ERROR ANALYSIS OF HYDROGRAPHIC POSITIONING
AND THE APPLICATION OF LEAST SQUARES

AM Kaplan

NAVAL POSTGRADUATE SCHOOL

Monterey, California

•

THESIS

ERROR .

ANALYSIS OF HYDROGRAPHIC POSITIONING

AND

THE APPLICATION OF LEAST SQUARES

by

Ali Kaplan

September 1980

Thesis

Advisor: Dudley Death

Approved for public release; distribution unlimited,

T1Q7fUl

SCCU WlTY CLASSIFICATION OP THIS »Af.E rW»«w» DgM tilWiX)

REPORT DOCUMENTATION PAGE

READ INSTRUCTIONS
BEFORE COMPLETING FORM

»f»0*T NUMiTR

2. OOVT ACCESSION NO.

1. REClRlENT'S CATALOG NUMREM

4. TlTLt r«n<(lu»(m.l

• • TYRE Or »tPO»T * RERIOO COVCRFD

Error Analysis of Hydrographic Positioning
and the Application of Least Squares

Master's Thesis;
September 1980

t. RERPORMINO ORG. RBRORT NUMIIN

7. AuTMOKrtJ

I. CONTRACT ON GRANT NUMBERTcj

Ali Kaplan

» »(«»o«uiNa organization mamc and adoress

Naval Postgraduate School
Monterey, California 93940

Tr "cwNYROLLirfo oV»rc^i»A«l<-lrirt^ooirlrii'

Naval Postgraduate School
Monterey, California 93940

tO. RRQOJtAM ELEMENT. RROjECT, T ask
AREA A MONK UNIT NUMBERS

12. HtPOUT DATE

September 1980

IS. NUMBER Of RAGES

147

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Naval Postgraduate School
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Rtm«r\)

IE SURRLEMENTARY NOTES

It KEY WORDS fCanrfnw* on r«»»»«« •<<*• II n«c«««arr ■«* ifntlty »r Meek RilBfJ

Repeatability Least Squares Adjustment

Predictability Observation Equations

Circular Error Approximations Error Ellipse

Root Mean Square Error

Accuracy Contours

20. ABSTRACT (Conilmt* an rrnvma* •/«• II noeooomwr «n« i««m<f* »r »<•«* i

•**0

Repeatable accuracy of hydrographic positioning was examined
in terms of the two-dimensional normal distribution function
which results in an ellipitical error figure. The error ellipse
was discussed, and two methods for conversion of elliptical
errors to circular errors were given. These methods are "circle
of equivalent probability" and "root mean square error" (d ) .

DO , :?:*„ 1473
(Page 1)

EDITION OP 1 NOV •» IS OBSOLETE

S/N 0103-OM- AftOI '

SECURITY CLASSIFICATION Of TNIS RAOE (Whmit Dmtm Kniottxt)

fneu«*lTV CLAUDICATION OW TMtS »>OIr*>»n n*fa tnlmnd

Using the drms error concept, repeatable accuracy of ranging,

azimuthal, and hyperbolic systems was evaluated, and
methods were developed to draw repeatability contours for
those systems.

A brief theoretical background was provided to explain
the method of least squares and discuss its application to
hydorgraphic survey positioning. For ranging, hyperbolic,
azimuthal, sextant angle, and Global Positioning System the
least squares observation equations were developed.
Specific examples were constructed to demonstrate the
capabilities of this data adjustment technique when applied
to redundant position observations.

DD Form 1473
S/N 0102-014-6601 sccu"itv claudication o' this **Gerw»»«« o«»« e«f«r«d)

Approved for public release; distribution unlimited,

Error Analysis of Hydrographic Positioning
and the Application of Least Squares

by

Ali Kaplan

Lieutenant /^Turkish Navy

Turkish Navy Academy, 19 74

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY (HYDROGRAPHY)

from the

NAVAL POSTGRADUATE SCHOOL
September 1980

■ \

ABSTRACT

Repeatable accuracy of hydrographic positioning was
examined in terms of the two-dimensional normal distribution
function which results in an elliptical error figure. The
error ellipse was discussed, and two methods for conversion
of elliptical errors to circular errors were given. These
methods are "circle of equivalent probability" and "root
mean square error" (d ) . Using the d error concept,
repeatable accuracy of ranging, azimuthal, and hyperbolic
systems was evaluated, and methods were developed to draw
repeatability contours for those systems.

A brief theoretical background was provided to explain
the method of least squares and discuss its application to
hydrographic survey positioning. For ranging, hyperbolic,
azimuthal, sextant angle, and Global Positioning System the
least squares observation equations were developed. Specific
examples were constructed to demonstrate the capabilities
of this data adjustment technique when applied to redundant
position observations.

TABLE OF CONTENTS

I. INTRODUCTION- -------- 10

II. REPEATABLE ACCURACY OF HYDROGRAPHIC

SURVEY POSITIONS- --------- 12

A. TYPES OF ERRORS ----- 12

1. Blunders- ------ 12

2. Systematic Errors -- ---12

3. Random Errors ---------13

B. ACCURACY AND HYDROGRAPHIC POSITIONS ----- i4

C. REPEATABLE ACCURACY - ----- 16

1. Elliptical Errors ------------ ig

2. Circular Error Approximations 23

a. Circle of Equivalent Probability- - - 23

b. Root-Mean-Square Error- -------31

D. REPEATABLE ACCURACY OF HYDROGRAPHIC
POSITIONING SYSTEMS ------ 37

1. Ranging Systems - -----37

2. Hyperbolic Systems- -----------38

3. Azimuthal Systems ---- __40

4. Sextant Angle Positions 43

E. REPEATABILITY CONTOURS- ----------- 43

1. Ranging Systems ----. ----43

2. Azimuthal Systems _______ 47

3. Hyperbolic Systems- -----------51

III. APPLICATION OF LEAST SQUARES TO HYDROGRAPHIC

SURVEY POSITIONS - - - - - 59

A. THE PRINCIPLE OF LEAST SQUARES 59

1. Weighted Observations ----------50

2. Method of Least Squares Adjustment- - - - 62

3. Higher Order Functions 76

4. Equations for the Precision

of Adjusted Quantities- ---------79

B. APPLICATION OF LEAST SQUARES TO
HYDROGRAPHIC POSITIONING SYSTEM ------- 85

1. Azimuth Angle Positions ---------85

2. Sextant Angle Positions -- -92

3. Range-Range Positions --------- -101

4. Hyperbolic Positioning Systems 109

5. Global Positioning System -113

C. USE OF THE ERROR ELLIPSE IN ANALYZING

THE ACCURACY OF HYDROGRAPHIC POSITIONS- - - -121

IV. CONCLUSION- - - - - _....._ -129

APPENDIX A - ANALYSIS OF RANDOM ERRORS 13 2

APPENDIX B - USEFUL GRAPHS FOR THE DETERMINATION OF

REPEATABILITY CONTOURS 140

LIST OF REFERENCES - - -142

INITIAL DISTRIBUTION LIST - - -144

LIST OF TABLES

Table Page

1-1 Values of constant h ________ 21

1-2 Circular error probabilities- -------- 25

1-3 Factors for conversion, K, of the error
ellipse to circle of equivalent
probability ------------ 28

1-4 Significant parameters of error ellipses

when a1 = a-- ---------------- 29

1-5 For 90% probability interval, significant

parameters of error ellipse when a, = a2- - - 30

1-6 Variations in probability as a function of

eccentricity ------------ 35

1-7 Relations between 23 and 0, or 8- ----- - 55

1-8 The angle 23 and 6X C©2^ defining the

4m d contour for several values of p - - - 58
rms

II - 1 For least squares solution, successive
iterations applied to azimuth angle
positions ------- _____ 93

II - 2 For least squares solution, successive
iterations applied to sextant angle
positions --- ____ 100

1 1 - 3 For least squares solution, successive
iterations applied to range-range
positions --------- 10°

II-4 For least squares solution, successive

iterations applied to GPS fixes ------- J-20

Al Linear error conversion factors for

several probability levels- --------- 135

LIST OF FIGURES

Figure Page

1-1 Position location at the intersection

of two lines of position- ---------- 17

1-2 Expanded view of the intersection of
two lines of position and associated
error ellipse ---------------- 18

1-3 Contours of equal probability areas ----- 22

1-4 Geometric dilution of precision for

CEP and 90% probability interval 32

1-5 Illustration of root mean square error- - - - 33

1-6 Variation in d with ellipticity (1 d s) - 35

1-7 A hyperbolic triad- _______ 39

1-8 Azimuthal system repeatability- ------- 41

1-9 Ranging system geometry ----------- 45

1-10 For ranging systems, the graph

of the drms/a and e/b" " " " ' ' 46

1-11 Repeatability contours of a ranging pair- - - 48

1-12 For azimuthal systems, the graph of

the d „«./«, .v and e/b- -- _____ 50

rms/a *b

1-13 Repeatability contours of an azimuthal

system- - - -- 52

1-14 In hyperbolic systems, for several

choices of p, d / curves- -------- 54

r' rms/crw

1-15 Repeatability contours of a hyperbolic
system (a - .01 lane width and
f • 2 Mhz)- ------- - - 57

I I - 1 Azimuth angle positions ----------- 87

1 1 1 - 2 Determination of a position for

azimuthal systems using the least

squares method- --------------- 89

1 1 1 - 3 Sextant angle positions ----------- 94

1 1 1 - 4 Determination of a position for
sextant angle fixes using least
square adjustment -------------- 97

1 1 1 - 5 Determination of a position for range-
range systems using least square
adjustment -_____-__.--_ 106

I I I -6 Determination of a position for hyperbolic
systems using the least square adjustment
method- ------------------- no

1 1 1 - 7 Error ellipses formed at the determined

positions - - 122

III -8 Error ellipse - -._.... 123

Al One dimensional normal distribution curve - - 134

A2 Equal probability density ellipses 137

A3 Constant probability density ellipse

for correlated errors ------- 139

Bl For ranging systems, the graph of the

drms/as and e/b 140

B2 For azimuthal systems, the graph of the

drms/a-b and e/b- -------------- 141

I. INTRODUCTION

Positioning of the survey vessel is equal in importance
with depth determination in the collection of hydrographic
survey data. Fundamental to an understanding of the accuracy
of position information is an analysis of the various errors
and their sources which must be either eliminated, compen-
sated for, or otherwise modeled. The result of this analysis
is that the reliability of position data can be evaluated
and used to estimate the overall accuracy of hydrographic
soundings .

Once these potential error sources are understood,
methods must be developed to quantify accuracy. Much
research has been conducted in this area in the past. One
purpose of this thesis is to collect and present useful
concepts of error theory which apply directly to hydrographic
survey. Simple graphical techniques were developed which
can be used to produce accuracy contours as a function of
the survey net geometry.

Conventional survey techniques rely on only two lines
of position (LOP) to determine a positioning fix. This
introduces the possibility of significant error.

In navigation, although inherently less accurate than
positioning due to the techniques and systems used to
determine the LOP's, three LOP's are required to produce a

10

fix. Position is adjusted graphically by placing the fix
in the center of the triangle formed by the three inter-
secting LOP's. This concept of taking one redundant
observation can lead to significant improvement in hydro-
graphic survey positioning data. Mathematical adjustment
techniques such as the method of least squares may be used
to determine the best estimate of position.

Least square adjustments are commonly performed on land
survey data where redundant observations are easily made.
With the advent of new positioning systems and computer
technology, making redundant observations at sea is no
longer impractical. The second purpose of this thesis
is to explain the basic method of least squares, and to
formulate examples of the least squares adjustment pro-
cedure applied to specific types of hydrographic survey
systems. This data adjustment technique not only provides
the best estimate of position but also may be used to deter-
mine the absolute positioning accuracy associated with each
data point in a hydrographic survey.

11

II. REPEATABLE ACCURACY OF HYDRQGRAPHIC
SURVEY POSITIONS

A. TYPES OF ERRORS

It is impossible to make measurements of physical data
without making errors. These measurement errors may be
classified in the following manner.

1. Blunders

These are mistakes which result from misreading
instruments, transposing figures, faulty computations, etc.
They may be large and easily observed, or smaller and less
detectable, or very small and indistinguishable in the
data. Blunders are usually detected through comparing repeated
measurements, careful editing, and procedural checks in the
data collection process. Physical measurements will contain
a constant bias if these errors are not removed from the data
set.

2. Systematic Errors

Uncalibrated instruments or environmental factors,
such as temperature and humidity changes which affect the
performance of the measuring instruments, will induce system-
atic errors into the observations. The occurrence of this
type of error may result in a pattern which can be recognized
and mathematically modeled. The simplest pattern to model
would be some observable trend in the data of constant
magnitude and direction. Such a trend can easily be

12

subtracted from the observations to remove the systematic
error.

If numerous systematic errors exist, or the errors
are such that they cannot be accurately modeled, then their
effect on the data must be estimated by calibration. Cali-
bration is the process of comparing the measuring instrument
against a known standard. The difference between the observed
and known value may be used as an estimate of the total effect
of all systematic errors present. Thus, calibration provides
a "corrector" which must be applied to the data set. Examples
of important systematic errors in hydrographic survey position-
ing include instrument errors, errors in positioning control
points, and variations in the propagation velocity of electro-
magnetic energy.

3. Random Errors

These errors result from accidental and unknown
causes. Their effect cannot be removed from the observations
and, therefore, must be quantified statistically. Random
errors have certain characteristics which facilitate such an
approach. Positive and negative errors occur with equal
frequency, small errors are more probable than large errors,
and extremely large errors rarely occur.

The frequency distribution of random errors can be
modeled mathematically by the normal distribution function.
Assuming all measurement errors are independent and random,

13

thereby conforming to the normal distribution, measurement
accuracy can be specified statistically by defining a con-
fidence interval around the best estimate of the measured
value. Procedures for computing these intervals are reviewed
in Appendix A.

B. ACCURACY OF HYDROGRAPHIC POSITIONS

The achievable accuracy of a hydrographic survey
positioning system is best described by defining the follow-
ing terms: repeatability and predictability.

Repeatability is a measure of the accuracy with which
the positioning system permits the user to return to a
specific point on the surface of the earth defined in terms
of the lines of position generated by the system. Included
in repeatability are the effects of random errors, errors
due to net geometry, and errors resulting from the angle
of intersection for the two lines of position that establish
a fix. Repeatable accuracy is therefore a measure of the
relative accuracy of a positioning system. Unresolved biases
exist in hydrographic positions due to the presence of
systematic errors that have not been subtracted from the
data or compensated for as a result of calibration.

