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SCHOOL
. CALIF. 93940

0 -^

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESIS

CALCULATION OF HYDROGRAPHIC POSITION

DATA BY LEAST SQUARES ADJUSTMENT

by

Francisco Castro e Silva

June 198 2

Thesis Advisor: Dudley Leath

Approved for public release; distribution unlimited

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REPORT DOCUMENTATION PAGE

READ INSTRUCTIONS
BEFORE COMPLETING FORM

1 «l»0»T NUMIIR

2. GOVT ACCESSION NO.

1 RECIPIENT'S CAT tLoc NUMBER

4. TITLE f«»« Submit)

Calculation of Hydrographic Position
Data by Least Squares Adjustment

S. TYPE OF REPORT A PERtOO COVERED

Master's Thesis
June 198 2

«. PERFORMING ORG ReRORT NUMBER "

7. *uThO»,'l|

Francisco Castro e Silva

S. CONTRACT OR GRANT NLMBERf.j "

• PERFORMING ORGANIZATION NAME ANO AOORESS

Naval Postgraduate School
Monterey, CA 93940

10. RROGRAM ELEMENT project task
AREA 4 WORK UNIT NUMBERS

II CONTROLLING OFFICE NAME ANO AOORESS

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12. REPORT DATE

June 1982

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Approved for public release; distribution unlimited

17 DISTRIBUTION STATEMENT (at tho mmmtrbd mtlormd In Slock 10, II dllUrortl from Roport)

IS SUPPLEMENTARY notes

IS. KEY WORDS Caniinum oft rmwrmo t,<io II n«ca«««IT •«« Imtnilty T Woe* numbor)

Least squares, Redundant observations, Fix by redundant
observations

20. ABSTRACT 'ConOnuo on rovmoo nam II noemmmmr 4R« Im—tHtr »r *lbch «•»•«)

When redundant observations are available, hydrographic position-
ing problems require the application of a data adjustment method
so that all information may be used for obtaining the most
reliable "fix". One of the oldest and best engineering tech-
niques developed for the purpose is based on the least squares
principle. The theoretical background is provided to explain
that principle and the technique for its application. Also, the

DO , jam "-1 ^473 COITION OF I NOV 4S IS OBSOLETE

■; N linl-dll- AAO I

UNCLASSIFIED

UNCLASSIFIED

t«tuwt» et«ni>ic»TioM 00 Twit xaifiw

analytical solutions, and respective computer programs implement-
ing them, are developed for the following hydrographic positioning
methods: a) fix by N azimuths, b) fix by N sextant angles,
c) fix by two range distances and one azimuth. For each method,
an illustrative application of the respective computer program is
presented.

The least squares adjustment method not only yields the most
likely values for the fix coordinates but also statistically
quantifies position accuracy. Relative accuracy achieved with
conventional survey methods is elevated to absolute accuracy ,
when redundant observations are made and adjusted using the
method of least squares.

DD ForTfl 1473

" an < 3 .

102-014-6601 ucuntv ciAMi'iCAfiOM d' r*i* **o«r**«« o«

"" J Jm .3 UNCLASSIFIED

Approved for public release; distribution unlimited

Calculation of Hydrographic Position Data by Least

Squares Adjustment

by

Francisco Castro e Silva
Lieutenant Commander, Portuguese Navy
Portuguese Naval Academy , 1967

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY (HYDROGRAPHY)

from the

NAVAL POSTGRADUATE SCHOOL
June 1982

MONTErtEir, CALIF. 93340

ABSTRACT

When redundant observations are available, hydrographic
positioning problems require the application of a data adjust-
ment method so that all information may be used for obtaining
the most reliable "fix". One of the oldest and best engineering
techniques developed for the purpose is based on the least
squares principle. The theoretical background is provided to
explain that principle and the technique for its application.
Also, the analytical solutions, and respective computer programs
implementing them, are developed for the following hydrographic
positioning methods:

a) fix by N azimuths

b) fix by N sextant angles

c) fix by two range distances and one azimuth

For each method, an illustrative application of the respective
computer program is presented.

The least squares adjustment method not only yields the
most likely values for the fix coordinates but also statistically
quantifies position accuracy. Relative accuracy achieved with
conventional survey methods is elevated to absolute accuracy
when redundant observations are made and adjusted using the
method of least squares.

TABLE OF CONTENTS

I. LEAST SQUARES ADJUSTMENT THEORY 10

A. INTRODUCTION 1]-

B. LEAST SQUARES PRINCIPLE FOR

UNWEIGHTED OBSERVATIONS 12

C. LEAST SQUARES PRINCIPLE FOR

WEIGHTED OBSERVATIONS 12

D. OBSERVATION EQUATIONS

E. LEAST SQUARES ADJUSTMENT METHOD 15

F. PRECISION OF OBSERVATIONS 18

G. PRECISION OF ADJUSTED VALUES 20

H. THE ERROR ELLIPSE 22

II. APPLICATION OF LEAST SQUARES ADJUSTMENT 24

A. FIX DETERMINATION BY AZIMUTHS 26

1. Solution for Azimuths from 3

Stations 26

a. Determination of Adjusted
Coordinates °

b. Precision of Observations

and Adjusted Values "

c. Error Ellipse 30

2. Numerical Example

a. Determination of Adjusted
Coordinates

b. Precision of Observations

and Adjusted Values

38

c. Error Ellipse

4 0

3. Solution for the General Case

a. Determination of Adjusted

41
Coordinates

b. Precision of Observations 46

B. FIX DETERMINATION BY SEXTANT ANGLES 48

1. Solution for 3 Sextant Angles

(between 4 Stations) 48

2. Numerical Example 51

3. Solution for the General Case 58

C. FIX DETERMINATION BY TWO RANGE

DISTANCES AND ONE AZIMUTH 72

1. Solution for Two Range Distances
and One Azimuth from 3 Different

Stations 72

2. Numerical Example 78

3. Solution for the General Case °^

III. RESULTS AND CONCLUSIONS 97

A. RESULTS 97

B. CONCLUSIONS "

APPENDIX A LEAST SQUARES PRINCIPLE AND

NORMAL DISTRIBUTION 102

APPENDIX B LEAST SQUARES PRINCIPLE FOR

WEIGHTED OBSERVATIONS 107

APPENDIX C NORMAL EQUATIONS IN ALGEBRAIC

NOTATION 109

APPENDIX D NORMAL EQUATIONS IN MATRIX

NOTATION 111

APPENDIX E A COMPUTATIONAL CHECK FOR THE

LEAST SQUARES ADJUSTMENT TECHNIQUE 113

APPENDIX F THE CONTROVERSIAL CRITERION FOR

ASSIGNING WEIGHTS 114

APPENDIX G PRECISION OF ADJUSTED VALUES 117

APPENDIX H ERROR ELLIPSE 12°

APPENDIX I ALGORITHMS 126

COMPUTER OUTPUTS 148

COMPUTER PROGRAMS 151

LIST OF REFERENCES 183

BIBLIOGRAPHY 184

INITIAL DISTRIBUTION LIST 185

LIST OF FIGURES

Figure

1-1 Error Ellipse 22

II-l Fix by 3 Azimuths 25

II-2 Error Ellipse 40

II-3 Fix by 3 Sextant Angles 47

II-4 Undetermined Fix by 2 Sextant Angles 60

II-5 Converting Angular Residual into Metrical Residual 74

II-6 Converting Angular Standard Deviation into Metrical

Standard Deviation 7 7

II-7 Fix by Two Range Distances and One Azimuth 79

II-8 Fix from 3 Stations 85

II-9 Fix from 2 Stations 86

11-10 Range Distances Not Intersecting 88

11-11 Undetermined Fix by 2 Range Distances and 1 Azimuth 91

A-l Normal Distribution 103

A-2 Fix by 3 Range Distances 104

A- 3 Residuals 104

A-4 Residuals and Normal Distribution 105

H-l Two-dimensional Normal Distribution 120

H-2 Error Ellipse 124

ACKNOWLEDGMENT

For those that producing more than consuming, allowed
me, during two years at least, to consume more than produce,
my best thanks.

Monterey, the 6th of September 1981

LCDR Francisco Castro e Silva
Portuguese Navy

I. LEAST SQUARES ADJUSTMENT THEORY

In hydrographic surveying, the determination of position
is as important as the measurement of depth. Conventional
survey methods rely primarily on two lines of position (LOP)
to establish a fix. These LOP ' s are obtained by measuring
angles and distances directly. Alternately, electronic
positioning systems are used to establish a pattern of LOP f s
(an electronic lattice) based on the propagation of electro-
magnetic energy.

Until recently it has been logistically unfeasable to
obtain redundant observations in hydrographic surveying.
However, with the advent of computers and miniaturized
electronic positioning systems, redundant observations are
now being made in order to increase fix accuracy and prevent
delays due to equipment malfunction.

Mathematical adjustment methods must be applied to the
redundant data sets in order to maximize the accuracy of
each fix. One such adjustment method is based on the
principle of least squares. It assumes that blunders and
systematic errors have been removed from the data so that
only random errors remain. This method yields not only the
best estimate of position for a given set of redundant LOP ' s ,
but also assesses the absolute accuracy associated with each
fix determination.

10

A. INTRODUCTION

In general, the redundant observations of any variables
in a physical system (such as in hydrographic position
determination) do not precisely satisfy the mathematical
model developed to represent that system. However, the
derivation of every mathematical model is based on the
assumption that the true values of the variables will satisfy
the model. The difference between the true and observed
value for any physical variable is called the residual;

residual = true value- observed value. (I-i)

In making physical measurements, true values can never
be determined. Considering the observed values as values
assumed by random variables following normal distributions,
every true value can be represented as the mean of a random
variable. Therefore, eq. (1-1) can be rewritten as

res idua I =• mea* of random van' a b(e- obs. value . (1-2)

The least squares principle establishes a criterion
for obtaining the best estimates of the true values. It
states that the true values will be such that the sum of
squared residuals is a minimum. For a further discussion of
this principle, see Appendix A.

11

B. LEAST SQUARES PRINCIPLE FOR UNWEIGHTED OBSERVATIONS
Measuring different parameters of a mathematical or
functional model, we associate with each parameter a random
variable, y^ . Designating by VV the value assumed by the
random variable (the observed value) and by u£ its mean
(the adjusted value) , the residual V£ is given by

vi »jh- ]f i. (i- a)

For Jl observed parameters, the least squares fundamental
condition is expressed as

U a V V'" f ^h " minimum
or, in matrix form

where V? > [ V, V% ... V^ ]

C. LEAST SQUARES PRINCIPLE FOR WEIGHTED OBSERVATIONS

If the n observations are unequally weighted, then the

least squares fundamental condition is expressed as
H

12

or, in matrix form

V WV = minimum ( J-5)

where Wis the f? * *? weight matrix. See Appendix B for a
more complete discussion on the concept of weighted
observations.

D. OBSERVATION EQUATIONS

In the expression for the residual, fr is a known value

(the observed value) and ji-i represents (from a deterministic
point of view) the true value, thus satisfying the relation-
ship between the variables as expressed in the functional
model. The model must define an analytical expression
relating the unknown values with the known ones. In general,

U^ may be expressed as a function of the unknowns;

where *i > **2 j • ♦ • > X»n are the unknowns. Therefore, eq. (1-3)
can be rewritten as

The above expression is referred to as an observation equation

13

If U is a linear function, the observation equation
may be written as

Si * Zlo * 3ii *i + ■ • • * *Lm *m ~JC ( l' 7)

where 3{o 7 <3ti ; • • • j ^Cm are coefficients. The least
squares method does not require that the observation
equations be expressed in linear form. However, the compu-
tations to determine the values of the unknowns are greatly
simplified if the observation equations are linearized.

If r^ is nonlinear, a Taylor's series expansion may be
applied to linearize the function. Since it is not
practical to work with all the terms of the expansion, only
the zero and first order terms are used. Thus, the linear-
ized form is an approximate analytical expression for £^ ;

Since the function Cj and its partial derivatives may be
evaluated given approximate "initial values" for the unknowns,
the observation equations can be expressed as linear functions
of the increments

j?£ • 3i0 + d£i .A*, + . . . + 2 l,n AX

m

14

where Bio - ft I

and

3* la dtfm

Therefore, the residual V^ will be stated as

V|> aio + au a^ +.. . + aL-^ ^xw- ^ , (1-8)

It must be emphasized that, using the approximate
expression for \£ , the least squares method will yield
adjustments ( ^^j »•-• ■> A^m ) which must then be applied
to the "initial approximations".

Therefore, an iterative solution is required to solve
for the final values of the unknowns. The first adjusted
results are used as the new initial values, and the obser-
vation equations must be formulated again. This process is
continued until the increments become vanishingly small or,
from a practical point of view, converge to within a
specified tolerance.

E. LEAST SQUARES ADJUSTMENT METHOD

Considering eq. (1-7), and combining the constant terms,
a new expression for the observation equation is obtained

15

VL- = ^iX, +...4 a^ym - Li [1-9}

where

ii. - yi - 3to .

Then, the n observation equations can be presented as the
following system of h equations with tn unknowns, where
/7>/n f or the case of redundant observations:

Yi ; en /i * .. . "** 3 i

m *m - L,

v»7 s 3 ni ><, -»• . . . +■ ah4, x^ - in

(I-70J

or, in matrix notation, as

V-AX - L

(Ml J

where

V-

v,

lV*j

A-

3« &\x ••« <3 1

3*, <3ni • »• 3

t1/7»

L>

I,
i.

16

Equations (1-10) and (1-11) express the general form of the
observation equations. By imposing the least squares prin-
ciple that V W V* minimum , a set of equations are obtained
which can be solved to find the best estimate of the unknown
values. These expressions are known as the normal equations.
For the observation equations as expressed in eq. (1-10) ,
they form a set of m equations and m unknowns :

• ^ (1-12)

[u>;a:*ac,]x,*... * [u>< aim*] Xm- [u*a;„, Ltjso

where the brackets have the usual meaning of sum ( L - lj Lj. . . n) t
In matrix notation, the normal equations are expressed as

(ATW A) X - /WL. (1-13)

The normal equations are used to solve for the values of .X ;

X=(ArWA)' (4TWL) (I- 14)

where J\ is the vector whose elements are the adjusted values
for the unknowns. For a more complete development of the
normal equations see Appendix C and Appendix D.

17

F. PRECISION OF OBSERVATIONS

When observing an unknown variable a finite number
of times f n , the value of <J* can be estimated by comput-
ing a sample standard deviation, 5 , according to the
following formula:

£r

- ^1

TZ ex,-*)

i.- 1

n-\

\

£ Vc:

Vt

u* i

rt- <

where V/ ( l - l,Z»«**«> W) represents the observed values and
X the average value, for a set of equally weighted observa-
tions .

Similarly, if m unknowns are (indirectly) observed
h times, the best estimator for (J is the sample standard
deviation, -S , represented by the expression iREF . lj

5:

U * I

n-m

J4

W* \

!4

assuming all observations are equally weighted. The value
r = n-m in the equation is known as the "degrees of
freedom" .

18

A set of ft observations with assigned weights

n
u>. ...j u>n is equivalent to a set of 5" *»UL observations

L-i

which are all equally weighted. Thus, an artificial set of
observed values is created in which UJj observations are
equal to the 1st actual observed value, u)^ observations
are equal to the 2nd real observation, and so on for the
remaining weighted observations.

Given an arbitrary set of weights, the set may be

scaled so that the smallest weight has a value of ONE. This

c2

scale factor is known as the variance of unit weight, o0 .

For a more complete discussion of this topic, see Appendix F
Therefore, in the case of weighted observations, the
best estimate for the value of the standard deviation of
unit weight is given by

7 Vl

s.

2 ">i*i

te l

H

(£>i)-»71

1= I

11-15]

where Yn is the number of unknowns.

In matrix notation, eq. (1-15) is written as

50, [Zk£

V h-m

(1-16)

19

where h , the number of unit weight observations, is given
by the trace of weight matrix

h= trace (, W) * £ ^ • t *- '7j

The standard deviation of the L observation (with
weight ^u) is given by

SiM

uj.

H

{1-13)

G. PRECISION OF ADJUSTED VALUES

The elements of vector /C ( x, . . . *„, ) given by

X- (ATWfi)" (ATU I)

represent the adjusted values of the unknowns. The matrix
( A W A J is known as the variance-covariance matrix Q, and
individual elements are identified by the term Q^j .

The standard deviation of each adjusted value X£ is

given by

Svi - Sa fw

(X- 19)

where J - i , so that the Pt'j terms are diagonal elements of
the matrix (/4 W H )

20

The covariance between adjusted values xt' and Vj is

given by

c-i - S

-V; v: — *-'<

'*L *j

1y

( 1-20;

For hydrographic position determination problems,
the adjusted coordinates x and y correspond respectively
to elements Xj and X^ of vector ^C (^1,^1).

Therefore, the standard deviation of adjusted coordi-
nates x and y is given as

I

(r-2i)

The covariance between adjusted coordinates x and y is
given by

S - s

1'*

(I-22 ;

where factors Q(< ^a^j and <3 - are elements of the
symmetric square matrix

Q-. (/fWA)" =

21

For a more complete discussion of precision of adjusted
values, see Appendix G.

H. THE ERROR ELLIPSE

Position errors are two dimensional and must be evaluated
in terms of the errors along the x and y axes. Since
the maximum and minimum errors do not necessarily occur along
these axes, the orientation of these maximum and minimum
errors must also be considered. Positioning errors may be
evaluated in terms of the error ellipse (See Fig 1-1) .

y

/^\

^T

X

Xs*

FIG 1-1

ERROR ELLIPSE

22

The greater errors occur along a line making an angle Vp
with the x-axis (measured anticlockwise) such that

CO

t 2.y0 --

li» " 1*2

(1-23)

2- T.z

The respective standard deviation is given by the semi-
major axis length of the error ellipse

Sa. 5,

2 ■?., gtj

CX-24J

where

The smaller errors occur along a line perpendicular to $3 ,
and the respective standard deviation is given by the semi-
minor axis length of the error ellipse

5b. 5

z <*«, 1

22

^ * ^2

D

%

C I- 25)

The derivation of these equations is presented in Appendix H

23

II. APPLICATION OF LEAST SQUARES ADJUSTMENT

The determination of a vessel's position at sea is a
typical hydrographic problem for which the least squares
adjustment is particularly well adapted. Various methods
can be used for fix determinations. In this thesis, the
following three methods will be presented:

a) fix by N azimuth angles

b) fix by N sextant angles

c) fix by two range distances and one azimuth angle.
For each method, the least squares adjustment is

applied in the following manner:

1st: solution of the problem for particular conditions
2nd: numerical example

3rd: solution of the problem for general conditions
4th: formulation of an algorithm for the general

conditions case
5th: implementation of the algorithm in Fortran language

24

( Mo**t h in 3 J

- A

P C *,j >

/

N

47.965[^

X (.Basttncj)

FIG II-l: FIX BY 3 AZIMUTHS

25

A. FIX DETERMINATION BY AZIMUTHS

1. Solution for Azimuths from 3 stations

a. Determination of Adjusted Coordinates

Given a positioning problem as diagramed in
FIG II-l, where:

fl^p — is the observed azimuth from station 1 to

vessel position i
Ajp " is the observed azimuth from station 2 to

vessel position IT
$3p - is the observed azimuth from station 3 to

vessel position Y
and

(*i#i ) - are the grid coordinates of station 1
tXi.^t) • are the 9rid coordinates of station 2
(*3,^3 ) - are the grid coordinates of station 3
the grid coordinates (xy) of vessel position P
will be determined.

