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Theses and Dissertations Thesis Collection

1984

A comparison of methods of least squares

adjustment of traverses

Aumchantr, Saman

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/19196

KNOX LIBRARY

*3f

NAVAL POSTGRADUATE SCHOOL

Monterey, California

T SISA Comparison of Methods

of

Least Squares Adjustment of Traverses

by

Saman Aumchantr

December 1984

Thesis Advisor: Rolland L. Hardy

Approved for public release; distribution unlimited

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A Comparison of Methods of Least Squares

Adjustment of Traverses

S. TYPE OF REPORT 4 PERIOD COVEREDMaster's ThesisDecember 1984

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Saman Aumchantr

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Naval Postgraduate School

Monterey, California 93943

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19. KEY WORDS (Continue on reverse aide If neceaaary and Identity by block number)

Adjustment, Approximate Method, Comparison, Condition Equation Method,

Least Squares Method, Traverse, UTM grid coordinates.

20. ABSTRACT (Continue on reverse aide If neceae-ary and Identity by block number)

Traverse is a method of surveyinq in which a sequence of lenqths and

directions of lines between points on the Earth are measured and used in

determining positions of the points. This method is one of several used to

find the accurate geodetic positions which various agencies use. Traversingis a convenient, rapid method for establishina horizontal control.

The theoretical background is provided here to explain the method of

traverse station position computations and adjustments in the Universal

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Transverse Mercator grid coordinates. Closed traverse station positionswere computed and adjusted using the Approximate Method and by the LeastSquares Method. The adjusted coordinates of both methods were transformedfrom the Universal Transverse Mercator grid coordinates to geoqraphiccoordinates and compared with the coordinates which were adjusted by the U.S.National Ocean Service.

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A Comparison of Methodsof

Least Squares Adjustment of Traverses

by

Saman AumchantrLieutenant, Royal Thai Navy

B.S., Royal Thai Naval Academy, 1976

Submitted in partial fulfillment of therequirements for the degree of

MASTER OF SCIENCE IN HYDROGRAPHIC SCIENCES

from the

NAVAL POSTGRADUATE SCHOOLDecember 1984

4 9C2

£ ' I OUDLEY KNOX LIBRARY

NAVAL POSTGRADUATE SCHOOL

MONTEREY, CALIFORNIA 93943

ABSTRACT

Traverse is a method of surveying in which a sequence of

lengths and directions of lines tetween points on the Earth

are measured and used in determining positions of the

points. This method is one of several used to find the

accurate geodetic positions which various agencies use.

Traversing is a convenient, rapid method for establishing

horizontal control.

The theoretical background is provided here to explain

the method of traverse station position computations and

adjustments in the Universal Transverse Kercator grid coor-

dinates. Closed traverse station positions were computed

and adjusted using the Approximate Method and by the Least

Squares Method. The adjusted coordinates of both methods

were transformed from the dniversai Transverse Mercator grid

coordinates to geographic coordinates and compared with the

coordinates which were adjusted by the U.S. National Ocean

Service.

DUDLEY KNOX LIBRARYNAVAL F03T IOOLMONTEREY, C."

TABLE OF CONTENTS

I. INTRODUCTION 10

II. TRAVERSE . . . 12

A. GENERAL 12

B. ANGLE AND DIRECTION MEASUREMENT 13

C. DISTANCE MEASUREMENTS 15

D. ACCURACY 16

E. DATA ACQUISITION 16

III. TRAVERSE COMPUTATIONS AND ADJUSTMENTS 2 1

A. INITIAL TRAVERSE COMPUTATIONS 21

5. COMPUTATION OF DISCREPANCIES 23

C. ADJUSTMENT OF A TRAVERSE BY AN APPROXIMATE

METHOD 2 6

D. ADJUSTMENT OF A TRAVERSE BY LEAST SQUARES

METHODS 28

1. The Principle of Least Squares 28

2. Least Squares Adjustment of Indirect

Observations 31

3. Least Squares Adjustment by the

Condition Equation Method 47

IV. DISCUSSIONS AND ANALYSIS OF RESULTS 59

V. CONCLUSIONS AND RECOMMENDATION 66

A. CONCLUSIONS 66

B. RECOMMENDATION : . . 67

APPENDIX A: PROGRAM 1 63

APPENDIX E: PROGRAM 2 89

APPENDIX C: PROGRAM 3 121

LIST OF REFERENCES 1U3

BIBLIOGRAPHY 149

INITIAL DISTRIBUTION LIST 150

LIST OF TABLES

II.