Predictability is the measure of accuracy with which the
system can define the location of the same point in terms
of geographic (or geodetic) coordinates rather than simply
the intersection of two lines of position. Thus, predictable

14

accuracy is an absolute accuracy. Using conventional hydro-
graphic survey techniques, predictability could be achieved
only if all systematic errors were removed from the data so
that only the effects of random errors, net geometry, and
intersection angle remain. For example, the lattice generated
by an electronic positioning system is distorted primarily as
a result of the variability in the propagation velocity of
electromagnetic energy. Ideally, if there was no distortion
of the electronic lattice, then the accuracy of a position,
corrected for any remaining systematic errors, could be
quantified statistically in terms of predictable accuracy.
However, since these distortions exist, the effective velocity
of propagation would have to be accurately modeled through-
out the survey area. Then it would be possible to subtract
the effects of this systematic error and derive positions
in terms of predictable geographic coordinates. Research
is currently being conducted to quantify the parameters which
affect propagation velocity in order to model these values
for such application [Ref. 19].

A second method to achieve predictable accuracy is by
making redundant observations to establish hydrographic
survey positions. If three intersecting lines of position
are available instead of the usual two, the resulting fix
is overdetermined, and data adjustment techniques must be
applied. The method of least squares is most useful in

15

adjusting such data. Through the application of least
squares adjustment techniques, the best estimate of position
is found and the position' s predictable accuracy is resolved.
A complete discussion of this procedure is presented in
Section III.

C. REPEATABLE ACCURACY

In the determination of hydrographic positions, blunders
are eliminated by observing strict survey procedures, and
system calibration is performed in an attempt to remove system-
atic errors. Because some systematic errors still remain,
the accuracy of hydrographic positions must be stated in
terms of repeatability.

The modeling of random errors is done by using the two-
dimensional normal distribution function. When the normal
distribution is applied to the positional errors, the result-
ing error figure is an ellipse.

1 . Elliptical Errors

Hydrographic positions are determined by the inter-
section of two lines of position (LOP). Because of the errors
in each LOP, the actual position may lie somewhere between
the error limits (shown as additional arcs either side of
LOP's in Figure II -1) .

The intersection of the two LOP's, together with the
standard errors associated with each, is drawn to an expanded
scale in Figure 1 1 - 2 . By applying the two-dimensional normal

16

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distribution to positional errors, it is seen that the contours
of equal probability density about such an intersection are
ellipses with their center at the intersection point.

For simplicity in the discussion, the following assump-
tions are made:

1. Only errors contributing to repeatable accuracy are
considered.

2. The random errors associated with each LOP are
assumed to be normally distributed.

3. The random errors in each LOP are assumed to be
independent, i.e., a change in the error of one
LOP has no effect upon the other.

4. The LOP's are assumed to be straight lines in the
small area in the immediate vicinity of their
intersection.

5. Errors of position are limited to the two-
dimensional case.

As shown in Figure II-2, the general case of the inter-
section of two LOP's at any angle and with different values
of errors associated with each LOP results in an elliptical
error figure.

It is readily seen from Figure II -2 that the exact shape
of the error ellipse varies with the magnitudes of both of
the one-dimensional LOP errors, a, and o?, as well as with
the angle of intersection, 3.

The values of the semi -major and semi-minor axes of the
error ellipse (using one a error) are given by the following
equations [Refs. 4 and 5].

19

Semi -major axis :

aS:n2pL Y J ' (II -la)

Semi-minor axis:

25

Sin p L J *

Generally <j1 = a2 = a, then equations (Il-la) and (Il-lb)
simplify to

OY s and (5M = g[ r 1 1 - 2 1

After computing the semi-major and semi-minor axes, the
probability of the error ellipse is given by the distribution
function

PU,y) = 1 - e"2 , ("-33

where

2 u2-

h2=_*L 4- »

ffj erj

(x and y are the errors in the direction of a and a ")

x y J

20

The solution of equation 1 1 - 3 with values of h for different
probabilities yields the results shown in Table II-l.

Probability (%) h

39.35 1.0000

50.00 1.1774

63.21 1.4142

90.00 2.1460

99.00 3.0349

99.78 3.5000

Table II-l: Values of constant h

For example, for 39.351 probability the axes of the

ellipse are 1.00 a and 1.00 a ; for 50% probability the axes

x y

are 1.1774 a and 1.1774 a Figure II-3 shows the error

x y

ellipses for different values of h.

The angle, 9, between the semi-major axis of the error
ellipse and the line of position which has smaller standard
error is given by [Ref. 5]

-rooie = ** Sin lft CII-4)

21

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where a is the smaller standard error. In the case of
a1 = a?, equation 1 1 - 4 simplifies to

e »JL . (n-5)

2

The importance of the angle 9 is that it specifies the
orientation of the error ellipse according to the lines of
position.

2. Circular Error Approximations

In general, the use of the error ellipse is compli-
cated by the problem of axis orientation and the propagation
of elliptical errors. Therefore, in order to simplify
probability calculations and avoid the above problems, the
elliptical errors are approximated by circular errors which
are easier to use and understand. The accuracy of a hydro-
graphic position may then be stated in terms of a circle of
specified radius about the point.

Note that when the angle of intersection is a right
angle and the two errors are equal, the error ellipse becomes
a circle and is described by the circular normal distribution.
Generally, this is not the case, and elliptical errors must
be converted to circular errors. This is done by using either
the circle of equivalent probability or the root-mean-square
error concept.

a. Circle of Equivalent Probability

A circle of equivalent probability is obtained
utilizing an existing table for the two-dimensional normal

23

distribution (Table II-2). This table is used with the

two standard errors along the semi-major and semi-minor axes

of the error ellipse (Equations Il-la, Il-lb or II-2). To

find the radius of equivalent probability, equations Il-la,

Il-lb or 1 1 - 2 must first be utilized to obtain the values of

a and a . To enter the table the following ratios are needed
x y

a
c * -£ where a is the greater standard error
°x x

and

K _ Radius of circle of equivalent probability
Greater standard error

where K is the conversion factor needed to solve for the

radius (R) of the circle of equivalent probability.

The table relates varying values of ellipticity

to the radius of circles of equivalent probability. Enter

the table with the computed values for c and K to determine

the probability for a circle of given radius, or alternately,

for a given value of probability, determine the radius of the

error circle.

EXAMPLE II -1: The two standard errors of a positioning

system estimated from field observations are a.. = a_ = 6 meters

To determine the probability of location within a circle of

10 m radius when the angle of intersection, 3, is 60°,

equation II-2 must be used to find a and a :

x y

24

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9978609

. 9084880

. 998936H

. 9992583

9994888

.9006506

. 9997833
. 9098412 |
. 9998945 |
. 0909305

. inWv-3^7

0000707 j
. 9999813
. 9990881 !
. 0999025 | .

. 99V99."4
. 999997 1
. 9999083
. 9999990 I
.9000994 .

. 9909906 I .

. IfUVVn

. 0990909 i
. 9090099
. 0000000 I

.0009377
. 0390103
. 0851535
. 1461808
.2152880

. 2914682
3699305

. 4474207
5213098
5000963

. 6524480
. 7079073
. 7507265
. 7989288
.8360816

. 8657559

. 8915536

. 9130680

9308015

9454646

9573205

. 9668H45
9745239

. 9805703
9853112

. 9977296
. 99M3892
9988677
,9992115
, 9994.159

9990281
9007482
99983 1 I
9998878
99092UI

9099961 i

9099970 !

0000082 I
9990"**9

9990993 j

9099**96
BBWMOft
900909*.
990009*4

00O4M '*U I

2548177 I

3280302 I

4025028 '

. 4750375 I

. 5461310

. 6116316

. 6714260

7249673

7720880

. 8120287

8478303
. 8773116
9019110
9222277
9388416

9522999

9631017

. 9716934

. 97840OI
9837560

. 9878.127
. 9900944
. 9933821
. 99.11798
. 996.1205

. 9975109
. 9982356
. 0987607
. 9991376
9994053

. 990. 938
9Wt72.11
90081.17
9098776
9909191

. 1909475
. 9999661
. 9999783
. 0909863
. 9099914

. 0999947
9999967
9999980 I
9999986 I
9909093 '

•I99900U

9009908 !

9099999 |

99't9999 '

WH8MXH* I

0071157 -

0281415

. 0621386

. 1070237

. 1020820 i

. 2251114 '

. 2025.154 j

. 3627122 I

4333028 i

5025700 |

. 5687467 I
8306168
6873122
7383089

. 7833062 |

8220246 '
8562471
8840624 !
9083009
9278799 ,

9437608
9565.122
9007306 |
9747495 ;
9810035

. 007134m
9979733 !
9085792 I
. 9990129 j
. 9003204

. 9905364 I
. 9996867 I
. 9997902
. 9998608
. 999008.1 !

9999404 I
9999610

. 99997.14

. 0000*45 |
9990902
I

. 9909939
9999063
9999977
9999986

. 9999992

. 9909995
. 9999907
. 9990998
9000099
9999990

0062299
0240824

054i. .198
00.1 1 >49.1
1443941

2009797
2620373
33*3453
30.13279
4621421

. 5272462
5803404
6474304
7007900

. 7480500

7017194
8291137
861 1-238
8881.731
9115762

930.1013
9459380
. 9583739
9682098
0700523

. 9821023
. 9867.130
. 9902888
. 0020483
. 9049274

. 9063i.1l
. 9074478
9982147
. 9987026
. 9991102

. 9904218
. 9006102
. 9997396
. 9908276
, 9098870

. 9090266

. 9909.127

. 0009608

. 9009800

9999881

I. OOOOOOO

. 0900004

9000007

. 9099908

. 0099009
. 0909999

I"
18)55400 :
02 19757 :
0487639 I
08.10321. ,
1200280 ;

1811783 j
2381183
2980700
362013.'
4257'.' I

4S8787 I
5408730
0079822

■.02303.1
7122.140

7.174708 I
7977882

833217.1 I

803014!) '

8901495 I

9122714 !
. 9301.821 '
94.18085 I
9.180804 I
9t.79l36 |

971*1969
. 9817837

9864876
. 0000803

9927925

. "4048108

. 9903105 l

99741*04 j

9981.108 I

9987480 j

9001443 ;
9994208

9996119
. 9997428
9998300 I

999891 KJ |

9099292 !

. 0090548 I

. 99*19715

. 9099622 j

I
. 0000889 j
9909932 I
99009.19
909007.1
9000085

9099001
9999905

0090007
9*109008
9999090

I. OOOOOOO

45392.10
5132477
5704426
0216889
0753475

7210627

7402.139
8021013
835525.5
8046647

. 889749.1
. 9110784
. 9289946
. 9438652
. 9500631

, 96.1952.1
. 9738786
. 9801.180
. 98.10792
9888010

9918113
9940240
99.16822
9969113
9978125

. 9984662
. 9989352
9992882
. 909.1020
. 990664.1

. 9907763
. 9998.123
. 9999034
. 999937.1
. 9990590

. 9999716
. 9999840
. 9999901
. 9999939
. 9000063

. 9990078
. 9900987
. 9099992
. 999990.1
9990097

. 999999S

9999990

I. (8)00000

Table II- 2 : Circular error probabilities
CBowditch, 1977).

25

fl Sin Wl <Z Sin 6O/2.

(To =

— if-9m

fi Cos P/2. >/ICos^.

Using the ratio, c = -Si. - I^j . =. . 5 B and

<3% 8-^6

IC s ro<^us of circle, _ 10 ^ J. . 2.

enter Table II-2 with K = 1.2 and c = .58 - .6. The proba-
bility is found to be approximately 67%. (The value in the
table is .6714269.)

EXAMPLE 1 1 - 2 : For the system described in example II-l,
the radius of the error circle with 90% probability may be
determined.

First, entering Table II - 2 with c = .6, for 90%
probability (the closest table value is .9019110), K is
found to be 1.8. The radius of the error circle is equal
to K times a :

1.8 x 8.48 * 15.3 meters

26

Table 1 1 - 3 is more convenient for solving
problems such as in example II-2 because the table is entered
by using values of c and probability, P, in order to solve
for the conversion factor, K. Note that the error circles
identifying the 50% probability area (circular error
probable, or CEP) and 90% area (circular map accuracy
standard, or CMAS) are the most frequently used probability
intervals.

For constant values of ff- and a , circular error
probabilities vary as a function of the angle of intersection,
3, of the lines of position. To simplify the investigation of
geometrical effects, the common case of a, = a? = a will be

considered. Under this condition, the equations for a and a

' x y

simplify to equation 1 1 - 2 . Taking the ratio of these two
values, c is found to be C =. ^*/#% =.\an(fi>/z) . Using

the simplified equations, significant parameters of the error
ellipse have been listed in Table 1 1 - 4 as a function of the
intersection angle, 3, for the 501 probability interval (CEP)
and in Table II -5, for the 901 probability interval. The
data shows that the radius of the error circle, R, increases as
the angle of intersection decreases. In the last columns,
the error factor is defined as

Frrnr fartnr = R (at an>" intersection angle)
trror tactor R (at g - 90u)

27

3

w— a

r* a »

— 3 —

CS3^-.
I- :»x
r» 3 ^

»S3

« 3 —

— — ?*

••C* rt

-*«?.-!

_.. ._

— ,

—

a

is 13

= 53

i§2

-r x 3

flirt £

'sis

— — Kt

— » iC

— — > ?•

?»S*e*

fl fl fl

a
3

3 = 3

«3 ■»
3«3

3»3

= 31

1 ---

M«e*

•"5?5«

St

— — n

3 i*1!*
3S»<3

x-s-e

a«s —

J5 fl IIS
»3-
— ^ c»

^ — ^»

= 53

a — fl

3--

r»e»e»

;«?»«

3'

3»«?

333

a — a
•«?»■»

ETC 3
Z 3 u-

3 — —

?■«*•

S»<m

•3

3

Tn 3
3tf>»

333

a»^s

3 — ^

X 3 -

3 — —

9*C*«

!■»«?»

si

* 3 ""

f»— <a

1 sna

— r-3

3»*a
fl-fl —
3«r»
flua —

ran

! 3--

-»-«c»

c*«n

! 333

, 532

r» — o

3«3 —

55*»

a x> 3
x a jo

?!33

; *--

— s»e»

?«««

1— — rjp.