Step 1) Formulation of observation equations

The analytical expression for the azimuth from station C
(Xt,2ft ) to P (*,*f) is given by

At i. Cra4ia»*>. \in' *~*£. _- P(x.y).

3 - 3 i

The function F(xy) must be expressed in a Taylor's
series around an "initial position", Vq , whose coordinates

26

are defined as xa and y^ . Evaluating the zero and first
order terms of the series, the following expression is
obtained:

Designating the distance and azimuth from station L ( X{, Yc )
to the "initial point" r0 i^o}Yo) by 5^0 and A? i0
respectively, then

Si0 = [(*<,- xl)1 + Ly0- yLi'1 \

and

Az,-

= fan

1

*o -

*L

' ' *" LO

Y° -

Yt

Therefore,

AtLS a

^■,o

+-

lo-

- Yt

ax . y* : Xd Ay

and the observation equations will be (for i=l,2,3)

V; = V°-y- 4X _ x° - Xi _ ( Aip-Aiio )
i^,)1 (5L0)2

27

where fl^p is the observed azimuth. In the matrix form
/\X""l-* * f the °bs« equations will be

yd-2»

(5,*)*

yo-y-i.

C$2o)Z

yo-y$

CS3*);

ax

. A-

to

v,

V.

V,

The angles must be expressed in radians.

Step 2) Normal equations

Forming the normal equations, the adjusted values for & X*
and Ay will be given by

X-- (A7W A)'' (fiTvJL )

where X •».

Step 3) With the values 4 X and £y a new "initial point"
?e> C*^;^*) will be obtained;

^0 r *o * & *

and the procedure may be repeated in an iterative way until
the increments £y and £N become vanishingly small. Then,
the most probable values for the coordinates x and y will
coincide with the coordinates of the last "initial point" obtained

28

b. Precision of Observations and Adjusted Values

Step 1) From observation equations A X'-'U = V , where JXT
has been obtained by the least squares method, the residuals
are obtained, i.e., the differences between the "true" and
observed values of the parameters. t

Then, the standard deviation of unit weight is given
by

O *

1 2 Z

to, ♦ uu»^ + UJ3 - m

Yx

where n*» is the number of unknowns observed. For that problem,
the unknowns observed (indirectly) are x and y (m=2) .

Therefore, in matrix notation, the above equation is
expressed as

O r

vTw v

Mi ~^ i^ MM ■MHMHP

trace ( W J - 1

where trace ( W ) = uj, -r uj^ * cu^ #

Step 2) The standard deviation of each observation (with
weight uJ^ ) is given by

L 2

5/

U»u

L Cs 1,1,3 }

29

Step 3) The standard deviations of adjusted values are given
by

$} = So >/lw .

The covariance is given by

<; ^ z

Step 4) The correlation coefficient P between x and y
is given by

p_ Ox^

12,

s* sy \|^"- *?«* •

c. Error Ellipse
Given the matrix

Q,carw4j

-:

/til *** .

and the standard deviation Sa of a unit weight observation,
the error ellipse parameters will be determined.

Step 1) Obtaining the value D,

X 1m \ /%

30

Step 2) Semi-major axis,

5

a s

1 %* q

•2.X

^••+ <Uz - D

a

Step 3) Semi-minor axis,

Sl. 5.

b:

-2 <?l. <=?

2Z

*=!«. * 1iz + P

!4

Step 4) Determining the angle Y^ (measured anticlockwise)
from x-axis to semi-major axis :

4.1) The angle Ya will satisfy

tan (2 Jo) s fan il .- 1 111

4.2) For computer applications, XL is defined to
fall within the following limits: - TT/2 ^ ^1 <" TT/2

Then,

a) if <^„- cj2a , choose y - ^ / 4

b) if ^,, ^qn and J7. ^, a , choose y- ^/i

c) if <iu # lu and SL. < o , choose 8% (_fL+i?Wz

4.3) The intersection of error ellipse,

with the straight line y — xtar.^ is given by x', and ^»;

31

such that

»l r

3?-

3 « <H* 5,

2 t 'i

4.4) Considering

Do* , [CSa t Sy) /2 ] ^

a) if D, > 00 , then ^ , #

b) if p* < p* , then ^ = 3" + ^ ' 2 •
2 . Numerical Example

a. Determination of Adjusted Coordinates

The U.T.M. grid coordinates of shore stations, in
FIG II-l, are:

COORDINATES

LUCES (#1)

MUSSEL (#2)

MB4 (#3)

x (EASTING)

y (northing;

595,794.5
4,055,042.7

597,967.3
4,053,453.2

603,425.2
4,053,917.2

For illustrative purposes, the standard errors
for azimuth observations made at each station were assigned
the following values: <T, * o.oz , <$\ - o°. a z 4 ^ 4%, , o° <2 i 3

32

The observed angles at each station were:

P - LUCES - MUSSEL = c*, =50. 164 ,

P - MUSSEL - LUCES - <*Z = 99. ° 360,

P - MB4 - MUSSEL = ^3 = 4 7." 8 65 .

Step 1) From the grid coordinates, the fpllowing azimuths
between stations are obtained:

Step 2) The observed azimuths will be
A\p = A, 2. - at , = 76* 017,

/tap i ^3*2. f- 0*3 s 3ld?oa5".

Step 3) Formulation of observation equations

3.1) The first "initial point" is determined by the
intersection of azimuth lines from stations #1 and #2 expressed
by the equations

Solving these equations simultaneously to find x and y ,
these values are used as the coordinates [Xo^o) of "initial

33

point", where

X

o z

600 .6 77.5

t y0 - ^056, 3 0 8. A .

3.2) Determining azimuths between stations and "initial
point" f0 (Xotfo) ,

A?l0 . tan"' [u0 * *<-> /(/• - y* )] = 76° 0 17,

A?z0 » U»[k*9 -x3)/ (Yo -Vs; ]="5I3° 195.

Then, evaluating the elements of the L matrix:

Ajg . /^2td x 0*060 - 0.0 rad^

^ae - ^^10 a- of. 00 1 ^ -0.000 017 5 ra<4,

^3f^^^3o s -O*. ISO =-0,003 141 6 f a«i.

3.3) Determining squared distances between stations
and "initial point" £q ,

t5{0) - (/„- x,)1^. (y0. y, ;2 - 27,438,895,
lS%o )2 r (*o- **)*♦ (%-YOx ^ 1^,^/8,511,

(S3o )* = (*o -^3)l+ (y*-^3 >* a \lf 103,613.

3.4) Then, evaluating the elements of the A matrix,
(y0-7.)/5lo -0.0000^61 -(*o-*)/5jd --0.0OO1952.

tYo-Y*)/5i0 s 0.000 171 9 -L*6-*2)/Su *-Aoaol7SJ

l*-1%)/5si 3o.fl«o 195^ -(*o-*3)/S30 * 0,000 203 7 ^

34

3.5) Therefore, in matrix form, the observation
equations are written as

0.000 0^6 J
0.000 8710
0.000 1^53

Step 4) Solution of normal equations

4.1) Determining the weight matrix ,

•a

O.OOOlQS 1

["ax]

0.000 1751
0.0001097

u

O.OOO O06

"v, ■

d, ooo on 5

X.

^

0,©O3 14 16

v3

d

I / 07 a 2Sdo
i / ff* * » 1736

0", 5 o. o I o — »

<T3 s 0. O I fc ► 1/cTgT s 3 0 0 6.

Considering the least weight equal to ONE, it will be obtained
that

u)i , I. od

O*"

W*

u)

3 c

1-7 8

1.4* o o

o \.0 0

o o J.78

4.2) The solution of normal equations is given by

X * iATV! A)'' (AT W L J;

4.2.1) obtaining matrix

AW- a ooo obbk

-Odoo Ibb 7

4.2.2) obtaining matrix

A7* A ,

0>000 000 \0
0.000 000 03

0. oooHlQ 0. 0003^67
~o,ooo 115 t 0.000111 5 |

0. 066 000 Co

0.000 000 10

35

T -I

4.2.3) obtaining matrix Q = (A >• A )

Q.

<;57o, 68i
5, 235, 601

4.2.4) obtaining matrix

A V L .-

0. 0 0O 0O\ \

OsCoo oo\ X

-* »

4.2.5) finally, vector X is evaluated by solving
the normal equations :

X,

-S.6
-4.6

1

Step 5) First adjusted values of x and y

With the increments Ax" and ^V a new "initial
point" is obtained:

[ Xo = 6oo, 877-5 - 3.6 = 600,367.3

Step 6) With the new values, for the "initial point", the
procedure indicated in steps 3.2, 3.3, 3.4, 3.5, 4.2 and 5
is repeated, and with the values now computed for &X and
A^j a "closer" initial point is obtained.

36

Step 7) This procedure must be repeated, in an iterative way,
until the increments A* and ^^/ become vanishingly small,
or, in practical terms, converging to within a specified
tolerance. Then, the last "initial point" obtained will
coincide with the most probable position for P(x,y).

b. Precision of Observations and Adjusted Values

Step 1) The residuals are obtained introducing £>X= -3.6
and Ay =. — 4.6 into the observation equations. Therefore,

V-

v,

V,

0.0 co A oe 4

-o.ooo 0 26 3

o.ooo 'boo 5

Step 2) Obtaining scalar y W V,

V W V - 0. ooo oo\ O&S .

Therefore, SQ = 0.000 6992 radians .

Step 3) Obtaining standard deviation of each observation,

5

u -

3,

<-o i

Then,

6, . o.ooo 58*2 7 TaA
S^ - o.ooo 6331 r^<£

^ <3 °0 3 3
- 0,040

37

Step 4) Standard deviation and covariance of adjusted values

x and y ,

5* r S0^77 = 2.26

5 y = S0^l =, \.6o
Sxy = Sj- «j,i a -0.768 .

Note, S and Sy are expressed in the same units
as the grid coordinates.

Step 5) Correlation coefficient ,

f=

*:>

Sx • S

- . 0.21

Given

c. Error Ellipse

So z o. ooo & 98 Z

^M r f^; ^7fj 2<3« .

the error ellipse parameters will be obtained
Step 1) Determining D,

D- . (In - <?2*H- ^ <? c* J s 6; <<^; 7og

Step 2) Semi-major axis ,

Sa=5,

2 «v 1l2

111 + Izz " D

^

2.36

38

Step 3) Semi-minor axis ,

Sfc. S,

1 <=!., ^U

H

57

Note, Jg and 5^ are expressed in the same units
as the grid coordinates.

Step 4) Determining the angle Y0 (measured anticlockwise)
between x-axis and semi-major axis 5 3
4.1) The solution of equation

tanU.- 11

\Z

°lu- ^Z1

IS

Si c -3oT5<54.

4.2) Since a ^^j. and JT < O , then

ya Jl +180% 7^° 52,
0 2

2 1

4.3) Obtaining X, and ^ ,

*,% 0.175

4.4) Obtaining P, and D,

*i * 2.28 ,

1!

ft t 1

Pi s X, 4 3, - 2. 4 6

Po ^ [^5a * Su;/2 3** 3.86 ,

4.5) Since J)1, < D2X / then

^,, ^^50°. I6 4..52 (See FIG II-2) .

39

FIG II-2: ERROR ELLIPSE

All of these computations may be compared with
those shown in the computer output section on page 148 .
Differences in the results are due to the fact that the cal-
culations illustrated on the preceeding pages were only carried
out for one iteration.

3 . Solution for the General Case

The solution will be presented in a way that can
easily be implemented by an algorithm satisfying a modular
design.

40

a. Determination of Adjusted Coordinates
Given:

a) the grid coordinates of N statons Si L*L2i. J.
where the x-coordinate represents EASTING'S
and the y-coordinate represents NORTHING'S

b) the azimuths /\ip C L* *>,.., A/ J from stations
D^ to vessel's position P ( x,^ )

c) and the standard deviations (J~u ( l? | | ...,AJ)
of observed azimuths,

the adjusted coordinates for P(xv) will be determined,

Step 1) Weight matrix ^v

1.1) Squaring the inverse of standard deviations *X~^ ,

U)C . J ( t . 1,2 ,..- , N ) ,

1.2) Designating by a>|^ the least uj£ , the

weights UJJ will be obtained;

u>«

1.3) The elements of square matrix fy7 will be such
that

^J * i ° , P" ^ J (!•', j « 1,2,..., A/)

41

Step 2) Observation equations

2.1) Determination of first "initial point"
2.1.1) Designate by Aj^p an observed azimuth Al?

(i=2, 3, . . . ,N) such that

tan AKP £ tan A \p .

If no such azimuth is available, then the vessel's
position is undetermined.

2.1.2) The intersection of the azimuth line /4 \$ p
from station K (X- y* ) with the azimuth line Aip from
station 1 (*i,jy.i) determines the first "initial point"
P0 (Xp,7o ) . Therefore,

a) if AxpznV (n=0/l) , then Va will be given
by

Vim*** **« (££• ~ \e )- (*-**>,

by

b) if /\$p x d TT (nso, L) i then ^> will be given

(

X,* Xk

>» yi+ i^n

2

-Aip ). (**-*i) ,

c) otherwise, *o will be given by

A0

>a 31+ mt (, x<? - Xi )

42

where m

m±- tah[C5tr/2)- AIS ]

2.2) Determination of azimuths At 'ue> between stations
SliXirfi) and "initial point" Pd

2.2.1) Two angles, A? ^Q and (fitl^^) satisfy the
equation

Also, r& 10 must be a positive angle between 0
and 2T. Since, in general, calculators give a solution
between (*• TT/2) and (+TT/2 ) , a criterion will be estab-
lished for selecting the valid solution.

2.2.2) Criterion :

a) if >--}/ and *o>*i , then At 'L0= 7* t*L

b) if J* s^f C and yo<^i > then A I ^0 = 317/2

c) if Yo = X<; and ^o»l r then y4?^0 = 0

d) if XosXt. and ^o<^c , then /4?^0='17
For X^ ^ *i and ^ ^ ^c* , designate by p^

the solution, given by a calculator, of

43

Therefore,

e) if «/t'o>° and ^e>Xu , then A2^Q = oi'LC

f) if <*Co<° and *o>*L * then ^io=^o^

g) if &lo>° and V© <XC , then ^ Co =<^t^)+IT
h) if cLl0<o and X* < >V , then A* La = ^*21T

2.3) Determination of elements u£ of matrix JU :

U = Aig- ^io U"s 1,1,..., N ).

2.4) Determination of squared distances between 3 £ (/£, JtJ
and P<> (**>):

*2. 1 1,

2.5) Determination of elements 2£j (Ls l;...; ^ : J-- \,1 '
of matrix A •

Step 3) Normal equations

3.1) Determine matrix A W (a matrix 2 x N) ,

3.2) Determine matrix A WA (a matrix 2x2).

3.3) Determine matrix i^W Ay (a matrix 2x2).

44

Since yAv/4 is a symmetric matrix, then its
inverse matrix will be Q = (A WA) , also symmetric,
such that

/mi ^h

nil Ai%

C3,» Q«*

1 O
O 1

It can be shown that

On* - Ait 1 ( A* - An. A^)

t?n .- Qj. * An/ (Asi - A, 1 . ^2zJ

3.4) Determine matrix <4 W L (a matrix 2x1).

3.5) Finally, determine

X,

r* "t
4}

-»

= C/fWAf (A' W L).

Step 4) First adjusted values

With the values Jb * and £ N the coordinates of the
new "initial point" Po ( *o , 3 '& ) are obtained:

Step 5) 2nd iteration

For obtaining a "closer" initial point repeat the
steps 2.2, 2.3, 2.4, 2.5, 3, and 4.

45

Step 6) Next iterations

Repeat Step 5 until £x and 6j become vanishingly
small, or, in practical terms, converging to within a speci-
fied tolerance.

Then, the adjusted values for x and y will coincide
with the coordinates of the last "initial point" obtained,
b. Precision of Observations

Given N (number of stations) and the matrices A, X,
W and L determine :
Step 1) Matrix of residuals V (a matrix Nxl)

Y.-4Y- L

Step 2) standard deviation $0 of the unit weight observation

2.1) Obtain VTW V (a scalar).

2.2) Obtain trace of weight matrix W;

trace ( W ) s "£ U^

u-«

where t-^££ is a diagonal element of weight matrix W .

2.3) Finally, $0 (in radians) will be given by

5,, /vTw V

Step 3) Standard deviation S t- of each observation (with
weight Oi^ ) ,

S £ x . 5° _- C i - I, 2 , • • • , N )

where 5^ is expressed in radians.

46

-j ( tfor t bung )

Luce 5

? (.X,? )
*JP4 »**»*> V-^

/ I

/

I

I

I

use

\

\

>

MB4
(ill)

X ^(rastfns )

FIG II-3: FIX BY 3 SEXTANT ANGLES

47

B. FIX DETERMINATION BY SEXTANT ANGLES

1. Solution for 3 Sextant Angles (Between 4 Stations)
Given a positioning problem as diagramed in Fig II-3
in which:

ct , a * - is the observed sextant angle from P between

stations 1 and 2
&2£3 "" -"-s t^le °kserve(3 sextant angle f rom F between
stations 2 and 3

0C^„. - is the observed sextant angle f rom .P between
Oik

stations 3 and 4
and

(xi/^» ) "" are grid coordinates of station 1

C*x 2z1 "~ are 9r^ coordinates of station 2

(X3 2a \ "* are 9rid coordinates of station 3

(*4Jki) - are grid coordinates of station 4

the grid coordinates (xy) of a vessel's position P

will be determined.

Step 1) Formulation of observation equations

The analytical expression for the sextant angle
o(Lp (uH) ' from tne vessel's position Z (x,y) , between
stations L (X£ty£) and L-H ( V^*- t 2V+ ■ ' is 9iven bv

48

The function F(x,y) must be expressed in a Taylor's
series around an "initial position", »© , whose coordinates
are defined as X Q and y0 . Evaluating the zero and first
order terms of the series, the following expression is
obtained:

-~(*3 ) =: At f (U,J - A**i ~

tan'' v"' - y« - tan

Jc + i - 2/ o

^o - 3 t ♦ I

„l

3C - 3*

}*~ 3 J

O- 2fi+»)* + (^-^♦.)1 (y.-^J1-* l** -x*)*

x* - Xc'

X© — XJ

ri

4*

^y

Designating by 5<jC and S^'^,) , and by A?0^ and /t20^,),

the distances and azimuths between "initial point" ^(Xw>;^©j
and stations L ( Xi N i} and t-fi ( Vjf( ^^» ) ^ respectively,
then

•Soft*!; ^ [lx\:*.« -**)% c ^^i *>•'*' 1

/t?0[; - tan"' [ {*: -xo) / ^t - >) J

49

and,

[lx.-«)/(5,c )l- (%- ^■♦•)/(s.a«.)),]<lJf .

Therefore, the observation equations may be expressed as
(for i = 1,2,3)

V; .

