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The Statist

Wei oars E2a Kren, CLE TP

TP 76-10
(At>- A029 ©38)

ical Anatomy
of
Ocean Wave Spectra

by

Leon E. Borgman

TECHNICAL PAPER NO. 76-10
JULY 1976

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The findings in this report are not to be construed as an official
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THE STATISTICAL ANATOMY OF OCEAN WAVE SPECTRA

7. AUTHOR(s)

DACW72-72-C-0001

Leon E. Borgman

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‘Laramie, Wyoming 82071

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- SUPPLEMENTARY NOTES

KEY WORDS (Continue on reverse side if necessary and identify by block number)

~”-Gulf of Mexico “Ocean wave spectra
-AHurricane Carla “Spectral analysis
Hurricane waves

ABSTRACT (Continue on reverse side if necesaary and identify by block number)

The statistical variations in wave energy spectral estimates for hurricane
waves are examined empirically for 12 separate intervals of wave records
measured during Hurricane Carla in September 1961. Measurements were made on
a Chevron Oil Company platform in South Timbalier Block 63, Gulf of Mexico,
at a water depth of 100 feet. Hurricane waves were chosen for the analysis
because these waves show, in exaggerated form, the effects of departures from
linearity on the statistical variability in spectral estimates.

continued

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This report gives the analysis for Hurricane Carla and develops certain
implications and consequences uf the empirical results.

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PREFACE

This report is published to provide coastal engineers with the empir-
ical results of statistical variations in wave energy spectral estimates
for 12 intervals of wave data measured during Hurricane Carla in
September 1961. The measurements were made on a Chevron 0il Company
platform in the Gulf of Mexico at a water depth of 100 feet. The work
was carried out under the wave mechanics program of the U.S. Army Coastal
Engineering Research Center (CERC).

This report is published, with only minor editing, as received from
the contractor; results and conclusions ere those of the author and do not
necessarily represent those of CERC or the Corps of Engineers.

The report was prepared by Leon E. Borgman, University of Wyoming,
Laramie, Wyoming, under CERC Contract No. DACW72-72-C-0001. Dr. D. Lee
Harris, Chief, Oceanography Branch, was the CERC contract monitor for
the report, under the general supervision of Mr. R.P. Savage, Chief,
Research Division.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th
Congress, approved 31 July 1945, as supplemented by Public Law 172,
88th Congress, approved 7 November 1963.

Des

OHN H. COUSINS
Colonel, Corps of Engineers
Commander and Director

VIII

IX

XI

XII

CLITA

XIV

XV

APPENDIX
A

B

CONTENTS

INTRODUCTION .

THE SPECTRAL COMPUTATION .

ESSENTIAL EQUIVALENCE OF THE FFT AND COVARIANCE METHODS.

FFT AVERAGING NEEDED TO GIVE SPECTRA EQUIVALENT TO
THOSE FROM THE COVARIANCE METHOD . Shicwla sw can

SUMMARY OF COMPUTATIONAL FORMULAS USED FOR THE
HURRICANE CARLA WAVE SPECTRA .

DISCUSSION OF HURRICANE CARLA WAVE SPECTRA .
STATISTICAL VARIABILITY OF THE SPECTRAL LINES

THE EMPIRICAL PROBABILITY LAW FOR THE SYMMETRICALLY
NORMED RESIDUAL

EMPIRICAL PROBABILITY INTERVALS FOR ie
BY SIMULATION api tees 0 .

COMPARISON WITH CHI-SQUARED PROBABILITY INTERVALS
FOR (fm)

COMPARISON OF THE EMPIRICAL DISTRIBUTION FUNCTION
OF THE SYMMETRICALLY NORMED RESIDUALS WITH THE
CHI-SQUARED VERSION

THE OUTLIER SPECTRAL LINES

WHY DOES CHI-SQUARED WORK FOR HURRICANE WAVES?
IMPLICATIONS FOR DIRECTIONAL SPECTRUM RELIABILITY
SUMMARY AND CONCLUSIONS

LITERATURE CITED .

EFFECTIVE WIDTH
EFFECT OF A DETERMINISTIC LINE ON THE SPECTRUM .

ASYMPTOTIC CHI-SQUARED PROPERTIES OF THE FFT SPECTRAL
LINES FOR NON-GAUSSIAN, M-DEPENDENT WAVE TRAINS

Page

15

16
18

18

50

50

64

75
78
81
82
82

86

87

88

93

18

19

Hurricane

Weights used in
to produce the

Characteristics

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Carla data analyzed

CONTENTS

TABLES

FIGURES

Carla spectrum record number

Carla data number 6878

Carla spectrum record number

Carla spectrum
Carla spectrum
Carla spectrum
Carla spectrum
Carla spectrum
Carla spectrum
Carla spectrum

Carla spectrum

record number

record number

record number

record number

record number

record number

record number

record number

Carla spectrum record number

the moving average of the spectral lines
estimates of the spectral density

of outlier spectral lines

6877

6879

6880

6881-1

6881-2

6882

6883

6884

6885

6886-1

6886-2

Residual analysis, Hurricane Carla data number 6877

Hurricane Carla data number 6878

Residual

Residual

Residual

Residual

Residual

analysis,
analysis,
analysis,
analysis,

analysis,

Hurricane

Hurricane

Hurricane

Hurricane

Hurricane

Carla data

Carla data

Carla data

Carla data

Carla data

number 6879
number 6880
number 6881-1
number 6881-2

number 6882

Page

16

7,

79

19
19
20
20
21
21
22
22
235
ZS
24
24
26
27
28
29
30
31

32

4]

42

CONTENTS

FIGURES —Continued

Residual analysis, Hurricane

Residual analysis, Hurricane

Residual analysis, Hurricane

Residual analysis, Hurricane

Residual analysis, Hurricane

Scatter diagram;

Scatter diagram;

Scatter diagram;

Scatter diagram;

Scatter diagram;

Scatter diagram;

Scatter diagram

a

Scatter diagram;

Scatter diagram;

Scatter diagram

’

Scatter diagram;

Scatter diagram;

Cumulative distribution

Cumulative distribution

Cumulative

Cumulative

Cumulative

Cumulative

distribution

distribution

distribution

distribution

data number

data number

data number

data number

data number

data number

data number

data number

data number

data number

data number

data number

Carla data

Carla data

Carla data

Carla data

Carla data

6877

6878

6879

6880

6881-1

6881-2

6882

6883

6884

6885

6886-1

6886-2

function; data
function; data
function; data
function; data
function; data

function; data

number 6883

number 6884

number 6885

number 6886-1

number 6886-2

number 6877

number 6878

number 6879

number 6880 .

number 6881-1

number 6881-2 .

Page
33

34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
51
51
52
52
53

53

43

44

45

46

47

48

49

50

Sil

SZ

53

54

55

56

S57

58

59

60

61

62

63

64

65

66

Cumulative

Cumulative

Cumulative

Cumulative

Cumulative

Cumulative

distribution

distribution

distribution

distribution

distribution

distribution

CONTENTS

FIGURES — Continued

function;
function;
function;
function;
function;

function;

data

data

data

data

data

data

number

number

number

number

number

number

Probability density; data number 6877

Density function; data number 6878

Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability
Probability

Probability

density;
density;
density;
density;
density;
density;
density;
desnity;
density;
density;
intervals
intervals
intervals
intervals

intervals

intervals

data

data

data

data

data

data

data

data

data

data

for

for

for

for

for

for

number 6
number 6
number 6
number 6
number 6
number 6
number 6
number 6
number 6
number 6
A; data
B; data
D; data
p; data
p; data

p; data

879
880
881-1
881-2
882
883
884
885
886-1
886-2
6877

number

number 6878

6882

6883 .

6884

6885

6886-1

6886-2

number 6879 .

number 6880
number 6881

number 6881

=

=2

Page

54
54
55
55
56
56
57
Sh
58
58
59
59
60
60
61
61
62
62
65
65
66
66
67

67

67

68

69

70

ol

72

73

74

Probability intervals
Probability intervals
Probability intervals
Probability intervals
Probability intervals

Probability intervals

for

for

for

for

for

for

CONTENTS

FIGURES — Continued

Ratio of probability interval

spectrum estimate .

Comparison of the distribution function; Hurricane

Carla data 6883 .

; data

data

data

data

data

data

number 6882

number 6883 .

number 6884

number 6885

number 6886-1

number 6886-2

bounds to the

Page
68

69
70
71
Wz

73

74

83

THE STATISTICAL ANATOMY OF OCEAN WAVE SPECTRA

by

Leon E. Borgman

I. INTRODUCTION

In the future, more and more ocean engineering design considerations
will involve the wave energy spectrum. Typically, various secondary
calculations will be made from the spectral and cross-spectral estimates
at and between various space locations. The statistical reliability of
the values obtained from the secondary calculations depend critically on
the inherent statistical variability of the spectral estimates.

The estimation of directional wave spectra is obtained by just such
secondary calculations on the auto- and cross-spectral densities or
corresponding finite Fourier transform coefficients for various wave
properties measured at one or more space locations. The reliability of
the directional spectrum depends both on the method of computation and on
the intrinsic statistical variability of the Fourier coefficients or spec-
tral estimates. Such characterizing quantities as the main direction of
wave travel for a given wave frequency or some measure of the arc of
directions from which waves of a given frequency are coming each have their
own confidence intervals which ultimately relate back, through the method
of calculation, to the spectral and Fourier coefficient variability.

Over the years various theoretical probability relations have been
derived which apply to linear waves (Pierson, 1955; Goodman, 1957;
Blackman and Tukey, 1958). However, engineers are usually concerned with
wave heights large enough to make linear assumptions questionable. Waves
in hurricanes and other severe storms are prime examples of this situation.
Yet it 1S just in such situations where probability confidence statements
for the wave spectra or for derived secondary quantities are needed.

In the following report, the statistical variations in wave energy
spectral estimates for hurricane waves are examined empirically for 12
separate intervals of wave record measured during Hurricane Carla (Septem-
ber 1961). The measurements were made on a Chevron 0il Company platform
in South Timbalier Block 63, Gulf of Mexico, in a 100-foot water depth.
Hurricane waves were chosen for the analysis because they would illustrate,
in exaggerated form, the effects of departures from linearity on the
Statistical variability in spectral estimates.

Various aspects of the study were reported at the 13th International
Conference on Coastal Engineering in Vancouver, B.C., 10 to 14 July 1972,
for one 20-minute record (Borgman, 1972). This study gives the analysis
for the whole storm and develops certain implications and consequences of
the empirical results.

II. THE SPECTRAL COMPUTATION

Two basic methods have been used in the past for computing the wave
spectrum. The earlier one was based on the covariance function which was
then numerically Fourier transformed and smoothed. That is, if Nn
(n = 0,1,2..., N-1) is the water level elevation above mean water level,
then the covariance function is:

N-k-1

Neuter
Gh SS ear Ss "A"n+k ° @)

(The quantity, N, is 4,096 for the analysis of Hurricane Carla.) Usually

ey would be negligible for k larger than some value, say k,. Thus,
the N numbers would be adequately summarized in the k,+1 values of Cy,

Ordinarily, km is selected to be around one-tenth of N for most
analysis of this type. The spectral density would be obtained from the

numerical transform of the Cy:

a Sie R mn
p(fy) = At [& > DO De oF cos a Cs cos rn | , (2)
q=1
where
fT (3)
for r=0,1,3,..., k, (Blackman and Tukey, 1958). The quantity, At, is the

timelag between successive measurements of n,(At = 0.2 second for the
Hurricane Carla data).

The computation of equation (1) entails a loss of information (N values

replaced by k, values). This causes p(f,) in equation (2) to be a smoothed

version of the true spectral density and distorts or eliminates features
of the spectrum. The method of computation prevents the user from seeing
aspects of the spectrum which may be important.

The second method for computing the spectral density, which has come
into wide favor during the last few years, is based on the application of
the fast Fourier transform computing algorithm to the water level eleva-
tions (Borgman, 1973). Complex-valued Fourier coefficients, A,, are
obtained (or ™m—OF 25a N=) bye:

N-1 N-1
277 ; .. 2T7mn :
A, = At ye ky COS coo iAt » Mn) SOD sa We aD (Ct)
n=0 n=0

Thus, Um is the cosine transform while V, is the sine transform of the
water level elevations. The spectral lines are then computed by the
formula,

P(fm) = (Um + Vin)/(NAt) , (5)
where
m
fn = WE (6)
for m=0,1,2,3,..., N-l. These N spectral lines can be computed with

great rapidity on a digital computer with the fast Fourier transform (FFT)
procedure (Cooley and Tukey, 1965; Robinson, 1967).