33S
33^

3 — — — e»« j»»m

»»l-

§£3

3--

m ■ p*

333
3S3

«3n

— s»e» finn

!Z3

^-- -C^otf

323

133

3>-3

a 3 =
3 a a

t/J •

•H t»*
iH t>.
rH C*
O (H

>-• •

j- o
o o

Stfj

o «-*

•H -H

en XJ

f-i cO

> O

fi f-i

o Q«
u

+->
*-. c

0 o

CO
en >

M-H
O 3
■M O*
U O
CO

o

.. <o
tn rH

1 o

i-r ^i

t-4-H

o

(U
»H O

cO

28

u o

O 4-»
^ U

o

o

CM

o

vO

I-t

\o

««fr

en

o

oo

to

O

o

CM

cm

to

cm

CM

en
vo

CM

II

"

fi

a>

/"*>

X

a.

o

D

D

o

t>

D

o

5

w

to.

vO

CO

CM

o

OO

o

O

D

O

u

to

00

CM

Cn

CM

O

CM

rH

LO

CM

c/»

v '

iH

I-l

CM

CM

»3-

LO

vO

O

OO

LO

<u

•

•

•

•

•

•

•

•

•

•

to

<*

I-l

<-l

t-l

t-l

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t-l

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CM

CM

LO

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to
to.

«

oo
to.

o

vO

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en

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cn

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<3-
to
to.

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cn
to

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to.

to.
LO

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vo

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vO

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cn

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to.

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to.

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OO

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$-.
<L>
+J

s

c3

■M

cd

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c

oo

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to

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CM

to.

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00

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PQ

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I— 1

E-

29

u

u o

O -P

O rH
O O

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to

LT>

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eg
eg

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to

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r^.

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rr

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rg

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<H

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00

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rH

cd

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>

10

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a

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c

rH

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XH

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•H

f-i •

tH

U /— >

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0) r-

XI

r-^

n}«HOi

43

O rH

O

*-•

tf) •>

&. 5-

4) O

o\»

■M +J

O

<U -H

<y>

S'G

rt 3:

u

r-l O

o

<rt pq

tu JV

m

o

a

J3

©

o

o

o

0

o

o

o

o

o

a

o

o

o

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in

o

o

o

o

e-«

C>

00

r^

vO

m

■*t

•^i-

to

rg

rH

30

or a multiplier by which the error circle radius, R, at any
intersection angle may be computed from the radius of the error
circle at 3 = 90°. For example, from Table II -4 it is seen
that at a 50° intersection angle, R is 1.206 times greater
than the radius at 3 = 90°.

As seen in Tables 1 1 -4 and 1 1 - 5 , the optimum
accuracy is obtained when the intersection angle, 3, is 90°.
It can be said that the geometric dilution of precision (GDOP)
is minimum for a 90° intersection angle. Thus, the error
factor defined in Tables II -4 and II -5 is commonly known
as GDOP. Effects of geometric dilution are shown in Figure
II-4 for CEP and 901 probability interval (CMAS) . Acceptable
intersection angles for LOP's used in fixing hydrographic
positions usually range between the limits of 30° and 150°.
As seen in Figure II -4, the radius of the 90° probability
interval circle is increased by a factor of two near the
acceptable limits for hydrographic fix angles. Correspond-
ingly, positioning accuracy is decreased by a factor of two.

b. Root -Mean Square Error (d )

^ v rms

The root -mean-square error, d , is defined as

n ' rms ■

the square root of the sum of the squares of the error

components along the major and minor axes of the error ellipse.

To calculate the d error, first equations Il-la, lb or

rms

II -2 are utilized to obtain the values of a and a . Then

x y

the definition of d is used,

rms '

31

«0 -

10

10

Cu
O
Q

a

1.0

«V*o-,

C£f-S0%

J L

' ' I I L

!_l I I L

O A tO U « M N TO N N IM l» 120 IM 140 ISO IM 170

Angle of intersection

Figure II -4:

Geometric dilution

of precision for CEP and 90% probability-
interval (Bowditch, 1977).

Tms

= i£7tT = feFZ

«3

(II-6)

where a » cr is the semi -major axis of the error ellipse
and b s a is the semi-minor axis.

Alternately, formulas II -la and II -lb are sub-
stituted into the definition of d error (equation II-6)

rms

and a more useful form of d „ is obtained in terms of

rms

a.., a- and the angle of intersection, 0:

6 - 1

uF+

rf

(II-7)

SCn£

Figure II -5 illustrates the definition of d error.

32

drms = 7 ^2 I crv2

Figure II -5: Illustration of root mean square error.

One d is defined as the radius of the error circle

rms
obtained using one ax and one ay as the semi-major and semi-
minor axes of the error ellipse. Two drms is defined as the
radius of the error circle obtained using two times the

a and a values .
x y

The value of d does not correspond to a fixed
probability interval for given values of ol and o2. It
corresponds to a fixed probability interval only when
3 = 90° and a. ■ a- so that the resulting probability figure
is a circle. In the elliptical cases, the probability
associated with a fixed value of drms varies as a function of

33

the eccentricity of the error ellipse. This can easily be

seen with an example using Table II - 2 .

First, consider a = 15 m, a - 10 m,

a y

drm, = VOS^TOO^ =!6m,

C=5L=.G66 ond ^ ro<Kos cf err°r c;rol& >i§- sl.1 .

For c = .666 table values must be interpolated. Enter
Table II -2 with c = .6 and c = .7 for K = 1.2. The correspond-
ing table values are found to be .6714269 and .6306168.
Thus the probability of 18 m d is found to be 64.78%.
Secondly, consider a = 17 m, a = 6 m^

C= V|7-.= .353 and *=iL.±-059.

IT

The interpolated probability from Table II -2 is 67.41.

As seen above for the two cases, d errors are equal but

' rms

c values (eccentricity) are different. As a result the

corresponding probabilities are 64.78% and 67.4%.

Table II-6 shows the variations in probability

associated with the values of 1 d„m~ and 2d „ as a

rms rms

function of eccentricity (a /a ), and Figure 1 1 - 6 shows the

y 3c

same information graphically for 1 d error.

° r ' rms

34

Probability

1 d

0

.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0

rms

.683
.682
.682
.676
.671
.662
.650
.641
.635
.632
.632

2 d

rms

.954
.955
.957
.961
.966
.969
.973
.977
.980
.981
.982

Table II -6: Variations in probability as a function of
eccentricity (Bowditch, 1977).

aMO

aMO

Id

rma

O.fTO

O.MO

1 o.«so

1 0.4*0
0.«30

I.I

a«zo

O.CI0

.1

»

Figure I I -6:

0 0.1 0.2 0.3 0.4 0.9 04 07 O.i 0.9 1.0

Variation in d with ellipticity (1 drmc.)
CBowditch, 1977).

rms

35

As seen in Table 1 1 -6 , the probability that the
position will be within the 1 d error circle ranges from
68.3% when a = 0.0 to 63.2% when a = a and two d
ranges from 95.4% to 98.2%, respectively.

In Equation II -7, the d error was given
assuming the errors in each line of position are independent.
If the measurement of line of position #1 is related to
measurement of line of position #2, then there is correla-
tion between a., and a • e.g., a, is dependent on a2»or a
change in a, produces a corresponding change in a2. In
this case, the equation for the root mean square position
error is given as

(II-8)

where o is the correlation coefficient between a- and o?.
Two different derivations of this equation are presented in
the following papers: Bigelow (1963) and Heinzen (1977)
[Refs. 2 and 10] .

In summary, root mean, square error is easy to
obtain mathematically, and it yields relative values of
accuracy which are normally understood. Therefore, in subse-
quent sections, d will be used to explain the repeatable
accuracies of hydrographic positioning systems.

36

D. REPEATABLE ACCURACY OF HYDROGRAPHIC POSITIONING SYSTEMS

1. Ranging Systems

In ranging systems, the lines of positions are drawn
as circles centered about each control station. The repeat-
ability of this type of system is a function of the inter-
section angle, 3, and the random errors associated with each
line of position.

The two ranges are independent of each other. There-
fore, the correlation coefficient, p, is zero, and d is

' ' v* ' rms

given by Equation II -7 which is repeated here :

(III-9)
5wp

drms = i \£*7-

Usually, the standard errors of the two shore stations are
equal. The system standard error, a , of a time measuring
positioning system is given as

°1 = °2 = as *

The system standard error, a , of a phase comparison
positioning system is computed as a fraction of the lane width
so that a1 = a • = aw = a , where a is the standard error of
range in fractions of a lane (i.e., a = .1 lanes) and w is
lane width. Then Equation II -9 reduces to

37

d_ = JE g* (n-io)

where a is the system standard error.

s '

As seen from the above formula, the d is
smallest at a 90° intersection angle and becomes large as
$ approaches 0° or 180°.
2. Hyperbolic Systems

As in ranging systems, the repeatable accuracy of
hyperbolic systems is a function of intersection angle and
random errors. Because landwidth is not constant for hyper-
bolic systems, the change in lane width must also be quanti-
fied. As the user moves away from the base line between the
master and a slave unit, the lane becomes wider due to the
divergence of the hyperbolic LOP's [Ref. 8], This divergence
is expressed as an expansion factor, E ;

%- 1/sin C9±/2) 3

where 0. is the angle between the radius vectors from the
position at p to the master and the respective slave station
CFigure II- 7) . Then the standard error of one line of
position at p is

_ CW Ca)

2

38

i

t

f s

r,y

^^f*

^^^/fl

o^\'/'

s\iAstJ

''*/N

* / **i

S\

h / i i

\/ / /

S1 \

s

s

s

i /

s

t /

•

i i

•

i /

s

s

i /

^s

s

i /

*

■ #

*

M

_i d\

\

\

\

Figure II-7: A hyperbolic triad

where a is standard error in the base line in fractions of
a line, w is lane width and E expansion factor.

The hyperbolic LOP's bisect the angle between the
radius vectors from p to master station and the respective
slave station. Therefore, the angle of intersection, 6, is

P =

2

(b)

Substituting equations (a) and (b) into II -8, d becomes

Pms =

C*NX/

1 _i lpCosfo*Q*/Q

Sin S! . SJo *£

(II-1D

39

In a triad (three-station net) one range is common to both
lines of position. Therefore, the correlation coefficient
is not zero. Bigelow (1963) [Ref. 2] assumes a value for
the correlation coefficient, ©, of 0.33 while Swanson (1963)
[Ref. 14] gets £ = 0.4. Since the determination of this
value is based on observations comprising a statistical
sample, the most conservative value of d may be obtained
by using £ = 0.4.

3. Azimuthal Systems

In an azimuthal system, whether it is optical or
electronic, the lines of position are radial vectors eminating
from each of the shore stations. The repeatability of such
systems is dependent upon the angular resolution of the
system, and the angle of intersection of the radial vectors.
The errors of position depend on:

(1) the distance, r, along the radial,

(2) the angular resolution, a, in degrees,

(3) the angle of intersection, 3.

The angular error may be expressed as an arc distance perpen-
dicular to the respective radial at p as

Cl = rH°- (a)

57-29fe '

where r is the distance along the radial , a is angular resolu
tion, and 57.296 is conversion factor from degrees to radians

40

tfi

- hiL^ ^\

<*v

1 w / Ls^ ^^

C ^^

• 1 / NOvN

! ' X^

My 9l Is

t b

— »<

Figure II -8: Azirauthal System Repeatability

The two shore stations are independent; therefore, the corre-
lation coefficient, £, is zero.

Substituting Equation (a) into H-7,

51-236 *infc

(H-12)

Applying the sine law to the triangle shown in Figure I I -8,
it is seen that

41

where Sin [180° - (9 + e2)]= Sin 3 = Sin (e'.j + Q 2) and
b = baseline distance. And Equation 11-12 may be written
as

drms =. 2dfe . — \/Sm2e, ^Svn^Ga' . (11-13)

5-F.196 s\oa(e^©^

Bigelow (1963") [Ref . 2] approximates Equation 11-13 by-
letting

Then Equation 11-13 reduces to

dr,r,s - — 2=i — CII-14)

Equation 11-14 is the approximate form of Equation 11-13.
However, Equation 11-14 is easier to compute and the error
introduced is negligible. For a = .03°, b = 8000 meters,
9-, s 80°, 92 = 30°^ comparing the equations 11-13 and 11-14:

using 11-13

<4 _ (-o3Ueooo) . __£ „ ^SJna80fl+ Sin^O0' - 5.2m .

rms

using 11-14

57- 19* Sv**(fco°*3fl

A _ (-03U8O0O) 1 _ s e _

urms — — * — - -*• j <Ti .

\FL * 5-?- 296 Stn TO0 Svn T0°/a

42

it is seen that the difference between Equation 11-13 and
Equation 11-14 is negligible.

4. Sextant Angle Positions (Three Point Fixes)

The evaluation of repeatability for sextant angle
positions is difficult. The mathematics involved in the
computation are quite complex. Thus, repeatability of
sextant angle positions is more easily evaluated by a graphi-
cal analysis. For the development of an analytical solution,
see Heinzen (1977) [Ref. 10].

As will be seen in later sections, it is much easier
to derive the accuracy of sextant angle positions by applying
the method of least squares.

E. REPEATABILITY CONTOURS

Using the root mean square error concept, one can con-
struct a family of curves to display convenient values of
d in terms of the system geometry.

1. Ranging Systems

For ranging systems, the d is given by
Equation 11-10 as

j _ ^T5 g-s (ii-io)

Sin p

Note that the intersection angle, 3, is the only

controlling geometric factor of d . Figure II -9 shows an

rms

example of suitable geometry for a ranging system. Mathemati-
cally, it can be proven that the intersection angle, 3,

43

is equal to the angle formed by radius vectors from p to
the slaves and also the angles S,OD and S-OD. The locus of

points having a constant d and constant 3 describes a

r ° rms

circle of radius r ■ b/2 Sin 3 with the two shore stations
as points on the circle.

The distance, e, along the perpendicular bisector of
the line connecting two shore stations to the center of the
circle is

h / ?

(since, from Figure II -9, tan 3 = -—-) where b is the distance
between shore stations S^ and S».

Using 2 o error (approximately 95% probability
interval), Equation 11-10 may be written in the following

form

6pmS _ 1^
<rs S0o£

Writing Equation 11-15 as

b ltaop>

Figure 11-10 was constructed to show the relationship of
drms/a and e/b as a function of intersection angle, 3.
Using this graph, selected contours of constant d may
be drawn as in Figure 11-11. First, plot the location

44

Figure II -9: Ranging system geometry.

of the two shore stations at a convenient scale. Draw a
perpendicular bisector to the line joining them. Using
Figure 11-10 determine the values of e/b for the desired

d „„ contours. From the known value of b, determine distance

rms '

e for each contour. Lay off distance e along the perpen-
dicular bisector to define the center, 0, of the desired

constant d„„,„ circle. The radius of the selected contour
rms

is the distance from the center, point 0, to the shore
stations.