III.

IV.

V.

VI.

VII.

VIII

IX.

X.

XI. •

XII.

Classification, Standards of Accuracy, and

General Specifications for Horizontal Control ... 17

Coordinates of Known Stations and Azimuths .... 19

Grid Distances and Standard Deviations 20

The Observed Angles and Standard Deviations .... 20

Data for Iritial Traverse Computations 23

The Azimuth Calculation 24

Initial Traverse Computations 25

Initial Traverse Computations (Approximate

Method with Adjusted Azimuth) 27

Adjusted Coordinates (Approximate Method) 29

The Comparisons Between the Computer Storage

Area and CPU Time of Programs 1 , 2, and 3 60

Geographic Coordinates 64

The Comparisons Between the Standard Deviations

of Least Squares 65

LIST OF FIGQRES

2. 1 Closed Traverse 12

2.2 Closed-loop Traverse 13

2.3 Horizontal Angle 15

3. 1 A Closed Traverse 22

ACKNOWLEDGEMENTS

I express sincere gratitude to my Thesis Advisor, Dr.

Rolland L. Hardy, and Second Reader, Cdr. Glen R. Schaefer,

foe their suggestions and assistance. Finally, I thank Ms.

Tamara M. Hayling, Lt. Nicholas E. Pengini, and Messrs.

Mark I. Faye, Peter J. Rakowsky, and James R. Sherry who

made this thesis possible.

I. INTRODUCTION

Hydrographic surveying includes many branches of science

for the purpose of the production of nautical charts

specially designed for use by the mariner. The determina-

tion of position in hydrographic surveying is as important

as the measurement of depth. Before determining an accurate

hydrographic position, accurate geodetic positions 1 for

shore control must be established. There are many methods

available to establish geodetic control in the survey area.

These methods are:

1. Triangulation

2. Trilateration

3. Traverse

4. Intersection

5. Resection

The process of making a proper nautical chart consists,

first of all, in setting up a framework of marks on the

ground. Before 1950 the main framework, of a geodetic survey

almost always consisted of triangulation, which was replaced

by traverse if the topography made triangulation impracti-

cable. During the last decade, the introduction of elec-

tronic distance measuring (EDM) equipment has made both

trilateration and traverse economical, and an acceptable

substitute for triangulation. In fact, it appears probable

that these new methods will replace triangulation as the

main framework for new geodetic surveys [Ref. 1], It is

evident that triangulation and traverse are the main methods

used for establishing control. There seems to be no

X A position of a point on the surface of the Earthexpressed in terms of geodetic latitude, geodetic longitude-ana geodetic height. A geodetic position implies an adoptedgeodetic datum.

10

agreement among the various agencies as to which of these

two methods is mostly used. The U.S. National Ocean Service

(NOS) does the majority of its horizontal control surveys

for hydrography with traverse (about 90^) [Eef. 2]. The

main factor for the selection of one or the other method

depends on the geographical configuration of the survey

project area and the availability of good EDM equipment

.

Traversing is a convenient, rapid method for establishing

horizontal control. It is particularly useful in densely

built areas, when the coastline tends to be even, along a

railroad track, and in heavily forested areas where lengths

of sight are short so that neither triangula tion nor

trilateration is suitable.

The objectives of this thesis are to show methods of

traverse station position computation and a comparison of

methods of least squares adjustment. Closed traverse

station positions were computed and adjusted using the

Approximate Method and by the Least Squares ilethod in the

Universal Transverse Hercator (UTM) grid coordinates. The

computer programs were written to calculate the adjusted

traverse station positions. Test data included those data

obtained during the Geodetic Survey Field Experience course

at the Naval Postgraduate School (NPS) in October 1933.

