands above the 1.35 cm line, 73, is described by: 



73 
P 



0.05 Av4 

X2 



(7.3) 



ATTENUATION BY ATMOSPHERIC GASES 



273 



where Au^ is the effective Hue width of the absorption bands above the 
1.35-c'm Hne. The nonresonant term has been increased by a factor of 
4 over the original Van Vleck formula in order to better satisfy experi- 
mental results [7]. 

Although Van Vleck gives estimates of the various line widths, more 
recent experimental determinations were used whenever possible. The 
line-width values used in this chapter are summarized in table 7.1. 

Table 7.1. Line width factors used to determine atmospheric absorption 



Line 
width 


Temperature 


Value 


Sources 


Al'2 

Ai's 

Aj/4 


293 °K 
300 °K 
318 °K 
318 °K 


0. 018 cm-i atm^i 
. 049 cm~i atm-i 
. 087 cm^i atm-i 
. 087 cm-i atm-i 


Birnbaum & Maryott [8] 
Artman & Gordon [9] 
Becker & Autler [7] 
Becker & Autler [7] 



The preceding expressions for gaseous absorption are given as they 
appear in the literature and do not reflect the pressure and temperature 
sensitivity of either the numerical intensity factor or the line widths. 
This sensitivity must be considered for the present application since it is 
necessary to consider the manner in which the absorption varies with 
temperature and pressure variations throughout the atmosphere. The 
dependence of intensity factors upon atmospheric pressure and tempera- 
ture variations was considered to be that given by the Van Vleck theory. 
The magnitude and temperature dependence of the line widths is a 
question not completely resolved. Both theory and experiment indicate 
the line width to vary as 1/7'^, x <0. Different measurements on the 
same line of oxygen have given values of x ranging from 0.71 to 0.90 
with differences in the magnitude of Ai^ of about 2 percent [10, 11]. Ex- 
periments have also clearly indicated that the line width changes from 
line to line, with maximum fluctuations of about 15 percent. In the 
frequency region considered in this chapter (10 to 45 Gc/s) the centroid 
frequency approximation for oxygen is valid and a mean line width can be 
used with good accuracy, but in the region of the resonant frequencies of 
oxygen, the line-to-line line width variations must be taken into account. 
The expressions used to calculate the absorptions are given in table 7.2. 
The reference temperatures given are those at which the appropriate 
experimental determinations were made, and the pressures are to be 
expressed in millibars. A detailed discussion of the theoretical aspects of 
the pressure and temperature dependence is given by Artman [12]. 

Experimental measurements on the absorption of microwaves by the 
atmosphere (performed after our original work) show different values of 
the loss than those obtained by theoretical prediction methods. There 
is reasonably good agreement between the predicted and measured loss 
for oxygen, but the measured loss of water vapor is considerably greater 



274 ATTENUATION OF RADIO WAVES 

Table 7.2. Values used in the calculation of atmospheric absorption 



Absorption 


Intensity factor Line width 


dB/km 

71 


0. 34 / P \ /293\ 2 
X2 \1013. 25/ \ T J 


cm ' 

\1013. 25/ \ T / 
and 

\1013.25/ \ T / 


72* 
P 


0. 0318 /293\ 5/2 / 644\ 
X2 \ T / \ T / 


/ P \ /318\ "2 

Ai-a ( ) { ) (1+. 0046p) 

\1013. 25/ \ T / 


73* 

p 


0. 05 /293\ 
X2 \ T/ 


/ P \ /318\>/2 

Ar4 ( ) ( — ) (1+. 0046p) 

\1013. 25/ \ T / 



*p is water vapor density in g/m^ 

than that of the predicted amount, particularly above 50,000 Mc/s [13]. 
These observed discrepancies have little effect upon the present study, 
which is confined to frequencies less than 50 Gc/s. The results of the 
present study, for the frequency range 100 to 50,000 Mc/s, agree with 
those reached by Tolbert and Straiton [14] in their field experiments at 
Cheyenne Mountain and Pikes Peak, Colo., at altitudes of 14,000 ft. 

The above approach represents that presented by Bean and Abbott 
[3]. The following treatment was given by Gunn and East [15] and 
based on Van Vleck's two papers [4, 5]. This latter presentation is only 
valid when single line absorption with no appreciable overlap from 
adjacent lines is considered. 

By taking into account the temperature and pressure dependence of 
the line widths, it is seen that for a given quantity of water vapor, the 
attenuation is proportional to 

P- and T- exp (- f ) , 

at the resonance line, to 

P and T~' exp (- ^) 



at the sides of the curve, and to P and T"3/2 ^g][ away from resonance. 
In applying the above considerations to absorption approximations it 
also must be remembered that for a given relative humidity, the density 
will vary considerably with temperature. Table 7.3 shows attenuation 
by water vapor at various temperatures and wavelengths. 



ATTENUATION BY ATMOSPHERIC GASES 275 

The behavior of water vapor attenuation near the resonant line is very 
remarkable, as can be seen by inspecting (7.2). Since Av^ is small com- 
pared to 1/X, it may be neglected in the denominator of (7.2) for non- 
resonant wavelengths. The attenuation per unit density is thus directly 
proportional to /^v^. and hence to the total pressure for these frequencies. 
But at the resonant frequency, the dominant term in the expression is 
proportional to l/A^s, and thus inversely proportional to the pressure. 
In the atmosphere, the water vapor density is proportional to the total 
pressure. Therefore, the attenuation is independent of pressure at the 
resonant frequency and now depends only on the fraction of water vapor 
present. For practical purposes, this means that attenuation can occur 
at high altitudes with the same effectiveness as in the lower, denser layers 
if the mixing ratio is the same. 

On the other hand, oxygen absorption occurs because of a large number 
of lines around 60 Gc/s. In the region from 3 to 45 Gc/s the attenuation 
is proportional to P- and to T-n/^ ^5j ^g ^j^g temperature decreases the 
attenuation increases gradually. At —40 °C oxygen attenuation is about 
78 percent higher than at 20 °C due to increased density at low tempera- 
tures. Table 7.4 shows the pressure and temperature corrections for 
oxygen attenuation at wavelengths between 0.7 and 10 cm. 

Figure 7.2 shows the attenuation at a pressure of 1 atmosphere and 
20 °C as a function of wavelength [15]. The solid lines represent values 
of attenuation measured by Becker and Autler [7]. The dashed line 
shows values calculated from Van Vleck's theory. The water vapor 
absorption curve, c, corresponds to a water content of 1 g/m^. 

Since absorption is so sensitive to the absolute humidity level, it is 
helpful to have information on the climatic variation of absolute humidity 
throughout the 1 to 99 percent range of values normally used in radio 
engineering. Estimates of the values of absolute humidity at the surface 
expected 50 percent of the time for the United States for February and 
August are given in figures 7.3 and 7.4 respectively [16]. It is evident 
that for either month the coastal regions display greater values of absolute 
humidity than do the inland regions. Note that for any location the 
August values are consistently greater than the February values. Figures 
7.5 through 7.8 show the values of absolute humidity expected to be 
exceeded 1 and 99 percent of the time throughout the United States in 
both summer and winter. 

In addition to oxygen and water vapor, there are a number of other 
atmospheric gases which have absorption lines in the microwave region 
from 10 to 50 Gc/s. These gases normally constitute a negligible portion 
of the general composition of the atmosphere, but could conceivably 
contribute to attenuation. Table 7.12 shows the resonant freciuencies, 
maximum absorption coefficients at 300 °K (attenuation coefficient if the 
fraction of molecules present were equal to unity), expected concentration 



276 



ATTENUATION OF RADIO WAVES 



Table 7.3. Water vapor attenuation (one way) in dB/km 

[After Gumi-Eastl. 

P, pressure in atmospheres; W, water vapor content in g/m"^ 



T 

(°C) 


X(cm) 10 


5.7 


3.2 


1.8 


1.24 


0.9 


20 



-20 

-40 


0.07X10-3PW 

0. osxio-^pw 

0.09X10-3PW 

0. loxirspw 


0. 24X10-3P\V 
0.27X10-3PW 
0.30X10-3PW 
0.34X10-3PW 


0.7X10~3PW 
O.SXIO-'PW 
0.9X10-3PW 
1.0X10-3PW 


4.3X10-3PW* 

4. 8X10-3PW* 

5. OXlO-spW 
5. 4X10-3PW 


22.0X10-3PW* 
23.3X10-3PW* 
24.6X10-3PW' 
26. 1X10-3PW* 


9. 5X10-3PW 
10.4X10-3PW 

11. 4X10-3PW 

12. 6X10-3PW 



*The pressure dependencies shown are only approximate. Near the 1. 35 cm water vapor absorption line 
(between 1. cm and 2. cm) no simple power dependency of P and W is accurate. 



Table 7.4. Pressure and temperature correction for oxygen attenuation for wavelengths 

between 0.7 and 10 cm 
[After Qunn-East] 



n°c) 


Factor 




(P is pressure in atmospheres) 


20 


1.00P2 





1. 19 P2 


-20 


1. 45 P2 


-40 


1. 78 P2 



o 

O I0-' r 

CVJ 

< 



I 10-2 



CD 



8 10-3 
< 

Z 

UJ 



^-4 



fV 




\ 


A ">' 


7 


\i , . 


:^>s^ 


"^^^(<') 


- 


\ 
S 


. 


\ 




\ 


- 


X (C) 


* 


\ 


• 


s 




v 




\ 




\ 


■ 


\ 




V 


1 1 


1 1 1 . i .. I 1 'Ik 1 



0.8 



.5 2 



6 8 10 



A (cm) 



Figure 7.2. Attenuation of microwaves by atmospheric gases. 
(After Guim and East, 1954). 



ATTENUATION BY ATMOSPHERIC GASES 



277 




Figure 7.3. Estimate of the value of absolute humidity expected 60 percent of the time 

for February. 




Figure 7.4. Estimate of the value of absolute humidity expected 50 percent of the time 

for August. 



278 



ATTENUATION OF RADIO WAVES 




Figure 7.5. Values of absolute humidity expected to be exceeded 1 percent of the time 

for February. 




Figure 7.6. Values of absolute humidity expected to be exceeded 1 percent of the time 

for August. 



ATTENUATION BY ATMOSPHERIC GASES 



279 




Figure 7.7. Values of absolute humidity expected to be exceeded 99 percent of the time 

for February. 




Figure 7.8. Values of absolute humidity expected to be exceeded 99 percent of the time 

for August. 



280 ATTENUATION OF RADIO WAVES 

in the atmosphere, and expected absorption coefficients due to these 
trace constituents. The data on molecular absorption coefficients was 
taken from Ghosh and Edwards [33], that on concentrations from the 
Glossary of Meteorology [29]. It is readily seen that the attenuation due 
to these sources is negligible compared to the high absorption due to 
oxygen and water vapor. 



7.4. Estimates of the Range of Total 
Gaseous Absorption 

The range in gaseous absorption can be seen by considering the data 
for the months of February and August at Bismarck, N. Dak., and 
Washington, D.C., two stations with very different climates. The values 
of total gaseous absorption (defined as the sum of 71, 72, and 73, where 
7i = oxygen absorption in decibels per kilometer, 72 = water vapor 
absorption arising from the 1.35 cm line and 73 = additional absorption 
arising from absorption lines whose frequencies are considerably higher 
than that corresponding to the 1.35 line) at each station and elevation up 
to 75,000 ft are shown in figures 7.9 and 7.10 for each of the four station 
months for the frequency range of 100 to 50,000 Mc/s. Above 75,000 ft 
the absorption values for all four station months are identical and are 
given for each frequency in figure 7.11. The absolute humidity was 
calculated using the upper air monthly average values of temperature, 
pressure, and humidity as reported by Ratner [17]. Readings for the 
relative humidity are not generally given in this report for altitudes 
greater than about 15 km due to the inability of the radiosonde to meas- 
ure the small amount of humidity present at these altitudes. It is 
believed that the climates represented by these station months encompass 
the range of those of the majority of the continental United States radio 
propagation paths. 

An interesting property of the annual range of absorption as a function 
of the frequency may be seen in figures 7.9 and 7.10. For the first 5,000 
ft above the surface, it is noted that in the frequency range of 10 to 32.5 
Gc/s, the summer values are greater than the winter values due to in- 
creased humidity of the summer months. Outside of this frequency 
range, however, the winter values of absorption are greater due to the 
increased oxygen density. 

In the frequency range 6 to 45 Gc/s, atmospheric absorption, 7;,, at 
a frequency v, arises primarily from oxygen absorption, jdi,, and water 
vapor absorption, juw', i.e., 

7^ = 7rf. + Jwv. (7.4) 



RANGE OF TOTAL GASEOUS ABSORPTION 



281 





7 
5 

3 
2 

0.1 

a? 

0.5 

a3 

0.2 

0.1 
0.07 

ao5 

0.03 
0.02 

0.01 
0.007 
0005 

0003 

0.002 

ODOI 






: : : .. 






















BISMARK, N D. 




































' 

























\ 
























-\ 
































































^ 


-50,000 Mc/s 






















\ 




HJ 








- 




ebruary 
ugust 








A 






\ ^ 
























%v 
























\ N 






















\ 


\ 1 1 ._! ._ 


\, 






















\ 


1 i 


\. 




















vV ^ 


^x-U 22,000 Mc/s 




^^ 


















\r\\^ ^ 


'v 






\ 


















32,000 Mc/ 


\\ 


\ 
\ 

\ 








\ 


\:- 














q3 




\ 

\ 
\ 








\ 


\ 


\ 
\ 










E 


\^r 


\ X \ 






















HO,OOOMc/s 


\ 


x\ 












s ^ 










— 




\ 


^■^ 












\ 


\ 








i^ 


^ 




N 




*^ 












\\ 








a> 


^""C"^ 


\ 






\ 


\s 












\N 










\\ 


Iv 




\ 
















^N 








3,000 Mc/s^^ 


\ 




I % 












\ 






o 

0) 

Q 


§^ 






\ 

\ 
\ 


^- 


^. 










\ 




o 


^ 


tg^ 


00 Mc 


/s ^ 


--^ i ^c^ 




\ 
\ 

\\ 














\ 


S 








'^--'-''^' "S."^ 




^,, 


\ V 












^ 











^'V 


>^ ^^J^. 




\ 


\ 












< 












SA^ 


,^ 


^ N 


N 


\ 


























C'^v 


N ^ 




\ ^. 
























^.\^'^ 


\ 


\ 


\' 


V 
























NV\x 


\ ^ 


\ ^v 




y\ 


























is^ 


A 


\ 


\v \ 












' 












\^ 


\ 


\ 








-T_^ 


— ^ — ^ — 


■~>=». 








\ 


-100 w 


c/s 






^^ 


..,__2l"^-> 








^v 


k 


\ 














































X-^XXX \,\ ! N 


























k^^^^^ >^^ 






OBOOS 

0M03 
QD002 

onooi 
























X^Sy^ 


V 




























>!&5-, 


s 


X 




























•^ 


^ 


'x 




























^ 




s 
































^ 
































^ 



5 10 15 20 25 30 35 40 45 50 55 

Height Above the Surface in Thousands of Feet 



65 70 75 



Figure 7.9. Total gaseous atmospheric absorption from the surface to 75,000 feel: 

Bismarck, N. Dak. 



282 



ATTENUATION OF RADIO WAVES 



1 








1 


1 1 1 : 
















■ 










1 


0.T 
















^\J 1 W 


'1 *-'• 






































0.5 




























vx. 




'50,000 Mc/s 
























^ 


























0.2 


^\ 






























'\ 








is. 


















- 


August 
















■*v 




















^^ 




\ 






s V 




















\ \ 












V 














] 




\v 




\ 








V 














1 




s \ 




^^ 




IZ,Z0(. 


Mc/s 


^ 


N 


















\ 


\, 




^</' 




\^ 
















neter 


s 


n\ 


^ / 


' < 








\ 


















N 


?>^ 


\ 


\ 








\ 


s. 












32,50 


DMc/s 


K^ 


^J^ 












\ ""n 


\'^^ 










V ^s 




000 M 




vV 












\ 












S ^ 






^VS 












\ 


\ 










I \' 


r 






s ^ 


^ \ 












s N 












^\ 






\ 


N. ^ 












\, ^ 








oi 


^•,.\ 


N> 








\ >! 












\ 


\ 












^ 






\^ 


\. 












NN. 






•"§ aoo3 

.£ 0.002 
o 

Q. 

_§ 0.001 


"^ 


^ 






\ 


\ 


V 

\^'^ 










\ 


V 




3,a 


DO Mc/ 


S^ 


^ 


V, 




\ 


'^ 


^ '^S. 










\\ 




^^^ 


:r7f^ 


-300 r 


Ic/S 


^ 


k 


\ 


N 

\ 


\ 












\ 














vN> 


\ 




\ 






















^- 


-^S^,^ 


S\- 




N, 




s ^, 










0.0007 












"^ 


^^ 


\^ 


N 


\ 


s 






















f'^ 




\ 


\ 


s 


^\ 






















"s 


^S\\ 




\ 




N '\ 
























^\Si\"~"^v 


\ 


\ 


\^ 
























N 


r^^ 




s^ \^ 


\ 


\\ 










// 














^ 


V V^ 


\^\ 


\ 
\ 


X 


. 




00002 


















^^^"^ 


•--.^ 








\^ 


\ 
























\ ^ 




\ 




X 
























N 


x^ 


































^^\ 






























\^ 


^V^^ 


\; 






























X >^. i 


aoooo5 


























^ 




V 
































> 
































%s 


0.00001 






























^ 



5 10 15 20 25 30 3a 40 45 50 55 

Height Above the Surface in Thousands of Feet 



65 ID 75 



Figure 7.10. Total gaseous atmospheric absorption from the surface to 75,000 feet: 

Washington, D.C. 



TOTAL RADIO PATH ABSORPTION 283 

Zhevankin and Troitskii [31] have indicated that 7^^ and 7^,^ can be repre- 
sented as exponential functions of height, Z, above the earth's surface, 



7d. = 7d.o exp [j^~), Iw. = Tu-.o exp \j[—)> (7-5) 



where 7<f„o and 7^;,o are the values of jdv and jwy, respectively, at the 
earth's surface, and H dv and //„,;, are called the "scale heights" of 7^^ and 
7^,„. This model is known as the "bi-exponential" model of absorption, 
and ydy and 7,^,^ are often called the "dry" and "wet" terms of y^. The 
scale height for the dry term in the frequency range 6 to 45 Gc/s can be 
written as [32] 



Hd. (m km) ^ -—V (7.6) 



where To is the surface temperature in °K, a is the temperature lapse rate 
with height in °K/km, and b, c, are constants determined from thermo- 
dynamic considerations. Because of the hump in the Hy,^ curves as op- 
posed to the flat H dp curve in figure 7.28, such a handy expression as (7.6) 
for H „,^ is not possible in the 6 to 45 Gc/s frequency range (fig. 7.28 was 
determined from actual radiosonde data at Verkhoyansk, U.S.S.R.). 



7.5. Total Radio Path Absorption 

The total path absorption is determined by calculating the various 
absorption coefficients as functions of the heights along the ray path and 
then numerically integrating the values along the entire path using stand- 
ard ray tracing techniques outlined in chapter 3. The values of total path 
integration over a 100-km path thus obtained are presented in figure 7.12 
for Bismarck, N. Dak., and Washington, D.C. The difference between 
the two climates is evident principally at the higher frequencies, where 
the Washington absorptions are consistently above the Bismarck values. 
This is apparently due to a combination of generally greater humidities 
and greater refractive effects. These two effects are related. The in- 
creased humidity at Washington enhances the water vapor absorption 
and increases the refraction causing the radio ray to travel consistently 
through lower levels of the atmosphere with consequent increase in total 
path absorption. 



284 



ATTENUATION OF RADIO WAVES 



aoooooo3 

00000002 



\ 






























\ 
































S 






























\ 






























\ 






























\ 




























\ 


s 




























\ 


^50,0( 


X) Mc/ 


s 


























\ 


\ 






























V 
































k 


















\ 












\ 


















N 












\ 


















\ 














\, 


















\ 












\ 
















\ 


\ 


v^^ 


2,500 Mc/s 








\ 














\ 


^ 


\ 


^ 










\ 


\ 












\ 


^^ 


^22,2 


oo\ 

:/s 


\ 










\ 


\ 












■^x 


V 


\ 














\ 










xv.\. 




\ 




\ 












N, 










^ 


\ 


S 




\ 












N 












^N 


\ 




N 












N 












^> 




\ 




\ 












\ 










^ 


k 


\ 




\ 












\ 








lopc 


)OMc/s 


\n 


\ 




\ 












\ 










3,000 


Mc/s ' 


^ 




\ 






















300 £ 


ilOOM 


c/.'^ 


k 


\ 


\ 


\ 


\ 










\ 














V\, 


\ 




\ 
























X^ 




\ 


\ 
























X 




\ 




\ 


























\ 




\ 


























\ 


\ 


\ 
























\ 


\ 


\ 




\ 
























\\ 


\ 


^ 


\ 
























\ 


^ 


\ 




\, 
























\ 


\ 


\ 


\ 
\ 




s 



Heigiit Above the Surface in Thousands of Feet 

Figure 7.11. Common values of total gaseous atmospheric absorption for elevations 

greater than 75,000 feet. 



TOTAL RADIO PATH ABSORPTION 



285 



CD 20 



CD I 




500 700 1000 



2000 5000 7000 10000 

FREQUENCY (MC/S) 



20000 



50000 70000 100000 



Figure 7.12. Total path absorption over a 1000 km propagation path with the climate 

of Bismarck, N. Dak. 



286 ATTENUATION OF RADIO WAVES 

7.6. Derivation of Absorption Estimate for Other Areas 

The values of total path absorption given above are for two specific 
locations. For this particular study to be of practical use, a means 
should be provided for arriving at estimates of geographic and annual 
variations of total path absorption for various surface distances and 
frequencies. The method chosen utilizes the correlation between total 
path absorption and the surface value of the absolute humidity, expressed 
in grams of water vapor per cubic meter, which appears explicity in 
Van Vleck's water vapor absorption formulas [4,5]. The basis for expect- 
ing a correlation to exist between the absolute humidity and the total 
path absorption is that the absorption at those frequencies for which 
water vapor absorption is dominant (approximately 10 to 32 Gc/s) varies 
directly as the absolute humidity while, for those frequencies at which 
oxygen absorption is dominant, it varies inversely as the absolute humid- 
ity due to the inverse relationship of oxygen density and water vapor 
density. That is, during the warm seasons of the year the total atmos- 
pheric pressure tends towards its yearly minimum (as does the oxygen 
absorption), while the absolute humidity tends towards its yearly maxi- 
mum (as does the water vapor absorption). Conversely, during the 
colder seasons of the year, the pressure tends towards its maximum value 
while the absolute humidity tends towards its minimum value [3]. 

As an example of the correlation method, the surface absorption was 
calculated at a water vapor-dominated frequency (22.2 Gc/s) and 
oxygen-dominated frequency (50 Gc/s) for each month throughout the 
year for both Washington and Bismarck. These values are plotted on 
figure 7.13. The term surface absorption is used for the values of ab- 
sorption calculated from standard ground level weather observations. 
The water-dominated 22.2 Gc/s data fall on a smooth curve despite the 
pressure and temperature differences of the two stations. The oxygen- 
dominated 50 Gc/s data, however, display an interesting separation of 
points for each station, although the distribution of points at the two 
locations display similar slopes. The 50 Gc/s absorption is more sensi- 
tive to the atmospheric density difference between the stations. If the 
pressure differences were taken into account, the Bismarck data would 
increase about 12 percent and the two curves would be distributed along 
a common line w^ith the same slope as the original two curves. This 
figure, then, indicates that the absorption is correlated with the absolute 
humidity. The above illustration is for surface values rather than for 
integrated propagation path values. Variations in the upper air meteor- 
ology that are not reflected in the surface values will tend to diminish 
the correlation. (For comparison of percentage absorption over a 300-km 
path, the first few hundred feet contribute about 5 percent at 100 Mc/s 
increasing to 42 percent at 10 Gc/s and remaining constant to 50 Gc/s.) 
Keeping these reservations in mind, one may utilize the method of least 



ABSORPTION ESTIMATE FOR OTHER AREAS 



287 



o 

Q. 
O 



CO 



0.07 



0.05 



0.03 



0.02 



0.01 



0.007 



0.005 



0.003 



0.002 



0.001 













































































































"— 




-^ 




& 
































--^ 


-^ 




> 


>o 


























i» 






















/ 


/ 


















/ 


/ 


/ 


% 


















/ 


























/ 
























/ 
























r/ 


^ 
























/ 
























/ 






















/ 


























</ 




































o Washington 0^ at 50,000 Mc/s 
• Washington H^O at 22,200 i^c/s 
9 Bismar/< 0^ at 50,000 l\4c/s 
« BismariK H^O at 22,200 /l/c/s 

1 1 1 1 1 1 



0.2 0.3 0.4 0.5 OB 0.7 0.8 0.9 



1.2 1.3 



Log of Absolute Humidity (gm/m ) 

Figure 7.13. Surface value of 22.2 and 60 Gc/s absorption versus absolute humidity at 
Washington, D.C., and Bismarck, N.Dak. 



288 



ATTENUATION OF RADIO WAVES 



Table 7.5. Values of m and b in the regression line y = mx + b, 
where y is the logarithm of the total path attenuation, y, b is the value of y when the absolute humidity 
is 1 g/m^ and x is the logarithm of the absolute humidity. 
[After Bean and Abbott] 



Distance 


100 km 


300 km 


1000 km 


Freq. (Gc/s) 


m 


b 


m 


6 


m 


6 


100 .. 


-0. 07263 
-. 08212 
-. 10203 
+. 06996 
+. 26022 
+. 48097 
+. 62034 
+. 75045 
+. 77630 
+.77153 
+. 74979 
+. 54155 
+. 46305 
+. 25936 
+. 18460 


-3. 69822 

-1.72139 

-0. 49427 

-. 24077 

-. 14038 

+. 20891 

+. 65324 

+1. 41086 

+1. 67001 

+1. 69105 

+1. 62838 

+1.34408 

+1. 51053 

+2. 46361 

+2. 91681 


-0. 06324 
-. 06363 
-. 07078 
+. 06872 
+. 23447 
+. 44214 
+. 58044 
+. 70693 
+. 72874 
+. 72508 
+. 70659 
+. 50126 
+. 42416 
+. 23331 
+. 16668 


-2. 60042 
-0. 73037 
+.31846 
+. 54167 
+. 64637 
+. 98393 
+1. 42477 
+2. 25807 
+2. 57270 
+2. 57734 
+2. 46633 
+2.11161 
+2. 28497 
+3. 25085 
+3. 70472 


-0. 13298 
-. 15276 
-. 16932 
-.04731 
+. 10831 
+. 31506 
+.46112 
+. 60558 
+. 62885 
+. 62963 
+. 60458 
+. 37634 
+. 29646 
+. 10712 
+. 04360 


1 83711 


300 


17792 


1000 


+ .75687 
+ .96517 
+ 1.06172 
+1. 36994 
+1. 77604 
+2. 56360 
+2. 87766 
+2. 87575 
+2. 77175 
+2. 48455 
+2. 67460 
+3. 66659 
+4. 12531 


6000 

10,000 


15,000 

18,000 


21,000 

22,406 


23,076 . 


24,000 

30,000 


33,000 


42,000_ . 


45,000 






SWOO 70CWO ICBOM 



Figure 7.14. Values of h and m in the regression line y = mx + b, where y is the 
logarithyn of the total path absorption, x is the logarithm of the absolute humidity, b is 
the value ofy when the absolute humidity is 1 g/m^. 



ABSORPTION ESTIMATE FOR OTHER AREAS 



289 



squares to obtain the total path absorption as a linear function of the 
absolute humidity. The parameter is a statistical regression line at each 
frequency for path propagation distances of 100, 300, and 1,000 km as 
given in table 7.5, and also plotted on figure 7.14, allowing the reader to 
estimate the regression line for frequencies other than those of the 
present study [3]. 

Statistics for the variation of absolute humidity have already been 
given, and contours of the mean values of absolute humidity for the world 
are given on figures 7.15 and 7.16 for the months of February and August 
respectively. It is noted that the absolute humidity for either month is 
greater in the coastal regions than in the continental interiors and that 
the windward sides of continents have larger values than do the leeward 
sides. 

By reference to figures 7.15 and 7.16 [16] for the United States, and 
figure 7.14, estimates of radio power loss due to absorption may be 
obtained. As an example, figure 7.17 gives the values expected to be 
exceeded 1 percent of the time over a 300 km propagation path at 10,000 
Mc/s for the United States during the month of August. 




»• lOO- nr \vr i60't»si iwiEsi lec no* i!0' lOO* sc 



«• 20ESI 0'£«SI W «!■ 



Figure 7.15. Average absolute humidity (g/ni^), February. 



290 



ATTENUATION OF RADIO WAVES 



20* 140' 160'iASI I80'*ESI leC HO* 120' W eo* 60' «• WKSI 0*E«SI JIf w 




100' OT IW lSO'f«ST IJO'IESI ISO- HO- l!0' 100' IC »• 40* !<ri£ST 0*E»ST 20' 40* 



Figure 7.16. Average absolute humidity {g/m}), August. 




Figure 7.17. August values of 300-mi propagation path absorption loss {in dB) to be 
exceeded 1 percent of the time for a frequency of 10,000 Mc/s. 



ATTENUATION IN CLOUDS 



7.7. Attenuation in Clouds 



291 



Cloud droplets are regarded here as those water or ice particles having 
radii smaller than lOO/i or 0.01 cm. For wavelengths of incident radiation 
well in excess of 0.5 cm, the attenuation becomes independent of the 
drop size distribution. The generally accepted equations for attenuation 
by clouds usually show the moisture component of the equations in the 
form of the liquid water content (g/m^). Observations indicate that the 
liquid water concentration in clouds generally ranges from 1 to 2.5 g/m^ 
[18], although Weickmann and aufm Kampe [19] have reported isolated 
instances of cumulus congestus clouds with a reading of 4.0 g/m^ in the 
upper levels. In ice clouds, it will rarely exceed 0.5 and is often less than 
0.1 g/m^. The attentuation of cloud drops may be written as: 

K = KiM, 

where K = attenuation in dB/km, 

Ki = attenuation coefficient in dB/km/g/m', and 

M = liquid-water content in g/m^. 

Values of Ki by ice and water clouds are given for various wavelengths 
and temperatures by Gunn and East in table 7.6. 



Table 7.6. One-way attenuation coefficient, Ki, in clouds in dB/km/gm/m^ 



Temperature 




Wavelength (cm) 


(°C) 


0.9 


1.24 


1.8 


3.2 


Water cloud 
Ice cloud 


20 
10 


-8 


-10 
-20 


0.647 
.681 
.99 

1.25 

8. 74X10-3 
2.93X10-3 
2. XlO-3 


0.311 
.406 
.532 
.684 

6.35X10-3 
2.11X10-3 
1.45X10-3 


0.128 

.179 

.267 

.34 

(extrapolated) 

4.36X10-3 
1.46X10-3 
1.0 xio-3 


0.0483 

.0630 

.0858 

.112 

(extrapolated) 

2.46X10-3 
8. 19X10-3 
5.63X10-" 



Several important facts are demonstrated by table 7.6. The decrease 
in attenuation with increasing wavelength is clearly shown. The values 
change by about an order of magnitude for a change of X from 1 to 3 cm. 
The data presented here also show that attenuation increases with de- 
creasing temperature. Ice clouds give attenuations about two orders of 
magnitude smaller than water clouds of the same water content. The 
attenuation of microwaves by ice clouds can be neglected for all practical 
purposes [20]. The comprehensive works of Gunn and East [15] and 
Battan [20] on attenuation offer excellent sources of detailed information 
on this subject. 



292 ATTENUATION OF RADIO WAVES 

7.8. Attenuation by Rain 

Ryde and Ryde [21] calculated the effects of rain on microwave propa- 
gation and showed that absorption and scattering effects of raindrops 
become more pronounced at the higher microwave frequencies where the 
wavelength and the raindrop diameters are more nearly comparable. In 
the 10 cm band and at lower wavelengths the effects are appreciable, but 
at wavelengths in excess of 10 cm the effects are greatly decreased. It is 
also known that suspended water droplets and rain have an absorption 
rate in excess of that of the combined oxygen and water vapor absorp- 
tion [3]. 

In practice it has been convenient to express rain attenuation as a 
function of the precipitation rate, R, which depends on both the liquid 
water content and the fall velocity of the drops, the latter in turn depend- 
ing on the size of the drops. 

Ryde studied the attenuation of microwaves by rain and .showed that 
this attenuation in dB/km can be approximated by: 



Ku=K RirYdT {1.1) 



where Kr = total attenuation in dB 

K = con.stant 
R{r) = rainfall rate 

r = length of propagation path in km, and 
a = constant. 

Laws and Parsons [22] observed the distribution of drop sizes for 
various rates of fall on a horizontal surface. The higher the rainfall rate, 
the larger the droits, and also the greater the range in size of the drops. 
However, in order to derive the size distributions occurring while the 
drops are falling in the air, each rainfall rate must be divided by the 
particular velocity of fall appropriate to the corresponding drop diameter. 
Figure 7.18 shows the resulting distribution in air, expressed as the rela- 
tive mass of. drops of given diameter, for three chosen representative 
values of precipitation rate, p, namely, 2.5 mm/hr, 25 mm/hr, and 100 
mm/hr [23]. The excess path loss per mile, according to Ryde, for the 
three carrier frequency bands of 4, 6, and 11 Gc/s is shown on figure 7.19. 
Figure 7.20 [24] is a scatter diagram showing transmission loss versus rain- 
fall rate. For comparison, Ryde's equation is plotted on figure 7.20. 

The greatest uncertainty in predictions of attenuation due to rainfall, 
using theoretical formulae as a ba.sis for calculation, is the extremely 
limited knowledge of drop size distribution in rains of varying rates of 
fall observed under differing climatic and current weather conditions. 



ATTENUATION BY RAIN 



293 



w 
w 

<; 
'^ 

w 

> 
I— I 

H 

<: 

w 
Pi 




O-G 



Figure 7.18. Relative total mass of liquid water in the air contributed by rain drops of 

diameter D for various precipitation rates. 

(Derived from Laws and Parsons' distributions for a horizontal surface by dividing by the appropriate 

terminal velocities.) 



294 



ATTENUATION OF RADIO WAVES 



40 



20 



1 
0.8 
0.6 

0.4 



0.2 

O.t 
0.08 

0.06 
0.04 

0.02 



O.Ot 































/ 


























/ 


/ 


























/ 




























/ 






























/ 




























/ 


/ 




/ 


/ 




















/ 


/ 


/ 


^ 


r 




















/ 






/ 


















Lfv 


f 






/ 




















( 
<;\ 


r 








/ 






^ 




















<0 


y 




/ 


/ 












/ 




J 


/ 


/ 




f 


r 
















/ 




/ 






/ 


/ 
















J 


f 


j\ 


/ 






/ 
















/ 


f 




/ 








/ 












> 


/ 


' 




/ 


/ 


/ 


/ 


/' 














/ 






/ 


/ 


/ 


/ 














_ 



0.01 0.02 0.04 0.1 0.2 0.4 0.6 1 2 

RAINFALL RATE IN INCHES PER HOUR 

Figure 7.19. Rain attenuation versus rainfall rate. 
(Theoretical, after Ryde and Ryde). 



6 8 10 



There is little evidence that a rain with a known rate of fall has a unique 
drop-size distribution though studies on this problem seem to indicate 
that a most probable drop size distribution can be attached to a rain of 
given rate of fall [22]. Table 7.8 shows the percentage of total volume 
of rainfall occupied by raindrops of different diameters (cm) and varying 
rainfall rates (mm/hr). Tables 7.8 and 7.9 are offered as an aid to 
estimating, through a qualitative approach, the attenuation of radio 
waves by raindrops [25]. 

Table 7.8 gives the decibel attenuation per kilometer in rains of different 
rates of fall and radio wavelengths between 0.3 and 10 cm. In table 7.9, 



ATTENUATION BY RAIN 



295 



similar to table 7.8 an additional set of results is contained for rains of 
measured drop size distribution. 

Since the total attenuation cross section [26] depends on the temperature 
(because of its effects on the dielectric properties of water), it is important 
to evaluate the attenuation of rains whose drops are at different tempera- 
tures from those in the preceding tables. Table 7.10 contains the neces- 
sary data relative to the change of attenuation with temperature and is 
to be used with table 7.8. For example, in table 7.8, with a p^fcipitation 
rate of p = 0.25 mm/hr, temperature of 18 °C, X = 1.25 cm, the attenua- 
tion is 0.0215 dB/km. Using the correction factors obtained from table 
7.10, for the same general conditions of precipitation and wavelength, 
for a temperature reading of °C, the attenuation reads 0.02043 dB/km; 
for a value of 30 °C an attenuation of 0.019350 dB/km is noted; and for 
a temperature of 40 °C the attenuation is 0.01742 dB/km. 



6.0 



5.0 



a 3.5 





























y 




























• > 


/ 




























• y * 


• 
























• 


/ 


' • • 






















t ' 


< 


y 
























• 




^ 


V 


















; 


• 




:># 


V" 

• 


















' 


• 




/^ 


^.^ 


K^ 


















c 




y 




















■ 




• 


^ 


V 


!> 




















y 


/ 




•( 


^ 























2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0 10 

TRANSMISSION LOSS IN DECIBELS PER MILE 



Figure 7.20. Transmission loss in decibels per mile versus rainfall in inches per hour. 
(After Hathaway and Evans, 1959). 



296 



ATTENUATION OF RADIO WAVES 



Table 7.7. Drop size distribution 

( Burrows-A ttwood ) 



Drop diameter, 

D, in 

centimeters 



0.05 
.10 
.15 
.20 
.25 
.30 
.35 
.40 
.45 
.50 
.55 
.60 
.65 
.70 



Precipitation rate, p, in mm/hr 



0.25 



1.25 2.5 



Percentage of a given volume containing 
drops of diameter, D 



28.0 


10.9 


7.3 


2.6 


1.7 


1.2 


1.0 


50.1 


37.1 


27.8 


11.5 


7.6 


5.4 


4.6 


18.2 


31.3 


32.8 


24.5 


18.4 


12.5 


8.8 


3.0 


13.5 


19.0 


25.4 


23.9 


19.9 


13.9 


0.7 


4.9 


7.9 


17.3 


19.9 


20.9 


17.1 




1.5 


3.3 


10.1 


12.8 


15.6 


18.4 




0.6 


1.1 


4.3 


8.2 


10.9 


15.0 




.2 


0.6 


2.3 


3.5 


6.7 


9.0 






.2 


1.2 


2.1 


3.3 


5.8 








0.6 


1.1 


1.8 


3.0 








.2 


0.5 
.2 


1.1 

0.5 

.2 


1.7 
1.0 

0.7 



1.0 

4.1 

7.6 

11.7 

13.9 

17.7 

16.1 

11.9 

7.7 

3.6 

2.2 

1.2 

1.0 

0.3 



Table 7.8. Attenuation in decibels per kilometer for different rates of rain precipitation 

Temperature 18 °C 
(Burrows-Attwood) 



Precipi- 








Wav 


elength, X in cm 








tation 




















rate, p, 




















in mm/hr 


X = 0.3 


X=0. 4 


X = 0. 5 


X = 0. 6 


X = 1.0 


X = 1.25 


X=3.0 


X = 3.2 


X = 10 


0.25 


0.305 


0.230 


0.160 


0.106 


0.037 


0. 0215 


0. 00224 


0.0010 


0.0000997 


1.25 


1.15 


0.929 


0.720 


0.549 


0.228 


0.136 


0. 0161 


0.0117 


0. 000416 


2.5 


1.98 


1.66 


1.34 


1.08 


0.492 


0.298 


0. 0388 


0. 0317 


0. 000785 


12.5 


6.72 


6.04 


5.36 


4.72 


2.73 


1.77 


0.285 


0.238 


0.00364 


25.0 


11.3 


10.4 


9.49 


8.59 


5.47 


3.72 


0.656 


0.555 


0. 00728 


50. 


19.2 


17.9 


16.6 


15.3 


10.7 


7.67 


1.46 


1.26 


0. 0149 


100. 


33.3 


31.1 


29.0 


27.0 


20.0 


15.3 


3.24 


2.80 


0.0311 


150. 


46.0 


43.7 


40.5 


37.9 


28.8 


22.8 


4.97 


4.39 


0.0481 



Table 7.9. Attenuation in rains of knoivn drop size distribution and rate of fall 

(decibels per kilometer) 
(Burrows-Attwood) 



Precipitation 










Wavelength X in cm 










rate, p, in 






















mm/hr 


























1. 