45

Example II -3: A phase comparison range -range
positioning system has standard error as " .01 w (lane width)
It operates at 2 Mhz frequency. The distance, b, between two
shore stations is 20,000 m.

t -j*u v „ 300,000 7C „
Lane width, w = ^ = 2 x^QoO * 75 m i

a » aw s 7S x .01 a .75 m .

s

For the 2 m d „ contour, d^/a ■ 2/. 7 5 s 2.66.

rms rms s

Enter Figure 11-10 with drms/as - 2.66 which intersect the
d /a curve at 85°. Follow the 85° line vertically to

</)

3

o

<D
3

i-H

>

100

50
40

30
20

10

1

N

^ —

— rH

-±-

\

I

I

kt

i i

\

1 i

\|

N

s

\

J

b

>

\

^

l

^

\

^

s»^

vj

s*_

rr

•

^v

^

•^

9K

J

.6
.5
.4

.3

.2

.1

.06
.05
.04

10°
170°

20°
160°

02

30° 40°50o60370o80o90°
150° 140° 130° 120° 110° 100°

Figure 11-10:

.£3

O

O

tn

<u

3

.03 2

05
>

Intersection angle, 3, in degrees

For ranging systems, the graph of the
drms/as and e/b. (For enlarged figure,

see Appendix B.)

46

the e/b curve. It intersects at e/b = .043. For this
specific pair where b = 20,000 m,

e = b (.043) = 860 m

Using the described technique, the 2m d contour

can be drawn, and the result is shown in Figure 11-11.

Thus between the 2m d contour for 3 = 85c and the 2m dmi.

rms rms

contour for 3 = 95°, the d„m error for the described system

' rms

will be < 2m. 95% of the time.

Note that when the angle of intersection, 3 = 90°,
d error is minimum. Therefore, as 3 increases toward

180° or decreases toward 0°, d becomes larger. Because

' rms

the tangent of the angles greater than 90° is negative,
e values will be negative as well. Thus, the center of

the constant d,.^ circles, for angles greater than 90°,

rms ' ° ° *

will be on opposite side of the baseline. As shown in

Figure 11-11, d m_ error increases as the baseline is approached.

' rms r

Contours for d „ values of 3, 4 and 5 meters may be constructed

rms ' J

by following the procedures outlined above.
2. Azimuthal Systems

For azimuthal systems, d error is given by

' rms

Equation 11-14 as

rrns TZ — T7^ 1

^ a 5f - 296 S»n £ ^o £ /%

47

/ /^^ ^"^\ \5m

/ / \4m \

/ / / ^m(^m3m \

'^^/\\2m / /

<\^\X \\(£-85") / / /

\ Nv \\ Vrlm / / /

\ N. M l/*=95-V / /

Figure 11-11: Repeatability contours of a ranging pair.

48

where a is the angular resolution, measured in degrees,
b is the distance between two azimuth stations and 3 is
the intersection angle of radial vectors which is defined by
the equation 8 = 180° - (6, + 9-) (Figure II-8). As with
ranging systems, the intersection angle, 8, is the only
geometric factor contributing to d . Constant error con-
tours are obtained in a similar fashion.

Writing the Equation 11-14 with 2a error (approximately
95% probability interval) as

irms

a'b nTT (5?. 296) S(n £ ■ Svo P/x

where a = angular resolution and b = baseline distance.
Since e/b ■ 1/(2 tan 8), a graph is drawn showing drms/a*b

and e/b as a function of intersection angle, 3 (Figure 11-12)

Figure 11-12 provides a convenient means to obtain

the values of e distance for a selected d if a and b

rms

are known.

Example 1 1 -4 : The distance, b, between two azimuth
shore stations is 2,000 m. Angular resolution of the station

is ax = a2 = .01°.

For the 2 meters d contour

rms

rms 2 .,

~a7F ' .01 x 2000 ' -1

49

.1

etf

s

.05

0.04

M-4

o

.03

(A

3

.02

r-(

Ctt

>

.01

"V

_.......__ ... m .

->

st

1

V

f

s

I

I

-1 -

!\

7

N.

\i

I

\

1

dfms

a»b

V

*

V*

LX-e

! p* b /

)

\i

1

/ • /
1 /

+», — s

10

20

30 40 50

100

Values of angle 3 (in degrees)

.5
.4

.3

.2

.1

tf

.04

«M

.03

3

U)

.02

,i

d

>

.01

200

Figure 11-12: For azimuthal systems, the graph of r£21i and" -i.

a-*

CFor enlarged figure, see Appendix B.)

50

Enter Figure 11-12 with d /a»b = .1 which intersects

the d /a-b curve at 44°. Follow 44° line vertically to the

rms '

e/b curve, which intersects at e/b = .52. For this pair where

b = 2,000 m, e = 2,000 x .52 = 1040 m. Using the technique

as described for ranging systems , the 2m d error contour

may be drawn (Figure 11-13). Other contours are computed

in same manner.

Note that when 6 > 90°, the tangent value is negative

and the center of constant d circle will be on the

opposite side of the baseline. For azimuthal systems, the

minimum d_„ error is found at 3 = 109°.
rms

3. Hyperbolic Systems

The root mean square error for hyperbolic systems
is given by Equation 11-11 as

rms - \/ + + — £

s™ P V Sm1 ©» S»'oa 5> Svn Si . Sin ®i

x i a. ±

(11-11)

where a is the standard error along the baseline between the
master and respective slave station in fractions of a lane,
w is the lane width and 8 is the intersection angle which is

equal to

91 + 92
3 = _J^ — £ (Figure II-7).

51

Figure 11-13: Repeatability contours of an azimuthal system

52

Equation 11-11 is written with 2a error as

<Ws _ 2. \ / 1 + ± + 2 (,H)Cosfi

where the correlation coefficient, o, was taken as .4
[Ref. 14]. Figure 11-14 was produced to show the d s/aw
values as a function of the angle subtended by the two slave
stations , i.e., 23 .

A parameter, p, defined as

p _ Q> + ei _ _x_£_ or p= a + ea _ n$
e. e, e2 8a.

The parameter, p, is computed with the smaller of the two

angles, a, or a-, in the denominator. Thus when p = 2

the master station is positioned on the bisector of the angle

subtended by two slave stations, p = 3 places master station

on one of the two trisectors, and so on (Table II-7).

Knowing the angle subtended by the two slave stations

at a particular point, Figure 11-14 may be used to develop

contours of constant d_ „. First determine the d /aw ratio

rms rms

for a selected d__ . Enter Figure 11-14, for several values

rms & '

of parameter p, and read the corresponding values of angle 28
Using the relation between 26 and 6, (8-) (Table 1 1 -7) , plot
these angles, 26 and 0, (9-) , on a conveniently scaled chart

53

C/V

e

o
en

3

>

200.

300.

2$, in degrees
Figure 11-14: In hyperbolic systems, for several values of p,

d /crw curves,
rms

54

p

91 °r 92

2

(2S)/2

3

(26)/3

4

C23D/4

5
6
7
8
9
10

(23)/5

(23)/6
(28)/7
(23)/8
(2S)/9
(23)/10

Table II -7 : Relation between 23 and 6.. or 6

55

with a three arm protractor. Interpolating between the points,

draw the d contour. The curve thus determined defines the
rms

location of a selected d contour for the specific conditions

rms

of triad configuration.

Example 1 1 - 5 : A hyperbolic system has standard
error, a, equal to .01 lanes along the base line. It
operates at a frequency of 2 Mhz. Triad configuration is
as seen in Figure 11-15 :

i -j^u v 300,000 -c
lane width, w = ^ = 2 x'2>00o = 75 m

aw = 75 m x .01 = .75m,

For the 4 m d „ contour ,d_/aw = 4/. 75 = 5.32. Enter

rms ' rms

Figure 11-14 with d /aw = 5.32. For several values of p,

read the corresponding values of angle 2$. Determine the

values of angles 9 or 6 according to Table 1 1 - 7 . For the

4m contour, these values are shown in Table II-8. Using a

three arm protractor, the points defining the 4m d contour
r ' r & rms

may be plotted. The other contours are drawn in a similar
manner (Figure 11-15).

56

vO

e . .

^/^

■*-
0

c
c

•

u tn

Figure 11-15:

Repeatability contours of a hyperbolic system
Co - .01 lane width, and f ■ 2 mhz)

57

o

o

U"J

m

•

m

oo

<ji

O

vO

CM

o

oo

II

II

t--

CM

/ — \

/~\

CM

rH

II

CD

CD

v '

\^

ca

I— 1

CM

CM

<x>

CD
o

0

in

LO

•

•

in

oo

<5f

"*

rH

o

^t

II

II

ct>

!-H

f~\

i — \

"3-

CM

rH

ii

CD

CD

II

\mJ

S_^

ao.

rH

CM

D<

CM

O

CD

O

o

0\

o\

CD

o>

rH

O

00

II

II

C\

CM

/"~\

r"*"»

CM

rH

11

CD

CD

v_/

v_/

CQ

i— 1

CM

CM

CD

CD
O

o

00

in

o

m

rH

O

*1-

II

II

vO

i-H

i — \

r— \

ro

CM

rH

II

CD

CD

II

\_/

v— '

CO-

rH

CM

a,

CM

CD

CD

•

O

O

tn

to

t^-

t^.

o

vO

ir

ir

«*

rH

rmm \

r— \

CM

CM

rH

II

CD

CD

11

v— /

«>_s

CO.

rH

CM

a

CM

CD

CD

Pi

O

CO

O

3

rH

>

rH
>

to

o
m

rH

3
O
■P

c
o
o

to

s

rH

<U

00

•H

e

•H

«4h:

CD

T3

CM

CD

CD

c

crj
CO.

CM

10
<U

rH

00

c

CTJ

<D
A
H

oo
i

CD

rH
JO

ed

E-

58

III. APPLICATION OF LEAST SQUARES TO
HYDROGRAPHIC SURVEY POSITIONS

A. THE PRINCIPLE OF LEAST SQUARES

Given a set of unknown parameters to be computed from
measured physical quantities such as distance or azimuth,
the least squares method provides a mathematical procedure
by which the best values for the unknown parameters may be
obtained.

Equations must be written to define the relationship
between the observed and the unknown parameters. If the
number of equations that can be written is equal to the number
of unknowns, then a unique solution may be computed.
However, no statement can be made about the accuracy of
the solution. In the least square method, the number of
equations must be greater than the number of unknowns.
As a result of this over-determined solution, the best values
for the unknown parameters are estimated.

This computational procedure is referred to as a least
squares adjustment. In application, corrections are com-
puted and applied to observed quantities and these quantities
are then said to be adjusted.

For a given set of equations, the fundamental condition
of the least square technique requires that the sum of the
squares of the residuals be minimized. A residual is defined

59

as the difference between an oberved value of a quantity
and the arithmetical mean value of that quantity obtained
from a number of observations. If the arithmetical mean value
is stated by x. and observed value by Xj, the residual, v,
is expressed as

V - 3Ci - X± . (III-l)

Suppose a set of observations were taken having residuals

v, , v?, v_ , v . Then in equation form, the fundamental

condition of least squares is expressed as

J. lV^)a= W»^+(Ma>a+(s/^+ + Wasmin\«im1 (III-2)

T
or in matrix form: V V = minimum.

1. Weighted Observations

In general, some of the observed values may be more
precise, and, therefore, entitled to have greater influence
upon the result. Observations are assigned values called
weights corresponding to their quality or worth.

The assignment of weights to observed values is
largely a matter of judgment. For example, if one set of
measurements of a distance was made with four repetitions
and another was made with eight repetitions, the mean of the
second set of observations may be given twice the weight

60

of the first set. Or, when measuring angles in azimuth angle
positions, the atmosphere may be so unsteady during one obser-
vation that the observer arbitrarily assigns a weight of
one half.

As a general rule, if a standard error, a, has been
computed for a set of observations, then weights are usually
estimated according to the equation

i*J£- , (III'3J

\jvr ■=

<rf

whe

re w. is the weight of the ith observed quantity, a-

2
is the standard error of that observation ,and k is any

number which has the same value for all observations.

Equation III-3 states that weights are inversely proportional

to the square of the standard error. Usually, the weight

corresponding to the least accurate measurement is assigned

a value of 1 (a unit weight). Then the value of k can be

found and the other weights computed accordingly.

For example, consider the standard error for two

observations where a, = 3, and o? = 1.5. Assigning w, = 1 %

1 = Jl. „ fc = s

*

61

it is found that the second observation has a weight of 4 relative
to the first observation.

If measured values are to be weighted and used in a
least squares adjustment, then the condition is that the sum
of the weight times their corresponding squared residuals
must be minimized,

X wi.Ncf'-Nx/.Mi'-t-^aVi +_ V*V<?; -romvrnum , (III-4)

or in the matrix form, \j \X/ V — minimum
2. Method of Least Squares Adjustment

In the "observation equations" method of least
squares adjustment, the observed quantities are related to
the desired unknown quantities through formulas or functions
which are called observation equations.

One observation equation is written for each measure-
ment, and it is assumed that observations are independent
of each other. In order to solve for the best value of each
unknown parameter, at least one redundant observation equation
must be written. That is, the number of observations must
be greater than the unknowns.

Observation equations may be linear or higher order
functions. Linear observation equations can be written in
general as follows :

62

Q, * +V>, y + C» "i + -h ki = £j

(III-5)

Qo* +tny + cni

+ kn -= Gn 4

where a's, b's, c's, etc. are coefficients of unknowns x, y,
z, etc. and the k's are constants.

Because the observations (G, , G_ G ) are not

v 1 ' 2 n

free from random errors, each G- must be corrected by a
residual value, v., in order to obtain a mathematically
correct equation system. Thus,

Qo^ + ^>n9 + C-nt-t- -- + kn = Gn-fVn

(III-6)

Introducing a new notation &, = G.. - k1 , £ = G2 - k2
etc., the following equation is obtained;

63

+■ -

— -«. -v.

(III-7)

ortx + b

ny -t- cn^ t-

-•?« *v,

or in the matrix form, V = AX - L

(III-8)

This equation is called the observation equation or observa-
tion equation matrix, where

nX =

n A

n m

Oi , Vm , Ci

Ql > ba , Cx --

rr»

t

i

i

Qn j «n ^ ^n rnn

x =

X

y

n 'x»

„L,

4.

64

In the above matrices, the subscript n denotes the number of
observations and m denotes the number of unknowns.