The frequency
fyy = 1/2At (7)
is called the Nyquist frequency. A symmetry relation
Pen) = DCENem) (8)

holds because of intrinsic mathematical properties of equations (4), (5),
and (6). Hence, it is only necessary to specify PGES) LOT SO <msN/- Zi Gane,
for frequencies between zero and the Nyquist frequency). The spectral
density defined in equation (5) is defined by analogy to the conventions
used by Blackman and Tukey (1958) to be a two-sided spectral density in
which only the right-hand side is reported. That is, the total variance

Nevers
a Capt.

of the water level elevations would be equal to 2

The second procedure, involving the fast Fourier transform, was
selected for the spectral computations for Hurrican Carla because it
incurred less loss of information than the covariance procedure. The FFT
method permits the inspection of all 2,048 spectral lines for frequencies
up to the Nyquist frequency before the lines are averaged to yield a

smooth estimate, p(f), of the spectral density. The covariance method
gives only the smoothed spectral density.

However, conclusions concerning statistical confidence for spectral
estimates based on data computed with the FFT method are valid also for
spectral estimates based on the covariance method. The two procedures
for computing the spectral density give essentially the same result.

III. ESSENTIAL EQUIVALENCE OF THE FFT AND COVARIANCE METHODS
The covariance procedure gives spectral estimates which approximate a

smoothing of the population or "true'' spectral density. The effective
width of the smoothing is approximately 1/(2k,At), where k,At is the

maximum lag used in the covariance estimate (i.e., k, is the maximum value

of k used in computing equation (1)) and At is the time increment between
water level measurements (Blackman and Tukey, 1958). (See App. A for the
definition of effective width.) If "hamming'' smoothing,

pC) = WAC) = CAC) = CAC aaa) . (9)

is then made, the effective smoothing width for the estimates, DACEDE
is changed to 1/(k,At) (Blackman and Tukey, 1958).

The FFT method also yields estimates which are smoothed versions of
the true underlying population spectral density. This can be seen from
the following derivation.

The true or population spectral density is defined as the integral
Fourier transform of the theoretical covariance function. That is,

Ge) = JailinGe)) MCE sy (10)
where E[*] denotes the expectation operator and

oo) .
-127£T

p(f) =| C(t) e dt (11)

(Blackman and Tukey, 1958). Since C(t) is symmetric about T = 0 for
stationary stochastic processes, it follows that equation (11) reduces to:

foe)

p(f) - f C(t). Cos, (Zee) Ge. (12)

Equations (1) and (2) are obvious analogues to equations (10) and (12).
It is not immediately obvious that the same thing can be said for equa-
tions (4) and (5), although it is true there also. To see this, let

N-1

a it
Cy =. N Le Nn Nn+k ’ (13)
n=0

where N = 4,096 in the context of the Hurricane Carla records and the n
for n>N and n<O required for the computation of equation (13) are
defined by periodicity as:

= Thay. (14)

Equation (13) will give almost the same estimate of the covariance function
as equation (1) for O<k<N/2. There is, of course, a little distortion

related to the effects of equation (14). However, this will be over-
whelmed by the averaging against the other "within sequence" lag products
provided water level observations separated by more than k, time incre-

ments are essentially independent of each other and k, is small.

The periodicity introduced in equation (14) causes a corresponding
periodicity in CL.

CMC euCyay. for y.N/2<k<N (15)

An approximation to equation (12) would be
N/2
Pea) tye NC Hicos) (27kkAe)y ar! (16)

k=t +]

With the introduction of equation (15) and the definition of f, (equa-
tion 6), this becomes:

N-1 N-1

nw n~ m an

p(fm) = At » Ck cos (2nsRe at) = At ye Ge cosm(2imk/N)) ven) l7)
k=0 k=0

/

The last equation is algebraically equivalent to equation (5) although
it appears somewhat different. The equivalence can be seen after equation

(13) is substituted for Cy, and the transform is shifted to exponential

form. Thus,

Z|

N-1 N-1

fs 1 a

Ge) = We », ( Ny Mea exp (-i21mk/N)
k=0 n=0

N-1 N-1
= ot yy In, exp (+4 2nmn/) [oe exp (-i2mm(ntk)/N]
k=0 n=0
N-1 Neal
= = In, exp c2nan/X) i n exp (-idnmk/n| (8)
n=0 k=0

after the introduction of the periodicity assumed in equation (14). It
follows that

Gen) = (ny Aa) /NiNe S (UE WE)/NieY (19)

where Am denotes the complex conjugate of Am. This is the desired con-
clusion and equations (5) and (17) have been shown to be equivalent.

Returning to the question of the amount of spectral smoothing involved

in the estimate, DiGawe define the finite Dirac comb (Blackman and Tukey,
1958) as:

c-1

Se At ‘

Vo (t3at) = 5 6 (t + ¢ At) + at De S(t - kat) + 5 6(t - ¢ At) ,
k=-c+l1 (20)

where 6(x) is the Dirac function which has the properties:
if SCgeb<e = i (21)

SG) sO | if x#0 (22)
and for any bounded function g(x):

f ste - etdax = afc) - (23)

Now suppose the sampling questions involved C, are ignored and C(kAt)
1s inserted in) placenot ) Cy. an) equation) i@o)e- Dheny,

foe}

p(Eq) =i Ga) Yayo Cents) Ge Nake (24)

The right-hand side of this equation is the Fourier transform of
iG) Vn/2 (CegAe)l Thus, by Fourier transform theory p(f,) is the convo-

lution of the transforms of the separate functions or

eS le (25)

=-0O

P(fm) ~ p(£m) *
k

In the above p(f,) is the transform of C(T) by equation (11) and
© k J !
He Qo (F - ie) is the transform of Vn/2 (t;At) (Blackman and Tukey,

1958). The function Qo(f) is (Blackman and Tukey, with Ty = NAt/2):

_ sin(Tf£N At)
Zo = ——s (26)

Hence,

2

B(f_) ;

2, i Qh Eni x - y)p(y)dy . (27)

This shows that DiC) to the extent that Cy behaves like C(kAt), is
the result of smoothing with the Qjo(f) function and then summing or
aliasing as indicated by the summation.

The function Qo(f) has an effective width of (1/NAt). Hence, 9 (Ge)

represents a smoothing approximately over a frequency interval of (1/NAt).
This is the spacing between the f, in equation (6). It follows that
2ach FFT spectral estimate represents approximately a smoothing of p(f)

WereheIncernval t= A @y/2NAE):

It should be noted that problems related to the side lobes of Qo(f),
the aliasing, and the sampling variability have been ignored in the above
jiscussion. These effects will now be discussed briefly. The distortion
due to aliasing can be mitigated by choosing the Nyquist frequency suf-
ficiently large. The sampling variability shows up in the statistical
fluctuations of the estimates and can be examined from that perspective.
If the true spectra are relatively smooth and linearly changing, the side
lobes of Qg(f) will compensate for each other somewhat. However, if the
true spectra are not smooth, there will be unavoidable leakage of large
spectral departures or spikes into neighboring spectral estimates.

IV. FFT AVERAGING NEEDED TO GIVE SPECTRA EQUIVALENT TO THOSE
FROM THE COVARIANCE METHOD

From the previous section, the covariance spectral estimates (after
hamming) represent a smoothing over the frequency interval of width,
(1/ky At), while the FFT spectral estimates involve a smoothing over a
frequency interval of width, (1/N At). Hence, the number of FFT spectral
lines that should be averaged together to yield an estimate with approxi-
mately the same smoothing as the covariance spectral estimates is:

: (1/km At)

number =
(1/N At)

Psi (28)

V. SUMMARY OF COMPUTATIONAL FORMULAS USED FOR THE HURRICANE CARLA
WAVE SPECTRA

Twelve pieces of record at various times in the storm (Table 1) were
chosen for analysis. Each piece consisted of N = 4,096 values of water
level elevations taken at a time increment of At = 0.2. Thus, each of
these records was NAt = 13.65 minutes long. The frequency scale incre-
ment would be Af = 1/NAt = 0.00122 sec.~!. The Nyquist frequency for
this choice is fy = 1/2At = 1/0.4 = 2.5 sec.~} which corresponds to a

period of 0.4 second. Energy in waves with periods smaller than 0.4 second
would thus be aliased into lower frequencies. An examination of the final
spectra produced little evidence of much energy past this selected Nyquist
frequency.

Table 1. Hurricane Carla data analyzed.

Date Data code

8 Sept. 1961

9 Sept. 1961

10 Sept. 1961

The 4,096 water level elevations were transformed by the fast Fourier
transform algorithm to yield Fourier coefficients, for O<m<N:

4,095
A, = At So. Tn eo 12mmn/N (29)
n=0
The spectral lines were then computed from

DCE) S |Anle/ CG1uOs2). 5 (30)

when | Am | denotes the complex modulus of A, . The spectral density was
estimated by a moving average of the spectral lines:

B(fm) = Ei) B(f-5)/ ae (31)

where

W5 = exp (Gas /ats) ae (32)
These weights are, thus, Gaussian smoothers with a standard deviation of

o = 3Af = 0.00366 sec.~!. The numerical values of the Wj are given in
Table 2. In the vicinity of zero frequency, the subscript j in equation
(31) was summed over the possible values and the divisor normed the weights
appropriately. At m= 6, for example, j was summed over -6<j<13 (the
left tail of the moving average was truncated off). we

Table 2. Weights used in the moving average
of the spectral lines to produce
the estimates of the spectral density.

au
=
Cat,

0)
1
2
3
4
5
6
ii
8

SOLO OOO Oo Oroiore or
(>)
lon
ea)
—N

Note: w_; = w:.

The effective width of the Gaussian smoother in equation (32) is V2T oO
where oO = 3Af, if lags are measured on the frequency scale and o =3,
if lags are measured relative to the number of spectral lines. The effec-
tive width in terms of number of spectral lines encompassed is thus

3 V2n = 7.52, or rounding to the nearest integer, 8 spectral lines. By
equation (28), the FFT spectral density estimates so produced would be
comparable to covariance spectral density estimates with maximum covariance
lag derived from:

N
se 8} (33)
kn
or
4,096
ky = Se = S12 (34)

VI. DISCUSSION OF HURRICANE CARLA WAVE SPECTRA

The spectral lines (shown as dots) and the spectral density estimates
averaged from the lines (shown as a line or a string of pluses) are given
in Figures 1 through 12.

Close examination yields two very interesting facts. First, the peak
values of the spectral densities are related closely to one or two spec-
tral lines. Only in record 6883 is the peak related to four exceptionally
large lines. In the other cases it is always one or two. Second, these
exceptionally large spectral lines do not seem to persist. The two
members of record pairs (6881-1 and 6881-2; 6886-1 and 6886-2) are sepa-
rated from each other by only 20 minutes. Yet in both cases, exception-
ally large spectral lines are present in one member of the pair but not
in) the others.

A general examination of the spectral lines versus the spectral
density estimates cannot help but develop a sense of healthy skepticism
concerning the general reality of the fine structure in the spectral
density. JAlson ite ais tellit thats the texceptionaliliy larce spectralaglamiess
often twice as large as the nearest other line value, must belong to
another population from the rest of the lines. Perhaps some sort of
resonant phenomenon is creating a main wave train with the rest of the
lines functioning as superimposed noise.

VII. STATISTICAL VARIABILITY OF THE SPECTRAL LINES

The spectral density was subtracted from each spectral line for
6<m<305 to provide 300 residual values, R, ,

Rn = P(t) - B (£m) : (35)

A positive and negative standard deviation for the residuals were computed
with the weights introduced in Table 2. Let J, be the values of the
index m for which the residuals are positive while J_ are the values
of m for which the residuals are negative. The positive and negative

variance of the residuals are defined as:

2) ye -R2 . :
OF eet 2 wees) / » wi (36)
JES 5, jeJ,
625 a w5Ra-3 / ee (37)
Ain aM J m-) 4 J
jeT_ jeJ_
where Mev emeansm bDelongs Comune Seta Oteuus wisi O) will be a moving
+,m

average estimate of the root-mean-square (rms) of the positive residuals
in the vicinity of the frequency f,. A similar statement relative to

O_ ,, and the negative residuals will also hold.

Spectral Lines

400

Raw Spectral Lines ccceece
Smoothed Spectrum
Water Level Variance = 9.7ft®

8

ao
Variance Approx. Equals 2fsitidt
0

Spectral Density ( ft®s)
nm
8

100

(0) 005 0.10 0.15 0.20 ; 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 1. Hurricane Carla spectrum record number 6877,
0600 hours, 8 September 1961.

400
SOON CaM ten MMI We nh es PY SARL ve Gaussian Smoothing by Weighted
i Values (o = 3Af)
—— Smoothing by Blocked Averaging
Water Level Variance = 12.5 ft?
2)
Variance Approx. Equals 2fs(tdt
200 0

100

Oe Oy ep = a7 -
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Frequency (c/s)

Figure 2. Hurricane Carla data number 6878, 1200 hours,
8 September 1961. Various smoothings of spectral data.