^ - 3 ;

■f *

X - 3i!

(S0^,)) (SoO*

Xo - XV.

'Np - Xu-» I

43-

£* +

*ieiM* ^<>i -A*o <*.i

where o( £ 9 C&fO ^s t^ie observed sextant angle. In matrix
form, the above equation is expressed as

V - A X - L

where

A m

Vo- 7i

*•-:*•

Xa -Xl

*© -X2

(S62)' l£«>* (So,)11 CS^iJ

V*-73

C502)1

<S*2)Z

Xo-*J

6S03)2

(So*)2 (Sat?

50

L-

°<3P4 + ^03 "At

-04

x.

AX

Ay

V-

Vi

IV3J

In that result it should be noted that the angles
are expressed in radians.
Step 2) Normal equations

Forming the normal equations, the adjusted values for
AX and j£y will be given by

X = MTWAf' (ATWU.

Step 3) With the values Ax and A>j a new "initial point"
» o (Xo Vo) is obtained,

'■ 3« r Jo*-*? ,

and the procedure will be repeated in an iterative way until
the increments /l* and Ay become vanishingly small.

Then, the most probable values for the coordinates
x and y will coincide with the coordinates of the last
"initial point" obtained.
2 . Numerical Example

Referring to FIG II-3, the U.T.M. grid coordinates

51

of shore station are ;

Coordinates MB4 (#1) USE (#2)

MUSSEL (#3)

LUCES (#4)

x (EASTING)
y (NORTHING)

600,425.2
4,053,917,2

600,372.0
4,051,216.9

597. 967.8
4,053,453.2

595,794,5
4,055,042,7

The observed sextant angles are equally precise; thus, the
weights will be

and the weight matrix W is the identity matrix

W>

I

o
o

o
I
o

o

c

I

The following sextant angles were measured:
MB4 - P - USE = d jp2 = 49°. 927
USE - P - MUSSEL = ot2? 3 = 33*.130
MUSSEL - P - LUCES = ^3P^ = 30*. 396

Step 1) Formulation of observation equations

1.1) Determination of first "initial point" vQ (X0i^t
1.1.1) The first "initial point" Pa will be the

point determined by the two sextant angles o((?l and <*iP3

such that

X7- *<

tan

P2

■ t*i(*Lt-Ai#-), _I«i*

3.- 1*

52

and

*3 - >o yz-^o

tat, 4in..U«{fy*ot 42ot), *L±* ?i^*

I*

1.1.2) After some algebraic manipulation, it will be
obtained that

3# • C *« *- £>

where

lull _ lilS * x, - x3

■ — » - ■■■ ■ - J < T J3

and

^_ tanot,p^ tan »^iP

V2 - X3 X. - X2

- J 1 + j 3

"id* olxt% tan oi ig 2

1.1.3) The value Xd will be a solution of equation

U ** + R *> * S .- o ( IT - i )

53

and

where 7 r 4. Z

U = Lan siipi . (C H)

R - iah <Vlfi [2 CD- C(y,+fr)- (x(+yzj] -
5 * ta«o(if2 [ £>- D [^.o*) tx, ** +>f, 2ft J -

1.1.4) Two solution sets, (JC#J , 3*1 ) a^d (x0j(j^)
satisfy eq. (II-l) . The valid solution corresponds to the
solution set that, introduced into the following expression,
yields the value that best approaches tan 0(^2. •

1.1.5) Using the numerical values

tan Q£\p2 = 1.1887 tan ^^^3 = 0.7850

y, = 603,425.2 ^,= 4,053,917.2

Xz= 600,372.0 ^= 4,051,216.9

y5= 597,967.8 ^= 4,053,453.2
it will be obtained

C - 11.109096 D = -2,613,387.9

U = 147.885 403 R = -1.776434 x 108
S = 5.334 734 3 x 1013 .

54

The two solution sets satisfying eq (II-l) are

)(0, = 600,833
J0{ = 4,056,325

and

#oz = 600,390
L ^2.= 4,051,405
Introducing the first solution set (X0{ yff ) into expression
(II-2) the value 1.2923 will be obtained. Introducing (^,
^dl) into the same expression , the value -0.9944 is obtained.
Since the first set is the one that best approaches the value
of "tan o^ = 1.1887, the coordinates of first "initial point"
are

Xt> = Xa = 600,833
Jo = >t - 4,056,325 .
1.2) Determination of azimuths between "initial point"
("^(X*^) and stations Si (x't^t) > (i=l,2,3,4) •

A?0 I

A?

At

03

0U =

tan' [(*■-**>/ (},-:/•) ] = I32°609

tan""1 [ix2-kp; /[ y%-y.) J - (B5?i5?
tan"1 [(**-*•) I (*-*) ]* 224*934

tan"' [(*-*•)/{**->) ] --255° 721

55

Then,

- O. 04* 075" * ^3^

- O. $2$ JUS 6 fa<J

- O. 006 8 2H Z tj<]

1.3) Determination of squared distances between \0
and stations :

(5., ) = (*,-*») + (*j, -3») = 12,517,002

[5^2' = (*»•*•>* + (j**-?*)* =26,305,207

( 603 1 = (^-y»)2 + (yj-^o)1 = 16,456,606

[5oA ) = (*H-**f + (**-*»>* " 27,030,776.

1.4) then,

f° - *** _ Vj« . 0. 000 ool 8 *°'x' m X9-*x m _ ^).^o 22 U 6

(S^)1 {S6,)z (So.)* LSaX)z'

jfr^jfe - -d.odoo/9 7

>£-/? Xo -^ __ o, 000 oil. 3

1.5) Therefore, in matrix form, the observation
equations will be expressed as

♦ 0. 000 001 8

- 0. 000 0J<? 1
- 0. 000 Vll 1

0,000 1Z<4 6

0 000 If 6 &

o.ooo o\% 3

J

r

- o.o^o &1S I)
-0.026 IMS o
-6.006 8 2^*2.

V,

I/,

56

Step 2) Normal equations

The solution of normal equations is given by

_ i

X * (AW A) lA'vJL) -,

2.1) obtaining matrix A W,

A W = 0-oco ooi 8 -0.0000137 -o.oooi27»
-O.Ooo 22*» 6 -0.0001S66 -o. ooocmz

2.2) obtaining matrix A W/A ,

aV/A.

-8
-9

4.1^ x \o

-3

^.2^ y IO
2.3) obtaining matrix Q = [/| W4) ,

7. 5" I y 1 £>
- i

-8

J .

Q. 6I;^8,1"78 -3,<»72>o"?a

2.4) obtaining matrix A W (— ,

ATWL:

. /.3 77 v Id

-6 1

-5

2.5) finally, vector X it will be obtained

X:

35.8
181.3

Step 3) First adjusted values of x and y

With the increments AX and AN a new "initial

point" is obtained;

j y0-6oo;833 «■ 3S.9 * 6ao;869.8

1 y0 = ^os6;325 + /8I-3 = UjOS 6;5o6. 3 .

57

Step 4) With the new values for the "initial point", the
procedure indicated in steps 1.2, 1.3, 1.4, 1.5, 2 and 3 is
repeated, and with the values now obtained for ^Jx and &y
a "closer" initial point is obtained.

Step 5) That procedure must be repeated, in an iterative way,
until the increments AX and /w become vanishingly small,
or, in practical terms, converging to within a specified
tolerance. Then, the last "initial point" obtained will
coincide with the most probable position for P(xy).

These computations may be compared with those shown
in the computer output section on page 149 . Differences
in the results are due to the fact that the calculations
illustrated on the preceeding pages were only carried out
for one iteration.

3 . Solution for the General Case

The solution will be presented in such a way that
easily can be implemented by an algorithm satisfying a modular
design.
Given:

a) the grid coordinates of M=N+1 stations S^ ( *t,Jt) ,
ordered in a clockwise sense around vessel's
position ,

b) the N sextant angles O^lPCi.*!) between stations
5(1 (Xi,y£) and Si4i (*k, r}U, )f

58

c) and the standard deviations 0^ (i-1,2, . . . ,N)

of observed sextant angles,
the adjusted coordinates for P(x,y) will be determined.

Step 1) Weight matrix W

Obtained as indicated on Step 1 of subsection II. A. 3. a

Step 2) Observed equations

2.1) Determination of first "initial point" F£ (>*,^o)
2.1.1) The first "initial point" fj ( Yo,Jo) will be

the point determined by the two sextant angles oA»* an^ Q^2f3
such that

Vz- *o

tan oL[Pn s tatj(4aLM- Aiol ) s 3*- J<

IP2L z a ' ^"'"Ol

■ + ^^

x.-

Xo

^. -

>

x\-

■ yQ

32- 3* :>.- 3<

and

Xa - .Xa

tah ciU3s tah (/Uo3- 4?02; .. ^-^ ^J.

Xj - .Xo x^ - xc

I +

?3 - % ^Z - X

2.1.2) Test for undetermined initial position
See FIG II-4) .

59

FIG II-4: UNDETERMINED FIX BY 2 SEXTANT ANGLES

If To rSx, 5j». an<3 S3 belong to the same
circumference, then

/a - iso°- c c<lfz + ^fs).

Therefore, the dot product of vectors U and V
(FIG II-4) will be

60

So, the condition for an undetermined fix by two sextant
angles will be

2.1.3) Of the initial position is not undetermined,
then its coordinates can be obtained as follows:

2.1.3.1) 1st case: 0^ jpj aJ: 3d and o/2£3 £ 9<J
Let

/i ss tan o£ i g ^
8* tan oil? 3

F* [t^o,)//4 J* [(*-?*)/ B] * x,- x3

a) If F=0 , then

3^ G/E" ^ p.

Let

Then, from

Lf yo 4- R y0 + S - o obtain

x.=

-Ri \/rz-^ U5

1 u

61

From solution sets (v*,/-^) and (X^^y*) , choose the one
that best satisfies

tah u _ (Xi-xQf?,-?,)- (x,-y«) (jx-Jo) (n-3)

l}*-» f^|-^o)+ (Xt-X*) (x'.-X*)

b) If E=0, then
^ = - S/P 3 H

Let

R -- A (:*,* ^a) - *i + xz

Then, from

"U^^R^o+S^O obtain

^ -Rt V Rz- A MS

]o=

62

From solution sets (Vtf.^oj ) and (X0 y07_ ) choose the
one that best satisfies equation (II-3).

c) If E 7* 0 and F ^ 0 , then

}*= ( F/E) x* + &! S = CXo + D.

Let

Then. from

Vy/ + R x„ + 5 = o

obtain

Xo -. ~

Rt 7 Ra- u \J 3

1 V

From solution sets (x„, , y . ) and (x „ , v ) choose the
one that best satisfies equation (II-3) .

63

2.1.3.2) 2nd case: <* [£2 = 90 and °*2?l ** 90

Let

B * tan *Z£ 3
F a B ( *» -**) +- }i- ^3

G> s B C>*}3 +***3 ~:?»7l - >fXl.> 4-Xz. }3 - X3 }z ,

a) If F = 0 , then

3„ * £ / £ r p .

Let

K=r - t*i + Xa.)

Then , from

P<o •+• K Xo t* 5 » <s> obtain

0 2

-Rt Vft^Ts

From solution sets (.y Na ) and (y«« sA) choose
the one that best satisfies

tano*ir3 - jx^^ig^.Jt)- (yi.-/#) Os-^) fix -4)

b) If E = 0 , then

X* - - <9/F = H .

64

Let

Then , from

^o + R 3o t S ■ o obtain

^

0 =

_ -Rt \/r^5

From solution sets Cx*;^*« ) and (^, ^0l) choose the one
that best satisfies equation (II-4) .

c) If E ? 0 and F ^ 0 , then

Let

"Us C -M

3= D - P t3|-0*}+ ^»^2+ * *** .

Then, from

Xfx0 + R X© + S - O obtain

1 _ — — -,M, , , , | T —

2. O

*c_-

From solution sets Cx^, ,^a<) and (X0)>^}) choose the
one that best satisfies equation (II-4) .

65

2.1,3.3) 3rd case: ^2j?3 =^° and °^/F2. * 90"
Let

A - taw ol,g2
F» A (X3-X1; + ». -3t

a) If F = 0 , then
Let

R-- c *- 4 x* ;

Then, from

X/d +• R *© -f S - o obtain

V* *

- R± \/Ra-

4 S

From solution sets {Xo\ >^o) and (X01m,^o) choose the one
that best satisfies equation (II-3).

b) If E = 0 , then

Yo^ -6/F-- H .

Let

5 r H*- H(^^3)-»^^^i^.

66

Then, from

Jo + R ^0 + ^ = * obtain

yOJS -R± >/g2- ^S

From solution sets (X* / "}o\ ) and ( x0 , j02 ) choose the one
that best satisfies equation (II-3).

c) If E ? 0 and F ? 0 , then

Ja; (F/e)y* + [ ^/E) = CXctD,
Let

5r P*- P ( ;>*+ 33) + *z*3 + ^i^? .

Then, from

U *0 + P> yo+ S s o obtain

xg, -Rt n/R*- ^US

2 \J

From solution sets (>d( ,^d<) and (yol , ^t) choose the one
that best satisfies equation (II-3).

2.1.3.4) 4th case: ^ipl= 90° and ^wC ^°°

Let

F = **- *>

6 ^ X. Xi + 3« ^i. -"^a.^3 - *z *3 .

67

Let

a) If F = 0 , then

b) If E = 0 , then

c) If E ^ 0 and F ^ 0 , then

Then, from

Ux^+Rx^f S--

obtain

V* -

- R+ V R*_ ^ u S

1 U

From solution sets (Xpt ry0l ) and i^ot,t^t,) choose the one
that best satisfies

C}i-:W (3i-:W + (x2-^; (x,- x*) = o . (31-5 )

2.2) Determination of azimuths Alt qI between "initial
point" Pf (Xo'jo) and stations Si (XlJC)

2.2.1) Two angles, A^0i and H^ ^ + 180 / satisfy

68

the equation

Alois tan' *:-*«, (i-.i;2r.;A/*«).

Also, '♦^ftt, must be a positive angle between 0 and iTf .
Since, in general, computers give a solution for the above
equation between (-TT/21) and (+V/i.) r then a criterion will
be established for selecting the valid solution.
2.2.2) Criterion :

a) If y0=yC and Xt>> XL' , then 4^0^= 3tf/Z j

b) If Jo=yi and y*< vL' ,then A*oi= **/* -
For ^jo 7* ^l designate by 0^o£ the

solution given by a computer of

^L = tahJ .x^'-**. (L= i,t;...; AJ+«) .

Then :

c) If #0,;^' D and Xey^C, then

d) If ^oi^° and ^o 4 Xl » then

e) If ^^ <o and Xo> X C s then

f) If O^OL ^° and ** ^ **•' , then

A*oLz oiOL -i rr.

69

2.3) Determination of elements Le* of matrix L, ;

Note, Ujl must be expressed in radians.

2.4) Determination of squared distances between
?Q (* f«j0) and St (yt', tf) :

2.5) Determination of elements 3 £ j (i=l , 2 , . . . ,N; j=l,2)
of matrix A:

Step 3) Normal equations

f

3.1) Determine matrix AW (a matrix 2 x N).

3.2) Determine matrix AWA (a matrix 2 x 2).

3.3) Determine matrix (flW<*)" (a matrix 2x2)

as indicated in Step 3.3 of subsection II. A. 3. a,

3.4) Determine matrix /I WL (a matrix 2x1).

3.5) Finally, determine

X- (ATW fiT1 ( fi1 W L ).

Step 4) First adjusted values

As indicated in Step 4 of subsection II.A.3.a.

70

Step 5) 2nd iteration

As indicated on Step 5 of subsection II. A. 3. a
Step 6) Next iterations

As indicated on Step 6 of subsection II. A. 3. a

71

C. FIX DETERMINATION BY TWO RANGE DISTANCES AND ONE AZIMUTH

This problem illustrates how to deal with observations
of different kinds (distances and angles) . The procedures
for obtaining the residuals and the weight matrix are more
complex.

1. Solution for Two Range Distances and One Azimuth from
3 Different Stations.

Given a positioning problem as diagramed in FIG II-7,
in which:

R, — is the observed range distance from station #1
^2 - is the observed range distance from station #2
/\ - is the observed azimuth from station #3
(Xi3i) ~ are the grid coordinates of station #1
(*i St) " are the 9rid coordinates of station #2
t**,t)3) " are tne grid coordinates of station #3
07 - is the standard error of R, (in meters)
(T^ - is the standard error of Rx (in meters)
OT - is the standard error of A (in degrees)
the grid coordinates of vessel's position l (x,y) will be
determined.
Step 1) Formulation of observation equations

1.1) The analytical expression for the range distance
between station i (i— 1,2) and vessel's position P (xy) is
given by

rL (meters) = [(X-*i)*+ C^-IK)1] = F (*») [C* [,!)

72

The function F(xy) must be expressed in a Taylor's
series around an "initial position" , P , whose coordinates
are defined as x and yQ . Evaluating the zero and first order
terms of the series, the following expression is obtained:

?C s [(Yo- Xc)% (>-3t)lI^ +

Then, designating by sio the distance from station i (i=l,2)
to "initial point" P0 (X0'Y0^ ' t*ie following expression is
obtained:

The observation equations are given by

5 to ^^°

where R-j_ is the observed range distance.

In this result it should be noted that the residuals,
v-j_ (i=l,2), are expressed in meters.

1.2) The analytical expression for the azimuth
between station 3 and P(x,y) is given by

AB (radians) ■ tan x~ **

73

Therefore, the observation equation is expressed as

tan

- 1

2 - 23

where A is the observed azimuth angle. In that result, it
should be noted that 1/3 is expressed in radians. Therefore,
it will be necessary to obtain V3 expressed in meters.
(See FIG II-5) .

w.

>A/*rfcb

.'>

^3 (*3 2&>

FIG II-5: CONVERTING ANGULAR RESIDUAL INTO
METRICAL RESIDUAL

From the FIG II-5 is concluded that

Vx(meters) = sin V* (radians) x distance between P and 5 3

or

V3 (m*t*rs) - sin UV £2 - A

30i

t*-*s) * (3-35)

54

74

Expressing the function V3 (xy) as a Taylor's
series around the "initial point" Sq ( to^y* ) / and taking
only the zero and first order terms, the following is obtained

V5- sin! W x«-x> „ A

?0- ^5

CO 5

taiT *» : y* _ A

3* -33

^0-^3) * c >-?*>'

^o-^

+

r 1 * 7 y^

0 +

5it7

.1

tan" jfe-niS.