The results of comparative computations are shown to

more significant figures, in this thesis, than are normally

considered desirable in production work. The same observed

data are used with each of three computational methods. It

is important to recognize that what is being compared is not

observational precision but computational precision. Hence,

it is considered necessary for a rigorous comparison of

computational precision, including round off error, to show

results to several more decimal places than is justifiable

based on observational precision alone.

1 1

II. TRAVERSE

A. GENERAL

Traverse is a method of surveying in which a series of

straight lines connect successive established points along

the route of a survey. An angular measurement is taken

using a theodolite at each point where the traverse changes

direction. Distances along the line between successive

traverse points are determined by EDrf equipment. The points

defining the ends of the traverse lines are called turning

points, traverse points, or traverse stations. Each

straight section of a traverse is called a leg or a traverse

line.

A closed traverse originates at a point of known posi-

tion and is closed or. another point of known horizontal

position. Traverse I-1-2-3-F originates at point I with a

backsight along line IA of known azimuth and closes on point

F, with a foresight along line FB also of known azimuth

(Figure 2. 1) . This type of traverse is preferable to all

Figure 2. 1 Closed Traverse.

12

others since computational checks are possible which allow

detection of systematic errors 2 in both distance and

direction.

A closed-loop (closed polygon) traverse is a special

case of a closed traverse in which the originating and

terminating points are the same point with a known hori-

zontal position. Traverse I-1-2-3-4-I, originates and

terminates on point I (Figure 2.2). This type of traverse

permits an internal check on the angles, but there is no

check on the linear measurements. Therefore, there is a

possibility that an error proportional to distance may occur

and not be detected.

Figure 2.2 Closed-loop Traverse.

B. ANGLE AND DIRECTION MEASUREMENT

Angles and directions may be defined by means of bear-

ings, azimuths, deflection angles, angles to the right, or

interior angles. These quantities are said to be observed

2 Systematic errors may be caused by faulty instrumentsor factors such as temperature or humility changes whichaffect the performance of measuring instruments.

13

when obtained directly in the field and calculated when

obtained indirectly by computation.

A theodolite is an instrument designed to observe hori-

zontal directions and measure vertical angles. It consists

of a telescope mounted to rotate vertically on a horizontal

axis supported by a pair of vertical standards attached to a

revolvable circular plate containing a graduated circle for

observing horizontal directions. another graduated arc is

attached to one standard so that vertical angles can be

measured

.

Repeating and direction theodolites have features that

are common to both types of instruments. Repeating theodol-

ites are read directly to 20" or 01' and by estimation to

one-tenth the corresponding direct reading. Direction theo-

dolites are usually read directly to 01" and can be esti-

mated to tenths of seconds [Ref. 3, p. 215]. In general,

direction theodolites are more precise than are repeating

theodolites.

The direction theodolite observes directions only and

angles are computed by subtracting one direction from

another. Assume that the horizontal angle AI3 (Figure 2.3)

is to be measured with a direction theodolite. The theo-

dolite is set over point I, leveled and centered, and a

sight is taken on point A. The horizontal cirzle is then

viewed through the optical- vie wing system and the circle

reading is observed and recorded. Assume the reading is

450 02' 40". The telescope is then sighted on point B. The

horizontal circle is then viewed through the op tical- viewing

system and the circle reading is observed and recorded.

Assume the reading is 124° 11' 59". Ihese two observations

constitute directions which have a common reference direc-

tion that is completely arbitrary. The clockwise horizontal

angle is

i = (124° m 59") - (450 02' 40") = 79° 09' 19"

14

Figure 2-3 Horizontal Angle.

C. DISTANCE MEASUREMENTS

There are several methods of determining distance, the

choice of which depends on the accuracy required and the

cost. For example, tacheometry, taping, and ED.1 equipment

can be used. The general availability of EDM equipment has

practically eliminated the use of taping for the measurement

of traverse lengths. Accuracies are comparable, or

superior, to those obtained with invar tapes.

Distances measured using EDM equipment are subject to

errors arising from the instrumental components, calibration

of the equipment, inaccuracies in the meteorological data,

elevation discrepancies, and centering of the instruments or

reflectors. The reduction of measured distances involves

converting the slope distance to a horizontal distance,

converting the chord distance to an equivalent arc distance,

and reducing the arc distance to the ellipsoid. The reduc-

tion of slope distance to horizontal distance is necessary

to compensate for the difference in elevation of the end

points of the measured line. The horizontal distance is

15

reduced to tha geodetic distance ty applying a sea level

corrector and a chord-arc corrector to the horizontal

distance [Ref. 4, pp. 124-125].