400

Row Spectral Lines DODCINA0
300

Smoothed Spectrum

= Water Level Variance = 13.6 ft?
oof @o
os Variance Approx. Equals 2fs(t rat
iS 0
= 200
oa
2
i
a
wo
100
0 o LZ a rf be
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)
Figure 3. Hurricane Carla spectrum record number 6879,
1800 hours, 8 September 1961.
400
Raw Spectral Lines ceceeee
Smoothed Spectrum
300 Water Level Variance = 12.7 t®
= o
2 Variance Approx. Equals 2fs(t)at
= 0
= 200
a
‘2
Ss
a
n”

100

0 0.05 0.10 0.15 0.20 0.25 03 0.35 0.40
Frequency (c/s)

Figure 4. Hurricane Carla spectrum record number 6880,
0000 hours, 9 September 1961.

20

Spectral Density ( ft®s)

Spectral Density ( ft®s)

400 » Gee

300 .
9 Raw Spectral Lines
Smoothed Spectrum
° Water Level Variance = 19.8 ft?
@o
200 Variance Approx. Equals eyfetiss
100
0 5 om “A iS esis”! A Din ye oe
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)
Figure 5. Hurricane Carla spectrum record number 6881-1,
0600 hours, 9 September 1961.
500
Raw Spectral Lines ceereee
400 Smoothed Spectrum
Water Level Variance = 16.1 ft?
oO
oer Variance Approx. Equals 2fscryat
0
300
200
100

te) 0.05 0.10 0.15 0.20 0.25 0.30 0.35, 0.40

Frequency (c/s)

Figure 6. Hurricane Carla spectrum record number 6881-2,
0620 hours, 9 September 1961.

2

Spectral Density ( ft?s)

Raw Spectral Lines
Smoothed Spectrum

= 300 is Water Level Variance = 23.3 ft?
ok [--}

= Variance Approx. Equals 2f s(f)df
= 0

<

J

a

= 200

2

nw

100

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 7. Hurricane Carla spectrum record number 6882,
1200 hours, 9 September 1961.

606 o—a
742 o—_ =
73So—_
606°

400 rs
Raw Spectral Lines eeeee
cs Smoothed Spectrum
7 Water Level Variance = 25.4ft*
ao
300 Variance Approx. Equals 2fslt )df
0
200

100

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 8. Hurricane Carla spectrum record number 6883,
1500 hours, 9 September 1961.

22

Spectral Density ( ft®s)

500 as.

400
: Raw Spectral Lines -ccceece
Smoothed Spectrum

2 Water Level Variance = 19.1 ft?
= 300 o
> F Variance Approx. Equals 2f s(tyat
= 0
2
J
a
2
3 200
a
wn

100

ty Lol n
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 9. Hurricane Carla spectrum record number 6884,
1800 hours, 9 September 1961.

om ||

1123" * 868
Raw Spectral Lines e°°°°
5 Smoothed Spectrum

Water Level Variance = 22.6ft®

300 o
Variance Approx. Equals 2fs(tat

0
200

100

{0} 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 10. Hurricane Carla spectrum record number 6885,
2100 hours, 9 September 1961.

(2)

Spectral Density ( ft?s)

Spectral Density ( ft®s)

500

400 Raw Spectral Lines ec cee
Smoothed Spectrum

Water Level Variance = 22.1 ft?

@
ae Variance Approx. Equals 2fsitlat
0)

wo
fo}
fo}

n
fo}
Oo

100

0 0.05 0.10 0.15 0.2 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 11. Hurricane Carla spectrum record number 6886-1,
0000 hours, 10 September 1961.

i |

1026° °581

Raw Spectral Lines ececece
Smoothed Spectrum

300 Water Level Variance = 22. Oft?

@o
Variance Approx. Equals 2fsirver

200

100

Shs: rh See ao UL
10) 0.05 0.10 0.15 0.20 0.25 030 0.35 0.40
Frequency (c/s)

Figure 12. Hurricane Carla spectrum record number 6886-2,
0020 hours, 10 September 1961.

24

Symmetrically normed residuals (SNR) are then defined as:

a SO, m > ae I >

SNR = (38)
Lain rer aii iea (allen

Thus, the symmetrically normed residuals are ratios of the residuals to
the rms positive or negative residuals, as the case may be.

Originally, the analysis was made with rms residual, disregarding
whether the residuals were positive or negative. This, however, led to
ridiculous results particularly in the subsequent use in simulation. The
use of different norming divisors for negative and positive residuals
avoided these peculiarities completely.

The residuals and the corresponding SNR's are plotted in Figures 13
through 24. The symmetrical normalization seems to very adequately pro-
duce a uniform cloud of points. There does not seem to be any tendency
for the SNR's to be systematically large or small at any particular
frequency. Generally about 60 percent of the SNR's fall below zero.

As a check against sequential dependence among the SNR's, neighboring
pairs of SNR's were plotted on a scatter diagram in two-dimensional space.
The first member of the pair of SNR's was the x-coordinate while the
second member was the y-coordinate for the plotted point. Any tendency fo
big values to follow big values (or the reverse) would show up as a clus-
tering tendency on such a plot. Complete independence, on the other hand,
would show up as a uniform cloud of points.

The plots of this type for the 12 Hurricane Carla data sets are given
in Figures 25 through 36. The point scatter is really quite uniform for
all of the records with no obvious dependencies showing up. However, it
should be emphasized that this type of examination only reveals overall
average dependencies. There may be dependencies between neighboring SNR's
at certain frequencies which are counteracted by opposing dependencies
at other frequencies. However, the earlier graphs (Figs. 13 through 24)
would show any strong dependencies tied to frequencies if they were
present.

Everything considered, the analyses strongly support the conclusion
that the spectral fluctuations have been successfully decomposed into a
smoothed spectrum plus a constant (either O, or O_) times independent
random noise. The smoothed spectrum and the constants are frequency
dependent. The noise apparently does not depend on frequency. In symbols,
this decomposition can be written:

B(fm) = B(Em) + ¢ ° (SNR) , (39)

25)

Symmetrically Normed Residuals
(dimensionless)

100

g
=

2 0

i=]

>

3

yn

a

ae

- 100

-200

) 0.05
Paleubes) IG

0.10 0.15 0.
Frequency (c/s)

Residual analysis,

26

20 0.25 0.30 0.35 0.40

Hurricane Carla data number 6877.

Symmetrically Normed Residuals

Normed Residuals
(dimensionless )

Residuals

Residuals (ft2/s)

{e) 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 14. Hurricane Carla data number 6878, 1200 hours,
8 September 1961. Residual analysis, Gaussian
smoothing on spectral lines.

Call

Symmetrically Normed Residuals
(dimensionless)

200

100

Residuals (f12/s)

-100

fe) 0.05

Figure 15.

Arrows indicate off plot points which lie
in the designated directions

0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Residual analysis, Hurricane Carla data number 6879.

28

Symmetrically Normed Residuals
(dimensionless )

200

100

Of ve -- ee

Residuals (ft?/s)

- 100

9 0.05

Fugue lor

0.10

Residual

Arrow indicates an off plot point which lies
in the designated direction

So ee cee e Spl Acs) Sam ho, oe

0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

analysis, Hurricane Carla data number 6880.

29

Symmetrically Normed Residuals
(dimensionless)

100

fo)

Residuals (ft2/s)

-100

Figure 17.

0.05 0.10 0.15

Residual analysis,

Arrows indicate off plot points which lie
in the designated directions

0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Hurricane Carla data number 6881-1.

30

fo)

Symmetrically Normed Residuals
(dimensionless)

200

Arrow indicates an off plot point which lies
in the designated direction

100

Bw Pane em msey* S 5 0 SY A LA ii A eh en et Tonto

Residuals (ft/s)
°

'
(o}
oO

{0} 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Figure 18. Residual analysis, Hurricane Carla data number 6881-2.

3|

Symmetrically Normed Residuals
(dimensionless)

100

Residuals (ft/s)
°

-100

0 0.05

Fatoumen eS

0.10 0.15

Arrows indicate off plot points which lie
in the designated directions

OEE CO Ooo a0 0 ampere iceoy mn 04
Mate Ohne cette at peste cme gett e nee a seme pal” Nye 40 oc east

0.20 0.25 0.30 0.35

Frequency (c/s)

Residual analysis,

Hurricane Carla data number

32

0.40

6882.

Symmetrically Normed Residuals
(dimensionless)

200

100

Residuals (ft?/s)
°

'
fo)
°o

0 0.05

Figure 20.

Arrows indicate off plot points which lie
in the designated directions

5
uote AS Th DAV ARe Metabo.

0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Residual analysis, Hurricane Carla data number 6883.

38

Symmetrically Normed Residuals
(dimensionless )

100

Residuals (f12/s)
[o)

'
fo}
fe)

ed eee Bote

0.05

Figuae 721"

Arrow indicates an off plot point which lies

in

the designated direction

0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Residual analysis,

34

Hurricane Carla data number 6884.

Symmetrically Normed Residuals
(dimensionless )

200

100

© meme ow og

Residuals (ft2/s)
fo)

- 100

(0) 0.05

Figure 22.

Arrows indicate off plot points which lie
in the designated direction

os fe ANS QV TN eet Re ee tee

0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Residual analysis, Hurricane Carla data number 6885.

35

Symmetrically Normed Residuals
(dimensionless )

100

Residuals (ft2/s)

i}
(oe)
(oe)

-200
te)

Figure

0.05

Dore

Arrow indicates an off plot point which lies
in the designated direction

(0)) re

0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Residual analysis, Hurricane Carla data number 6886-1.

36

Symmetrically Normed Residuals
(dimensionless )

200

100

Bea ey,

Residuals (ft?/s)
(eo)

'
{o)
fe)

-200
O

Figure 24.

Arrows indicate off plot points which lie
in the designated directions

ehd ed pens ptleoens Gime bee wne

j 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Frequency (c/s)

Residual analysis, Hurricane Carla data number 6886-2.

Si,

Subsequent Symmetrically Normed Residuals (dimensionless )}

Figure 25.

-1.0 -0.5 ce) 0.5 1.0 IES) 2.0
Antecedent Symmetrically Normed Residuals (dimensionless)

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6877.

38

Subsequent Symmetrically Normed Residuals (dimensionless)

Scatter Diagram (SNR), Versus (SNR) m4,

-4

Figure 26.

-3 =e, -I (0) I 2 3
Antecedent Symmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6878.

39

Subsequent Symmetrically Normed Residuals (dimensionless )

Cie

-1.0 -0.5 (0) 0.5 1.0 15 2.0
Antecedent Symmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6879.

40

Subsequent Symmetrically Normed Residuals (dimensionless )

-2.0

Figure

28.

-1.0 -0.5 10} 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6880.

4

Subsequent Symmetrically Normed Residuals (dimensionless )

2.0

0.5

=0!5

-2.0
-2.0 I)

Figure 29.

=1.0 0:5 fe) 0.5 1.0 1.5 2.0
Antecedent Syimmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6881-1.

42

Subsequent Symmetrically Normed Residuals (dimensionless )

Figure 30.

-1.0 -0.5 {e) 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless)

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6881-2.

43

Subsequent Symmetrically Normed Residuals (dimensionless )

~2.0 -1.5

JBlal feqblaetsy | Spd

-1.0 -0.5 0 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless }

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6882.

44

Subsequent Symmetrically Normed Residuals (dimensionless )

Sr.

-1.0 {0} fe) 0 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless }

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6883.

45

Subsequent Symmetrically Normed Residuals (dimensionless )

-2.0 -1.5

Figure 33.

-1.0 -0.5 (o} 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6884.

46

Subsequent Symmetrically Normed Residuals (dimensionless )

34.

-1.0 -0.5 te) 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6885.

47

Subsequent Symmetrically Normed Residuals (dimensionless )

35.

-1.0 -0.5 (e) 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless )

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6886-1.

48

Subsequent Symmetrically Normed Residuals (dimensionless )

Figure 36.,

-1.0 -0.5 fo) 0.5 1.0 1.5 2.0
Antecedent Symmetrically Normed Residuals (dimensionless }

Scatter diagram: Sequential pairs of symmetrically
normed residuals, data number 6886-2.

49

where
oO, 5 abe GR S @

Cir { , (40)
o if SNR < 0

and SNR denotes the SNR regarded as a random variable.
VIII. THE EMPIRICAL PROBABILITY LAW FOR THE SYMMETRICALLY NORMED RESIDUAL
The cumulative distribution function for the SNR is defined as:

FoNR (w) = P [SNR < w] , (41)

where P[°] is the probability of the event specified within the brackets.
This distribution function can be estimated from the 300 values of the
SNR's for each record of Hurricane Carla. Let (SNR), be SNR's ranked

in order of increasing size:
(SNR) 1 < (SNR), < (SNR), Oe < (SNR) 369 . (42)

A statistically reasonable estimate of Fonr (Wy) for

We = (SNR), (43)
is
E (wap piso (44)
SNR* k 301

(Gumbel, 1954). Thus, a graph of Pon (wy) versus w, for k = 1,2,3,°**,300

gives the distribution function estimate. The graphs are shown in Figures
37 through 48.