25 


3 




5 


8 




10 




15 




2.46 


1.93 


10-1 


4.92 


10-2 


4. 24 10-3 


1.23 


10-3 


7.34 


10-* 


2.80 


10-* 


4.0 


3.18 


10-1 


8.63 


10-2 


7.11 10-3 


2.04 


10-3 


1.19 


10-3 


4.69 


10-4 


6.0 


6.15 


10-> 


1.92 


10-1 


1. 25 10-2 


3.02 


10-3 


1.67 


10-3 


5.84 


io-< 


15.2 


2.12 




6.13 


10-1 


5. 91 10-2 


1.17 


10-2 


5.68 


10-3 


1.69 


10-3 


18.7 


2.37 




8.01 


10-1 


5. 13 10-2 


1.10 


10-2 


6.46 


10-3 


1.85 


10-3 


22.6 


2.40 




7.28 


10-1 


5. 29 10-2 


1.21 


10-2 


6.96 


10-3 


2.27 


10-'' 


34.3 


4.51 




1.28 




1. 12 10 1 


2.32 


10-2 


1.17 


10-2 


3.64 


10-3 


43.1 


6.17 




1.64 




1. 65 10-1 


3.33 


10-2 


1.62 


10-2 


4.96 


10-3 



RAINFALL ATTENUATION CLIMATOLOGY 297 

Table 7.10 Correction factor {multiplicative) for rainfall attenuation 

(Burrows-Attwood) 



Precipitation 














rate, p, in 


X cm 


0°C 


10 °C 


18 °C 


30 °C 


40 °C 


mm/hr 














0.25 


0.5 


0.85 


0.95 


1.0 


1.02 


0.99 




1.25 


.95 


1.00 


1.0 


0.90 


.81 




3.2 


1.21 


1.10 


1.0 


.79 


.55 




10.0 


2.01 


1.40 


1.0 


.70 


.59 


2.5 


0.5 


0.87 


0.95 


1.0 


1.03 


1.01 




1.25 


.85 


.99 


1.0 


0.92 


0.80 




3.2 


.82 


1.01 


1.0 


.82 


.64 




10.0 


2.02 


1.40 


1.0 


.70 


.59 


12.5 


0.5 


0.90 


0.96 


1.0 


1.02 


1.00 




1.25 


.83 


.96 


1.0 


0.93 


0.81 




3.2 


.64 


.88 


1.0 


.90 


.70 




10.0 


2.03 


1.40 


1.0 


.70 


.59 


50.0 


0.5 


0.94 


0.98 


1.0 


1.01 


1.00 




1.25 


.84 


.95 


1.0 


0.95 


0.83 




3.2 


.62 


.87 


1.0 


.99 


.81 




10.0 


2.01 


1.40 


1.0 


.70 


.58 


150. 


0.5 


0.96 


0.98 


1.0 


1.01 


1.00 




1.25 


.86 


.96 


1.0 


0.97 


0.87 




3.2 


.66 


.88 


1.0 


1.03 


.89 




10.0 


2.00 


1.40 


1.0 


0.70 


.58 



7.9. Rainfall Attenuation Climatology 

The above paragraphs have been concerned with a descriptive presen- 
tation of the theoretical and technical background of the problem of 
power loss due to attenuation by rain and atmospheric gases. In an 
attempt to circumvent the difficulties of the above methods of attenua- 
tion prediction it was considered important to try a climatological ap- 
proach to this problem. However, the results of such a study were 
disappointing due to the fact that the problems of a systematic climato- 
logical estimation of rainfall attenuation are many and varied. Answers 
are needed for such questions as; how often do various rainfall rates and 
drop sizes occur in geographical areas and over how large an area do 
these rate and drop size statistics apply? Furthermore, to what height 
in the atmosphere do these data apply [1]? 

Unfortunately, the present state of meteorological knowledge concern- 
ing these problems in such that no conclusions can be drawn on a syste- 
matic and climatological basis. In view of these facts, it appears 
prudent at this time to provide only engineering estimates of the com- 
bined gaseous and rain absorption. 

In this regard, Bussey [1] has shown that the absorption due to rainfall 
exceeds that of gaseous constituents about 5 percent of the time for 
frequencies around 6,000 Mc/s. The 5 percent figure was obtained by a 
study of the rainfall rate distribution for various locations in the United 
States. Figure 7.21 shows the combined rain and gaseous absorption 
to be exceeded 1 per cent of the time. 



298 



ATTENUATION OF RADIO WAVES 



bl).Ol)l) 






















n 


n 










~ 




"T 








— 


- 




^ 


p=n 




a 


^ 


TTTl 






















































/ 




< 








zaooo 




















































/ 


y 
































' 










II 


^ 




=»» 


«• 


z. 


U-- 


=^ 


^ 










10.000 


\tOOmite\^ 
















































































































7.000 






























^ ' : 






















































/^ ■ >>- 




















































y 




Vi'^ 




















































y 






<ry 


f 1 


II 
























3,000 


























/ 






/I 


-^303/7,,*! 
















































/ 


// 
































2.000 
1.000 


1,000 mile] 
























/ 


/i 


^ 


























































































































































100 


























II 
























































j 




// 




































500 






















' 




// 






















































/ 






'/ 




































500 
200 

100 


















/ 






/ 


/ 


















































y 


/ 




/ 




Y 












































^ 


-^ 






./■ 


^ 


:^ 


^ 











































OjOI OjO! 01)3 01)5 007 0.1 02 0.3 05 0.7 I 2 3 5 7 10 20 30 50 70 100 200 300 500 TO IMC 

Total Path Attenuation in Decibels 

Figure 7.21. Combined rain and gaseous absorption to be exceeded 1 percent of the time. 



7.10. Rain Attenuation Effects on Radio 
Systems Engineering 

Attenuation due to rainfall is obviously a dominating factor in deter- 
mining the reliability of a communications system, especially at frequen- 
cies in excess of 30 Gc/s. Rain varies greatly in frequency of occurrence 
from one region to another, so it is important to have an effective method 
of predicting the performance of a radio system in any region in order 
that the communications engineer will be able to gain the widest possible 
application and degree of reliability, consistent with cost, in any system 
of his design. 

This section will be concerned, in the main, with the results of the 
Bell System's [24] field experiment in the Mobile, Ala., area, which was 
designed to establish a relationship between excess path attenuation and 
instantaneous rate of rainfall and to seek out any relationship between 
the profile of rate of rainfall along a radio path and rainfall measured 
at a point. 

The main problem concerns the ability to predict outage time due to 
rainfall (time the system noise exceeds the system objective) at 11 Gc/s 
in all areas of the country. This is obviously a difficult problem since, 
due to reasons of cost, it is not feasible to measure rainfall attenuation 
in all parts of the country. Therefore, it is desirable to be able to use 
what rainfall data are available and to couple the data, through what are 
thought to be reasonable assumptions, to the relationships between rain- 
fall and attenuations. 



RAINFALL ATTENUATION CLIMATOLOGY 



299 



In approaching the problem of predicting the outage time due to rain- 
fall, it has been assumed that the annual distribution of one-hour point 
rainfall rates is indicative of the instantaneous values over 30-mi radio 
paths [1] and that the frequency of severe rainfall of the type measured 
in the Mobile area will be reduced in other parts of the country in propor- 
tion to the distribution of annual point rates of 1 in. or more rainfall 
per hour. 

Figure 7.22 indicates the expected outage time due to rainfall for 
various path lengths in different rain areas of the United States. The 
curves A through H correspond to the areas contoured in figure 7.23 
which illustrates contours of constant path length for fixed outage times 
for different areas of the United States. The longer paths have been 
somewhat weighted to take into account the less severe rainfall covering 
larger geographical areas than the intense storms typical of the Gulf 
region. 

In a complete 11-Gc/s Bell System point-to-point relay system, the 
rain outages of the individual hops must be added to obtain the perform- 
ance of the system. It is desirable that the individual hops meet the 
same objective, but this is not always possible. Sometimes one or more 















// 




1 


/ 


/ 


/ 


/ 














/ / 


/ 


/ 


/ 


/ 


A 


/ 












a/ 


/ ^ 


/, 


/ 


'./ 


7 


/ 
















/ 


/ 


/ 


/ 


1 


/ 
















/ 


/ 


A 


// 


/ 












y^ 






/ 


/ 


/ 


^/ 


/ 






^ 


^ 


c 








^^ <« 


^A 


^ 


/ 









15 20 26 

PATH LENGTH IN MILES 



35 40 45 50 



Figure 7.22. Expected outage time in hours per year versus path length in miles for 

various areas of the United States. 

(After Hathaway and Evans, 1959). 

The curves A through H correspond to the areas contoured in figure 7.23. 



300 



ATTENUATION OF RADIO WAVES 




Figure 7.23. Contours of constant path length for fixed outage time. 
(After Hathaway and Evans, 1959). 



hops of a system are electrically long; they will have insufficient fading 
margin (the number of dB the receiver input level can be reduced before 
the noise exceeds the system objective) and hence contribute more than 
their share of the outage time. So, this excess must be made up by 
imposing tighter requirements on the remaining hops. To meet the 
overall objective of the Bell System, it is necessary to know the contri- 
butions of the long hops — those having a fading margin less than 40 dB. 
Figure 7.24 shows the excess path loss due to rain, versus hours per year 
for the Mobile area study. Since the shape of this curve is nearly 
identical to Bussey's curve of cumulative distribution for point rates in 
Washington, D.C. (if we assume the shape of this curve to be representa- 
tive for other areas of the country), then the additional outage time for 
path lengths given by figure 7.23 can be estimated for hops having a 
fading margin less than 40 dB. The data shown on figure 7.24 have been 
rationalized and are shown in figure 7.25 as an estimate of additional 
outage time. 

Sometimes it is practical to shorten a proposed path to bring the fading 
margin up to 40 dB. An approximation of the necessary reduction path 
length can be made if uniform rainfall rate is assumed over the path. 



RAINFALL ATTENUATION CLIMATOLOGY 



301 



^43 

40 

35 
If) 

_i 

LU 
ID 

U 30 

LU 
Q 

Z 

If) 25 

If) 

O 

_l 

I 20 

t- 

in 
If) 15 

LU 

u 

X 

LU 

10 

5 






































^ 


y^ 




























^\^^ 




























\ 


V) 




























V 


V, 




























N 


\ 


V 




























^^ 


N 


N 


s 






























NJ 


\ 































6 7 8 9 10 20 30 40 50 60 70 80 100 

HOURS THAT THE ORDINATE VALUE WAS EXCEEDED 



200 



Figure 7.24. Excess path loss due to rainfall versus hours per year at Mobile, Ala. 
(After Hathaway and Evans, 1959). 



40 



35 



Z 

o 
cr 
< 30 



o 

2 25 

Q 

< 

^ 20 



15 






20 



30 



40 50 60 70 80 100 



200 



300 400 



Figure 7.25. Additional outage time expected for 11 Gc/s systems having a fading 

margin less than 40 dB. 



302 



ATTENUATION OF RADIO WAVES 



Under this condition the attenuation due to rainfall should be directly 
proportional to the path length. Thus the path length in figure 7.24 can 
be shortened to correct for in.sufficient fading margin. 

Bell System results indicate that for their microwave relay links in the 
extreme southeastern region of the United States rainfall will limit 11 
Gc/s radio systems having a 40 dB fading margin to path lengths of 
approximately 10 to 15 mi, depending on the number of hops, if normal 
reliability objectives are to be met. Path lengths of 20 to 30 mi should 
be acceptable in the central area and paths as long as 35 mi should be 
acceptable in the northwestern part of the country. However, in their 
existing point-to-point radio relay systems, the paths average about 23 
mi due to other considerations than those of propagation. It appears 
that the 11 Gc/s systems will not be penalized unduly except in the 
southeastern part of the United States. 

An illustration of the correlation between the rainfall and path loss on 
March 15, 1956, of the ]\Iobile study is presented in figure 7.26 in support 
of the above conclusions. 



CALCULATED 
(RAINFALL DATA) 
MEASURED 




1200 
TIME OF DAY 



Figure 7.26. Correlation between rainfall and path loss, March 15, 1956. 
(After Hathaway and Evans, 1959). 



7.11. Attenuation by Hail 

Ryde concluded that the attenuation caused by hail is one-hundredth 
that caused by rain, that ice crystal clouds cause no sensible attenuations, 
and that snow produces very small attenuation even at the excessive rate 
of fall of 5 in. an hour. However, the scattering by spheres surrounded 
by a concentric film of different dielectric constant does not give the same 
effect that Ryde's results for dry particles would indicate [24, 27]. For 
example, when one-tenth of the radius of an ice sphere of radius 0.2 cm 
melts, the scattering of 10 cm radiation is approximately 90 percent of 
the value which would be scattered by an all-water drop. 



ATTENUATION BY FOG 



303 



At wavelengths of 1 and 3 cm with 7 = 0.126 (7 = 2a/X; a = radius 
of drop) Kerker, Langleben, and Gunn [27] found that particles attained 
total-attenuation cross sections corresponding to all-melted particles when 
less than 10 percent of the ice particles were melted. When the melted 
mass reached about 10 to 20 percent, the attenuation was about twice that 
of a completely melted particle. These calculations show that the attenu- 
ation in the melting of ice immediately under the °C [28] isotherm can 
be substantially larger than in the snow region just above, and under some 
circumstances, greater than in the rain below the melting level. Further 
melting cannot lead to much further enhancement, apparently, and may 
lead to a lessening of the reflectivity of the particle by bringing it to 
sphericity or by breaking up of the particle. This effect, combined with 
the fact that hail has greater terminal velocities than rain, gives rise to 
the so-called "bright band" near the 0° isotherm. 



7.12. Attenuation by Fog 

The characteristic feature of a fog is the reduction in visibility. Visi- 
bility is defined as the greatest distance in a given direction at which it 
is just possible to see and identify with the unaided eye (a) in the day- 
time, a prominent dark object against the sky at the horizon, and (b) at 
night, a known, preferably unfocused moderately intense light source [29]. 

Although the visibility depends upon both drop size and number of 
drops and not entirely upon the liquid-water content, yet, in practice, 
the visibility is an approximation of the liquid-water content, and there- 
fore, may be used to estimate radio-wave attenuation [28]. On the basis 
of Ryde's work, Saxton and Hopkins [30] give the figures in table 7.11 
for the attenuation in a fog or clouds at °C temperature. The attenua- 
tion varies with the temperature because the dielectric constant of water 
varies with temperature; therefore, at 15 and 25 °C the figures in table 
7.11 should be multiplied by 0.6 and 0.4 respectively. It is immediately 
noted that cloud or fog attenuation is an order of magnitude greater at 
3.2 cm than at 10 cm. Nearly another order of magnitude increase 
occurs between 3.2 cm and 1.25 cm. 



Table 7.11. Attenuation caused by clouds or fog [SO] 
Temperature=0 °C 



Visibility 


Attenuation (dB/km) 




X = 1.25cm 


X=3. 2 cm 


X = 10cm 


m 
30 
90 
300 


1.25 
0.25 
0.045 


0.20 
0.04 
0.007 


0.02 
0.004 
0.001 



304 ATTENUATION OF RADIO WAVES 

7.13. Thermal Noise Emitted by the Atmosphere 

General laws of thermodynamics relate the absorption characteristics 
of a medium to those of emission. Good absorbers of radiation are also 
good emitters, and vice versa. Thus, in the microwave region, the atmos- 
phere is also a good emitter, as well as a strong absorber, of radiation. 
We may, therefore, describe quantitatively both emission and absorp- 
tion by the same parameter; namely, the absorption coefficient. 

The emission characteristics of any real body at a fixed frequency may 
be compared to those of a blackbody at the same temperature. In the 
microwave region, the noise intensity emitted by a blackbody is given by 
the Rayleigh -Jeans law: 



i(.) = 2kT[-J (7.8) 



where \l/{v) = emitted blackbody flux density per unit frequency 
V = frequency 
T = absolute temperature, °K 
c = the velocity of light, and 
k = Boltzmann constant (1.38054 X 10-i« erg/°K) 

The emission per unit length along an actual ray path may now be ex- 
pressed as 

Biu) = yiu) 4^ip) (7.9) 

where 7(1') = attenuation per unit length. Remembering that the fraction 
of energy absorbed in a path length, ds, is given by the optical depth, 
dr = y{v)ds, we may obtain the differential equation for transmission of 
radiation through the atmosphere: 



^ = -/(.) + V(«^) (7.10) 



where liu) is the flux density per unit frequency. 
The solution to this radiative transfer equation is 

liv) = i:lm{v) exp (- / "W) 

+ / 4^{v) exp {- \ dr) dr (7.11) 



THERMAL NOISE EMITTED BY ATMOSPHERE 305 

where the summation extends over all discrete noise sources which may 
be present, I m{v) is the unattenuated flux density transmitted from the 
mth discrete source located at position r^, s is the point of reception of 
energy, and the other symbols have their previous meaning. It should 
be recognized that the above integrals extend over a ray path determined 
by the refractive properties of the medium and cannot be evaluated 
unless these refractive properties are known. 

In analogy to the temperature dependence of the noise energy as by 
the Rayleigh-Jeans law, we may, in the microwave region, relate the 
intensity of radiation received from a particular direction, I{v), to an 
equivalent temperature, T n{v), by the following relation 



/(^) = ?^ (7.12) 



or, from (7.8) 

TM = ^TnAv) exp(- / "'rfr) 

+ / T^expf-/ dr) dr. (7.13) 



This equivalent temperature is called the thermal noise temperature. 

It is apparent that the thermal noise temperature of the atmosphere, 
as measured by an antenna, will depend explicitly upon the antenna angle 
and the frequency, and implicitly upon the atmospheric conditions along 
the ray path giving rise to attenuation and emission of energy. It seems 
plausible, therefore, that one could exploit this dependence of thermal 
noise on atmospheric conditions as a probe of atmospheric structure. 

Thermal noise is equally important in communications receiving, since 
it represents the lowest possible noise level that can be attained by an 
antenna immersed in the atmosphere. This minimum noise level will, 
of course, vary, depending on atmospheric conditions, the frequency, and 
the antenna orientation. For example, in the microwave region, the 
antenna noise temperature at vertical orientation may be as low as 1 °K, 
and in a horizontal position, where more of the lower layers are "seen" 
the noise temperature may be of the same order as the actual tempera- 
ture of the lower atmosphere; i.e., around 280 °K. Figure 7.27 shows 
sky temperature as frequency for various anteinia angles for mean atmos- 
pheric conditions at Bismarck, N. Dak., during February 1940-43. 



306 



ATTENUATION OF RADIO WAVES 




Figure 7.27. Thermal noise versus frequency for mean profile conditions 
at Bismarck, N.Dak. 



THERMAL NOISE EMITTED BY ATMOSPHERE 
Table 7.12 



307 



Gas 




ymax 


Percent by 

volume 
at ground 


7 at ground 




Mc/s 
12, 258. 17 


(dB/km 
1.9X10-1 


(Oto l)X10-6 


(dB/km) 
(0-1.9)X10-' 




12, 854. 54 


8.7X10-1 


(0-8. 7) X 10-7 




23, 433. 42 


1.2X10-1 


(0-1. 2) X 10-' 


SO2 


24, 304. 96 


2.3 


(0-2.3)X10-6 


25, 398. 22 


2.1 


(0-2. l)X10-« 




29, 320. 36 


3.3 


(0-3.3)X10-« 




44, 098. 62 


5.2 


(0-5.2)X10-6 




52, 030. 60 


9.5X10-1 


(0-9. 5)X10-' 




24, 274. 78 


2.5 


0.5X10-6 


1.25X10-6 


N2O 


22, 274. 60 


2.5 


1.25X10-6 


25, 121. 55 


2.5 


1.25X10-6 




25, 123. 25 


2.5 


1.25X10-6 


NO2 


26, 289. 6 


2.9 


(0 to 2) X 10-8 


(Oto 5. 8) XI 0-8 




10, 247. 3 


9. 5X10-2 


Summer 
(0 to. 07) X 10-6 

Winter 
(0to.02)X10-« 


(0 to 6. 3) X 10-9 


O3 


11,075.9 


9. 1X10-2 


(0 to 6. 3) X 10-9 




42, 832. 7 


4.3X10-1 


(0 to 2. 8) X 10-8 



X 

C2 

UJ 
X 

LlI 

_1 
< 
o 
(f) 



4 












10 



20 



30 



40 



AUGUST. H, 



FEBRUARY, H^ 



FEBRUARY. H, 



AUGUST, H^ 



50 



60 



FREQUENCY IN Gc/S 

Figure 7.28. Variations with frequency of the scale heights of the bi-exponential 
absorption model at Verkhoyansk, U.S.S.R. 



308 ATTENUATION OF RADIO WAVES 

7.14. References 

[1] Bussey, H. E. (Aug. 1950), Microwave attenuation statistics estimated from 

rainfall and water vapor, Proc. IRE 38, No. 7, 781. 
[2] Davidson, D., and A. Pote (Dec. 1955), Designing over-horizon communication 

links. Electronics, 28, 126. 
[3] Bean, B. R., and R. Abbott (May-Aug. 1957), Oxygen and water vapor absorp- 
tion of radio waves in the atmosphere, Geofis. Pura Appl. 37, 127-144. 
[4] Van Vleck, J. H. (Apr. 1947), Absorption of microwaves by oxygen, Phys. Rev. 

71, 413-424. 
[5] Van Vleck, J. H. (Apr. 1947), The absorption of microwaves by uncondensed 

water vapor, Phys. Rev. 71, 425-433. 
[6] Van Vleck, J. H. (1951), Theory of absorption by uncondensed gases. Book, 

Propagation of Short Radio Waves, pp. 646-664 (McGraw-Hill Book Co., Inc., 

New York, N.Y.). 
[7] Becker, G. B., and S. H. Autler (Sept. 1, 15, 1946), Water vapor absorption of 

electromagnetic radiation in the centimeter wavelength range, Phys. Rev. 70, 

Nos. 5 and 6, 300-307. 
[8] Birnbaum, G., and A. A. Maryott (Sept. 15, 1955), Microwave absorption in 

compressed oxygen, Phys. Rev. 99, 1886. 
[9] Artman, J. O., and J. P. Gordon (Dec. 1954), Absorption of microwaves by oxygen 

in the millimeter wavelength region, Phys. Rev. 96, No. 5, 1237-1245. 
[10] Tinkham, M., and M. W. P. Strandberg (July 15, 1955), Line breadths in the 

microwave magnetic resonance spectrum of oxygen, Phys. Rev. 99, No. 2, 

537-539. 
[11] Hill, R. M., and W. Gordy (Mar. 1954), Zeeman effect and line breadth studies 

of the microwave lines of oxygen, Phys. Rev. 93, 1019. 
[12] Artman, J. O. (1953), Absorption of microwaves by oxygen in the millimeter 

wavelength region, Columbia Radiation Lab. Rept. (Columbia Univ. Press, 

New York, N.Y.). 
[13] Straiton, A. W., and C. W. Tolbert (May 1960), Anomalies in the absorption of 

radio waves by atmospheric gases, Proc. IRE 48, No. 5, 898-903. 
[14] Tolbert, C. W., and A. W. Straiton (Apr. 1957), Experimental measurement of 

the absorption of millimeter radio waves over extended ranges, IRE Trans. 

Ant. Prop. AP-5, No. 2, 239-241. 
[15] Gunn, K. L. S., and T. W. R. East (Oct.-Dec. 1954), The microwave properties of 

precipitation particles. Quart. J. Roy. Meteorol. Soc. 80, 522-545. 
[16] Bean, B. R., and B. A. Cahoon (Sept. 1957), A note on the climate variation of 

absolute humidity, Bull. Am. Meteorol. Soc. 28, No. 7, 395-398. 
[17] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative 

humidity over the United States and Alaska (U.S. Weather Bureau). 
[18] Donaldson, Ralph J., Jr. (June 1955), The measurement of cloud liquid-water 

content by radar, J. Meteorol. 12, No. 3, 238-244. 
[19] Weickmann, H. K., and H. J. aufm Kampe (June 1953), Physical properties of 

cumulus clouds, J. Meteorol. 10 204-221. 
[20] Battan, L. J. (1959), Radar Meteorology, Book, p. 43 (Univ. of Chicago Press, 

Chicago, 111.). 
[21] Ryde, J. W., and D. Ryde (1945), Attenuation of centimeter waves by rain, hail, 

fog, and clouds (General Electric, Wembly, England). 
[22] Laws, J. O., and D. A. Parsons (Apr. 1943), The relationship of raindrop size to 

intensity. Trans. Am. Geophys. Union, 24th Annual Meeting, 452-460. 
[23] Ryde, J. W. (1946), The attenuation and radar echoes produced at centimetre 

wave-lengths by various meteorological phenomena. Meteorological Factors in 

Radio Wave Propagation, pp. 169-188 (The Physical Society, London, England). 



REFERENCES 309 

[24] Hathaway, S. D., and H. W. Evans (Jan. 1959), Radio attenuation at 11 IcMc and 

some implications affecting relay systems engineering. Bell Syst. Tech. J. 38, 

No. 1. 
[25] Burrows, C. R., and S. S. Atwood (1949), Radio wave propagation, Consolidated 

Summary Technical Report of the Committee on Propagation, NDRC, p. 219 

(Academic Press, Inc., New York, N.Y.). 
[26] Mie, G. (1908), Beitrage zur Optik triiber Medien, speziell Kolloidaler Metal- 

lasungen, Ann. Physik, 25, 377. 
[27] Kerker, M., M. P. Langleben, and R. L. S. Gunn (Dec. 1951), Scattering of 

microwaves by a melting spherical ice particle, J. Meteorol. 8, 424. 
[28] Best, A. C. (1957), Physics in Meteorology (Pittman and Sons, London, England). 
[29] Glossary of Meteorology (1959), Am. Meteorol. Soc. 3, 613. 
[30] Saxton, J. A., and H. G. Hopkins (Jan. -Feb. 1951), Some adverse influences of 

meteorological factors on marine navigational radar, Proc. IRE 98, Pt. Ill, 26. 
[31] Zhevankin, S. A., and V. S. Troitskii (1959), Absorption of centimetre waves in 

the atmosphere, Radioteknika i Elektronika 4, No. 1, 21-27. 
[32] Button, E. J., and B. R. Bean (June 1965), The biexponential nature of tropo- 

spheric gaseous absorption of radio waves, Radio Sci. J. Res. NBS 69D, No. 6, 

885-892. 
[33] Ghosh, S. N., and H. D. Edwards (Mar. 1956), Rotational frequencies and 

absorption coefficients of atmospheric gases. Air Force Surveys in Geophysics 

(Air Force Cambridge Research Center, ARDC, USAF). 



Chapter 8. Applications of Tropospheric 
Refraction and Refractive Index Models 

8.1. Concerning the Bi-Exponential Nature of 
the Tropospheric Radio Refractive Index 

8.1.1. Introduction and Background 

The recent explosive growth of space science and telecommunications 
has spurred the development of new models of the tropospheric radio 
refractive index to account for the systematic refraction of radio waves 
and the calculation of theoretical radio field strengths at satellite heights. 
The simple exponential model has been found to represent, to a first 
approximation, the average refractive index structure within the first few 
kilometers above the ground for the United States [1],^ France [2], and 
Japan [3]. All of the above investigations have reported varying degrees 
of departure of the atmosphere from this model and Misme [4] has en- 
deavored to delineate the regions of the world where the exponential 
model is most applicable, although subsequent analysis of several types 
of data has shown this model to be more generally applicable than at first 
sight and not unreasonable for use even in arctic and tropic locations [5]. 

If one considers that A^ is composed of a dry term, 



and a wet term, 



(8.1) 



,^ = -^^g X 10'. (g^2) 



then one may consider the height variation of each term separately. We 
shall examine the possible advantages of a model of the form 

N(z) = Do exp {-^} + Wo exp {-^} (8-3) 

to describe the average decrease of A'^ with height, where Do and Wo are 
the values of the dry and wet comi)onents at the earth's surface and Hd 



* figures in brackets indicate the literature references on p. 373. 

311 



312 REFRACTION AND REFRACTIVE INDEX MODELS 

and H^, are the scale heights of D and W, respectively. This particular 
form has been found useful by Katz [6], in his derivation of the potential 
refractive modulus, and by Zhevankin and Troitskii [7] in their treatment 
of atmospheric absorption. It would be well for the reader to recall 
that scale height, as used in this study, is merely the height at which the 
value of the atmospheric property has decreased to 1/e of its surface value. 
Typical values of Do, Wq, and A^o are listed for arctic, temperate, and 
tropical locations in table 8.1. It is seen that the contribution of W to 
the total value of A^ is nearly negligible in the arctic but becomes greater 
as one passes from temperate to tropical climates. There is, of course, 
generally an inverse correlation between the magnitude of D and W since, 
at sea level, where P '^ 1,000 mbar, the low arctic temperature increases 
the D term and, combined with low atmospheric water vapor capacity, 
decreases the wet term. Conversely, the higher temi^eratures of the 
temperate and tropical climates depress the D term and provide a greater 
water vapor capacity with the result that W may have a sizable contribu- 
tion to the total A^. 



Table 8.1. Typical average values of the dry and wet components of N 



station and climate 


Do 


Wo 


No 


Isachsen (78°50' N), arctic 

Wasliington, D.C. (38°50' N), temperate 

Canton Island (2°46' S), tropic 


332.0 
266.1 
259.4 


0.8 
58.5 
111.9 


332.8 
324.6 
371.3 



8.1.2. N Structure in the I.C.A.O. Atmosphere 

One may examine A^ structure in a standard atmosphere as a guide to 
its general distribution in the free atmosphere. On this basis the I.C.A.O. 
standard atmosphere [8] (fig. 8. 1) was examined. The conditions specified 
for this atmosphere are an approximately exponential pressure decrease 
with respect to height and a linear temperature decrease from ground level 
to the tropopause. It is evident, then, that in this atmosphere D de- 
creases in an exponential fashion with height. 

When these data are converted into refractive index and plotted on 
semilogarithmic paper, as on figure 8.2. both D and W are seen to display 
an approximately exi^onential distribution from the surface to the tropo- 
pause. This conclusion is based upon the observation that the distribu- 
tion is nearly linear as one would expect if one inverted the function 

y = A exp i-h/c) (8.4) 



N STRUCTURE IN I.C.A.O. ATMOSPHERE 



313 



50 



40 - 



MESO7PEAK 

OZONOPAUSE 
H = 47gpkm 




400 800 1200 210 230 250 270 290 
P(mbar) T(°K) 



100 200 300 



D = 77.6 Y 



Figure 8.1. The U.S. extension to the I.C.A.O. Standard Atmosphere. 



into 



\n y = — h/c + In ^, 



(8.5) 



which is the equation of a straight Hne on semilogarithmic paper. The 
exponential distribution of W with height in this atmosphere follows 
naturally from the definition of constant relative humidity, since the 
saturation vapor pressure, e^, is itself, to a first approximation, an ex- 
ponential function of temperature. It is evident that the value of W can 
significantly affect the surface values of A^ but has no appreciable effect 
upon the value of A'' at the tropopause. 

An examination of long term means from observations in the actual 
atmosphere shows that this same general bi-exponential trend is observed 
in practice, for temperate climates at least. Examples are given on 
figures 8.3 and 8.4 for Bismarck, N.Dak., and Brownsville, Tex. 

Bismarck is typical of the high, dry great plains region of North 
America which is frequently subjected to strong intrusions of arctic air, 
while Brownsville represents the humid periphery of the Gulf of Mexico. 
Even in these very dissimilar climates one finds a strong tendency towards 
a bi-exponential distribution of A^, particularly when W is large. 



314 



REFRACTION AND REFRACTIVE INDEX MODELS 



1,000 
700 
500 



N 



RH = 100% 
RH = 67 % 

RH = 33% 

RH =0% (DRY) 




TROPOSPHERE 



STRATOSPHERE 



W(h),RH=IOO% 



W(h),RH = 67 % 
W(h),RH= 33% 



4 6 8 10 12 

H,gpkm 

Figure 8.2. N distribution for the I.C.A.O. standard atmosphere. 



14 



8.1.3. Properties of the Dry and Wet Term Scale Heights 

The average values of D and W versus height were determined for 
22 U.S. radiosonde stations located about the country. The data used 
were the published values of mean pressures, temperatures, and humidities 
for the United States [9] which may be converted into mean values of the 
refractive index with negligible error [10]. The method of least squares 
was then used to determine the scale heights of the wet term, H^, and of 
the dry term, Hd. Examination of these scale heights did not reveal any 
simple method of predicting their geographic and seasonal behavior, other 
than simply to map them. Such maps were prepared for the United States 



DRY AND WET TERM SCALE HEIGHTS 



315 



for both winter (February) and summer (August). The immediate con- 
clusion that one reaches from these maps, figures 8.5 and 8.8, is that H^ 
has a year round, country-wide average value of perhaps 2.5 km while 
H d has an average value of about 9 km. 

Since the scale height is the height at which the value of an atmos- 
pheric property has decreased to \/e of its surface value it reflects the 
degree of stratification of the property. For example, cold arctic air is 
very stratified with very little vertical motion, with the result that its 
density scale height would be expected to be low. By contrast, tropical 
maritime air that has moved over land is characteristically unstable with 
convective activity thoroughly mixing the original moist surface air 



N 



1,000 
700 
500 

300 
200 

100 
70 
50 

30 
20 

10 
7 
5 

3 
2 



2 4 6 8 10 12 14 

ALTITUDE ABOVE M.S.L. IN km 

Figure 8.3. N distribution for Bismarck, N.Dak. 

Note that altitude rather than geopotential height is used here to facilitate the eventual calculation of radio- 
ray bending through actual atmospheric layers. 



! 1 i 1 1 


1 


ELEVATION =505m 


— 




: 


- AUGUST 7^"''"'^^ 
A^ ^D(h) 

- \ 

\ 

\ ^AUGUST 






- 


\ \-^W(h) 

\ *^ 
\ \ 

- FEBRUARY^A. \ 


- 



316 



REFRACTION AND REFRACTIVE INDEX MODELS 



1,000 
700 
500 



N 



ELEVATION = 7m 
- ^FEBRUARY 




2 4 6 8 10 12 14 

ALTITUDE ABOVE M.S. L. IN km 

Figure 8.4. N distribution for Brownsville, Tex. 



throughout the entire troposphere, with the result that the density scale 
height is relatively larger than in the case of arctic air. The dry term 
scale height on figures 8.5 and 8.7 show a distinct tendency to be larger 
during the warm summer months when the atmosphere is well mixed to 
great heights. Consequently D decreases slowly with height. 

A slight geographic pattern is observed in the Hd maps: the coastal 
regions display somewhat higher values than the inland regions. The 
north-south direction of the isopleths along the west coast on the February 
Hd map definitely reflects the uniform onshore advection of low-density 
maritime air. By contrast, the east coast shows an east-west isopleth 
pattern, the high values in Florida reflecting the well-mixed nature of sub- 
tropical air, and the lower values in New England inaicating the presence 



DRY AND WET TERM SCALE HEIGHTS 



317 




Figure 8.5. Dry term scale height, Hd, in kilometers for February. 




Figure 8.6. Wet term scale height, Hw, m kilometers for February. 



318 REFRACTION AND REFRACTIVE INDEX MODELS 




Figure 8.7. Dry term scale height, Hd, in kilometers for August. 




Figure 8.8. Wet term scale height, Hw, in kilometers for August. 



DRY AND WET TERM SCALE HEIGHTS 319 

of the more dense and stratified continental air that customarily flows 
offshore during the winter months. The same pattern is repeated on the 
summertime map along the west coast but is less i^ronounced on the east 
coast due to the combination of more uniform heating and also onshore 
advection of maritime air |)roduced by the circulation pattern of the 
Bermuda high-pressure area. The high value of H d = 10.5 km observed 
in the southwest during the summer appears to be due to the intense heat- 
ing with resultant convective mixing to great heights so common in that 
desert area. A somewhat opposite pattern is evidenced by the //„, maps. 
For example, the coastal areas generally have the lowest values and thus 
reflect the characteristic strong humidity stratification of maritime air. 
The smaller humidity gradients of the inland regions produce somewhat 
larger scale heights for that area. The summer Hy, map is quite sur- 
prising in that very little variation is shown, perhaps indicating uniform 
vertical convection of the available moisture at all locations throughout 
the country. The strong convection indicated in the southwest on the 
summer H d map is again reflected by the high value of //,„ = 3.0 km for 
that same area. 

It is quite evident from figure 8.9 that within the troi)osphere, /i < 10 
km, the bi-exponential model has a lower rms error for the common, 
near-zero angles of departure used in tropospheric propagation of radio 
waves. Both models yield about a 12 percent error in determining 
T for dt) — Q and h = W km. At ^o = 100 mrad, however, the percentage 
error decreases to 4 percent for the bi-exponential model and 7 percent for 
the exponential reference atmosphere. The rather marked errors of the 
single exponential model at 10 km simply reflect that that model is 
deliberately fitted to the average A'^ structure over the first few kilometers 
with the result that this model systematically departs from the average 
atmospheres in the vicinity of the tropo pause. This is particularly 
apparent at the higher values of ^o where the integral tends to become a 
function of the limits of integration. That is, using the theorem of the 
mean for integrals. 



cot d dn ^ -cot da dn ^ [N^ - N] 10-" cot ^o (8.9) 



under the assumption that cot 6 may be replaced by cot ^o over the interval 
of integration. For do < 20 mrad this assumption introduces less than 
a 10 percent error for the interval < /i < 10 km. It is ai)parent from 
(8.9) that the error in j^redicting r for large do is then simj)ly a matter of 
how closely the model approaches the true value of n in the atmos}ihere. 
At ^0 near zero, however, the integral for r is very heavily weighted to- 
wards the effect of n gradients near the earth's surface [1]. Since the 
values of do commonly used in tropospheric propagation are as near as 



320 



REFRACTION AND REFRACTIVE INDEX MODELS 



practically possible to zero, and both models show comparable rms errors 
for do = 0, one concludes that there is no clear advantage to the bi- 
exponential model for this application. This conclusion is furthered by 
the fact that charts of Hd, Hy,, Do, and Wo are not available, while to use 
the single exponential model one need only to refer to existing regional 
or worldwide maps [10, 12]. 



4.0 



(T 



or 
o 
(r 
(r 

UJ 
CD 

Q 

LxJ 
CD 



^ 0.3 





0.4 



= 



EXP. REF 



I -EXP. 



9„ = ?0mr 





9n-- 50 mr 



EXP REF 



BI-EXP 



n 




0n=IOOmr 



-EXP REF 



-BI-EXP 



0,5 

0,4 

0.3 

0.2 

0.1 


0.20 

0.16 

0.12 

0.08 

0.04 



cr 
o 
cr 
cr 

UJ 



LU 
CD 



CD 



cr 



3 10 15 70 3 10 15 70 

HEIGHT IN KILOMETERS 

Figure 8.9. Rool-mean-square errors of predicting bending for both the bi-exponential 

refractive index model and the C.R.P.L. (single) exponential atmosphere. 

Height is used here to indicate the actual thickness of atmosphere traversed by the radio ray 



REFRACTION IN BI-EXPONENTIAL MODEL 321 

8.1.4. Refraction in the Bi-Exponential Model 

The test of a model of atmospheric refractive index is the degree to 
which it represents the average A^ structure of the atmosphere. A 
further critical test is the degree to which the refraction, or bending, of a 
radio ray is represented by this atmosphere. The bending is given as 
the angular change of a radio ray as it passes from rii to n-i in a spherically 
stratified atmosphere and is adequately represented by 

Ti,2 = - / cot ddn (8.6) 

where 6 is the local elevation angle and is determined at any point from 
Snell's law, 

nr cos 6 = rioro cos ^o, (8.7) 

where r is the radial distance from the center of the earth and zero sub- 
script denotes the initial conditions. It is customary to evaluate (8.6) 
numerically [11] since the integral is intractable for all but the most 
simple models of n versus r. The bending was obtained for the mean A^ 
profiles for one-half of the 22 U.S. weather stations mentioned earlier. 
The values of r predicted by the bi-exponential model for these same 
stations were obtained by preparing U.S. maps of Hd and H,j, from the 
other half of our data, selecting values oi Hd and Hu, for the test stations, 
calculating the bending and obtaining rms differences between these 
values and those obtained from the mean A'' profiles. These rms differ- 
ences are shown on figure 8.9. Also, for comparison, the rms errors 
obtained from the CRPL exponential atmosphere are shown. This 
atmosphere, based upon a single exponential curve passing from the 
surface value, N s, to the value at 1 km, A^i, is founded upon the expression 

A^i = A^, - 7.32 exp {0.005577 A^^} (8.8) 

which has been found to be applicable in the United States [1]. 

An obvious advantage of the bi-exponential model is that the scale 
heights do reflect the physical properties of the atmosphere in a much 
clearer way than does the single exponential model. 



8.1.5. Extension to Other Regions 

The present study is based upon data from the continental United 
States and one wonders if the same approach might be of utility in other 
regions. As a brief check, the D and W term scale heights were deter- 
mined for conditions typical of the long arctic night (Isachsen, Northwest 



322 REFRACTION AND REFRACTIVE INDEX MODELS 

Territory, Canada, for February) and for humid tropical areas (Canlon 
Island, South Pacific, February) from 5-yr means of A'^ versus height 
and are listed in table 8.2. 

The extreme meteorological differences of these two locations are quite 
evident. The Ha = 9.4 km and 7/^, = 2.0 km at Canton Island indicate 
a warm atmosphere with a strong humidity gradient, while the value 
of H d = 6.3 km at Isachsen indicates very stratified air with high surface 
density and a strong density decrease with height. The value of //„, = 
6.5 km at Isachsen reflects a very low humidity gradient; in fact, at no 
point in the troposphere does W exceed 3 A^ units for this location and 
season. 

Table 8.2. Hd and Yi^ifor arctic and tropical locations 



Station 


Hrf 


Hu. 


Canton Island 


km 
9.4 
6.3 


km 
2.0 


Isachsen, N.W.T., 


Canada.. 


6.5 



The value of //„, = 2.6 km reported for the characteristic altitude of 
water vapor for the middle belt of the U.S.S.R. [7] also appears to be in 
agreement with the conclusions of the present study. 



8.2. Effect of Atmospheric Horizontal 
Inhomogeneity Upon Ray Tracing 

8.2.1. Introduction and Background 

It is common in ray tracing studies to assume that the refractive index 
of the atmosphere is spherically stratified with respect to the surface of 
the earth. Thus, the effect of refractive index changes in the horizontal 
direction is normally not considered, although Wong [13] has considered 
the effect of mathematically smooth horizontal changes in airborne propa- 
gation problems. 