For a group of equally weighted observations, recall
that the following condition must be enforced in order to
perform a least square adjustment:

2_ Cv^) — minimum ,
Ul

or in the matrix form,

T . .
V V = minimum

Substituting the value for the V matrix from the observation
Equation III -8 where V = AX - L ,

VT\/ = (AX-L^IAX-L)

= UTAT- C) I M.-0 (-from motrv* al9ebra)

= KTATAX - KTATL - LTAX + Cl_

and from matrix algebra, L7A^ -= )OATl- •>
then

VTV = XTA'AX - 2 X'A'L +LtL .

65

The minimum of this function can be found by taking
the partial derivatives of the function with respect to each
unknown or with respect to the X matrix (which contains all
of the unknowns) and equating it to 2ero, i.e.:

_(VT\M =2^A^-2^i- =0.
-a*

Dividing by 2, the following result is obtained:

A7A*- ATi_ - O

11-9 )

This is called the normal equation. In conventional notation,
the normal equation (III-9) becomes

Coa] * + tafcly + Zcxcl-t + CaJ] -0

[fcal * + CWbly +Ctc3z + -Lbf] -0

fcalx 4 Ccb]y +Cccl-J v _Cei3 ^0

[nalx + CnV>3y +tncli^ -Cnil *0 ,

where the symbol [ ] denotes the sum of the products, i.e.,
[aa] = aiax + a^ ♦ a3a3 + *. anan, [ba] = b^

+ b2a2 + bea3 +

+ b a .
n n

66

T
In Equation III-9, A A is the matrix of normal

equation coefficients of the unknowns. Multiplying Equation

T -1
I I I -9 by (A A) and reducing, the solution is obtained.

(ATAVA(ATMK - (*TAfVTL ^0,

Equation III -10 is the basic least squares matrix equation
for equally weighted observations. The matrix X consist of
best values for the unknowns x, y, z, etc.

For a system of weighted observations the funda-
mental condition is

ft

^_ sV C L^l) = rruoimocn ,

or in the matrix form 3 V >CwV s minimum.

The normal equation matrix is derived similarly to
the unweighted case.

AT^KX - AT^/L *0 , (in-ii)

or in conventional notation }

67

C\Naal* + C^qVjI _>. _\LvA/afl

L^bal* + Cv/WV>l + _Lwb/l

; ; i

! ! I

I J I

Lwnolx + Lv/nbl ^ ..twnfl

= o

= 0

In Equation III-ll the matrices are identical to those of
the equally weighted equation, with the addition of the
matrix, \^/, which is a diagonal nxn matrix.
In detail, W becomes

\N =

0
0
0

Wa 0

0

0 W3 0

0 0 VJl+

(111-12)

where according to Equation III-3,

Wo »

k*

NA/ x a

©Vc

The best values of unknowns are obtained by solving
Equation III-ll as

68

* = ( A^nWAV1 ATWL. (hi-13)

From the combination of Equations III - 8 and III-9 or 1 1 1 - 8
and III-ll, it is seen that

AT ( V -r O - AXL = O

At\n (v+O - at\a/l -0.

or

Therefore,

ATV=0 or ATWV=0. (111-14)

Equation 111-14 can be used as a check on the computation.

Example III-l1: As an elementary example illustrating
the method of least squares adjustment by the observation
equation method, consider the following equally weighted
observations :

-*l - *2. - ** =-6.

^he numerical values of this example problem were taken
from Ref. 17, page 517.

69

These four equations relate the three unknowns x,, x? and
x, to the observations.

By including residuals, the equations may be
rewritten as observation equations as follows:

1 *t + 3 %a -i- X* = \0 + v»
or in matrix form,

^

= «<\A-^i. '

where

A-

13 1

10

v,

T 1 -1

J *=

**

j L^=

5

J

V =

Mi

-i -l -i

L

*3

-6

%

70

The normal equation is ATA X. "" ATL — O;

K* =

2

i 1

-1

1

3

-2 X

-i

1

3 -1

-i

-i

ATL =

2 i

-1

3 -2 i _1
1 B -2 -i

3 1

-2 3

i -2

—

-i -1

55

11

-8

12

15

"^

-8

-<t

IS

-

-

to

52

s

3

=

23

-t>

25

-

• -

ATAs*-ATL =

55 |l -Q
12 is -4
-* "^ 15

-

-
52

Ka

—

29

*3

25

— —

= 0

-1

And the solution is X = (ATA') A^L- ;

(ATA^-

.022859 -.OI6187 -0o?87^8
-.016197 .083233 .0135613

.007875 * 01*562} .074^83

71

*«

.022853 -.oi6lS? , OOlZl-h*

-. 0i4*S> .083233 . C»3S623

.00?8?5 .OI35&23 * 07*1*1 83

Si
29
25

X =

- 9i tl

1-9H09
2.^88

Thus the best values for the unknown parameters x, , x? and
x_ are Xl = .9161, x2 = 1.91109 and x3 = 2.66488.

This computation was performed by requiring that

T
V V = minimum. Thus, when the best values are used in the

equation V = AX - L, the resulting minimized residuals can

be found. If the minimized residuals are applied to the

observations then the observations are said to be adjusted,

V^ A*-L »

X 3 i

1 -2 3

1 i -1

-i -i -i

-9161

10

1.91109

—

5
3

—

2*6649

-6

„

.

, 23035
■ 08856
-<0S9?
.SOT93

72

. 23035
V = . 08856

. 50733
Then adjusted observations are G., = 10.23035, G2 = 5.08856,

G3 = 2.9403 and G4 = -5.49207.

T
Computational check: A V must be equal to zero

according to Equation 111-14 .

ATV =

2

i

T

-i

- 23 035

3

-2

i

-1

. 08856
--OS9T

—

1

3

-2

-L

- S0t9J

»000
.000

.ooo

According to the theory of probability, the
above values of x, , x~ and x^ have the highest probability
of occurrence.

Example III -2: Suppose the constant terms 10, 5,
3, and -6 of the observation equations of Example I I I - 1
represent measurements having relative weights of 1, 2, 2,
and 3, respectively. Using weighted least squares, best
values for x, , x2 and x^ will be calculated.

73

The observation equations in Example III-l were
1*1 -*■ 3*2 t *3 s 10+Vi

f X, + %2 - 2 X3 = 3> -t ^i

or in the matrix form>

V = AX - L

where

* =

2 3 1

i -1 3

T i -2

"A -i -1

,**

*2
X3

, L-

10

5

3

-6

V =

v,

*3

The normal equation for weighted observations is

ATvVAX -AT\A/L =0,

where weight matrix VV is a diagonal matrix of weights as
follows :

w =

i o o o

0 2 0 0

0 0 2 0

0 0 0 3

J.

74

AT>A/A»

1 1 ? -1

3 -2 i -1

i 3-1-1

.

1 0 O 0
0 2 0 0

oo2o

0 0 o 3

2 3 A

1 ^2 3

7- 1 -2
-i -1 -1

—

19 21 -10
-ff -10 30

h?\NL-*

a i "f -i

3 -1 i -1
1 3 -1 -i

10 0 0

\o

9o

0 2 c o

0 o 1 o
0 o o 3

—

5
3

-6

=■

3^

■* •>

am

A"\n/A*- At\A/L =

10? 19 -17
13 22 -10

-a -io io

■1

X.

-

90

Xl

-

Zh

-N

*3

*6

= o

The solution is t = (ATvVA ) A\VL ;

* -

.0113839 --0OSI3I4 ,00^74

-.oo3\3^ ,0592795 ,015185
.00*74 .0\5>S54 . 04 0 51 47

90
3f

X =

. Sioi
i-9856
2.H66

►«■ %i*.91-0i , K1= 1.3654 qoA *3 = 2..TU6

75

Residuals are found using Equation 1 1 1 - 8:

V=AX-L=

2 3 i

1-2 3

* i -2

-1 -i -1

.9201
1,9356

2.?lt6

_ —

\o

5

3

-6

.

Computational check: ATW\/

.5136

-.oo69
. mi
must be equal to

zero.

ATWV =

X 1 * -i

3-2 1 -1

i 3-2-1

1 0 0 0

0 2 0 0

0 0 2 0

0 0 0 3

- 5136

.005

•098?

— oo63

3

.000

.sm

- ooo

3. Higher Order Functions

The observation equations presented by Equation III-8
are linear equations. If this relationship is nonlinear,
thus defined by a higher order function, then the observation
equations must be linearized in order to apply the least
square adjustment method.

Defining the general observation equation as
G = f(x, y) , where f represents a non-linear function.
The function must be linearized by Taylor series expansion
or by some other method. The best values of x and y can be

76

regarded as the sum of an approximate value x , y and a

small correction Ax, Ay. Therefore x = xQ + Ax and

y = y + Ay and the above function is written in. the following

form

G n f(xQ + Ax, yQ + Ay).

Using Taylor series expansion the observation equations may
be linearized.

Gt sf (x0J \j0) i- M_ A* + ILL- Aj + \Al9her orcier Urms .

(111-14)

The higher order terms in the series are neglected and only
the zero and first order terms are maintained.

After linearization, the observation equations
become

Vi*

■ay0,

-S-

(111-15)

^n =

77

or in the matrix form, V = AX - L, where

A^

■ — ) ~J —

D%0

5£a

"OUo

r- -\

* =

N/ =

L =

V,

!
i
i

The remainder of the least square procedure is the same as
indicated by Equations III-9, 111-10 or III-ll, 111-13.

In the linearization process, the higher order terms
were neglected. For this assumption to be valid, Ax and
Ay should be small so that their products in the series
expansion approach zero. (Ax • Ay = 0) . This can be achieved
only if the values of x and y are very close to the values
of x and y. Therefore, x and y must be precomputed, or
the original assumed x and y must be improved by successive
iterations until the adjusted observations equal the measured
values .

78

4. Equations for the Precision of Adjusted Quantities
After calculating the best values of the unknowns,
or X matrix, the V matrix, or the adjustments to the observations,
can be computed from the observation equation which is
V = AX - L, whether the observations are weighted or not.

Using the V matrix, the standard error of an obser-
vation of unit weight is given by the following equations
[Ref. 17] :

for unweighted observations,

"° = #fe =\l^k 5 ^-^

for weighted observations,

(To ^

(III-16b)

where

a0 is the standard error of an observation which has unit

weight,
n is the number of observations,
m is the number of unknowns.

79

Standard errors of the best values for the unknowns are
then given by the following equation:

^C = cr0 sj^~i t

(111-17)

where

a. is the standard error of the ith adjusted quantity,

e.g., the quantity in the ith row of the X matrix,
a is the standard error of unit weight as found by

Equation III-16a or III-16b,

T - 1
q.. is an element of, for unweighted case, (A A)

T -1
or, for weighted case, (A WA) matrix.

T -1 T -1
If the (A A) or (A WA) matrices are written in detailed

form as

Ru Rn Ri3 • —

<W S". 9*3 -'

! ' '

-I

( AtaV or (AT\NA)~ =

J

then the standard errors of the best values of the individual
adjusted quantities are:

Oi s. CT0n/ ^h ' ,

80

Example 1 1 1 - 3 : The standard errors of the best
values for x.. , x? and x_ in Example III-l.

Standard error of unit weight for unweighted
observation is

do =

VTV

n-m '

MTM =

. 23035 .08956 -.00597 .50193>

\

.23035
• 08B56

_. 0059T
. 50T93

J *

VTV- .319 ,

crv si

.319
<f-3

= . 5£>S

81

The standard errors of the best values are given by-
Equation III -17 as

\ = yf%ii •

For unweighted observations, q. .'s are the elements

T -1
of [k A) which was calculated in Example III-l.

(AWT1-

• 02X859 -„o»6l8fl. .00~FS>5
--OlfeiSi ,083233 . 0135&1B
. 00?8^5 , 0t2>5fc23 . 0?44-S3

— Co n q 33

565 V ,021855 s -085,

= . 5&5 \Z.0?4zfS'il = -ie>i+.

T -1
In the (A A) matrix, off diagonal terms are used

to find the covariances of unknowns. Covariance a

12

is equal to

crlx •= <rQ \/^~l<

From covariances, the correlation coefficients of variables
are obtained. Correlation coefficient g, - is given as

e.2 -

cr,

tx

o-, o-j.

82

The interpretation of the standard errors computed
above is that there is a 68% probability that the adjusted
values for x, , x? and x_ are within ±.085, ±.163 and ±.154
of their true values, respectively.

Example III-4: The standard errors of x, , x? and
X. in Example III- 2.

The standard error of unit weight for weighted
observations is

(Jo =\/^T^V
n-m

\TvAW -

.5136 .0981--. 0069 -3>7T-

1 0 0 0

,5\^>

oioo

J09S>

0 0 2 0

-.0069

0 0 0 3

. vm

L

MTWV =: .Tll^ ,

\

<Tn -

n-m

4-3

= • 8«

83

The standard errors of best values are given as

crL = o-0 \/s CI ' .

T -1
For weighted observations, q.^'s are the elements of (A WA)

which has been already calculated in Example III- 2.

(AT\NA^-

,011 38 39 -00 8\3\i4- -00374
-0O8\3l4 .0593T-95 .0IBI8SH
.DO 3^+ , 0\Sl854 . 0<+OS\<+J

CT. ^O-o^qT1 =-8« \/.sDU3839' =-0899,

cr*

fc=o-D^T -.8HaV-o59»S5 =-20S45

crM - ^o^ ^.ew>fo555i3r*.»m.

84

B. APPLICATION OF LEAST SQUARES TO HYDROGRAPHIC
POSITIONING SYSTEMS

If redundant data are available, the least square adjust-
ment method may be used to compute the coordinates of hydro-
graphic survey positions. Observation equations may be
written for various types of survey methods. By expressing
these equations in matrix notation and using successive approxi
mations of the unknowns, the best values for the coordinates
of survey positions may be determined. The predictable
accuracy of these best values may also be found. Thus,
redundant observations, coupled with mathematical data
adjustment techniques, produce a viable method of system
calibration for hydrographic survey data. This method of
calibration is referred to as auto calubration.

1. Azimuth Angle Positions

The working range of azimuthal systems is limited to
line of sight distances, i.e., 5-15x nautical miles, depending
upon the height of the observing instrument.

Because of this range limit, the Universal Transverse
Mercator (UTM), or other plane coordinate systems, may be used.
Let

y, , y2, y, ■ Northings of the shore stations 1, 2 and 3,
respectively,

x1 , x2, x_ = Eastings of the shore stations 1, 2 and 3,
respectively,

P = The position of the survey vessel.