D. ACCURACY

In general, the accuracy of a traverse is judged on the

basis of the resultant error of closure of the traverse.

This resultant closing error is a function of the accuracies

in the measurement of directions and distances. The classi-

fication, standards of accuracy, and general specifications

foe horizontal control have been prepared by the Federal

Geodetic Control Committee (FGCC) [ Ref . 5] and have been

reviewed by the American Society of Civil Engineers, The

American Congress on Surveying and Mapping, and the American

Geophysical Onion (Table I) . The third-order traverse class

I and class II are of particular interest because these are

the orders of accuracy the hydrographer is usually required

to accomplish.

E. DATA ACQUISITION

The data for this thesis were acquire! at Moss Landing,

California, by NPS Hydrographic students during the Geodetic

Survey Field Experience course in October 1983. All of the

known station positions and azimuths (Table II) were

adjusted by the Coast and Geodetic Survey by third-order

methods [Ref. 6 ].

The distances were observed by using a Kern DM102 and a

Tellurometer HRA5. The Kern DM102 is an electro-optical

distance meter. The measuring accuracy was ±(5 mm + 5 ppm) .

The Tellurometer MRA5 is a microwave instrument. Precision

in terms of probable error, in the temperature range of

-32°C to +U40C, is ± (5 cm + 100 ppm) for a single

16

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18

TABLE II

Coordinates of Known Stations and Azimuths

Station

Moss 2

HoIe

UTM grid coordinatesGrid Northing Gri.3 Easting

( m. ) ( m. )

4,072,555.35206

4,079,258.31754

Grid azimuth clockwisefrom North

From Moss 2 to Pipher stations

From Holm to Moran stations

608, 279.04404

612, 238.85256

Azimuth

o » ii

100 16 23.778

136 33 26.334

determination. The distances were observed in the field,

corrected by temperature and pressure for propagation error.

Unfortunately, meteorological data are generail/ accuired

only at the end points of a measured line. Using the mean

of these meteorological values only approximates the actual

conditions of the entire measured line and does not

completely correct for errors in the propagation velocit7 of

electromagnetic radiation. By applying the elevation, sea

level, and scale factor corrections they were reduced to UTM

grid distances [Ref. 4, pp. 124-125]. The UTM grid

distances and standard deviations of the distances were

determined in meters (Table III) .

The angles were observed by using a Wild T-2 theodolite.

To ensure the correctness of the beginning and ending

azimuths, a check azimuth to a second station of known posi-

tion was observed. The angles were observed at stations

Moss 2, Mossback, Dune Temp, and Holm (Table 17). All

observations were made by NPS students and conform to

specification for a third-order class I traverse.

19

TABLE III

Grid Distances and Standard Deviations

StandardGrid distances between Distances deviation

stations ( m.) ( m. )

Moss 2 and Mossback 1,424.004 0.001

Mossback and Dune Temp 365.744 0.001

Dune Temp and Holm 6,476.271 0.003

TABLE 17

The Observed Angles and Standard Deviations

Eacksight Center Foresight Observed Standard|

station station station angles deviationlI II it

Pipher Moss 2 Mossback 246 05 43.200 01.984

Moss 2 Mossback Duns Temp 222 51 08.600 01.405

Mossback Dune Temp Holm 190 15 02.600 01.203

Dune Temp Holm Moran 277 05 17.000 01.614

20

III. TRAVERSE COMPUTATIONS AND ADJUSTMENTS

A. INITIAL TRAVERSE COMPUTATIONS

A traverss (Figure 3.1) originates at station 1 (known

position) and terminates at station 4 (also known position)

(Table V). To compute the forward azimuth (Table VI) of an

unknown leg, the angle is added to the back azimuth of the

previous leg (Equation 3. 1) .

AZj = Az

Where

y dj - (i-1) 180'

j =1

the number of legs,

(3.1)

AZj = the forward azimuth of an unknown leg, and

Az = the fixed initial azimuth.