Neglecting the effect of horizontal gradients seems quite reasonable in 
the tropospheric because of the relatively slow horizontal change of 
refractive index in contrast to the rapid decrease with height. In fact, 
examination of climatic data indicates that one must compare sea-level 
stations located 500 km from each other on the earth's surface in order to 
observe a difference in refractive index values which would be comparable 
to that obtained by taking any one of these locations and comparing its 
surface value with the refractive index 1 km above the location. Although 
the assumption of small horizontal changes of the refractive index appears 



CANTERBURY 323 

to be true in the average or climatic sense, there are many special cases, 
such as frontal zones and land-sea breeze effects, where one would expect 
the refractive index to change abruptly within the 80-odd kilometers of 
horizontal distance traversed by a tangential ray passing through the 
first kilometer in height. 

It is these latter variations that are investigated in this section. Two 
cases of marked horizontal change of refractive index conditions were 
studied, one which occurred over the Canterbury Plain in New Zealand, 
and the other at Cape Kennedy, F)a. Although these particular sites 
were chosen for several reasons, such as land-to-sea paths and subtrojMcal 
location (where marked changes in refraction conditions are common), the 
major consideration was that detailed aircraft and ground meteorological 
observations were available for prolonged periods. 

These detailed measurements allow a quantitative evaluation of the 
error apt to be incurred by assuming that the refractive index is hori- 
zontally stratified. The procedure used was to determine the refractive 
index structure vertically over the transmitter and assume that this same 
structure vertically described the atmosphere everywhere. Rays were 
then traced through this horizontally laminated atmosphere. These ray 
paths were then compared with those obtained by the stcp-by-step ray 
tracing through the detailed convolutions of refractive index structure in 
the two cases under study. 

In the sections that follow we will discuss the two cases chosen for 
study, the methods of calculation used to evaluate refraction effects, and 
the degree of confidence to which standard prediction methods may be 
used under conditions of horizontal inhomogeneity. 

8.2.2. Canterbury 

The Canterbury data were compiled by a radio-meteorological team 
working from September 1946, through November 1947, on the South 
Island of New Zealand under the leadership of R. S. Unwin [14]. This 
report proved invaluable in this investigation, as it was very carefully 
prepared, giving minute details of the experiment on a day-to-day basis. 
Anson aircraft and a trawler were used for meteorological measurements 
over the sea, and three mobile sounding trucks for observations on land. 
The trucks and the trawler carried wired sonde equipment, whereby 
elements for measuring temperature and humidity up to a height of from 
150 m to 600 m (depending on wind conditions) were elevated by means 
of balloons or kites. Standard meteorological instruments provided a 
continuous record of wind, surface i)ressure, temi)erature, and humidity 
at stations at the coast and 14 km and 38 km inland. The headtiuarters 
of the [)roject were at Ashburton Aerodrome, and the observations ex- 
tended out to sea on a fine })erpendicular to the coastline of Canterbury 



324 REFRACTION AND REFRACTIVE INDEX MODELS 

Plain. Aircraft were equipped with a wet-bulb and dry-bulb psychrom- 
eter, mounted on the })ortside above the wing. Readings were taken 
three or four times on each horizontal flight leg of 2 or 3 min duration. 
Special lag and airspeed corrections were applied, resulting in accuracy 
of ±0.1 °C. It was found that, under the variety of conditions in which 
observations were made, the aircraft flights were more or less parallel to 
the surface isobars; hence, the sea-level pressure as recorded at the beach 
site was considered to hold over the whole track covered by the aircraft. 
The relationship used for calculating the pressure, P, in millibars at a 
height h in feet was: 

P(h) = Po - h/30 

where Po is the surface pressure. This approximation (determined by 
averaging the effect of the temperature and humidity distributions on 
pressure in a column of air) resulted in a maximum error in the refrac- 
tivity of 0.5 percent at 900 m. Radiosonde ascents at Hokitika on the 
west coast of South Island and Paraparaumuo and Auckland on North 
Island were used to supplement the aircraft measurements, particularly 
in the altitude levels above 1 km. 

The observations, diagrams, and meteorological records were studied, 
and a profile of unusually heterogeneous nature was chosen. The 
synoptic situation for the morning of November 5, 1947, was selected, as 
it revealed a surface-ducting gradient near the coast with an elevated layer 
about 100 km off shore. A cross section of the area from Ashburton to a 
point 200 km offshore was plotted with all available data, and isopleths of 
modified refractive index, AI, were drawn to intervals of 2.5 units 

M = N -^ {Keh)lO', (8.10) 

where Ke = (15.70) (10-8)/m, and A^ is as defined in chapter 1, (1.20). A 
simplified version of the lower portion of this cross section with the corres- 
ponding M curves is accompanied by a sketch of the general location of 
the experiment in figure 8.10. Some smoothing was necessary, particu- 
larly near the sea surface and in those areas where aircraft slant ascents 
and descents caused lag errors in altimeter readings and in temperature 
and humidity elements. Isopleths over land were plotted above surface 
rather than above sea level, with an additional adjustment in the scale 
ratios of height and distance in an attempt to simplify the reading of 
values from the diagram. 

8.2.3. Cape Kennedy 

The second area studied was the Cape Kennedy to Nassau path for the 
period of April 24 to May 8, 1957. This material was supplied by the 
Wave Propagation Branch of Naval Research Laboratories and the 



CANTERBURY 



325 



METERS 
900-1— 



ISOPLETHS OF MODIFIED REFRACTIVE INDEX 



4IO-X-rr4JO 

0^--^ ^^- - :r_i~ -400 



FEET 

5000 




METERS 
300+-- 



200- - 



20 20 

—LAND SEA— 



—LAND SEA 



60 80 100 120 140 
RANGE FROM COAST (KILOMETERS) 



20 10 20 40 



M CURVES 
80 100 



120 140 160 



100 



340 



i 



\k 



t 



W 



T 



I 



w 

V 



340 340 340 340 340 340 340 340 340 
HORIZONTAL SCALE= ONE DIVISION -10 M UNITS 



340 340 



800 
«00 

y1440o 

V--200 

-lU-0 
340 



NOTE: VERTICAL SCALE CHANGE ABOVE 300 KM 



(ADAPTED FROM CAHTEBBURy PROJECT) 



-NEW ZEALAND- 




X DENOTES RAOB SITES 
LAND OVER 1500 FEET 



36 S 



40 S 



44 S 



Figure 8.10. Isopleths and curves of refractive index for November 5, 1947, Canterbury, 

with a map locating sources of meteorological data. 

(Courtesy of the Canterbury project). 



326 REFRACTION AND REFRACTIVE INDEX MODELS 

University of Florida. The particular case chosen for study was the 
meteorological profile of May 7, 1957 (2000 L.T.) due to its heterogeneous 
nature, showing a well-defined elevated layer at about 1 ,500 m. Fourteen 
refractometer soundings from aircraft measurements taken at various 
locations along the 487-km path (fig. 8.11) and six refractive index 
profiles (deduced from radiosonde ascents from Cape Kennedy, Grand 
Bahama Island, and Eleuthera Island) were read in order to plot a cross 
section of the atmosphere which would represent as closely as possible 
the actual refractive conditions at that time. Unfortunately, the data 
near the surface (up to 300 m) were quite sparse compared to those 
recorded in the Canterbury Project, and calibration and lag errors had 
not been noted as carefully in this preliminary report; therefore, some 
interpolation and considerable smoothing of refractive index values were 
necessary when drawing isopleths. 

8.2.4. Ray Bending 

The classic expression for the angular change, t, or the bending of a 
ray passing from a point where the refractive index is Wi to a second point 
where the refractive index is rii is given in chapter 3 as 

r,^, = - Z'^cot^, (8.11) 



n 



where 6 is the local elevation angle. Equation (8.11) was evaluated by 
use of 

ni,2 

where 

- gi + 02 

The value of 6 at each point was determined from Snell's law: 

niTi cos di = 712^2 cos 02 = constant, (8.13) 

where r is the radial distance from the center of the earth and is given 
by a + h, where a represents the radius of the earth and h the altitude 
of the point under consideration. For simplicity one may rewrite 

(8.13) as 

(1 + A^i X 10-«) (a + hi) cos Oi 

= (1 + A^2 X 10-6) (q + /j^) cos 02. (8.14) 



RAY BENDING 



327 



ISOPLETHS OP REPRACTIVE INDEX 




CAPE CANAVERAL 



220 280'^ 340 

DISTANCE IN KILOMETERS 



REPRACTOMETER PLIGHT PATH 



ELEUTHER A ISLAND 

■p =X=Sj^ SAN SALVADOR 'X^ 




X DENOTES RAOB SITES 



Figure 8.11. Isopleths of refractive index and map of refractometer flight path for 
May 7, 1967, Cape Kennedy to Nassau. 



328 REFRACTION AND REFRACTIVE INDEX MODELS 

Then, when 6 is small, one may expand (8.14) [as in chapter 3, (3.50) to 
(3.58)] and obtain the convenient expression: 



1^2 ^ 2(h, -h,) _ 2(^^ _ ^^) ^ jo-ejy^ 



(8.15) 



where all values of 6 are in milliradians. 

After obtaining r by use of (8.12) and (8.15), one may determine the 
distance, d, along the earth's surface that the ray has traveled from: 

rfi,2 = a [ri,o + (62 - di)]. (8.16) 

Thus by successive application of the above formulas, one may trace 
the progress of the radio wave as it traverses its curved path through 
the atmosphere. Normally the use of these equations is quite straight- 
forward. When considering horizontal changes in n, however, one must 
satisfy these equations by iterative methods. In the present application, 
since n had to be determined by graphical methods, it was felt to be 
sufficient to assume a constant distance increment of 250 to 500 m, solve 
for appropriate height increment from 



Ah = Ad tan di 



1 + ' 

a. 



(8.17) 



graphically determine N for the point di + Ad, hi + Ah, and then deter- 
mine do and Ti,2. 

This latter type of ray tracing was done for various rays of initial 
elevation angles between 261.8 mrad (15°) and 10 mrad (^0.5°). The 
calculations were not carried to smaller elevation angles, since this type 
of ray tracing is not valid within surface ducts for initial elevation angles 
below the angle of penetration [16]. 

8.2.5. Comparisons 

Although both of the calculated ray paths consisted of an oversea 
itinerary with coastal transmission sites, they are quite different in other 
aspects. Canterbury Plain is located southeast of the 10,000-ft chain 
of the Southern New Zealand Alps at a latitude of 44°S (the equatorward 
edge of the westerly belt of winds in November). Cape Kennedy is 
located on a sea-level peninsula at 28°N (the poleward edge of the north- 
east trade circulation in May). While the Canterbury profile showed 
superrefractive tendencies, the Cape Kennedy profile illustrated sub- 
refraction at the surface counterbalanced by an elevated trade wind 
inversion layer, indicating that the total bending values of Canterbury 



COMPARISONS 329 

would be higher than normal, while the Cajje Kennedy example would 
have values near or lower than normal. 

These differences are illustrated by figures 8.12 and 8.13 where the 
bending, r, in milliradians is plotted versus altitude in kilometers. The 
effect of horizontal changes is most pronounced for rays with initial eleva- 
tion angles of 10 mrad. On these figures the term "vertical" ray is used 
to designate the ray path through the horizontally homogeneous n struc- 
ture determined from the refractive index vertically over the station. 
The term "horizontal" ray designates the ray path through the complex 
actual n structure. It is quite evident that a consistent difference in 
bending of about 1 mrad exists between the "vertical" and "horizontal" 
rays at Canterbury above 1 km for ^o = 10 mrad. This would be expected 
since the vertical M profile (fig. 8.10) at the beach (our hypothetical 
transmitter site) is nearly normal in gradient while as little as 10 km to 
sea a duct exists, thus indicating a near maximum difference between the 
"horizontal" and the "vertical" rays at any initial elevation angle small 
enough to be affected by the duct. This is in contrast, however, with 
the case of Cape Kennedy where, except for the region of the elevated 
duct centered at about 1,500 m, the "vertical" and "horizontal" rays are 
in c^uite close agreement. These two examples illustrate that horizontal 
variations must be near the surface to be most effective. The importance 
of the altitude of the variation is due to the fact that refraction effects 
are very heavily weighted toward the initial layers [16]. 

Also shown on figures 8.12 and 8.13 are the values of the bending which 
would be predicted from the Central Radio Propagation Laboratory 
corrected exponential reference atmosphere [1]. The values shown are 
obtained from the value of A'' at the transmitter site as corrected by the 
vertical gradient over the first 100 m. It is noted that, for do = 10 mrad 
at Canterbury, the value of bending predicted by the model is in essential 
agreement with the "vertical ray" bending but under estimates the 
"horizontal ray" (which has the largest variation of n with horizontal 
distance) by about 1.25 mrad. For Cape Kennedy at ^o = 10 mrad, 
the model atmosphere overestimates the bending by about 1.25 mrad for 
altitudes in excess of 2 km. It should be emphasized that, although the 
model exponential atmosphere appears to represent the average of the 
two specific cases studied, the departure from this average arises from 
quite different causes in each case. The differences in the Canterbury 
case arise from the marked effect of horizontal variation of n as is indi- 
cated by the agreement of the vertical ray bending with the model atmos- 
phere. The disagreement in the Cape Kennedy case is due to the 
presence of a very shallow surface layer of nearly normal gradient topped 
by a strong subrefractive layer; therefore, it represents a shortcoming of 
the model rather than an effect of horizontal changes of n. 



330 



REFRACTION AND REFRACTIVE INDEX MODELS 



if) 8 
cr 

bJ 

I- 

UJ 
9 6 



UJ 
Q 

3 



< 2 



(J) 8 
UJ 



q 6 



Ld 4 
Q 

Z) 

< 2 









j 








7^ 








/ / 






/ 






















/ I 




/ / 


/ 










o 
o 

E 






V 








x,lf 




4 


/ 








00 

io 

00 




/ 








Hi 


1 


/if y 
$ / 










1 






/ 








/ // 
/ // 

/ ^/ 


1 


1 
1 

/ 


/ 










1 






// 






// 


/ // 


/ 


1/ 


/ 
















^ 










/ 


























Y / / 














/ 


y 












^^00 Ho 

,^^\0 mrad " 


1 

Vert. 








1 


/ 




^ 


l^. 


^' 




17.4 mrac 






/? -^ ^ 










(^ 


i J^^ y 




J c 






o^.r, „^. _ 1 


i 




fe:^^^ 




r^ 


Corrected Exponential 



-2 2 4 6 8 10 

BENDING IN MILLIRADIANS 

Figure 8.12. Canterbury, to 10 km, altitude versus ray bending. 

















'/ 






/ 


// 




/ 
/ 

/ 










' 


T3 




/ 


; 

/ 






/ 


// 


1 
1 
1 












ti 




/ / 


e 








/ 


^1 












!«> 




-V^ 






t-7 


/ 


^1 












s 




1 1^- 

1^ 








' .( 


/ 












^ 






i 1 










/ 














1 
















/ 














1 














/ / 


/ 














! 












i 1 




/ 














I 




j 


1 






f 1 




/ 












































i 




1 


1 




// 


/ / 




/ 














I 




Jl 






/ 


/ / 




/ 














1 




It f 






/ 


/ / 


/ 
















1 




jl / 






/ 


' / 


/ 




















! 






^"'/ 


/ 


/ 
/ 














J 




i 






/'' 


/ 


00 Hor 


Vert. 














j^ / 


■'y' 




10 mrad '- 






111 // 


J 














^ 




^^ 


y-y^ y' 






17.4 mrad <^ 


° 












/>-^ 


"-^^s 




^^ 




52.4 mrad -^ » 
















































81.8 mrad" ° 


































<s 




'"'"^ 


'^ 




Corrected Exponential 







2 4 6 

BENDING IN MILLIRADIANS 



Figure 8.13. Cape Kennedy, to 10 km, altitude versus ray bending. 



EXTENSION TO OTHER REGIONS 



331 



The preceding analysis of bending throws the refractive differences in 
each case into sharp rehef. The effect of refraction, of course, is to vary 
the ray path. Figures 8.14 and 8.15 show the ray paths corresponding 
to the bendings of figures 8.12 and 8.13. Note that for Canterbury at 
^0 = mrad, the effect of the horizontal variation of n is to produce a 
difference in estimation of about 1 km in height or 20 km in ground dis- 
tance at 300 km from what would be obtained from considering the verti- 
cal n profile as a representation of the entire path. The effect of the 
subrefractive layer at Cape Kennedy is not so large, but it does cause an 
overestimation of the ground distance by about 5 km and an underesti- 
mation of the height by less than one quarter of a kilometer at a ground 
distance of 300 km by assuming that the vertical profile may be used 
throughout the entire ray path. 



8.2.6. Extension to Other Regions 

It should be pointed out that the ducting case at Canterbury represents 
an extreme refraction condition and is not necessarily typical of conditions 
observed in other regions or, indeed, at Canterbury. The Canterbury 
project was purposely restricted to a study of ducting conditions with the 
result that less than 20 percent of the total observations for the fifteen 




Corrected exponential reference 
atmosphere 

Path of Canterbury vertical ray 

Path of Canterbury horizontal ray 



120 160 200 240 

DISTANCE IN KILOMETERS 
Figure 8.14. Canterbury, to 10 km, attitude of ray versus distance. 



332 



REFRACTION AND REFRACTIVE INDEX MODELS 






LJ 

O 

_J 



3 



1 






/ 






/ 




^ 








i>/ 




f. 


^-^ 








^1 


V 




.-> 


Ko-^ 


r 








4i? 




r 


>^ 






j 




/ 






■ >^ 


■/' 






/ 


/ 


^ 


^ 


r ^ 








/ 


/ 




y^ 


^^ r„..l.t„^ 


' 


y^ ^ 


Y^ 


atm 
■ Path 




/ 


/ \^^\ 


^sphere 

of Canaveral vertical ray - 
of Canaveral horizontal ray 


^ 






o 


5 Pnth 


^ 


^ 








1 1 1 1 







40 



240 



320 



360 



DISTANCE IN KILOMETERS 
Figure 8.15. Ca-pe Kennedy, to 10 km, altitude of ray versiis distance. 



months are reported. Therefore, because one of the more extreme cases 
is represented by the November 5 example, one might conclude that much 
less than 20 percent of the observations would show the same degree of 
horizontal n change as the profile studied. 

If one further hypothesizes that the greatest horizontal n change would 
be associated with ducting conditions, then the percentage incidence of 
ducts as evaluated from radiosonde observations, listed for various 
stations in table 8.3 would indicate that the effects of horizontal changes 
of n sufficient to cause variations in the ray path as large as those of the 
present study would be observed less than 15 percent of the time, regard- 
less of geographic location. 

The probable importance of subrefractive layers upon the prediction 
of refraction effects has emerged as a secondary result of the present study. 
Although subrefraction is normally neglected, it is potentially a very im- 
portant refractive factor for distances of, say, less than 40 km. Even 
though the percentage occurrence of subrefractive layers can be as large 
as 6 percent (see table 8.4), this effect is frequently offset by the con- 
current occurrence of an adjacent superrefractive layer, as is illustrated 
by the Cape Kennedy example. 



CONCLUSIONS 333 

Table 8.3. Percentage occurrence of surface ducts during the years 1952 to 1956 



Station 


Percent incidence 




February 


May 


August 


November 


Fairbanks, Alaska 

Columbia, Mo 

Washington, D.C. 


9.4 
0.7 
0.7 
10.0 
0.7 


0.4 
2.5 
4.8 
9.2 
3.5 


0.4 
8.4 
4.3 
12.4 

8.5 


6.2 
1.3 
1.4 


Canton Island 


11.5 


Miami, Fla 


2.7 



Table 8.4. 



Percentage occurrence of surface subrefractive layers during the years 
1952 to 1956 



Station 


Percent incidence 




February 


May 


August 


November 


Fairbanks, Alaska 

Columbia, Mo 

Washington, D.C 


0.0 
0.0 
0.9 
0.0 
0.7 


0.0 
1.6 
2.2 
0.0 
0.3 


1.2 
0.6 
5.8 
0.0 
0.9 


0.4 
4.0 
2.7 


Canton Island 


0.3 


Miami, Fla 


0.7 



8.2.7. Conclusions 

The conclusions of the present study could be considerably modified 
by the analysis of many more examples, although it is evident that hori- 
zontal variation of n near the earth's surface produces the most marked 
deviations from the ray paths obtained by assuming horizontal stratifica- 
tion of n. The effect of horizontal changes occurring more than a kilom- 
eter above the surface appear from our present examples, to have little 
effect. Further, the effects of horizontal changes appear to be most 
pronounced in the presence of surface ducts and at small elevation angles. 
The tentative conclusion is reached that the effect of horizontal n change 
is normally small, since ducting will occur less than 15 percent of the time. 



8.3. Comparison of Observed Atmospheric Radio Re- 
fraction Effects With Values Predicted Through 
the Use of Surface Weather Observations 



8.3.1. Introduction 

The atmospheric radio refraction effects considered in this section are 
of two general types: errors in measuring distance by means of timing 
the transit of radio signals between two points, known as radio range 
errors, and errors in estimating the elevation angle of a target by means 
of measuring the angle of arrival of radio signals from the target, known 



334 REFRACTION AND REFRACTIVE INDEX MODELS 

as elevation angle errors. Many methods have been proposed to take 
into account these refraction effects for the purpose of improving measure- 
ments by removing systematic bias. One of these involves the use of 
the surface value of the radio refractivity, A''^, a quantity which can be 
measured directly with a microwave refractometer, or calculated from 
the ordinary meteorological variables of temperature, pressure, and 
humidity, to predict values of either range error or elevation angle error; 
this method has been shown theoretically to be useful, with the accuracy 
increasing with increasing initial elevation angle [17, 18, 19]. It is the 
purpose of the present note to compare recent experimental determina- 
tions of atmospheric refractive effects with values estimated theoretically 
from surface meteorological conditions. 

8.3.2. Theory 

The operation of a radio tracking system depends on the measurement, 
in some manner, of radio signals received from the target. The radio 
signals are transmitted in the form of radio waves which travel from the 
target to the tracking system. The form of these radio waves is distorted 
by the presence of the earth's atmosphere. Since solutions of the wave 
equation are extremely difficult to obtain for the case of general atmos- 
pheric propagation over a spherical earth, it is common practice to 
evaluate refraction effects by means of ray tracing, a process which is 
based on the use of Snell's law. 

One of the two types of refraction errors considered in this appendix 
is the elevation angle error, c, which is the difference between the apparent 
direction to a target, as indicated by the angle of arrival of a normal to 
the radio wave front, and the true direction. This error is primarily a 
function of the refraction, or bending, of the radio ray. For targets 
beyond the atmosphere, the two quantities are asymptotically equal (with 
increasing range). . The values of e and r at any point on the ray path 
obey the following inequality: 

r/2 ^ e ^ T. 

Recalling that (chapter 3) the bending of a radio ray may be expressed 
by an equation of the form 

T = a -\-hNs (8.18) 

where a and b would be functions of the initial elevation angle of the ray, 
do, and the height (or range) along the ray path at which the bending is to 
be calculated. Such an assumption can be checked by examining the 
behavior of values of r, ray traced for a number of observed height profiles 
of radio refractive index, plotted against the corresponding values of N s- 



THEORY 335 

Such a plot is shown in figure 8.16 for a small initial elevation angle, 
50 mrad (about 3°), and a "target" height beyond the atmosphere, 70 km. 
The family of A^ profiles used in ray tracing this sample of bending values 
is referred to as the CRPL Standard Sample. ^ It can be seen from inspec- 
tion of figure 8.16 that the assumption of linearity expressed in (8.18) is 
justified for this case. A similar conclusion can be reached from examina- 
tion of data for other cases, including low target heights and elevation 
angles down to zero degrees, although for these extremes the degree of 
correlation between t and N s is not as marked as that shown in figure 8.16. 
The other refraction variable treated in this section is the radio range 
error, ARe, which is here defined as being that error incurred in measuring 
the distance between two points by means of timing the transit of radio 
signals between the points, and assuming that the velocity of propagation 
is equal to that of free space. For the case of a radio ray, this error is 
composed of two parts: the difference between the curved length of the ray 
path, called the geometric range, Rg, and the true slant range, Ro; and 
the discrepancy caused by the lowered velocity of propagation in a refrac- 
tive medium. The geometric range is given by 



Rg = / CSC 6 dh, 





and the apparent, or radio, range by 

J n 

Jo 

Thus the total radio range error, ARe — Re — Ro, is given by 

fht 

ARe = I n CSC 6 dh — Ro, 

Jo 



Re ^ I n esc d dh. 



or 



ARe ^ 10 ^ f N CSC ddh -\- I esc d dh - Ro. ,^ ,^. 

Jo (o.iyj 



^Meaning explained in section 8.3.3. 



336 REFRACTION AND REFRACTIVE INDEX MODELS 

Table 8.5. Typical and extreme values of range errors for targets beyond the atmosphere 





Typical N,= 


= 320 


Extreme Ns= 


=400 


Maximum 
percent 


00 


ARg 


ARw 


ARe 


ARg 


A/e.v 


ARe 


ARg/ARe 




meters 

















10 


100 


110 


60 


165 


225 


27 


20 mrad 


2.5 


62.5 


65 


4.5 


73 


77.5 


6 


50 mrad 


0.7 


38.1 


38.8 


1.0 


43 


44 


2.3 


100 mrad 


0.14 


22.26 


22.4 


0.2 


24.8 


25 


0.8 


200 mrad 


0.02 


11.9 


11.9 


0.03 


13.0 


13.0 


0.23 


500 mrad 


0.001 


5.01 


5.01 


0.002 


5.50 


5.50 


0.04 



The first term on the right-hand side of (8.19) is the "velocity" or 
"refractivity" error, ARn', the last two terms represent the geometric 
range error, ARg, which is the difference in length between the straight 
path, Ro, and the curved ray path, Rg. Table 8.5 gives some typical and 
extreme values of range errors ray traced for observed A'^ profiles. 



7.0 



T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 




-T=-0,685476 + 0.01804535 Ng (mrad) 



J I I L 



J I I L. 



Figure 8.16. Total refraction at do equals 50 mrad, h equals 70 km, for the CRPL 

standard sample. 



THEORY 337 

From table 8.5 it can be seen that the geometric range error, ARg, does 
not represent a significant portion of the total range error except at very- 
small initial elevation angles, between zero and about 3°. This being so, 
the behavior of the total range error will be primarily a function of the 
first integral in (8.19) for elevation angles greater than about 3°. The 
integral expression 

ARe ^ 10~' / ' N csed dh, 



may be rewritten as 



ARe ^ CSC ^0 X 10"' 



N dh 



1 - 2 sin' ( — T-^ ) + cot 00 sin {d - do) 



or 



ARe ^ CSC do N dh 



+ Z (-l)''-"^ N 



cot 00 sin (0 - do) -2 sin' 



^)] 



dh, (8.20) 



for sin d < 2 sin do, < d < 7r/2. This expression is analogous to that 
derived for the case of ray bending, (3.6) of chapter 3, and, similarly, the 
integral series on the right-hand side of (8.20) contributed only 3 percent 
or less to the value of ARe for do larger than about 10°. From (8.20) one 
would thus suspect that the radio range error might be well e.stimated as 
a linear function of the integral of N with respect to height. In treating 
this integral, it is informative to note that any given N(h) profile may be 
"broken up" into three primary components: 

N(h) = N'{Ns, h) + N"ih + hs) + 8N(h) 

where A'^' is that part of the profile which can best be expressed as a 
function of A^^ and height, N" is a standard distribution of refractivity 
with resjiect to altitude above mean sea level (h -\- hs) which is independ- 
ent of N s, especially above the tropopause, and 5N represents a random 
component of the profile which cannot in general be accounted for 
a priori. 



338 REFRACTION AND REFRACTIVE INDEX MODELS 

The A'^' component is generally effective over the first few kilometers, 
while above 6 or 7 km altitude, the A'^" component forms the bulk of the 
profile [1]. Thus the integral of the A^ profile with respect to height may 
be written as: 



Ndh = I N' (Ns, h)dh-\- / N" {h + h,)dh + / 8N{h) dh, 

Jo Jh, Jo 



or. 

Ndh ^ F, (Ns, hd + F2 (K, ht) + 6F (ht) 



where 8F is the random contribution to the integral. For any particular 
ht then 



Ndh = Fy (Ns) + F2 (hs) + dF 



or 



where 



N dh = F, (hs = 0) + Fi (A^^) - F3 (hs) + 8F (8.21) 



F3 = / N"(h + hs) dh, and F2{hs = 0) is a constant. 



It was found empirically, from integrated N{h) profiles, that 



N dh^a + biNs - hihs ^ S.E. (8.22) 



The analogy between (8.21) and (8.22) is plain (the standard error of 
estimate of (8.22), SE, represents the standard deviation, SF, of (8.21)). 
The results of such an empirical study are shown in figure 8.17 for the 
CRPL Standard A'^ Profile Sample, for ht beyond the atmosphere. 

For any particular application of (8.22) at a single location, the term 
62/1 s will be absorbed into the constant a, since hs does not vary. How- 
ever the introduction of this term is necessary to explain the station 



THEORY 



339 



2,5 



CO 

or 



I^^ iO-y°° N(h) dh (m 




I^= 1.4588 + 0.0029611 Ng (m) 
r = 0.982 
S.E.= 0.0368nn 



I I I 



250 



350 



Figure 8.17. Integrated refractive index -profiles for the CRPL Standard Sample. 



elevation dependence of integrated N{h) profiles when taken from a 
sample containing stations at widely differing elevations, such as the 
Standard Sample. 

It is thus apparent that radio range errors, at least at the higher eleva- 
tion angles, are primarily a linear function of N s- That this is also true 
at comparatively low angles is shown in figure 8.18, for ^o = 50 mrad 
(about 3°) for the same profile sample. The reader should especially note 
the similarity of the distributions of the points about the regression lines 
between figures 8.17 and 8.18, showing that the range errors at about 3° 
are still primarily a function of the integral of A'^ with respect to height, 
or the range error at 90°. 

It has thus been demonstrated that, theoretically, it should be possible 
to estimate both the angle of refraction of radio rays and errors in radio 
range measurements from measurement of the refractive index at the 
surface of the earth. This should be true for targets in or beyond the 
atmosphere, at elevation angles down to, and possibly lower than, 3°. 
In addition, if the behavior of refractive index profiles is similar in differ- 
ent parts of the world, it should be possible to specify "universal" values 



340 



REFRACTION AND REFRACTIVE INDEX MODELS 



45 



en 



40 - 



35 



30 



1 1 1 1 1 1 


1 1 1 1 1 1 I 1 1 1 1 1 


1 


_ AR'e = l8.2968 + Q063l05 Ns^m) 


REGRESSION OF 
ARg ON Ns AND hg 


'^\ °V 


'^- 


- AR'e= ARg + 1.962 


hs ^„ ^^° °° 


- 






n POINTS FOR DENVER, COLO., AND 






° 


ELY, NEV. ; SHOWN ALSO AS: 




t 


Q 


• ADJUSTED FOR ELEVATION 


- 


1 1 1 J 1 1 




DEPENDENCE (ARg) 


1 


1 , 1 


1 1 1 1 1 1 1 1 1 



210 



250 



350 



400 



Figure 8.18. Total range error at do equals 50 mrad, h equals 70 km, for the CRPL 

standard sample. 



of the coefficients in (8.18) and (8.22) and to predict these values in ad- 
vance by analysis of a large heterogeneous sample of refractive index 
profiles. In the succeeding subsections it will be shown how this has 
been done, and a comparison will be made between the results so derived 
and the results of some measurements over actual radio paths. 

8.3.3. The CRPL Standard Atmospheric Radio 
Refractive Index Profile Sample 

In the preceding section it was shown that, theoretically, it should be 
possible to estimate either radar elevation angle errors or radio range 
errors at any particular location by means of a system of linear equations 
in N s, where the coefficients are functions of the target position. The 
target position can be specified by either the apparent elevation angle and 
target height, or the apparent range and target height (or as a third possi- 
bility, the apparent range and elevation angle), each having advantages 
in different situations [19]. The equations recommended are 



= ai (00, hi) + bi {do, ht) Ns ± S.E.i {do, he), 



(8.23) 



REFRACTIVE INDEX PROFILE SAMPLE 341 

and 

ARe = a. (Re, ht) + 62 [Re, ht) Ns ± S.E.2 (Re, ht), (8.24) 

where e is the elevation angle error, AR e is the radio range error, ^0 is the 
apparent elevation angle, hi is the target height, Re is the apparent radio 
range, and S.E. is the standard error of estimate about the regression 
line of € or ARe on N s- Values of the coefficients may be obtained by 
performing linear regressions of e or ARe, as ray traced for an appropriate 
sample of radio refractive index profiles, ujjon A^^, for a large number of 
target positions. As a byproduct of these calculations, one also obtains, 
for each target position, a value of the residual error (the standard error of 
estimate) to be expected for the particular type of profile sample used. 

In order to obtain a general set of eciuations to be useful under arbi- 
trary conditions of location, climate, and weather, a large sample of 
N profiles has been assembled which is believed to be representative of 
both mean climatic and geographic trends and the larger synoptic varia- 
tions which may be encountered. This was done by choosing 13 radio- 
sonde stations representative of the major geographic and climatic types 
of the world, and then choosing from each station six A'' profiles of particu- 
lar types, two of which are typical of the extremes of monthly mean 
conditions for that location, and the other four of which are typical of 
some of the variations which are found at that location [18]. The result 
is a sample of 77 N profiles,^ which has been found over a period of years 
to be a sound cross section of general refractive conditions and has thus 
been named the CRPL Standard Atmospheric Radio Refractive Index 
Profile Sample, hereafter referred to as the CRPL Standard Sample. 
Although the locations chosen for this sample are heavily weighted towards 
the United States, it has been found that the general behavior of the 
refractive index structure as inferred from the standard sample is typical 
of conditions experienced in most parts of the world [5]. 

The remainder of this section will be devoted to some comimrisons of 
observed radio refraction data with the predictions supplied by the 
CRPL Standard Sample, as derived from the linear regressions men- 
tioned above. 

Since the refraction measurements re[3orted here consist of samples 
taken at particular locations over comparatively short i)eriods of time, 
they should provide a test for the general set of coefficients derived from 
the Standard Sample; not only is the general theoretical api)roach tested 
against measured values, but the measurements coming from places of 
more or less homogeneous nature, they provide a check as to whether 01 
not coefficients derived for a large heterogeneous sample of data are 
applicable also to individual places and times; i.e., they should reveal 



One of the types could not be found for one of the stations used. 



342 REFRACTION AND REFRACTIVE INDEX MODELS 

how much of the observed correlation of the heterogeneous sample is 
derived from correlation between "classes" of data (in the statistical 
sense). For a more thorough treatment of the CRPL Standard Sample 
and the associated regression coefficients for range error and elevation 
angle error, the reader is referred to Bean and Thayer [19]. 

For the Standard Sample, the standard error of estimate is equal to 
the standard prediction error within ±1 percent over the range of A^s from 
200 to 470, and will be used interchangeably with the latter. 

8.3.4. Comparison With Independent Data 

Before turning to an examination of the experimental refraction data 
and the degree of success realized in applying the theoretical prediction 
model to those data, it seems appropriate to examine the accuracy of the 
prediction model when ai^plied to some independent theoretical (i.e., ray- 
traced) data. For this purpose, four check stations were selected which 
were not only independent in the sense of not having been included in 
the original 13-station Standard Sample, but were from locations widely 
differing from the region of selection of the original sample. It was de- 
cided to select one station representative of an arctic type climate, one 
temperate, one tropical, and one from a "problem" climate area. 

Amundsen-Scott station at the South Pole (lat. 90°S) was chosen as the 
arctic type ; this station was expected to present the most rigorous test of 
the prediction model (as based on the Standard Sample) that could be 
obtained anywhere in the world. In the first place the extreme arctic- 
continental climate, with almost no water-vapor contribution to the re- 
fractive index and the nearly incessant temperature inversion, is more 
alien to the Standard Sample than any other type ; in the second place the 
station elevation is 2800 m, which is 900 m in excess of the highest station 
(Ely, Nev., 1908 m) included in the Standard Sample. These two effects 
were expected to augment each other as regards refraction. 

Dakar, Senegal, on the western coast of Africa, was selected as a 
"problem" climate station; an inverse relationship exists there between 
N s and AN (the A'-gradient over the first kilometer above the surface). 
A Congo basis station, Bangui, in what was French Equatorial Africa, 
was selected as the tropical location, and Moscow, U.S.S.R., was selected 
as the temperate location. 

In order to combine brevity with comprehensiveness, ray tracings were 
done of the total refraction (bending at 70 km target height) at two 
elevation angles, 20 mrad and 100 mrad, for six profiles from each loca- 
tion. The six profiles were selected as representing roughly the range of 
Ns in winter (February), summer (August), and spring-fall (May and 
November), two profiles being selected from February, two from August, 
and one from each of May and November. The 20-mrad elevation angle 



COMPARISON WITH INDEPENDENT DATA 



343 



— I 1 1 I 1 

AMUNDSEN -SCOTT STATION 
SOUTH POLE, ANTARCTICA 

LAT = 90° S 

LONG. - N A. 

ELEV = 2800 m 



OBSERVED FIT- 
•=-5.89»tt0520 Ns 
S E =«006mtad^^ |8„ = 20n 




^STANDARD PREDICTION 

ERROR LIMITS 

(67% CONFIDENCE) 



^STANDARD SAMPLE 
T = -OI40t000963N5 
(PREDICTION ERROR: lOOl r 




DAKAR, SENEGAL, AFRICA 
LAT= 14° 43' N 
LONG =17° 14' W 
ELEV. = 40 m 



-OBSERVED FIT 
l56>003l6Ns 
-'\ SE = «033iT 

!£l STANDARD SAMPLE 

r'-287>00358Ns 

^STANDARD PREDICTION 

ERROR LIMITS 

(67% CONFIDENCE) 



^—STANDARD SAMPLE 

T = -QI40. 0,00963 Ns 

(PREDICTION ERROR = '0 01 



200 210 220 230 240 250 260 2?0 



320 330 340 350 360 370 




MOSCOW, USSR 
LAT =55° 49' N 
L0NG = 37°37'E 
ELEV = 186m 



20 mrad| ,,---C-i: 



^OBSERVED FIT 
r=-4l9.00390Ns 
SE ='0 ISmrad 



STANDARD SAMPLE 
T=-0l40>000963Ns 
(PREDICTION ERROR. >0 01 r 




BANGUI, CENTRAL AFRICAN REPUBLIC 
LAT = 4° 23' N 
LONG = 18° 34' E, 
ELEV =385 m 



"■STANDARD PREDICTION 

ERROR LIMITS 

(67% CONFIDENCE) 



OBSERVED FIT 
= -6 l7-00457Ns 
S E = t0 34rnrod 



STANDARD SAMPLE 

T = -0 I40'000963N5 

(PREDICTION ERROR = 'OOImrod) 



290 300 310 320 330 340 350 360 



310 320 330 340 350 360 370 3«0 



Figure 8.19. Comparison of predicted refraction at h equals 70 km for Oqs of 20 mrad 
and 100 mrad from regressions of the CRPL standard sample and ray-traced values 
from four independent locations. 



344 REFRACTION AND REFRACTIVE INDEX MODELS 

was selected as representing roughly the lower limit of elevation angles 
for which the bending is expected to be strongly correlated with N s (say 
r > 0.9), while at 100 mrad (about 6°) the correlation is expected to be 
extremely high (say r > 0.99) and the refraction should be reasonably 
free of random profile effects. 

The results of the ray tracings and the comparison with predicted 
values are shown in figure 8.19. As expected, the results from the South 
Pole seem to depart significantly from the predicted values, at least for 
the 20-mrad elevation angle. At the 100-mrad elevation angle some of 
the calculated points lie more than one standard deviation from the pre- 
dicted line (the theoretical prediction error is too small to show on the 
graph clearly) ; however, in all four cases the differences are less than 50 
^trad, a figure which as shall be seen may represent the limit of accuracy 
obtainable from the atmosphere in actual practice. At angles over 100 
mrad the errors would be smaller; in fact they should tend to decrease in 
inverse proportion to the square of the initial elevation angle, as indeed 
they do between 20 and 100 mrad. 

A conclusion which may be drawn from the above results is that any 
regions where the prediction model based on the Standard Sample would 
not be expected to provide the theoretical accuracy are probably regions 
of climatic extremes, and at least for the case of angular errors the effects 
will be negligible for elevation angles of a few degrees or more. As an 
interesting aside it can be noted that apparently the Antarctic may be a 
desirable area for tracking systems location, at least with respect to 
atmospheric refraction effects, since (most likely because of the lack of 
substantial water vapor and the relatively homogeneous conditions) the 
prediction error for ^o = 20 mrad in figure 8.19 is only about one-fifth as 
large as for temperate climates, indicating a possibly more stable atmos- 
phere (even 90 percent confidence limits for the SE in figure 8.19 yield a 
value less than half of the theoretical temperate value of ±0.286 mrad). 

8.3.5. Comparison With Experimental Results 

Before comparing the theoretical and experimental results, it is appro- 
priate at this point to examine what one would expect to observe on the 
basis of propagation theory. In the case of angular errors it is expected 
that i^ropagation through the real, turbulent atmosphere will produce 
random variations in the shape of the incoming wavefront, so that meas- 
urements made with systems in which the receiving antenna is alined with 
the incoming signal will have random variations introduced in addition to 
the ordinary refraction effects. Since these variations will probably not 
be a function of elevation angle to any great extent, this implies that the 
residual variance in predicting the elevation angle errors will probably 
always be greater than predicted from theoretical (static) considerations, 
and that there will probably be some minimum value of this variance for 



COMPARISON WITH EXPERIMENTAL RESULTS 345 

very large elevation angles. Thus, in some cases, the residual errors will 
probably not decrease steadily with increasing elevation angle, but will 
tend to flatten out at some point and assume a more or less constant value 
above that point. These effects will be complicated in comparing one set 
of data with another by such things as differences in the location or time 
of day or season in which data are taken, and instrumental effects such as 
aperture averaging. 