J«03

.Sm

tan' *»-»

-/\

-/*

-A

Xa-X*

J ft*.-**)** C3» -3 ft/]

4X +

"^-Vo

[ (X0-^J? *(3o-33)_

IT" *

it-ys

[t*i-i4),W3.-j»)T

m ^

Designating by -^30 and r^^o the distance and azimuth
between station S-x (Xj/Jj) and "initial point" r© (yo >2
then

a / 1

SO r

l*©-*3) + l]0 - ^3)2

1

and

A*,., tan

- »

3o:

Xo-X*

3. -7*

75

Therefore, the observation equation may be written as

\43 Sin {A-itz0-A)% S50 +

53o 330

.AX +

[ ^3o 5 3o

1.3) Finally, the observation equations in matrix
notation are expressed as

AX- L - V

where the elements 3^ '. and (,£ of matrices A and L
are given by

a „ - (Xo-*o/ S i,

«3»2L= (J0-J1) / -S

\c

a 4, a (*o-*2)/s2o

«3iZ*fo»-Oi) 7^20

a»lje C05 (A*3*-A). ^'^ * *'« t/4^3^-^)' ^"^

53

S3

a32 = cos (Az3o - A). %dl!L+s'*« lA*3a-A). h^L

S30 S 3c

L, -. R, - 5

10

12 s R2 - 5

20

L-. - 5«n ( A\3o- A) . S3

76

Step 2) Determination of weight matrix W

The standard errors #7 and Oz of range observations
are expressed in meters; the standard error (f$ of the ob-
served azimuth angle is expressed in degrees. Therefore, it
will be necessary to obtain (J3 expressed in meters. (See
FIG II-6) .

FIG II-6: CONVERTING ANGULAR STANDARD DEVIATION
INTO METRICAL STANDARD DEVIATION

From FIG II-6 is concluded that

($3 (meters) - SinQ*3 ( m degrees) *- S3o t

Having obtained £Jg expressed in meters, the pro-
cedure for obtaining the weight matrix W is as indicated in
Step 1 of subsection II. A. 3. a.

77

Step 3) Normal equations

Forming the normal equations, the adjusted values
for /$•< and Ay are given by

- J

X = UTW A f UTW L) .

Step 4) With the values Ax, and A^/ a new "initial point"

Ol I <

Xo ( *o,ya) is obtained;

Step 5) For the new coordinates ( X © ^J o ) of "initial point",

the value of <T£ (in meters) is recomputed and the weight

matrix W readjusted.

Step 6) The procedure will be repeated, in an iterative way,

until the increments ^K and ^V^j become vanishingly small,

or, in practical terms, converging to within a specified

tolerance.

Then, the most probable values for the coordinates
(xy) will coincide with those obtained for the last "initial
point" .

2 . Numerical Example

Referring to FIG II-7, the grid coordinates (U.T.M.)
of shore stations are
COORDINATES LUCES (#1) MB4 (#2) MUSSEL (#3)

595,794.5
4,055,042.7

603,425.2
4,053,917.2

597,967.8
4,053,453.2

78

WN/0ftTH

N g

(#3)

£r 45.54*/

/

3*0/
/

'vA

&

\

WS4

FIG II-7: FIX BY TWO RANGE DISTANCES AND ONE AZIMUTH

79

The following angle has been measured by theodolite:
P - MUSSEL - LUCES = 99^.360

For illustrative purposes, the instrument is assigned
a standard error Cf^ = 0 .024.

The following distances were measured:

P - LUCES = 5233 m

P - MB4 = 3515 m

Their standard errors are assumed to be (Tj = (J^ r 10 m
(fictitious values)
Step 1) The azimuth between MUSSEL and LUCES is given by

6

A23{ * tanV *-**. - 306. (Si

3i- 33

Therefore, the following data are available:

R, - 5233 m. 07 , \0 m

R2- 3515 rr> ^Tz - \0 m

A r 45*5 4 i ^rj5 o?oz4

Step 2) Formulation of observation equations

2.1) Determination of first "initial point"

The first "initial point" will be the point
determined by range distances R< and Kg, (for which the
azimuth from station 3 is closer to A). Therefore, the point
Yq {Xo,2)o) will satisfy the following system of equations:

\

30

Introducing numerical values, the following
solution sets for the above equations are obtained '•

X9I: 6oo} $67. 1 and Jv0? s 6oo?2&c.\

The azimuth from station #3 to (V^, ^jol ) and ^oi ,^oz^ '
respectively, are obtained: n^^0{ - 45. 24 and A^ 30Z -
115 .5 . Therefore, the valid solution is the one corres-
ponding to At^oi , i.e.,

j *<s>= *<?, = 600,867.2
I 2om ?o* = 4,056,328.0

2.2) Determination of azimuth between station 3 and

Then, A^3o -/Is- 4° 297.

2.3) Determining distances between stations and »o ,

r * 1 1 l/^

5 |d . [(*.-**> + O,- Jo1) J ^ 5U3-0

-52o s [(yx- y*)% (3l - j<>>z ]* r 35u5. a

81

2.4) Therefore, the elements 3u.'
A and L will be

and Ll of matrices

3u - (Xo-x.) / 5l0 z 0.16 3 3 68 a21 s Lj0-3,)/Slo = o.2<iS 6tH

S30 5*0

^o

S30 s3<:,

ti. R.-5

I ' 10 5 O

L: R7 ' 5

2~ ^2d> - *

L5r -Sin (b*3o-fi). S30- 2/' '**

2.5) The observation equations in matrix notation
may be written as

-0.127 138
0.1 00 lj o o

»

0. U±5 6 14

4x

r

0

v,

oMs 861

A}

■»

0

s

v2

-0.1(1 75 5 _

I

/2U6S\

. v3 J

32

Step 3) Normal equations

3.1) Determination of weight matrix W:

<37s 10 m

Then

=> ^3* (meters)* Sjo . 5in(o!e^)

^ 1.7IO m

Setting the least weight equal to one, it will be obtained

that

1

ujz s i or ~W =

U)3 = 34

I 0 O

O I o

by

3.2) The solution of normal equations is given

Xs <A WAI (AWL).

ATW =

3.2.1) Determination of A W :
,o. 24S 6/4 0.-6 85*86 l

23- S«3 £oo
-24. 2<S7-6 7o

83

3.2.2) Determination

af ATWA:

ATWA

10. Me

L-H25S

17. 2 s e

17. SSI

a .

3.2.3) Determination of (A W/4 f :

(ATW fifl

» * •

3.2.4) Determination of (A^L) :

(AWL).

5 o4. o \S
.-SJ3. £25

3.2.5) Finally,

x>

5.1
23.$

J .

Step 4) First adjusted values

With the values <£X =5.1 and ZV^ = - 2 3.8 a new
"initial point" is obtained:

Xo = 600,867,2 + 5.1 - 600,872.3

^o = 4,056,328.0 - 23.8 = 4,056,304.2
Step 5) With the new values for the "initial point" the
procedure indicated in steps 2.2, 2.3, 2.4, 2.5, 3 and 4 is
repeated and a "closer" initial point is obtained.

84

Step 6) That procedure must be repeated, in an iterative way,
until the increments ^X and Ay become vanishingly small,
or, in practical terms, converge to within a specified
tolerance. Then, the last "initial point" obtained will
coincide with the most probable position for P.

These computations may be compared with those shown
in the computer output section on page 150. Differences
in the results are due to the fact that the calculations
illustrated on the preceeding pages were only carried out for
one iteration.

3. Solution for the General Case

The solution will be presented in such a way that
easily can be implemented by an algorithm satisfying a
modular design.

a. Two cases will be considered:
1st case) Two range distances and one azimuth from three
stations (See FIG. II-8)

>J

i\

' f

V\ 0

V \Rl

/

\H \

/

\ J

i

&.

Sj

-*\

A

/ \
1
t /

\

\

\ **J

FIG II-8: FIX FROM 3 STATIONS

35

Designate by S ( and S^ the stations from which range
distances are observed; the station from which an azimuth
is observed will be designated by Sj.

2nd case) Two range distances and one azimuth from just two
stations. (See FIG II-9)

FIG II-9

FIX FROM 2 STATIONS

Designate by St tne station from whch an azimuth and a
range distance were measured ( S]_concides with Sa ) ; the
remaining station will be designated by 5«.
b. Given

(xl,Yj_) - grid coordinates of station S-j_
(X2'Y2) - grid coordinates of station S2
(X3/Y3) - grid coordinates of station S3
(in the 2nd case, they coincide with the coordinates

(xl,Yl) of SX)

Rj_ - range distance from station Si
1*2 - range distance from station S2

A - azimuth from station S->

86

and
Ui - standard error of R]_ (in meters)
C12. - standard error of R2 (in meters)
(13 - standard error of A (in degrees)
the coordinates of vessel's position P(x,y) will
be determined.
Step 1) Formulation of observation equations

1.1) Determination of first "initial point" iQ (X0jy0\
The first "initial point" will be:

a) one of the intersection points of circum-
ferences centered at Z>\ and $2. with radius ranges of R|
and R2 respectively .

b) the intersection point which lies closer to
the azimuth line through Z)-x .

Therefore, the following equations must be
satisfied:

(x,- to) f ( },- ^ - R
(X2-x«) «■ (3,. - jo? - &i

Let

E * *i - Rx * 3a - J. + ** - *■

Then,

X* s

2 Cx-2- x,)

87

1.1.1) For X-2. ^ X\ proceed as follows:
Let

7_

E.wJlirJ!!. ) *

I

2(x2-Xi) ' x2-x,

EA= ( e«)*- 4(£.) <E*> •

Then,

o =

1 E ,

a) If E^<Oj, then the two circumfer-
ences don't intersect; therefore, choose point Q as first
"Initial point (See FIG 11-10).

FIG 11-10: RANGE DISTANCES NOT INTERSECTING

88

Then,

*o = x. +

lA

}° -- 3, t

Ri ( 3? - :M

b) If E4 = 0 , then the two circumferences
are tangent. Therefore

%* -

Is

2 E

Vnz

2 ( /2-->M **- *l

c) If E4 > o , then the two circumferences
intersect at 2 points; therefore, the following intersection
points are obtained :

(

3*i

[- Fa + VE4 ] /2 E,

2 ( *2 - y, )

, 111 ^ -i
x, - yt

>i =

<

01

[- lx - /IT ] /z £

£ 3 1- 3z

4- ^Jol

2(X7-y, ) y7 . x,

and

89

and
3o*= [- Ft - v/T^ ]/iE.
*oz = Er/[tCxz-x.)]+[(^l-^)/(xa.^l^]^z b

c.l) If X3 = *<?i and 2J3 =^<?»/ then

X o — X01
Jaz J0z .

C.2) If X3 = Xo2 and ^3=%2' then

* les Jot .

c.3) Otherwise, determine azimuths between

station Sr$^,-fr) and O^r^i.) (for i=l,2).

Criterion:

J) if 3*l = ^3 and XoL>*% , then

II) If Jtf^ =^3 and xai <; X3 ,then

^*3*c = 31T /2. .

For ^„l 7^ ^3 designate by o< 3 c
the solution given by a computer of

90

Then:

III) If 0^3t >,o and XolZ x3 , let

IV) If ot^i<° and Xai < X3 ,let

V) Otherwise, A 23o|_= ^3^ + TT .
Having determined azimuths Z^^, and n^j^^'
check which one is closer to the observed
azimuth A .

I) If ^?30,= ^23<91 = A , then the

solution will be undetermined (See
FIG 11-11) .

\ '

>N

— • /

FIG 11-11: UNDETERMINED FIX BY 2 RANGE DISTANCES
AND 1 AZIMUTH

91

1 -L

II) If (A\t-A) = (-4 23tfl-/») , then choose

III) if (A^i6,-/A) > (Af3ox-/4 ) , then choose

Xo = Xoi
}<? - Jot .

IV) If (A? -A) < (A*, -A) .then choose

1.1.2) For X2. =X| /

the following equation is obtained:

> = E/ j> ( Ik-JM ] .

Let
Then ,

t /f

92

a) If F ^ o , the two circumferences
do not intersect or are tangent;
then

Xa - *\ .

b) If r 'J o , the two circumferences
intersect at points {xaf ,^0\) and

(*aa.#2aiJ t tnen

x0[ - x, h- v F

and
/

-WF

b.l) if *3 =X^, and ^3 = ^01 , then

Xo - X01.

b.2) If X3 = X02- and ^3=^

tf-2- 1

then

X«r X

o 1

b.3) Otherwise, determine azimuths /n"2 3^,

and /A"S 3^2 between station

S3 (X3, 2>3 ) and (Xot ,^«»t ) (i=l/2)

using criterion presented in step 1.1.1

Having determined ^i^oi, and ^^307. .

determine which one is closer to A .

I) If A^3oi - /^23o2 = Ajthen

the solution is undetermined.

93

2.

II) If (fal0x-A) = (/**3«,x-d)*
then choose

in) if (A^3oi-A) > lA*Soz-A)Z^

then choose

iv) if tAt^-A)' < (AHj^-^)1,

then choose

XU a Xc, |

1.2) Determining the azimuth n 2»c between
station $3 (^3^3) and (X^o)
Criterion:
I ) If "jo = ^3 and X* > x3 , then

A*3o, IT/ Z .

II) If 20=^3 and Xd<x3)then

A^0 = 3ir / z .

For *Jo ? ^3 designate by 0^3 the
solution given by a computer of

94

Then:
III) If (*3£o and Xa %, x3 ,then A^3o = °^3 .
IV) If ^3 <o and y0< >3 jthen /te;?o = v3 * * tr.

V) Otherwise, • Al-$0= <*3 + **** .

1.3) Determining distances between stations and Xo

5 |0 i C Ui«*a> 4 {****•?}'*

1.4) Determining elements of matrix A ,

33|J mCA«9»-4>. 3£i*5 4 *•*(*>.-*). *!&.

1.5) Determining elements of matrix L ,

l3: si* ( A- A?3*) • 53<,

Step 2) Solution of normal equations
2.1) Obtain weight matrix W.

2.1.1) Determine standard error of observed azimuth
angle expressed in meters,
(C (meters) = sin O3 (radians) x S33

95

2.1.2) With (J3 expressed in meters, the procedure
for obtaining the weight matrix W is as
indicated in Step 1 of subsection II. A. 3. a

2.1.3) Finally, (T~$ is again expressed in radians;

(HCra^ians) =: Si'tf' (V3 (meters) / Szo\

2.2) Determine matrix /n W .

2.3) Determine matrix /\ V*A.

2.4) Determine matrix (A WA)

2.5) Determine matrix /♦ r L •

2.6) Finally, determine X = ( /\T V /0~ ( ATW L ) (
Step 3) First adjusted values

With the values AX and A-^) obtain new "initial
point" Pq (tf'o So) 5

3d - ^0+ *} -

Step 4) 2nd iteration

For obtaining a "closer" initial point, repeat steps
1.2, 1.3, 1.4, 1.5, 2 and 3.
Step 5) Next iterations

Repeat step 4 until AX and ^^ become vanishingly
small, or, in practical terms, converge to within a specified
tolerance. Then, the adjusted values for x and y will
coincide with the coordinates of the last "initial point"
obtained.

96

Ill . RESULTS AND CONCLUSIONS
A. RESULTS

From the general case solutions developed for the selected
positioning methods, algorithms were written in a structured
programming format. All algorithms are presented in appendix
I.

These modular algorithms were translated into Fortran
language for implementation on the NPS computer, an IBM 303 3.
Program listings are provided in the Computer Programs Section
beginning on page 151.

Data sets given in each Numerical Example section were
input into the corresponding computer program, and the output
of each run is given in the Computer Output Section starting
on page 14 8.

Additionally, the programs were tested using several fic-
titious data sets to insure their performance in handling the
various initial conditions which were modeled for each fix-
ing method.

In applying these programs to real positioning data the
following points should be considered:

1. The presence of blunders and systematic errors in
the observations will be reflected in the dimensions of the
error ellipse. If all blunders are removed by careful
editing and all systematic errors are eliminated by modeling
or calibration, then the size of the error ellipse will

97

represent the positioning error due to net geometry and
random errors.

2. When the information about the standard errors of the
observations is reliable (for example, determined by field
calibration procedures) , then the estimates obtained for the
standard deviations of the observed values will be close to
the a priori values (see example of fix by 3 azimuth angles
in Computer Output section on page 148).

3. When no a priori values are given for the standard
deviations of the observations (it is assumed that the
observations are equally weighted) , then the application of
the least squares method will provide estimates of instrument
(or observation) accuracy (see example in computer output
section, on page 149).

4. When the correlation coefficient is close to one,
the error ellipse becomes flatter approaching a straight
line (see example of fix by two range distances and one
azimuth angle in the Computer Output section on page 150).

5. When the correlation coefficient is negative, the
major axis of the error ellipse runs through the 2nd and
4th quadrants. Thus, the angle from the x-axis to the major
axis measured counterclockwise lies between 90° and 180°. If
the correlation coefficient is positive, the major axis runs
through the 1st and 3rd quadrants, and the angle from x-axis
to the major axis measured counterclockwise is between 0° and
90°.

98

B.

The most significant result of this thesis is that well
documented programs are now available which can be used for
the analysis of hydrographic positioning data. These programs
may be employed to process and analyze hydrographic survey
data that have been collected using one of the three position-
ing methods discussed. Ideally, such software should be
adapted to run in a mini computer aboard a survey vessel or
launch. This capability would allow "real time" analysis of
positioning accuracy.

In addition to processing actual survey data, the programs
may assist in survey planning. By scaling observations from
existing charts of a survey area, sample data sets may be
formed to test net geometry. This information can be used
to establish the best location for shore control stations.

The programs are written in modular form so that they
may be adapted for use by other types of positioning systems.
The significant differences between all the programs lie in
the modules dealing with the computation of the "initial
point" and formulation of the observation equations.

It should be noted that the accuracy of the geodetic
control stations has not been specifically considered in
these formulations. However, any survey error in the station
coordinates will be reflected in the dimensions of the error
ellipse of the adjusted hydrographic position.

99

All of the programs were developed using a plane
coordinate system model. Thus, they are primarily applicable
to nearshore hydrographic positioning problems. Application
to offshore hydrography would require a geodetic coordinate
system model based on a selected spheroidal datum surface.
Obviously j the use of a geodetic coordinate system would yield
more complex analytical expressions relating the unknowns.
But, once these were obtained and linearized, then the pro-
cedures for computing adjusted survey coordinates and the
statistical values defining their precision are identical to
those developed in this thesis.

Whether the existing programs are used in their current
form or modified to accomodate other variables, one final
point should be made. The most significant contribution of
the least squares method to hydrographic position adjustment
is its ability to quantify errors statistically. When
programs are operated aboard the survey vessel in "real time ",
relative accuracy achieved with conventional survey methods
is elevated to absolute accuracy if redundant observations
are made and adjusted using least squares.

Monitoring the size and orientation of the error ellipse
alerts the user to the presence of gross blunders and inord-
inately large systematic errors. The need for electronic
positioning system calibration can be realistically evaluated,
and calibration may be performed on an as needed basis. With

100

sufficient redundant observations, electronic positioning
systems can, in fact, become self calibrating.

As the trends in electronic and computer technology
continue to decrease the cost of collecting and processing
redundant observations, conventional two LOP ' s survey
positioning will be relegated to the historical equivalent
of lead line hydrography.

101

APPENDIX A. LEAST SQUARES PRINCIPLE AND NORMAL DISTRIBUTION
When measuring a parameter, the outcomes of that experi-
ment can be considered as values assumed by a random variable
following a normal distribution. For a random variable X
following a normal distribution, the value most likely to
occur is its mean U*. . The true value, from a deterministic
point of view, of an observed parameter is, in a
stocastical sense the mean of the random variable associated
with the experiment. Therefore, when using the least squares
technique for the adjustment of a redundant number of obser-
vations, not only a set of "consistent" values are ob-
tained but also the most probable values for the means of
the random variables considered. Therefore, the adjusted
values are also the best estimates for the "true" values of
the parameters considered.