For the computation of UTM grid coordinates, let AEj

and ^N- be designated as the departure and latitude for leg

i (Figure 3.1) so that the general formulas to compute the

departure and latitude are

A E . = dj

s in A zj

AN. = d. cos Az.i i i

(3.2)

(3.3)

where i=1,2,3,...n,n = the number of the observed angles.

The algebraic signs of the departure and latitude for a

traverse leg depend on the signs of the sine and cosine of

the azimuth of that leg. The algebraic sign of the depar-

ture and latitude is determined by the following rules:

1. UTM grid azimuths are refered to grid north.

2. For azimuths between 0° and 180°, the departure is

plus; for all other azimuths, the departure is minus.

3. For azimuths between 90° and 270°, the latitude is

minus; for all other azimuths, the latitude is plus.

21

Figure 3. 1 A Closed Traverse.

The calculators and computers with internal routines for

trigonometric functions yield the proper sign automatically

given the azimuth. The major portion of traverse computa-

tions consists of calculating departures and latitudes for

successive legs and cumulating the values to determine the

coordinates for consecutive traverse stations.

22

Leg

1 - 2

2-33-4

TABLE V

Data for Initial Traverse Computations

Distance( m. )

= 1,424.004

365.744

= 6,476.271

Station 1

DTM grid Easting

<*1

ex

The initial UTM grid azimuth kzf

UTM grid Northing M1

Angle

246 05 43. 200

222 51 08.600

190 15 02. 600

= 100 16 23.778

= 4,072,555.85206 m

.

608,279.04404 m.

The coordinates of station j (j = i + 1) are E;

and

N. , the coordinates of station j are

E, = E

= H-

AE(

An.

(3.4)

(3.5)

where i = 1 , 2 ,3 ,

The values of departures, latitudes, and coordinates are

shown in Table VII and were computed from the data in Tables

V and VI by application of Equations (3.2) through (3.5).

B. COMPUTATION OF DISCREP& NCIES

A closed traverse at iloss Landing (Figure 3.1) origi-

nated at station 1 (Moss 2) and closed at station 4 (Holm).

The closing errors' for a traverse are caused by observation

errors in the observed angles and the measured distances.

The closing errors may be computed by applying Equations

(3. 1) through (3.6) .

23

TABLE VI

The Azimuth Calculation

1

ForwardAzimuth

BackAzimuth

ObservedAngle

i

i

ii

AzF

100 16 23.778

«i 246 05 43.200

Az1

346 22 06. 978

- 180 00 00.000

A22-1

166 22 06. 978

°2 222 51 08.600

Az2

389 13 15.578

- 130 00 00.000

AZ 3-2 209 13 15. 578

<*3 190 15 02.600

Az3

399 28 18. 178

- 180 00 00.000

AZ4-3

219 28 18. 178

i

The angular error of closure may be compute! by applying

Equation (3.1) becomes

Az r + 2oL- (n-1) 180° - Az, = H

I =1

(3.6)

where Az. is the fixed closing azimuth and ff-, is the

angular error of closure. Az is the grid azimuth from

stations Holm to Moran (AzL

= 136° 33' 26.334") (Table II).

The computed closing azimuth is equal to Az plus the

observed angle at station Holm = (219° 28' 18.178") +

(2770 05' 17") = 4960 33' 35.178" = 136° 33 f 35.178". From

Equation (3.6) , the angular error of closure is

(1360 33' 35.178") - (1360 33' 26.334") = + 08.844".

24

TABLE VII

Initial Traverse Computations

Aw.

An,

An.

Values in meters

4,072,555.85206

1,383.89267 A E,

4,073,939.74473

319.20061 AE2

4,074, 258.94534

4,999.28234 AE3

4,079, 258.22818

Values in meters

608,279.04404

- 335. 60166

607,943.44238

178.54871

608, 12 1. 99109

4, 1 16. 94755

612,238.93864

To compute the errors in closure in position for a

closed traverse. Equations (3.2) to (3.5) are applied to

obtain:

n-1

or

i ; 1

n-1

*3 " 2^ Ni

" ( NT

- N1

)

i =1

= E,

= N.