The case of range errors is more straightforward. The effects of 
turbulent atmospheric inhomogeneities are expected to average out over 
regions of abnormally high or low density, when considering the transit 
time of particular points on the wave front. Hence the effect on the 
residual range errors is expected to be small, and the observed values are 
expected to compare rather well with the predicted (theoretical )values. 

Turning first to the comparison of observed and predicted elevation 
angle errors, figure 8.20 shows some data on the mean refraction of 1.85- 
cm radio waves received from the sun, a target at essentially infinite range 
so that the elevation angle error is identical with the total angular bending 
of the radio ray, r. The data shown in figure 8.20 were obtained by 
tracking the sun with a precise radio sextant developed by the Collins 
Radio Company, and were collected in August through December of 1959 
at Cedar Rapids, Iowa [20]. These data represent essentially instan- 
taneous measurements. The mean of all observations at each elevation 
angle is plotted for elevation angles ranging from 2 to 65°, and the mean 
value of N s associated with each point is about 332; the curve for the 
mean bending of the CRPL Standard Sample corresponds to the mean 
value of A^s of 334.6 for that sample and hence the data should be com- 
parable. The standard deviation "wings" refer to the standard deviation 
of the individual "instantaneous" data, not to the standard error of 
estimate of the mean value. The close agreement observed for elevation 
angles between 2 and 35° constitutes not only a confirmation of the use- 
fulness of the Standard Sample, but also a verification of the accuracy 
of ray-tracing theory in estimating radio wave refraction in the actual, 
and thus heterogeneous, atmosphere. The standard deviation of the 
Collins data (shown on the lower part of fig. 8.20) is generally lower than 
for the standard sample, but this is to be exjjected in view of the larger 
range of climatic variation contained in the CRPL Standard Profile 
Sample. The apparent discrepancies in the measurements made at eleva- 
tion angles over 40° are apparently due to some slight inaccuracies in the 
calibration procedure used on the radio sextant during the period of data 
acquisition.'* In fact, the data shown in figure 8.20 are almost precisely 

'' The data for the highest elevation angles in figure 8.20 were necessarily collected 
during the early part of the period when the sun was higher in the sky. In a private 
communication, Anway states that the mean A'^s applicable to the data at 60° to 65° 
was 358 rather than 332; this difference would account for about one-third of the 
discrepancies noted, reducing the residual bias to a maximum of about 40 Mi'ad. 



346 



REFRACTION AND REFRACTIVE INDEX MODELS 

ELEVATION ANGLE, 9^, DEGREES 



26° 40" 55° 65° 




Figure 8.20. Comparison of measured total atmospheric refraction of 1.85 cm radio 
waves at Cedar Rapids, Iowa, with values predicted from Ns. 



what one would expect to observe if all of the measured values of refrac- 
tion were increased by a systematic calibration error of about 50 /xrad 
over their correct values. The standard deviations in figure 8.20 tend 
to flatten out at high elevation angles, an effect which is to be expected 
theoretically as pointed out previously. At any rate, the largest differ- 
ence between the observed data and the predicted curve in figure 8.20 
at elevation angles over 30°, is only about 50 /urad or 10 sec of arc (the 
angular diameter of the planet Mars at its average distance from the 
earth is 10 sec of arc, an angle not discernible to the naked eye). Al- 
though this discrepancy might be significant militarily, it is only about 
0.5 percent of the diameter of the target sun and is probably near the 
limit of accuracy of the equipment used. 



COMPARISON WITH EXPERIMENTAL RESULTS 



347 



Figure 8.21 shows the results of the specific measurements reported by 
Anway for the radio sextant for all cases at an elevation angle of 8° zt 
0.09°; each point represents an "instantaneous" reading. The soHd hne 
represents the Hnear regression of the measured refraction data on the 
values of A''s; the dashed line shows the predicted linear relationship 
derived from least squares fits to the CRPL Standard Sample ray traced 
refraction data. The mean bias between the two lines is about 40 jurad, 
interestingly close to, and in the same direction as, the apparent cali- 
bration error noted in the mean refraction data at high elevation angles. 
The standard error of estimate is considerably higher than predicted ; how- 
ever, the rms uncertainty of ±0.052°, or ±0.91 mrad, in the apparent 
elevation angle would be sufficient by itself to increase the standard error 
of estimate to about ±0.017 mrad, which is 4 times larger than the pre- 
dicted value. It is not known how much of the total standard error of 
±0.12 mrad is due to measurement errors as opposed to unforeseen 
fluctuations in actual atmospheric refraction. 



< 

< 



2.5 



< 

Li_ 
LJ 



1.5 



;LEAST SQUARES FIT 



-0.216 + 0.00754 Ns (mrad) 
0.87, S.E. = 0.12 mrad $ Ij 





\^ 



MEASURED REFRACTION OF 1.85 cm RADIO WAVES 
AT 8° ± 0.052° ELEVATION ANGLE , WITH 
UNCERTAINTY WINGS DUE TO INEXACT 
ELEVATION ANGLE : rdX.] „„ = 0.0163 



^LINEAR REGRESSION OF C.R.P.L. STANDARD 
N- PROFILE SAMPLE; PREDICTED VALUES; 
T = -0.058 + 0.00698 Ns (mrad) 
r = 0.999, S.E. = 0.004 mrad 

\ \ \ \ 



290 300 



310 



320 



330 



340 



350 360 370 380 390 



Figure 8.2 L 1.85 cm radio refraction at an elevation angle of 8 degrees, at Cedar Rapids, 

Iowa. 

(After Anway, 1961). 



348 



REFRACTION AND REFRACTIVE INDEX MODELS 



T3 
O 



^ \ ^ \ ^ \ 

CAPE KENNEDY, FLA 

MEAN APPARENT ELEVATION ANGLE =0.7 mrad 
RANGE = 17. 1 km 
1EAN TARGET HEIGHT = l3 7m 




Figure 8.22. Elevation angle fluctuations from phase differences taken across a 24-ft 
vertical baseline, at Cape Kennedy, Fla. 



Figure 8.22 shows some results of measurements taken at Cape 
Kennedy, Fla., on November 1-3, 1959, at a very low elevation angle, 
about 0.7 mrad or 0.04°. These are "instantaneous" measurements, 
taken at half-hourly intervals, of the phase difference fluctuations between 
the signals from a beacon as they arrived at the upper and lower terminals 
of a vertical 24-ft baseline, thus being very closely equivalent to a meas- 
urement of the fluctuations in the angle of arrival of the wave front at the 
centerpoint of the baseline (the altitude difference between this point and 
the target beacon is referred to as the "mean" target height). Since only 
the fluctuations and not the total phase differences were measured, only 
the slope and scatter of the elevation angle errors as a function of the 
observed N s data can be compared with the predicted values from the 
CRPL Standard Sample. The zero point on the graph is set by the pre- 
dicted mean value for the sample. The correlation coefficient is, as ex- 
pected, only 0.57. In this case the scatter of the observed data is well 
inside the limits of the standard error of estimate of the regression for the 
standard sample, even at this very small elevation angle where horizontal 
changes in the A'^ profile can exert a large effect on elevation angle errors. 



COMPARISON WITH EXPERIMENTAL RESULTS 



349 



Figure 8.23 shows the results of a comparison between predictions of 
elevation angle errors estimated from the CRPL Standard Sample and 
some measurements made with a 6-cm radar at Tularosa Basin, N. Mex. 
[21]. Each point represents the mean of five "instantaneous" readings 
made at 1-min intervals over a period of 4 min. The standard deviation 
of each five-reading group averaged 0.16 mrad, and the maximum range 
in any one group was 0.58 mrad. The radio energy was propagated over 
a 45-mi path at a mean apparent elevation angle of 18 mrad; the target 
was a beacon located on a mountain peak 5610.5 ft higher than the desert 
floor where the radar was located. The data in figure 8. 18 show that even 
for this rather extreme case, where the degree of correlation between N s 
and € is expected to be only 0.4, agreement is obtained between: 

(a) the predicted and observed mean refraction, 

(b) the observed and predicted slopes of the e versus N s relation, and 

(c) the observed and predicted residual errors of predicting e from N s 
alone. 



O 



a: 
o 
cr 

or 



< 

O 

h- 
< 
> 

LaJ 



I I I \ 

TARGET HEIGHT=I7I km 
RANGE -72.5 krn 
MEAN APPARENT 
ELEVATION ANGLE = 19.0 mrad 



STANDARD ERROR 
h OF ESTIMATE 



\ \ \ 

PREDICTED FROM 
ORP.L, STANDARD SAMPLE 
6p=0OI02Ns-l,50 
SE = + 0.608mrad 




OBSERVED REGRESSION: 
eQ=O.OI03Ns-l.65 
r=0 427 



STANDARD ERROR 
OF ESTIMATE 



230 240 250 260 270 280 290 300 310 320 330 

Ns 



Figure 8.23. Measured refraction of C-band radar at Tidarosa Basin, N .Mex. 



350 



REFRACTION AND REFRACTIVE INDEX MODELS 



C/5 

ttr 

LlI 



1.5 



TARGET HEIGHT =3.046 km 
RANGE =25, 1 km 
ELEVATION ANGLE = 7° 




STANDARD ERROR 
OF ESTIMATE 



J I \ \ L 



340 



350 360 370 

Ng-PUUNENE AIRPORT U.S.W.B. 



380 



Figure 8.24. Range error fluctuations observed on Maui path. 



The small discrepancy between the intercepts (i.e., between the mean 
refraction) of the observed and predicted e versus A'^ « lines may be perhaps 
attributed to, for example, antenna lobe pattern distortion caused by 
differential refraction, or defocusing [22]. 

The remaining data which are examined were of necessity taken in 
such a manner as to have a rather high degree of autocorrelation (trends). 
Such data are not as suitable for confirming the accuracy of a regression 
prediction process as are independent data. A discussion of this is in- 
cluded at the end of this section. 

Turning to examination of radio range errors, figure 8.24 shows the 
results of some measurements of apparent radio range fluctuations over a 
25-km path on the island of Maui, Hawaii, on November 9-11, 1956 [23]. 
These measurements were made at 1-hr intervals, and are essentially 
"instantaneous" values. The target beacon was situated on the summit 
of Mount Haleakala at an elevation of 10,025 ft, while the "ground" 
station was near Puunene Airport at an elevation of 104 ft, thus yielding 
a target height of 3.046 km, in a region of critical target heights for pre- 
diction of radio range errors in tropical climates [19]. The measured 
range fluctuations (absolute errors not measured) are plotted against 
values of N s taken at about the same time (mostly 15 to 20 min later) by 



COMPARISON WITH EXPERIMENTAL RESULTS 351 

U.S. Weather Bureau personnel at the Puunene Airport weather station. 
The agreement between observation and prediction is fairly good, espe- 
cially when one considers that only 32 of the 86 points lie outside of the 
predicted standard error of estimate limits, while chance would indicate 
that 29 points would exceed these limits. Also, it should be kept in mind 
that in this case, as for all except the Collins data, the target beacon is 
located on the surface of the earth, whereas the predictions from the 
CRPL Standard Profile Sample are derived for targets in the free atmos- 
phere; there is undoubtedly some bias introduced in this way. 

As a part of a continuing investigation into the atmospheric limitations 
imposed on electronic distance measuring equipment, some measure- 
ments have been made recently by the Tropospheric Physics Section, 
NBS, of both range errors and range difference errors (across a phase- 
differencing baseline) over a propagation path near Boulder, Colo. 
Figures 8.25 and 8.26 are based on some of the preliminary results of 
these measurements [24]. Figure 8.25 shows the results of measurements 
of the fluctuations in apparent range, made at half-hour intervals on 
May 9-11, 1961, over a 15.5-km path between a transmitting beacon 



r 

LINEAR REGRESSION 

r =0.86 

SLOPE : 0,865 ppm/N 

PREDICTED SLOPE, 0879ppm/N 



THE STANDARD ERROR OF ESTIMATE 
OF EITHER LINEAR FIT IS 5.4ppm, 
THE PREDICTED VALUE = 5.8ppm. 



E 

Q. 

a. 




Figure 8.25. Range error fluctuations observed over the Boulder Creek-Green Mountain 

path, Colo. 

3-day run, half-hourly readings, range = 15.5 km, target height 088 m, approx elevation angle 26° May 
9-12, 1961 



352 REFRACTION AND REFRACTIVE INDEX MODELS 







1 




1 







1 1 






120 


- 












- 


o 


















Q 










o 








H 



















Q 




^^ 


„ ° 


, ° 










D- 






^^ ° 


a> 






o 




3 


no 


- 


^"°i^^^ 









„ 


- 


i 






„ o° 


T^^W^ ° 




° 






o 










>==- 


'*?*^''S2<. 


^^-^^ OS 




I 








° 


8 


* o° ° 


°^-C^^^^° ^^OBSERVED SLOPE 




z 
< 


100 


~ 








° CO 


PREDICTED SLOPE— -^^^^^^^^.^ 


~ 


tn 














^-~~^^ 


~' 














BOTH LINES HAVE ^ 


^ 


E 
£ 












° 


SE = 3.5mm, r =0.625 


^^ 


< 


90 


1 








1 


I 1 


- 



N,-AT TWO ANTENNAS 



Figure 8.26. Range difference fluctuations observed over a 460-m in-line baseline, 

Boulder Creek-Green Mountain path, Colo. 

4.15-hr run, 460 meter in-line baseline, target height = 688 m apparent range 15.5 km, May 9-11, 1961 



on Green Mountain at an elevation of 2242 m and a receiving antenna 
located near Boulder Creek at an elevation of 1554 m, the true target 
height thus being 688 m. 

The apparent range fluctuations, expressed in parts per million of the 
15.5-km path length (with an arbitrary zero since the total range was not 
measured), are plotted as a function of the surface value of the refractive 
index taken at a point quite close to the lower terminal. Quite good 
agreement is seen between the simple linear regression of the observed 
AT^e values on N s and the predicted linear relationship obtained from 
the CRPL Standard Sample. Note that both hues have statistically 
equal standard errors of estimate with respect to the observed data. 

Figure 8.26 shows the results of the range difference measurements 
made over a 460-m baseline essentially in line with the transmission path, 
where the second antenna was farther from the target beacon than the 
primary antenna. Here the range difference fluctuations (again with an 
arbitrary zero) have been plotted as a function of the mean value of N s 
measured at each end of the baseline. The zero point on the graph is set 
by the predicted mean of the sample. In this case there seems to be some 
discrepancy between the regression of the data and the predicted slope; 



DISCUSSION OF RESULTS 353 

however, note that the standard errors of estimate for the two hnes are, 
to two significant figures, equal, indicating that the difference in the 
slopes is probably statistically insignificant. 

There are some data points in figure 8.26 having a rather large devia- 
tion from the regression lines. Statistical theory (using the "Student" 
^-distribution for 84 deg of freedom) shows that, if the data points are 
drawn from a normally distributed population, there should be only one 
point having a deviation of more than ±9 mm from the observed regres- 
sion line. There are in fact five points in figure 8.26, four above and 
one below the line. If these five points are thrown out, on the grounds 
that they weight too heavily the extremes of the distribution of data points 
(this is especially true when using least squares regression), and the regres- 
sion is then redone using the remaining 81 data points, the resulting value 
of the slope is —0.385 mm/A'^-unit with r = 0.77 compared to the pre- 
dicted slope of —0.381 mm/A'^-unit, a rather close agreement. 

8.3.6. Discussion of Results 

As a summary of the results of the experimental versus theoretical 
comparisons given in the preceding section, a statistical analysis has been 
run on the significance of the differences between the slopes of the ob- 
served and predicted regression lines. In order to make the tests more 
stringent, it was assumed that the slopes derived from the Standard Sam- 
ple should be taken to be the slopes of the population regression lines (/3), 
thus yielding an estimate of the significance of the departure of the ob- 
served slope from the assumed population value. 

A value of t was first calculated for each case using the relation [25] 



_ I b - ^0 I V2(x. - xY 



where b is the observed slope, ^o the assumed population, or theoretical, 
slope, X refers to the independent variable in each regression, N s, SE is 
the standard error of estimate, and ij_2 is the value of t for j — 2 deg of 
freedom. From tj-2, confidence limits for /3 at the 100(1— a) percent 
level can be calculated from [25] 



lj^,aSE < ^ < ^ ^ tj^2,.SE ^g_26) 



VS(Xi - x)2 \/2(Xi - 

The probability that the observed value b would have fallen outside of 
these limits by chance is a. Many statisticians consider a value of tj^o 



354 REFRACTION AND REFRACTIVE INDEX MODELS 

falling below the 100a = 5 percent level to be not significant, between the 
5 percent and 1 percent levels to be of questionable significance, and over 
the 1 percent level to be significant [25]. An observed slope b falling 



Vi:(xi - xY 

would thus be taken to represent a significant departure from the value 
/So, and would thus imply the possibilities 

(a) j3o does not represent /3, or 

(b) h represents the regression of data from a population different than 
that used in determining ^So, or 

(c) both. 

Before making the significance tests, however, the value of j, the num.- 
ber of independent observations going into the determination of h, must 
be known. In general, data of the type presented here are more or less 
highly autocorrelated, and hence not all independent. The data pre- 
sented here, with the possible exception of the Collins data and the 
Tularosa Basin data for which the calculations could not be performed, 
have autocorrelation coefficients Vk, for lag k {k = 1, 2, 3 units of time be- 
tween successive measurements) that can be approximately described by 

and for this type of data the effective number of pieces of independent 
data, j, is given by [26] 



■fr 



— r 
J = n 



+ r'J 



(8.27) 



For the data treated here weighted mean values of r' were calculated 
from 

r. + ^rl-^9rl-+-^.^kM- 
"^ - 1 + 4 + 9+-.. A;2 ' ^'^■^''^ 

where k was the largest lag for which the autocorrelation coefficient was 
calculated, usually 4 or 5. No special justification is offered for the use 
of (8.28) other than the obvious fact that Vk is to be approximated by 
the kth. power of r', and hence a function of k would seem to be the most 
logical weighting function to use; the use of k"^ as a weighting function 
seemed to give the best overall fit to the series of r^ encountered from 
these data. 



DISCUSSION OF RESULTS 



355 



Table 8.6 shows the results of the significance tests on the slopes of the 
various experimental and theoretical (predicted) regression lines. The 
number of pieces of data is shown in the first column, the observed slope 
b and theoretical slope /3 in the second and third columns, the autocorrela- 
tion coefficient for lag of one time unit in the fourth column, and the 
weighted mean r' as defined in (8.28) in the fifth column. In column 6 
the effective number of independent pieces of data, j, is shown, while in 
column 7 the value of tj^o is shown for the difference between b and /S. 
The next column shows the value of ^_,_2, 0.5, the value for the 50 percent 
significance level for j — 2 degrees of freedom. 

Only one of the t values turns out to be significant at the 50 percent 
level, which means that there was a better than even chance that such 
differences would have occurred by chance in the other cases. In the case 
of the Colhns data at ^0 = 8°, the value of t = 1.01 would not be significant 
at the 25 percent level; the value ^46 = 1.01 corresponds to a = 0.34, or 
a 34 percent chance that the observed deviation |6 — /3| is of a random 
nature, and thus not significant. 



Table 8.6. Experimental versus theoretical slopes 





















Is [6-/3] Sig- 


e VS N, 


n 


6 





Tl 


r' 


i 


tj-2 


t(0. 50, 
>-2) 


nificant at 

the a =50 

percent level? 


Collins data 8°.. . 


48 


0. 00754 


0. 00698 






(48) 


1.01 


0.68 


Yes 


Tularosa Basin_.. 


161 


0. 0103 


0. 0102 








(161) 


0.031 


0.676 


No 


Cape Canaveral.. 


86 


0. 01356 


0.00648 


.870 


.860 


6.5 


0.708 


0.73 


No 


ARe VS N, 




















Maui data 


86 


0. 02833 


0.01610 


.974 


.950 


2.2 


3.56 


7.6 


No 


Boulder Creek- 




















Green Mt 


155 


0.865 


0.879 


.944 


.950 


4.0 


0.32 


0,82 


No 


A(Ai?) VS Ns 




















Boulder Creek- 




















Green Mt 


86 


-0. 344 


-0. 381 


.957 


.946 


2.4 


0.60 


2.0 


No 



From the point of view of a statistician, the results of these tests are 
such that no significance can be attached to any of the apparent discrep- 
ancies between theory and observation, and given reason to believe that 
the values of /3 are theoretically sound, one could say that the results are 
significantly positive in nature. The significance of the differences be- 
tween the predicted and observed slopes of the regression lines for ^0 = 
20 mrad for the independent data check of subsection 8.3.4 were tested 
using the same method as the preceding tests, except that the six observa- 
tions in each case were assumed to be independent. The results are 
summarized in table 8.7 and confirm the general use of the Standard 
Sample for do > 20 mrad. 



356 



REFRACTION AND REFRACTIVE INDEX MODELS 



From the experimental data which are available at the present time 
it may be concluded that: 

(a) Radio range and elevation angle errors can be predicted from the 
surface value of the radio refractive index, and the accuracy obtained will 
be generally commensurate with the estimates of residual errors made 
from theoretical ray-tracing considerations. 

(b) The functional dependence of either angular refraction or range 
errors on the surface value of the refractive index as derived from the 
CRPL standard A'' profile sample may be applied to arbitrary locations 
or climates without noticeable decrease in accuracy over that obtained 
with a sample from the location under consideration. 

(c) The effects of horizontal inhomogeneities of the refractive index, 
which certainly must have been prevalent over the transmission paths for 
which experimental data have been presented, do not appear to introduce 
any bias or additional residual variance into the values of observed re- 
fraction variables over those j^redicted from surface observations. 



Table 8.7. Comparison of slopes for independent check 
Predicted slope at 0o=2O mrad: 0. 358 mrad/iVs 





Observed 
slope 


Difference 
6-/3o 




t 


100a 




Station 


V^(N, 


-NsV- 


Significance 


Amundsen-Scott 

Dakar 

Bangui 


0. 0520 
.0316 
.0457 
.0390 


+0. 0162 
-. 0042 
+.0099 
+.0032 


37.9 
56.5 
47.2 
44.3 


10.2 
0.72 
1.37 
1.18 


<0. 1 percent 
52 percent 
25 percent 
31 percent 


very high 
none 
very low 


Moscow 


very low 






-Nsr- 



S. E. 



8.4. Correction of Atmospheric Refraction 
Errors in Radio Height Finding 



8.4.1. Introduction 

As a radio ray passes through the atmosphere, the length and direction 
of its path varies with the radio refractive index. Uncorrected radar 
output determines the position of a target by a straight line path at con- 
stant velocity. The difference between the straight path and the actual 
path results in an error which becomes increasingly significant as the 
distance to the target increases. The height error (the component of the 
position error normal to the surface of the earth) constitutes over 95 per- 
cent of the total error. Until recently, the range of height finding equip- 
ment was sufficiently limited so that the refraction errors could be either 
neglected, or approximated by the effective (four-thirds) earth's radius 
correction [27]. 



RAY THEORY 357 

Bauer, Mason, and Wilson [28] obtained an equation for accurately 
estimating radar target heights in a specific exponential atmosphere. 
Beckmann [29] presented a i)robability estimate of the height errors with- 
out meteorological measurements. 

The purpose of the study is to investigate the correlation between 
available meteorological ])arameters and height errors for targets of 
interest in terminal air traffic control and to develop height error correc- 
tion procedures using these parameters. 

8.4.2. Refractive Index 

The radio refractive index, n, of a propagation medium is the ratio of 
the free space velocity of light, c, to the velocity in the medium, v, (i.e., 
n = c/v). Since the propagation velocity of the atmosphere is only 
slightly less than the free space velocity, it is often convenient to use the 
scaled up difference between the refractive index and unity. This ciuan- 
tity is the refractivity. 

The refractivity is obtained from (1.20). Normally, the equation for 
N is dominated by the first term so that the refractivity can be approxi- 
mated by an exponential function of height as shown in section 3.8. 

8.4.3. Ray Theory 

If the refractive index is assumed to satisfy (for a spherically strati- 
fied atmosphere) 

I > - 7- («-^«) 



then, for frequencies greater than 100 kc/s, the path of a radio ray is 
determined by Snell's law for polar coordinates (3.1) of chapter 3 (see 
fig. 8.27) and the bending angle, r, is determined from (3.2). The dis- 
tance, d, along the surface of the earth is obtained from (3.62). 

The length of the path is called the geometric range and is obtained by 



R = I CSC dr, (8.30) 



and the apparent or radio range is found by 



Re = I n CSC e dr = R -\- N X 10"^ esc d dr. (8.31) 



358 REFRACTION AND REFRACTIVE INDEX MODELS 




Figure 8.27. Geometry of radio ray refraction. 



RAY THEORY 



359 




Fku're 8.28. Effective earth's radium; (leoineli 



(/eontetnj. 



360 REFRACTION AND REFRACTIVE INDEX MODELS 

Because the difference between Re and the true slant range, Rq, is small 
compared to the height error, the slant range and radio range are assumed 
to be identical to the geometric range, R. 

The apparent height of the target, in figure 8.28, is obtained by solving 

(ro + haY = ro^ + R"" + 2r, R sin 6^ (8.32) 

for ha. The following form is useful for numerical calculations: 

K = ^(^ + 2rosin^o) _ ^g 33^ 

ro + Vro^ -\- R{R -\- ro sm do) 

The height error for a target at height h is found by 

€, = ha - h (8.34) 

which will always be positive if n decreases with height. 

If the refractive index is known as a function of height the foregoing 
procedure is useful for determining the height error when the true height 
and the arrival angle of the ray are hypothesized. Unfortunately, it is 
not applicable for obtaining the height error from the apparent position 
of the target. 

8.4.4. Use of the Effective Earth's Radius 

The inaccuracy of the "four-thirds earth" correction stems mainly 
from the assumption that all radio rays have the same constant curvature. 
The accuracy would be greatly enhanced if an "average" effective radius 
could be determined for each ray path. 

The following expression, with the effective earth's radius denoted by 
Te in figure 8.28, 

(r, + hy = r,2 -\- R^ -\- 2re R sin ^o, (8.35) 

can be combined with (8.32) and (8.34) to obtain 



« + ^ - f = (~^)(- - -). (8.36) 

Ve 2re \ 2 /Vo Ve/ 



The difference between the curvature of the actual earth and the "average" 
effective earth for the ray path represents the "average" curvature of the 
ray. Thus, if the ray curvature can be determined as a function of the 



USE OF EFFECTIVE EARTH'S RADIUS 361 

target position and the refractive index structure, (8.36) will provide a 
simple formula for approximating the height error, 

since for target heights (h < 70,000 ft) at the maximum range (R c^ 150 
mi) to be considered 

I ^ - -^ I < 20 ft, 

Ve 2re 

or less than one percent of the total height error. 

If the curvature of a ray, 1/p, at any point on the path is called K, then, 
from (8.29) and (3.20) 

_ Wo cos 00 dn 
^ ~ 71^(1 + h/ro) dK ^^-^^^ 

From (8.32) ignoring terms of the order l/r^f, one obtains 

so that (8.38) becomes 

J. no {R" - hiy " V "^ Vo) dn_ (8.40) 

~ n' R / , \ dh ' 



To/ 



1 + 

ro 
The refractive index usually decreases with height so that the quantity 

1/2 



(-6" 



no ^ . 

n2 1 + ^ — ^ 

To 

varies only slightly with height, and the curvature at a point on the ray 
path can be approximated by 



i^r 



K^('-^^)"'\'i\. (8.41) 



362 REFRACTION AND REFRACTIVE INDEX MODELS 

Therefore, (8.37) becomes 

e.^ ^^-^Q (8-42) 

where g represents the "average" gradient on the ray path (i.e., g = dn/dh 
at an intermediate point on the path). 

Since g depends upon the meteorological conditions along the path, 
the basic problem is to determine g for a given target from the conditions 
at and/or near the surface. 

8.4.5. Meteorological Parameters 

Measurement of the refractivity at the radar site will provide an esti- 
mate of the gradient if a model of the refractive index structure is assumed. 
In the exponential model, for example, 

n{h) = 1 + A^, exp (-c/i) X 10-« 
where N ^ is the surface refractivity and c is a constant, then 

-77 = —cNs exp ( — c/i) X 10~^ 

For a target at a height, ht, the average gradient along the ray path is 

g = -^[1 - exp i-cht)] X 10-', (8.43) 

lit 

but since ht is not known, g must be approximated as a function of the 
apparent height. 

Additional meteorological measurements at a sufficient height above 
the surface to obtain values significantly different from the surface values 
can be used to determine the initial gradient, 

^-1 '- = !'" (If) xi«'' («■«' 

where Nh is the refractivity at the height, H, of the above surface meas- 
urements. The initial gradient provides a boundary condition for esti- 
mating gf as a function of the apparent height. For convenience the initial 
gradient of refractivity. Go = go X 10^ with H in kilometers, was used 
as a prediction parameter. 



ESTIMATION OF AVERAGE GRADIENT 363 

For the purposes of this study, the average (per kilometer) gradient of 
the first kilometer of the atmosphere is the only prediction parameter 
used which will require upper air measurements. The average 1 km 
gradient, 

AN = Ni - Ns, (8.45) 

where A'^i is refractivity at 1 km above the surface, was selected because 
climatological summaries, as in chapter 4, can be used to estimate the 
height error when meteorological measurements are unobtainable. 

8.4.6. Calculation and Correlation of Height Errors 

Bean, Gaboon, and Thayer [18] selected refractive index profiles, deter- 
mined from radiosonde observations at 13 climatically distinct locations, 
which represent a wide variety of mutually exclusive profile types. The 
ray paths at arrival angles varying from 0° to near 90° were determined 
for each profile by numerical evaluation of (8.29) through (8.33) using 
methods similar to those described by Bean and Thayer [1]. The height 
errors were calculated with (8.33) and (8.34) at selected height intervals 
of 70,000 ft for each ray path. Newton's method of interpolation was 
used to determine height errors for fixed ground distances to 150 mi. The 
limits of height and distance were chosen to extend beyond the current 
needs in terminal air traffic control, but are sufficiently restricted to allow 
some of the previous assumptions. 

The prediction parameters, N s, Go, and AA'^, were obtained from each 
of the refractive index profiles. Linear and multiple regression analysis 
were employed to obtain least squares estimates of the height error at 
each height and distance for each prediction parameter and for various 
combinations of the parameters. 

8.4.7. Estimation of the Average Gradient 

Based on the correlations the following forms, suggested by (8.43), were 
selected for approximating g: 

?■ = J^^fniha), (8.46) 

92 = ^/2l(/la) -^-—-Mha) (8.47) 

or 

^3 = ^Mha) + ^Mha) + -J^Miha) (8.48) 

depending upon the availability of Go and AA'^. 



364 



REFRACTION AND REFRACTIVE INDEX MODELS 



To obtain a direct estimate of the height error, (8.46) through (8.48) 
were combined with (8.42) and the functions /ij {i > j = 1, 2, 3) were 
determined as least squares polynomials. 

8.4.8. Regression Analysis 

The volume of data processed is of sufficient magnitude that it is im- 
practical to include it all in this report. Therefore, certain information 
obtained from the regression analysis was selected as being the most 
significant. 

The mean height error provides the best general estimate obtainable 
if meteorological data are not available. The standard deviation (about 
the mean) of the height errors determines the reliability of this estimate, 
since 68 percent of the observed height errors are within ±1 standard 
deviation of the mean height error if the observations are normally dis- 
tributed. In figure 8.29, the mean height error was plotted for each 
target position, then contour lines were drawn to display the mean height 
error as a function of true height and distance. By similar construction 
the standard deviation of the height error as a function of target position 
is displayed in figure 8.30. 

The standard error of estimate establishes the same confidence limits 
for prediction with a regression equation as the standard deviation does 
for the mean. Thus, comparison of the standard error to the standard 
deviation indicates the improvement in accuracy of prediction with 



500 1750 2000 




40 150 IGO 



DISTANCE (MILES! 



Figure 8.29. Mean height errors in feet. 



REGRESSION ANALYSIS 



365 



2 5 10 20 30 



250 300 




00 no 120 liO 140 ISO 160 



DISTANCE (MILES) 



Figure 8.30. Standard deviation of height errors. 

meteorological parameters. The standard error as a function of target 
position for 



eh = hiNs + a 
is shown in figure 8.31; for 

eh = biNs + hiGo + a 
in figure 8.32; and for 

eh = biNs + b2Go + 63 AA^ + a 



(8.49) 



(8.50) 



(8.51) 



in figure 8.33. 

The regression equations, (8.49) through (8.51), were used because the 
respective figures demonstrate how each additional parameter enhances 
the accuracy of the estimate. The parameters Go and AA'^ applied indi- 
vidually, that is, 



and 



€h = bOo + a 



eh = bAN + a, 



were of significant value only for targets at low heights (h < 10,000 ft). 
Examination of the figures shows that prediction of eh with A'^., provides 
significant improvement over the mean for target heights above 15,000 ft. 
The addition of Go improves the estimate for heights below 15,000 ft 
and the addition of AA'^ provides a slight overall improvement. 



366 



REFRACTION AND REFRACTIVE INDEX MODELS 



In figures 8.29 through 8.33 the contours do not extend below 15,000 ft 
for distances greater than 120 mi and 10,000 ft for distances greater than 
80 mi. Correlations were not calculated for these target positions, be- 
cause for certain refractive index profiles they are beyond the radio 
horizon and for certain other profiles the arrival angle is too low for the 
ray to penetrate a trapping layer. If conditions exist, without violating 
the assumptions of sections 8.4.2. through 8.4.4. such that a target at 
5000 ft height and 150 mi distance would be visible to radar, the resulting 
height error would be about 10,000 ft. 




Figure 8.31. Standard error of eh versus Ng 




40 150 160 



Figure 8.32. Standard error of th versus Ng and Go. 



REGRESSION ANALYSIS 



367 




120 150 HO 150 160 



Figure 8.33. Standard error of «h versus Ns, Go, and AN. 



As an aid to further studies the coefficients for (8.49) through (8.51) 
are Hsted in tables 8.9 to 8.17. 

Table 8.8. The coefficients sm for {8.62) through {8.57) 



-19.596 
-17.849 
-15.319 



. 014096 
.011202 
. 006388 



0. 77906 X10-< 
. 13665X10-3 
. 18549X10-3 



0.67545X10-6 
. 58925X10-2 
.39074X10"' 



-0.64975X10-2 
-.55818X10-2 



0. 12340X10-3 
.12671X10-3 



-0. 023980 



-0.22547X10-* 





Table 8.9 


Constant term, a, in the regression equati 


on {8.62) 




Distance, 


Height, kft 


mi 


5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5.... 

10.... 

15 

20 

25 

30 

35 


-3 

-14 
-31 

-55 
-86 

-123 
-166 
-215 
-269 
-327 

-454 
-586 
-709 


-3 
-13 
-29 
-53 
-83 

-119 
-162 
-211 
-267 
-329 

-470 

-633 

-812 

-1000 

-1186 

-1352 
-1473 


-2 
-10 
-24 
-44 
-69 

-100 
-136 
-178 
-226 
-280 

-404 
-552 
-722 
-913 
-1122 

-1342 
-1565 
-1774 
-1946 
-2048 


-1 
-8 
-19 
-35 
-56 

-81 
-111 
-146 
-186 
-231 

-335 
-461 
-607 
-776 
-965 

-1176 
-1404 
-1644 
-1889 
-2123 


-6 
-15 
-28 
-45 

-66 
-91 
-120 
-153 
-190 

-277 
-381 
-505 
-648 
-812 

-997 
-1203 
-1429 
-1672 
-1928 


-4 
-12 
-23 
-37 

-54 

-74 

-98 

-125 

-156 

-228 
-315 
-418 
-539 
-678 

-836 
-1014 
-1215 
-1434 
-1674 


1 

-3 

-9 

-18 

-29 

-44 

-60 

-80 

-102 

-127 

-187 
-259 
-345 
-446 
-563 

-697 

-850 

-1022 

-1214 

-1428 


1 

-2 

-7 

-14 

-24 

-35 
-49 
-65 
-83 
-104 

-153 
-212 

-284 
-367 
-465 

-578 

-707 

-852 

-1019 

-1203 


2 

-4 

-9 

-15 

-23 

-32 
-43 

-55 
-70 

-103 
-144 
-193 
-252 
-320 

-399 
-491 

-597 
-718 
-855 


2 

1 

-2 

-5 

-10 

-15 
-22 
-30 
-39 
-49 

-73 
-102 
-137 
-179 

-228 

-286 
-352 
-429 
-618 
-620 


2 

1 

-1 

-3 

-7 

-11 
-16 
-22 
-29 
-36 

-54 

-76 

-103 

-134 

-171 

-214 
-264 
-322 
-389 
-465 


40 . 


45 

50. . 


60.--. 


70 

80. - 


90.... 

100 

110 

120 

130 

140 

150 



368 



REFRACTION AND REFRACTIVE INDEX MODELS 



Table 8.10. Constant term, a, in the regression equation (8.53) 













Height, kft 










Distance, 
























mi 


























5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 


-3 


-3 


-2 


-1 






1 


1 


2 


2 


2 


10 


-12 


-12 


-10 


-8 


-6 


-4 


-3 


-2 


— 


1 


1 


15 


-28 


-29 


-24 


-19 


-15 


-12 


-9 


-7 


-4 


-2 


-1 


20 


-50 


-51 


-43 


-35 


-28 


-22 


-18 


-14 


-9 


-5 


-3 


25 


-77 


-80 


-67 


-55 


-45 


-36 


-29 


-23 


-15 


-10 


-7 


30 


-110 


-115 


-97 


-80 


-65 


-53 


-43 


-35 


-23 


-15 


-11 


35 


-148 


-156 


-133 


-110 


-90 


-73 


-60 


-48 


-32 


-22 


-16 


40 


-190 


-204 


-174 


-144 


-118 


-97 


-79 


-64 


-43 


-30 


-22 


45 


-235 


-257 


-221 


-183 


-151 


-124 


-101 


-82 


-55 


-38 


-28 


50 


-284 


-316 


-273 


-227 


-187 


-154 


-126 


-103 


-69 


-48 


-36 


60 


-383 


-449 


-394 


-329 


-273 


-225 


-185 


-151 


-102 


-72 


-54 


70 


-473 


-600 


-536 


-452 


-376 


-311 


-257 


-210 


-143 


-101 


-76 


80 _._ 


-537 


-762 
-925 


-700 
-880 


-595 

-797 


-497 
-637 


-413 
-531 


-341 
-441 


-281 
-363 


-191 
-249 


-136 
-178 


-102 


90 


-133 


100 




-1076 


-1075 


-940 


-797 


-667 


-556 


-460 


-317 


-226 


-170 


110 




-1189 


-1275 


-1141 


-976 


-822 


-687 


-571 


-395 


-283 


-212 


120 




-1236 


-1468 


-1355 


-1174 


-996 


-837 


-697 


-486 


-349 


-262 


130 






-1637 


-1576 


-1390 


-1189 


-1005 


-840 


-590 


-425 


-319 


140 






-1748 


-1794 


-1619 


-1400 


-1192 


-1003 


-708 


-512 


-385 


150 






-1765 


-1989 


-1855 


-1630 


-1399 


-1183 


-843 


-612 


-460 



Table 8.11. Constant term, a, in the regression equation {8.54) 













Height, kft 










Distance, 






















mi 


























5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 




-2 


-1 


-1 






1 


1 


2 


2 


2 


10 


-2 


-8 


-8 


-6 


-5 


-3 


-2 


-2 


— 


— 


1 


15 


-4 


-18 


-18 


-15 


-12 


-10 


-8 


-6 


-3 


-2 


-1 


20 


-7 


-33 


-33 


-28 


-23 


-19 


-15 


-12 


-7 


-5 


-3 


25 


-9 


-51 


-51 


-44 


-37 


-31 


-25 


-20 


—13 


-9 


-6 


30 


-12 


-73 


-74 


-64 


-54 


-45 


-37 


-30 


-20 


-13 


-10 


35 


-13 


-98 


-100 


-87 


-74 


-62 


-51 


-42 


-28 


-19 


-14 


40 


-13 


-125 


-130 


-114 


-97 


-81 


-67 


-55 


-37 


-26 


-19 


45 


-10 

-5 

21 
70 
152 


-156 
-187 

-254 
-318 
-369 


-164 
-201 

-285 
-378 
-478 


-144 

-178 

-256 
-346 

-448 


-123 
-152 

-221 
-302 
-395 


-104 
-129 

-187 
-257 
-338 


-86 
-107 

-156 
-216 

-285 


-71 

-88 

-129 
-179 
-237 


-48 
-60 

-88 
-123 
-164 


-33 
-42 

-62 
-87 
-117 


-25 


50 


-31 


60 


-47 


70 


-66 


80 


-88 


90. 