1. Normal distribution

The density function associated with a random variable
X following a normal distribution is expressed by (see
FIG A-l) .

f

X

_ [(*-/>x) /i <m ]

i»i = — ! e

1 02

pod

>

ll ^^••te*

-S 2AX

FIG A-l: NORMAL DISTRIBUTION

Then, the probability of occurrence of values between
Xp-AX and 3f0+ A/ will be given by

V*"

<0-d«*

i^r q*> ax

Therefore, it can be concluded that the probability of
occurrence of values "around" %0 is proportional to the
density function value at that point, i.e.,

? { x,-a* 4 X 4

7C-+ &X.

} ,« f ix.) _ (A- I)

2. Probability of occurrence of a set of values assumed
by independent random variables

103

FIG A-2: FIX BY 3 RANGE DISTANCES

Suppose that the distances between a vessel and stations A,
B and C are measured and the results are, respectively,
x, y, and z (see FIG A-2).

FIG A-3

RESIDUALS

If the vessel is situated at 0, then the means of the random
variables X, Y and Z, associated with the range distances
AO, BO and CO, are at distances V ,1^, and V^ from, respec-
tively, observed values x, y and z. (See FIGS A-3 and A-4).

104

FIG A-4: RESIDUALS AND NORMAL DISTRIBUTION

The probability of occurrence of a set of observed values
■{x y zA is given by

If the observations are equally weighted then the standard
deviations <J^ , ^ and tf^ have the same value, say, (f .
Then, from (A-l) , it will be obtained

3 - [jL_ [(*-/**?♦ ^-/y)% (3-^*?]]
Pl-|xy x})^ (tC/o-v/^) e L2<r (£-2)

Recalling that

it will be obtained from (A-2)

r(|xtjz|) = CONSTANT x e

2<r

[ v^v,2* v3* ]

(/A- 3}

The means l>^ , U ,. and D -j are values such that the observed
values x, y and z have a probability as high as can be

105

expected to occur.

Therefore, from (A-3) it will be concluded that the
residual values maximizing the probability of occurrence of
event j X ^ 2 \ will be the set of values minimizing the

2. 'Z. 2

expression ( Vt + Yz + Y3 ) , i.e., those that minimize the
sum of the squared residuals. That is the reason why the
least squares technique yields the most probable values for
the means of the random variables considered, i.e., the best
estimates for the "true" values of parameters being observed

106

APPENDIX B. LEAST SQUARES PRINCIPLE FOR WEIGHTED OBSERVATIONS
1. If an observed value XL' has the weight uJ , then the
observed value ~Xi is worth as much as <J^ observed values X/
with weight equal to unity.

A usual criterion for establish weights is to consider
the weights inversely proportional to the squared standard
deviations, i.e.,

Then, given a set of observed values XL' (i=0 , 1 , 2 , . . . ) , un-
equally precise, with standard deviations (/£ , if the least
precise value x0 is considered to have a weight equal to
unity ("k=D / the different weights U)L" will satisfy

LU^

: " t L*o,\,zr..).

<nz

2. A set of observed values x. , x , . . . , x , with respective

weights equal to u>, , u^ , . . . , u)n , is equal to a set of
W]_ values equal to xt , UJj values equal to X^ , . . . , and u>„
values equal to xh / all with unity weight. Therefore, the
sum of the squared residuals will be given by

and the basic least squares principle will be expressed as

n n

L W

/ Lai

107

or, in matrix form, as

where

V W V - mi'nimiin

O ^2 O ' O

I

o o oj3 : ;

o o

U>

1C8

APPENDIX C. NORMAL EQUATION IN ALGEBRAIC NOTATION
Below are the observation equations in algebraic notation

V, * afl x, t atl t± + .. . +almxm-L,= o
Vnran, *, * an2 xa * . . . * a„m xm - Ln =°

The unknown values X, ,*t ,» -• i^m that satisfy the basic
least squares principle

n

t -2

t-vjti V^ s, mincmun

are those that satisfy the following expression:

u\ (a» «i +afi xx+... +aimxtn- L, ) +

^a CaZ( *,+ 3u Xx+ ... -*3^m xm- l*)S

^nUn, *,-*<3nz S. + •' 43^xm-U) = F (x, «»••• «tn ) -

= mLnCrnun .

The values xi,*?,... x^ minimizing P (x^ , X2 » • • • %) are those
such that

J F ■ = O

( j » •; 2 ; ' ' •

IG9

Considering that

dXr

the following normal equations are obtained :

l«i 3u. aCl J X, ♦ . . . +[uLaLi acmivm - L^^ul L 1 - O

<

to,; a^ai

tJx, ♦ - • - *M a^ i** -LwiaLmUjs

110

APPENDIX D. NORMAL EQUATIONS IN MATRIX NOTATION
Taking the observation equations in its matrix notation,

AX-L= V,

the unknown vector X satisfying the basic least squares
principle ,

V W V = rniLnunvjn ,

is the one such that

(AX- l)T W IAX- L) =

X1 AJVJ AX-Z XTATWL + LT W L = minim un .

The vector X satisfying the above expression will be such

that

<5 ( XVWAX-2XVWL t LTW L.\ =o . (Q-1)

2>X

Considering that

JL (XTATWAX) = 1 XT AT VJ A

ax

1_ (XT/\TW L). LT Vi^A
it- ( C W L ) -. 0 ,

a*

it will be obtained from (D-l) that

1 XVWA - 2 L' WTA * o .

in

Therefore, the normal equations are of the form

(ArW A) X - ATW L,

and the solution will be given by

X • iaVa)"' i/\tw L ).

112

APPENDIX E. A COMPUTATIONAL CHECK FOR THE LEAST SQUARES

Adjustment Technique
By taking the normal equations

(ATVI A) X = ATW L ( E - 1)

and recalling that

A X - V+ L,

then the following can be obtained:

(flTV/A)^ATWV* AtWL. (E-2)

Therefore, from (E-l) and (E-2) it is obtained that

ATWV=o. fe-3]

Equation (E-3) provides a check on the computations for least
squares adjustment.

113

APPENDIX F. THE CONTROVERSIAL CRITERION FOR ASSIGNING WEIGHTS
1. The usual criterion for assigning weights is stated as:

a) the weights are inversely proportional to the squared
standard deviations, i.e.,

r L

ojl 5C = K

(F-1J

where K is an arbitrary constant;

b) the least precise observation has the unity weight,
i.e. ,

or

u0o S/ = K => K= O) , 5/

tC= 5,

IF-!}

where Sd is the standard deviation of least precise obser-
vation.

2. For the moment, consider only equation (F-l) . The
influence of the value assigned to k on the computations
for obtaining the adjusted values and standard deviations of
adjusted values will be determined. From eq (F-l) it will be
obtained that

U0L = K I Si .

Then, the weight matrix W will be

W*

x/s,-1

kr/s,

K/S

= K

•/S,1

Vsl

VsZ

:KW

r '

114

and the trace TR (W) - X / $? t . . . + * I 5 «* s K TR (W') .

a) Computing adjusted values,

X -- ( a7\ja)-' (ATWL) *

M'/K)(ATW*A)"! K(ATW'L)= IATW'A)"' (aVl).

Therefore, it is concluded that, for the adjusted
values X, the value assigned to K is, in fact, arbitrary.

b) Computing standard deviations of adjusted values,

--^. = "50 * III where Q^i is an element of matrix Q
Recalling that

Q. (ATW/w"'= W«) MTw'Af '= (l/IOQ1,

then

Next, by considering

50, /vTwv s yfc . / vrwv

it follows that

KTR(W')-m '

Therefore, as should be expected, the standard deviations of
adjusted values are affected by the value assigned to K.

115

3. In fact, eq (F-2) imposes a constraint on the K value:

K - S0Z .

To illustrate the consequences of accepting that kind of
constraint, suppose that, given 100 observations, 99 are
equally precise and one is less precise, say, with a standard
deviation 40 0 times greater than the standard deviation of the
remaining 99 observations. Then, according to the usual
criterion, the least precise observation has the unity weight
and the other 99 observations have the weight 20. That dis-
tribution of weights does not seem "good," and it is the
author's opinion that there should be a better constraint
minimizing the disturbances introduced by the assignment of
different weights.

116

APPENDIX G. DECISION OF ADJUSTED VALUES
Given the observation equations

I

- I, = V,

a, * + b, ^ - lj - v,

the standard deviations of adjusted values (by least squares
method) for x and y will be determined |^REF.2j.
1. Assuming the observations were equally weighted, and
solving the normal equations, the following is obtained:

y z

3«

£»»]CbL]-(yn>L3

[ab3a- MO]
[ab]*-[a*][b*]

te-n

where the brackets have the usual meaning of sum,
Rearranging (G-l) ,

X = <*. Li * oi2 Lz «■ <*3 L3
3 s^, L, ♦ /4a L2 + /i3 L3

where

0(. Q]a, -fab] k
[a»ir^] -D»bl*

L =

A- [av] K -[ab] bl
L" [*][»] -Ob]*

117

Consider L\ , *-2. and L3 as values assumed, respectively,
by independent random variables Lj , L^ and |_ 3 • Since
the observations were equally weighted, then L,jLi and
L3 present the same standard deviation, say, SQ . Then.

3 r i i

L - I

3

<

VAR(Y) -- £ AN/»R ( Li) = \P\b? (6-2)

1= I
3

coWR(x,y) - H <*LA w*( Lt) = [<*& I 5* .

L * J

T -1

2. Determining the matrix Q - (A W A ) , the result is

Q=

Cb*]

[»*JM -Cab]*

-fab]

-Cab]

[a*] 0*3-

■ [a

b]»

[a*]

[a«] [bM - [a b] * MM -&" J

3. Since

[«i«]._I*2

[a«]Cb*]-[ab]

<=| „

02L_M

[a-lCb^-Lab]

= 1

21

[a/S]._^±1

t~ .12.

G>l)[tl]-[ab]

- ^

12.

it may be concluded from eq. (G-2) that

5„-

Is-,.

50 . /«! „

5a ■ Ri*

4. Finally, the correlation coefficient Q between random
variables x and y is given by

P-

x y

i%

S* • S* Q<=u -^]^

LI 9

APPENDIX H. ERROR ELIPSE
1. The position i (x y) of a vessel at sea is a two-
dimensional random variable; its density function is the
joint density function of the random variables x and y ;

(Tx gy

^x--2

Z.(Gx <Jy -<J^y )

<T7Z J,

<Z

<r*<Ts

2.7T VG^CTy*- <wr

Since Ux and u^ are the adjusted coordinates i^o^o) ,
if the origin of the coordinate system is positioned there,
it will be obtained that

<£ cS

mH<rS<ri*-<&?) L 5

err1-

-l<Qy

X3

frp (T7z

Txy

2 TT >/<fc2 (f)'2 - <S7~:j'L

0^*

7

>

sy

o

A PC**)

2ax

(*«#>)

Sx

*

FIG H-l: TWO-DIMENSIONAL NORMAL DISTRIBUTION

120

Then, the probability of occurrence of the vessel's position in
a small area (2d*. 16"}) aroundP(x,y) will be given by
(see FIG H-l)

P ( x= ~t**, Y=*±A3)* J ) £ uv a? <**

2. Now, it will be determined in what kind of line points with
equal probability of occurrence are situated.

These points will present the same value f for the
density function. Therefore, letting

|fa.[. Ln (2*7^ f }(2 fc>] / (5^2^^
and K-3= - *^2 ^2<57^ p then

(Ty2 x1- 2„ (7^ s, X ^ + IQ vj* 4- fr3 - o ( H - 1 J

That quadratic equation in x and y represents a conic; the
existence of the xy-term indicates that the conic is rotated
out of its standard position.

a. Before determining what kind of conic equation (H-l)
represents, check if points (0^ © ) and (o}Gy) are both
over the same contour line for a constant density function .
Inserting point (T^o ) into (H-l) results in

Inserting point {c}(Ty) into (H-l) results in

K-3 - - fr* &* *.

Therefore, the points (Oy;a) and (0,(Jy) are over the

mtour line (corresponding to K3 = -(F* (7j ) .

same coi

121

b. The analytical expression for the specific contour
line containing points «J^;0 ) and (0}G~y ) is

T. _2

0^ x - z (J^ xj +(Hi J - (Ji (PJ =0

Considering

3 = -z(f^

C^ (Tic
p - o

E = o I

it is concluded that the discriminant is less than or equal
to zero, i.e.,

Bz- 4 Ac <C o .

Therefore, if d-^Ac<o then equation (H-2) represents an

ellipse; if B>~^Ac = o then it will represent a straight

line (a degenerate ellipse corresponding to a perfect

correlation between random variables x and y) .

3. The equation of the error ellipse in standard position:

Consider

(ATWA)"-. Q -

Recalling that
and

13

^3

122

then equation (H-2) is equivalent to

<\**- * «,, v;, ♦ <,, «,z _«,1<lt 5,.*,. . (H-3)

Consider

If the x-axis is rotated an angle V^ such that

cot 2 g-0 - 6liSl 9 liii? f H - 4 )

then the equation of the ellipse (H-3) in its standard
position is

+

, ^, <H So v

12 COS* Jo - 2 ^3 COS #0 Sin flo + <), Sl'n1 Jo

^ 1 (H-5!

After some algebraic and trigonometric manipulation, the
following expression is obtained from (H-5) I

X

1 ^z

* _2! ,1 (H-6)

l^j^z S0^ v j t^qr So2, >v

123

where

D =

t^i -q«)*+ 4 =|3

/«

(H-7J

4. If the positive value of D satisfying eq. (H-7) is
choosen, then the semi-major axis of the error ellipse is

positioned along the "new" x-axis. Therefore, from the

a
two solutions C^ = d. and 0^ - o<-+ 90 satisfying eq. (H-4) the

valid one must be choosen (considering o( as the smallest

positive angle satisfying (H-4) ) .

FIG H-2: ERROR ELLIPSE

The only way to solve that ambiguity is to test either with

0(1 or o<2-
For that purpose, it is recommended to

a) obtain the point of intersection (either P, or t\ )

124

of the line y ■ xtanoi with the ellipse before rotation
(expressed by eq (H-3));

b) determine the distance _d between the origin and P.
(orf^);

c) if d = semi-major axis a , then the major axis makes
an angle )£q =oi{ (measured counterclockwise) with the "old"
x-axis; if not, then the angle will be J o = °^2. = ol\ + 90 ,
that is, the major axis runs through 2nd and 4th quadrant.

125

APPENDIX I - ALGORITHMS

MODULE 1

ALGORITHM FIX-BY-N-AZIMUTHS
INPUT N

PI-3. 141592653589793
OUTPUT' NUMBER OF STATIONS =',N
DO FOR I*- 1 TO N

INPUT TABLE-INPUT(I,1),TABLE-INPUT(I,2),TABLE-INPUT(I,4)
OUTPUT ' ST# ' , I , ' EAST= ' , TABLE-INPUT (1,1),' NORT= * ,
TABLE-INPUT ( 1 , 2 ) , ' ST ERROR= * , TABLE-INPUT (1,4)
END DO
DO FOR I - 1 TO N

INPUT TABLE-INPUT (I, 3)
END DO

DO WHILE TABLE-INPUT (1 ,3)7*400.0
OUTPUT' OBSERVED AZIMUTHS'
DO FOR I «*■ 1 TO N

OUTPUT 'AZIMUTH FROM STATION# ' , I , ' = ' , TABLE-INPUT (I , 3 ) ,

'DEGREES*
ALGORITHM CONVERSION-DEGREES-RADIANS (TABLE-INPUT (1,3))

TABLE-INPUT (1,3 )*TABLE-INPUT ( 1 , 3 )* (PI / 180 . 0 )
END CONVERSION-DEGREES-RADIANS (TABLE-INPUT (1,3))
END DO

MODULES 2,3,4,5,6,7
DO FOR I*-l TO N

INPUT TABLE-INPUT (I, 3)
END DO
END DO
END FIX-BY-N-AZIMUTHS

MODULE 2

ALGORITHM WEIGHT-MATRIX (N , TABLE-INPUT (I , 4) )

ALGORITHM ZERO (TABLE-WEIGHT)

DO FOR I— 1 TO 10

DO FOR J —1 TO 10

TABLE-WEIGHT ( I , J ) — 0 . 000

END DO

END DO

END ZERO (TABLE-WEIGHT)

ALGORITHM SQUARE (N .TABLE-INPUT(I , 4) , TABLE-WEIGHT)

DO FOR I*- 1 TO N

TABLE-WEIGHT (1,1)— TABLE-INPUT ( I , 4)**2

END DO

END SQUARE (TABLE-WEIGHT)

ALGORITHM NORMALIZE (TABLE-WEIGHT)
GREATEST - TABLE-WEIGHT (1,1)

DO FOR 1-2 TO N

IF TABLE-WEIGHT (I, I) > GREATEST THEN

GREATEST <- TABLE-WEIGHT (1,1)

END IF

END DO

126

DO FOR I«-l TO N

TABLE-WEIGHT (I, I )«-GREATEST /TABLE-WEIGHT (I, I)
END DO
END NORMALIZE (TABLE-WEIGHT)
END WEIGHT-MATRIX (TABLE-WEIGHT)

MODULE 3

ALGORITHM FIRST-INITIAL-POINT (TABLE-INPUT ,N)
ALGORITHM SELECT-AZIMUTHS (TABLE-INPUT(I , 3) ,N)
I«-2

DO WHILE TANGENT(TABLE-INPUT(I,3))=TANGENT(TABLE-
INPUT(1,3))
I«-I+l
IF 1 >N THEN

OUTPUT* POSITION IS UNDETERMINED FOR THAT DATA SET'
PICK UP ANOTHER DATA SET
END IF
END DO
END SELECT-AZIMUTH(I)
ALGORITHM INITIAL-COORDINATES (TABLE-INPUT, I)

IF TABLE-INPUT (1,3) =0.0 OR TABLE-INPUT(1 ,3 )=PI THEN
MK *- TANGENT ( ( 5 . /2 . )*PI-TABLE-INPUT (1,3))
XO *- TABLE-INPUT (1,1)

YO 4- TABLE-INPUT (1,2) +MK* (XO-TABLE-INPUT (1,1))
ELSE IF TABLE-INPUT(I,3)=0.0 OR TABLE-INPUT(I , 3)=PI THEN
MI *- TANGENT ( ( 5 . / 2 . )*PI-TABLE-INPUT (1,3))
XO«-TABLE-INPUT (1,1)

YO «•- TABLE-INPUT (1,2) +MI* (XO-TABLE-INPUT (1,1))
ELSE

MI *• TANGENT ( ( 5 . /2 . )*PI-TABLE-INPUT (1,3))
MK «- TANGENT ( ( 5 . / 2 . )*PI-TABLE-INPUT (1,3))
XO *• (TABLE-INPUT (1,2) -TABLE-INPUT (1,2) +MI*TABLE-

INPUT ( 1 , 1 )-MK*TABLE-INPUT (1,1))/ (MI-MK)
YO +-TABLE-INPUT (1,2) +MI* (XO-TABLE-INPUT (1,1))
END IF
END INITIAL-COORDINATES (XO,YO)
END FIRST-INITIAL-POINT(XO,YO)