- E

- N

Where : n = the number of observed angles,

»2 , W

3= precalculat ed discrepancies,

E1

, N1

= the fixed initial coordinates,

Ej , N = the fixed closing coordinates, and

E n , N n= the computed closing coordinates.

The result of the errors in closure in position of a

traverse is

(3.7)

(3.8)

(3.7a)

(3.8a)

25

£d = ( W| + W| )W2 (3.9)

The computed closing coordinates of this traverse are

E„ = 612,238.93864 m and N. = 4,079,258.228 18 m. The fixed4 4

closing coordinates are ET

= 6 12,238.85256 m and N =

4,079,258.31754 m. The precalculated discrepancies can be

computed by using Equations (3.7a) and (3.8a). The results

are W"2

= 0.08608 m and W3

= -0.08936 m. By applying

Equation (3.9), Ad is 0.12408 m . From Table III, the total

measured distance is 8,266.019 m. The ratio of the distance

error of closure to the total distance is 0.12408/8,266.0 19

or 1 part in 66,618. The ratio is an indication of the

goodness of this traverse.

C. ADJUSTMENT OF A TRAVERSE BY AN APPROXIMATE METHOD

The angular error of closure of traverse may be distrib-

uted equally among the observed anjles. The angular error

of closure of this traverse is +8.844", which corresponds to

a correction for each angle of -2.211". The observed angles

were correctei by this value. The adjusted azimuth was then

computed. The departures, latitudes, and coordinates were

recomputed ty using the adjusted azimuth (Table VIII) . The

closure corrections in E and N coordinates are +-0. 09635 m

and -0.04326 m, respectively. The resultant closure is

0.10562 m. The ratio of the distance error of closure to

the total distance is 0.10562/8,266.0 19 or 1 pact in 78,262.

The adjustments of a closed traverse by the Approximate

Method are completed by the compass rule which proportions

the errors in E and N coordinates according to the distance

of the course [Ref. 4, p. 354]. The corrections are applied

to the departures and latitudes prior to computation of

coordinates.

26

TABLE VIII

Initial Traverse Computations (Approximate Method withAdjusted Azimuth)

CL

d

Ant

An2

An,

Observed angle Correction

246° 05' 43.2" - 2.211

2220 sit 08.6 - 2. 211

OU 1900 15' 02. 6" -2.211"

CL 2770 05' 17.0" - 2.211"

dN

Values in meters

4 ,072,555.85206

1,383.88907

4 ,073, 939.741 13

319.20444

4,074, 2S8-.94557

4,999.41523

4 ,079, 258.36080

4 ,079,258.31754

0.04326

Az,

Az

Az 2-1

Az.

Az 3-2

Az.

Az4-3

AZ4-L

Ae,

Ae2

Ae,

£.2

Adjusted azimuth

o » '1

100 16 23.778

dE

246 05 40. 989

346 22 04. 767

180 00 00. 000

166 22 04. 767

222 51 06. 389

389 13 1 1. 156

180 00 00. 000

209 13 1 1. 156

190 15 00. 3 89

399 28 11. 545

180 00 00. 00

219 28 11. 545

277 05 14. 789

136 33 26. 334

Values in meters

608, 279. 04404

- 335.61649

607,943. 42755

178.54137

608, 121. 96942

4, 1 16. 78679

612,238. 75621

612,238. 85256*

r

- 0.09635

27

6 E. = ( dE / D ) . d;

(3.10)

8n- =( dN / D ) . d. (3.1 1)

Where: Se. = correction to A 2 ,/

8 N. = correction to An< ,

dE = total closure correction in the S coordinate,

dN = total closure correction in the N coordinate,

and

D = total distance.

The adjusted E and N coordinates (Table IX) were

different from the fixed E and N coordinates (Table II) by

0.00001 m due to round off error. The calculations and

adjustments were illustrated in sections A, B, and C by

using the hand calculator (TI-59) and rounded off at 5

decimal places. The computer program was written to calcu-

late the adjusted traverse station positions by the

Approximate Method by using 16 decimal places (Appendix A).

Ths approximate traverse adjustment is based on an assumed

condition that the angular precision equals the precision in

linear distance.