-396 
-383 


-578 
-669 


-560 
-677 


-499 
-615 


-431 
-536 


-366 
-457 


-306 
-384 


-213 
-269 


-152 
-194 


-115 


100 


-146 


110 




-310 


-741 


-795 


-739 


-652 


-560 


-473 


-334 


-241 


-182 


120 




-163 


-776 


-906 


-867 


-776 


-673 


-573 


-408 


-296 


-224 


130 






-754 


-998 


-996 


-909 


-797 


-682 


-491 


-359 


-272 


140 






—649 


-1055 


-1117 


-1044 


-929 


-804 


-485 


-430 


-327 


150 






-436 


-1057 


-1217 


-1179 


-1068 


-933 


-690 


-5)0 


-389 



REGRESSION ANALYSIS 



369 



Table 8.12. Coefficient of Ns, bi, in the regression equation {8.62) 



Distance, 


Height, kft 


mi 


5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

60 

70 

80 

90 


0. 0318 
.1153 
.2545 
.4495 
.7005 

1.0078 
1.3714 

1. 7917 

2. 2688 
2. 8030 

4. 0426 
5.5119 
7. 2127 


0. 0313 
.1059 
.2303 
.4046 
.6291 

.9040 

1. 2295 

1. 6060 

2, 0336 

2. 5129 

3. 6268 

4. 9494 
6. 4794 
8. 2133 

10. 1439 

12. 2529 
14. 5240 


0. 0296 
.0944 
.2025 
.3541 
.5494 

.7889 

1. 0728 
1. 4018 
1. 7762 
2. 1968 

3. 1787 
4.3513 
5. 7236 
7. 2966 
9. 0736 

11.0533 
13. 2284 
15. 5836 
18. 0866 
20. 7053 


0. 0281 
.0848 
.1795 
.3123 
.4835 

.6934 
.9424 
1.2311 

1. 5599 

1. 9295 

2. 7938 
3. 8300 

5. 0453 

6. 4464 

8. 0412 

9. 8363 
11.8360 
14. 0399 
16. 4440 
19. 0333 


0. 0267 
.0770 
.1608 
.2784 
.4300 

.6159 
.8366 

1. 0923 

1. 3837 
1.7113 

2. 4780 

3. 3981 

4. 4788 

5. 7273 

7. 1529 

8. 7649 
10. 5679 
12. 5772 
14. 7931 
17.2171 


0. 0257 
.0705 
.1454 
.2504 
.3858 

.5519 
.7489 
.9774 

1. 2377 

1. 5305 

2. 2155 

3. 0380 

4. 0046 

5. 1225 

6. 4005 

7. 8482 
9. 4723 

11. 2895 
13. 3002 
15. 5220 


0. 0248 
.0652 
.1326 
.2271 
.3489 

.4983 
.6756 
.8812 
1.1154 

1. 3788 

1. 9952 

2. 7355 

3. 6053 
4.6119 
5. 7634 

7.0684 
8. 5370 
10. 1777 
12. 0051 
14. 0251 


0. 0243 
.0608 
.1218 
.2074 
.3177 

.4531 
. 6136 
.7998 
1.0119 

1. 2503 

1.8085 

2. 4788 

3. 2666 

4. 1779 

5. 2207 

6. 4028 

7. 7337 
9. 2192 

10. 8816 
12. 7148 


0. 0236 
.0543 
.1056 
.1776 
.2704 

.3841 
.5190 
.6754 
.8536 

1. 0539 

1. 5223 

2. 0845 

2. 7442 

3. 5078 
4. 3804 

5. 3687 

6. 4808 

7. 7254 
9. 1125 

10. 6504 


0. 0231 
.0498 
.0943 
.1568 
.2373 

.3359 
.4529 
.5885 
.7429 
.9163 

1.3216 

1. 8072 

2. 3769 

3. 0342 

3. 7846 

4. 6329 

5. 5863 

6. 6512 

7. 8354 
9. 1488 


0. 0225 
.0462 
.0858 
.1413 
.2127 

. 3003 
.4042 
.5244 
.6613 
.8150 

1. 1738 

1. 6032 

2. 1060 
2. 6854 


100 

110 

120 - 

130 

140 

150 


3. 3456 

4. 0909 

4. 9257 

5. 8573 
6.8904 
8. 0339 



Table 8.13. Coefficient of Ns, bi, iri the regression equation (8.53) 



Distance, 


Height, kft 


mi 


5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

60 

70 

80 


0. 0285 
.1014 
.2226 
.3915 
.6072 

.8686 

1. 1741 
1. 5217 

1. 9089 

2. 3325 

3.2717 

4. 2984 

5. 3577 


0. 0303 
.1015 
. 2202 
.3862 
.5997 

.8603 
1. 1680 

1. 5225 

1. 9235 

2. 3701 

3. 3976 

4. 5945 

5. 9423 
7.4110 
8. 9558 

10. 5049 
11.9716 


0. 0292 
.0922 
.1972 
.3444 
.5430 

.7661 
1.0411 
1.3591 
1.7204 

2. 1254 

3. 0671 

4. 1841 

5. 4790 

6. 9435 
8. 5676 

10. 3305 
12. 1954 

14. 1047 

15. 9660 
17. 6672 


0. 0279 
.0835 
.1763 
.3065 
.4742 

.6797 
.9234 

1. 2056 
1. 5268 

1. 8873 

2. 7287 

3. 7339 

4. 9073 
6.2511 
7. 7696 

9. 4588 
11.3115 
13.3104 
15.4244 
17.6007 


0. 0267 
.0762 
.1588 
.2746 
.4239 

. 6069 
.8240 
1.0755 
1.3619 
1. 6836 

2. 4355 

3. 3361 

4. 3907 

5. 6047 

6. 9854 

8. 5356 
10. 2579 
12. 1560 
14. 2189 
16. 4316 


0. 0257 
.0701 
. 1441 
.2479 
.3817 

.5457 
.7403 
.9658 

1. 2227 
1.5114 

2. 1864 

2. 9957 

3. 9449 

5. 0400 

6. 2887 

7. 6975 
9.2713 

11.0200 
12. 9405 
15. 0401 


0. 0249 
.0649 
.1317 
.2253 
.3459 

.4939 
.6694 
.8729 

1. 1046 

1. 3651 

1.9743 

2. 7052 

3. 5629 

4. 5537 

5. 6849 

6. 9635 

8. 3974 

9. 9945 
11.7614 
13. 7056 


0. 0244 
.0607 
.1212 
.2061 
.3155 

.4497 
.6089 
.7935 
1.0037 

1. 2400 

1. 7928 

2. 4560 

3. 2348 

4. 1346 

5. 1625 

6. 3258 

7. 6323 
9. 0868 

10. 7087 
12. 4899 


0. 0237 
.0543 
.1053 
.1768 
.2690 

.3821 
.5162 
.6716 
.8486 
1.0475 

1.5126 
2. 0705 
2. 7249 
3.4816 

4. 3456 

5. 3231 

6. 4215 

7. 6489 
9. 0139 

10. 5245 


0. 0232 
.0498 
.0942 
.1563 
.2365 

.3346 
.4511 
.5860 
.7396 
.9121 

1.3152 

1. 7981 

2. 3643 

3. 0171 

3. 7620 

4. 6035 

5. 5484 

6. 6027 

7. 7737 
9. 0705 


0. 0227 
.0463 
.0857 
.1410 
.2122 

.2994 
.4028 
.5226 
.6589 
.8119 

1. 1691 

1. 5965 
2 0968 


90 

100 

110 

120 

130 

140 

150 


2. 6730 

3. 3294 

4.0699 

4. 8989 

5. 8234 

6. 8477 

7. 9804 



370 



REFRACTION AND REFRACTIVE INDEX MODELS 



Table 8.14. Coefficient of Ns, bi, in the regression equation {8.54) 













Height, kft 










Distance, 
























ml 


























5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 


0.0091 


0. 0231 


0. 0259 


0. 0265 


0. 0265 


0. 0262 


0. 0260 


0. 0258 


0. 0255 


0. 0252 


0. 0248 


10 


.0217 


.0689 


.0744 


.0720 


.0687 


.0652 


.0619 


.0589 


.0540 


.0504 


.0474 


15 


.0418 


.1447 


.1551 


.1478 


.1391 


.1303 


.1218 


.1141 


.1016 


.0923 


.0850 


20 


.0682 


.2497 


.2677 


.2539 


.2378 


.2214 


.2058 


.1915 


.1683 


.1511 


.1378 


25 


.0991 


.3830 


.4119 


.3902 


.3646 


.3386 


.3138 


.2912 


.2542 


.2268 


.2058 


30 


.1325 


.5430 


.5873 


.5567 


.5198 


.4821 


.4462 


.4133 


.3595 


.3196 


.2891 


35 


.1660 


.7281 


.7933 


.7532 


.7033 


.6520 


.6030 


.5579 


.4843 


.4295 


.3878 


40 


.1971 


.9358 


1. 0291 


.9795 


.9151 


.8483 


.7843 


.7253 


.6287 


.5567 


.5020 


45 


.2232 


1. 1641 


1. 2939 


1.2354 


1. 1553 


1.0712 


.9903 


.9155 


.7929 


.7014 


.6318 


50 


.2418 


1.4090 


1. 5863 


1. 5206 


1. 4239 


1. 3209 


1.2213 


1. 1290 


.9772 


.8637 


.7775 


60 


.2501 


1. 9361 


2. 2485 


2. 1771 


2.0459 


1. 9009 


1. 7588 


1. 6263 


1. 4070 


1.2423 


1. 1170 


70 


.2231 


2. 4816 


2. 9993 


2. 9439 


2.7804 


2. 5892 


2. 3986 


2. 2192 


1. 9202 


1. 6944 


1. 5223 


80 


.2005 


3.0016 


3. 8207 


3.8126 


3. 6248 


3. 3863 


3. 1423 


2. 9101 


2. 5191 


2. 2224 


1. 9953 


90 




3. 4452 


4. 6780 


4. 7689 


4. 5741 


4. 2912 


3. 9913 


3. 7012 


3. 2075 


2. 8288 


2. 5384 


100 




3. 7639 


5. 5316 


5. 7994 


5. 6251 


5. 3042 


4. 9474 


4. 5952 


3. 9877 


3. 5170 


3. 1545 


110 




3. 9201 


6. 3291 


6. 8710 


6. 7588 


6. 4182 


6.0094 


5. 5936 


4. 8629 


4.2900 


3.8464 


120... 




3. 9312 


7.0067 


7. 9475 


7. 9627 


6. 6279 


7. 1738 


6. 6971 


5. 8369 


5. 1516 


4. 6173 


130 






7. 4937 


8. 9769 


9. 2092 


8. 9193 


8. 4423 


7. 9028 


6. 9123 


6. 1053 


5. 4713 


140 






7. 7293 


9. 8930 


10. 4569 


10. 2723 


9. 7952 


9. 2175 


8. 0909 


7. 1551 


6.4118 


150 






7. 7099 


10. 6160 


11. 6542 


11.6633 


11.2325 


10. 6218 


9. 3770 


8. 3049 


7. 4432 



Table 8.15. Coefficient of Go, bj, in the regression equation {8.53) 













Height, kft 










Distance, 






















mi 


























5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 


-0. 0065 


-0.0019 


-0. 0008 


-0. 0003 


-0.0000 


0.0001 


0.0002 


0. 0002 


0.0003 


0.0003 


0.0004 


10... 


-.0268 


-.0084 


-. 0043 


-.0024 


-.0015 


-.0009 


-. 0005 


-.0003 


-.0000 


.0001 


.0002 


15 


-.0617 


-.0196 


-. 0102 


-.0060 


-.0039 


-.0026 


-. 0018 


-. 0012 


-.0006 


-.0003 


-.0001 


20 


-.1123 


-. 0356 


-.0187 


-.0112 


-. 0073 


-.0050 


-. 0035 


-.0026 


-.0015 


-.0008 


-. 0005 


25. .- 


-. 1807 


-. 0571 


-.0299 


-.0179 


-.0118 


-.0081 


-. 0057 


-.0043 


-.0025 


-. 0016 


-.0011 


30 


-. 2695 


-.0846 


-.0441 


-.0264 


-.0175 


-.0120 


-.0086 


-.0064 


-. 0039 


-. 0025 


-.0018 


35 


-. 3820 


-.1191 


-.0615 


-.0368 


-.0244 


-. 0168 


-.0120 


-.0091 


-.0055 


-.0036 


-.0026 


40 


-. 5228 


-. 1616 


-. 0827 


-. 0493 


-.0326 


-.0224 


-.0161 


-.0122 


-. 0075 


-.0048 


-.0035 


45 


-. 6968 


-. 2132 


-. 1080 


-.0642 


-.0423 


-. 0291 


-.0209 


-.0158 


-.0097 


-.0063 


-.0047 


50 


-.9019 


-. 2765 


-. 1382 


-. 0817 


-. 0537 


-. 0369 


-.0265 


-.0200 


-.0123 


-.0081 


-.0059 


60 


-1.4924 


-.4437 


-. 2160 


-. 1260 


-. 0822 


-.0564 


-.0404 


-.0305 


-.0187 


-. 0123 


-.0090 


70 


-2. 3492 


-. 6870 


-. 3238 


-. 1860 


-. 1201 


-. 0820 


-. 0586 


-. 0441 


-.0270 


-.0177 


-. 0130 


80 


-3.5911 


-1.0399 


-.4737 


-. 2672 


-.1704 


-.1156 


-.0822 


-.0616 


-.0374 


-.0245 


-.0179 


90 




-1. 5533 


-.6835 


-.3782 


-. 2375 


-. 1596 


-.1128 


-.0839 


-. 0507 


-.0330 


-.0239 


100 




-2. 3002 


-. 9795 


-. 5258 


-.3243 


-. 2163 


-. 1520 


-. 1125 


-. 0674 


-.0437 


-.0314 


110 




-3. 3840 


-1.3993 


-. 7309 


-. 4438 


-. 2917 


-. 2030 


-. 1491 


-.0884 


-. 0568 


-.0406 


120 




-4. 9414 


-1.9997 


-1.0155 


-.6002 


-. 3891 


-.2703 


-. 1962 


-.1148 


-. 0734 


-. 0518 


130 






-2. 8631 


-1.4124 


-.8152 


-. 5219 


-.3547 


-. 2562 


-. 1482 


-. 0939 


-.0658 


140 






-4. 1054 


-1.9740 


-1.1116 


-.6963 


-.4717 


-.3347 


-. 1909 


-.1194 


-. 0827 


150 






-5. 8817 


-2. 7735 


-1.5206 


-. 9329 


-. 6185 


-. 4353 


-. 2437 


-.1515 


-. 1037 



REGRESSION ANALYSIS 



371 





Table 8.16. 


Coefficient of Go, b2, 


in the 


regression equation (8.54) 




Distance, 


Height, kft 


mi 


5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 


-0. 0032 
-. 0133 
-.0309 
-. 0573 
-. 0942 

-. 1442 
-.2105 
-.2974 
-.4101 
-. 5553 

-. 9784 
-1.6559 
-2.7138 


-0. 0060 
-.0029 
-. 0067 
-. 0124 
-. 0202 

-. 0306 
-.0442 
-.0618 
-. 0840 
-.1130 

-. 1951 
-.3276 
-. 5396 
-. 8786 
-1.4169 

-2. 2638 
-3. 5736 


-0. 0002 
-.0013 
-.0031 
-. 0056 
-.0091 

-. 0136 
-.0194 
-.0265 
-. 0354 
-. 0464 

-. 0767 
-. 1222 
-.1916 
-. 2981 
-. 4631 

-.7186 
-1.1171 
-1.7384 
-2. 7042 
-4. 1878 


-0. 0001 
-.0005 
-.0012 
-. 0022 
-.0036 

-.0055 
-.0079 
-.0109 
-.0146 
-.0193 

-.0322 
-. 0516 
-.0809 
-. 1261 
-. 1907 

-.2906 

-. 4432 

-.6752 

-1.0330 

-1.5853 


-0.0000 
-. 0002 
-.0005 
-.0011 
-. 0017 

-. 0027 
-.0038 
-.0053 
-.0072 
-.0095 

-.0159 
-. 0256 
-.0401 
-. 0622 
-.0929 

-.1416 
-. 2097 
-.3139 
-.4716 
-. 7079 


0.0000 
-.0001 
-. 0002 
-.0004 
-. 0008 

-.0012 
-.0017 
-. 0024 
-.0034 
-.0045 

-. 0078 
-.0129 
-.0206 
-. 0323 
-.0488 

-.0741 
-. 1095 
-. 1646 
-. 2424 
-.3584 


0.0000 
-. 0000 
-. 0001 
-.0002 
-. 0003 

-.0005 
-.0007 
-.0010 
-.0015 
-.0020 

-. 0037 
-. 0065 
-.0107 
-.0171 
-. 0266 

-. 0406 
-.0622 
-.0906 
-. 1372 
-. 1978 


0.0000 
-.0000 
-.0000 
-.0001 
-.0001 

-. 0002 
-.0004 
-.0006 
-.0008 
-.0012 

-. 0022 
-.0038 
-.0063 
-.0102 
-.0160 

-.0246 
-.0371 
-.0548 
-.0810 
-.1175 


-0. 0000 

.0000 

-.0000 

-.0000 

-.0000 

-.0001 
-.0001 
-.0002 
-.0002 
-.0004 

-.0007 
-. 0014 
-.0024 
-.0040 
-.0065 

-.0101 
-.0153 
-. 0229 
-. 0339 
-.0485 


-0. 0000 
.0000 
.0000 
.0000 
.0001 

.0001 
.0001 
.0001 
.0002 
.0002 

.0001 
-.0000 
-.0004 
-.0010 
-.0020 

-. 0035 
-.0059 
-.0093 
-.0142 
-. 0212 


0000 


10 

15 

20 

25 

30 

35 

40 

45 

50 

60 

70 

80 

90 

100 

110 

120 

130 

140 

150 


.0000 
.0000 
.0000 
.0000 

-0000 
-.0000 
-.0000 
-.0001 
-.0001 

-.0002 
-.0003 
-.0006 
-.0010 
-.0017 

-.0026 
-.0039 
-.0059 
-.0085 
-. 0123 



Table 8.17. Coefficient of AN, hs, in the regression equation {8.54) 



§1 


Height, kft 


'c« p 
























Q 


5 


10 


15 


20 


25 


30 


35 


40 


50 


60 


70 


5 


-0. 0806 


-0. 0301 


-0. 0136 


-0. 0060 


-0.0011 


0. 0021 


0.0043 


0.0057 


0.0073 


0.0083 


0.0087 


10 


-. 3321 


-. 1360 


-.0739 


-.0481 


-.0313 


-. 0201 


-.0127 


-.0075 


-.0012 


-.0024 


-.0044 


15 


-. 7533 


-.3146 


-. 1754 


-.1189 


-.0819 


-.0575 


-.0411 


-. 0296 


-.0154 


-.0077 


-.0029 


20 


-1.3470 


-. 5687 


-.3196 


-.2191 


-. 1535 


-.1104 


-.0813 


-.0608 


-. 0355 


-.0218 


-.0132 


25 


-2. 1168 


-.9026 


-. 5084 


-.3498 


-. 2468 


-. 1793 


-. 1336 


-.1015 


-.0616 


-.0401 


-. 0265 


30 


-3. 0664 


-1.3217 


-. 7448 


-. 5125 


-. 3628 


-. 2648 


-. 1986 


-.1519 


-. 0940 


-.0628 


-.0430 


35 


-4. 1993 


-1.8327 


-1.0320 


-. 7091 


-. 5028 


-.3679 


-.2768 


-. 2126 


-. 1330 


-.0901 


-.0627 


40 


-5. 5178 


-2.4440 


-1.3744 


-.9818 


-.6680 


-. 4894 


-. 3690 


-.2842 


-. 1788 


-. 1222 


-.0860 


46 


-7.0221 


-3. 1634 


-1.7769 


-1.2136 


-.8604 


-. 6308 


-.4761 


-.3672 


-. 2320 


-. 1592 


-.1127 


50 


-8. 7091 


-4. 0036 


-2. 2458 


-1.5276 


-1.0819 


-. 7935 


-. 5992 


-. 4625 


-. 2929 


-. 2016 


-. 1434 


60 


-12.5871 


-6. 0883 


-3.4103 


-2. 2978 


-1.6230 


-1. 1893 


-.8979 


-.6937 


-. 4402 


-.3038 


-.2171 


70 


-16.9766 


-8. 8019 


-4. 9354 


-3. 2912 


-2.3149 


-1.6931 


-1.2773 


-.9865 


-.6263 


-.4320 


-.3092 


80 


-21.4838 


-12.2504 


-6. 9080 


-4. 5605 


-3. 1905 


-2.3271 


-1.7521 


-1.3526 


-. 8573 


-. 5909 


-.4227 


90 




-16.5206 


-9. 4378 


-6. 1745 


-4. 2931 


-3.1194 


-2. 3426 


-1.8054 


-1.1417 


-. 7843 


-. 5608 


100 




-21. 6282 


-12.6475 


-8. 2077 


-5.6667 


-4. 1012 


-3. 0720 


-2. 3636 


-1.4910 


-1.0207 


-.7284 


110 




-27.4311 


-16.6690 


-10.7799 


-7. 4021 


-5.3291 


-3. 9747 


-3. 0501 


-1.9168 


-1.3061 


-.9311 


120 




-33. 4941 


-21.6151 


-14.0135 


-9. 5614 


-6. 8460 


-5.0972 


-3. 8960 


-2. 4355 


-1.6531 


-1.1731 


130 






-27. 5400 


-18.0521 


-12.2761 


-8. 7510 


-6. 4662 


-4. 9325 


-3. 0686 


-2.0720 


-1.4666 


140 






-34.3121 


-23. 0424 


-15.6717 


-11.1154 


-8. 1908 


-6.2123 


-3. 8447 


-2. 5769 


-1.8159 


150 






-41. 4796 


-29.0964 


-19.9016 


-14.0671 


-10.3024 


-7. 7822 


-4. 7800 


-3. 1896 


-2. 2378 



372 REFRACTION AND REFRACTIVE INDEX MODELS 

8.4.9. Height Error Equations 

The equations for approximating e^ were determined as 



eh, = aio jrj + -B~ 9i + k, (8.52) 



D D"^ 

eh, = a2o -7-7 + -jB" ^2 + k, (8.53) 



and 



en, = 030 7— H — ^ (?3 + A; (8.54) 



where 

D = R'' - 0.03587 /i^'. 

Values for gi from (8.46) through (8.48) are found from 

fiiiha) = a,i + a^-Jia + a,3/ia^ , («' = 1, 2, 3) , (8.55) 

/i2(/ia) = an + a,5/ia , (z = 2, 3) , (8.56) 



and 



fzzQia) = age + a^^ha (8.57) 



for R in miles and /la in thousands of feet. The term in D/h^ was intro- 
duced to account for, in part, a large negative constant term which tended 
to produce negative height errors for ranges less than 30 mi. Further- 
more, the inclusion of this term increased the accuracy of the estimate of 
thi by about 2 percent. An additional term in hj' for (8.55) increased 
in the accuracy by about 1 percent but introduced a fictitious minimum 
near 60,000 ft, while a term in ha~ for (8.56) and (8.57) increased the 
accuracy of (8.53 and (8.54) by less than 0.1 percent. The relative im- 
provement of (8.53) over (8.52) is about one percent. The coefficients 
aij are listed in table 8.8. The constant term, k, which would vanish 
if the equations were exact, is about —70 for a least squares approxima- 
tion. 



REFERENCES 373 

8.4.10. Conclusions 

Height-error correction can be significantly imi^roved by accounting 
for the surface refractivity at the radar site. The use of the initial grad- 
ient, in addition to the surface refractivity, yields a significant improve- 
ment only for targers beyond about 60 mi and below 15,000 ft. In this 
case, Go is important not only to improve the accuracy but to deter- 
mine if the assumption in section 2.2 has been violated, namely, if 
Go < — lOV^'o- The still further improvement obtained with the use of 
AA'' would not, in general, justify the trouble and exi:)ense of measuring 
this parameter. 

If the distance to the target exceeds about 50 mi, the normal decrease 
with height of the gradient should be accounted for in a height error 
correction. 

8.5. References 

[1] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric refrac- 
tive index, Proc. IRE 47, No. 5, 740-755. 
[2] Misme, P. (Nov.-Dec. 1958), Essai de radiocHmatologie d'altitude dans le nord 

de la France, Ann. Telecommun. 13, Nos. 11-12, 303-310. 
[3] Tao, K., and K. Hirao (1960), Vertical distribution of radio refractive index in 

the medium height of the atmosphere, J. Radio Res. Lab. 7, No. 30, 85-93. 
[4] Misme, P. (Nov.-Dec. 1960), Quelques aspects de la radio-climatologie, Ann. 

Telecommun. 15, 266. 
[5] Bean, B. R., and G. D. Thayer (Aug. 1960), Rebuttal to P. Misme's comments on 

"Models of the Atmospheric Radio Refractive Index," Proc. IRE 48, No. 8, 

1499-1501. 
[6] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient 

of refractive modulus in homogeneous air, potential modulus, Book, Propaga- 
tion of Short Radio Waves, ed. D. E. Kerr, pp. 198-199 (McGraw-Hill Book 

Co., Inc., New York, N.Y.). 
[7] Zhevankin, S. A., and V. S. Troitskii (1959), Absorption of centimeter waves in 

the atmosphere, Radioteknika i Electronika 4, No. 1, 21-27. 
[8] Minzner, R. A., W. S. Ripley, and T. P. Condron (1958, U.S. extension to the 

ICAO standard atmosphere, Tables and Data to 300 Standard geopotential 

kilometers, U.S. Government Printing Office, Washington, D.C. 20402. 
[9] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative 

humidity over the United States and Alaska (U.S. Weather Bureau). 
[10] Bean, B. R., and J. D. Horn (Nov.-Dec. 1959), The radio refractive index near 

the ground, J. Res. NBS 63D (Radio Prop.), No. 3, 259-273. 
[11] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere, 

Proc. IRE 40, No. 5, 554-561. 
[12] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1900), CUmatic Charts and 

Data of the Radio Refractive Index for the United States and the World, 

NBS Mono. 22. 
[13] W(mg, M. S. (Sept. 1958), Refraction anomalies in airborne propagation, Proc. 

IRE 46, No. 9, 1628-1639. 
[14] Report of Factual Data from the Canterbury Project (1951), Vols. I-III, (Dept. 

of Scientific and Industrial Research, Wellington, New Zealand). 



374 REFRACTION AND REFRACTIVE INDEX MODELS 

[15] Freehafer, J. E. (1951), Tropospheric refraction, Book, Propagation of Short 

Radio Waves, ed. D. E. Kerr, pp. 9-22 (McGraw-Hill Book Co., Inc., New 

York, N.Y.). 
[16] Bean, B. R. (July-Aug. 1959), Climatology of ground-based radio ducts, J. Res. 

NBS 63D (Radio Prop.), No. 1, 29. 
[17] Bean, B. R., and B. A. Cahoon (Nov. 1957), The use of surface weather observa- 
tions to predict the total atmospheric bending of radio waves at small elevation 

angles, Proc. IRE 45, No. 11, 1545-1546. 
[18] Bean, B. R., B. A. Cahoon, and G. D. Thayer (1960), Tables for the statistical 

prediction of radio ray bending and elevation angle errors using surface values 

of the refractive index, NBS Tech. Note 44. 
[19] Bean, B. R., and G. D. Thayer (May-June 1963), Comparison of observed 

atmospheric radio refraction effects with values predicted through the use of 

surface weather observations, J. Res. NBS 67D (Radio Prop.), No. 3, 273. 
[20] Anway, A. C. (1961), Empirical determination of total atmospheric refraction at 

centimeter wavelengths by radiometric means, Collins Res. Rept. CRR-2425 

(Collins Radio Company, Cedar Rapids, Iowa). 
[21] Anderson, W. L., N. J. Beyers, and R. J. Rainey (Aug. 1960), Comparison of 

experimental and compeutd refraction, IRE Trans. Ant. Prop. AP-8, 456. 
[22] Wilkerson, R. E. (July-Aug. 1962), Divergence of radio rays by the troposphere, 

J. Res. NBS 66D (Radio Prop.), No. 4, 479. 
[23] Norton, K. A., J. W. Herbstreit, H. B. Janes, K. O. Hornberg, C. F. Peterson, 

A. F. Barghausen, W. E. Johnson, P. I. Wells, M. C. Thompson, Jr., M. J. 

Vetter, and A. W. Kirkpatrick (1961), An experimental study of phase varia- 
tions in line-of-sight microwave transmissions, NBS Mono. 33. 
[24] Thompson, M. C, Jr. (1962), Private communication. 
[25] Bennett, C. A., and N. L. Franklin (1954), Book, Statistical Analysis in Chemistry 

and the Chemical Industry (John Wiley & Sons, Inc., New York, N.Y.). 
[26] Brooks, C. E. P., and N. Carruthers (1953), Handbook of Statistical Methods in 

Meteorology (Her Majesty's Stationery Office, London). 
[27] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave 

propagation, Proc. IRE 21, No. 3, 427-463. 
[28] Bauer, J. R., W. C. Mason, and F. A. Wilson (1958), Radio Refraction in a Cool 

Exponential Atmosphere, Tech. Rept. 186, Lincoln Laboratory, Massachusetts 

Institute of Technology, Cambridge, Mass. 
[29] Beckmann, P. (1958), Height errors in radar measurements due to propagation 

causes, Acta Technika, 3, No. 6, 471-488 



Chapter 9. Radio-Meteorological Charts, 
Graphs, Tables, and Sample Computations 

9.1. Sample Computations of Atmospheric 
Refraction 

The following problem will serve to illustrate the a})plication of the 
various methods of calculating bending of a radio ray as described in 
chapter 3. A particular daily set of RAOB readings from Truk in the 
Caroline Islands yields the following data: 

Height above the 
surface (km) 
0.000 
0.340 
0.950 
3.060 
4.340 
5.090 
5.300 
5.940 
6.250 
7.180 
7.617 
9.660 
10.870 

What is the total bending up to the 3.270-km level at initial elevation 
angles of 0°, 10 mrad, 52.4 mrad (3°), and 261.8 mrad (15°) by (a) Schul- 
kin's approach, (b) the exponential model, (c) the initial gradient method, 
(d) the departures from normal method, (e) the use of regression lines, 
and (f) the graphical method of Weisbrod and Anderson? Since the 
gradient between the ground and the first layer is 



AA^ 365.0 - 400.0 ,nooAr •. /i 

71 = rv nAr. = — 102.9 A' umts/km, 

An 0.340 



and this is a decrease of N per kilometer that is less than the —157 N 
units/km required for ducting, no surface duct is {)resent. However, 

375 



N value 


(A units) 


400.0 = N 


365.0 


333.5 


237.0 


196.5 


173.0 


172.0 


155.0 


152.0 


134.0 


125.5 


98.0 


85.0 



376 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

should a surface duct have been present, it would have been necessary to 
calculate the angle of penetration, 



dp = V'2 [N, - Nh - 156.9 (Ah) (in km)], 

to find the smallest initial elevation angle that yields a non-trapped ray. 
Any initial elevation angle less than 9p cannot be used in bending calcula- 
tions. 

Schulkin's approach (a) of (3.13) yields the results shown in table 9.19 
for mrad, table 9.20 for 10 mrad, table 9.21 for 52.4 mrad, and table 
9.22 for 261.8 mrad, where dk+i is determined from (3.58) using 
Tk = a -\- hk, and a is the radius of the earth, 

a = 6370 km. 

It should be remembered that dk = 0, 10, 52.4, or 261.8 mrad only for the 
first-level calculation, and that thereafter 6k is equal to the dk+i computed 
for the preceding layer; e.g., for the second layer of table 9.19 
(^0 = Omrad), dk = 6.15 mrad,which is the ^t+icalculatedforthe first layer. 
The exponential model solution (b) may be found by using tables 
9.10 through 9.17. Interpolation will usually be necessary for A''^, do, and 
height; this interpolation may be done linearly. In practice, one of these 
three variables will often be close enough to a tabulated value that inter- 
polation will not be necessary, thus reducing from 7 to 3 the number of 
interpolations necessary. Since in the problem forA'^s = 404.9, h = 10.0 km 
and 00 = 10 mrad 

To. 10. 0(10 mrad) = 15.084 uirad 



and at A = 20.00 km, do = 10 mrad. 

To, 20.00(10 mrad) = 15.946 mrad, 

and thus by linear interpolation for h = 10.870 km, do = 10 mrad, 

TO, 10. 870, (10 mrad) = 15.084 + (15.946 - 15.084) 20 00 - 10 00 
= 15.159 mrad. 



COMPUTATIONS OF ATMOSPHERIC REFRACTION 377 

Similarly for N s = 377.2 in the exponential tables, 

To, 10. 870 (10 mrad) ~ 13.120 Hirad. 

Again using linear interpolation, but now between the N s = 377.2 and 
A^, = 344.5 atm, the desired value of r at 3.270 km for A^, = 360.0 and 
^0 = 10 mrad is obtained. 

Thus 

400 — 377 2 

T0.10.870(10 n.rad) = 13.120 + (15.159 - 13.120) 404 Q _ 377 2 

= 14.798 mrad. 

For the ^o = 0, 52.4, and 261.8 mrad cases, by similar calculations, using 
linear interpolation : 

7"o,io.87o, (0 mrad) = 21.386 mrad. 

To, 10. 870, (52.4 mrad) = 5.816 mrad, 
To, 10. 870, (261.8 mrad) = 1.270 mrad. 

The initial gradient correction method (c) may be used if one deter- 
mines the A^s* which corresponds to the observed initial gradient and then 
applies (3.45). The initial A'' gradient is —102.9 N units/km, which, as 
can be seen from table 3.17, corresponds to the N s = 450.0 exponential 
atmosphere. Therefore, using the exponential tables of Bean and 
Thayer [1]^ and (3.45) to determine the bending for the ^o = mrad case, 
one finds by linear interpolation 

Tio, 000(0) = Tio,ooo (400.0, mrad) + [rioo (450.0, mrad) 
- Tioo (400.0, mrad)] 
= 21.309 mrad + [5.908 - 3.657] mrad = 23.560 mrad. 

The Tioo (400.0, mrad) is determined by linear interpolation between the 
404.9 and 377.2 atm. At h = 20.0 km as given in the tables, 

T2o,ooo (0 mrad) = T2o,ooo (400.0, mrad) + [tioo (450.0, mrad) 
- r,oo (400.0, mrad)] 
= 22.191 + [5.908 - 3.657] mrad 
= 24.442 mrad. 



^Figures in brackets indicate the literature references on p. 423. 



378 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

Hence by linear interpolation, 

.,...,. ,. „„. = 23.560 + [24.442 - 23.5601 ^oiooO I loioOO 
= 23.637 mrad. 

The bendings for ^o = 10, 52.4, and 261.8 mrad are as given below: 

do = 10; no, 870 (10) = 15.053 mrad, 
do = 52.4; 710,870 (52.4) = 5.864 mrad, 
do = 261.8; Tio, 870 (261.8) = 1.280 mrad. 

To use the departures-from-normal method (d) of determining bending, 
it is first necessary to know the atmosphere which must be used for the 
calculation. In the problem, 

dN I 
_ -^ = 102.9 A^ units/km, 

Cl" I initial 

which is within the range of the N ^ = 450.0 exponential atmosphere, as 
can be seen from table 9.18. Thus one will use table 9.17 to determine 
the d's and the r's in the N s = 450.0 exponential atmosphere, or one can 
use the exponential atmosphere tables [1]. 

For an A^s = 450.0 atmosphere to a height of 10.870 km: 

TNsio mrad) = 30.776 mrad, 

TNsiio mrad) = 19.414 mrad, 

rArs(52.4 mrad) = 7.024 mrad, 

T7Vs(26i.8 mrad) = 1.506 mrad. 

Equation (3.58) should be used for the 6 interpolation in preference to 
linear interpolation, although, if no tables or other facilities are present 
at the engineering site for easy acquisition of square roots, linear inter- 
polation will suffice. Proceeding in table 9.17 with (3.58) for the first 
layer at /i = 0.340 km and the ^o = mrad case: 



V^2 ^ 2(n - ro) ^ ^q6 _ 2(^^ _ ^^) 



ro 



= 6.388 mrad. The remaining d's for the various layers are shown in 
table 9.23. To determine the value of A at the bottom and top of the 



COMPUTATIONS OF ATMOSPHERIC REFRACTION 



379 



layer, one makes use of (3.46) or figure 3.17. First, however, one must 
determine the value of c in (3.46) to be used. Usually interpolation will 
be necessary in table 9.18, but in the N s = 450.0 case it is not possible, 
and thus the straight N s — 450.0 exponential atmosphere values are used. 
From (3.46) 

^(A^„ h) = N{h) + .¥, [1 - exp (-ch)], 

and for the layer running from h = io h = 0.340 km, figure 3.17 yields 

377.2 (1 - exp (cO)] = 0.0 

450.0 [1 - exp (-C X 0.340)] = 32.8, 

A (450.0, 0) = 400.0 + = 400.0 

A (450.0., 0.340) = 365.0 + 32.8 = 397.8, 
whence 

AA = A (450.0, 0.340) - A (450.0, 0) = 397.8 - 400.0 = -2.2 A^ units. 
Therefore, the departure term of (3.23), 



and 

and, therefore, 

and 



+ 



7/c+l L 



AAiNs) 






becomes 



+ 6.388 L 



2.2 



= +0.689 mrad. 



The remaining calculations are tabulated in table 9.23 for the ^o = mrad 
case, in table 9.24 for the ^o = 10 mrad case, in table 9.25 for the ^o = 52.4 
mrad case, and in table 9.26 for the ^o = 261.8 mrad case. The sum of 
the departures for the mrad case is 






+ 



/fc+i L 



iN 



AA (N.) 



k+i 



N, 



= —5.335 mrad. 



Determination of the bending is required in part (e) of the problem by 
using regression lines. By (3.10), using table 9.7 and 9.8, it is found for 
the 00 = mrad case that at 10.0 km (from table 9.7) 

TO, ,0.0 = (0.1149) (400.0) - 18.5627 ± 7.5227 
= 27.3973 ± 7.5227 mrad 



380 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

and at 20.0 km (from table 9.8) 

To.20.0 = (0.1165) (400.0) - 17.9573 ± 7.5131 
= 28.6427 ± 7.5131 mrad. 

Thus, by linear interpolation 

Ti.2 = To.10.87 = 27.3973 + (28.6427 - 27.3973) 9q qq I |q qq ± 7.5227 

^ .„ _<^„„ „_.„.. 10.87 - 10.00 , 
T (7.5227 - 7.5131) 20.00 - 10.00 

Ti,2 = 27.5056 ± 7.5218 mrad. 

Similarly, for the remaining 0's, 

ri,2(io mrad) = 13.9548 ± 0.9701 mrad, 

Ti,2(53.4 mrad) = 5.2186 ± 0.0817 mrad, 

Ti, 2(261. 8 mrad) = 1.2695 ± 0.0158 mrad. 

Determination of the bending by means of the graphical method (f) 
of Weisbrod and Anderson yields, from figure 3.18 for 500 tan 6, for the 
first layer. 

At do At ^0 At ^0 At do 

h(m.) = mrad = 10 mrad = 52.4 mrad = 261.8 mrad 



0.000 0.0 5.0 26.2 134.0 

0.340 3.0 5.8 27.0 134.0 

which yields for the bending in the first layer, 

At do At ^0 At ^0 At do 

= mrad = 10 mrad = 52.4 mrad = 261.8 mrad. 



11.67 3.24 0.66 0.13 

Similarly, the bending for the entire profile may be obtained, and shown 
to be 

At do At do At ^0 At do 

= mrad =10 mrad =52.4 mrad = 261.8 mrad. 

24.42 14.00 5^32 iTlT 



COMPUTATIONS OF ATMOSPHERIC REFRACTION 



381 



The answers to the several parts of the problem are summarized in the 
next table of this chapter (immediately preceding the main body of 
tables). Bending values for the assumed profile, from a method which 
exponentially interpolated layers between given layers and then integrated 
between resulting layer^, assuming only a linear decrease of refractivity 
between interpolated layers, are included for the sake of comparison. 
The comi^utations were performed on a digital computer. 

The reason that the answers to i)art (e) vary so radically from the 
remaining answers for the ^o == mrad case and not so much for the 
^0 = 261.8 mrad case is the fact that the accuracy of the regression line 
method increases with increasing initial elevation angle, ^o- It must be 
remembered that the statistical regression technique, like the exponential 
model, is an adequate solution to the bending problem for all ^o's larger 
than about 10 mrad, and all heights above 1 km. 

The reason that the answers in part (f) and part (a) agree more closely 
than with any other of the answers is because (3.49) is, as mentioned 
before, Schulkin's result with only the approximation, tan dk = dk for 
small angles, omitted. For this individual profile, the bending obtained 
from an exponential atmosphere does not give particularly accurate bend- 
ings; however, for 22 five-year mean refractivity profiles, figure 3.9 shows 
that exponential bending predicts accurately within 1 percent of the 
average bending for these five-year means. Figure 3.19 shows the rms 
error in predicting bending at various heights as a per cent of mean bend- 
ing (not including superrefraction). 

In summary, it is recommended that the communications engineer 
either use the statistical regression technique or the exponential tables 
[1] without interpolation (i.e., pick the values of height, N s, and Oq that 
are closest to the given parameters) for a quick and facile bending result, 
keeping in mind the restrictions on these methods. However, as men- 
tioned before, use of Schulkin's method is recommended if accuracy is 
the primary incentive. 



Summary of refraction results for the sample computation 



Prob- 
lem 
part 


Method used 


Bending 
in mrad at 
6o==0 mrad 


Bending 
in mrad at 
So = 10 mrad 


Bending 

in mrad at 

00=52.4 mrad 


Bending 

in mrad at 

00=261.8 mrad 


a. 


Schulkin's method 


24. 248 


14.008 


5.341 


1.196 


b. 


Exponential model 


21.386 


14. 798 


5.816 


1 270 


c. 


Initial gradient correction method. 


23. 637 


15. 053 


5.864 


1.280 


d. 


Departures from normal method.. 


25. 441 


14. 858 


5.350 


1.143 


e. 


Statistical regression method 


27. 506 

d=7. 522 


13. 955 
±0. 9701 


5. 2186 
±0. 0817 


1. 2695 
±0. 0158 


f. 


Graphical method 


24.42 
24 171 


14.00 
14. 104 


5.32 
5.343 


1 168 








Compa 
inter 


rison (exponential layer 

polation) bending... 


1. 168 







382 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



Tables 9.1 through 9.9 are tables of coefficients, a and h, standard errors of 
estimate, SE, and correlation coefficients, r, for use in the statistical method 
of refraction prediction. 



Table 9.1. 