MODULE 4

ALGORITHM ITERATIONS (TOLERANCE)

DO UNTIL TOLERANCE < 1.0

MODULES 8,9,10,11,12,13,14

END DO
END ITERATIONS (XO,YO)

MODULE 5

ALGORITHM FINAL-ADJUSTED-POSITION (XO,YO)

OUTPUT* ADJUSTED COORDINATES X= ' ,XO , ' Y= ' ,YO
END FINAL-ADJUSTED-POSITION (X,Y)

127

MODULE 6

ALGORITHM PRECISION (TABLE-A, TABLE-WEIGHT ,TABLE-Q,LIST-L ,

DELTAX,DELTAY,N)

MODULES 15,16,17,18,19
END PRECISION (SU,SX,SY,SXY,RO)

MODULE 7

ALGORITHM ERROR-ELIPSE (TABLE-Q , SU )

OUTPUT 'ERROR ELIPSE SEMI-AXIS AND ORIENTATION'

MODULES 20, 21, 22, 23, 24, 25, 26
END ERRO-ELIPSE(SA,SB,GAMAO)

MODULE 8

ALGORITHM INITIAL-AZIMUTHS (XO , YO , TABLE-INPUT ,N)
DO FOR I«-l TO N

IF Y0=TABLE-INPUT(I,2) AND XO > TABLE-INPUT (I , 1 ) THEN

LIST-A0(I)«-PI/2.
ELSE IF Y0=TABLE-INPUT(I,2) AND XO < TABLE-INPUT (I , 1 ) THEN

LIST-AO(I)«-(3.0*PI)/2.
ELSE IF X0=TABLE-INPUT(I,1) AND YO > TABLE-INPUT (I , 2 ) THEN

LIST-AO(I) 4-0.0
ELSE IF X0=TABLE-INPUT(I,1) AND YO <TABLE-INPUT(1 , 2 ) THEN

LIST-AO(I)*- PI
ELSE

ALFA( I ) 4- ARC TANGENT ( (XO-TABLE-INPUT (1,1))/ (YO-

TABLE-INPUT(I,2)))
IF ALFA(I) >0.0 AND XO > TABLE-INPUT (I , 1 ) THEN
LIST-AO(I)*-ALFA(I)
ELSE IF ALFA(I)>0.0 AND XO < TABLE-INPUT (1,1) THEN

LIST-AO(I)<- ALFA(I)+PI
ELSE IF ALFA(I)<0.0 AND XO > TABLE-INPUT (I , 1 ) THEN

LI ST-AO ( I ) <r ALFA ( I ) +PI
ELSE

LIST-AO ( I ) «-ALFA ( I ) +2 . 0*PI
END IF
END IF
END DO
END INITIAL-AZIMUTHS (LIST-AO)

MODULE 9

ALGORITHM MATRIX-L (TABLE-INPUT (I , 3 ) , LIST-AO, N)
ALGORITHM ZERO(LIST-L)
DO FOR I «- 1 TO 10

LIST-L(I) «- 0.000
END DO
END ZERO(LIST-L)
DO FOR I*-l TO N

LIST-L( I)*- TABLE-INPUT (I, 3) -LIST-AO (I")
END DO
END MATRIX-L(LIST-L)

123

MODULE 10

ALGORITHM INITIAL-SQUARED-DISTANCES (XO ,YO ,N , TABLE-INPUT)
DO FOR I 4-1 TO N

LIST-SO(I)*-(XO-TABLE-INPUT(I, 1) )**2+(YO-TABLE-
INPUT(I,2))**2
END DO
END INITIAL-SQUARED-DISTANCES (LIST-SO)

MODULE 11

ALGORITHM MATRIX-A (N, TABLE-INPUT, XO,YO, LIST-SO)
ALGORITHM ZERO (TABLE-A)
DO FOR I«-l TO 10
DO FOR J «- 1 TO 2

TABLE-A(I,J)*-0.000
END DO
END DO
END ZERO (TABLE-A)

ALGORITHM ELEMENTS (TABLE-A , N , TABLE-INPUT , XO , YO , LIST-SO )
DO FOR I *-l TO N

TABLE-A(I,l)<*-(YO-TABLE-INPUT(I,2))/LIST-SO(I)
END DO
DO FOR I*-l to N

TABLE-A (1,2) «- (TABLE-INPUT ( 1 , 1 ) -XO ) /LIST-SO ( I )
END DO
END ELEMENTS (TABLE-A)
END MATRIX-A (TABLE-A)

MODULE 12

ALGORITHM NORMAL-EQUATIONS (TABLE-A, TABLE-WEIGHT, LIST-L)
ALGORITHM ATW(TABLE-A, TABLE-WEIGHT)
DO FOR I*-l TO 2

DO FOR J *-l TO 10

TABLE-ATW(I,J)*- 0.00
DO FOR K*- 1 TO 10

TABLE-ATW(I,J) ^TABLE-ATW(I , J )+TABLE-
A(K , I STABLE-WEIGHT (K , J )
END DO
END DO
END DO
END ATW(TABLE-ATW)

ALGORITHM ATWA (TABLE-ATW, TABLE-A)
DO FOR I*- 1 TO 2
DO FOR J *-l TO 2

TABLE-ATWA(I,J) *-0.00
DO FOR K*-l TO 10

TABLE-ATWA ( I , J ) ^TABLE-ATWA ( I , J ) +TABLE-
ATW(I,K)*TABLE-A(K,J)
END DO
END DO
END DO
END ATWA (TABLE-ATWA)

129

ALGORITHM INVERT-ATWA(TABLE-ATWA)

BETA*-TABLE-ATWA(1,2)**2-TABLE-ATWA(1,1)*TABLE-ATWA(2,2)

TABLE-Q (1,1)*- -TABLE-ATWA (2,2) /BETA
TABLE-Q ( 1 , 2)*- TABLE-ATWA (1 , 2 ) /BETA
TABLE-Q (2,1)*- TABLE-Q (1,2)
TABLE-Q (2,2)*- -TABLE-ATWA (1,1) /BETA
END INVERT -ATWA( TABLE-Q)
ALGORITHM ATWL(TABLE-ATW,LIST-L)
DO FOR I*- 1 TO 2

LIST-ATWL(I)«-0.0
DO FOR K*-l TO 10

LIST-ATWL(I)*-LIST-ATWL(I)+TABLE-ATW(I,K)*LIST-L(K)
END DO
END DO
END ATWL(LIST-ATWL)

ALGORITHM ADJUSTED-INCREMENTS (TABLE-Q ,LIST-ATWL)
DELTAX*- TABLE-Q ( 1 , 1 )*LIST-ATWL( 1 ) +TABLE-
Q(1,2)*LIST-ATWL(2)

DELTAY*- TABLE-Q ( 2 , 1 )*LIST-ATWL( 1 ) +TABLE-
Q(2,2)*LIST-ATWL(2)
END ADJUSTED-INCREMENTS (DELTAX, DELTA Y)
END NORMAL-EQUATIONS (DELTAX, DELTA Y)

MODULE 13

ALGORITHM NEW-INITIAL-POINT (XO ,YO , DELTAX ,DELTAY)

XO*-XO+DELTAX

YO *-YO+DELTAY
END NEW-INITIAL-POINT(XO,YO)

MODULE 14

ALGORITHM TOLERANCE (DELTAX, DELTAY)
TOLERANCE «- DELTAX**2+DELTAY**2

END TOLERANCE (TOLERANCE)

MODULE 15

ALGORITHM RESIDUALS (TABLE-A , LIST-L , DELTAX , DELTAY , N )
LIST-X(1)«- DELTAX
LIST-X( 2)*- DELTAY

ALGORITHM LIST-AX(TABLE-A, LIST-X,N)
DO FOR I *- 1 TO N
LIST-AX(I)*-0.0
DO FOR J*-l TO 2

LIST-AX(I)*-LIST-AX(I)+TABLE-A(I,J)*LIST-X(J)
END DO
END DO
END LIST-AX(LIST-AX)

130

ALGORITHM LIST-V(LIST-AX,LIST-L,N)
DO FOR I *■ 1 TO N

LIST-V(I)*-LIST-AX(I)-LIST-L(I)
END DO
END LIST-V(LIST-V)
END RESIDUALS(LIST-V)

MODULE 16

ALGORITHM ST-DEVIATION-OF-UNIT-WEIGHT-OBS(LIST-V , TABLE-WEIGHT ,N)
ALGORITHM LIST-VTW(LIST-V, TABLE-WEIGHT ,N)
DO FOR I*-l TO N

LIST-VTW(I)«-LIST-V(I)*TABLE-WEIGHT(I,I)
END DO
END LIST-VTW(LIST-VTW)
ALGORITHM VTWV(LIST-VTW,LIST-V)
VTWV v- 0 . 0
DO FOR I«-l TO N

VTWV*-VTWV+LIST-VTW(I)*LIST-V(I)
END DO
END VTWV (VTWV)

ALGORITHM TRACE (TABLE-WEIGHT)
TRACED 0.0

DO FOR I «-l TO N

TRACE «-TRACE+TABLE-WEIGHT (1,1)
END DO
END TRACE (TRACE)
ALGORITHM SU ( VTWV , TRACE)

CHARLIE «e- (VTWV/ (TRACE-2 . 0 ) )
SU ^SQRT( CHARLIE)
END SU(SU)
END ST-DEVIATION-OF-UNIT-WEIGHT-OBS ( SU )

MODULE 17

ALGORITHM ST-DEVIATION-OF-EACH-OBS (SU , TABLE-WEIGHT)
OUTPUT' PRECISION OF OBSERVATIONS"
DO FOR I*-l TO N

S <-(SU/SQRT(TABLE-WEIGHT(I,I)))*(l80.0/PI)
OUTPUT' ST DEVIATION OF OBS ', I , ' = ' , S ,' DEGREES '
END DO
END ST-DEVIATION-OF-EACH-OBS

MODULE 18

ALGORITHM ST-DEVIATIONS-AND-COVARIANCE-OF X~AND-Y(SU ,TABLE-Q)

SX «- SU*SQRT (TABLE-Q (1,1))

SY *- SU*SQRT (TABLE-Q (2,2))

SXY «^ ( SU**2 )*TABLE-Q (1,2)

OUTPUT ' SX= ' , SX , ' SY= ' , SY , ' SXY= ' , SXY
END ST-DEVIATIONS-AND-COVARIANCE-OF-X-AND-Y ( SX , SY , SXY )

131

MODULE 20

ALGORITHM D(TABLE-Q)

D -e-SQRT ( (TABLE-Q (1,1) -TABLE-Q (2,2) )**2+4 . 0* (TABLE-
Q(l,2)**2))
END D(D)

MODULE 21

ALGORITHM SEMI -MAJOR-AXIS (SU , TABLE-Q)

SA<- SU*SQRT ( 2 . 0*TABLE-Q (1,1 ) *TABLE-Q ( 2 , 2 )/( TABLE-Q (1,1) +
TABLE-Q(2,2)-D))

OUTPUT' SEMI -MAJOR AXIS SA= ' , SA
END SEMI-MAJOR-AXIS(SA)

MODULE 22

ALGORITHM SEMI-MINOR-AXIS (SU , TABLE-Q)

SB «-SU*SQRT ( 2 . 0*TABLE-Q (1,1) *TABLE-Q (2,2 )/( TABLE-Q (1,1) +
TABLE-Q(2,2)+D))

OUTPUT' SEMI-MINOR AXIS SB=',SB
END SEMI-MINOR-AXIS (SB)

MODULE 23

ALGORITHM GAMA (TABLE-Q)

IF TABLE-Q(1,1)=TABLE-Q(2,2) THEN
GAMA «- PI/4.0

ELSE

OMEGA *• ARC TANGENT ( 2 . 0 -TABLE-Q (1,2)/ (TABLE-Q (1,1)-
TABLE-Q(2,2)))

IF OMEGA > 0 . 0 THEN

GAMA IOMEGA/ 2.0
ELSE

GAMA «- ( OMEGA+PI ) / 2 . 0

END IF
END IF
END GAMA (GAMA)

MODULE 24

ALGORITHM INTERSECTION ( SU , TABLE-Q , GAMA)

X10 <r-( SU**2 ) •'•'TABLE-Q (1,1 )*TABLE-Q (2,2)

XI 1*— TABLE-Q ( 2 , 2 ) -2 . 0-TABLE-Q ( 1 , 2 ) '-'TANGENT (GAMA ) +
(TANGENT (GAMA** 2 )*TABLE-Q (1,1)

X1<-X10/X11

Y 1 *- X 1 * ( TANG ENT ( GAMA ) ** 2 )
END INTERSECTION (XI, Yl)

MODULE 25

ALGORITHM AVERAGE (SA, SB)

AVER*-((SA + SB)/2.0)**2
END AVERAGE(AVER)

132

MODULE 26

ALGORITHM SELECTION (AVER, XI ,Y1 )

D1«-X1+Y1

IF Dl 7 AVER THEN
GAMAO *r GAMA

ELSE

GAMAO *-GAMA+P 1/2.0

END IF

GAMAO*- GAMAO* (180.0 /PI)

OUTPUT' ANGLE FROM X-AXIS TO SA ANTICLOCKWISE= ' , GAMAO
END SELECTION (GAMAO)

MODULE 19

ALGORITHM CORRELATION-COEFFICIENT ( SX , SY , SXY )
RO«-SXY/(SX*SY)

OUTPUT' CORRELATION COEFFICIENT RO=',RO
END CORRELATION-COEFFICIENT(RO)

133

MODULE 30

ALGORITHM FIX-BY-N-SEXTANT-ANGLES
PI *-3. 141592653589793
INPUT N

OUTPUT'NUMBER OF SEXTANT ANGLES=',N
M«-N + l
DO FOR I «-l TO N

INPUT TABLE-INPUT (1,1), TABLE-INPUT (1,2), TABLE-
INPUT (I, 4)
OUTPUT' ST#' ,1, 'EAST=* , TABLE-INPUT (1 , 1 ),'NORTH= ' ,TABLE-
INPUT(I,2) , 'ST ERROR=' ,TABLE-INPUT(I , 4)
END DO

INPUT TABLE-INPUT(M,1),TABLE-INPUT(M,2)

OUTPUT 'ST#' ,M, 'EAST=' ,TABLE-INPUT(M, 1 ) , 'NORTH=' , TABLE-
INPUT^, 2)
DO FOR I<-1 TO N

INPUT TABLE-INPUT (I, 3)
END DO

DO WHILE TABLE-INPUT (1,3) 7*400.0
OUTPUT' OBSERVED SEXTANT ANGLES'
DO FOR I<-1 TO N
J«-I + l
OUTPUT 'SEXTANT ANGLE BETWEEN ST#',I,'AND ST#',J,

'=' , TABLE-INPUT ( 1,3 ) , 'DEGREES'
ALGORITHM CONVERSION-DEGREES-RADIANS (TABLE-INPUT (1,3))

TABLE-INPUT (1,3) TABLE-INPUT ( I , 3 )* (PI / 1 80 . 0 )
END CONVERSION-DEGREES-RADIANS (TABLE-INPUT (I , 3 ) )
END DO

MODULES 2,31,32,5,6,7
DO FOR I*-l TO N

INPUT TABLE-INPUT(I,3)
END DO
END DO
END FIX-BY-N-SEXTANT-ANGLES

MODULE 31

ALGORITHM FIRST-INITIAL-POINT-FOR-FIX-BY-N-SEXTANT-

ANGLES (TABLE-INPUT)

MODULES 33,34,35,36
END FIRST-INITIAL-POINT-FOR-FIX-BY-N-SEXTANT-ANGLES(XO,YO)

MODULE 32

ALGORITHM ITERATIONS (TOLERANCE)

DO UNTIL TOLERANCE < 1.000

MODULES 37,38,39,40,12,13,14

END DO
END ITERATIONS (XO,YO)

134

MODULE 33

ALGORITHM SELECT-SEXTANT-ANGLES (TABLE-INPUT)
J*-l

DO UNTIL FRAC 1 f FRAC 2
J*-J + l
IF J>/M THEN

OUTPUT' SOLUTION UNDETERMINED FOR THAT DATA SET'
PICK UP ANOTHER DATA SET
END IF
IW-1
K+-J+1
ANGUL «- TABLE-INPUT (1,3) +TABLE-INPUT ( J , 3 )
FRAC 1 «- COSINE (ANGUL )*SQRT(( (TABLE-INPUT (I ,1 )-
TABLE-INPUT (J, 1))**2+ (TABLE-INPUT(I , 2 )-TABLE-
INPUT (J , 2 ) )**2 )* ( (TABLE-INPUT (K , 1 ) -TABLE-INPUT (J , 1 ) )**2+
(TABLE-INPUT (K , 2 )-TABLE-INPUT ( J , 2 ) )**2) )
FRAC 2 —(TABLE-INPUT (1 , 1 )-TABLE-INPUT( J , 1 ) )* (TABLE-
INPUT ( J , 1 ) -TABLE-INPUT (K , 1 ) ) + (TABLE-INPUT (1,2)
-TABLE-INPUT (J , 2 ) )* (TABLE-INPUT (J , 2 )-TABLE-INPUT (K , 2 ) )
END DO
END SELECT-SEXTANT-ANGLES (I, J, K)

MODULE 34

ALGORITHM INTERCHANGE-DATA(TABLE-INPUT , I , J ,K )

STORE ( 1 ) *■ TABLE-INPUT (1,1)

STORE (2)*- TABLE-INPUT (1,2)

STORE ( 3 ) — TABLE-INPUT (1,3)

STORE (4) ♦- TABLE-INPUT (2,1)

STORE ( 5 ) — TABLE-INPUT (2,2)

STORE ( 6 ) ^-TABLE-INPUT (2,3)

STORE ( 7 )*— TABLE-INPUT ( 3 , 1 )

STORE ( 8 )*- TABLE-INPUT (3, 2)

STORE ( 9 ) —TABLE-INPUT (1,1)

STORE ( 10)+- TABLE-INPUT (1,2)

STORE( 11)*- TABLE-INPUT (1, 3)

STORE ( 12 )«- TABLE-INPUT ( J, 1)

STORE (13)— TABLE-INPUT (J, 2)

STORE (14)— TABLE-INPUT (J, 3)

STORE (15)— TABLE-INPUT (K,l)

STORE (16)— TABLE-INPUT (K, 2)

TABLE-INPUT (1,1) -STORE(9)

TABLE-INPUT (1,2) — STORE ( 10 )

TABLE-INPUT (1, 3 )— STORE ( 11)

TABLE-INPUT (2,1) — STORE (12)

TABLE-INPUT (2, 2)*- STORE (13)

TABLE-INPUT (2, 3)— STORE (14)

TABLE-INPUT (3,1)*- STORE (15)

TABLE-INPUT (3, 2)— STORE (16)
END INTSRCHANGE-DATA(TABLE-INPUT, STORE)

135

MODULE 35

ALGORITHM INITIAL-COORDINATES (TABLE-INPUT)