The corresponding probability densities may be obtained by differen-
tiating the distribution function numerically. For the present study,
this was done by selecting a band on the SNR axis which is 0.5 unit wide
and fitting a least square line to all the ranked points lying within the
band. The slope of the line is the probability density estimate assigned
to the midpoint SNR value for the band. This was repeated for all 300
possible midpoints on the SNR axis. The resulting probability densities
are given in Figures 49 through 60.

IX. EMPIRICAL PROBABILITY INTERVALS FOR p (fm) BY SIMULATION

Suppose a new spectral density estimate, io 2.) is developed by

simulation from equation (39) by the following procedure. A random num-
ber, uniformly distributed on the interval, (0,1), is generated in the
digital computer. One of the pieces of record is selected for study and

50

Cumulative Distribution Function

0.8

0.2

0.8

0.6

Uniform Random Number = 0.532

10
SS
10
tt

0.4 H uw
13
12
te
I's
12
|
1

0.2 13
16
ja
!
‘a
2
He)

(0)
-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 15 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 37. Cumulative distribution function, Hurricane Carla data
number 6877, 0600 hours, 8 September 1961.

OOO STOO OS

FP
oe
ef

Empirical Distribution Function of
SNR

4 90% Cont. Int on True Distribution
D003 Function (Kolmogorov Nonporametric
Method )

-2.0 -1.5 -1.0 -0.5 fe) 0.5 1.0 15 2.0 2.5

SNR realization = x (dimensionless)

Figure 38. Cumulative distribution function, Hurricane Carla data
number 6878.

S|

Cumulative Distribution Function

Cumulative Distribution Function

06

0.2

-20 1.5 -1.0 -0.5 fo) 0.5 1.0 1.5 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 39. Cumulative distribution function, Hurricane Carla data
number 6879, 1800 hours, 8 September 1961.

0.8

06

0.2

-20 “1.5 -1.0 -05 10) 0.5 1.0 5 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 40. Cumulative distribution function, Hurricane Carla data
number 6880, 0000 hours, 9 September 1961.

ae

Cumulative Distribution Function

Cumulative Distribution Function

0.8

0.6

0.4

0.2

-2.0 -15 -1.0 -0.5 0 0.5 1.0 15 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 41. Cumulative distribution function, Hurricane Carla data

number 6881-1, 0600 hours, 9 September 1961.

0.8

0.6

0.4

0.2

0
-20 -1.5 -1.0 -0.5 (0) 0.5 1.0 15 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 42. Cumulative distribution function, Hurricane Carla data
number 6881-2, 0620 hours, 9 September 1961.

53

Cumulative Distribution Function

Cumulative Distribution Function

0.8

0.6

0.4

0.2

-2.0 -15 -1.0 -0.5 0 0.5 1.0 1.5 20

Symmetrically Normed Residuals ( dimensionless )

Figure 43. Cumulative distribution function, Hurricane Carla data
number 6882, 1200 hours, 9 September 1961.
1.0

0.8

0.6

0.2

-2.0 “1.5 -1.0 -0.5 0 0.5 1.0 15 2.0

Symmetrically Normed Residuals ( dimensionless)

Figure 44. Cumulative distribution function, Hurricane Carla data
number 6883, 1500 hours, 9 September 1961.

54

Cumulative Distribution Function

Cumulative Distribution Function

0.8

0.6

0.4

0.2

0 :
-2.0 -1.5 -1.0 -0.5 0 05 1.0 1.5 2.0

Symmetrically Normed Residuals (dimensionless )

Figure 45. Cumulative distribution function, Hurricane Carla data
number 6884, 1800 hours, 9 September 1961.

0.8

0.6

0.4

0.2

fe)
-2.0 “1.5 -1.0 -0.5 (0) 0.5 1.0 1.5 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 46. Cumulative distribution function, Hurricane Carla data
number 6885, 2100 hours, 9 September 1961.

59

Cumulative Distribution Function

Cumulative Distribution Function

0.8

0.6

0.2

-2.0 -1.5 -1.0 -0.5 0) 0.5 1.0 1.5 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 47. Cumulative distribution function, Hurricane Carla data
number 6886-1, 0000 hours, 10 September 1961.

0.8

0.6

0.4

0.2

io)
-2.0 1.5 -1.0 -0.5 ie) 0.5 1.0 1.5 2.0

Symmetrically Normed Residuals ( dimensionless )

Figure 48. Cumulative distribution function, Hurricane Carla data
number 6886-2, 0020 hours, 10 September 1961.

56

Probability Density

0.6

0.5

0.4

0.3

Probability Density

0.2

O
-2.0 -1.5 -1.0 -0.5 O 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 49. Probability density, Hurricane Carla data number 6877,
0600 hours, 8 September 1961.

-2.5 -2.0 -1.5 -1.0 -05 fe) 0.5 1.0 1.5 2.0 25
Symmetrically Normed Residuals (dimensionless)

Figure 50. Density function, Hurricane Carla data number 6878.

Sf,

06

0.5

Probability Density
[e) [o}
w p

oO
rN)

-2.0 -1.5 -1.0 -0.5 O 05 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 51. Probability density, Hurricane Carla data number 6879,
1800 hours, 8 September 1961.

0.6

0.5

©)
an

Probability Density

0.2

ie)
-2.0 -1.5 -1.0 -0.5 (0) 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 52. Probability density, Hurricane Carla data number 6880,
0000 hours, 9 September 1961.

58

0.6

0.5

0.4

0.3

Probability Density

0.2

O
-20 -1.5 -1.0 -0.5 oO 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 53. Probability density, Hurricane Carla data number 6881-1,
0600 hours, 9 September 1961.
06
0.5

0.4

0.3

Probability Density

0.2

-2.0 -1.5 -1.0 -0.5 0 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 54. Probability density, Hurricane Carla data number 6881-2,
0620 hours, 9 September 1961.

59

0.6

0.5

° °
(e) fh

Probability Density

°
rN)

(@)
-2.0 -1.5 -1.0 -0.5 (0) 0.5 1.0 1.5 2:0
Symmetrically Normed Residuals (dimensionless)

Figure 55. Probability density, Hurricane Carla data number 6882,
1200 hours, 9 September 1961.

0.6

0.5

oO
'b

°
w

Probability Density

0.2

0
-2.0 -1.5 -1.0 -0.5 O 0.5 1.0 re) 2.0

Symmetrically Normed Residuals (dimensionless)

Figure 56. Probability density, Hurricane Carla data number 6883,
1500 hours, 9 September 1961.

60

0.6

0.5

0.4

0.3

Probability Density

0.2

Oo
-2.0 -1.5 -1.0 -0.5 (o) 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 57. Probability density, Hurricane Carla data number 6884,
1800 hours, 9 September 1961.

0.6

0.5

0.4

0.3

Probability Density

0.2

(0)
-2.0 -1.5 -1.0 -0.5 (0) 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 58. Probability density, Hurricane Carla data number 6885,
2100 hours, 9 September 1961.

6l

06

0.5

0.4

0.3

Probability Density

0.2

(0)
-2.0 -1.5 -1.0 -0.5 (0) 0.5 1.0 1.5 2.0
Symmetrically Normed Residuals (dimensionless)

Figure 59. Probability density, Hurricane Carla data number 6886-1,
0000 hours, 10 September 1961.

0.6

0.5

0.4

0.3

Probability Density

0.2

0
-2.0 -1.5 -1.0 -0.5 O 0.5 1.0 1.5 2.0

Symmetrically Normed Residuals (dimensionless)
Figure 60. Probability density, Hurricane Carla data number 6886-2,
0020 hours, 10 September 1961.

62

the corresponding cumulative distribution function figure is picked from
Figures 37 through 48. The coordinate value on the vertical axis of the
distribution function which equals the uniform random number is located.
Reading horizontally from this coordinate value to the empirical distrib-
ution curve and then down vertically to the SNR axis yields a random SNR
value. This procedure is illustrated in Figure 37 by the dotted line.
The uniform random number is 0.532. The corresponding random SNR value
is -0.16.

The distribution function for the SNR values obtained by this proce-
dure will be identical to the graphed empirical distribution function
Fonr(w). This follows from the following argument. The SNR, so developed

are less than or equal to w if, and only if, the uniform random number
is less than or equal to Foyp(w). This is true because the two numbers

are tied together via the graphed curve. Hence,

(45)

>

P [random SNR < w] = P [U < Fonp(w)]

where U denotes the uniform random number. But by definition, the dis-
tribution function for a uniform random number is:

Ey Cy eek LU Se Se (46)

Hence, returning to equation (45),

P [random SNR < w] -Ip (U

[A

The above procedure is repeated for 300 independent uniform random
numbers to obtain 300 random SNR values. These 300 SNR values are just
as likely to have happened as the originally occurring values, provided
the decomposition in equation (39) is accepted as valid and provided the
independence assumption truly holds.

e Hence, equation (39) can be used with the 300 SNR values and the
DCE Ne O+,m and O_ im frequency functions to create a new set of
spectral lines, pace) These spectral lines might just as well occurred

as the original set if the random spectral fluctuations had accidentally
gone that way.

Finally, the 300 simulated spectral lines, Da Geen) are smoothed
according to equation (31) to produce new simulated spectral densities,

B* (fm) -

The above procedure in its entirety was repeated 900 times for each
of the 12 pieces of Hurricane Carla data. Thus, 900 statistically equiva-
lent spectral densities were generated by simulation for each hurricane
data record.

63

n
a

How much do the p*(f,) as a group differ from the spectral density,
D(E_) > used in the simulation? The answer to this question was developed
for the 10 frequencies, 0.070, 0.072, 0.074, 0.077, 0.083, 0.088, 0.101,
0.132, 0.168, and 0.243 sec.~!,. The 900 values of p* (fm) were ranked for

each frequency and the two values with ranks 45 and 855 were selected as
estimates of the 5th and 95th percentiles for p*(f,). These percentile

estimates are plotted versus frequency as dots in Figures 61 through 72.

X. COMPARISON WITH CHI-SQUARED PROBABILITY INTERVALS FOR A (£m)

If the sea surface is Gaussian, the spectral density will follow a
probability law closely related to a chi-squared random variable with 16
degrees of freedom (for the Hurricane Carla estimates) (Borgman, 1972).
Symbolically, i

nS a) |e
mew Tie ee

where p(f,,) denotes the true or population spectral density. Thus, if
denote the 5th and 95th percentiles for a chi-

2 2
SG 0.05 8) 2 Sa20.05
squared random variable with 16 degrees of freedom, then:

2 2
x Ge.) x PGE)
; | 16,0.05 ae ait) < 16,0.95 m
16 16

= 0.90 5 G49)

. D 2 .
The interval levanes DG) ye 16 , X16 0.95 p(£) i] 16 ) thus provides

a 90 percent probability interval for CGE DE However, p(fm) is not
known. As an approximation, f(f,,) may be substituted for p(f,). An
analogous approximation was made in the simulations. The resulting upper
and lower limits are plotted as asterisks in Figures 61 through 72 versus
each of the selected frequencies. The spectral density values p(fm) are

shown in the figures as pluses.

As was noted in an earlier paper based on an analysis of part of the
data (Borgman, 1972), the chi-squared probability intervals do not differ
excessively from the simulated probability intervals. They are both about
the same, although there are substantial variations from record to record.
The upper bounds show appreciably more scatter than do the lower bounds.
Another comparison of the two kinds of probability intervals is given in
Figure 73. The two ratios,

simulation upper bound / (£m)

and,

simulation lower bound / pice) ,

64

Spectral Density (ft2/s)

Spectral Density (ft2/s )

300

200

100

Figure 61.

300

200

100

Chi- Squared Probabilities *
Probabilities from Simulation °
Estimated Spectral Density , f ny

TRS ee
0.05 0.10 OnIS 0.20 0.25
Frequency (c/s)

Probability intervals for i, Hurricane Carla data
number 6877.

Chi- Squared Probabilities *
Probabilities from Simulation .

Estimated Spectral Density, p 4

0.05 0.10 0.15 0.20 0.25
Frequency (c/s)

Probability intervals for p, Hurricane Carla data
number 6878.