D. ADJUSTMENT OF A TRAVERSE BY LEAST SQUARES METHODS

The method of least squares provides a rigorous

adjustment and best estimates for positions of all traverse

stations. The Least Squares Method is used to

simultaneously eliminate closing errors in azimuths and

coordinates of traverses.

1 • The Principle of Lea st S qua res

The fundamental condition of the least squares tech-

nique in surveying requires that the sum of the squares of

the residuals be minimized. A residual is defined as the

difference between the true and observed values. In making

28

TABLE IX

Adjusted Coordinates (Approximate Method)

Departure Correction

AE1

-335.61649 + 0.01659

AE2

178.54137 + 0.00426

AE3

4,116. 78679 + 0.07549

Latitude Correction

AN, 1,383.88907 - 0.00745

AN2

319.20444 - 0.00191

AN 4,999.41523 - 0.03389

Grid

E1

Grid

N1

Values in meters

608,279.04404

- 335.59990

607,943. 44U14

178.54613

608, 121.99027

4, 1 16.86228

612,238. 85255

Values in meters

4,072,555.85206

1,383. 88162

4,073,939.73368

319.20253p

4,074,258.93621

4,999. 38134

4,079,258.31755

physical measurements, the true values can never be deter-

mined. The least squares principle establishes a criterion

for obtaining the best estimates of the true values.

If the best estimates of the true values are stated

by x. and observed values by x. , the residuals are expressed

as

v. = X: - X;I

I "I

The fundamental condition of least squares, for

uncorrelated observations with equal precision are expressed

as

29

]> ( v. )2 = (v^ 2 (v

2 )2 (v

3 )2 + . . . . (v

n )2 = Minimum

i z 1

or in matrix form

VT V = Minimum (3.12)

In general, the observed values are of unequal

precision. The observed value of high precision has a small

variance. Conversely, a low precision of the observed value

has a large variance. Since the value of the variance goes

in opposite direction to that of the precision, the observa-

tion is assigned a value called weight corresponding to a

quality that is inversely proportional to the observation's

variance.

For uncorrelated measurements x. , x , x -, , . . .222 2x n , with variances

1, tf M <$ , . . . C , respectively, the

weights of these uncorrelated measurements are,2 2 2 2 ^2 2

P «0./5,P = ft/ <S , p = Q/ ... .

11] 22 2 33 3

n n

^ 2 2

= <J/C5 n (3.13)

where (Jo is the proportionally constant of an observation

of unit weight [ Eef - 7, p. 67]. These weights may be

collected into a corresponding diagonal P matrix, called the

weight matrix:

P =

p .c1

1

p .

2 2

p .

33

0.

.

.

• pn n

30

The weight matrix consists of weighted observations

of a traverse with mixed kinds of measurements. They are

distances and angles- The variance of a measured distance

is expressed in meters and an angle in radians. For

uncorrelated observations of unequal precision, the funda-

mental condition of least squares is expressed as

I : 1

n.v.2 =p V 2 +P V 2 +P V 2 +

= Minimum

+ p v 2nn n

or in matrix form

V'PV = Minimum (3.14)

In the problems involving the adjustment of observed

values, all observed quantities are expressed by functions

of the quantities to be determined. In simple cases these

relations are linear, but when this is not the case, the

relations must" be converted into the linear form by

expanding them into Taylor's series. The terms of higher

'order are neglected, so as to obtain linear relations,

solving the resulting linear equations, then iterating until

the effect of the neglected higher order terms are

minimized.

2- Least Squares Adjust me nt of Indirect Observati ons

The observed values are related to the desired

unknown values through formulas or functions which are

called observation equations. One observation equation is

written for each measurement. To solve for the best value

of each unknown parameter, at least one redundact observa-

tion equation must be written. That is, the number observa-

tions must be greater than the number of unknowns. The

linear observation equations can be written in the general

form as follows:

31

aiO+ a n X

i

+ a!2

X2 • • •

+ a im X m= G

1

+ 71

a20

+ a2i

xi

+ a22

x2

* 2mm I 2

a + a x + a «x «nO n1A

l n2 2+ a orT, x m = G„ + v nn m m n n

where n is the number of observations; m is the number of

the unknowns; a^, a 20 # • • - an Q are constants; a n , a i2'

. a are coefficients of the unknowns x. , x ,

x m ; and v , v2 , . . . v n are the residuals.