85

Then, the azimuth (from north) of the survey vessel from
shore stations can be written in terms of coordinates as

A^ion-1 J*^L , A^tofT' ggl*E , A^p = ta^^, J*zM_

*?-**. xp-^x *p-*3

In these equations, x^ and y are the best estimate of the

P P

survey vessel coordinates which are to be determined.

These equations are non-linear. Thus, in order to
form observation equations, they must be linearized.
Letting x = x + Ax and y = y + Ay, where x and yQ are
the approximate coordinates of the vessel's position, and
using Taylor series expansion for linearization,

fujy> •=. -f ( *0 > sic) + ^£-A* + 5Laj +

where f (*o>vj^ = /\ LO - tan"* ^gZMi .

The partial derivatives are

^f _ 9o-VK "&£ - *q-*l

■ x

j

The observation equations now may be written in the follow-
ing detailed form:

86

Figure III-l. Azimuth angle positions.

V,

Vi

v3

»e

Sib

3o-2i

^

JO

where

si

A*

&j

L

Aip - kio

and Aip , Aap, A3p

87

are the measured azimuths, and g ■ 57.2958, the conversion
factor from radians to degrees.

Having obtained the observation equation, the normal
equation may be formed and solved by following the procedures
outlined in Section II. A.

Using the computed values of Ax and Ay, new trial
point coordinates may be formed as follows:

The values are substituted in the observation equation for
the initial x , y coordinates. The least square solution
is iterated until the Ax and Ay values approach zero.

Example III -5: Referring to Figure III-2, the
coordinates of the shore stations are

Luces (#1) Mussel (#2) MB4 (#3)

x 4,055,042.7 m 4,053,453.2 m 4,053,917.2 m

y 595,794.5 m 597,967.8 m 603,425.2 m

The standard errors for the azimuth observations are a. ■ .02°,

a2 = .024° and a3 = .018°. The following angles were

measured:

p - Luces - Mussel = a, = 50?164

p - Mussel - Luces ■ a- - 99?360

p - MB4 - Mussel = a3 = 47?865

88

Figure III-2

Determination of a position for azimuthal
systems using the least square method.

89

The least square method will be used to determine the best
values for the coordinates of the survey vessel.

Given: A = 126?180
Measured: a-, « 50?164

Given

A * 76?016
A32 = 265?140

Given: A£1 = 306?180
Measured: a2 = 99?360
A = 45?540

Measured: a = 47?865

A,^ = 313?005
3p

First, assume xQ = 4,055,000
y0 = 600,000

The observation equation, V = AX - L, is then

01362 -,000138
■ 0W8S .013 58?
OIS208 -00<f8CR-

^

-"K183
25 .Wl

90

Using the standard errors of each station, a weight matrix
is formed:

\A/ -

0
0

a;

x 0

*/

0
0

/cr2.

Let k - 0.024, then

V7 =

0 i 0
0 0 138

Using the A, L and W matrices, the components of the

T T
normal equation, A WAX - A WL = 0, are written

. 0OO997H -.000IO9T

.-0001093 -00021S^

i.\o32

= o

91

The solution, X = (ATWA) _1CATWL) , is then

X^

1059.2 51*. 6

514. G W9.M

j

1,1031

.1132

1-231.8

1144.0

Now, the new trial point coordinates are found.

W*o+&* =4,055,000 -H23U8 « 4,056,231. 8 ,
3 = 9o-i-^y =600,000 -+±144-0 = 601,14^-0.

Using new trial point coordinates, solutions are repeated
until Ax and Ay vanish. Table I I I - 1 shows the data for other
trial Ax and Ay values and the new trial point coordinates.
The best estimate of the coordinates for the survey
vessel is

x = 4,056,302.9

y = 600,867.4
P

2. Sextant Angle Positions

In sextant angle positions, similar to the resection
problem in geodetic work, the measured quantities are the
included angles at the sounding vessel between the shore
stations as shown in Figure III-3.

92

vO

00

o

"3-

o

t-^

r^

o

^

t^

vO

vO

^

o

1—1

o>

CO

oo

o

1— 1

o

o

o

o

o

o

o

o

vo

o

VO

vO

vO

to

o

p

cO

C

•H

H3

'-H

o

o

u

00

CM

CM

Ov

•

•

•

•

o

1—1

LO

«=t

CM

o

to

cr>

o

o

X

o

CM

CM

to

to

LO

vO

vO

vO

vO

in

LO

LO

LO

to

o

o

o

o

o

tO

CD
P

cd

S

•H
T)
P.
O
O
U

c

•H

CD

C

CO X

u

to

i— I
O

I

ri-

O

00

•

•

•

to

CT>

iH

VO

ro

CM

tO

71

e

o

•H

P

CO

fH

CD

P

•H

CD

>

•H

to

•

V)

CO

CD

c

O

o

U

■H

3

P

to

•H

to

»

o

fi

Ph

o

•H

<D

P

i—l

3

60

rH

fl

O

c«

to

J3

to

P

CD

3

P.

£

CO

•H

3

N

o* co

to

O

p

P

to

CO T3

CD

CD

I—l

•H

rH

P.

P-

O

Q*

tk

co

l

CD
CO

o

•H

P •
CO O

CD

CM

to

93

Q 1 .

if >— **V

7 \

ll \

'/ \

' / 1

// ^2

' / y^ "**"' "^w

. / // ^v

/ / // \

' / ' /

, / ' /

1 / /

. / ' /

' / ' /

' / / /

/ / ' /

/ ' /

1 \ ' / JEl'z

fefcry >^ \

o T~~/- — >^ \

/ /Ov^ ^— --,~~=*

i 4

\/^j£z-~ — " "

p

Figure III-3: Sextant Angle Positions

Plane coordinates are again used because of the
visual range limitation.

It can be seen from Figure I II -3 that

V " V = °i «

V ■ V * a3 •»

94

or in terms of the coordinates,

ton"1 «*!«£ _ too"' *3£. = <* , ,

ton"' *3~9p - WV,^-^ «<**,

totT1 %^p - "tan"1 ^-yp = <*3

Letting x = x + Ax and y = y + Ay and linearizing with
Taylor series expansion,

where

92.-^0 9*-^a

>20

si

af.

"Dy©

j^i"" *© % a— >^

c1-

95

The observation equations may then be expressed as

Vt

va

^3

-e

y^-iio y»-ao

Si^

*t*

*z- Xp

y-i-vjo vh--a« **-*o *3-^©

*\o

**•

.)

£•

^-9o _

J " — :

-.©

5

3©

>3©

'30

"^-^

»4e

A

x

A^

where

£ * 57.2958 is the conversion factor from radians to

degrees,
A .., A A , A are the computed azimuths of lines
01, 02, 03, 04 using trial point coordinates x , y .
Once establishing the observation equations, the
solution is found as

T -IT
X = CA A) iAiL .

The process is repeated until Ax and Ay become very
small.

Example III -6: Referring to Figure III-4, the
coordinates of the shore station are

96

(#4) .

MB 4

Use
(#2}

Figure III -4

Determination of a position for sextant angle
fixes using least squares adjustment.

97

MB4 (#1)

Use (#2)

Mussel (#3)

Luces (#4)

x 4,053,917.2 4,051,216.9 4,053,453.2 4,055,042.7
y 603,425.2 600,372.0 597,967.8 595,794.5

Measured angles are
MB4 - p - Use = 49?927,
Use-p - Mussel = 38?130,
Mussel - p - Luces = 30.396.

The least square method will be used to determine
the best values for the coordinates of the survey vessel.
Let the first assumed position be x = 4,057,000 and y^ = 599,000
Using the first approximate position,

AOi = tan

1 Yl

AQ1 = 124?862, AQ2 = 165?712, A03 = 196?235 and AQ4 = 238?591
are obtained.

The observation equation is

^3

0O&366 -Oo383<t
00 6661 . 00 Wf
0O86S3 ..00 694-1

A*

b>

y

5\ 0T*

T. to*

•ir.160

J -5

T T
and normal equation, A AX - A L = 0, is

98

■* OOOlfeOG ,0000025
.000002.5 . C000S1T

-.00 301-
* 151-14

= 0

The solution, X = (A A)"1 A L, is

x«

6229.4 -lilt

-.0030?

-ifKO
AT3S.1

Then, the new trial point coordinates are

x = x0 + Ax = 4,057,000 + (-47) = 4,056,953,

y = y0 + Ay = 599,000 + 1793.1 = 600,793.1.

Using new trial point coordinates, the above steps are
repeated until Ax and Ay values become vanishingly small.
For every trial Ax and Ay value, new trial points coordinates
are tabulated in Table III-2.

The best values for the coordinates of the sounding
vessel are

x = 4,056,512.3,

y^ = 600,864.5 .

99

f-l

00

LO

LO

LO

•

•

•

•

•

o

to

CM

vO

LO

•*t

o

(J>

t-l

oo

vO

vO

o

t^

a\

00

CO

00

en

o

o

o

o

o

X

en

o

o

o

o

o

LO

\o

vO

vO

vO

vO

en

0)

■M

CO

e

•H

*a

00

to

*H

•

•

o

o

to

CM

00

i-H

eg

o

o

in

r-

LO

rH

r-t

u

o

Cn

rt

**

LO

LO

r^-

vO

vO

vO

vO

vO

LO

lo

LO

LO

LO

LO

X

o

o

O

©

o

o

to

■P

c

•H

-

o
o
u

(3

•H

O

c
cj

to

to

en

en

vO

iH

CM
i

CM
1

X

I

00

I

o

i

00

to

LO

LO

V)

C

O

•H

■P

CCJ

U

O

•M

•H

<D

>

•H

7)

V)

7J

o

e

o

o

u

•H

3

+->

V)

•H

V)

#\

O

C

a.

o

•H

CD

+■>

i— 1

3

00

tH

fl

o

cfl

7)

4->

V)

(3

<D

nS

U

-M

c3

X

3

<u

a* c/>

w

O

4->

4J

(0

a

T3

o

CD

t—i

•H

i— l

?H

ft

O

ft

tL.

cU

■ •

CM

1

CD

iH
-O

cd

E-

O

CCj O

u -z.

cm

to

LO

100

3. Range -Range Positions

In range-range positioning, when the distances are
short (i.e., line of sight type equipment, less than 20
nautical miles) , a plane coordinate system may be used. Let
x.. , x x_ = Eastings of the shore stations #1, 2, 3

y-, , y7» y? = Northings of the shore stations #1, 2, 3

x = Easting of the survey vessel

y = Northing of the survey vessel

Then, the distance between the ith shore station and
the survey vessel, in plane coordinates, is

This function is non-linear and has to be linearized.
Introducing approximate coordinates (x , y ) for x and y ,
then

x^ = x_ = Ax

y/> = y* + &y

'O o

Using Taylor series for linearization, the result becomes

101

After linearization, the observation equation is

V3

*©— Xi

>io

£.2.0

^3o

*20

*30

A*

^

S,p-

Sio

£>ip .

-S10

S3p-

-S30

-* 5

or in the matrix form, V = AX - L,

where :

S,^, S0^, S__ are the measured distances,
IP 2p* 3J>

S,0, S20, S ft are the computed distances using x and y ,
In ranging systems, when the distances are long,
coordinate computations must be carried out on the appro-
priate ellipsoid using rigorous geodetic formulas. Let

h>\

<K> Xc

o

Oi

Oi

- Computed geographical coordinates, latitude

and longitude, of the survey vessel,
= Latitude and longitude of the ith shore station,
= Azimuth from ith shore station to approximated

position Oj
= Azimuth from O to ith shore station ,
= Distance between O and ith shore station.

102

Although the computed observations must utilize rigorous
geodetic solutions, the differential equations of the observa-
tions may be approximated using spherical trigonometry [Ref . 3] :

ds0L = Sin 1" C-£oC0sAolHo- RtCos Md<1<J>l

where d<J> and ^ are in seconds of arc, dS in meters. Sin 1"

is the 'conversion factor from seconds to radians. RA and R.

O* l

are the radius of curvature in the plane of meridian at
point O and ith shore station, respectively, defined as
[Ref. 18]

where a is the semi major axis of the datum ellipsoid,

2
and e is the eccentricity of the datum ellipsoid.

N^ and N. are the radius of curvature in the plane of prime

vertical at point O and at ith shore station, respectively,

defined as [Ref. 18]

(J0 is the latitude of point O and 0£ .is the latitude of the ith shore
station.

103

The partial derivatives of computed observations
with respect to parameters are

~5i*L= _ Sinl"R0Cos Aol,

"&S

"fcXi

5l = Sml"MtCos9tS»a/\io.

Then, observation equation is

1

V,

L^J

* Slnl'

_ £0C©s Aot , Hi Cos$| SirtAi©
_ £o Cos /W > Hi G>s<^2 SJo-At*
_£© Cos Ao3 » ^3 Cos(t>3 Svn A30

A (J)

AX

Sp,

-So*

Spa-

-S<tf-

5p3

^Soi

J 5

where Sfi, , S-2 , Sft_ are computed distances using inverse

distance and azimuth formulas (these formulas could be found

in any geodesy text); and S , , S - , S , are measured distances

> pi P^ P>j

between point p and the respective shore station.

After forming the observation equation, the normal
equation is found and solved as in previous examples. This
process is repeated until Acj> and AX become smaller than the
resolution of the positioning system.

104

Example 1 1 1 - 7 : Referring to Figure III-5, the coordi
nates of the shore stations are

Luces (#1)
x 4,055,042.7
y 595,794.5

Mussel (#2)

4,053,453.2

597,967.8

MB4 (#3)
4,053,917.2
603,425.2

Using the least squares procedure, best values of
coordinates of the vessel may be found. Let the first
assumed position xq = 4,056,000 m and y = 598,000 m.
Measured distances are p - LUCES = 4350 m, p - MUSSEL = 4506 m,
and p - MB4 = 5267 m. For the first approximate position of
the vessel, the observation equation is written:

v,

.999

, Ollis

A*

IS03oO

Va.

=

.333

-sn-

-

2lol,>

^1

.362

-.533

h

-5HM.3

T T
Normal equation, A AX - A L = 0, is

1.28T- -<Wi
.043 l.^ft

A*
A3

J

1S3>^ .8

rtn.f

= o.

105

Figure II 1-5

Determination of a position for range-range
systems using least squares adjustment.

106

T -1 T
And solution, X « (A A) A L, is

X-

A*

--C19

_„019
.584

283 V- 8

<2\83 ,1

Then, new trial point coordinates are:

x = x0 + Ax = 4,056,000 + 2183.2 = 4,058,183.2

y = y + Ay = 598,000 + 787.4 = 598,787.4

Using the new trial point coordinates, the above steps are
repeated until Ax and Ay values become smaller than the
system resolution. For every trial, the change in coordi-
nates and the coordinates of new trial points are tabulated
in Table III-3.