65

Spectral Density ( ft2/s)

Spectral Density ( ft2/s )

300

Chi- Squared Probabilities *
. Probabilities from Simulation °
° a
* Estimated Spectral Density, p 4
*
200 °
100
O
O OOS 0.10 0.15 0.20 (0).25)
Frequency (c/s)
Figure 63. Probability intervals for A, Hurricane Carla data
number 6879.
300
Chi- Squared Probabilities *
Probabilities from Simulation D
A
Estimated Spectral Density,p A
200
100

0) 0.05 0.10 ORS 0.20 0.25
Frequency (c/s)

Figure 64. Probability intervals for p, Hurricane Carla data
number 6880.

66

Spectral Density ( ft2/s )

Spectral Density (ft?/s)

400

*
Chi-Squared Probabilities *
ooo e Probabilities from Simulation °
Aa
Estimated Spectral Density, p 4
% A
200 i
CON ae
100 “f z *%
a a
K
0) 0.05 0.10 0.15 0.20 0.25
Frequency (c/s)
Figure 65. Probability intervals for 6, Hurricane Carla data
number 6881-1.
300
Chi- Squored Probabilities 36
Probabilities from Simulation °
* Estimated Spectral Density, p 4
200
100

¥
0) 0.05 0.10 0.15 0.20 0.25

Frequency (c/s)

Figure 66. Probability intervals for A, Hurricane Carla data
number 6881-2.

67

Spectral Density ( ft2/s )

600

500

Chi- Squared Probabilities *
Probabilities from Simulation e
400

Estimated Spectral Density , p A
300

200

100

B
O 0.05 0.10 0.15 0.20 0.25
Frequency (c/s)

Figure 67. Probability intervals for p, Hurricane Carla data
number 6882.

68

500 *

Chi-Squared Probabilities x
400 g Probabilities from ae °
x Estimated Spectral Density, p A

300

200

Spectral Density ( ft2/s

100

O 0.05 0.10 0.15 0.20 0.25
Frequency (c/s)

Figure 68. Probability intervals for 6, Hurricane Carla data
number 6883.

69

Spectral Density( ft2/s )

400

300

200

100

Figure 69.

Chi- Squared Probabilities *
Probabilities from Simulation °

“A
nN

Estimated Spectral Density, p A

0.05 0.10 0.15
Frequency (c/s)

Probability intervals for p,
number 6884.

70

0.20 0.25

Hurricane Carla data

Spectral Density ( ft2/s)

600

500

400

300

200

100

Figure 70.

0.05

Probability
number 6885.

Chi-Squared Probabilities
Probabilities from Simulation

Estimated Spectral Density §

0.10 0.15 0.20
Frequency (c/s)

*

A

0.25

intervals for p, Hurricane Carla data

al

Spectral Density ( ft2/s)

400

300

200

100

Evounei/ 1.

Chi-Squared Probabilities *
Probabilities from Simulation

aA
Estimated Spectral Density, p A

0.05 0.10 0.15
Frequency (c/s)

Probability intervals for D,
number 6886-1.

72

0.20 0.20

Hurricane Carla data

500

*

%

° Chi-Squared Probabilities *
400 Probabilities from Simulation °

PAY

Estimated Spectral Density, p A

Ww
(oe)
‘o)

ip’)
oe)
(e)

Spectral Density ( ft2/s)

100

@) 0.05 0.10 0.15 0.20 0.25
Frequency (c/s)

Figure 72. Probability intervals for p, Hurricane Carla data
number 6886-2.

Ud

Ovo GeO o¢'0 S20 Orage) GIO OlO GOO
(pete SE Iclch duo enon @ ns On ne te eae EY dso0¢ -
s etic’ Vv v
: ; ae
; u ; mi wt
See ee ar eh ea (sigewonTonon. = oe or ae dycccod
sabpsaAy ; ari M Mi

"oqzeuwTyso wnIzd0ds ay 03 SpuNog TeALOJULT ATT TGeqoid jo oTIeYyY “SZ 9aANBTY

(s/9) Aouanbesy

O14DY

74

are plotted versus frequency for all 12 data records. The average over
the 10 frequencies for each record is plotted on the right of the graph
above the frequency value, 4.0 sec.~!. For the chi-squared distribution
the corresponding values are theoretically,

95th percentile 6 (£m) / p(fm) = ‘Gis 0 95 / 16 = 1.64 (50)
Sth percentile p(f,) / P(f_) = X26 5 af) i648 0.80 2 Giy

These values are shown as dashlines in the figure.

On the average the two types of probability intervals agree fairly
well.

XI. COMPARISON OF THE EMPIRICAL DISTRIBUTION FUNCTION OF THE SYMMETRI-
CALLY NORMED RESIDUALS WITH THE CHI-SQUARED VERSION

If the chi-squared distribution is reasonably valid for (fm) » it

may also hold for the spectral lines, at least approximately. The theore-
tical relation to be checked for validity is:

SC / DD 296 0 2 (52)
The chi-squared analogy to o? in equation (38) is:

E | | Stem) = p (fn) } ; B(fm) > p (én) |

theoretical of

P(f,) 2 pig)
= pty) & [{ sey - 1} Tt
eh ie hae
By exact analogy:
theoretical o2 = ptm) | {x3 i at | x3 < 2 | ; (54)

Since the probability density for Ne random variable is:

ea W/2
5 A acon ny 2\\(0)
2 (w) = { i (55)
2 0 , for w <0

Us)

and,

vu
ra |

><

Nod

Vv

i)
a

i}

i -w/2
f —=— dw = Sa (56)
2

iT]
a

i}
i)

p | <2 | ane (57)

it follows that the conditional densities needed to evaluate equations
(53) and (54) are:

eo W/2
5. one iy &
2 oom a
f 2| 255 (Ww) = (58)
X51 X57
D2
0 » otherwise
oe W/2
Si » for O<w<2
P Zier =)
Nivea, (W)) = (59)
x2] x32
0 , otherwise

Thus, the quantities in equations (53) and (54) are given by:

co

2 f -w/2
theoretical On ae (fm) if (w - D2 Ses dw

Age 2 ze
(60)
2
ef 279-62.) = {1.414 p(n) }
2
2 -w/2
a
theoretical o% = a / (we=2)) 2 e dw
mt adices .) 2
0
(61)

oil 2
Y (+s) P?(fq) = { 0.6465 p(n}

l-e

76

Finally, the chi-squared analogy to equation (38) will be:

D(fm) - fy
theoretical SNR = P(fm) = p(fm) . (62)

cp (fy)

where

We ANA” esse GSGee)) eS tales)

QO
tl

(63)
0.6465 , bts pice) <p)
In terms of the 1G random variable, equation (62) becomes:
a B (fm)
theoretical SNR = c i | Bem - i
p (fm)
2
= -1 Let
ee) Wye 2 (64)
Ae 2

Hence, the theoretical distribution function for the SNR variables
(consistent with the chi-squared assumption) is:

: i ‘dan fan
theoretical FoNR (w) = P We {x3 - 2} <w

Hl
ae)
(SS)
><
NO NO
[A
NO
fe)
=
+
No
eed

= F 9 (2cw + 2) = 1 -exp(-cw - 1)
X9

l= exp(-1.414 w- 1) ; df w > 0
= . (65)
1 - exp(-0.6465 w- 1), if w< 0

The theoretical F our) is graphed in Figures 37 through 48 as a solid
line.

UY

By inspection, it can be seen that distribution function derived from
the chi-squared probability law is reasonably close to the empirical
distribution functions although there are systematic differences. In
general, the empirical curves tend to lie below the theoretical curve for
most argument values. The Kolmogorov confidence interval for the dif-
ference between the true distribution function and the empirical one is
drawn in on record 6878. The chi-squared curve exceeds the upper boundary
in the midranges but the two distribution curves are in fair agreement in
the vicinity of the 5th and 95th percentiles. This is probably why the
probability intervals agree fairly well. It should be noted that record
6878 is one of the more extreme cases and most of the other records
attain closer agreement between the two curves.

A numerical error was made in the earlier report (Borgman, 1972, Figs.
7 and 8) relative to the chi-squared related distribution function and
probability densities. The theoretical curves in those figures should be
ignored.

XII. THE OUTLIER SPECTRAL LINES

In 7 of the 12 records, one or more spectral lines loom high above
the others. These are ordinarily associated with the largest spectral
density values and determine where the peak of the spectral density will
occur in most cases. The spectral density decreases appreciably if these
extreme or outlier spectral lines are deleted from the averaging process
in the density determination.

A list of all the outliers is given in Table 3. Spectral lines which
exceeded 500 are shown and one value which went to 477 is included. The
spectral density value at that same frequency is tabled as well as the
spectral density which results if the outlier spectral lines for that
record were all deleted from the. averaging. The spectral density without
the lines is usually about 60 percent of what the density is with the lines
included in the averaging.

Two other measures of the "extremeness" of the lines are listed in
Table 3. The first is the ratio of the spectral line value to the density
computed with the outliers deleted. If chi-squared theory holds, this

ratio should behave like a y2 / 2 random variable. The 99.5 percentile
X9

for a x5 / 2 random variable is 5.3. Six of the 14 outliers listed

exceed 5.5 in) values “However, itis ditticullt to antexrprets chisem Ones
examining the larger members of 300 lines. Hence, extremal statistical
theory needs to be introduced. Straightforward application of the theory
of extremes would say that the probability that the largest such ratio

in a record would be less than 5.3 is:

O98)! = 0.22, (66)

providing the chi-squared interrelation and the independence assumptions

78

Table 3. Characteristics of outlier spectral lines.
Outlier
Record No. Freq. line
(a)
6878 0.0732 477 148 96
0.0842 525 WILY 54
6879 0.0757 620 141 67
6881-1 0.1025 652 231 166
6882 0.0720 557 299 203
0.0745 660 338 251
6883 0.0696 586 i2 102
0.0781 741 248 132
0.0830 733 302 141
0.0854 606 266 126
6885 0.0745 Is Nas 273 128
0.0818 868 281 Leal
6886-2 0.0769 1,026 298 7a
0.0830 582 215 124

BPD UM|S HFUMNMN NM W WO WOM

Sere) te) (02) LS) On C2) ony Sy Koy RY ee) )

PD OD WWE HP WHY WN OF

WAN NO NUWD WO WA NY CF

lRatio a/b refers to the ratio of the numbers indicated by

a and b.

*The SNR is computed from Oo, and 0_ based on averages with
the outliers deleted.

79

are both accepted. Two of the seven records have maximum outlier ratios
less than 5.3. This is 28 percent which is remarkably close to the 22
percent derived above.

The second measure of extremeness is the value of the SNR as based on
o, and O_ computed without the outliers present. Other than noting
that a three-sigma bound is often used in reliability to indicate unusual
extremeness, no attempt will be made to interpret these results at this
point in the research.

One nonstatistical observation may be more significant than all the
Statistical computations. This is the twofold fact that (a) the outliers
always seem to fall in the maximum energy frequency range, and (b) the
outlier lines are exceptionally separated from their neighbors, i.e.,
there is not a smooth transition with lots of small lines, some moderately
large lines, and a few very large lines. It is more like two separate
populations with no moderate range values between the two. This situation
occurs in statistics in "gross error" or outlier questions. However,
there is really insufficient statistical data to draw any firm conclusions.

From an oceanographic viewpoint, the one or several outlier spectral
lines might well dominate the waves present so that an aerial photo would
show waves of that frequency proceeding in their particular direction.
This is, of course, just conjecture since aerial photos for Hurricane
Carla at that space-time location are not available. The other spectral
lines present might be contributing noise and making the waves highly
short-crested.

The spectral line outliers may be some sort of resonant phenomena
within the storm waves whereby energy tends to be concentrated on certain
frequencies. Again, this is beyond the present research and is only
noted in passing.

In the situations where several outlier lines are present in the
record, an investigation was made as to whether there could really be only
one wave train present with the other lines showing up as leakage due to
purely mathematical manipulations. Appendix B gives a derivation of the
FFT leakage for a single cosine wave. It is shown that, at least in a
gross sense, the leakage is delineated by the following formulas. Let
the wave profile be given by:

2T7mon
Mea aces ( N - Ik (67)

where Mo is not necessarily an integer. The approximate FFT spectral
Finesse ate 20 <M +m << N, are given by the formula:

2 sint(my - m) 2
~ a“NAt { WG) } (68)

T™(mpg - m)

80

From this, leakage would always be to the immediate neighboring lines.
The outlier lines in Table 3 are always separated by one or more small
lines in each case. Hence, one has to conclude that the cases with two
or more outlier spectral lines cannot be explained by a single wave train
with FFT leakage.

XIII. WHY DOES CHI-SQUARED WORK FOR HURRICANE WAVES?

One of the mysteries arising from the data is the surprisingly good
probabilities arising from the chi-squared derivations. If linear wave
theory was holding and the seas were Gaussian, this would be expected
(Borgman, 1972, 1973). However, the hurricane waves were decidedly non-
linear. The waves in many of the records have been plotted by computer
and examined visually. The nonlinearities are really there. Why does the
the chi-squared work so well?