IF TABLE-INPUT (1 ,3)^(PI/2 . 0) AND TABLE-INPUT(2 ,3)?fc(PI/ (2 .0 )
THEN

AB *- TANGENT (TABLE-INPUT (1,3))
BA *- TANGENT (TABLE-INPUT (2,3))
E *-(TABLE-INPUT(2,l)-TABLE-INPUT(l,l))/AB+(TABLE-

INPUT (2,1) -TABLE-INPUT (3,1)) /BA+TABLE-INPUT (3,2)-
TABLE, INPUT (1,2)
F «- (TABLE-INPUT ( 2,2 ))-TABLE-INPUT(l, 2) )/AB+ (TABLE-INPUT (2. 2)-
TABLE-INPUT (3,2))/ BA+TABLE-INPUT (1,1)-
TABLE-INPUT(3,1)
G *- (TABLE-INPUT ( 1 , 2 )*TABLE-INPUT (2,1) -TABLE-INPUT (2,2)*

TABLE-INPUT (1,1)) /AB+ (TABLE-INPUT (2,1 )*TABLE-INPUT (3,2)-
TABLE-INPUT(3,1)-TABLE-INPUT(2, 2)) /BA+TABLE-
INPUT (2,1 )*TABLE-INPUT (3,1) +TABLE-INPUT ( 2 , 2 ^TABLE-
INPUT (3,2) -TABLE-INPUT (1,1 )*TABLE-INPUT (2,1)-
TABLE-INPUT ( 1 , 2 )*TABLE-INPUT (2,2)
IF F=0.0 THEN
DAO«-G/E
Y01«-DAO
Y02 *-DAO
U+-AB
R +- TABLE-INPUT ( 1 , 2 )-TABLE-INPUT (2,2) -AB* (TABLE-INPUT (1,1) +

TABLE-INPUT(2,1))
SAL*- AB* (DAO** 2 -DAO* (TABLE- INPUT (1,2) +TABLE- INPUT (2,2)) +
TABLE-INPUT (1,1 )*TABLE-INPUT (2,1) +TABLE-INPUT (1,2)*
TABLE-INPUT (2,2)) +DAO* (TABLE-INPUT (2,1) -TABLE-INPUT (1,1))+
TABLE-INPUT (1,1 )*TABLE-INPUT (2,2) -TABLE-INPUT (2,1)*
TABLE-INPUT (1,2)
DISC *-SQRT(R**2-4 . 0*U*SAL)
X01 *-(-R+DISC)/(2.0*U)
X02 *-(-R-DISC)/(2.0*U)
ELSE IF E=0.0 THEN
H*-(-G/F)
X01«-H
X02*-H
U ♦- AB
R «- TABLE-INPUT(2,1)-TABLE-INPUT(1,1)-A3*(TABLE-INPUT(1,2)+

TABLE-INPUT(2,2))
SAL*- AB* (H**2-H* (TABLE-INPUT ( 1 , 1 ) +TABLE-INPUT ( 2 , 1 )) +

TABLE-INPUT ( 1 , 1 )*TABLE-INPUT (2,1) +TABLE-INPUT (1,2)*
TABLE-INPUT (2,2)) +H* ( TABLE-INPUT (1,2) -TABLE-INPUT (2,2))+
TABLE-INPUT (1,1 )*TABLE-INPUT (2,2) -TABLE-INPUT (2,1 )*
TABLE-INPUT (1,2)
DISC*-SQRT(R**2-4.0*U*SAL)
Y01 — (-R+DISC)/(2.0*U)
Y02 *- (-R-DISC)/(2.0*U)
ELSE

C--F/E

136

DAO— G/E

U «- AB*(C**2+1.0)

R <- AB* ( 2 . 0*C*DAC~C* (TABLE-INPUT (1,2) +TABLE-INPUT (2,2))
-TABLE-INPUT (1,1) -TABLE-INPUT (2,1)) +C* (TABLE-
INPUT (2,1) -TABLE-INPUT (1,1)) +TABLE-INPUT (1,2)-
TABLE-INPUT(2,2)
SAL— AB* (DA0**2-DA0* (TABLE-INPUT (1,2) +TABLE-INPUT (2,2))+
TABLE-INPUT (1,1 )*TABLE-INPUT (2,1) +TABLE-INPUT (1,2)*
TABLE-INPUT (2,2)) +DAO* (TABLE-INPUT (2,1) -TABLE-
INPUT (1,1)) +TABLE-INPUT ( 1 , 1 )*TABLE-INPUT (2,2)-
TABLE-INPUT (2,1) *TABLE-INPUT (1,2)
DISC— SQRT(R** 2-4. 0*U* SAL)
XOl— (-R+DISC)/(2.0*U)
X02 — (-R-DISC)/(2.0*U)
YOl — C*X01+DAO
Y02— C*X02+DAO
END IF

ELSE IF TABLE-INPUT(l,3) = (PI/2.0) AND TABLE-INPUT(2 , 3 )t£
(PI/2.0) THEN

BA — TANGENT ( TABLE-INPUT (2,3))
E — BA*(TABLE-INPUT(3,2)-TABLE-INPUT(1,2))+

TABLE-INPUT (2,1) -TABLE-INPUT (3,1)
F -r- BA*(TABLE-INPUT(1,1)-TABLE-INPUT(3,1))+TABLE-INPUT(2,2)-

TABLE-INPUT (3, 2)
G — BA* (TABLE-INPUT ( 2 , 2 )*TABLE-INPUT (3,2) +TABLE-INPUT (2,1)*
TABLE-INPUT (3,1) -TABLE-INPUT (1,2) *TABLE-INPUT (2,2)-
TABLE-INPUT ( 1 , 1 )*TABLE-INPUT (2,1))+

TABLE-INPUT ( 2 , 1 )*TABLE-INPUT (3,2) -TABLE-INPUT (3,1)*
TABLE-INPUT (2, 2)
IF F=0.0 THEN
DAO+-G/E
YOl-DAO
Y02 — DAO

R *- -TABLE-INPUT(1,1)-TABLE-INPUT(2,1)
SAL *-DAO**2-DAO* (TABLE-INPUT ( 1 , 2 ) +TABLE-INPUT ( 2 , 2 )) +
TABLE-INPUT ( 1 , 2 )*TABLE-INPUT (2,2) +TABLE-
INPUT(1 , 1 )*TABLE-INPUT(2 , 1)
DISC — SQRT(R**2-4.0*SAL)
X01*-(-R+DISC)/2.0
X02— (-R-DISC)/2.0
ELSE IF E=0.0 THEN
H-(-G/F)
X01-H
X02 *-H

R *- -TABLE-INPUT (1,2) -TABLE- INPUT (2,2)
SAL «- H**2-H* (TABLE-INPUT (1.1) +TABLE-INPUT (2.1) +
TABLE-INPUT (1,1) )*TABLE-INPUT (2,1) +TABLE-
INPUT ( 1 , 2 )*TABLE- INPUT (2,2)
DISC-SQRT(R**2-4.0*SAL)
YOl— (-R+DISC)/2.0
Y02 «-(-R-DISC)/2.0

137

ELSE

C — F/E

DAO *-G/E

U*-C**2+1.0

R«-2.0*C*DAO-C*( TABLE-INPUT (1, 2) +TABLE-INPUT (2, 2))

-TABLE-INPUT (1,1) -TABLE-INPUT (2,1)
SAL *- DAO**2-DAO* (TABLE-INPUT (1,2) +TABLE-INPUT (2,2)) +

TABLE-INPUT ( 1 , 2 ) +TABLE-INPUT (2,2) +TABLE-INPUT (1,1) *

TABLE-INPUT (2,1)
DISC*- SORT (R**2-4.0*U*SAL)
XO 1 *- ( -R+DISC ) / ( 2 . 0*U )
X02*-(-R-DISC)/(2.0*U)
Y01*-C*X01+DAO
Y02*-C*X02+DAO
END IF

ELSE IF TABLE-INPUT(1,3)t£(PI/2.0)AND TABLE-INPUT(2 , 3 ) =
(PI/2.0) THEN

AB «- TANGENT (TABLE-INPUT (1,3))
E — AB* (TABLE-INPUT (1,2) -TABLE-INPUT (3,2)) +TABLE-

INPUT( 1,1) -TABLE-INPUT (3,1)
F — AB* (TABLE-INPUT (3 , 1 ) -TABLE-INPUT ( 1 , 1 ) +TABLE-
INPUT (1,2) -TABLE-INPUT (2,2)
G *-AB* (TABLE-INPUT (1 , 1 )*TABLE-INPUT( 2, 1 )+TABLE-INPUT( 1,2)*

TABLE-INPUT (2,2) -TABLE-INPUT (2,1 )*TABLE-INPUT (3,1)-

TABLE-INPUT (2,2 )*TABLE-INPUT (3,2)) +TABLE-INPUT (1,1) *

TABLE-INPUT (2,2) -TABLE-INPUT ( 2 , 1 )*TABLE-INPUT (1,2)
IF F=0.0 THEN

DAO*-G/E

Y01*-DAO

Y02 — DAO

R — -TABLE-INPUT (2,1) -TABLE-INPUT (3,1)

SAL — DAO**2-DAO*(TABLE-INPUT(2 , 2)+TABLE-INPUT(3 , 2) ) +
TABLE-INPUT ( 2 , 2 )*TABLE-INPUT (3,2) +TABLE-INPUT (2,1)*
TABLE-INPUT(3,1)

DISC *- SQRT (R**2-4 . 0*SAL)

X01*-(-R+DISC)/2.0

X02*-(-R-DISC)/2.0
ELSE IF E=0.0 THEM
H*-(-G/F)
X01- H
X02*-H

R«- -TABLE-INPUT (2,2) -TABLE-INPUT (3,2)
SAL*- H**2-H* (TABLE-INPUT (2,1) +TABLE- INPUT (3,1)) +

TABLE-INPUT ( 2 , 1 )*TABLE-INPUT (3,1) +TABLE-INPUT (2,2) *
TABLE-INPUT (3, 2)
DISC 4- SQRT(R**2-4 . 0*SAL)
YOl^-(-R+DISC)/2.0
YO2*-(-R-DISC)/2.0
ELSE

C*-F/E
DAO*-G/E

U*-C**2+1.0

R «- 2 . 0*C*DAO-C* (TABLE-INPUT (2,2) +TABLE-INPUT (3,2))-

TABLE-INPUT (2,1) -TABLE-INPUT (3,1)
SAL «- DA0**2-DA0* ( TABLE-INPUT (2,2) +TABLE-INPUT (3,2)) +TABLE-

INPUT ( 2 , 1 )*TABLE-INPUT (3,1) +TABLE-INPUT (2,2 )* TABLE-

INPUT(3,2)
DISC*-SQRT(R**2-4.0*U*SAL)
X01*-(-R+DISC)/(2.0*U)
X02 — (-R-DISC)/(2.0*U)
Y01*-(C*X01+DAO
Y02 «-C*X02+DA0
END IF
ELSE

E*- TABLE-INPUT (1,2) -TABLE-INPUT (3, 2)
F <-TABLE-INPUT (3,1) -TABLE-INPUT (1,1)
G <~ TABLE-INPUT ( 1 , 1 )*TABLE-INPUT( 2 , 1 ) +TABLE-

INPUT ( 1 , 2 ) STABLE- INPUT (2,2) -TABLE-INPUT ( 2 , 2 ) *TA3LE-

INPUT (3,2) -TABLE-INPUT (2,1 )*TABLE-INPUT (3,1)
IF F=0.0 THEN

DAO«-G/E

Y01«-DAO

Y02*-DAO

X01 4- TABLE-INPUT (1,1)

X02 «- TABLE-INPUT (1,1)
ELSE IF E=0.0 THEN

H*-(-G/F)

X01*-H

X02*-H

YOl +- TABLE-INPUT (1,2)

Y02 ^-TABLE-INPUT (1,2)
ELSE

C^F/E

DAO *-G/E

U«- C**2+1.0

R — 2 . 0*C*DAO-C* (TABLE-INPUT (2,2) +TABLE-INPUT (3,2))-
TABLE-INPUT( 2,1) -TABLE-INPUT (3,1)

SAL *-DA0**2-DA0* (TABLE-INPUT (2, 2 +TABLE-
INPUT (3,2)) +TA3LE-INPUT ( 2 , 2 ) -TABLE-
INPUT (3 , 2 ) +TA3LE-INPUT (2,1 )*TABLE-INPUT (3,1)

DISC *-SQRT (R**2-4 . 0*U*SAL)

X01*-(-R+DISC)/(2.0*U)

X02*-(-R-DISC)/(2.0*U)

YOl «-C*X01+DAO

Y02«-C*X02+DAO
END IF
END IF

ALGORITHM SELECTION (TABLE-INPUT ,X01 ,X02 , YOl ,Y02 )
IF TABLE-INPUT(l,3)?i(PI/2.0) THEN

VALOR 1*-((TABLE-INPUT(2,1)-X01)*(TABLE-INPUT(1,2)-
Y01)-(TABLE-INPUT(1,1)-X01)*(TABLE-INPUT(2,2)-

139

Y01 ) ) / ( (TABLE-INPUT (2,2) -YOl )* (TABLE-INPUT (1,2)-
Y01)+(TABLE-INPUT( 2, 1)-X01)* (TABLE-INPUT (1,1)-
X01))
VALOR 2 *-((TABLE-INPUT(2,l)-X02)*(TABLE-INPUT(l,2)-
Y02 )- (TABLE-INPUT (1,1 )-X02)* (TABLE-INPUT (2,2)-
Y02 ) ) / ( (TABLE-INPUT (2,2) -Y02 )* (TABLE-INPUT (1,2)-
Y02 )+ (TABLE-INPUT ( 2, 1)-X02)* (TABLE-INPUT (1,1)
-X02))
MODUL 1*-ABS (TANGENT (TABLE-INPUT (1, 3 ))-VALOR 1)
MODUL 2 <^ABS (TANGENT (TABLE-INPUT (1, 3 )) -VALOR 2)
IF MODUL 1 < MODUL 2 THEN
XO<-X01
YO<-Y01
ELSE

XO «-X02
YO *-Y02
END IF
ELSE IF TABLE-INPUT(2,3)^(PI/2.0) THEN

VALOR 1 «-((TABLE-INPUT(3,l)-X01)*(TABLE-INPUT(2,2)-
YO 1)- (TABLE-INPUT (2,1 )-X01)* (TABLE-INPUT (3,2)-
YO 1 ) ) / ( (TABLE-INPUT ( 3 , 2 ) -YO 1 )* (TABLE-INPUT (2,2)-
YO 1 ) + (TABLE-INPUT ( 3 , 1 ) -XO 1 )*TABLE-INPUT ( 2 , 1 ) -XO 1 ) )
VALOR 2 «- ((TABLE-INPUT(3,1)-X02)*(TABLE-INPUT(2,2)-
Y02 )- (TABLE-INPUT (2,1 )-X02)* (TABLE-INPUT (3,2)-
Y02 ) ) / ( (TABLE-INPUT ( 3 , 2 ) -Y02 )* (TABLE-INPUT (2,2)-
Y02 ) + (TABLE-INPUT (3,1) -X02 )* (TABLE-INPUT ( 2 , 1 )-X02 ) )
MODUL 1 *-ABS (TANGENT (TABLE-INPUT (2, 3 )) -VALOR 1)
MODUL 2 *-ABS (TANGENT (TABLE-INPUT (2,3) )-VALOR 2)
IF MODUL 1 < MODUL 2 THEN
XO+-X01
YO «-Y01
ELSE

XO *-X02
YO «-Y02
END IF
ELSE

VALOR 1 *-(TABLE-INPUT(2,2)-Y01)*(TABLE-INPUT(l,2)-

YO 1 ) + (TABLE-INPUT (2,1) -XO 1 )* (TABLE-INPUT (1,1) -XO 1 )
VALOR 2 *-(TABLE-INPUT(2,2)-Y02)*(TABLE-INPUT(l,2)-

Y02)+ (TABLE-INPUT ( 2, 1)-X02)* (TABLE-INPUT (1,1)-X02)
MODUL l*-ABSr VALOR 1)
MODUL 2 *-ABS (VALOR 2)
IF MODUL 1 < MODUL 2 THEN
X0*-X01
YO^YOl
ELSE

XO *-X02
YO *-Y02
END IF
END IF

140

END SELECTION (XO,YO)
END INITIAL-COORDINATES (XO,YO)

MODULE 36
ALGORITHM

TABLE-INPUT (1
TABLE-INPUT(1
TABLE-INPUT (1
TABLE-INPUT (2
TABLE-INPUT (2
TABLE-INPUT (2
TABLE-INPUT (3
TABLE-INPUT (3
TABLE-INPUT (I
TABLE-INPUT(I
TABLE-INPUT (I
TABLE-INPUT (J
TABLE-INPUT (J
TABLE-INPUT (J
TABLE-INPUT (K
TABLE-INPUT (K

RESTORE-INITIAL-DATA(TABLE-INPUT, STORE)

STORE(l)
STORE(2)
<r-STORE(3)
*-ST0RE(4)
*-STORE(5)
*-STORE(6)
*- STORE (7)
*- STORE (8)
*-STORE(9)
*- STORE(IO)
*-STORE(ll)
«-STORE(12)
*-STORE(13)
*-STORE(14)
*-STORE(15)
*-STORE(16)

END RESTORE-INITIAL-DATA(TABLE-INPUT)

MODUDULE 37

ALGORITHM INITIAL-AZIMUTHS (XO , YO , TABLE-INPUT ,M)
DO FOR I«-l TO M

IF YO=TABLE-INPUT(I,2) AND XO > TABLE- INPUT(I ,1 ) THEN

LIST-AZ(I) <-(3.0*PI)/2.0
ELSE IF YO=TABLE-INPUT(I,2) AND XO < TABLE-INPUT (I , 1 ) THEN

LIST-AZ (I )<*- PI /2.0
ELSE

LIST-ALFA(I)*-ARC TANGENT ( (TABLE-INPUT(I , 1 )-XO) /

(TABLE-INPUT (1,2) -YO ) )
IF LIST-ALFA(I) ^0.0 AND XO < TABLE-INPUT(I , 1 ) THEN
LIST-AZ(I)<-LIST-ALFA(I)
ELSE IF LIST-ALFA(I) < 0.0 AND XO > TABLE-INPUT(I . 1 ) THEN

LIST-AZ(I)*-LIST-ALFA(I)+2.0*PI
ELSE

LIST-AZ(I) «-LIST-ALFA(I)+PI
END IF
END IF
END DO
END INITIAL-AZIMUTHS (LIST-AZ)

MODULE 38

ALGORITHM MATRIX-L (TABLE-INPUT (1,3), LIST-AZ , N )
ALGORITHM ZERO(LIST-L)
DO FOR I*-l TO 10
LIST-L(I) 4-0.0
END DO

141

END ZERO(LIST-L)
DO FOR J«-l TO N

J*-I+l

LIST-L ( I ) ^-TABLE-INPUT (1,3) +LIST-AZ ( I ) -LIST-AZ (J )
END DO
END MATRIX-L(LIST-L)

MODULE 39

ALGORITHM SQUARED-DISTANCES (TABLE-INPUT , XO , YO )
DO FOR I*-l TO M

LIST-SO(I) «-(TABLE-INPUT(I,l)-XO)**2+(TABLE-INPUT(I,2)
YO)**2
END DO
END SQUARED-DISTANCES (LIST-SO)