Variables in the statistical method of refraction prediction 
for h — hs = 0.1 km 



So 


r 


b 


a 


SE 


0.0 

1.0 

2.0 


0. 2665 
.2785 
.2881 
.3048 
. 1915 

.2070 
.2100 
.2105 
.2105 
.2107 
.2108 


0. 0479 
.0257 
.0162 
.0073 
.0053 

.0025 
.0009 
.0005 
.0002 
.0001 
. 00004 


-8.7011 

-4. 1217 

-2. 2732 

-.9181 

-. 6085 

-.2639 
-.0973 
-. 0507 
-. 0250 
-.0120 
-.0040 


6. 7277 
3. 4363 
2. 0960 


5.0 


0. 8792 


)0.0 

20.0 

52.4 


1.0551 

0. 4555 
. 1688 , 


100. 


. 0879 


200.0 

400.0 

900.0 


.0435 
.0208 
.0070 



Table 9.2. 



Variables in the statistical method of refraction prediction 
for h - ha = 0.2 km 



00 


r 


b 


a 


SE 


0.0 


0. 2849 
.2979 
.3104 
.3415 
.2306 

.2550 
.2604 
.2610 
.2613 
.2613 
.2604 


0. 05801 
.0348 
.0239 
.0117 
.0073 

.0035 
.0013 
.0007 
.0003 
.0002 
.00005 


-10.4261 
-5. 6431 
-3. 5707 
-1.5287 
-0. 7436 

-. 3184 
-. 1162 
-.0603 
-. 0299 
-. 0143 
-.0047 


7. 5726 


1.0- 

2.0. 


4. 3330 
2. 8357 


5.0 .- 


1.2512 


10.0 


1. 1990 


20.0 

52.4. 


0. 5122 
.1890 


100.0 


.0983 


200.0 


.0486 


400.0 


.0233 


900. 0... 


.0078 







Table 9.3. Variables in the statistical method of refraction prediction 
for h — ha = 0.5 km 



flo 


r 


b 


n 


SE 


0.0 


0.3615 
.3997 
.4369 
.5205 
.3933 

.4563 
.4731 
.4753 
.4760 
.4761 
.4764 


0. 0769 
.0510 
.0384 
.0228 
.0140 

.0071 
.0027 
.0014 
.0007 
.0003 
.0001 


-14.6443 
-9. 0567 
-6. 5408 
-3. 6605 
-1.9055 

-0. 8926 
-.3308 
-. 1721 
-. 0851 
-. 0408 
-.0137 


7. 6170 


1.0 

2.0 


4. 4954 
3. 0395 


5.0 


1. 4376 


10.0 


1. 2733 


20.0 


0. 5365 


52.4 


.1966 


100.0 


.1022 


200.0 

400. . 


.0505 
.0242 


900.0 


.0081 







COMPUTATIONS OF ATMOSPHERIC REFRACTION 



383 



Table <).4. 



Variables in the statistical method of refraction prediction 
for h — he = 1.0 km 



Bo 


r 


b 


a 


SE 


0.0... 


0. 3936 
.4620 
.5238 
.6348 
.5718 

.6598 
.6823 
.6851 
.6859 
.6860 
.6864 


0. 0840 
.0607 
. 04918 
.0337 
.0224 

.0124 
.0049 
.0026 
.0013 
.0006 
.0002 


-15. 1802 
-10.3739 
-8. 2066 
-5. 4816 
-3. 2378 

-1. 6959 
-0. 6495 
-.3388 
-. 1676 
-. 0803 
-. 0270 


7 6151 


1.0 

2.0 


4.5217 
3 1040 


5.0 


1 5931 


10.0 


1 2574 


20.0 


5531 


52.4 


2071 


100.0 . . 


1080 


200.0 


0534 


400. - 


0256 


900.0 


0086 







Table 9.5. Variables in the statistical method of refraction prediction 
for h - he = 2.0 km 



Bo 


r 


6 


a 


SE 


0.0 


0. 4524 


0. 0985 
.0752 
.0636 
.0475 
.0345 

.02111 

.0089 

.0047 

.0023 

.0011 

.0004 


-17.7584 
-12.9451 
-10. 7566 
-7. 8969 
-5. 3712 

-3. 1571 
-1.2770 
-0. 6705 
-0. 3323 
-0. 1593 
- .0535 


7 5391 


1.0. - 




5490 
6316 
7707 
7634 

8515 
8668 
8679 
8681 
8682 
8684 


4 4420 


2.0. 


3 0277 


5.0 


1 5234 


10.0. 


1 1421 


20.0 


5086 


52.4.. 


2003 


100.0 


1057 


200.0... 


0524 


400. 0. . 


0252 


900.0 


0084 











Table 9.6. 



Variables in the statistical method of refraction prediction 
for h — hs = 5.0 km 



Bo 


r 


6 


a 


SE 


0.0 


0. 4962 
.6101 
.7030 
.8504 
.8674 

.9484 
.9674 
.9695 
.9701 
.9702 
.9703 


0.1115 
.0881 
.0764 
.0601 
.0464 

.0308 
.0139 
.0075 
.0037 
.0018 
.0006 


-19. 1704 

-14.3543 

-12.1589 

-9. 2514 

-6. 6445 

-4. 0706 
-1.6236 

-0. 8348 
-.4098 
-. 1960 
-.0658 


7 5676 


1.0 

2.0.. 


4. 4401 
3 0001 


5.0 


1 4422 


10.0 


1 0420 


20.0 


4028 


52.4.. 


1426 


100.0 


0739 


200.0 


0365 


400.0 


0175 


900.0- 


0059 







384 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



Table 9.7. 



Variables in the statistical method of refraction prediction 
for h - hs = 10.0 km 



So 


r 


b 


a 


SE 


0.0 


0. 5099 
.6290 
.7250 
.8734 
.8950 

.9723 
.9907 
.9927 
.9931 
.9932 
.9932 


0. 1149 
.0915 
.0799 
.0635 
.0498 

.0338 
.0157 
.0085 
.0043 
.0020 
.0007 


-18.5627 

-13.7469 

-11.5514 

-8. 6434 

-6.0729 

-3. 5012 

-1. 1441 

-0. 5084 

-.2310 

-. 1078 

-.0359 


7. 5227 


1.0 

2.0 


4. 3895 
2 9443 


5.0 


1.3733 


10.0 


0.9713 


20.0 


3179 


52.4 


.0844 


100.0 


.0406 


200.0 


.0197 


400.0 


.0094 


900.0 


.0032 







Table 9.8. 



Variables in the statistical method of refraction prediction 
for h - ha = 30.0 km 



e. 


r 


b 


a 


SE 


0.0 


0. 5155 
.6367 
.7336 
.8815 
.9028 

.9785 
.9968 
.9984 
.9986 
.9986 
.9986 


0.1165 
.0931 
.0814 
.0651 
.0514 

.0353 
.0169 
.0093 
.0047 
.0023 
.0008 


-17.9573 

-13.1413 

-10.9463 

-8. 0397 

-5. 4747 

-2. 9228 
-0. 6738 
-. 1802 
-.0467 
-.0161 
-.0048 


7. 5131 


1.0 

2.0 


4. 3763 
2. 9281 


5.0 


1.3521 


10.0 


0. 9573 


20.0 


.2909 


52.4 


.0535 


100.0 


.0203 


200.0 


0096 


400. 0- - 


.0046 


900.0 


.0016 







Table 9.9'. 



Variables in the statistical method of refraction prediction 
forh - h, = 70.0 km 



e. 


r 


b 





SE 


0.0 


0. 5174 
.6391 
.7361 
.8837 
.9051 

.9797 
.9979 
.9997 
1.0000 
1.0000 
1.0000 


0. 1170 
.0936 
.0820 
.0656 
.0519 

.0358 
.0173 
.0096 
.0048 
.0024 
.0008 


-17.9071 

-13.0912 

-10.8960 

-7.9895 

-5. 4209 

-2.8696 
-0. 6246 
-. 1402 
-.0212 
-.0027 
-.0002 


7.5113 


1.0 


4. 3738 


2.O.. . - 


2. 9251 


5.0 


1.3481 


10.0. -. 


0. 9539 


20.0 


.2862 


52.4 


.0445 


100.0 

200.0 


.0095 
.0013 


400.0 


.0002 


900.0 


.0001 







COMPUTATIONS OF ATMOSPHERIC REFRACTION 



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COMPUTATIONS OF ATMOSPHERIC REFRACTION 



387 



3 
II 


t- 


0. 0020 
.0040 
.0100 
.0200 
.0397 
.0969 

.1864 
.3455 
.6960 
1.0107 
1.2171 
1.2698 


«> 


261.803 
261.807 
261.819 
261.838 
261.877 

261. 995 

262. 198 
262. 623 
264.018 
266. 587 
272. 053 
298. 561 


II 


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0. 0103 
.0206 
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.2021 
.4910 

.9365 

1. 7076 

3. 2972 

4. 5676 

5. 2725 
5.4173 


a> 


52. 380 
52. 399 
52. 458 
52. 557 

52. 754 

53. 352 

54. 362 
56. 420 
62. 725 
72. 997 
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0. 0180 
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1. 5801 

2. 8020 
5.1119 

6. 7691 

7. 6053 
7. 7650 


51 


30. 034 
30. 069 
30. 171 
30. 342 

30. 684 

31. 701 

33. 373 
36. 633 
45. 755 
59. 064 
80. 627 
148. 250 


o 


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0. 0537 
.1068 
.2624 
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2. 1433 

3. 6454 
5. 7374 
8. 9749 

10. 9677 
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31 


10. 102 
10. 204 
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10. 985 
11.895 
14.316 

17.715 
23. 282 
35. 970 
51.858 
75. 513 
145. 546 




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0. 3926 
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1. 2341 
1.8997 

2. 8453 
4. 6739 

6. 6046 
9. 0096 
12. 4665 

14. 5172 

15. 4524 
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Cfc 


1.749 
2.263 
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6.519 
10. 294 

14. 657 
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«> 


1.435 
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14. 623 
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^CM to O OO 



COMPUTATIONS OF ATMOSPHERIC REFRACTION 



389 



Table 9.18. Initial N gradients, ANe, in the CRPL 
Exponential Reference Atmosphere* 



Range of ANe 
(N^ units/km) 


N{h) 


ANe <, 27. 55 
27. 55 < ANe <, 35. 33 
35. 33 < ANe ^ 47. 13 
42. 13 < ANe <, 49. 52 
49. 52 < AA^€ <, 59. 68 
59. 68 < ANe <. 71. 10 
71. 10 < ANe ^ 88. 65 
88. 65 < ANe 


200 exp(-0. 1184/1) 
252.9exp(-0.1262h) 
289 exp(-0. 1257/1) 
313 exp(-0. 1428/1) 
344. 5 exp(-0. 1568/i) 
377. 2 exp( -0.1732/1) 
404. 9 exp(- 0.1898/1) 
450 exp(-0. 2232/1) 



*Note height, h, is in kilometers. 



390 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



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COMPUTATIONS OF ATMOSPHERIC REFRACTION 



391 



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392 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



Table 9.23. Departures method sample computation for 60 = 10 mrad 



h 


N(h) 


A(N,h) 


A.4 


e 


2 


De- 
parture 
term 


TNsih) 


'"1.2 





400.0 
365.0 
333.5 
237.0 
196.5 
173.0 

172.0 
155.0 
152.0 
134.0 
125.5 
98.0 
85.0 


400.0 
397.8 
419.5 
459.7 

475.7 
478.5 

484.1 
485.6 
490.5 
493.4 
493.3 
495.9 
495.2 


-2.2 
+21.7 
+40.0 
+ 16.0 

+2.8 

+5.6 
+1.5 
+4.9 
+2.9 

-.1 
+2.6 

-.7 




6.388 
11.230 
22. 683 
28. 195 

31. 383 

32. 191 
34. 654 
35.811 
39. 123 
40. 581 
47. 236 
50. 645 


0.3131 
.1135 
.0590 
.0393 
.0336 

.0315 
.0299 
.0284 
.0267 
.0251 
.0228 
.0204 


+0. 689 
-2. 463 
-2. 359 
-0. 629 
-.094 

-.176 
-.045 
-.139 
-.077 
+.003 
-.059 
+.014 
S = -5. 335 






0.340 






0.950 






3.060 






4.340 






5.090 






5.300 






5.940 






6. 250 






7.180 . 






7.617 






9.660 






10. 870 


30. 776 


25. 441 







Table 9.24. Departures method sample computation for do = 10 mrad 



h 


N(h) 


A(N,h) 


^A 


e 


2 


De- 
parture 
term 


TN>(h) 




Ok+Bk+i 







400.0 
365.0 
333.5 
237.0 
196.5 
173.0 

172.0 
155.0 
152.0 
134.0 
125.5 
98.0 
85.0 


400.0 
397.8 
419.5 
459.7 
475.7 
478.5 

484.1 
485.6 
490.5 
493.4 
493.3 
495.9 
495.2 


-2.2 

+21.7 

+40.0 

+16.0 

+2.8 

+5.6 
+1.5 
+4.9 
+2.9 

-.1 
+2.6 

-.7 


10.000 

11.873 
15. 038 
24. 789 
29. 947 

32. 938 

33. 712 

36. 069 

37. 181 
40. 383 
41.801 
48. 282 
51. 630 


0. 0914 
.0743 
.0502 
.0365 
.0318 

.0300 
.0287 
.0273 
.0258 
.0243 
.0222 
.0200 


+0. 201 

-1.613 

-2. 008 

-.585 

-.089 

-.168 
-.043 
-.134 
-.075 
+.002 
-.058 
+.014 
2 = -4. 556 






0.340 






0.950 






3. 060 






4.340 . - 






5.090 






5 300 






5 940 






6.250 






7.180 






7.617 






9 660 






10.870 


19.414 


14. 858 







Table 9.25. Departures method sample computation for do = 52.4 mrad (3°) 



h 


N{h) 


A(N,,h) 


A.4 


e 


2 


De- 
parture 
term 


w,(/l) 




Bk+Bk+i 







400.0 
365.0 
333.5 
237.0 
196.5 
173.0 

172.0 
155.0 
152.0 
134.0 
125.5 
98.0 
85.0 


400.0 
397.8 
419.5 
459.7 
475.7 
478.5 

484.1 
485.6 
490.5 
493.4 
493.3 
495.9 
495.2 


-2.2 
+21.7 
+40.0 
+ 16.0 

+2.8 

+5.6 
+ 1.5 
+4.9 
+2.9 
-.1 
+2.6 


52.4 

52. 750 

53. 550 
57. 061 
59. 567 
61.045 

61.477 

62. 792 

63. 433 

65. 368 

66. 277 
70. 512 
72.920 


0. 0190 
.0188 
.0181 
.0171 
.0166 

.0163 
.0161 
.0158 
.0155 
.0152 
.0146 
.0139 


+0. 042 
-.408 
-.724 
-.274 
-.046 

-.091 
-.024 
-.078 
-.045 
+.002 
-.038 
+.010 
2 == -1.674 






0.340 





0. 950 






3 060 






4.340 






5.090 






5.300 






5. 940 






6 250 






7 180 






7.617 






9 660 






10.870. 


7.024 


5.350 







EXPONENTIAL REFERENCE ATMOSPHERE 393 

Table 9.26. Departures method sample computation for do = 261.8 mrad {15°) 



h 


mh) 


AiNs,h) 


AA 


e 


2 


De- 
parture 
term 


TN.ih) 




dk+eic+i 







400.0 
365.0 
333.5 
237.0 
196.5 
173.0 

172.0 
155.0 
152.0 
134.0 
125.5 
98.0 
85.0 


400.0 
397.8 
419.5 
459.7 
475.7 
478.5 

484.1 
485.6 
490.5 
493.4 
493.3 
495.9 
495.2 


-2.2 
+21.7 
+40.0 
+16.0 

+2.8 

+5.6 
+1.5 
+4.9 
+2.9 

-.1 
+2.6 

-.7 


261.8 

261. 880 

262. 035 

262. 758 

263. 308 
263. 633 

263. 733 

264. 034 

264. 184 
264.647 
264. 872 

265. 934 

266. 592 


0.0038 
.0038 
.0038 
.0038 
.0038 

.0038 
.0038 
.0038 
.0038 
.0038 
.0038 
.0038 


+0.008 
-.083 
-.152 
-.061 
-.011 

-.021 
-.006 
-.019 
-.011 
+.000 
-.010 
+.003 
S = -0. 363 






0.340— 






0.950 






3.060 






4.340 






5.090 






5.300 






5.940 






6.250 






7.180 






7. 617_ 






9.660 






10.870 


1.506 


1 143 







9.2. Tables of Refraction Variables for the 
Exponential Reference Atmosphere 

The following table of estimated maximum errors should serve as a 
guide to the accuracy of the tables. 

Errors in elevation angle, 6: 

do < 4: mrad ± 0.00005 mrad ) nearly 
4 mrad < ^o < 100 mrad ± 0.000005 mrad > independent 
do < 100 mrad ± 0.00004 mrad ) of A^,. 

Errors in r, e (in milHradians) : 

N, = 450 404.8 377.2 344.5 313 252.9 200 

00 = ± 0.001 0.00065 0.0005 0.0004 0.0003 0.0002 0.0002 
do = 1° 0.0003 0.00015 0.0001 0.00008 0.00006 0.00005 0.00005 
00 = 3° 0.00004 0.000025 0.00002 0.000017 0.000015 0.000013 0.000012 

Errors in Ro, R, Re, or Ah (in meters): 



Ns = 450 404.8 377.2 344.5 313 252.9 200 

?o = ±5.0 2.7 1.8 1.2 0.8 0.65 0.6 

?o = 1° 0.4 0.3 0.25 0.2 0.17 0.15 0.14 



Assume that the error in AR or ARe is ±0.5 percent or ±0.1 m, whichever 
is larger. 



394 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



In tables 9.27 to 9.29 the following equations were used in determining 
the various profile variables: 



AA^ = 7.32 exp (0.005577 A^.), 



Ce = hi 



N.. 



Ns + AN 



-dN, = CeNs, 



k = 



1 - -^ (ceNs) X 10-' 

Us 



ris = 1 + A^, X 10-«, 

where ro is taken as 6373.024987 km for all values of N s in these tables. 
Table 9.29 was prepared using an iterative method for solution of 
transcendental equations involved. In all these tables the accuracy may 
be tak(m as ±1 in the last digit listed. 



Table 9.27. Parameters for exponential refraction with height for various values of Na 



A^, 


-AN 


Ce 


-dA^o 


k 


200 


22. 3317700 

23. 6125966 
24. 9668845 

26. 3988468 

27. 9129385 

29. 5138701 

31. 2066224 

32. 9964614 
34. 8889558 
36. 8899932 

39. 0057990 
41. 2429556 
43. 6084233 
46.1095611 
48. 7541501 

51. 5504184 
54. 5070651 
57. 6332884 
60. 9388149 
64. 4339281 

68. 1295015 
72. 0370324 
76. 1686780 
80. 5372922 
85. 1564647 
90. 0405683 


0.118399435 
. 119280212 
. 120458179 
. 121916361 
. 123642065 

. 125626129 
. 127862319 
. 130346887 
. 133078254 
. 136056720 

. 139284287 
. 142764507 
. 146502381 
. 150504269 
. 154777865 

. 159332141 
. 164177379 
. 169325150 
. 174788368 
. 180581312 

. 186719722 
. 193220834 
.200103517 
. 207388355 
. 215097782 
. 223256247 


23. 6798870 

25. 0488444 

26. 5007993 

28. 0407631 

29. 6740955 

31.4065323 
33. 2442030 
35. 1936594 
37.2619112 
39. 4564487 

41. 7852861 
44. 2569972 
46. 8807620 
49. 6664087 
52. 6244741 

55. 7662495 
59. 1038565 
62. 6503054 
66. 4195799 
70.4267116 

74. 6878887 
79. 2205420 
84. 0434770 
89. 1769927 
94. 6430240 
100.4653113 


1. 17769275 


210 


1. 18991401 


220 

230 


1.20315637 
1. 21752719 


240 

250.. 


1. 23314913 
1. 25016295 


260 


1.26873080 


270 


1. 28904048 


280-- 


1.31131073 


290 


1. 33579768 


300- - 


1. 36280330 


310 


1.39268608 


320 

330 

340 

350 

360 


1.42587494 
1. 46288731 
1. 50435338 

1. 55104840 
1. 60393724 


370 


1. 66423593 


380 

390 


1. 73349938 
1. 81374807 


400 

410 

420 

430 

440 

450 


1. 90765687 

2. 01884302 
2. 15232187 
2.31525447 
2. 51823286 
2. 77761532 



EXPONENTIAL REFERENCE ATMOSPHERE 395 

Table 9.28. Parameters for exponential refraction with height for various values of — AN 



-AN 


Ns 


Ce 


-dNo 


k 


20 


180. 226277 


0.117626108 


21. 1993155 


1. 15617524 


22 


197.316142 


. 118216356 


23. 3259953 


1. 17457412 


24 


212. 917967 


. 119594076 


25. 4637276 


1. 19366808 


26 


227. 270255 


. 121491305 


27.6113599 


1.21348565 


28 


240. 558398 


. 123746115 


29. 7681671 


1.23406110 


30 


252. 929362 


. 126255291 


31.9336703 


1. 25543336 


32 


264. 501627 


. 128950180 


34. 1075325 


1. 27764560 


34 


275. 372099 


. 131783550 


36. 2895127 


1. 30074523 


36 


285. 621054 


. 134721962 


38. 4794288 


1. 32478398 


38 


295. 315731 


. 137741207 


40. 6771452 


1.34981825 


40 


304. 513148 


. 140823306 


42. 8825481 


1. 37590934 


42 


313.261483 


. 143955014 


45.0955611 


1.40312414 


44 


321. 602888 


. 147125889 


47.3161107 


1.43153539 


46 


329. 573439 


. 150328075 


49. 5441405 


1. 46122250 


48 


337. 204713 


. 153555418 


51. 7796106 


1.49227226 


50 


344. 524418 


. 156803056 


54. 0224815 


1. 52477960 


52 


351. 557000 


. 160067149 


56. 2727266 


1. 55884863 


54 


358. 324138 


. 163344614 


58. 5303179 


1. 59459364 


56 


364. 845143 


. 166633002 


60. 7952415 


1.63214058 


58 


371. 137293 


. 169930326 


63.0674811 


1. 67162830 


60 


377. 216108 


. 173234984 


65. 3470266 


1.71321044 


62 


383. 095581 


. 176545680 


67. 6338699 


1. 75705732 


64 


388. 788373 


. 179861358 


69. 9280046 


1. 80335830 


66 


394. 305974 


. 183181171 


72. 2294300 


1.85232456 


68 


399. 658845 
404. 856538 


. 186504431 
. 189830583 


74. 5381454 
76.8541525 


1. 90419225 


70 


1. 95922635 


72 


409. 907798 


. 193159183 


79. 1774555 


2.01772514 


74..._ 


414. 820650 


. 196489873 


81. 5080567 


2. 08002556 


76 


419. 602477 


. 199822385 


83. 8459677 


2. 14650999 


78 


424. 260086 


. 203156494 


86. 1911914 


2. 21761358 


80 


428. 799768 


. 206492043 


88. 5437400 


2. 29383429 


82 


433. 227348 


. 209828917 


90. 9036251 


2. 37574437 


84 


437. 548229 


. 213167031 


93. 2708570 


2. 46400458 


86 


441. 767432 


. 216506335 


95. 6454475 


2. 55938222 


88 


445. 889634 
449. 919193 


. 219846812 
. 223188453 


98. 0274147 
100. 4167688 


2. 66277367 


90 


2. 77523207 


92 


453. 860184 


. 226531281 


102. 8135290 


2. 89800399 


94 


457. 716416 


. 229875327 


105. 2177108 


3. 03257531 


96.. 


461.491458 
465. 188659 
468.811163 


. 233220637 
. 236567271 
.239915290 


107. 6293319 
110.0484114 
112.4749663 


3. 18073184 


98 


3. 34463902 


100 


3. 52694820 



396 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

Table 9.29. Parameters for exponential refraction with height for various values of k 



fc 


N, 


-AN 


Cc 


-dNo 


1.0 

1.2 

1.3 

1.4 

1.5 


0.0 

217. 689023 
275. 037959 
312. 297111 
339. 003316 
359. 298283 

375. 341242 
388. 391792 
399. 243407 
408. 424907 
416. 304322 

423. 146728 
429. 148472 
434.458411 
439. 191718 
443. 438906 

447. 272272 
450. 750273 

523. 299600 

289. 036274 


0.0 

24. 6471681 
33. 9367000 
41. 7747176 
48. 4839018 
54. 2941700 

59. 3759008 
63. 8586055 
67. 8426334 
71.4070090 
74. 6148487 

77. 5171828 
80. 1557288 
82. 5649192 
84. 7734613 
86. 8054237 

88.6811886 
90. 4181120 

135. 5109159 

36. 6922523 


0.0 

. 120160519 
. 131692114 
. 143600133 
. 154339490 
. 163827653 

. 172203063 
. 179626805 
. 186242834 
. 192172034 
. 197514185 

. 202351472 
. 206751820 
. 210771674 
. 214458304 
. 217851443 

. 220984823 
. 223887193 

. 299693586 

. 135758874 


0.0 

26. 1576259 
36. 2203304 
44. 8459068 
52. 3215987 


1.6 ..- 


58. 8629945 


1.7 


64.6349115 


1.8 


69. 7655765 


1.9 


74. 3562234 


2.0 


78. 4878451 


2.1 


82. 2260091 


2.2 


85. 6243634 


2.3 .. 


88. 7272277 


2.4 


91. 5715264 


2.5 


94. 1883108 


2.6 


96. 6038056 


2.7 


98. 8403838 


2.8 


100. 9172131 




156. 829534 


4/3 


39. 2392391 







9.3. Refractivity Tables 

Tables 9.30 through 9.35 have been prepared on a digital computer to 
assist the reader in making an easy determination of the refractivity, N, 
value from pressure, temperature, and relative humidity data. Standard 
pressure values of 700, 850, and 1000 mbar have been used for the tables. 
The tables are at every 10 percent relative humidity and for every degree 
centigrade ranging for —45 to +45 °C. 

Interpolation for the A^ tables is accomplished in the following manner. 
First, round the temperature to the nearest degree, then interpolate 
linearly for the values of pressure, P, and relative humidity, RH, used. 
No distortion is introduced by this method as the wet term, W (as de- 
scribed in chapter 4), is a linear function of RH, and the dry term, D, is 
a linear function of P. 



Example: 



and knowing 



then 



Given P = 980 mbar 
RH = 47 percent 
T = 22 °C 

D = D(P, T) 

N = N(P, T, RH) 
W = W{T, RH), 



Z)(850, 22) = A^(850, 22, 0) = 223.6 A^ units. 
Z)(980, 22) = A^(980, 22, 0) = 223.6 (980/850) 



257.8 N units. 



REFRACTIVITY TABLES 397 

Now interpolating for humidity, 

N{850, 22, 50) = 280.3 N units. 
Since 

A^(850, 22, 50) - Z)(850, 22) = T^(22, 50), 

thus 

TF(22, 50) = 280.3 - 223.6 = 56.7 N units, and 
TF(22, 47) = 56.7 (47/50) = 53.3 A^ units. 

Adding 

T^^(22, 47) to i)(980, 22) yields 

A^(980, 22, 47) = D(980, 22) + IF(22,47) 

= 257.8 + 53.3 = 311.1 N units. 

Substitution directly into (1.20) and use of the Smithsonian Meteor- 
ological Tables [2] for vapor pressure yields the identical result to four 
significant figures. 

A^(980, 22, 47) = 311.1 A^ units. 

Ambient recording thermometers generally read to a tenth of a degree 
centigrade. Rounding the temperature to the nearest whole degree 
introduces an extraneous error into the interpolation process. At 45 °C, 
1000 mbar, and at a relative humidity of 100 percent, this error will have 
a maximum of 7.7 A^ units. However, these conditions are an extreme 
rarity climatologically speaking, and at normal temperature ranges an 
error of more than 2 N units is unlikely. 



398 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

Table 9.30. Refractivity for a pressure of 700 mhar 













Relative humidity 










Temper- 
























ature 





























10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


°C 
-45 


238.2 


238.3 


238.4 


238.5 


238.6 


238.6 


238.7 


238.8 


238.9 


239.0 


239.0 


-44 


237.2 


237.3 


237.4 


237.5 


237.6 


237.6 


237.7 


237.8 


237 9 


238.0 


238.1 


-43 


236.2 


236.3 


236.4 


236.5 


236.6 


236.7 


236.8 


236.9 


237.0 


237.1 


237.1 


-42 


235.2 


235.3 


235.4 


235.5 


235.6 


235.7 


235.8 


235.9 


236.0 


236.1 


236.2 


-41 


234.1 


234.3 


234.4 


234.5 


234.6 


234.7 


234.8 


235.0 


235.1 


235.2 


235.3 


-40 


233.1 


233.3 


233.4 


233.5 


233.7 


233.8 


233.9 


234.0 


234.2 


234.3 


234.4 


-39 


232.1 


232.3 


232.4 


232.6 


232.7 


232.9 


233.0 


233.1 


233.3 


233.4 


233.4 


-38 


231.1 


231.3 


231.5 


231.6 


231.8 


231.9 


232.1 


232.2 


232.4 


232.6 


232.7 


-37 


230.2 


230.3 


230.5 


230.7 


230.9 


231.0 


231.2 


231.4 


231.5 


231.7 


231.9 


-36 


229.2 


229.4 


229.6 


229.8 


230.0 


230.1 


230.3 


230.5 


230.7 


230.9 


231.1 


-35 


228.2 


228.4 


228.6 


228.9 


229.1 


229.3 


229.5 


229.7 


229.9 


230.1 


230.3 


-34 


227.3 


227.5 


227.7 


228.0 


228.2 


228.4 


228.6 


228.9 


229.1 


229.3 


229 5 


-33 


226.3 


226.6 


226.8 


227.1 


227.3 


227.6 


227.8 


228.1 


228.3 


228.6 


228.8 


-32 


225. 4 


225.7 


225.9 


226.2 


226.5 


226.7 


227.0 


227.3 


227.6 


227.8 


228.1 


-31 


224.5 


224.8 


225.1 


225.3 


225.6 


225.9 


226.2 


226.5 


226.8 


227.1 


227.4 


-30 


223.5 


223.9 


224.2 


224.5 


224.8 


225.1 


225.5 


225.8 


226.1 


226.4 


226.8 


-29 


222.6 


223.0 


223.3 


223.7 


224.0 


224.4 


224.7 


225.1 


225.4 


225.8 


226.1 


-28 


221.7 


222.1 


222.5 


222.9 


223.2 


223.6 


224.0 


224.4 


224.8 


225.1 


225.5 


-27 


220.8 


221.2 


221.6 


222.1 


222.5 


222.9 


223.3 


223. 7 


224.1 


224.5 


225.0 


-26 


219.9 


220.4 


220.8 


221.3 


221.7 


222.2 


222.6 


223.1 


223.5 


224.0 


224.4 


-25 


219.0 


219.5 


220.0 


220.5 


221.0 


221.5 


222.0 


222.5 


222.9 


223.4 


223.9 


-24 


218.2 


218.7 


219.2 


219.7 


220.3 


220.8 


221.3 


221.9 


222.4 


222.9 


223.5 


-23 


217.3 


217.9 


218.4 


219.0 


219.6 


220.2 


220.7 


221.3 


221.9 


222.5 


223.0 


-22 


216.4 


217.0 


217.7 


218.3 


218.9 


219.5 


220.2 


220.8 


221.4 


222.0 


222.7 


-21 


215.6 


216.2 


216.9 


217.6 


218.3 


218.9 


219.6 


220.3 


221.0 


221.6 


222.3 


-20 


214.7 


215.4 


216.2 


216.9 


217.6 


218.4 


219.1 


219.8 


220.6 


221.3 


222.0 


-19 


213.9 


214.6 


215.4 


216.2 


217.0 


217.8 


218.6 


219.4 


220.2 


221.0 


221.8 


-18 


213.0 


213.9 


214.7 


215.6 


216.4 


217.3 


218.1 


219.0 


219.9 


220.7 


221.6 


-17 


212.2 


213.1 


214.0 


215.0 


215.9 


216.8 


217.7 


218.6 


219.6 


220.5 


221.4 


-16 


211.4 


212.4 


213.4 


214.3 


215.3 


216.3 


217.3 


218.3 


219.3 


220.3 


221.3 


-15 


210.5 


211.6 


212.7 


213.8 


214.8 


215.9 


217.0 


218.0 


219.1 


220.2 


221.3 


-14 


209.7 


210.9 


212.0 


213.2 


214.3 


215.5 


216.7 


217.8 


219.0 


220.1 


221.3 


-13 


208.9 


210.2 


211.4 


212.7 


213.9 


215.1 


216.4 


217.6 


218.9 


220.1 


221.4 


-12 


208.1 


209.5 


210.8 


212.1 


213.5 


214.8 


216.1 


217.5 


218.8 


220.2 


221.5 


-U 


207.3 


208.8 


210.2 


211.6 


213.1 


214.5 


216.0 


217.4 


218.8 


220.3 


221.7 


-10 


206.5 


208.1 


209.6 


211.2 


212.7 


214.3 


215.8 


217.4 


218.9 


220.4 


222.0 


-9 


205.8 


207.4 


209.1 


210.7 


212.4 


214.1 


215.7 


217.4 


219.0 


220.7 


222.3 


-8 


205.0 


206.8 


208.5 


210.3 


212.1 


213.9 


215.7 


217.4 


219.2 


221.0 


222.8 


-7 


204.2 


206.1 


208.0 


29.9 


211.8 


213.8 


215.7 


217.6 


219.5 


221.4 


223.3 


-6 


203.4 


205.5 


207.5 


29.6 


211.6 


213.7 


215.7 


217.8 


219.8 


221.9 


223.9 


-5 


202.7 


204.9 


207.1 


209.3 


211.4 


213.6 


215.8 


218.0 


220.2 


222.4 


224.6 


-4 


201.9 


204.3 


206.6 


209.0 


211.3 


213.7 


216.0 


218.3 


220.7 


223.0 


225.4 


-3 


201.2 


203.7 


206.2 


208.7 


211.2 


213.7 


216.2 


218.7 


221.2 


223.8 


226.3 


-2 


200.4 


203.1 


205.8 


208.5 


211.2 


213.8 


216.5 


219.2 


221.9 


224.6 


227.3 


-1 


199.7 


202.6 


205.4 


208.3 


211.2 


214.0 


216.9 


219.8 


222.6 


225.5 


228.4 





199.0 


202.0 


205,1 


208.2 


211.2 


214.3 


217.3 


220.4 


223.4 


226.5 


229.6 



REFRACTIVITY TABLES 
Table 9.31. RefractivUy for a pressure of 700 nibar. 



399 













Relative humidity 










Temper- 
























ature 





























10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


° C 



199.0 


202.0 


205.1 


208.2 


211.2 


214.3 


217.3 


220.4 


223.4 


226.5 


229.0 


1 


198.2 


201.5 


204.8 


208.0 


211.3 


214.6 


217.8 


221.1 


224.4 


227.6 


230.9 


2 


197.5 


201.0 


204.5 


208.0 


211.5 


214.9 


218.4 


221.9 


225.4 


228.9 


232.3 


3 


196.8 


200.5 


204.2 


207.9 


211.7 


215.4 


219.1 


222.8 


226.5 


230.2 


233.9 


4 


196.1 


200.1 


204.0 


208.0 


211.9 


215.9 


219.8 


223.8 


227.7 


231.7 


235,6 


5 


195.4 


199.6 


203.8 


208.0 


212.2 


216.5 


220.7 


224.9 


229.1 


233.3 


237,5 


6 


194.7 


199.2 


203.7 


208.1 


212.6 


217.1 


221.6 


226.1 


230.5 


235.0 


239,5 


7 


194.0 


198.8 


203.5 


208.3 


213.1 


217.8 


222.6 


227.4 


232.1 


236.9 


241,7 


8 


193.3 


198.4 


203.4 


208.5 


213.6 


218.7 


223.7 


228.8 


233.9 


238.9 


244,0 


9 


192.6 


198.0 


203.4 


208.8 


214.2 


219.6 


224.9 


230.3 


235.7 


241.1 


246,5 


10 


191.9 


197.7 


203.4 


209.1 


214.8 


220.5 


226.3 


232.0 


237.7 


243.4 


249,1 


11 


191.3 


197.3 


203.4 


209.5 


215.6 


221.6 


227.7 


233.8 


239.8 


245.9 


252,0 


12 


190.6 


197.0 


203.5 


209.9 


216.4 


222.8 


229.2 


235.7 


242.1 


248.6 


255,0 


13 


189.9 


196.8 


203.6 


210.4 


217.3 


224.1 


230.9 


237.7 


244.6 


251.4 


258,2 


14 


189.3 


196.5 


203.7 


211.0 


218.2 


225.5 


232.7 


239.9 


247.2 


254.4 


261,7 


15 


188.6 


196.3 


204.0 


211.6 


219.3 


227.0 


234.6 


242.3 


250.0 


257.6 


265,3 


16 


188.0 


196.1 


204.2 


212.3 


220.4 


228.6 


236.7 


244.8 


252.9 


261.1 


269,2 


17 


187.3 


195.9 


204.5 


213.1 


221.7 


230.3 


238.9 


247.5 


256.1 


264.7 


273,3 


18 


186.7 


195.8 


204.9 


213.9 


223.0 


232.1 


241.2 


250.3 


259.4 


268.5 


277,6 


19 


186.0 


195.6 


205.3 


214.9 


224.5 


234.1 


243.7 


253.3 


262.9 


272.6 


282,2 


20 


185.4 


195.6 


205.7 


215.9 


226.0 


236.2 


246.4 


256.5 


266.7 


276.9 


287.0 


20 


184.8 


195.5 


206.2 


217.0 


227.7 


238.4 


249.2 


259.9 


270.6 


281.4 


292,1 


22 


184.1 


195.5 


206.8 


218.1 


229. 5 


240.8 


252.2 


263.5 


274.8 


286.2 


297,5 


23 


183.5 


195.5 


207.4 


219.4 


231.4 


243.3 


255.3 


267.3 


279.2 


291.2 


303,2 


24 


182.9 


195.5 


208.1 


220.8 


233.4 


246.0 


258.6 


271.3 


283.9 


296.5 


309,1 


25 


182.3 


195.6 


208.9 


222.2 


235.5 


248.8 


262.2 


275.5 


288.8 


302.1 


315.4 


26 


181.7 


195.7 


209.7 


223.8 


237.8 


251.8 


265.9 


279.9 


293.9 


308.0 


322.0 


27 


181.1 


195.9 


210.6 


225.4 


240.2 


256. 


269.8 


284.6 


299.3 


314.1 


328,9 


28 


180.5 


196.0 


211.6 


227.2 


242.7 


258.3 


273.9 


289.5 


305.0 


320,6 


336,2 


29 


179.9 


196.3 


212.7 


229.0 


245.4 


261.8 


278.2 


294.6 


311.0 


327.4 


343,8 


30 


179.3 


196.5 


213.8 


231.0 


248.3 


265.5 


282.8 


300.0 


317.3 


334.5 


351,8 


31 


178.7 


196.8 


215.0 


233.1 


251.3 


269.4 


287.6 


305.7 


323.8 


342.0 


360,1 


32 


178.1 


197.2 


216.3 


235.3 


254.4 


273.5 


292.6 


311.7 


330.7 


349.8 


368.9 


33 


177.5 


197.6 


217.6 


237.7 


257.7 


277.8 


297.8 


317.9 


337.9 


258.0 


378.1 


34 


176.9 


198.0 


219.1 


240.1 


261.2 


282.3 


303.4 


324.4 


345. 5 


366.6 


387.6 


35 


176.4 


198.5 


220.6 


242.7 


264.9 


287.0 


309.1 


331.3 


353.4 


375.5 


397.6 


36 


175.8 


199.0 


222.3 


245.5 


268.7 


291.9 


315.2 


338.4 


361.6 


384.9 


408.1 


37 


175.2 


199.6 


224.0 


248.4 


272.7 


297.1 


321.5 


345.9 


370.2 


394.6 


419.0 


38 


174.7 


200,2 


225.8 


251.4 


277.0 


302.5 


328.1 


353.7 


379.2 


404.8 


430.4 


39 


174.1 


200.9 


227.7 


254.5 


281.4 


308.2 


335.0 


361.8 


388.6 


415,4 


442,3 


40 


173.5 


201.7 


229.8 


257.9 


286.0 


314.1 


342.2 


370.3 


398.4 


426,5 


454.6 


41 


173.0 


202.4 


231.9 


261.4 


290.8 


320.3 


349.7 


379.2 


408.6 


438.1 


467,5 


42 


172.4 


203.3 


234.1 


265.0 


295.9 


326.7 


357.6 


388.4 


419.3 


450.1 


481,0 


43 


171.9 


204.2 


236.5 


268.8 


301.1 


333.4 


365.7 


398.0 


430.3 


462.6 


494,9 


44 


171.4 


205.2 


239.0 


272.8 


306.6 


340.4 


374.2 


408.1 


441.9 


475.7 


509.5 


45 


170.8 


206.2 


241.6 


277.0 


312.3 


347.7 


383.1 


418.5 


453.9 


489.2 


524,6 



400 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

Table 9.32. Refractivity for a pressure of 850 mbar 



Temper- 


Relative humidity 


ature 





























10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


° C 

-45 


289.3 


289.4 


289.5 


289.5 


289.6 


289.7 


289.8 


289.9 


289.9 


290.0 


290.1 


-44 


288.0 


288.1 


288.2 


288.3 


288.4 


288.5 


288.6 


288.7 


288.7 


288.8 


288.9 


-43 


286.8 


286.9 


287.0 


287.1 


287.2 


287.3 


287.4 


287.5 


287.6 


287.7 


287.8 


-42 


285.5 


285.6 


285.8 


285.9 


286.0 


286.1 


286.2 


286.3 


286.4 


286.5 


286.6 


-41 


284.3 


284.4 


284.5 


284.7 


284.8 


284.9 


285.0 


285.1 


285.3 


285.4 


285.5 


-40 


283.1 


283.2 


283.4 


283.5 


283.6 


283.7 


283.9 


284.0 


284.1 


284.3 


284.4 


-39 


281.9 


282.0 


282.2 


282.3 


282.5 


282.6 


282.7 


282.9 


293.0 


283.2 


283.3 


-38 


280.7 


280.8 


281.0 


281.2 


281.3 


281.5 


281.6 


281.8 


281.9 


282.1 


282.3 


-37 


279.5 


279.7 


279.8 


280.0 


280.2 


280.4 


280.5 


280.7 


280.9 


281.0 


281.2 


-36 


278.3 


278.5 


278.7 


278.9 


279.1 


279.3 


279.4 


279.6 


279.8 


280.0 


280.2 


-35 


277.1 


277.3 


277.6 


277.8 


278.0 


278.2 


278.4 


278.6 


278.8 


279.0 


279.2 


-34 


276.0 


276.2 


276.4 


276.7 


276.9 


277.1 


277.3 


277.6 


277.8 


278.0 


278.2 


-33 


274.8 


275.1 


275.3 


275.6 


275.8 


276.1 


276.3 


276.6 


276.8 


277.1 


277.3 


-32 


273.7 


274.0 


274.2 


274.5 


274.8 


275.0 


275.3 


275.6 


275.9 


276.1 


276.4 


-31 


272.6 


272.9 


273.2 


273.4 


273.7 


274.0 


274.3 


274.6 


274.9 


275.2 


275.5 


-30 


271.4 


271.8 


272.1 


272.4 


272.7 


273.0 


273.4 


273.7 


274.0 


274.3 


274.7 


-29 


270.3 


270.7 


271.0 


271.4 


271.7 


272.1 


272.4 


272.8 


273.1 


273.5 


273.8 


-28 


269.2 


269.6 


270.0 


270.4 


270.8 


271.1 


271.5 


271.9 


272.3 


272.7 


273.0 


-27 


268.1 


268.5 


269. 