Investigation of this question led to a central limit theorem for
dependent random variables which showed that the chi-squared relations
hold exactly as N tends to infinity even for a non-Gaussian sea surface.
The primary limitation is that water level elevations at the recorder
separated by more than a certain constant time interval should be statis-
tically independent of each other. The amount of the separation required
to achieve independence is unimportant in the validity of the theorem
although it will affect the speed of convergence to the asympototic
result. Time sequences with above dependency properties are said to be
"m-dependent ,"" in statistical terminology.

A sufficient condition, then, for the central limit theorem to hold
is that the probability density of the water level elevation measured
from mean water level satisfies at least one of two "'tail'’ conditions.
The density, f(m), should be such that there exists positive constants
a, b, and c, and a positive integer n, such that:

-b
E)mcaalyy|inneg Wide Wbg| eas un: (69)
or alternately, there exists a positive constant A _ such that:

(iq) Ss OS Re sy] SA (70)

The second condition is a special case of the first since if the second
condition holds then c= A and any a, b, and n values will permit the
first condition to be satisfied.

Equation (69) is not an unreasonable restriction. For low seas, the
sea surface has been found to be normally distributed. As the wave heights
increase, a gamma density might be a reasonable guess as to the proper
probability law. Both of these densities satisfy equation (69). In fact,
from a practical viewpoint, no one seriously suggests that water level
elevations can be infinite as required by the normal or the gamma densities.
In fact, there will be a large value of A (e.g., A = water depth) such
that equation (70) will hold.

8|

The full details of the derivation for the m-dependent central limit
theorem and the asymptotic chi-squared properties of the spectral esti-
mates are given in Appendic C. One useful secondary result, summarized
in item (C-8) of the appendix, is that the normed FFT coefficients,

( Um f Vin

Pubes AN | pee WMG) airs Lh Si oe ae (71)
VNp,, At/2 VNp,, At/2

asymptotically follow a multivariate probability law with zero mean and
covariance matrix equal to an identity matrix.

As an illustration of the asymptotic normality, 100 water level ele-
vations were selected from the Hurricane Carla data number 6883. The
elevations were taken 6 seconds apart from the beginning of the data
until 100 elevations were obtained. Also, the 50 pairs of Fourier coef-
ficients centered around the frequency associated with the largest spec-
tral density value (Fig. 8) were tabulated from the computer listings.
The water level elevations and the FFT coefficients were plotted on
normal probability paper (after norming them) as shown in Figure 74. An
examination of the figure shows that the FFT coefficients follow a rea-
sonably straight line while the water level elevations exhibit a concave
upward curve. The skewness of the water level elevations was computed to
be 0.55 while the FFT coefficients only showed a skewness of 0.03. Thus,
the FFT coefficients were reasonably close to normality even though the
water level elevations were decidedly nonnormal.

It would be interesting to explore the normalizing influence of the
Fourier transform relative to the other wave records in Hurricane Carla.
However, time did not permit that to be included in this investigation.

XIV. IMPLICATIONS FOR DIRECTIONAL SPECTRUM RELIABILITY

The asymptotic normality of the finite Fourier transform coefficients
are extremely important relative to the estimation of the reliability of
the directional spectrum and of other quantities such as energy diffrac-
tion computed from the FFT coefficients. The FFT coefficients, at least
for large data sets, can be taken as normally distributed and independent.
These properties can be carried through the various formulas used in
computing the particular quantity to obtain reliability measures for the
quantity.

XV. SUMMARY AND CONCLUSIONS

1. The statistical properties of the spectral lines for 12 pieces of data
measured during Hurricane Carla (8 to 10 September 1961) were examined
in detail. The spectral lines were found to show negligible serial
correlation. The spectral lines could be reduced to what appears to
be random noise by subtracting from the lines the spectral density
obtained by smoothing the lines with a moving average and then

82

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83

dividing the positive and negative deviations by the local root-mean-
square poSitive and negative deviations respectively.

A simulation procedure based on the probability behavior of the random
noise of the spectral line deviation was used to generate probability
intervals for the spectral density estimates.

The simulation probability intervals were found to agree reasonably
well with that predicted from the chi-squared probability law. Al-
though the chi-squared probabilities are expected to be valid for
"low wave'' spectral densities, it was somewhat surprising to have
them work for the nonlinear hurricane waves.

An investigation of the above item led to the discovery that the
finite Fourier transform coefficients will be approximately independ-
ent and normally distributed for large data sets even though the water
level elevations are not normally distributed. This face was derived
from probability theory after assuming (a) m-dependence of the water
level elevations about mean water level, and (b) a water level proba-
bility density which is bounded for large y by the function

aly|Me-bly| (a, b, and n are arbitrary constants). A central limit
theorem was derived from these assumptions which led to the finite
Fourier transform coefficients being asymptotically normally distrib-
uted and independent as the data length tended to infinity.

Under the conditions of the item above, the smoothed spectral esti-
mates were found to be asymptotically chi-squared distributed pro-
viding the true spectrum is constant over the smoothing interval. If
the true spectrum is not constant, the spectral density estimates
would still be reasonably close to having chi-squared behavior
although there would be some deviations as evidenced by the deviations
of the simulation results from the chi-squared results.

The wave spectral lines exhibit members which appear unusually large
as compared with the rest of the spectrum. The "outlier" lines

always appear associated with the "peak energy" part of the spectral
density. These unusual lines of energy might predominate the wave
record so that a satellite picture would show waves of those frequen-
cies riding on top of the rest of "noisy" wave combinations. However,
this is just conjecture.

The outlier spectral lines mentioned above do not persist for long
periods of time. In the two cases where Hurricane Carla wave data
were available 20 minutes apart, the outliers were found in one set
of the data pair but not in the other.

The conclusion of the study relative to reliability of directional
spectrum estimation is that the finite transform coefficients may be
taken as being serially uncorrelated and approximately normally
distributed with zero mean, variance p,N At/2, and the appropriate

84

cross-variances. This asymptotic normality can be carried forward
to provide reliability measures for quantities, such as the direc-
tional spectrum, which are computed from the Fourier coefficients.

85

LITERATURE CITED

BLACKMAN, R.B., and TUKEY, J.W., The Measurement of Power Spectra, Dover
Publications, New York, 1958.

BORGMAN, L.E., "Confidence Intervals for Ocean Wave Spectra," Proceedings
of the 13th International Conference on Coastal Engineering, Vancouver,
BeGeml Omer

BORGMAN, L.E., "Statistical Properties of Fast Fourier Transform Coeffi-
cients Computed From Real-Valued, Covariance-Stationary Periodic
Random Sequences,'' Research Paper No. 23, College of Commerce and
Industry, University of Wyoming, Laramie, Wyo., 1973.

COOLEY, J.W., and TUKEY, J.W., "An Algorithm for the Machine Calculation
of Complex Fourier Series," Mathematics of Computattons, Vol. 19, Apr.
NIGS PP eaeZ O75 Ole

FREUND, J.E., Mathematical Statistics, 2d ed., Prentice-Hall, Englewood
ClatESMuNEJe 197Ie

GOODMAN, N.R., "On the Joint Estimation of the Spectra, Cospectrum and
Quadrature Spectrum of a Two-Dimensional Stationary Gaussian Process,"
Scientific Paper No. 10, Engineering Statistics Laboratory, College of
Engineering, New York University, New York, 1957.

GUMBEL, E.J., "Statistical Theory of Extreme Values and Some Practical
Applications," Series 33, National Bureau of Standards Applied Mathe-
matics, U.S. Government Printing Office, Washington, D.C., 1954.

LOEVE, M., Probability Theory, Van Nostrand, Princeton, N.J., 1960.

PIERSON, W.J., Jr., 'Wind-Generated Gravity Waves ,'' Advances tn Geophy-
S105, WO 45 LOSS, joo SSo178.

RAO, C.R., Linear Statistical Inference and Its Appltcattons, 2d ed.,
Wiley and Sons, New York, 1973.

ROBINSON, E.A., Multtchannel Time Sertes Analysts with Dtgttal Computer
Programs, Holden-Day, San Francisco, Calif., 1967, pp. 62-64.

ROSEN, B., "On the Central Limit Theorem for Sums of Dependent Random

Variables," Z. Wahrscheinlichkettstheorte und Verw. Gebtete, Vol. 1,
1967, pp. 48-82.

86

APPENDIX A

EFFECTIVE WIDTH

The effective width of a moving average smoothing function is defined
to be the width of square pulse required to have the same area and same
middle height. For example, consider the pulse:

W(X)

This pulse has area

A -f w(x) dx

and middle height w(0). The effective width would be the value of e.w.
such that

(e.w.) x w(0) =A
or

(e.w.) = A/w(0)

87

APPENDIX B

EFFECT OF A DETERMINISTIC LINE ON THE SPECTRUM
Suppose that i for O<n<N-1 is a cosine curve,

i, = i eos [2m a | 5 (B-1)

where mg is not necessarily an integer. What will be the effect on the
spectral density? The fast Fourier transform of it gives:

An = At y a cos [2m 7 )

ul
|@
>
ct
a)
HH.
con
N
3
3
S
Z| ~
\
o-
——
+
(0)
|
ii
se
N
=
=}
S
2/3
1
cS
——
|
oO
'
He
N
3
~N
Zz

nN
Sale [e"** » cata |

n
rio “SS {1240-06} (B-2)

n=0

Since, by the geometric series,

DG rears Seely Sonia Nile d= Gs yWiG Ss se):

it follows that:

; i2m (mp-m) : ah
hy = 25% [ete Le « gio 2

e127 (mg+m)
2 1 - et2™(mp-m)/N 97 12m (mp+m) /N

(B-3)

88

ee aAt a oT (mg-m) et 7 (mp-m) eit (mp-m)
n gi™(mg-m)/N =i (mp-m)/N _,im(mp-m)/N

oi (my+m) iT (mg+m) _eit (mptm)

eo LT (mg+m) /N ei ™ (mp+m) /N ti 7 tM (mgtm) /N

aAt je my(1-q) - o} - ifm m1-4y} Sin MOmp-m)
are sin ™(m)-m)/N

e endl mp(1-<) - 6) - ifn ma-4y} Sin TMmp*m)

sin ™(my+m)/N

Soe sin T(mo-m) Ee sin 1(mp+m)
. ant [er Byes 0 4 pCR) 0 (B-4)
Sin ™(m-m)/N sin ™(mp+m)/N
with
d= TGs) 36
= Tg (1-5
1
B = m™ (1-5) 5 (B-5)
Hence, A, can be written in terms of real and imaginary parts as:
aAt = T (mp-m) (a-B) Sin ™(mg+m) (a+B)
= = | -———_ cos (a- + ————+——-_ cos (a+
An 2 \sriin 7 (mg-m) /N i sin ™(mg+m)/N
. aAt fsin ™(mp-m) Sin 7 (ing+in) :
+i = |= sin(a-8) - ——__" — _ ot : B-6
2 E ™ (mj-m)/N one sin ™(m)+m)/N a 8)] Oe)
It follows that:
lan 2 r (Ger sin* 1(mg-m) i sin* T(mg+m)
sin? 7 (mp-m) /N sin? 7 (mp+m) /N

89

> sin [7 (mp-m) | sin [1 (mg+m) |

sin [ (mp-m) /N] sin [7 (mp+m) /N] {cos(a-B) cos (a+B)

- sin(a-B) sin(a+8)}]

ale 2 =. T (Mo-m) sin? T (mo+m)
= — Ree oe

sin? 7 (mp-m) /N sin? 7 (mo+m) /N

2 sin [7 (mg-m) | sin [7 (mp+m) |
ee eee 2 es
sin [7 Gmg-m) /N] sin [7 (mp+m) /N] io «| ae
4 N? sin? m(mp-m)/N N* sin? t(mg+m)/N

ae sin [m(mo-m)] sin [7 (mp+m) |

1
N2 sin [7 (mp-m) /N] sin [ (mp+m) /N] cos {2mmg (1-5) - 203 . (B-8)

If mg is an integer with O<mp<N/2,

ao NGA 5) alien Ss My
P(fm) = (B-9)

0 » otherwise.