MODULE 40

ALGORITHM MATRIX-A(N , TABLE-INPUT ,XO ,YO , LIST-SO )
ALGORITHM ZERO(TABLE-A)
DO FOR I«-l TO 10
DO FOR J*-l TO 2

TABLE-A(I,J)«-0.0
END DO
END DO
END ZERO(TABLE-A)
DO FOR I *-l TO N
J*-I+l

TABLE-A(I,1) -<r-(YO-TABLE-INPUT(J}2))/LIST-SO(J)-
(YO-TABLE-INPUT (I , 2 ) ) /LIST-SO (I )
END DO

DO FOR I*-l TO N
J*-I+l

TABLE-A(I,2) *-(YO-TABLE-INPUT(1 , 1 ) ) /LIST-SO (I )-
(XO-TABLE-INPUT(J,l))/LIST-SO(J)
END DO
END MATRIX-A(TABLE-A)

142

MODULE 50

ALGORITHM FIX-BY -TWO-RANGES-AND-ONE-AZIMUTH
N <-3

PI «-3. 141592653589793
DO FOR I*-l TO 3

INPUT TABLE-INPUT(I,1) ,TABLE-INPUT(I,2) ,TABLE-INPUT(I,4)
END DO
DO FOR I *- 1 TO 2

OUTPUT ' ST# ' , I , ' EAST= ' , TABLE-INPUT (1,1),' NORT= ' , TABLE-
INPUT(I,2), 'ST ERROR=' ,TABLE-INPUT(I ,4) , 'METERS'
END DO
OUTPUT 'ST#3EAST=' ,TABLE-INPUT(3 , 1 ) , 'NORT=' ,TA3LE-

INPUT(3,2) , 'ST ERROR=' ,TABLE-INPUT(3 ,4) , 'DEGREES'
ALGORITHM CONVERSION-DEGREES-RADIANS (TABLE-INPUT (3,4))

TABLE-INPUT (3, 4) <- TABLE-INPUT (3 ,4) *(PI/ 180.0)
END CONVERSION-DEGREES-RADIANS(TABLE-INPUT(3,4))
INPUT TABLE-INPUT(1,3),TABLE-INPUT(2,3),TABLE-INPUT(3,3)
DO WHILE TABLE-INPUT (1 ,3)^0.0

OUTPUT 'OBSERVED RANGE DISTANCES AND AZIMUTH ANGLE'

0UTPUT'R1=* ,TABLE-INPUT(1,3) , 'METERS'

OUTPUT' R2=* , TABLE-INPUT (2, 3) , 'METERS'

OUTPUT 'A=' , TABLE-INPUT (3, 3) , 'DEGREES'

ALGORITHM CONVERSION-DEGREES-RADIANS (TABLE-INPUT (3,3))

TABLE-INPUT (3,3)*- TABLE-INPUT (3 , 3 ) * (PI / 180 . 0 )
END CONVERSION-DEGREES-RADIANS(TABLE-INPUT(3,3))
MODULES 51,52,5,53,7

INPUT TABLE-INPUT (1,3), TABLE-INPUT (2,3), TABLE-INPUT (3,3)
END DO
END FIX-BY-TWO-RANGES-AND-ONE-AZIMUTH

MODULE 51

ALGORITHM FIRST-INITIAL-POINT (TABLE-INPUT)

E *- TABLE-INPUT (1,3 )**2-TABLE-INPUT (2,3 )**2+
TABLE-INPUT ( 2 , 2 )**2-TABLE-INPUT ( 1 , 2)**2+
TABLE-INPUT (2,1 )**2-TABLE-INPUT ( 1 , 1 )**2
IF TABLE-INPUT (2,1) ^TABLE-INPUT (1,1) THEN
El *-( (TABLE-INPUT (1,2) -TABLE-INPUT (2, 2))/

(TABLE-INPUT (2 ,1 )-TABLfi-INPUT(l , 1 ; ) )**2+l . 0
E2 «- ( E* (TABLE-INPUT (1,2) -TABLE-INPUT (2,2)))/
( (TABLE-INPUT ( 2 , 1 ) -TABLE-INPUT (1,1) )**2 ) -
2 . 0*TABLE-INPUT ( 1 , 1)* ( (TABLE-INPUT (1,2) -TABLE-
INPUT(2,2))/(TABLE-INPUT(2,l)-TABLE-INPUT(l,l)))-2.0*
TABLE-INPUT (1,2)
E3 «- (E/ (2 . 0* (TABLE-INPUT (2,1) -TABLE-INPUT (1,1))) )**2-
( EATABLE-INPUT (1,1)) / (TABLE-INPUT (2,1) -TABLE-
INPUT (1,1)) -TABLE-INPUT ( 1 , 3 )**2+TABLE-INPUT ( 1 , 1)**2+
TABLE-INPUT (1,2 )**2

143

E4 «- E2**2-4 . 0*E1*E3
IF E4 < 0.0 THEN

XO «- TABLE-INPUT (1,1) +TABLE-INPUT ( 1 , 3 )* (TABLE-INPUT (2,1)
TABLE-INPUT (1,1))/ SQRT ( ( TABLE-INPUT (2,1)-
TABLE-INPUT (1,1 )**2+(TABLE-INPUT (2,2)-
TABLE-INPUT (1,2) )**2)
YO *- TABLE-INPUT (1,2) +TABLE-INPUT ( 1 , 3 )* (TABLE-INPUT (2,2)
TABLE-INPUT (1,2))/ SQRT ( ( TABLE-INPUT (2,1)-
TABLE-INPUT(1,1))**2+(TABLE-INPUT(2,2)-
TABLE-INPUT (1,2) )**2 )
ELSE E4=0.0 THEN
YO«--E2/(2.0*E1)

XO «- E / ( 2 . 0* (TABLE-INPUT (2,1) -TABLE-INPUT (1,1)))+
YO* (TABLE-INPUT (1,2) -TABLE-INPUT (2,2))/
(TABLE-INPUT (2,1) -TABLE-INPUT (1,1))
ELSE

Y01<-(-E2+SQRT(E4))/2.0*El)

X01 <- E/ ( 2 . 0* (TABLE-INPUT (2,1) -TABLE-INPUT (1,1))) +
YOl* (TABLE-INPUT (1,2) -TABLE-INPUT (2,2))/
(TABLE-INPUT (2,1) -TABLE-INPUT (1,1))
YO2<-(-E2-SQRT(E4))/(2.0*El)

X02 -e E/ ( 2 . 0* ( TABLE-INPUT (2,1) -TABLE-INPUT (1,1))) +
Y02* (TABLE-INPUT (1,2) -TABLE-INPUT (2,2))/
(TABLE-INPUT (2,1) -TABLE-INPUT (1,1))
IF TABLE-INPUT(3,1)=X01 AND TABLE-INPUT(3 , 2 )=Y01 THEN
XO«-X02
YO«-Y02
ELSE IF TABLE-INPUT(3,1)=X02 AND TABLE-INPUT(3 , 2 )=
Y02 THEN
XO*-X01
YO*-Y01
ELSE

CALL CRITERIUM(TABLE-INPUT (3,1), TABLE-INPUT (3,2),

X01,Y01,A301)
CALL CRITERIUM(TABLE-INPUT(3,1) ,TABLE-INPUT(3 ,2) ,

XO2,YO2,A302)
IF A301=TABLE-INPUT(3,3) AND A301=A302 THEN

OUTPUT' SOLUTION UNDETERMINED FOR THAT DATA SET'
PICK UP ANOTHER DATA SET
ELSE IF ((A301-TABLE-INPUT(3,3))**2)=

( ( A302-TABLE-INPUT (3,3) )**2) THEN
XO*-(XQ1+X02)/2.0
YO*-(Y01+Y02)/2.0
ELSE IF ((A301-TABLE-INPUT(3,3))**2) >

( (A302-TABLE-INPUT (3,3) )**2) THEN
XO *-X02
YO *Y02

144

ELSE IF ((A30l-TABLE-INPUT(3,3))**2) <

((A302-TABLE-INPUT(3,3))**2) THEN
XO+-X01
YO «-Y01
END IF
END IF
END IF
ELSE

YO *- E / ( 2 . 0* (TABLE-INPUT (2,2) -TABLE-INPUT (1,2)))
F *-TABLE-INPUT(l,3)**2-(TABLE-INPUT(l,2)-YO)**2
IF F $0.0 THEN

XO <- TABLE-INPUT (1,1)
ELSE

X01 «-TABLE-INPUT(l , 1)+SQRT(F)

Y01 «^Y0

X02 *-TABLE-INPUT (1,1) -SQRT (F )

Y02 *-Y0

IF TABLE-INPUT(3,1)=X01 AND TABLE-INPUT (3 , 2)=Y01 THEN

XO *-X02
ELSE IF TABLE-INPUT ( 3, 1)=X02 AND TABLE-INPUT(3 , 2 )=
Y02 THEN
XO^-XOl
ELSE

CALL CRITERIUM(TABLE-INPUT(3,1) ,TABLE-INPUT(3 , 2) ,

X01,Y01,A301)
CALL CRITERIUM(TABLE-INPUT(3,1) ,TABLE-INPUT(3 , 2) ,

X02,YO2,A302)
IF A301=TABLE-INPUT(3,3) AND A301=A302 THEN

OUTPUT' SOLUTION IS UNDETERMINED FOR THAT DATA SET
PICK UP ANOTHER DATA SET
ELSE IF ((A301-TABLE-INPUT(3,3))**2)=

( ( A302-TABLE-INPUT (3,3) )**2) THEN

XO«r-XOl

ELSE IF ((A301-TABLE-INPUT(3,3))**2) >

((A302-TABLE-INPUT(3,3))**2)THEN
X0*-X02

ELSE IF ((A301-TABLE-INPUT(3,3))** 2) <

( (A302-TABLE-INPUT (3,3) )**2 ) THEN
X0<-X01

END IF
END IF
END IF
END IF
END FIRST INITIAL-POINT(XO,YO)

MODULE 52

ALGORITHM ITERATIONS (TOLERANCE)

DO UNTIL TOLERANCE < 1.0

MODULES 54,55,56,5 7,58,2,59,12,13,14

END DO
END ITERATIONS (XO.YO)

145

MODULE 53

ALGORITHM PRECISION (TABLE-A, TABLE-WEIGHT, TABLE-Q,LIST-L,DELTA-X,

DELTA Y,N,S30)
MODULES 18,19,60,21,22
END PRECISION (SU , SX , SY , SXY ,RO)

MODULE 54

ALGORITHM A30 (TABLE-INPUT, XO,YO)

CALL CRITERIUM (TABLE-INPUT (3,1), TABLE-INPUT ( 3 , 2 ) , XO , YO , A30 )
END A30(A30)

MODULE 55

ALGORITHM DISTANCES (TABLE-INPUT ,X0 ,Y0)

S 10 *-SQRT ( (TABLE-INPUT (1,1) -XO )** 2 + (TABLE-INPUT (1,2)-

Y0)**2)
S20 *-SQRT((TABLE-INPUT(2,l)-XO)**2+(TABLE-INPUT(2,2)-

Y0)**2)
S30 «- SQRT ( ( TABLE-INPUT (3,1) -XO )**2 + (TABLE-INPUT (3,2)-
Y0)**2)
END DISTANCES

MODULE 56

ALGORITHM MATRIX-A (TABLE-INPUT ,X0 , YO , S10 , S20 , S30 ,A30)
TABLE-A (1,1)*- (XO-TABLE-INPUT ( 1 , 1 ) ) / S 10
TABLE-A( 1 , 2 ) <- ( YO-TABLE-INPUT ( 1 , 2 ) ) / S 10
TABLE-A (2,1)*- (XO-TABLE-INPUT ( 2 , 1 ) ) / S20
TABLE-A( 2 , 2 ) —(YO-TABLE-INPUT ( 2 , 2 ) ) /S20
TABLE-A(3, 1 ) *- COSINE(A30-TABLE-INPUT(3 , 3))* (YO-
TABLE-INPUT (3 , 2 ) ) /S30+SINE (A30-TABLE-INPUT (3 , 3 ) )*
(XO-TABLE-INPUT (3 , 1 ) ) /S30
TABLE-A( 3,2) -r-COSINE(A30-TABLE-INPUT(3 , 3))* (TABLE-
INPUT (3,1) -X0)/S30+SINE (A30-TABLE-INPUT( 3,3))* (YO-
TABLE-INPUT (3 , 2 ) ) ) /S30
END MATRIX-A(TABLE-A)

MODULE 57

ALGORITHM LIST-L(TABLE-INPUT , S10 , S20 , S30 , A30 )

LIST-L(l)*- TABLE-INPUT (1, 3 )-SlO

LIST-L(2)<-TABLE-INPUT(2,3)-S20

LIST-L(3)*~SINE(TABLE-INPUT(3,3)-A30)*S30
END LIST-L(LIST-L)

MODULE 58

ALGORITHM BEFORE-WEIGHT-MATRIX(TABLE-INPUT (3 , 4) ,S30)

TABLE-INPUT (3, 4) *-S30*SINE (TABLE-INPUT (3, 4))
END BEFORE-WEIGHT-MATRIX(TABLE-INPUT(3,4))

146

MODULE 59

ALGORITHM AFTER-WEIGHT-MATRIX (TABLE-INPUT (3 , 4) , S30)

TABLE-INPUT (3, 4)*" ARC SINE (TABLE-INPUT (3 ,4) /S30)
END AFTER-WEIGHT-MATRIX(TABLE-INPUT(3,4))

MODULE 60

ALGORITHM ST-DEVIATION-OF-EACH-OBS ( SU , TABLE-WEIGHT , S30 )

OUTPUT' PRECISION OF OBSERVATIONS'

DO FOR I«-l TO 2

S*-(SU/SQRT (TABLE-WEIGHT (I, J)))

OUTPUT' ST DEVIATION OF OBS ', I ,'=', S , 'METERS '

END DO

S *- ARC SINE ( SU/ ( SQRT (TABLE-WEIGHT (3,3) )*S30 ) )* ( 180 . 0/PI )

OUTPUT 'ST DEVIATION OF OBS 3= ' , S , ' DEGREES '
END ST-DEVIATION-OF-EACH-OBS

MODULE 70

SUBROUTINE CRITERIUM(XS ,YS,XP,YP,ASP)
PI -3. 14159 26535 89793
IF YP=YS AND XP > XS THEN

ASP*- PI/2.0
ELSE IF YP=YS AND XP < XS THEN

ASP*-3.0*PI/2.0
ELSE

ALFA «- ARC TANGENT ( (XP-XS) /YP-YS) )
IF ALFA > 0.0 AND XP >/ XS THEN

ASP*- ALFA
ELSE IF ALFA < 0.0 AND XP < XS THEN

ASP«-ALFA+2.0*PI
ELSE

ASP«e-ALFA+PI
END IF
END IF
RETURN
END CRITERIUM

147

LIBER OF STATIONS= 3
V- 1 EAST = 595794.50

It 2 EAST= 597967.30
\i 3 EAST= 603425.20

NORT=

4055042.70

ST

ERROR= 0.020C

NCRT=

-♦053453.20

ST

ERRQR= 0.0240

NORT=

4053917.20

ST

£RROR= 0.0180

JIRVED AZIMUTHS

JMUTH FRGM STATION* 1 = 76.017 DEGREES

i.MUTH FRCM STATION* 2 = 45.541 DEGREES

ZjMUTH FRCM STATION* 3 =313.005 DEGREES

JSTED COORDINATES X= o00363.306 Y = 4056302.731

VISION OF OBSERVATIONS
I DEVIATION OF OBS 1 =0.031 DEGREES
{DEVIATION OF CBS 2 =0.038 DEGREES
"DEVIATION OF OSS 3 =0.023 DEGREES
!= 2.13 SY= 1.70 SXY= -0.362
■tRELATICN COEFFICIENT RC=-.24

13 R ELIPSE SEMI-AXIS AND ORIENTATION

II-MAJCR AXIS SA= 2.283

Il-MINCR AXIS SB= 1.636

LE FRCM X-AXIS TO SA ANT ICLOCKW ISE = 157 . ODEG

148

HBER CF SEXTANT ANGLES= 3

'4 1 EAST= 603425.20 N0RT= 4053917.20 ST ERR0R= i.0000

\t 2 EAST= 600372.00 NORT= 4051216.90 ST ERROR* 1.0000

t 3 EAST= 597967.30 NORT= 4053453.20 ST ERROR= 1.0000

'4 4 EAST= 595794.50 NORT= 4055042.70

SRVED SEXTANT ANGLES

KTANT ANGLE BETWEEN ST# 1 AND ST# 2 = 49.927 DEGREES

IKTANT ANGLE BETWEEN ST# 2 AND ST# 3 = 38.130 DEGREES

KTANT ANGLE 8ETWEEN ST# 3 AND ST# 4 = 30.396 DEGREES

JSTED CCCRDINATES X= 600864.536 Y = 4056512.323

"ISION CF G8SERVATIQNS

DEVIATION CF CBS 1 =0.007 DEGREES

DEVIATION CF CBS 2 =0.007 DEGREES

DEVIATION CF CJBS 3 =0.007 DEGREES

= 1.02 SY= 0.48 SXY= -0.097

RRELATICN CCEFFICIENT RQ=-.20

bft ELIPSE SEMI-AXIS AND ORIENTATION

il-MAJCP AXIS SA= 1.050

II-MINOR AXIS S3= 0.430

LE FRCM X-AXIS TO SA ANT ICLOC KW I S E = 173 . 3DEG

149

t 1

EAST =

595794.50

NORT=

4055042.70

i 2

EAST =

603425.20

.MORT =

4053917.20

i 3

EAST =

597967.80

N0RT=

4053453.20

ST £RR0R= 10. 000 METERS
ST ERROR= 10.000 METERS
ST ERROR* 0.024 DEGREES

iRVED RANGE DISTANCES AND AZIMUTH ANGLE

• 5233.00 METERS

• 3515. CO METERS

45.54 DEGREES

JSTED CCGRDINATES X= 600672.166 Y= 4056304.120

VISION CF CBSERVATIONS
DEVIATION CF OBS 1 = 3.45 METERS
DEVIATICN CF OBS 2 = 3.^5 METERS
DEVIATION OF CBS 3 = 0.008 DEGREES
= 2.86 SY= 2.88 SXY= 7.911
IRRELATICN CCEFFICIENT RC= 0.96

ROR ELIFSE SEMI-AXIS AND ORIENTATION

MI-MAJOP AXIS SA=14.265

MI-MINOF AXIS S3= 2.051

GLE FRCM X-AXIS TO SA ANTICLOCKW I SE= 45.2DEG

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LIST OF REFERENCES

1. Ewing, C. E. and Mitchell, M. M. , Introduction to Geodesy ,
Elsevier, 1976.

2. Wolf, P. R. , Elements of Photogrammetry , p. 513, McGraw-
Hill, 1974.

183

BIBLIOGRAPHY

Freund and Walpole, Mathematical Statistics , 1962.

Graham, N. , Introduction to Computer Science , A Structured
Approach, 1979.

Kaplan, A. , Error Analysis of Hydrographic Positioning and
the Application of Least Squares, M. S. Thesis, Naval Post-
graduate School, 1980.

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

134

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190

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