269.4 


269.8 


270.2 


270.6 


271.0 


271.4 


271.9 


272.3 


-26 


267.0 


267.5 


267.9 


268.4 


268.8 


269.3 


269.8 


270.2 


270.7 


271.1 


271.6 


-25 


266.0 


266.5 


266.9 


267.4 


267.9 


268.4 


268.9 


269.4 


269.9 


270.4 


270.9 


-24 


264.9 


265.4 


266.0 


266.5 


267.0 


267.6 


268.1 


268.6 


269.2 


269.7 


270.2 


-23 


263.8 


264.4 


265.0 


265.6 


266. 1 


266.7 


267.3 


267.9 


268.4 


269.0 


269.6 


-22 


262.8 


263.4 


264.0 


264.7 


265.3 


265.9 


266.5 


267.2 


267.8 


268.4 


269.0 


-21 


261.7 


262.4 


263.1 


263.8 


264.4 


265.1 


265.8 


266.5 


267.2 


267.8 


268.5 


-20 


260.7 


261.4 


262.2 


262.9 


263.6 


264.4 


265.1 


265.8 


266.6 


267.3 


268.0 


-19 


259.7 


260.5 


261.3 


262.1 


262.8 


263.6 


264.4 


265.2 


266.0 


266.8 


267.6 


-18 


258.7 


259.5 


260.4 


261.2 


262.1 


262.9 


236.8 


264.6 


265.5 


266.4 


267.2 


-17 


257.7 


258.6 


259.5 


260.4 


261.3 


262.3 


263.2 


264.1 


265.0 


266.0 


266.9 


-16 


256.7 


257.6 


258.6 


259.6 


260.6 


261.6 


262.6 


263.6 


264.6 


265.6 


266.6 


-15 


255.7 


256.7 


257.8 


258.9 


259.9 


261.0 


262.1 


263.2 


264.2 


265.3 


266.4 


-14 


254.7 


255.8 


257.0 


258.1 


259.3 


260.4 


261.6 


262.8 


263.9 


265.1 


266.2 


-13 


253.7 


2.54.9 


256.2 


257.4 


258.7 


259.9 


261.2 


262.4 


263.6 


264.9 


266.1 


-12 


252.7 


254.1 


255.4 


256.7 


258.1 


259.4 


260.7 


262.1 


263.4 


264.8 


266.1 


-11 


251.8 


253.2 


254.6 


256.1 


257.5 


258.9 


260.4 


261.8 


263.3 


264.7 


266.1 


-10 


250.8 


252.3 


253.9 


255.4 


257.0 


258.5 


260.1 


261.6 


263.2 


264.7 


266.2 


-9 


249.8 


251.5 


253.2 


254.8 


256.5 


258.1 


159.8 


261.5 


263.1 


264.8 


266.4 


-8 


248.9 


250.7 


252.5 


254.2 


256.0 


257.8 


259.6 


261.4 


263.1 


264.9 


266.7 


-7 


248.0 


249.9 


251.8 


253.7 


255.6 


257.5 


259.4 


261.3 


263.2 


265.1 


267.1 


-6 


247.0 


249.1 


251.1 


253.2 


255.2 


257.3 


259.3 


261.4 


263.4 


265.4 


267.5 


-5 


246.1 


248.3 


250.5 


252.7 


254.9 


257.1 


259.3 


261.5 


263.6 


265.8 


268.0 


-4 


245.2 


247.5 


249.9 


252.2 


254.6 


256.9 


259.3 


261.6 


264.0 


266.3 


268.6 


-3 


244.3 


246.8 


249.3 


251.8 


254.3 


256.8 


259.3 


261.9 


264.4 


266.9 


269.4 


-2 


243.4 


246.1 


248.8 


251.4 


254.1 


256.8 


259.5 


262.2 


264.8 


267.5 


270.2 


-1 


242.5 


245.4 


248.2 


251.1 


254.0 


256.8 


259.7 


262.6 


265.4 


268.3 


271.1 





241.6 


244.7 


247.7 


250.8 


253.8 


256.9 


260.0 


263.0 


266.1 


269.1 


272.2 



REFRACTIVITY TABLES 
Table 9.33. Re fr activity for a pressure of 850 mbar 



401 











Relative humidity 










Temper- 
























ature 





























10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


° C 




241.6 


244.7 


247.7 


250.8 


253.8 


256.9 


260.0 


263.0 


266.1 


269.1 


272.2 


1 


240.7 


244.0 


247.3 


250.5 


253.8 


257.1 


260.3 


263.6 


266.8 


270.1 


273.4 


2 


239.9 


243.3 


246.8 


250.3 


253.8 


257.3 


260.7 


264.2 


267.7 


271.2 


274.7 


3 


239.0 


242.7 


246.4 


250.1 


253.8 


257.5 


261.3 


265.0 


268.7 


272.4 


276.1 


4 


238.1 


242.1 


246.0 


250.0 


253.9 


257.9 


261.9 


265.8 


269.8 


273.7 


277.7 


5 


237.3 


241.5 


245.7 


249.9 


254.1 


258.3 


262.5 


266.7 


271.0 


275.2 


279.4 


6 


236.4 


240.9 


245.4 


249.9 


254.3 


258.8 


363.3 


267.8 


272.3 


276.8 


281.2 


7 


235.6 


240.3 


245.1 


249.9 


254.6 


259.4 


264.2 


268.9 


273.7 


278.5 


283.2 


8 


234.7 


239.8 


244.9 


249.9 


255.0 


260.1 


265.1 


270.2 


275.3 


280.3 


285.4 


9 


233.9 


239.3 


244.7 


250.1 


255.4 


260.8 


266.2 


271.6 


277.0 


282.4 


287.8 


10 


233.1 


238.8 


244.5 


250.2 


256.0 


261.7 


267.4 


273.1 


278.8 


284.5 


280.3 


11 


232.3 


238.3 


244.4 


250.5 


256.5 


262.6 


268.7 


274.8 


280.8 


286.9 


293.0 


12 


231.4 


237.9 


244.3 


250.8 


257.2 


263.6 


270.1 


276. 5 


283.0 


289.4 


295.9 


13 


230.6 


237.5 


244.3 


251.1 


258.0 


264.8 


271.6 


278.4 


285.3 


292.1 


298.9 


14 


229.8 


237.1 


244.3 


251.5 


258.8 


266.0 


283.3 


280.5 


287.7 


295.0 


302.2 


15 


229.0 


236.7 


244.4 


252.0 


259.7 


267.4 


275.0 


282.7 


290.4 


298.1 


305.7 


16 


228.2 


236.4 


244.5 


252.6 


260.7 


268.8 


277.0 


285.1 


293.2 


301.3 


309.5 


17 


227.4 


236.0 


244.6 


253.2 


261.8 


270.4 


279.0 


287.6 


296.2 


304.8 


313.4 


18 


226.7 


235.8 


244.9 


253.9 


263.0 


272.1 


281.2 


290.3 


299.4 


308.5 


317.6 


19 


225.9 


235.5 


245.1 


254.7 


264.4 


274.0 


283.6 


293.2 


302.8 


212.4 


322.0 


20 


225.1 


235.3 


245.4 


255.6 


265.8 


275.9 


286.1 


296.3 


306.4 


316.6 


326.7 


21 


224.4 


235.1 


245.8 


256.6 


267.3 


278.0 


288.8 


299.5 


310.2 


321.0 


331.7 


22 


223.6 


234.9 


246.3 


257.9 


268.9 


280.3 


291.6 


302.9 


314.3 


325.6 


337.0 


23 


222.8 


234.8 


246.8 


258.7 


270.7 


272.7 


294.6 


306.6 


318.6 


330.5 


242.5 


24 


222.1 


234.7 


247.3 


260.0 


272.6 


285.2 


297.8 


310.4 


323.1 


335.7 


348.3 


25 


221.3 


234.7 


248.0 


261.3 


274.6 


287.9 


301.2 


314.5 


327.8 


341.1 


354.5 


26 


220.6 


234.6 


248.7 


262.7 


276.7 


280.9 


304.8 


318.8 


332.9 


346.9 


360.9 


27 


219.9 


234.7 


249.4 


264.2 


279.0 


293.8 


308.6 


323.4 


338.1 


352.9 


367.7 


28 


219.1 


234.7 


250.3 


265.8 


281.4 


297.0 


312.6 


328.1 


343.7 


359.3 


374.8 


28 


218.4 


234.8 


251.2 


267.6 


284.0 


300.4 


316.8 


333.2 


349.6 


365.9 


382.3 


30 


217.7 


234.9 


252.2 


269.4 


286.7 


303.9 


321.2 


338.4 


355.7 


372.9 


390.2 


31 


217.0 


235.1 


253.3 


271.4 


289.6 


307.7 


325.8 


344.0 


362.1 


380.3 


398.4 


32 


216.3 


235.3 


254.4 


273.5 


292.6 


311.7 


330.7 


249.8 


368.9 


388.0 


407.1 


33 


215.6 


235.6 


255.7 


275.7 


295.8 


315.8 


335.9 


355.9 


376.0 


296.0 


416.1 


34 


214.9 


235.9 


257.0 


278.1 


299.1 


320.2 


341.2 


362.3 


383.4 


404.5 


425.5 


35 


214.2 


236.3 


258.4 


280.5 


302.7 


324.8 


346.9 


369.0 


391.2 


413.3 


435.4 


36 


213.5 


236.7 


259.9 


283.2 


305.4 


329.6 


352.8 


376.1 


399.3 


422.5 


445.8 


37 


212.8 


237.2 


261.5 


285.9 


310.3 


334.7 


359.0 


383.4 


407.8 


432.2 


456.5 


38 


212.1 


237.7 


263.2 


288.8 


314.4 


340.0 


365.5 


391.1 


416.7 


422.2 


467.8 


39 


211.4 


238.2 


265.0 


291.9 


318.7 


345.5 


372.2 


399.1 


425.9 


452.7 


479.6 


40 


210.7 


238.8 


267.0 


295.1 


323.2 


351.3 


379.4 


407.5 


435.6 


463.7 


491.8 


41 


210.1 


239.5 


269.0 


298.4 


327. 9 


357.3 


386.8 


416.2 


445.7 


475.1 


504.6 


42 


209.4 


240.2 


271.1 


302.0 


332.8 


363.7 


394.5 


425.4 


456.2 


487.1 


517.9 


43 


208.7 


241.0 


283.3 


305.6 


338.0 


370.3 


402.6 


434.9 


467.2 


499.5 


531.8 


44 


208.1 


241.9 


275.7 


309.5 


343.3 


377.1 


411.0 


444.8 


478.6 


512.4 


546.2 


45 


207.4 


242.8 


278.2 


313.6 


348.9 


384.3 


419.7 


455.1 


490.5 


525.8 


561.2 



402 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

Table 9.34. Refractivity for a pressure of 1,000 mbar 













Relative humidity 










Temper- 
























ature 





























10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


° C 

-45 


340.4 


340.4 


340.5 


340.6 


340.7 


340.8 


340.8 


340.9 


341.0 


241.1 


341.1 


-44 


338.9 


339.0 


339.0 


339.1 


339.2 


339.2 


339.4 


339.5 


339.6 


339.7 


339.7 


-43 


337.4 


337.5 


337.6 


337.7 


337.8 


337.9 


338.0 


338.1 


338.2 


338.3 


338.4 


-42 


335.9 


336.0 


336.1 


336.3 


336.4 


336.5 


336.6 


336.7 


336.8 


336.9 


337.0 


-41 


334.5 


334.6 


334.7 


334.8 


335.0 


335.1 


335.2 


335.3 


335.4 


335.5 


335.7 


-40 


333.0 


333.2 


333.3 


333.4 


333.6 


333.7 


333.8 


334.0 


334.1 


334.2 


334.3 


-39 


331.6 


331.8 


331.9 


332.1 


332.2 


332.3 


332.5 


332.6 


332.8 


332.9 


333.1 


-38 


330.2 


330.4 


330.5 


330.7 


330.8 


331.0 


331.2 


331.3 


331.5 


331.6 


331.8 


-37 


328.8 


329.0 


329.2 


329.3 


329.5 


329.7 


329.8 


330.0 


330.2 


330.4 


330.5 


-36 


327.4 


327.6 


327.8 


328.0 


328.2 


328.4 


328.6 


328.7 


328.9 


329.1 


329.3 


-35 


326.1 


326.3 


326.5 


326.7 


326.9 


327.1 


327.3 


327.5 


327.7 


327.9 


328.1 


-34 


324.7 


324.9 


325. 1 


325.4 


325. 6 


325.8 


326.0 


326.3 


326.5 


326.7 


326.9 


-33 


323.3 


323.6 


323.8 


234.1 


324.3 


324.6 


324.8 


235.1 


325.3 


325.6 


325.8 


-32 


322.0 


322.3 


322.6 


322.8 


323.1 


323.3 


323.6 


323.9 


324.2 


324.4 


324.7 


-31 


320.7 


321.0 


321.3 


321.5 


321.8 


322.1 


322.4 


322.7 


323.0 


323.3 


323.6 


-30 


319.3 


319.7 


320.0 


320.3 


320.6 


320.9 


321.3 


321.6 


321.9 


322.2 


322.6 


-29 


318.0 


318.4 


318.7 


319.1 


319.4 


319.8 


320.1 


320.5 


320.8 


321.2 


321.5 


-28 


316.7 


317.1 


317.5 


317.9 


318.3 


318.6 


319.0 


319.4 


319.8 


320.2 


320.5 


-27 


315.4 


315.9 


316.3 


316.7 


317.1 


317.5 


317.9 


318.4 


318.8 


319.2 


319.6 


-26 


314.2 


314.6 


315.1 


315.5 


316.0 


316.4 


316.9 


317.3 


317.8 


318.2 


318.7 


-25 


312.9 


313.4 


313.9 


314.4 


314.9 


315.4 


315.8 


316.3 


316.8 


316.3 


317.8 


-24 


311.6 


312.2 


312.7 


313.2 


313.8 


314.3 


314.8 


315.4 


315.9 


316.4 


317.0 


-23 


310.4 


311.0 


311.6 


312.1 


312.7 


313.3 


313.9 


314.4 


315.0 


415.6 


316.2 


-22 


309.2 


309.8 


310.4 


311.0 


311.7 


312.9 


313.5 


314.2 


314.2 


314.8 


315.4 


-21 


307.9 


308.6 


309.3 


310.0 


310.6 


311.3 


312.0 


312.7 


313.3 


314.0 


314.7 


-20 


306.7 


307.6 


308.2 


308.9 


309.6 


310.4 


311.1 


311.8 


312.6 


313.3 


314.0 


-19 


305.5 


306.3 


307.1 


307.9 


308.7 


309.5 


310.3 


311.0 


311.8 


312.6 


313.4 


-18 


304.3 


305.2 


306.0 


306.9 


307.7 


308.6 


309.4 


310.3 


311.1 


312.0 


312.9 


-17 


303.1 


304.0 


305.0 


305.9 


206.8 


307.7 


308.7 


309.6 


310.5 


311.4 


312.3 


-16 


301.9 


302.9 


303.9 


304.9 


305.9 


306.9 


307.9 


308.9 


309.9 


310.9 


311.9 


-15 


300.8 


301.8 


312.9 


403.0 


305.1 


306.1 


307.2 


308.3 


309.4 


310.4 


311.5 


-14 


299.6 


300.8 


301.9 


303.1 


304.2 


305.4 


306.5 


307.7 


308.9 


310.0 


311.2 


-13 


298.5 


299.7 


300.9 


302.2 


303.4 


304.7 


305.9 


307.2 


308.4 


309.7 


310.9 


-12 


297.3 


298.7 


300.0 


301.3 


302.7 


304.0 


305.3 


306.7 


308.0 


309.4 


310.7 


-11 


296.2 


297.6 


299.1 


300.5 


301.9 


303.4 


304.8 


306.2 


307.7 


309.1 


310.6 


-10 


295.1 


296.6 


298.1 


299.7 


301.2 


302.8 


304.3 


305.9 


307.4 


309.0 


310.5 


-9 


203.9 


295.6 


297.3 


298.9 


300.6 


302.2 


303.9 


305.5 


308.2 


308.9 


310.5 


-8 


292.8 


294.6 


296.4 


298.2 


299.9 


301.7 


303.5 


305.3 


307.1 


308.8 


310.6 


-7 


291.7 


293.6 


296.5 


297.5 


299.4 


301.3 


303.2 


305.1 


307.0 


308.9 


310.8 


-6 


290.6 


292.7 


294.7 


296.8 


298.8 


300.9 


302.9 


305.0 


307.0 


309.0 


311.1 


-5 


289.6 


291.7 


293.9 


296.1 


298.3 


300.5 


302.7 


304.9 


307.1 


309.3 


311.5 


-4 


288.5 


290.8 


293.2 


295.5 


297.9 


300.2 


302.5 


304.9 


307.2 


309.6 


311.9 


-3 


287.4 


289.9 


292.4 


294.9 


297.4 


299.9 


302.5 


305.0 


307.5 


310.0 


312.5 


-2 


296.3 


289.0 


291.7 


294.4 


297.1 


299.8 


302.4 


305.1 


307.8 


310.5 


313.2 


-1 


285.3 


288.2 


291.0 


293.9 


296.8 


299.6 


302.5 


305.3 


308.2 


311.1 


313.9 





284.2 


287.3 


290.4 


293.4 


296.5 


299.5 


302.6 


305.7 


308.7 


311.8 


314.8 



REFRACTIVITY TABLES 
Table 9.35. Refractivity for a pressure of 1,000 mhar 



403 













Relative humidity 










Temper- 
























ature 





























10 


20 


30 


40 


50 


60 


70 


80 


90 


100 


° C 




284.2 


287.3 


290.4 


293.4 


296.5 


299.5 


302.6 


305.7 


308.7 


311.8 


314.8 


1 


283.2 


286.5 


289.7 


293.0 


296.3 


299.6 


302.8 


306.1 


309.3 


312.6 


315.9 


2 


282.2 


285.7 


289.1 


292.6 


269.1 


299.6 


303.1 


306.6 


310.0 


313.5 


317.0 


3 


281.2 


284.9 


288.6 


292.3 


296.0 


299.7 


303.4 


307.1 


310.9 


314.6 


318.3 


4 


280.1 


284.1 


288.1 


292.0 


296.0 


299.9 


303.9 


307.8 


311.8 


315.7 


319.7 


5 


279.1 


283.3 


287.6 


291.8 


296.0 


300.2 


304.4 


308.6 


213.8 


317.0 


321.2 


6 


278.1 


282.6 


287.1 


291.6 


296.1 


300.5 


305.0 


309.5 


314.0 


318.5 


323.0 


7 


277.1 


281.9 


286.7 


291.4 


296.2 


301.0 


305.7 


310.5 


315.3 


320.0 


324.8 


8 


276.2 


281.2 


286.3 


291.4 


296.7 


301.5 


306.6 


311.6 


316.7 


321.8 


326.8 


9 


275.2 


280.6 


285.9 


291.3 


296.7 


302.1 


307.5 


213.9 


318.3 


323.6 


329.0 


10 


274.2 


279.9 


285.6 


291.4 


297.1 


302.8 


308.5 


314.2 


320.0 


325.7 


331.4 


11 


273.2 


279.3 


285.4 


291.5 


297.5 


303.6 


309.7 


315.7 


321.8 


327.9 


334.0 


12 


272.3 


278.7 


285.2 


291.6 


298.0 


304.5 


310.9 


317.4 


323.8 


330.3 


336.7 


13 


271.3 


278.2 


285.0 


291.8 


298.7 


305.5 


312.3 


319.1 


326.0 


332.8 


339.6 


14 


270.4 


277.6 


284.9 


292.1 


299.3 


306.6 


313.8 


321.1 


328.3 


335.5 


342.8 


15 


269.4 


277.1 


274.7 


292. 5 


300.1 


307.8 


315.5 


323.1 


330.8 


338.5 


346.1 


16 


268.5 


276.6 


284.8 


292.9 


301.0 


309.1 


317.2 


325.4 


333.5 


341.6 


349.7 


17 


267.6 


276.2 


284.8 


293.4 


302.0 


310.6 


319.2 


327. 8 


336.4 


344.9 


353.5 


18 


266.7 


275.8 


284.9 


293.9 


303.0 


312.1 


321.2 


330.3 


339.4 


348.5 


357.6 


19 


265.8 


275.4 


285.0 


294.6 


304.2 


313.8 


323.4 


333.1 


342.7 


352.3 


361.9 


20 


264.8 


275.0 


285.2 


295.3 


305.5 


315.8 


325.8 


336.0 


346.1 


356.3 


366.5 


21 


363.9 


274.7 


285.4 


296.2 


306.9 


317.6 


328.4 


339.1 


349.8 


360.6 


371.3 


22 


263.1 


274.4 


285.7 


297.1 


308.4 


319.7 


331.1 


342.4 


353.7 


365.1 


376.4 


23 


262.2 


274.1 


286.1 


298.1 


310.0 


322.0 


334.0 


345.9 


357.9 


369.8 


381.8 


24 


261.3 


273.9 


286.5 


299.1 


311.8 


324.4 


337.0 


349.6 


362.3 


374.9 


387.5 


25 


260.4 


273.7 


287.0 


300.3 


313.6 


327.0 


340.3 


353.6 


366.9 


380.2 


393.5 


26 


259.5 


273.6 


287.6 


301.6 


315.7 


329.7 


343.7 


357.8 


371.8 


385.8 


399.8 


27 


258.7 


273.5 


288.2 


303.0 


317.8 


332.6 


347.4 


362. 2 


276.9 


391.7 


306.5 


28 


257.8 


273.4 


288.9 


304.5 


320.1 


335.8 


351.2 


366.8 


382.4 


397.9 


413.5 


29 


257.0 


273.3 


289.7 


306.1 


322.5 


338.9 


355.3 


371.7 


388.1 


404.5 


420.9 


30 


256.1 


273.4 


290.6 


307.9 


325.1 


342.4 


359.6 


276.9 


384.1 


411.4 


428.6 


31 


255.3 


273.4 


291.6 


309.7 


327.8 


346.0 


364.1 


382.3 


400.4 


418.6 


436.7 


32 


254.4 


273.5 


292.6 


311.7 


330.7 


349.8 


368.9 


388.0 


407.1 


426.1 


445.2 


33 


253.6 


273.6 


293.7 


313.8 


333.8 


363. 9 


373.9 


394.0 


414.0 


434.1 


454.1 


34 


252.8 


273.8 


294.9 


316.0 


337.0 


358.1 


379.2 


444.3 


421.3 


442.4 


463.5 


35 


251.9 


274.1 


296.2 


318.3 


340.5 


362.6 


384.7 


406.8 


429.0 


451.1 


473.2 


36 


251.1 


274.4 


297.6 


320.8 


344.1 


367.3 


390.5 


413.7 


437.0 


460.2 


384.4 


37 


250.3 


274.7 


299.1 


323.5 


347.8 


372.2 


396.6 


421.0 


445.3 


469.7 


494.1 


38 


249.5 


275.1 


300.7 


326.2 


351.8 


377.4 


403.0 


428.5 


454.1 


479.7 


505.2 


39 


248.7 


275.5 


302.3 


329.2 


256.0 


382.8 


409.6 


436.4 


463.2 


490.1 


516.9 


40 


247.9 


276.0 


304.1 


332.2 


360.4 


388.5 


416.6: 


444.7 


472.8 


500.9 


529.0 


41 


247.1 


276.6 


306.0 


335.5 


364.9 


394.4 


423.9 


453. 3 


482.8 


512.2 


541.7 


42 


246.3 


277.2 


308.1 


338.9 


369.8 


400.6 


431.5 


462.3 


493.2 


524.0 


544.9 


43 


245. 6 


277.9 


310.2 


342.5 


374.8 


407.1 


439.4 


471.7 


504.0 


536.3 


568.6 


44 


244.8 


278.6 


312.4 


246.2 


380.0 


413.9 


447.7 


481.5 


515. 3 


549. 1 


582.9 


45 


244.0 


279.4 


314.8 


350.2 


385.5 


420.9 


456.3 


491.7 


527.1 


562.5 


597.8 



404 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 

9.4. Glimatological Data of the Refractive Index 
for the United States 

Charts of 8-year mean A''o values for 0200 and 1400 local time are given 
for the four seasons of the year on figures 9.1 to 9.8. 

Selected stations illustrate the range of seasonal and diurnal variations 
of N s in differing climatic regions across the country. These are plotted 
on figures 9.9 to 9.14. Note the large annual and diurnal ranges of means 
for humid subtropical Washington, D.C. (fig. 9.9), compared with the 
modest variations exhibited by maritime-dominated Tatoosh Island 
(fig. 9.14). The results of these analyses are consistent with those ob- 
tained from climatic studies of the refractive index structure over North 
America, described in section 4 of chapter 4. That is, large seasonal and 
diurnal ranges are exhibited by continental-climate stations (Colorado 
Springs) and small ranges by maritime-climate stations (Tatoosh Island). 

Cumulative distribution curves for A^^, useful in radio ray bending pre- 
dictions, are given in figures 9.15 to 9.20 for the example stations above. 

Figures 9.21 and 9.22, standard deviation of N s, provide a measure of 
the accuracy of the A^o charts of figures 9.1 to 9.8. As has been noted 
in chapter 4, a number of climatic features are apparent on the standard 
deviation maps. The climatic stability of various regions of the country, 
for example, is reflected in these charts. Small standard deviations 
characterize the maritime climate of the west coast. By comparison, the 
strong air mass changes of wintertime synoptic patterns sweeping across 
the southeastern United States are indicated by large standard devia- 
tions in that region (fig. 9.21). 



9.5. Statistical Prediction of Elevation Angle Error 

The elevation angle error, e, as defined in chapter 3, can, like bending, 
be predicted from surface refractivity, N s, by an equation of the form 



e = mNs + i. 

In tables 9.36 to 9.44 for a given height h, the coefficients m and t have 
been determined by a statistical regression performed on the Standard 
CRPL Sample, each having a unique N s- 

This process is analogous to the statistical bending prediction method 
described in chapter 3. 



ELEVATION ANGLE ERROR 



405 




Figure 9.L Mean No, February 0200. 




Figure 9.2. Mean No, February I4OO. 



406 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 




Figure 9.3. Mean No, May 0200. 




Figure 9.4. Mean No, May 14OO. 



ELEVATION ANGLE ERROR 



407 




Figure 9.5. Mean No, August 0200. 




Figure 9.6. Mean No, August I4OO. 



408 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 




Figure 9.7. Mean No, November 0200. 




Figure 9.8. Mean No, November I4OO. 



ELEVATION ANGLE ERROR 



409 





1 1 1 
0200 Lncal Time 
















390 
380 
3T0 
360 
350 

330 
520 
310 
300 
290 
280 




1 1 y - 
1400 Locol Time 




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\ 










---_ 


..'■\ 




/ 


/ 








\ 


\ 


"^S 










/ 


/ 


/-- 




\ 






_ 




y 


/ 




/ 




N. 






^ 










/ 








^, 




\ 












— - 






y 








\ 


N 


— -H 




"^ 




^ 


^,.. 


-' 


/ 


\ 




"-.^ 




"^ 


^ 




■^ 














" 






■~'- 


---. 


^,'' 






/ 




k 


s 


^^ 


— - 


-" 


























V 




• 




y 








\ 




/ 
















































A 


"^ 







SEPT 
Yeor 



MY Ur SfPT 

Month of the Year 





1,' 

February 












































































^ 


^ 














^ 




/ 








\ 


^ 


/ 






\ 






























— 1 


































— 








1 










r****- 


.... 


— 


.... 




-,, 








-^ 











y 






— ■ 


X 


k 





— - 


-"' 
























^ 


^ 










""""-^ 





Max 


August 




















380 

370 












^^ 

















-— 




... 


-., 










,--' 






A 








1 


''■ 


— 


—- ■ 










3tiU 
550 










\^ 












^ 




tS.. 





1 






■V 






y' 




,..- 




340 
330 
520 
310 














'^~> 


,^ 






, 








Min_ 












"--■ 


— - 


' 








■ ' 




K 


v^ 










^ 


~- 


^ 












X 






^ 


y 






300 































Hour of the Day 



0800 1200 e 

Hour of the Doy 



Figure 9.9. Annual and diurnal cycles of Ns /or Washington, D.C. 



410 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



O2OOL0C0I Time 


y 




_ 


^ 

















/ 







,^__ 






\ 








/ 




/ 

/ 


' 




V. 