If N is much larger than either Mg-™ and Mg+m, which is usually
the case in applications, then approximately,

N sin 1(mg-m)/N ~ 1(mg-m)
N sin T (my+m) /N x T (my+m) (B-10)
1/N =~ 0

90

Hence, equation (B-8) can be written approximately as:

B(E,) = ashe = T (™p-m) } ; 2 jee ied

7 (mp-m) (m +m)
Ags {= num {= noi) } cos (2 Tm - 26) . (B-11)
FE m™ (mg-m) 7 (mp+m)

If 20<mpgt+m << N, a reasonable assumption in many applications, then the

first term inside the square brackets will dominate the other two. This
fact is illustrated in the following computations, where

72
Hr are
t= ee (mg ~ (B-12)
1 7 (mg-m)
in w(mg+m) )
sin T7(Mpyt+m
— a {==} 2 HB) 2 0.0003 (B-13)
2 T (mg+m) = LOO. ie

fel ce ee onan

T (mp-m) 7 (mp+m)

il sin 7(mg-m)
“70 7 T (mp -m) (B-14)
My-M) = ORS Upper bound
m qT, T, T,
my 0.405 0.0003 0.020
m,-1 0.045 0.0003 0.0068
m,-2 0.016 0.0003 0.0041
m,-3 0.0083 0.0003 0.0029

9|

Mpy-M, = OS 25 Upper bound

m Th Dy TA

m,+3 0.0067 0.0003 0.0026
m +2 0.016 0.0003 0.0041
n+l 0.090 0.0003 0.0096
m a 0.0003 O.029

m-1 0.032 0.0003 0.0057
m,-2 0.010 0.0003 0.0032
m -3 0.0048 0.0003 0.0022

The main characteristics of equation (B-11) are determined by the first
term and in a gross sense,

2
A _ aeNAt {== “tno-n) | i
P(f,,) = mn Gian) (B-15)

if 20<mp+m << N and |mp-m| << N.

92

APPENDIX C

ASYMPTOTIC CHI-SQUARED PROPERTIES OF THE FFT SPECTRAL LINES
FOR NON-GAUSSIAN, M-DEPENDENT WAVE TRAINS

1. Basic Definitions and Assumptions.

Let gs j = 0,41,+2,+3,... be water level elevations about mean
water level. It will be assumed that {n,1 is a stationary second-order

stochastic process which is not necessarily Gaussian and that the random
sequence {njJ has the properties that, uniformly in n,

E [nal = 0 (C-1)

3 [ne SU (€=2)

LOG Wee Maeee store vi S Onn A, Syeda siete | INES dis OG) is a finite
sequence of the water level elevations starting at and terminating
with NatN-] The time interval between water level elevation values is
denoted by At. The finite Fourier transform coefficients are defined as:

N-1
a) = At » Y

-i2T™mn/N
nee

(C-3)

where i= v-l . The superscript N is attached to Am to indicate
that the FFT coefficients are computed on the basis of a sequence of
length N.

It will be assumed that the sequence n. is m-dependent. That is,

iy Tye ee eee Tyas? ang ifs 9 Merrersil? °°

independent sets of random variables if b - c > m (Rosen, 1967).

af no} are statistically

The probability density for mn. will be assumed to satisfy either
conditions (a) or (b) under item 5 given in the following. The FFT
coefficients will be said to be degenerate if Dea 0 and hence

Us = he 210:

we)

2. Motivation.

Consider the set of nondegenerate coefficients:

LOO 2) OD OD OD es GED GOD
So eee Vingtl? Uingt2? Ving t2? 7 Mot Mo+T

where Mo = goNS and <x» denotes the largest integer less than or
equal to x. In the above, the constant a satisfies the inequality
O<a<0O.5 and r is an integer constant with r>l. It will be shown that
under the assumptions of item 1, the set S asymptotically is multi-
variate normal as N+. If the true spectrum is constant over this set
of r Fourier coefficients, then pP(fm)/p(fm) for the spectral lines

for the frequencies spanned by the band will be asymptotically distributed
as ol2 and the spectral density based on the average over the whole band

will be asymptotically distributed as Ne |f Pies
ZAG

The first step, then, is to prove that the set S is asymptotically
multivariate normal as N+. This requires the next two listed items.

3. A Central Limit Theorem for m-Dependent Sums (Rosen, 1967).
Consider the double sequence of random variables:

(1) (1) (1)
Xie haces Xe

(2) (2) (2)
Mg Mey booby Oe

(N) (N)
X5 Gsat0'o 8 xX

(N)
X
l N

>

That is, the nth line of the array consists of a sequence of ky random
variables. Let g (N be defined as:

Ky
eS Fa
k=1

94

(i.e., the sum of the nth row). The central limit theorem is concerned
with the conditions under which the probability law of 5 (N) converges to

the normal probability law as N+. Other quantities which will be
used in the theorem are:
o2(s™) ) = Variance of g WN) (C-5)
o2 = Variance of x ON) (C-6)
k
kN
Ey d = probability density for ae 5 (Gey)

Theorem: If the random variables in the same row of the array are
m-dependent and if:

(a) E(x) =0) VEonallyyke rand) IN);

(b) aise SLM Nein Oe
Ky
(c) lim ewe eGo) ake S10) 4 soe GiGi, GSO. .
ae kN
kT ee
and ky
: 2
(d) lim y ON <i Comm
N?>o
k=1
then the probability law for 5 (N) converges to a normal probability law

having zero mean and unit variance k tends to infinity.

Comments: Rosen gives the theorem in a more general form by stating
condition (c) in terms of the distribution.function and a Stieltjes inte-
gral. However, the above form of condition (c) in terms of the probabil-
ity density is sufficient for the present use.

Proof: Given by Rosen (1967).
4. Multivariate Central Limit Theorem (Rao, 1973).
Let F, denote the joint distribution function of the k-dimensional

random variable (a ae), Brana He Za mS Ueayoas eiaval JP the

An

95

uP 'G 6.0

distribution function of the linear function aT ae + yd

n n
+ Apa Also let F be the joint distribution function of a
k-dimensional random variable a) z(2) ee 7 (kK) If for each vector
elk > F,, the distribution function of i 71) + i z (2) algo SN z(k) 5
An r 1 2 k
then) Fa) aE.

Proof: See Rosen (1967).
5. Some Conditions for which (c) in the Theorem of Item 3 Holds.

Suppose that c are constants uniformly bounded in n and k,

nk

eS ee il

and the probability density of Ue is denoted by ee Og) Tifa:

k

(a) there exists positive constants a, b, and c such that
=D) gy
>
eu Opus ee 2) en llbalene

uniformly in n and k, or

(b) there exists a positive constant A_ such that
el) SO 5 ae yea

uniformly in n and k.

Then for any e€>0,

RS 2 2 pas
L = lim if xe £2 dx = 0
k= ed | ele

Proof: The probability density for Mo expressed in terms of £2) is:

$0) = ( lene? le (ene 7 7 |

96

Hence, L may be written in terms of y integration as:

n
L = lim c 2 |c | fs “nk d
nk” |°nk| *nk Tx
DAY k=1 Ca 5S¥ n z
nk Yn
—|>e
vn
n
— hoe ik 2 2
= ee = », cat | Pon OGY
k=1 ly[peyn/[c., |

Let B be the uniform bound on {e_,} :

(a) Proof of the conclusion under hypothesis (a) above: for a>0,
and fixed n,a, and b, define G(a) as:

foo} foe)

G(a) -f yeaynen olay, = — i oe ae
b /
a ba

Clearly G(a) is a monotone decreasing function of a and G(a)*0 as
aro,

Hence, including both tails of eewy) 5

a n
1 2
n » Ck a (O) chy S = \: ee nee bY ay
evn 0 = eal
k=1 Palate i k=1 yo
nk [cx

as n tends to infinity.

(b) Proof of the conclusion under hypothesis (b) above: BY)
Satisfies hypothesis (a) with C =A, n= 0, a = 1, and b = 1.

SI,

6. <A Useful Lemma.

etn iee - C5 > Cyy-ss be a sequence of constants and Wi> W,, Wz,
be a sequence of random variables. Suppose that

lim
Co Ses
I: rene Tip)

and that there exists a random variable Z with finite variance such that
the probability law for c,W, converges to the probability law for Z as
n-* © . Then the probability law for cW, also converges to that for Z

als) IM =e 2" g

Proof: The proof is a straightforward application of a relation given by
Rao (1973). In his terminology, the above lemma may be stated as:

L L
CpW,> Z implies cW,> Z

His relation states that this holds provided, for any e€ > 0,

iam P(|c W. - cW [>] = 0
nn n

no7o

P
(In Rao's notation this would be stated as |coaWn = cW, | > 0). Now

See W W
F(lenk, = Mal > | = &|[m,) > =| < a
nn n n [en-¢| 22

by the Tchebichev inequality (Loeve, 1960). Since Var(W,) > Var(Z)
and (c, - c)2 >0 as n+o© , it follows that:

lim
aN P| lentin = cWnll > | = 0
as required. Hence, the lemma is proven.

7. Asymptotic Normality of Linear Combinations of Nondegenerate FFT
Coefficients.

Let Ajo tor s = 1,2 sand) mp + 1 semis mp + © betany sequencesor

bounded constants and define:

roy, y ON) POnt y ON)
(N) _ m \ m
ui iz » AS ‘ » m2
Ty NPAGp a2 = yNeepeyi2
m=Mo+1 m=Mg+1

98

where the various terms are defined in item 1, and the assumptions listed
there hold. Let o2(T(N)) be the variance of T(N) , Then,

My+r

ee co eu) ale ce i 2, )

m=Mg+1

The probability law of

converges to a normal probability law with zero mean and unit variance as
N tends to infinity.

Proof: The terms wee and ve are asymptotically uncorrelated with

each other and with other pairs (yu 5 vod ) provided 0 <m< N/2
and 0 < m' < N/2 (Borgman, 1973). The asymptotic variance of both

mee and Vinge is PN At/2 (Borgman, 1973). Hence, U,/ WN Atp_/2 and

vA Atp,/2 have unit variance. It follows that:

Motr

san) 0 A ee

m=M +1

as N > o (Freund, 1971).

After substitution for Un and V in terms of Y_, 7) can be
expressed as: ui u

i k-1 : k-
7 ON) : 5 mop 5 28 oa cos ann) +d, sin eimkol))
o(7)) k=l YN o(1)) m=mg+1 yp, At/2

39

Let

mtr (a ee 27m (k-1) Be eat aro) )
5 ml m2 N
eae
Nk N
m=m+1 o(r' )) [> At/2
N
ne Snk vas )
ee qh way
and
ky = N
Then,
7 WN)

1:0) ) Dm

is in the form required for the application of the theorem in item 3.

Since Eve) = Elna] = 0 , condition (a) is satisfied. The
division by o(T(N)) satisfied condition (b). The variance of X(N)

is given by:

kN N
But ,
Myjt+r
0
| | Z [nal i nal im
“nkl = », (N) een
m=mp*l o(t ) /P.. At/2
so,
N 5 var(y.)) N (N)
2 —_—S 2 = 2
», OrN 5 Ser var( yf ) B
k=11 k=1
which is bounded as N>*©. This verifies condition (d).

100

Finally condition (c) will be satisfied if the probability density
for the water level elevations obey the bounding conditions (a) and (b)

in item 5. This was assumed in item 1. (Note: B= lim By + 1 will
> ©
provide the uniform bound for cok needed in item Sal

Thus, all conditions are satisfied and POND 75 cp ON) converges in
law to a zero mean, unit-variance normal probability law. Since

Mp)tr
2 2 A \ ae
PE) Ces) 0
m=M+1

as N>+o , it follows from item 6 that the probability law of TN) jg
converges to zero mean, unit-variance normal probability law. This
completes the proof.

8. Asymptotic Normality of the FFT Coefficients.

Let (ue ‘ vo) for m= Mot, Mot2, slots INO +e De asiet ot:

nondegenerate FFT coefficients. Then the multivariate probability law
for

y™ y)
m

( Beech Van eee ae
Jwp, at/2 Np, t/2

) pe casa

will be a multivariate normal with covariance matrix equal to an identity
matrix and mean vector having all components equal to zero.

Proof: In items 4 and 7.

9. Asymptotic Chi-Squared Distribution for Spectral Estimates.

For the range of m-values, mp+l < m < mp+r, suppose that Pease

is a nonzero constant. Let:

+

p=s y (u2 + v2) /n ae

m=Mg+1

10|

Then, 2rp/p is asymptotically a chi-squared random variable with 2r
degrees of freedom.

Proof:
My*r U 2 V 2
rir Spoon} (igen not rears
At At
mM=Mg+1
or
Mg*r U 2 V 2
P oy LS YNP At/2 WNp At/2
M=Mo

By item 8, the terms squared in the sum are asymptotically independent,
zero mean, unit-variance normal random variables. Hence, the right-hand

side is asymptotically chi-squared with 2r degrees of freedom (Freund,
1971).