The best values for the coordinates of the sounding
vessel are

*« = 4,057,501.2
P

y = 599,567.7

The standard errors in the northing and easting may
also be calculated :

\i — K^ -L ( ^-or A anci L 1 Lost \Ve-ratCca viaLues are usee* ) y

1

\l =

L

-32>0 „3b"r
• 5Hb .831-

-.8

-3.1
.5

2.4
-1.5

107

<y>

o

r^

r-

t^

t^

o

co

■<*

vO

NO

o

r-

vO

LO

LD

>s

CO

00

o>

o>

o>

en

o>

Q>

os

OS

t-n

10

lO

LO

LO

CO

o

■M

co

c

•H

n3

*

O

o

u

CM

•

vO

eg

•

o

to

LO

eg •

rH

o

CO

CO

o

O

o

rH

LO

LO

LO

*

A

a

•k

»>

SO

CO

t^»

r^

t-~

X

LO

10

LO

to

10

o

o

o

o

o

CO

^-

LO

OS

<0

•

•

•

-P

X

1 l*»

O

o

cO

< 1

1 CO

vO

CO

C

r-

CO

1

•H

T3

fi

O

O

u

c

•H

<u

00

c

CO

x:

CM

vO

vO

u

•

•

•

to

r-»

to

CO

o>

co

X 1

1 iH

LO

i

< 1

1 CM

1

CO

tO

c

o

•H

■P

co

u

cd

•m

•H

<D

>

•H

W

CO

0

•

U

to

O

C

3

o

in

• H

•M

Ml

■H

fi

to

O

o

•H

a,

•M

3

CD

i— (

oo

O

a

V)

co

rH

CO

1

CD

CD

r-<

00

CO

fi

3

CO

cr fi

to

o

■M

■M

CO

CO T3

CD

CD

rH

•r-t

i— 1

?-i

ft

O

ft

PL,

CO

• «

to

l

o

•H

+-> •

CO O

CD

rg

to

CD
rH
Xl

CO
E-

108

\P\/ =

2.1- 2.6 -i-5

"i.r

2.-6

-1.5

- Ifc.^i

(To =

n-m

3>-l

VI - cr0 Nfq[T

T -1
where q-'s are the elements of (A A) which has been

calculated as

wr1-

^ —14

— IH

.ft

j 3

4. Hyperbolic Positioning Systems

Hyperbolic positioning systems measure the difference
in distance from a vessel to the two shore stations. In
Figure III-6, station number 2 is the master station, and

109

Figure III-6:

Determination of a position for hyperbolic
systems using least squares adjustment
method.

110

numbers 1, 3 and 4 are slaves. Point p is the vessel's
position, and its coordinates are designated as <j>p and Xp.
Point O is the first approximate position, with <j>^ and XQ
representing its coordinates.

The differential equations of the computed distances
to each station, Sft., may be written as [Ref. 3]:

ds<>C « SCol" C-Ro CosAoLo0o- RlCos Aloi^C

r NC Cos^tS»aA(o UX0- AXi^l

where i represents the shore station number.

A. represents the azimuth from the ith shore station
10 r

to approximate position 0.

R is the radius of curvature in the plane of
meridian at point O (as defined in Section B.3.).

N. is the radius of curvature in the plane prime
vertical at the ith shore station (as defined in Section B.3.)

The partial derivatives of the function with respect

to d>^ and X^ are
o o

^Soi »_ SC(LlMCosAoC
"2)00

Oc ,

IS^L = SwU"Cos0c.$ih.ALo.
"oXo

ill

The range difference between the distance from the
vessel to the master and the distance from the vessel to
the respective slave station is expressed in the equations
below.

^£°i _ *S°L - SCo. 1" Ro t Cos Aoc - Cos fc0O
^00 ^<f>o

Note that station number 2 is the master station, and
the range difference is stated in terms of the partial
derivatives .

With this information, the observation equation is
written as :

V,

= Siol"

&> (CosAot-CosAoO, Hv Cos$, l$;cLk2o-Si*aA\o}

1

M

( Spj.- Sp\^ — (So^- S0i)
( Spz - Sp^V ( S01.- S03)
( Spa -Sp^-(So2.-So^

J 1

112

where ;

Aft, , Aft2,... are computed azimuths from approximate
position 0 to shore station 1, 2,...

A, ft, A.ft, . . . are computed azimuths from shore station
number 1, 2,... to approximate position 0.

(S - - Sjyj) » (SD2 - sdt) , ••• ar© measured range
differences.

*-S02 " S01^ • (-S02 ' S03') » "• are comPuted range
differences .

Sin 1" is conversion factor from second to radian.

After writing the observation equation, the normal
equation is solved and the best estimate of the coordinate
values is found as previously discussed.

The process is iterated until A<f> and AA become
smaller than the standard error of the specific hyperbolic
system be'ing used.

5. Global Positioning System (GPS)

Global Positioning System fixes are obtained
utilizing the computed distances from the position of GSP
satellites to a GPS receiver. The receiver measures the
arrival of a timing pulse from every satellite within
acquisition range. The transmit time of each pulse is
encoded in the received signal. Thus, distance is computed
using the one way travel time between each satellite and the
receiver multiplied by the propagation velocity of electro-
magnetic energy. Three such satellite to receiver ranges

113

may then be applied to solve for the coordinates of the
receiver.

Using three satellites to determine a fix results in
a unique solution for the position coordinates (x, y, z) .
However, significant error may be induced due to drift in
the receiver clock. This additional unknown, receiver clock
bias CE) , may be resolved by processing four satellite
ranges.

For position fixing at sea, it is likely that the
z coordinate may be input as a known value based on a given
antenna height above sea level. Thus, the number of unknowns
will be reduced to three. By using four or more satellites,
redundant observations are then available so that the data
can be adjusted by the method of least squares.

Introducing the following variables, observation
equations may be written in a straightforward manner:

R- , R2, R, , . . . = Measured distances from receiver
to satellites S,, S S_, ...

(x, , y-, z, ) , (x2, y2, z2), ... = Known positions of
satellites S-,, S2, S.,...

x, y, z = Unknown position of the observer, p.

E = Receiver clock bias (unknown) .

Then, the basic equations are

- E -+ vtv-*iVL + lij-v^%- t^-ti^

114

Rx =* E + Vu-^f + (u-yzf + ( i-^T

^3> •= E + \/U-^f Uy-^h) H"*-"*^

x«

1

'2-

Rn = E Wu-OVy-OVu-T.-A

Here, the ranges R, , R~ , R.,, ... , R include the actual

satellite to receiver distance plus some offset due to

receiver clock error. In the above equations, the

satellite positions are known, and the four unknowns are the

user position (x, y, z) and user clock error.

Since the observation equations are non-linear, the

Taylor series must be applied to form equations suitable for

use with the method of least squares. Let

x = X- + Ax z = z^ + Az

o o

y = v~ + Ay E = E^ + AE .

Using Taylor series,

where R. is the distance between a satellite and the user
position, p.

115

R. is the distance between a satellite and approxi
mated user position.

Observation equations are written in the following
detailed matrix form:

Y*

Vn

*o-*t ^o-^i ^-Im 1

*w

R

VO

^ Rao

AC

£-2o

I
I

t
I

>*»-*n 9o~Hn "2o-"Z:o \

1 i — — i . •*

i j

-r\o

^oo

A*

Rip-&o

tft

Ae

Ra.p-£

2.0

£np-Ro<

J >

where

Rlp' R2p
R10' R20

are the measured distances,

are computed ranges from the formula

Rio - E© +\/(^o-^u^+(yo-^L^-t-("to-^c^:L ,

From this matrix, the normal equation may be formed
and solved as previously discussed. The process is repeated
until the values of Ax, Ay, Az and AE approach zero.

116

Example III -8 : At 0800 Zulu, May 1, 1980, a satellite
fix was taken using a GPS receiver aboard USNS ACANIA in
Monterey Bay. The measured distances between the satellites
and the receiver were

Spl = 20,640,380.8 m

S 2 = 20,357,184.1 m

S 3 ■ 23,287,346.8 m

S . = 21,699,908.4 m

Sp5 = 25,416,133.6 m

The satellite coordinates were

S #1 S #2 S #3

xl = 6,097,294.4 x2 = 1,819,274.3 x3 = 9,268,094.7

y1 = -4,364,543.9 y2 = -2,240,846.4 y3 = 13,290,138.0

zl = 22,658,876.2 z£ = 23,721,192.7 z3 = 13,622,934.4

S #4 S #5

x4 = -8,198,461.9 x5 = -21,419,309.7

y4 = -18,813,603.1 y5 = 12,865,351.8

z4 = 19,040,626.8 zg = 4,832,143.1

Applying the method of least squares to determine the
user position, first assume:

117

xo =

-2,640,000

^o =

-4,235,000

zo =

3,960,000

B« =

10,000.

For the first iteration, the observation equation
is written as

V,

Vx

Vi

—

Vn

Vs

.

-, 2198 -09*9

-. M231 .00 611 -.90 55

-.3lo3
-.7522 -.4W

- . MM
_ . 0343

s -.5111

. 2«5to , 6>\5
,73&56 -.41-26

1

L

A*

L

&*

—

L

fcfc

1

AE

.

-35 09.4

- 8810^

- VOVOfc.S

_33hx-o

_"B<t3.8

- 4

T T

and the normal equation, A AX - A L = 0 ,

1.Q39 .0^83 . bOM3

.0183 J.4T8T- -.0421

. feo43 -.o4i\

'!• I HI

-.1584 -.B4H9 -3-019

-.15 Bit
_,8^*t9

-3.109
S.O

Ail

3^9b•*9

}o75.3
28091,7

-451V 8-1

= 0.

118

The solution, X = (A1/*.)"1 ATL, is

X-

30^0,9
-21-05.2

New trial point coordinates are

xO = "2,636,959.1

y0 = -4,236,646.6

z = 3,957,294.8
o

-840.9

Using new trial point coordinates, the above steps
are repeated until Ax, Ay, Az and AE values become vanish-
ingly small. For other trials (iterations), the Ax, Ay, Az
and AE values and new trial points coordinates are tabulated
in Table III-4.

The final user coordinates are

x = -2,636,937.1

y = -4,236,666.2

z = 3,957,250.8 .

And receiver clock bias is E = -869.8 m.

119

en

to

lo

CM

00

w

Cn

o

t^

o

oo

Cn

o

•*t

o

vO

r*-

vO

vO

o

oo

rH

oo

00

oo

oo

o

i

i—(

i

1

1

1

o

1

oo

oo

oo

M

•

•

•

CM

•

•

o

^t

en

to

vO

cm

o

o

cn

en

LO

vO

lo

in

o

CM

en

CM

cm

cm

CM

o

t^

vO

t^

r-»

r*~

r--

vO

lo

LO

to

to

LO

in

en

cn

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en

en

en

to

to

to

to

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cm

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1

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1

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3
tn

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•H
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3

rH •
O V)

X
(A »H

<d m

rH

n3 CO
3Cu

era
<a

o
+j +-*

M Cm
O &.

PL, Oj

I

rH
rQ

erj
E-

cn

en

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CM

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to

LO

vO

120

C. USE OF THE ERROR ELLIPSE IN ANALYZING THE ACCURACY
OF HYDROGRAPHIC POSITIONS

In the least square adjustment process, the positional

errors are found in the direction of the x and y (<j> and X)

coordinate axes. These a and a values indicate the

x y

expected displacement of the fix in the direction of the
coordinate axes, but they do not necessarily define the maxi-
mum and minimum errors associated with the axes of error
ellipse (Figure III -7] .

Maximum and minimum standard errors are found by defining
the orientation of the error ellipse in terms of the x,y
coordinate system. Let the coordinate system defining the
semi-major and semi-minor axes of the error ellipse be u
and v as indicated in Figure III-8.

The following relationship exists between the ellipse
(u and v) and the ground (x and y) coordinate system.

01 = * Sin 0 fj Cos e

(111-18)

V » % Cos9 -y SinS

In these transformation equations, the angle 0 is the
rotational angle between the y and u axes (measured clockwise
from y axis to the u axis).

The lengths of the semi-major and semi-minor axes are
given by :

121

-.-„- 50 % probabiUKj error elUps*
_____ 3 0/o pro^ooiUty error *U.ip*c

Figure III-7

Error ellipses formed at the determined
positions CD. B. Thomson and D. E. Wells,
1977).

122

Figure III-8: Error Ellipse

Cu « cr0 vJ quu
0*v ' « (To ^fq^7 .

(111-19)

In above equations, a , the standard error of unit weight,
is known from the least square adjustment of point p, q and

Vv are siven °yl ♦

^or detailed derivation of these equations see Ref. 16,
pages 181-183.

123

9uu =^lq^+qMS) + A-(qw-<?»w)Cos2-& + q%3^n2© (Hl-20)

Sw^^l^x^SysW^l^-R^^CosaG-^SmlG, (111-21)

T -1
where q , q and q are the elements of the (A A)
xx yy xy

T -1
(for unweighted observations) or (A WA) (for weighted

observations), i.e.,

(ATAfl or tA^KV' =

L

Equation 1 1 1 - 20 reaches its extreme value, and q is
maximum, when

uie =_-2b

*y

(111-22)

Inserting Equation III- 22 into III- 20 and III -21 , and
defining D as .

D =. C ( qw -9^ -+■ 4 Ccf^f 1 ,

Va

(111-23)

124

q and q may be written as

S*» = ^(r** + 9^ -+D^

(111-24)

<U ^ (q*> +qyy -t> };

(111-25)

and from Equation I I I - 19 , the semi-major and semi-minor axes are

ef-lo-oMq^+q^ +*)

(111-26)

k1™ o-oUq^ + q^-DV

(111-27)

Using these expressions, the error ellipse can be con-
structed at any point whose coordinates were determined by

T -1 T -1
least square adjustment if the (A A) or (A WA) matrixes

are known .

Example 1 1 1 - 9 : In order to determine the error ellipse

parameters for the range-range example problem (example III-7),

T -1
recall that (A A) was determined as

(ATM^

.64 -.14

-.14 .ft

125

and the standard error of unit weight was cr = 4.04.

The semi-major and semi-minor axes of the error ellipse,
according to Equations 111-26 and 111-27, are found by first
solving Equation I II -23 :

a -, vAl

Then,

t -r Xh.

b ^Z (.Li* -.T6) + 4 t-.i^ I ^.

305.