\ 




\ 


^ 


~~~" 




• 


■' 


/ 


_^ 




\ 




\ 








^ 


/' 


/ 






^i 


V 


\ 


\ 






"" — 






'' 


,' 






''\ 


N 


V 


'^N 


,'' 






/ 


/ 




/ 


S, 






\ 












/ 




/ 


s 


V 


\ 




^ 


— 










/ 






\ 




\ 














/ 






\ 








— 


N 






/ 










\ 


^ 












k 




/ 

































































1400 Locol Time 


/ 


\ 






















/ 








■^ 


\ 








^ 




■^ 












\ 


/ 




/ 








— H 


-^^ 








\ 


/ 


\ 














^ 


— 


X 












/ 




y 








"\ 












/ 






'v^ 




\, 




— 


-'' 








y 


/ 


-H 


^ 




s 


s. 








' 




/ 






— I 


V 




-^ 








t 










\ 














.-^ 




A 






N 




^•-_ 


— - 








f 


/ 








\ 








\ 


y 


\, 


/ 
















"^ 



























JAN KAR MAT JULY SEPT 

Month of the Yeor 



JM NAD HAY 



JULY SEPT 

Month of the Yeor 




Ns 





A 


igust 






























^ 








^ 








'^~ 


IT- 


.._ 








--^ 




^ 






'' 








\ 












X 


.^'" 


^ 


^ 






\' 


\ 








,\ 




^ 


,-.' 


h-***" 






\ N 


\, 


^v^ 





— - 


f 


/ 












> 


^ N 


N, 






/ 




'' 




/ 


■ — 


■^ 


I 


N 
\ 


V 






r 


f 




/ 


/ 






\ 










/ 














\ 






^■^ 


-— ■ 






/ 














\, 






/ 


/ 
















^-^ 




y 







oaoo 1200 

Hour of the Doy 



OeOO 1200 I6< 

Hour of the Doy 



Figure 9.10. Annual and diurnal cycles of Ns /or San Antonio, Tex. 



ELEVATION ANGLE ERROR 



411 





1 1 1 

0200 LdcoI Time 


























y 




s 


















/ 






\ 
















/ 




y 


■-, 




\, 














/ 




/ 




''•. 


\ 








\ 


-- 


y 




'' / 




-.,. 


S: 




k 


/" 


" 


--, 


.... 


.... 


■''y 


/ , 








\^ 






.,.- 


'ZI- 




— 






^ 




s 








' 










y 






\ 






y 


■ 





























































1 1 1 

1400 Local Tra 


j 


\ 
























s 


^ 


















^ 


1 




\ 
















/ 








\ 
















/ 


y 


^~^> 


\ 












/ 






/ 


\ 


\ 




\ 






N 





1 


--' / 






\ 




V 




^ 




-^ 


. 


K> ^ 


/ 


-'-'H ^ 


^— n 


'^ 


— 


■-tx.^ 


' 








\ 


-. ..■'■■■' 






\ 


L '^ 




\ 














^ 


^1 






■^ 


\ 


^ 


/ 





IW) MAY JUUf SEFT 

Month of the Year 



IW) NAT JULY S£PT NW 

Month of the Year 





February 




































































































^ 












_^ 


— ■ 






■ 






.— , 


.... 






__ 




























i::; 


in: 








— 




— . 













..... 










~\ 










^ 














' 
















"~~^ 


-^ 



























































oeoo I2X leoo 

Hour of the Day 





1 
August 














/ 


\ 




Max. 


-^ 










/ 


k 




/ 


\ 


1 












^'^ 




\ 


y 








.1?L. 






















— 

















"~-.. 


— 










-o- 






F 


^ 


^ 








y 






Mm 








■'- 


-, 










-" 




^ 




^ 


s 




'^',^ 


— 


-■■ 






^ 












s 


\ 








y 


/ 
















\ 




/" 


y 































0400 oeoo 1200 K 

Hour of the Day 



Figure 9.11. Annual and diurnal cycles of Ns /or Bismarck, N.Dak. 



412 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 






1 1 1 

1400 Local Time 






























\ 




















/ 






\ 


s 














/ 








\ 












/ 


















— ' 






,,^ 


'' 


/ 










■"^ 




— 


— ■ 


''^ 


^ 





/ 




\ 




—- 




"■ 




1 






/--K 












-*■ 











^-- 


--' 






' ' 









1 


^ 




\ 


-^ 

































MM) Nn JiUr SEPT WV JM 

Month of the Yeor 



SO 
300 
290 
2M 

no 

260 

2S0 

240 h 

230 

220 

210 

200, 





1 

February 


















































































































^ 

















— ~ 

























>]^ 


1 


— 


■— 


'" 


^ 



















*^ 


■'--. 








,^- 


'-- 















-«^ 








y 


' 























































Moi. 






August 












^ 


-— 








■---^ 






y^ 










♦tr 

























-A- 

















_^- 








-<r 






\ 


N 












■^ 








""■"■ 


\ 








— - 


y 


^' 


^^' 








Min,i 








^v^ 








/ 













\ 






■ 


■''' 






^ 


















V 








/ 

























































owo can 1200 

Hour of the Day 



2000 2400 



ono 1200 It 

Hour of the Doy 



Figure 9.12. Annual and diurnal cycles of Ns /or Colorado Springs, Colo. 



ELEVATION ANGLE ERROR 



413 



N^ '"" - 




Month of the Year 



Month of the Year 





February 
























































































































J 


















^ 





















,__. 










^ 









--■, 


;^ 






^ 





— ■ 








._ 










-— 


\ 














1 












"^ 


' 


— 


y 








1 























1 
August 




















Max 


^ 






































-— 


\ 


^ 


— 


































+ (r 







— H 


















A 


__ 






^ 


x.^ 








/ 


— " 








\ 




'^ 


— 


r— 1 


/ 


— 


— ■ 


Z-— 


•"" 




"--. 


•^ 


\ 


■--. 


. 


M 


/ 





— 


Min 








s 


^N 








/ 


^ 


L. 


















\ 


X 






"" 


/ 




















'~~~ 


— 









oeoo 12 

Hour of the Day 



leoo 



ONO 1200 1600 

Hour of the Day 



Figure 9.13. Annual and diurnal cycles of Ng/or Salt Lake City, Utah. 



414 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 





1 1 1 

0200 Local Tltne 








































































y 




__.. 


._ 


'^ 








-^ 




__ 


^ 




•^ 


^. 


r~^ 


\ 


^-.^ 




^^ 


— 


-J 


-- 


>. 


^ 




^ 


\, 




^ 


^ 


-^ 


..... 




.-'■ 


' 


^ 


^ 




\ 




-s 


-- 


_. 






/ 










^ 




— 


._._._^ 











































































370 
360 
350 




1 1 1 
1400 Locol Time 
























































s 
















y^ 




— 


— 1 


\ 




\ 




340 










yy 


<rr- 


,^ 


n;-^- 




^ 


330 
320 
310 
300 
290 
280 











■ — 




^^ 


' / 


s 




^^v 


^ 


^~' 







■ 


I^'''. 




/ 


\ 




•-4- 










/ 










N 








^ 


\ 


^ 


/ 
















^ 



















































Month of the Year 



Mr MY SEPT 

Month of ttie Year 





fiebruory 




























































































Ma> 











, 




, — . 


^^ 








'W 








1 







' — 


' — ' 
































.-?_ 






,— 







r— 






1 


^ 





Min. 












1 

























































































1 

August 




















360 
3S0 
































I 
















" 




^ 


— . 


■— 









I 




T T 




[■ 




— 










\"' 




...A 







■ 1 


.-. 


— • 










\^ 




^ 


S 










^ 








310 
300 

290 










"^ 


y 


\ 


y 






-H 







































































































0800 I2W K 

Hour of the Doy 



0800 1200 l( 

Hour of the Day 



Figure 9.14. Annual and diurnal cycles of Ns/or Tatoosh Island, Wash. 



ELEVATION ANGLE ERROR 



415 




- 


^ 


a 




^ 






Moy 






i 










... 


|X 














— 


- =0800 n=2l7 J 




i 1 


-■ 


t^ 


^ 










=1400 n=2l7 1 

=2000 n=2l7 J 




1 ' ' 

i , 






> 


^ 


^ 
























1 








\ 







^ 
















— 






1 

i 










\ 


N 


^ 


N 
































x'^$^ 






























■4^^ 


















; 










"~— 





— = 0200 n=248 

— = 0800 n=248 
= 1400 n=248 

— =2000 n=248- 





90 I 9B|99.5| 99.99 0.01 | O.S| 2 10 

I 5 20 40 60 80 95 99 99 9 0.1 I 5 20 40 60 80 95 99 999 

P OF N, > ORDINATE VALUE 



Figure 9.15. Cumulative probability distribution of Ns, Washington, D.C. 



416 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



370 
350 
330 
310 
290 
2 70 







I 




Febnjoryl I 




! 


1 






"*« 


^^f 




r 


i 1 : 


=0800 n=2Z6 




1 r^'^!^ 1 


1 h — 


= 1400 n=Z26 

=2000 n=226 








•^; I i ! 1 




1 






' 


% 


jir 
















^ 


i ; ! : 1 


i 
















^^H 








i 


1 ' ' 






^^r 
















^Tn-^ 




s, 


















\ 


"s 

\ 


-^ 
















i 1 




i- 


-^ 





380 
360 
340 
320 
300 
280 





-^ 


:v 


Sx 








f 










i 


1 


1 






■\ 


f. 




^. 








=0600 n=248 1 








N 


K%^ 


^^ 


V 




=1400 n=24e 1 




■ 








\^ 


^^- 


]\ 


^J 


1 




i 




















Y 


V. 




M 


























-v 


^ 


\ 




























■^N^ 


\ 




























VNO 


4 


s 
























\-iS4> 


























\ 


1 


-■0 


























1 


^-. 






99.5 99.99 



0.1 I 5 20 40 60 80 95 99 99.9 0.1 I 5 20 4060 80 95 99 99.9 

P OF N, >0RDINATE VALUE 



Figure 9.16. Cumulative probability distribution of Ng, San Antonio, Tex. 



ELEVATION ANGLE ERROR 



417 



















R 


t> 




rt 




1 1 




































._ 


0800 n"223 




























=1400 n=226l 

=20O0 n=2?6 1 




=» 




> 




































- 


- 






^ 


^ 








































"~~ 






« 


• 


L? 


^ 


■«Bj 




^ 




































~ 


-^ 








^ 








































~- 



























































1 

Moy| 




1 1 


1 


1 






'^^ 


v\ 



















0600 n>24e 






s 


^ 




















200O n=?4a 


1 






^ 


^ 
































^^ml 






























^-^:^^ 


:::> 


^ 






























^■ 


^ 


^ 




::: 




























-4 


-iv 




















































Mill 
November 


1 1 




































. 


0800 n=240 




























=1400 n=240l 

=2000 n=24oJ 




^ 


? 


■J 


=5a 


fti^ 












































■»i 


= 


sj 




iS5 


^ 


H 






































■~- 




, 


■n 




^ 


\ 








































■" 


















































































01 





5 


' 




1 





3 





S 





7 





9 





9 


8 


9 


9.5 


99 



260 

99.3 99.99 0.01 

I 
5 20 40 60 80 95 99 999 0.1 I 5 20 40 60 80 95 99 999 

P OF N, > ORDINATE VALUE 



Figure 9.17. Cumulative probability distribution of N9, Bismarck, N.Dak. 



418 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 









! 








1 1 1 1 

February 


Ml! 




-J 


























- 


0800 n=225 

— - 1400 n = 226 
2000 n = 226 




s 




1 


















- 




i=> 


^***^ 


^ 


[?=? 












































5 


; 






































' 


-- 


-, 






^ 


S^ 


































~~- 


- 


-.. 




-= 










































1 























































May 






1 1 1 1 








\ 


\ 


k 




















0800 n = 248 




~~-$ 


^ 


5^ 
















2000 n = 248 










" 


-*i 


^ 












































\ 


































-H 


■^ 


s 


\ 




































s 




^> 


\ 




































-~, 


-> 


^ 


^ 


- 


































■ 


-■ 


5 





tl 0,5 I Z 5 10 20 30 40 50 60 ?ll 80 90 95 9« 99 935 
P of N,>Ordmote Value 



05 12 5 10 2030«)50EOTOiO 9095 90! 
P of N, > Ordmole Value 



















! 

August 








o 






^ 


!; 


^ 


















0800 n = 248 










-,N 


s 


^ 


^ 












2000 n = 248 












^.\ 





J 


S 


K 




































'^ 


k^ 




S 


k 
































■ 






^> 


^ 


\ 


\ 
































"i 


^ 


\ 






s 






























X 




V ' 

^ 


X 


L-i 






























"--. 

























Nc 


ve 


m 


be 








.J 




























- 


0800 n = 240 

1400 n=240 

2000 n = 240 






L 






















- 






3 


^ 


^ 


-- 










































""- 




^" 


^5 


s 


^ 


^ 




































- 


- 


-^ 


ife 


*~. 




































■~ 


- 






•^5; 





















































































I S K) ZO3O4OSOC0TD80 90 95 
P of N, > Ordmote Value 



2 5 10 20304050607090 9035 9!! 
P of N, > Ordinate Value 



Figure 9.18. Cumulative 'probability distribution of Ns, Colorado Springs, Colo. 



















Fe 


br 




ry 














































0200 n=226 

0800 0=^226 










ks 


:^, 


?:; 




» 


~1 










1400 = 226 

2000 = 226 




















"■ 


- 


- 






^ 








































~~ 




s 


^ 


::; 






































~^ 


- 















































^^ 










May 










1 


1 


1 






^5S 


?5 


^ 










- 


0200 = 248 

0800 = 248 












■•^ 


^^ 


"^ 




- 


1400 = 248 

2000 = 248 












^ 


^■-.^ 


■•>, 


e.. 


> 


-N 
























"x 


. 


~'' 




J^^ 


--. 


























--, 




■x 






























"- 









^ 


X. 


^ 










i 


ugu 


t 


















300 










:>^ 


V 


















0800 0=248 










"^.^ 


\ 
















1400 n = 248 

2000 n = 248 


280 
















> 


s 








































N 




V 


^ 


















260 


















^- 


^ 








S 


^ 










































^~ 


=*i 




- 




240 




























~" 


■~ 


,., 


















































0. 


31 





9 






1 


a 


3 





5 





7 





9 





9 


e 


a 


95 


99 



230 

99.99 0.01 

0.1 I 5 20 40 60 80 95 99 99.9 01 I 5 20 40 60 80 95 99 99 9 

P OF N, > ORDINATE VALUE 



















10 


ve 


n 


» 














































0200 = 240 

0800 = 240 




'i 


^ 


^ 


~s 




















2000 0=240 












-- 


■"^ 


G^ 


i^ 


S) 


^ 


s 




































" 


~ 




,_ 




V 


5SH 




































'^' 


-> 




V 


^ 


v! 




























































































































31 





5 






1 





3 





5 





7 





9 





9 


8 


9 


9.5 


99 



FiGURK 9.19. Cumulative probability distribution of Ns, Salt Lake City, Utah. 



ELEVATION ANGLE ERROR 



419 

















Febfuo 


'» 








II 1 
































III 1 
-0200 n = 242 




- 
























: 


0800n = 255n 

I400n = 255 j 












'«^ 


^ 




^ 












_._ 


-2 


a 


XJ 


■Tl 
























>« 


^ 












































^ 


"i; 




3 


^ 































































r-TT 
Moy 














































- 








" 








^ 














0800n = 248 

1400 n = 248 






















^ 


•< 


^ 


^ 


LJ 


-2000 

1 


"1" I 
































^ 


k 










































\ 
\ 















































O.oi I 0.5| 2 I 10 30 1 50 1 70 I 90 { 98 |99.S | 99.99 0.01 0.5 2 10 SO 50 70 90 98 99.5 99.99 

0.1 r 5 20 40 60 SO 95 99 99.9 0.1 I 5 20 40 60 80 95 99 99.9 

P OF N, > ORDINATE VALUE 



Figure 9.20. Cumulative probability distribution of Ns, Tatoosh Island, Wash. 




Figure 9.2 L Standard deviation of Nb, February 0200. 



420 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 




Figure 9.22. Standard deviation of Ns, August 0200. 



Tables 9.36 to 9.44 

These tables show values of m and i in the regression equation 
e = mN s + ^, where: 

e is the elevation angle error in milliradians to the height h, 

N s is the value of the radio refractivity at the earth's surface, 

m the change in elevation angle error per unit change in N s, 

t is the zero intercept, 

r is the correlation coefficient, 

SE is the standard error of prediction using the regression line, 

00 is the initial elevation angle in milliradians. 



Table 9.36. Variables in the statistical method of elevation angle error for h = 0.1 km 



do 


T 


m 


I 


SE 


0.0 


0. 2637 
.2760 
.2855 
.3017 
.1609 
.1777 
.1805 
.1824 
. 1815 
.1844 


0. 0243 
.0131 
.0083 
.0037 
.0027 
.0012 
.0005 
.0002 
.0001 
.0001 


-4. 3954 
-2. 1055 
-1.2185 
-. 4741 
-.2834 
-.1175 
-.0426 
-.0227 
-.0110 
-. 0055 


3. 4429 


1.0 

2.0 


1.7712 
1.0837 


5.0 


0. 4561 


10.0 


.6501 


20.0 


.2706 


52.4 


.0998 


100.0 


.0518 


200.0 


.0256 


400.0 


.0123 







TABLES 9.36 TO 9.44 



421 



Table 9.37. Variables in the statistical method of elevation angle error for h = 0.2 km 



So 


r 


m 


I 


SE 


0.0 


0. 2739 
.2838 
.2927 
.3146 
. 1919 
.2125 
.2168 
.2175 
.2170 
.2170 


0. 0310 
.0191 
.0132 
.0065 
.0044 
.0021 
.0008 
.0004 
.0002 
.0001 


-5. 5364 
-3. 0776 
-1.9685 
-. 8506 
-.4623 
-.1941 
-.0705 
-.0367 
-.0179 
-.0085 


4. 2249 


1.0 

2.0 


2. 4984 
1.6702 


5.0 

10. 0- 


.7628 
.8906 


20.0 


.3757 


52.4 


. 1385 


100.0 


.0721 


200.0 


.0356 


400.0 


.0170 



Table 9.38. Variables in the statistical method of elevation angle error for h = 0.5 km 



eo 


r 


m 


I 


SE 


0.0 

1.0 

2.0 


0. 3255 
.3457 
.3661 
.4197 
.2887 
.3304 
.3418 
.3430 
.3439 
.3408 


0. 0430 
.0289 
.0216 
.0125 
.0083 
.0041 
.0016 
.0008 
.0004 
.0002 


-8. 0534 

-5. 0198 

-3. 5807 

-1.9306 

-1.0741 

-. 4899 

-. 1803 

-.0934 

-. 0463 

-.0218 


4. 8023 
3.0131 
2. 1131 


5.0 


1.0427 


10.0 


1.0693 


20.0-.- 


.4568 


52.4 

100.0 

200.0- .. - 


.1679 
.0872 
.0431 


400.0 


.0208 







Table 9.39. Variables in the statistical method of elevation angle error for h = 1.0 km 



Bod 


r 


m 


I 


SE 


0.0 


0. 3371 
.3772 
.4180 
.5150 
.3983 
.4831 
.5129 
. 5167 
.5177 
.5180 


0. 0472 
.0339 
.0270 
.0179 
.0123 
.0066 
.0026 
.0014 
.0007 
.0003 


-8. 2579 

-5. 5583 

-4. 2916 

-2. 7591 

-1.6158 

-.8154 

-. 3070 

-. 1595 

-. 0787 

-.0379 


5. 1154 


1.0 

2.0 


3. 2281 
2. 2775 


5.0 


1. 1590 


10.0 


1. 1044 


20.0 


.4703 


52.4 


.1714 


100.0 


.0890 


200.0- -. 


.0440 


400.0 


.0211 







Table 9.40. Variables in the statistical method of elevation angle error for h = 2.0 km 



Oo 


r 


m 


I 


SE 


0.0 


0. 3795 
.4413 
.5020 
.6351 
.5642 
.7007 
. 7531 
.7608 
. 7633 
. 7643 


0. 0567 
.0427 
.0355 
.0258 
.0188 
.0113 
.0047 
.0025 
.0012 
.0006 


-9. 8908 
-7. 0696 
-5. 7509 
-4. 1251 
-2. 7184 
-1.5650 
-.6256 
-.3279 
-. 1627 
-. 0784 


5. 3659 


1.0 

2.0 


3.3704 
2. 3735 


5.0 


1.2198 


10.0 


1. 0773 


20.0 


.4499 


52.4 


.1615 


100.0 


.0835 


200.0 


.0412 


400.0 -. 


.0198 







422 



CHARTS, GRAPHS, TABLES, AND COMPUTATIONS 



Table 9.41. Variables in. the statistical method of elevation angle error for h = 5.0 km 



00 


r 


m 


I 


SE 


0.0 


0. 4324 
.5186 
.5983 
.7543 
.7424 
.8789 
.9248 
.9316 
.9336 
.9339 


0. 0704 
.0554 
.0477 
.0372 
.0290 
.0194 
.0089 
.0048 
.0024 
.0012 


-12.0857 

-9. 0682 

-7.6679 

-5. 9085 

-4. 3100 

-2. 7570 

-1.1721 

-.6154 

-.3043 

-. 1460 


5. 6994 


1.0 

2.0 


3. 5425 
2. 4766 


5.0 


1. 2570 


10.0 


1 0255 


20.0 


.4118 


52.4... 


. 1431 


100.0 


.0733 


200.0 


.0360 


400.0 


.0173 







Table 9.42. Variables in the statistical method of elevation angle error for h = 10.0 km 



do 


r 


m 


I 


SE 


0.0 


0. 4617 
.5602 
.6475 
.8057 
.8125 
.9301 
.9679 
.9742 
.9763 
.9766 


0. 0797 
.0637 
.0555 
.0443 
.0353 
.0245 
.0118 
.0065 
.0032 
.0016 


-13.2953 

-10.0839 

-8. 5935 

-6. 6912 

-4. 9338 

-3. 1637 

-1.3002 

-. 6607 

-.3208 

-. 1533 


5. 9448 


1.0 

2.0 


3.6560 
2.5360 


5.0 


1. 2652 


10.0 


.9913 


20.0 

52.4 


.3779 
.1195 


100. 0. 


.0584 


200.0. 


.0281 


400.0 


.0135 







Table 9.43. Variables in the statistical method of elevation angle error for h = SO.O km 



So 


r 


TO 


I 


SE 


0.0 

1.0 

2. 0-- 


0. 4805 
.5866 
.6780 
.8351 
.8497 
.9538 
.9861 
.9917 
.9934 
.9939 


0. 0874 
.0703 
.0616 
.0495 
.0397 
.0278 
.0136 
.0076 
.0038 
.0018 


-14.0318 

-10.5942 

-8. 9972 

-6. 9273 

-5. 0189 

-3. 0650 

-1.0682 

-.4672 

-.2067 

-.0947 


6. 1895 
3. 7658 
2. 5905 


5.0 


1. 2665 


10.0 


.9637 


20.0 


.3423 


52.4 


.0899 


100.0 . 


.0384 


200.0 


.0172 


400.0 


.0080 







Table 9.44. Variables in the statistical method of elevation angle error for h = 70.0 km 



Bo 


r 


TO 


I 


SE 


0.0 


0. 4999 
.6137 
.0783 
.8617 
.8811 
.9701 
.9948 
.9983 
.9992 
.9995 


0. 0985 
.0794 
.0698 
.0563 
.0451 
.0317 
.0158 
.0089 
.0045 
.0022 


-15.2202 

-11.3649 

-9. 5814 

-7. 2363 

-5.0975 

-2.9121 

-.8032 

-. 2595 

-.0822 

-.0319 


6.6214 


1.0 

2.0 


3. 9661 
2. 6984 


5.0 -- 


1. 2867 


10.0 


.9476 


20.0 


.3102 


52.4 


.0630 


100.0 


.0202 


200.0 


.0070 


400.0 


.0028 







REFERENCES 423 

9.6. References 

[1] Bean, B. R., and G. D. Thayer (1959), On models of the atmospheric radio refrac- 
tive index, Proc. IRE 47, No. 5, 740-755. 

[2] Smithsonian Meteorological Tables (1951), Table 63, Sixth Revised Ed. (Wash- 
ington, D.C.). 



Subject Index 



A-unit (potential refractivity), 16-20, 78-79, 166-170, 189-194, 197, 211, 213-227, 379 

Absolute humidity, 270, 272, 275, 277-280, 286-290 

Absorption coefficient (see also Van Vleck's absorption formulas), 272 

Adiabatic refractivity decrease, 258-262 

dry, 261 

wet, 262 
Adiabatic temperature lapse rate, 133, 258 
Advection, 132, 201, 205, 213, 319 

definition, 132 

relative to duct formation, 133, 135 
Air masses, 79, 102, 114, 119, 123, 163-164, 167-169, 173-176, 179-181, 191, 195- 

197, 201, 213, 221-222 

refractive characteristics of, 163, 169-170, 174-175, 211 
Angle of penetration (critical angle), 137, 140, 142, 376 
Apparent height, 360 
Apparent range, 51, 335, 357-360 
ARDC Standard Atmosphere, 64 
Attenuation of radio waves, 269-309 

by fog, 303 

by gases, 270-290 

by hail, 302-303 

by rain {see also Climatology), 292-297 

in clouds, 291 

B 

B-unit, 14-15, 17-20, 189-191, 195 

Barium fluoride strip, 26 

Bending of a radio ray (see also Refraction) 

definition, 49 

methods of computing and predicting, 53-81, 375-393 
Bi-exponential model 

for absorption, 283, 307 

for refractivity, 311-322 
Birnbaum refractometer, 31, 33-34 
Bright band, 303 

C 

Caustic, 148 

Climatic types, relation to Na, 102-104 

Climatology, 89-172 

of ground based radio ducts, 42-43, 132-145 

of refractivity, A^, near the ground (No), 89-109, 111-128, 404-420 

of rainfall attenuation, 297-302 
Clouds, attenuation in, 291 
Cold front (see Frontal zone) 
Composite parameter, 238 
Continental cross sections of No, 122-128 

Continental arctic air mass (see also Continental polar air mass), 102, 316 
Continental polar air mass, 79, 123, 163-164, 169, 174-176, 179-180, 196-197, 

221-222, 319 
Continental tropical air mass, 79, 167-169, 175, 179 
Convergence 

centers of, 182, 240-241 

influencing atmospheric refractivity, 182-185, 240 
Core climate, 129 

425 



426 SUBJECT INDEX 

Correlation of field strength with meteorological parameters, 105, 174, 229, 233-266 

Crain refractometer, 31-33 

Cross-over height, 248-249, 263-264 

CRPL Reference Atmosphere {see Exponential atmosphere model for refractivity) 

CRPL Standard A^-profile Sample, 335, 338-353, 355-356 

Curvature of radio ray, 13, 42, 56, 264, 360-361 

D 

Deam refractometer, 36-37 

Delta-A'^ (A N) {see also Refractivity, vertical gradient), 15-16, 61, 63-66, 73-76, 

106, 129-131, 139, 229-230, 342, 363, 365, 371, 373, 389, 394-396 

as a field strength predictor, 234-238, 240, 243-244, 246, 264-266 

usage for 24-hour horizontal changes, 201-204, 209-211 
Departures-from-normal method of computing refraction, 77-79, 167-168, 181, 190- 

191, 375, 378-379, 381, 392-393 
Dielectric constant of air {see also Refractive index), 2-8, 12 
Dispersion, 5, 8 

Diurnal index (diurnal ratio), 129-130 
Diurnal range 

of field strength, 105, 235-236 

of A^o, 111-121, 129-130, 204 
Divergence 

centers of, 182, 240-241 

influencing atmospheric refractivity, 182-185, 240 
Drop size distribution, 292, 294-297 
Dropsonde, 36 
Dry term 

of atmospheric absorption, 283 

of the Smith- Weintraub equation, 30, 91, 93-94, 136, 174-175, 311-322 
scale heights of (see Scale height) 
surface values of, 93-94, 311-312, 320 
Ducting, 42-45, 52, 89, 132-163, 241-246, 248, 324, 328, 331-333, 376 

E 

Effective earth, 13-15, 51, 56-62, 64-65, 106, 189, 195, 250, 264, 359-362 

curvature, 13, 56-57, 62, 106, 195, 361 

method of calculating refraction, 56-59 

radius, 13-15, 51, 56-57, 59-62, 64-65, 106, 189, 195, 250, 264, 359-362 
Elevated layer, 241, 244-258, 324-329, 331 

reflection coefficient of, 252-257 
Elevation angle (see also Refraction) 

approximation for, 82-83 

definition, 49-50 

errors, 49-50, 178, 333-335, 340-342, 344-345, 348-350, 356, 404, 420-422 
prediction of, 174, 334, 340-342, 344-345, 348-350, 356, 404, 420-422 
Epoch chart, 221-227 
Equation of continuity, 183 
Equivalent gradient, 236-239, 243, 246, 264-266 

as a field strength predictor, 236-239, 243, 246, 265 

method of calculation, 264 
Exponential atmosphere models 

for absorption, 283 

for refractivity, 15-20, 56, 61, 63-80, 92-93, 106, 110, 149, 153-156, 164, 166- 
167, 174, 176-177, 186, 189-190, 192-194, 244-245, 261-263, 311-322, 
329-332, 357, 362, 375-378, 385-389, 393-396 

in radiosonde correction process, 40-41 

method of computing refraction, 375-377, 385-388 
Eta {t)) parameter, 238-239 

F 

Fading (see also Field Strength), 146-163, 223 

caused by ducting, 146-163 

definitions, 147 

due to rainfall attenuation, 298-302 
Field strength 

prediction from radio meteorological parameters, 89, 105-106, 109, 174, 177-180, 
229-236, 238-251, 253-258, 263-266 



SUBJECT INDEX 427 

Fog attenuation, 303 

Four-thirds earth (see also Effective earth), 14-15, 20, 51, 57, 59-60, 62, 66-75, 

247, 360 
Fresnel discontinuity value, 252, 257 
Frontal zone, influence on refractivity, 176-177, 187-193, 195-224, 323 

G 

Geometric range, 335, 357 

Gradient of refractive index (see Refractive index) 

Gradient of refractivity (see Refractivity) 

regression expressions for, 363-364, 372 
Graphical method of computing refraction, 80-81, 375, 380-381 
Ground-based radio ducts (see also Ducting), 42-44, 132-163, 324, 328, 333, 375 

frequency of occurrence, 140-141 

thickness of, 140, 143-145 
Ground-based superrefractive layers, 250 

H 

//-parameters, 265 
Hail attenuation, 302-303 
Half-length, 150-154, 160 
Hay refractometer, 36-37 
Height errors, 356-357, 360-373 

definition, 356, 360 

regression equations for, 364-372 
High angle approximation for computing refraction, 53-54 
Horizontal homogeniety, assumption of, 49, 52, 163, 322 
Horizontal inhomogenieties, 52, 322-333, 356 
Horizontal ray, 329 

I 

ICAO Standard Atmosphere, 9, 11, 312-314 

Initial gradient of refractivity {see Delta-A'^ and Refractivity) 

Intensity factor, 273-274 

Inversions (see Ducting and Elevated layers) 

K 

ii:-unit, 180, 182, 239-240 

L 

Lag times, 29, 38-45 

in electrical thermometers, 26 

of relative humidity sensor in radiosondes, 39, 41-45 

of temperature sensor in radiosondes, 39, 41-45 
Land breeze, relative to duct formation, 134-135, 323 
Level of nondivergence, 183 
Lifting condensation level, 260 

Line widths of atmospheric gaseous absorption (see also Attenuation), 272-275 
Lithium chloride humidity strip, 26, 28-29, 39, 45 

M 

il/-parameter {see Thermal stability) 

Macroscale fluctuation of refractive index, 173, 223 

Maritime polar air mass, 79, 114, 169, 179, 197 

Maritime tropical air mass, 79, 102, 119, 125-126, 164, 167, 169, 174, 177, 179-181, 

201, 213, 221-222, 319 
Maximum wavelengths trapped by a duct, 141-143 
Mesocale fluctuation of refractive index, 173, 185, 223 
Microscale fluctuation of refractive index, 173 
Mixing ratio, 260 
Models 

of atmospheric absorption structure, 283 
of atmospheres, 9, 11, 16-17, 64, 92-93, 180, 186, 312-314 

of refractivity structure, 13-20, 40-41, 51, 56-80, 92-93, 106, 110, 149, 153-156, 
164, 166-170, 174, 176-177, 180, 186, 189-195, 197, 213-227, 244-245, 250, 
261-264, 311-322, 329, 331-332, 357, 360-362, 375-378, 381, 385-389, 
393-396 



428 SUBJECT INDEX 

Modified effective earth's radius method of computing refraction, 59-65 
Modified index of refraction (M-unit), 14-15, 17-20, 193, 195, 324 

N 

A'^-unit (see Refractivity) 

NACA Standard Atmosphere, 16-17, 92-93, 186 

Nocturnal duct formation (see also Ground-based radio ducts), 133, 135 

Nonpolar gases, dielectric constant of, 2-3 

O 

Optical depth, 304 
Outage time, 298-301 
Oxygen 

absorption properties of (see Attenuation by gases) 

refractivity at STP, 9 

P 

Permeability of air, 3-4 

Pi-factors (turbulence parameters), 262-266 

Polar air mass (see Continental polar air mass) 

Polar front (see also Frontal zone), 195-197, 213 

Polar wave, 193, 195-197, 213 

Polar gases, dielectric constant of, 2-3 

Potential refractive modulus (<p-unit), 17, 19-20, 166, 193 

Pound oscillator (see Deam refractometer) 

Precipitation rate, 292 

Profile (gradient) classification (see also Air masses and Refractivity), 248-250 

Profiles of refractive index (see Refractive index, vertical profile of) 

Profiles of refractivity (see Refractivity, vertical profile of) 

Psychrometric equation, 12, 24 

R 

Radar 

applications involving (see also Rain attenuation), 49, 51, 56, 59, 61, 77, 340, 357 

anomalous propagation in (see Ducting) 
Radiation, relative to duct formation, 133 
Radiative transfer equation, 304 
Radio 

duct (see Ducting) 

holes (see Ducting) 

horizon 

equation for, 59 

in ducting, 146-148, 152, 155, 157 

refractive index (see Refractive index) 
Radiosonde (RAOB), 10-11, 26-30, 36, 38-43, 53-55, 59, 61, 70-72, 74-75, 79, 110- 

111, 135, 138-140, 144, 179, 181, 194, 205, 211, 213, 243, 246, 247-249, 251, 253, 

255, 258, 264, 280, 283, 326, 363, 375 

sensors in, 26-27, 29-30, 39-45 
Rain attenuation, 292-297 
Range 

errors (radio), 12, 174, 335-342, 345, 350-353, 356 
definitions, 335-336 
prediction of, 341, 345, 350-353 
total, 335-337 

of a radio ray, 51, 335-336, 357-361 
Ray tracing (see also Refraction), 52, 132, 135-136, 148, 322-323 

limitations of, 52, 132 
Rayleigh-Jeans law, 304-305 
Reflection coefficient (of an elevated layer), 185, 252-257 

models of, 252-257 
Refraction (angular) of a radio ray, 14-15, 19-20, 49-87, 163-170, 178-181, 190, 192, 

236, 266, 311, 319-323, 326-337, 339, 341-350, 357, 375-396 

definition, 49 

derivation in terms of refractive index, 87 



SUBJECT INDEX 429 

Refractive index, atmospheric radio (see also Refractivity) 

applications of, 12-14, 31-34, 36, 49-87, 136-138, 150-152, 165-166, 174, 223, 

231-266, 311, 319-323, 326-329, 335-336, 357, 359-362, 394 

definition, 23 

dispersion, 5, 8 

expressions for, 3-7 

general discussion of, 1-8, 16-17, 19-20, 49, 89-90, 92, 103, 109-110, 125, 129, 

133, 135, 163-166, 169-170, 173-175, 178, 185-195, 211, 222, 311, 314, 332- 

333, 342-343, 356, 404 
vertical gradient of (see also Delta-A^), 13-14, 19, 36, 38, 52, 56-57, 64, 77, 106-107, 

132, 138-140, 142, 180, 182, 189, 195, 236, 240-241, 248, 264, 319, 357, 361-364 
vertical profile of, 14-15, 40, 42-45, 52, 59, 62, 135, 173-176, 179-180, 182, 187, 

211, 222, 229, 240, 247-252, 254, 257-258, 331-332, 334, 339-342, 363, 366 
Refractivity (see also Refractive index) 

applications of, 12-13, 19-20, 49-87, 105-107, 136-170, 178-181, 190, 229-266, 

319-373, 375-396 
as a function of height (see Models) 
definition, 4, 90, 357 
diurnal range (see Diurnal range) 
expressions for, 4-7 

constants in, 4-11 

errors in, 9-12 
measurement of, 23-45 

accuracy and errors in, 24-32, 34-36, 38-45 
near the ground (surface values), 15-20, 36, 42-43, 53-54, 58, 61, 63-80, 89-163, 

176-182, 186-187, 192, 194, 197-205, 223, 229-230, 233-237, 239, 243-246, 

258, 261-266, 311-312, 319-321, 333-356, 361-363, 365-367, 369-370, 373, 

375-388, 392-396, 404-422 
of atmospheric constituents 

carbon dioxide, 5 

dry air (see also Dry term), 9 

oxygen, 9 

water vapor (see also Wet term), 9 
potential (see ^4 -unit) 
scale heights of (see Scale height) 
synoptic variation of (see Frontal zone) 
vertical gradient of (see also Delta-AA), 9-12, 15, 20, 33, 36, 42-43, 61, 63-66, 

73-77, 79, 106, 114, 123, 129-131, 136, 143, 145, 147-148, 150, 152, 166, 

168, 229-230, 234-241, 243-244, 246, 248, 250, 264-266, 342, 362-363, 

365-367, 370-371, 373, 375, 378, 389, 394-396 
vertical profile of (see also CRPL Standard N-profile Sample), 15, 17-19, 42-45, 

54-55, 57, 59-68, 70-71, 73, 75, 77-79, HI, 115, 117, 122, 129, 135, 138- 

139, 146, 162-164, 167-170, 174-177, 194, 211-227, 235, 244-245, 263, 266, 

321, 329, 336-339, 341, 344, 375, 381 
Refractometer, 8, 23, 31-38, 334 
Resonant frequencies (absorption lines) 
of oxygen, 270-275, 280 
of water vapor, 270-275, 280 
Rocket Panel data, 64 

S 

Satellite telecommunications, 65, 175 

Saturation vapor pressure, exponential assumption for, 260 

Scale height 

of absorption, 283 

of refractivity, 15-17, 92-93, 108, 186, 261-263, 311-312, 314-322 

of dry term, 93, 261, 263, 311-3)2, 314-322 

of wet term, 93, 262-263, 311-312, 314-322 
Scattering by atmospheric constituents (see also Attenuation) 185, 247-248, 255, 

269-270, 292 
Schulkin's method of computing refraction, 54-55, 80, 179, 375-376, 390-391 
Sea breeze, relative to duct formation, 134-135, 323 
Seasonal index (seasonal ratio), 129-130 



430 SUBJECT INDEX 

Seasonal range 

of field strength, 174 

of No {see also Climatology), 111-121, 129-130 
Shadow zones, 148-150, 152-155, 157, 159-160 
Slant range, 335, 360 
Snell's law in polar co-ordinates, 49, 84-87, 136-137, 165, 326 

derivation, 84-87 
Snow attenuation, 302 

Space cross sections of refractive index, 211-217, 221 
Sprung's formula, 12 
Stability 

atmospheric, 238-241, 258, 260, 262-263, 266 

thermal (see Thermal stability) 
Standard atmosphere, 9, 11, 16-17, 64, 92-93, 180, 186, 312-314 
Statistical method of computing refraction, 54, 335, 342-350, 379-380, 382-384 
Stratopause, appoximate location, 313 
Stratosphere 

approximate extent, 313 

meteorological characteristics of, 313 
Subrefraction, 133, 248-249, 329, 332-333 
Subsidence, relative to duct formation, 133-135, 180, 182, 185 
Superior air mass, 79, 169 
Superrefraction {see also Ducting), 133-135, 146-148, 156, 169, 182, 248, 250, 328-329, 

332-333 
Surface duct (see Ground-based radio ducts) 
Synoptic illustration of A'' change (see Frontal zone) 

T 

Temperature dependence 

of gaseous absorption (see Van Vleck's absorption formulas) 

of refractivity, 1-8 
Thermal noise temperature, 304-307 

definition, 305 
Thermal stability, 239-241, 262, 265 
Tilted elevated layer, 249-251 

Time cross sections of refractive index, 213, 217-221 
Tracking system, radio, 334 
Transhorizon propagation mechanisms, 248 
Transition zone (see Frontal zone) 
Trapping of a radio ray (see Ducting) 
Tropopause, approximate location, 313 
Troposphere 

approximate extent, 313 

meteorological characteristics of, 313 
Turbulence 

atmospheric, 260 

parameter (see Pi-factors) 



U 



Unstratified layer, 248, 250-251 



Van Vleck's absorption formulas, 270, 272-275 

Velocity of radio propagation, 56 

Vertical 

gradient of A^ (see Refractivity) 

profile of A'^ (see Refractivity) 

ray, 329 
Vetter refractometer, 31, 34-35 

W 

Warm front (see Frontal zone) 
Water vapor 

absorption properties of (see Attenuation by gases) 

an average refractivity value of, 9 

density (see absolute humidity) 

dependence of refractivity, 1-8 



SUBJECT INDEX 431 

Weather influence on refractivity {see Frontal zone) 
Wet term 

of atmospheric absorption, 283 

of the Smith-Weintraub equation, 30, 91, 93-94, 136, 174-175, 311-322 
scale heights of {see Scale height) 
surface values of, 93-94, 311-312, 320 
Wind velocity 

influence on atmospheric stability, 240-241 

vertical component of, 182-184 
Wiresonde, 326 



Author Index 



Abbott, R. 270, 274, 288, 308 

Abild, V. B. 268 

Adey, A. W. 46 

Airy, G. B. 148, 172 

Akita, K. I. 30, 45 

Anderson, L. J. 22, 55, 80, 88, 171, 227, 

268 
Anderson, W. L. 374 
Anway, A. C. 347, 374 
Armstrong, H. L. 21 
Arnold, E. 268 
Artman, J. O. 273, 308 
Arvola, W. A. 180, 182, 226, 227, 240, 

266 
Atwood, S. S. 8, 22, 296-297, 309 
aufm Kampe, H. J. 291, 308 
Autler, S. H. 273, 275, 308 
Averbach, B. L. 46 

B 

Badcoe, S. R. 268 

Barghausen, A. F. 374 

Barrel), H. 5, 21 

Barsis, A. P. 267 

Battaglia, A. 8, 21 

Battan, L. J. 291, 308 

Bauer, J. R. 22, 255, 268, 357, 374 

Bean, B. R. 22, 29, 46, 54, 87-88, 170- 

172, 180, 225-228, 243, 266-268, 

270, 274, 288, 308-309, 342, 363, 

373-374, 377, 423 
Becker, G. B. 273, 275, 308 
Beckmann, P. 357, 374 
Bennett, C. A. 374 
Best, A. C. 309 
Beyers, N. J. 374 
Birnbaum, G. 5-7, 21-22, 31, 33-34, 46- 

47, 273, 308 
Boithias, L. 236, 267 
Bonavoglia, L. 230, 266 
Booker, H. G. 87, 171, 227, 267 
Born, M. 21 
Boudouris, G. 8, 21 
Brekhovskikh, L. M. 254, 268 
Bremmer, H. 136, 171 
Brooks, C. E. P. 374 
Buckley, F. 5, 21 
Bunker, A. F. 39, 47 
Burrows, G. R. 8, 22, 56, 88, 106, 171, 

175, 225, 296-297, 309, 374 
Bussev, H. K. 46, 297, 302, 308 
Byers, H. R. 167, 172 



Gaboon, B. A. 54, 88, 243, 266, 308, 363, 

374 
Carruthers, N. 374 
Chatterjee, S. K. 6-7, 21 
Clarke, L. C. 39, 47 
Cline, D. E. 46 
Clinger, A. H. 30, 46 
Cole, C. F., Jr. 36, 47 
Condron, T. D. 373 
Cowan, L. W. 171 
Craig, R. A. 8, 22, 267, 373 
Grain, C. M. 6, 8, 21-22, 31-33, 46-47 
Crawford, A. B. 8, 22, 225, 238, 256, 258, 

267 
Crozier, A. L. 30, 46 

D 

Davidson, D. 308 

Debye, P. 2, 4, 21 

Deam, A. P. 36-37, 46-47 

Doherty, L. H. 148, 172 

Donaldson, R. J., Jr. 308 

Dubin, M. 64, 88 

du Castel, F. 171, 253, 267-268 

Dunmore, F. W. 47 

Dutton, E. J. 29, 46, 88, 172, 228, 309 



East, T. W. R. 274, 276, 291, 308 
Edwards, H. D. 280, 309 
Englund, C. R. 8, 22, 225 
Essen, L. 5-6, 8, 10-12, 21-22 
Evans, H. W. 295, 299-302, 309 



Fannin, B. M. 172, 179, 226 

Fehlhaber, L. 22, 171, 268 

Ferrell, E. B. 8, 21, 56, 88, 106, 171, 225, 

374 
Flavell, R. G. 180, 182, 227, 240, 267 
Franklin, N. L. 374 
Eraser, 1). W. 46 
Freehafer, J. E. 87, 374 
Freethey, F. E. 46 
Friis, H. T. 238, 256, 258, 267 
Froome, K. D. 5-6, 8, 10-12, 21 



Gallet, R. M. 22, 171 

( larfinkel, S. 47 

(ierson, N. C. 170, 174-176, 224 

(;h(.sh, S. \. 280, 309 

Gordon, J. P. 273, 308 

Gordon, \V. E. 227, 267 



433 



434 



AUTHOR INDEX 



Gordy, W. 308 

Gossard, E. E. 227, 268 

Gozzini, A. 8, 21 

Gray, R. E. 105, 171, 178, 226, 267 

Grosskopf, J. F. 22, 171, 268 

Groves, L. G. 6 

Gunn, K. L. S. 274, 276, 291, 303, 308-309 

H 

Hasegawa, S. 47 

Hathaway, S. D. 295, 299-302, 309 

Hay, D. R. 36-37, 47, 171, 175, 224 

Herbstreit, J. W. 31, 46, 148, 172, 374 

Hewson, E. W. 268 

Hill, R. M. 308 

Hirai, M. 170, 225, 267 

Hirao, K. 30, 45, 171, 373 

Hogg, D. C. 238, 256, 258, 267 

Holden, D. B. 182-183, 185, 227, 240, 

266 
Holmes, E. G. 46 
Hopkins, H. G. 303, 309 
Horn, J. D. 22, 171, 180, 226, 228, 267, 

373 
Hornberg, K. O. 374 
Hughs, J. V. 21 
Hull, R. A. 174, 225 
Humphreys, W. J. 61, 88 
Hurdis, E. C. 6, 21 



I 



Ikegami, F. 148, 172 



Janes, H. B. 267, 374 

Jehn, K. H. 19, 22, 171-172, 179-180, 

193, 226-227, 239-240, 267 
Johnson, W. E. 45, 170, 374 
Jones, F. E. 45, 47 
Jung, P. 47 

K 

Katz, I. 17, 22, 166, 171, 180, 193, 226, 

239, 267, 312, 373 
Kerker, M. 303, 309 
Kerr, D. E. 141, 144, 171 
Kirby, R. S. 148, 172 
Kirkpatrick, A. W. 374 
Kitchen, G. F. 268 
Krinsky, A. 47 
Kryder, S. J. 5, 22 



Lane, J. A. 16, 22, 182, 227, 240, 266-267 

Langleben, M. P. 303, 309 

Laws, J. O. 292-293, 308 

Lement, B. S. 46 

List, R. J. 24, 45 

Longley, A. 266 

Longley, R. W. 268 

Lukes, G. D. 166, 172, 180, 226 

Lyons, H. 5, 22 



M 

Macready, P. B., Jr. 45 

Magee, J. B. 8, 42 

Martin, H. C. 47 

Maryott, A. A. 5, 21, 273, 308 

Mason, W. C. 22, 357, 374 

Meadows, R. W. 268 

Meaney, F. M. 266 

Megaw, E. C. S. 227 

Meyer, J. H. 255, 268 

Middleton, W. E. K. 40, 47 

Mie, G. 309 

Millington, G. 13, 21 

Minzner, R. A. 373 

Misme, P. 30, 46, 147, 170-172, 175, 224- 

225, 236, 238-240, 246, 265-267, 311, 

373 
Moler, W. F. 182-183, 185, 223, 227, 240, 

266 
Montgomery, R. B. 22, 267, 373 
Mumford, W. W. 8, 22, 225 
Murray, G. 47 

N 

Niwa, S. 170, 225, 267 
Northover, F. H. 268 
Norton, K. A. 62, 88, 105, 148, 171-172, 
250, 266-267, 374 

O 

Onoe, M. 170, 225, 267 
Ozanich, A. M., Jr. 171, 227, 373 



Palmer, C. E. 201, 228 

Parsons, D. A. 292-293, 308 

Perlat, A. 224 

Peterson, C. F. 374 

Phillips, W. C. 6, 21 

Pickard, G. W. 105, 170, 224, 230, 232, 

235, 266 
Pote, A. 308 

Pound, R. V. 32, 36-37, 46 
Price, W. L. 148, 172 

R 

Rainey, R. J. 374 

Randall, D. L. 176-177, 224 

Ratner, B. 168, 172, 280, 308, 373 

Rice, P. L. 88, 92, 250, 266-267 

Riggs, L. P. 22, 171, 180, 226, 228, 267 

Ripley, W. S. 373 

Roberts, C. S. 46 

Rubenstein, P. J. 22, 267, 373 

Ryde, D. 292, 294, 308 

Ryde, J. W. 292, 294, 302-303, 308 



Saito, S. 8, 21 

Sargent, J. A. 31, 34, 46 

Saxton, J. A. 1-2, 21, 182, 227, 238, 267- 

268, 303, 309 
Schelleng, J. C. 8, 21, 56, 88, 106, 171, 

175, 225, 374 
Schilkorski, W. 268 



AUTHOR INDEX 



435 



Schulkin, M. 88, 150, 164, 170, 172, 179, 

227, 373 
Sheppard, P. A. 224 
Silsbee, R. H. 47 
Sion, E. 47 

Smart, W. M. 49, 87, 172 
Smith, E. K. 4, 6-8, 10-12, 17, 21 
Smith, M. J. A. 47 
Smith-Rose, R. L. 8, 22 
Smyth, C. D. 21 
Smyth, J. B. 6, 21, 227, 267 
Snedecor, G. W. 171 
Spilhaus, A. F. 40, 47 
Spizzichino, A. 267 
Sprung, A. 12, 21, 24, 45 
Starkey, B. J. 268 
Stetson, H. T. 105, 170, 224, 230, 232, 

235, 266 
Stickland, A. C. 8, 22, 61, 88 
Stover, C. M. 45 
Straiton, A. W. 30, 46, 274, 308 
Stranathan, J. J. 6, 21 
Strandberg, M. W. P. 308 
Sugden, S. 6 



Tao, K. 171, 373 

Thayer, G. D. 22, 54, 87, 170, 225-226, 

228, 267, 342, 363, 373-374, 377, 423 
Thiesen, J. F. 30, 46 
Thompson, M. C, Jr. 46, 374 
Tinkham, M. 308 
Tolbert, C. W. 274, 308 
Trewartha, G. T. 172 
Troitskii, V. S. 309, 312, 373 
Trolese, L. G. 21, 227, 267 
Turner, H. E. 47 
Turner, W. R. 268 



U 



Unwin, R. S. 323 



Van Vleck, J. H. 270, 273-275, 308 

Vetter, M. J. 31, 34-36, 46, 374 

Villars, F. 267 

Voge, J. 238-239, 267 

Yogler, L. E. 62, 88 

W 

Wadley, T. L. 8, 12-13, 21 

Wagner, N. K. 38-39, 42, 47, 253, 268 

Walkinshaw, W. 87, 171, 267 

Waters, D. M. 46 

Watkins, T. B. 47 

Waynick, A. H. 8, 22 

Weickmann, H. K. 291, 308 

Weintraub, S. 4, 6-8, 10-12, 17, 21 

Weisbrod, S. 80, 88 

Weisskopf, V. F. 267 

W^ells, P. I. 374 

Wensien, H. 268 

Wexler, A. 28-29, 39, 41, 45-47 

Wilkerson, R. E. 374 

Willett, H. C. 172 

WiUiams, C. E. 46 

Wilson, F. A. 22, 357, 374 

Wolf, E. 21 

Wong, M. S. 322, 373 



Yerg, D. G. 45, 47, 174, 225 



Zhevankin, S. A. 309, 312, 373 



ft U.S. GOVERNMENT PRINTING OFFICE : 19«6 O— 77S-028 



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