102

i

ay,

bp he: ee

; ni
fi i vi

Dela ye

t eT
tn TO
, Pos ealbec 1h) ae tee
TH he

ty

Prat

Rye Teed a

Ss earl band Wr

eideig aiyopigtinn ile win bma
Ba ; ; ig =i , Shines

jen tars ; ; i f
iy i
Ve i yee
x f
A Mi fs
: ( i rel)
a ie : i ; He) il
rh } ‘
i
i
i]
bay)
i
i ih
APA i
pie eae
sc ,
i a eal
i i Ws f
7 aetaet Al
i 4 ee ee
ress a
wet re
J ile
i i f r ay
‘ ae L a s ss ay
ee oe 1 an
\ u \\ 2
i! t Uwe he:
} i My
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-oedg *h ‘eT Ie) sUeOTAIN]] “gE “SeARA BsUeDTIAAN]] *Z ‘eaqoeds saeM *|
*s}[TNsei TeoTatdue |y2 Jo saduanbesuod pue suoT edT{T dwt
utejie. sdoTeAep pue eTieg suedTIANY A0F stsATeue vy} seats Jioda1z
STUL ‘L96| tTequeqdes ut eTAeD sUeDTAAN}Y] BuTAnp peiansesau spzode1 aAeN
JO STBAIEUT a}eIedes Z| 10z ATTeOTATdwWa poutwexe sre saaemM suUeoTIANY

OJ sozeut}se Teijqoods AZiaua eAPM UT SUOTIETIPA TeITISTIEIS Ju]

‘9g *d : AyderzsotT qTE
(1000-9-Z/-ZZMOVa + 193UeD
yoieesey BuptiveuTsug TeJseog — 4OeP1IUOD) OSTY ([-E/ “OU £ JUeMZ1edaq
AZoToat) *8utwokm Jo AWtTsTeatTun — JAodei1 yOAeesey) OSTY (O1-9/ “OU

£ Tequep yorIeesey BuTiveutTZuq [ejseog — teded Teoruysey) “ITF : *d ZoL
“Ol/6| ‘1eqUeD YDTesSay BuTIseuTsuq TeIseoD *S*pn : “eA SATOATOG 3104

— ‘ueusiog *qg uoay fq / eajzoeds aaem ue|D0 Jo Awojeue TeITISTIEIS SUL
"gq uosyT ‘ueus10g

daiecn* Le9 Ofoye “IE da,gcn° £020L
"1000-0-Z/-ZZ/MOVA 20e8A}U0D ‘*1eqUeD YOIeesey BuTrseuTSuq [Te3seop *s'n
: SeTZeS “AI “L-€/ Jaoday yOreasey ‘jueuqiedeq AZoToes) *AITSAeATUn
*SuTwokmM : SeTleg “JII ‘OL-9/ ‘ou teded Teotuyser *19}UeD YyoIPaSeYy
Sutileveutsug Teqyseog *S*n : SeTTeS “IT ‘eTITL ‘I ‘“stTsATeue wnazq

-oedg *H ‘eTieQ oueOTIAN]] *€ ‘“SaAeM sUeDTIANY] *Z ‘*erZOedS BAeM *|
*sq[nsea Teotatdua ay Jo sao.uenbasuod pue suoTzeoTTdut
uteji90 sdo[TeAep pue e[reD) sue ITIANY 10J¥ sptsATeue vyq seats J10de1
STUL ‘°L96,| Jequeqdesg ut e[TieQ suRDTIAN]] 3uTAnp perAnseoall Sp10de1 BAeM
JO s[TeAtejUT aqeaedes Z| AoF ATTeOTATdWa pouTwexa vie saAeM oUeOTIINY

AO} soqzeUTISe Teiqoeds Adi9ue SALEM UT SUOTIETAPA TeOTISTIEIS JUL

"9g ‘d : Aydeasottqrg

(1000-9-Z7Z-zZMova § 19]uUAa9

yoieesey ZuTisvsutTsug [Teyseog — 39e1RU0D) OSTV (| -EZ *ou § Juewz1edaq

ASojToat) *3utwokm Jo ARTSTaATUQ — Jioda1r YOTeasey) OSTY (01-94 “OU

£ zeque9 yieesey ZutzveuTsug Tejzseog — teded ~Teotuysel) “TIT : *d ZOL
"9/6, ‘199UeD YOAeesey BuTAseuTsug TeIseoD *s*n : *eA SATOATEG 310g

— ‘uewdsiog *q uoay fq / eijoeds aaeM ue|D0 Jo AWOJeUR TPOTISTIEIS YL
*qy uosy Suews10g

d3igcn° L729 Ol-9L ‘ou dargcn* €0Z0L
"1000-0-Z/-ZZMOWd 32e81QU0D “*1ae}UeD YOTeeSey BZuTIseUTSsUg TeIsSeOD *S°n
: SeTI9g “AI ‘L-¢€2 J10day yoreesey ‘JueuqyIedeq AZoTOSe) *ARTSISATU
*Sutmokm : SeTI9S “TII “OlL-9/ ‘ou iteded TeotTuyoey, *19e}UeD YOIPesey
Butisveut3ug TeyseoD *S*h : seTzag “II “eTITL *I ‘“sEsATeue wnIzq

-oedg *h ‘eT Ie) oUBOTIANY *¢€ “SeAeM aUeDTIINY *Z ‘erqoeds oaemM *1
*sq[Nsel Teotatdwe sy, Jo saduenbasuod pue suoT eoTT dut
ufejieo sdoTenaep pue eTie9 vueoTAAny 10F stTsATeue ay seAT3 Airode1
STUL ‘°196| Jequeqdes ut eT1eD euUedTAINy BZuTInp peinseew spi0ode1 sAeM
JO sTeArequT ojeiedas Z| 10jF ATT eoTATdMa peuTWexe sie seAeM JUPOTIANY

Io} soJeut Ise Teijqoeds ABi|aue VAeM UT SUOTIETAPA TeOTISTIeIS JUL

‘9g ‘d : Aydeasortqrg
(1000-9-Z2-ZZMOVad + 193Ue0
yoieesey sutiseuz3uqg [TeqseoD — JOeIWUOD) OSTY ([-E€f ‘OU $ JUeURZIedaq
AZ0Toe) *3utMokm Jo AZTSreaTuQ — Jtodexr yoreesey) OSTY (OL-9/ ‘ou

$ 1equep yo1eesey SuTieeuy3uq Teqseog — teded [eotuydeL) “TTT : *d zoL
“O/6L S19}UaD YODIeeSey BuTisvsuTSuq Teqseog *S'n : “BA SAFOATOG 104

— ‘ueu3iog *q uoey fq / e1z0eds eAeM UPTO Fo AWOJeUP TeOTISTIEIS JUL
“J uoeT] ‘ueusi0g

dajecn* £79 Ol-92 ‘ou daresn* €0ZOL
*"L000-0-ZZ-ZLMOVd 29819U0D ‘AaqjUaD YDIeasey BuTIseuTSug TeyseoD *s'n
: SeTI9S “AI ‘“L|-€/ JAOdey yoreesoy ‘juewj,Iedeq AZ0ToOaey *ARTSTeATUn
*ZuTwoAM : SeTI9S “III “Ol-9/ ‘ou aaded [Tedtuyoey, ‘*aezUaD YyoLTeVsSaey
SuTiseuT3uq TeIseOD *S*N : SeTI9S “II ‘“eTITL “I “stTsATeue umIzq

-0odg ‘yh ‘eTaeD ouedTIINY *€ “SaAeM sUPOTIINY *Z ‘*eAQDedS sAeM ‘|
*sqynsei TeoTatdwe ay jo saouenbasuod pue suoTjeoTTdut
uteji90 sdoTeaep pue eTied) suedTIANY 1OF stsATeue vy saaTs A1ode1
STUL ‘L96L Jequeqjdes ut eT1eD sueoTIINW BuTAnp peanseeu spiodesI sAeM
JO STBATSQUT o}eIedes Z| AOZ ATTeOTATdWa pouTWwexe aie saAeM aueoTAAINY

JO} sajzeutzse Terqoeds AZi19ue VAEM UT SUOTIETIPA TeITISTIeIS JUL

"9g *d : Aydeasor{ qTg
(1000-9-7£-ZLMOVd $ 223uUe9
yorieesey BZuTiseutsuq TeqIseoD) — 49e17U0D) OSTY ([-€/2 ‘ou $ Juewqiedeg
AB0Toe) *ButwokmM jo AjTsizeATuyQ — Jsode1 yOreasey) OSTY (01-92 ‘OU

$ 1ejueD yoreesey BSuy~ieeuzsuq [Te3seoj — aeded [eodtuyoey) “TTF : *d ZoL
“9/6, ‘1e}UeD YDIeVsey BuTiseUTSUq TeIseOD *S'N : ‘eA SAFOATOG 310g

— ‘uewsiog *q uoay Aq / e1j0eds aAeM ue|D0 jo AwojJeue TeITISTIeIAS JUL
*g uoey fuewsi0g

d31gcn° Le9 OL-92 “ou d31gcn* £0¢OL
"L000-0-7Z-ZZMOVG 39BTRUO0D *TAaRUeD YDIeeSey BuTiseuT3uq TeqseoD *s*pA
> SeTZeS “AI “L-€/ Jtodey yOATessey ‘juewjiedaq ABoToey °*AITsisATUp
“BZuTmokM : SOTIOS “III ‘OL-9/ ‘ou toded Teotuyoay *1eque9 Yyoreasay
SuTiseutTsug TeyseoDg *S*n : SeTies “II ‘eTITL ‘I “sTsATeue wn1zq

eoedg ‘hp “‘eTIeD sUuedTAIN]] *E€ “SeABA SaUeDTIINY *Z ‘*eaqoeds onemM *|
*sq—Tnsei TeoTatdwe oy} Jo saouenbasuod pue suotjeotT dwt
uz~ej1e9 sdoTeAep pue eTieD suedTIAny AozZ stsATeue ay, saaT3 J10da1
STUL ‘L96, tequeqdesg ut eTAeD suedTIAN}y] BuTAnp peansesul spz0da1 ane
JO STPAIORUT ajertedes Z| IOJZ ATTeOTATdWSa poutTwexe vIe saneM |auRoTAANY

IOJF saqeutT Ise Te1qoeds AZisue eAPM UT SUOTJeTIeA TeOTISTISIS JUL

‘9g *d : Aydezsott ata
(1000-0-Z/-ZZMOVa * 103uUeD
yoieesey SutiseutTsug ~Tezseog — 39e19U0D) OSTY (|-E4 “OU £ JuewI1edaq
AZoToa *8utwofm Jo AAT~SsTeaz~un — JAo0de1 yOARasey) OSTY (O1-9L ‘ou

$ TequeQ YyoIeesey BuTiseuTsuq ~Teqyseog — taded Teotuyosey) “TIF : *d ZoL
“9/6, ‘ZeqUeD YOTeVseay BuTAsveuTsUq TeIseoD *S*n : “eA SATOATOG 104

— ‘ueusiog *q uoey Aq / eiqoeds oAeM ue|D0 Fo AwojZeUe TeITISTIEIS Syl
“J uosT, ‘ueuwsi0g

darecn- £79 OP=9250u daigsn* €0ZOL
"L000-0-ZZ-ZZMOVU 29eTQUOD *1aqzUeD YOARasey BuTivseuTsuq [Te3seoD *s'n
2 SeTzeg “AI ‘“L-€/ Jazodey yoreesey ‘*juewqaedaq AZoToes *AITSsieATUn
*SuTwokm : SeTileg “JII ‘Ol-9/ ‘ou teded Teotuyset ‘*zaqueD YyoIPeseay
SuTiveutTsuq Teqjseo) *s'n : Ssetres “IT ‘“eTITL “I “stsATeue wna

-oeds ‘yh ‘eTale@) oUeOTIAN]] *¢€ “SeAeM sUeOTIAN]Y] *Z ‘erROeds aaeN *|
*sq[nsea [eotatdwa ay Jo saduenbasuod pue suoTeoTTdut
uteqi90 sdoteaap pue eTreg sUeOTIIN}Y I0JF sftsATeue vyq seaTs J1sodea
STUL ‘L96| Jequejdesg ut eT reD suUedTIAN]] 3uTAnp peanseall Sp10d91 |BAeM
JO sTeAejUT oje1edes Z| A0F ATTeOTATdwe paeutTwexe eile seAeM oUPOTAINY

1OJ sazeuT se Terqjoeds Adisaue SAEM UT SUOTIETAPA TeOTISTIEIS JUL

‘9g *d : Aydea8o0tT qT
(1000-0-Z7£-ZZMOVa * 193Ue9D
yoreesey duTiveuTZug Teqseog — OeAIUOD) OSTY ([-Ef ‘OU $ JuewRIedeq
ABojToe) ‘3utTwokm Jo ARTSTeATUQ — jaoder YOTeasey) OSTY (O1-9/ “OU

£ zequep9 yoieesey ZutzseuTsug [Teqseog — aeded Teotuyoser) “TIT : *d ZOL
"9/6, ‘1a qUeD YOARasey BuTAveuTsug Teqjseog *s*pn : “eA SATOATIG 410g

— ‘uewdiog *q uoay fq / eijoeds aaen ueas0 Jo AwojJeue TeOTISTIEIS SUL
"J uosy Suews10g

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