0"* = T °"o*(<to+qj!» + D)

Qa --L (4.0*rta(.6*t+.76-K305) sB-9

a - 3.?3> m

ba *i-(riL(9w + q^-b^

2

b = 2.89 m.

The semi -major axis is 3.73 m, and semi -minor axis is 2.99 m
According to Equation III -22, the angle 9 is found:

126

ton 16 = Iaaa = - 3J=liit)

Why (*6<f-.7fc)

29 = 113?2
9 = 56?6 which defines the orientation of the error
ellipse.

1 . Some Characteristics of the Error Ellipse

A number of important properties of the error ellipse
can be obtained by analyzing the equations given in the previous
section. The existence of the error ellipse points out an
important fact that the accuracy of the location of a point
in question is not the same in every direction. An analysis
of equations 111-26 and I II - 27 demonstrates that the formulas
for the semi-major and semi-minor axes are composed of two

parts: the standard error of unit weight, which defines the

T -1
scale of the error ellipse, and the elements of the (A A) or

T -1
(A WA) matrix, which define its shape.

To reduce the error ellipse into a circle, where the
accuracy of position is equal in every direction, the follow-
ing condition must be met;

a/b = 1 .
According to Equations 1 1 1 - 26 and III - 27 this is possible
only if D = 0 :

D » L C q„ - q >f + <*( W ] =°.

127

which means that

q = q and q „ = 0 ,
Mxx yy nxy '

and, according to Equation 111-17,

a = a
x y •

Another important characteristic is that the sum of
the squares of the standard errors in x and y directions is
invariant to the rotation of the coordinate system, or

Q1 + bx - cri + of - <?? + Co2" .

(111-28)

Equation 1 1 1 - 28 leads to the concept of root mean

square error, d „ , as follows :

^ ' rms

drmci = \I<J? + O? =\l&£ + O- f = n/o^aF (111-29)

or, from Equations 111-26 and 111-27.

^rms - °*o V^** +-q^

128

IV. CONCLUSION

Conventional survey systems will provide the primary means
of hydrographic positioning for several years to come.
Thus, the concepts of d and the graphical approach to
developing error contours are very useful tools in survey
planning and execution.

Survey planners must exercise care in establishing the
position of navigation aids. The resultant net geometry

determines the accuracy, and thus the d „ error, of the fix

' rms '

positions. Accuracy requirements for the collection of
hydrographic survey data greatly limit the size of the
effective survey area. Through careful planning, the
number of navigation aid shore stations can be minimized
while still meeting position accuracy requirements for the
survey.

Currently, more research is needed to determine the
environmental factors which govern variations in the propa-
gation velocity of electromagnetic energy. If this important
parameter could be more accurately modeled throughout the
survey area, the effects of systematic errors due to these
velocity variations could be greatly minimized.

As shown in this paper, methods exist today by which the
accuracy of survey positions can be greatly improved through
the use of redundant observations and data adjustment techniques

129

The method of least squares adjustment provides a best esti-
mate of position, plus the size and orientation of the error
ellipse associated with- that position, for every point deter-
mined by the survey system. In addition, the error ellipse
quantifies the predictable accuracy (as shown in Figure II-7)
of each position as compared to the repeatable accuracy
available from conventional survey methods.

The application of these techniques will become more
widespread when the Global Positioning System is fully
operational. Observations of position from any number of
navigation and positioning systems (GPS, LORAN, hydro position-
ing systems, etc.) can be combined in a least square solution.
Observation equations may be written and weights can be
assigned as a function of accuracy for each system.

In preparation for these future improvements, hydrographers
must work to understand and implement the concepts discussed
in this paper. Data must be processed by computer. There-
fore, programs need to be written which can perform the
iterative least squares adjustment on the appropriate obser-
vation equations. Errors must be analyzed to assess the
improvement in accuracy resulting from redundant observations.
Additionally, standard hydrographic survey procedures need
to be reviewed to determine if more efficient methods may
be adopted when redundant observations and data adjustment
techniques are used. For example, the method of least
squares may be programmed into onboard computers so that

130

data may be adjusted and evaluated in real time. Alternately,
redundant data may be collected and recorded for later pro-
cessing ashore.

The technology is available today to employ data adjust-
ment methods in hydrography. This technology must be
analyzed and adapted to match the systems and requirements
unique to hydrographic survey.

131

ANALYSIS OF RANDOM ERRORS
APPENDIX A

(The information given in this section was taken directly
from References 1 and 20.)

ONE DIMENSIONAL ERRORS

An error in a measurement is the difference between the
"true" value of a quantity and the measured or derived value.
The "true" value can never really be determined because of
instrument limitations and human fallibility. In determining
the value of a quantity, only one measurement may be necessary
when an approximate value is sufficient. If, on the other
hand, the quantity is important enough to require a more
precise value, repeated measurements are made. Variations
will exist between the values obtained from several measure-
ments. Applying the theory of the normal distribution to
these measurements, the "best" value for the quantity is the
mean or average of all the observed values. The differences
between the mean and the observed values are the apparent
errors or residuals which are used to derive a statement of
precision for the measuring process. When the residuals are
randomly distributed about the mean, the precision of the
measurement is expressed by a single term, the standard
error, which is commonly designated by the Greek letter "sigma"
(a). For a one dimensional normal distribution, this value
is computed by squaring all the residual errors (v) , adding
the squared values, dividing by the number of errors less

132

one (if n independent direct measurements are taken of the
same quantity, then the first measurement establishes a
value for the unknown and all additional measurements,
(n-1) in number, are redundant), and taking the square root

cr~

where v . is the residual defined by the equation v- = x-

x

>

X .

1

observed value
x : mean value
n : the number of observations .
The normal distribution itself is represented by the
function:

The normal distribution curve and the meaning of the standard
errors are illustrated in Figure A-l. The central vertical
axis, p(v), represents the probability of zero error with
positive errors plotted to the right and negative errors
to the left. The height of the curve above a particular
point on the horizontal axis is proportional to the probability
of an error of that amount.

133

Figure Al : One dimensional Normal Distribution Curve

It can be observed from the normal distribution curve
that the total area under the curve is equal to unity. Also,
the area under the curve between any two values of v^
and v2 *s eq^l to the probability of an error occurring
between these limits. So, to find the probability of an
error between v, and v2, p(v) has to be integrated between
v, and Y-. The area under the curve between the limits of
v, = -a and v2 = +a is 68 .27% of the total area under the
curve. This means that there is 68.27% probability that
errors in any further measurements made under the same

134

conditions will not exceed the standard error, a, with a
68.27% probability. The standard error does not indicate
the probability that an error of a certain size will occur; it
only indicates that 68.27% of the errors will fall within
the specified limits of plus or minus one sigma.

If other probability levels are desired, the appropriate
conversion factor may be found in Table Al. For example,
for 95% probability, a should be multiplied by a linear
error conversion factor of 2.

Probability, I

50

68.27

90

95

99.7

Linear error
conversion factor

.6745
1.000
1.6449
2.000
3.000

Table Al : Linear error conversion factors for several
probability levels.

TWO-DIMENSIONAL ERRORS

A two-dimensional error is the error in a quantity defined

by two random variables. For example, consider the position

of a point referred to x and y axes. Each observation of the

x and y coordinates may contain the errors v and v .

x. y

135

If the errors are random and independent, each error has
a probability density distribution of

"The probability of two events occurring simultaneously
is equal to the product of their individual probabilities"
[Ref. 1]. Applying this rule, the two-dimensional probability
density function becomes:

PK,vO =_L_ e*W * >}

rearranging terms,

2 v/>a

p(vy)v^ 2 tt c+ c\, = e a ^ «? °a ' j

taking the logarithm,

(-a^lKCpKjVsi litff^^jL.^

For given values of p(v v ) [physical meaning of

x y

p (v , v ) is that the probability that two random variables
x y

v and v take values in the interval ±v and ±v 1 , the left
x y x yJ *

side of equation is a constant, k?, then

136

ka-

si

vi

For several values of p(v, v) , a family of equal probability
density ellipses are formed with axes kcx and koy (Figure A2) .

Figure A2 : Equal probability density ellipses.

In general, when the two errors are correlated, i.e.,
a change in the one error has some effect upon the other, the
probability density function, POx, vy) , becomes

137

g£g£

p (v*,v,) =

"J^loSof-tfir0*

2 0V*

_a 2,

1

c^

r

xu Vff,v_ff^

Then, the equation of constant probability density ellipses
(Figure A3) is

ie=_i.

d-ea)

(5? ^ OVffy

2.

where p = correlation coefficient of v and v and is given by

The probability density function integrated over a certain

region becomes the probability distribution function which

yields the probability that v and v will occur simultaneously

x y

within that region, or:

P lV*,Vy) -Jl p(V*,Vy^ cJv^ciVj,

138

Figure A3: Constant probability density ellipse for
correlated errors.

139

APPENDIX B
USEFUL GRAPHS FOR THE DETERMINATION OF REPEATABILITY CONTOURS

</i

en

S

u

o

V)

<0

3

ft
S3
>

100

50
40
30

'20

10

6
5
4

3

2

.6
.5

.4
.3

.2

.1

£>

.06

O

.05

(4-1
O

.04

in

o

.03

3
i—t

S3

>

.02

.01

10°
170°

20°
160°

30°
150°

40° 50°60o70o80o90°
140o130°120o110°100<>

Intersection angle, B, in degrees.
•Figure Al: For ranging systems, the graph of the <*rms/°"s

and e/b.

140

X>

C/T

s

o

0)

3
r-f

CO

>

>v

jT^

"

\

1

\

\ 1

V

"f

\

T

\

\

/

.1

cJrms
o.V>

-\

>

\.

1 e 1

\

W b /

1

»

\

1 /

.05

/ i /

.04

.03

.02
,01

10

20

30

40 50

100

Values of angle B in degrees

.5

.4

.3

.2

.1

200

.04
.03

.02

.01

Figure A2 : For azimuthal systems, the graph of <irms/a»b
and e/b.

J3

<D

(4-1
O

WJ

<0

3
i-t

>

141

LIST OF REFERENCES

1. ACIC Technical Report No. 96, Principles of Error "
Theory and Cartographic Applications, by Greenwalt , C . R .
and Shultz, M. E. , February 1962.

2. Bigelow, H. W. , Electronic Surveying: Accuracy of
Electronic Positioning Systems, A.S.C.E. Journal of the
Surveying and Mapping Division, Volume 89, No. SU3,

p. 37-76, October 1963.

3. Bomford, G. , Geodesy, Third Edition, The Clarendon Press,
Oxford, England, 1971.

4. Bowditch, N. , American Practical Navigator, p. 1204-
1237, Defense Mapping Agency Hydrograpnic Center, 1977.

5. Burt, W. A. and others, Mathematical Considerations
Pertaining to the Accuracy of Position Location anU
Navigation Systems, Part 1, Naval Warfare Research Center
Research Memorandum, NWRC-RM34, Stanford Research Institute,
Menlo Park, California, April 1966.

6. Hirvonen, R. A. Adjustment by Least Squares in Geodesy
and Photogrammetry, Frederic Ungar Publishing Co., 1971.

7. Ingham, A. E., Hydrography for the Surveyor and Engineer,
Halsted Press, John Wiley and Sons, 1974, pp. 117-123.

8. Launlo, S. H. , Electronic Surveying and Navigation,
pp. 85-101, Wiley, 1976.

9. Mikhail, E. M. , Observations and Least Squares, IEP, 1976.

10. Morris R. Heinzen, Hydrographic Surveys : Geodetic Control
Criteria, M.S. Thesis, Cornell University, December 1977.

11. Morris R. Heinzen, Hydrographic Surveys : Geodetic Error
Propagation, paper presented at American Congress on
Surveying and Mapping, 39th Annual Meeting, March 18-24,
1979.

12. P. A. Cross, A Review of the Proposed Global Positioning
System, The Hydrographic Journal, No. 4, pp. 15-17,
April 1979:

13. Schoenrank, R. U. , The Determination of Accuracy Lobes for
Electronic Positioning Systems, Lighthouse pp. 83-92,
March 1977.

142

14. Swanson, E. R. , Geometric Dilution of Precision,

Navigation: Journal of the Institute of Navigation, V. 25,
No. 4, pp. 425-429, Winter 1978-79.

15. Swanson, E. R. , Estimating the Accuracy of Navigation
Systems , Research Report 1188, San Diego, California,
U.S. Navy Electronics Laboratory, 24 October 1963.

16. Veress, S. A., Adjustment by Least Squares, American
Congress on Surveying and Mapping, Washington, D.C., 1974.

17. Wolf, P. R. , Elements of Photogrammetry , pp. 501-517k
McGraw-Hill, 1974.

18. Clair E. Ewing and Michael M. Mitchell, Introduction to
Geodesy, Elsevier, 1976.

19. D. B. Thomson and D. E. Wells, Hydrographic Surveying I ,
Lecture Note No. 45, Department ot Surveying Engineering,
Canada.

20. ACIC Technical Report No. 28, User's Guide to Understand-
ing Chart and Geodetic Accuracies, by Greenwalt, C^ R. ,
September 1971.

143

INITIAL DISTRIBUTION LIST

No. Copies

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Department of Meteorology

Naval Potgraduate School
Monterey, California 93940

5. LCDR Dudley Leath, Code 68 Lf 20
Department of Oceanography

Naval Postgraduate School
Monterey, California 93940

6. Professor T. Jayachandran, Code 53 Jy 1
Department of Mathematics

Naval Postgraduate School
Monterey, California 93940

7. LT Ali Kaplan 2
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Gonen/Balikesir/TURKEY

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TURKEY

144

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FPO

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145

22. Director

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23. Commander

Naval Oceanography Command

NSTL Station

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24. Commanding Officer

Naval Oceanographic Office

NSTL Station

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25. Director (Code PPH)
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26. Director (Code HO)

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Administration
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(NC-2)
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Administration
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30. Chief, Marine Surveys and Maps (C3)
National Oceanic and Atmospheric

Administration
Rockville, Maryland 20852

146

31. Director

Pacific Marine Center - NOAA
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32. Director

Atlantic Marine Center - NOAA
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33. Chief, Ocean Services Division
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Administration
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34. Commanding Officer
Oceanographic Unit One
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Fleet Post Office

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Oceanographic Unit Two
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36. Commanding Officer
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FPO San Francisco, California 96601

147

'

Thesis
K14244
c.l

190743

Kaplan . e

Error analysis of

on

hydrographic position- -
ing and the applica-
tion of least squares.

I nova*

es

Thesis 190/

K14244 Kaplan

c.l Error analysis of

hydrographic position-
ing and the applica-
tion of least squares.

thesK14244

Error analysis of hydrographic positioni

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