Because the observations G. (i = 1 , 2, . n) are

not free from random errors, each G- must be corrected by a

residual value, v. . let b- = G- - a. . Thus

a n x^

+ a i2 x2

+ * a lm x m " b1 = v

1

a21

X1

+ a22 X2

+ 2mm 2 2

an1

X1

+ an2 X2

+ a x - b = vn m m n n

or in the matrix form

V = AX - B (3.15)

This equation is called the observation equation or

the observation equation matrix.

For uncorrelated observations of unequal precision,

substitute the value for the V matrix from the observation

Equation (3.15) into Equation (3.14)

32

VT PV = (AX - B)

T P(AX - B)

=[ (AX)

T - BT ]P(AX - B)

= (XT AT - B

T) P(AX - B)

= (XTAT P - B T

P) (AX - B)

= X T AT PAX - XT AT PB - B

T PAX + BT PB

= XTAT PAX - X

T ATPB - X T A

T PB + BT PB

= XTAT PAX - 2X T

A T PB BT ?B (3.16)

[from matrix algebra, (AX)T

= XTAT and B T PAX = X

TAT PB]

The minimum of this function can be found by taking

the partial derivatives of the function with respect to each

unknown variable (i.e., with respect to the x]

, x2 # • • •

x m ) must equal zero. Hence,

3_(VT PV) = 2A T PAX - 2AT PB = (3.17)dx

Dividing Equation (3.17) by 2, the following result

is obtained:

AT PAX = ATPB

This represents the normal equations, and by multi-

plying this equation by (A PA) -1, the solution is obtained:

X = (AT PA)-*AT PB (3.18)

For uncorrelated observations with equal precision,

the weight matrix is an identity matrix, and equation (3.18)

becomes

X = (AT A)-iAT B (3.19)

This equation can be derived similarly to the

unequal precision case. Equations (3.18) and (3.19) are the

basic least squares matrix equations.i,

33

If the relations are nonlinear, the relations must

be converted into the linear form by using Taylor's series

expansion [Eef. 3, p. 919]. Let F = f (x1

, x2 , m )

be the general observation equation that is a nonlinear

function. The Taylor's series expansion is

F = f (x°. , x* . . xV ) + If 8x,+ df 5x~ +

OX. Ox,+ Higher order terms

3f 5 mm

where x° , x2

, . are the approximate values of thel

/ «* 2' " ' * ~mvariables at which the function is evaluated. For the

traverse problems, an approximate value can be the

precalculated value, and Ox, , 8x2 , . . . 0x m can be the

corrections. The higher order terms in the series are

neglected and only the zero and first order terms are

maintained, the approximate value must be improved by

successive iterations until the effect of the neglected

higher order terms is minimized. After linearization, the

observation equations become:

r

1= f, (*°, ) 3LJi 6 x

1+ 3f

t5 x

2+ . . d f

15 xm- G

v2

= f2(x- # x°

2. m ) * Q f2^ x

i* ^ f

2Sx

2^

3

m

Z f 9 5x - g

9x ™m

'

n= f

n (*\. x°2

, . . x°m )+ lfn 5x1+ ^fn 5x 2

+

3x°, 3x*^fn 5 ^m- G n

m


V = AX - B (3.20)

34

where

A =

If, ?£,

if 2 jf

"9 x , 3 x°

3f.

m

^ I.

m

V =

V1

B =

G1

~ £1 v

X! / X 2

'

G^ — to (X. / x o '

G n " f n (x °» ^z

X' )m '

•*°m >m

X =

5 x1

The remainder of the least squares procedure is the

same as indicated by Equations (3.16) to (3.19).

There are two types of observation equations for the

adjustment of the traverse. They are the angle and distance

observation equations. Both of them are nonlinear, and they

must be linearized using the Taylor's series.

To derive the observation equation for the angle ct.i

(Figure 3.1), first write the two azimuth observation

equations for the azimuth Az-kand Az

;

. , where Az;l,

is the

back azimuth and- Az.. is the forward azimuth.U

E w - E:

Az.,, = arc tan -

ik

ik

Az.. = arc tan

Nk-N;

N« - N.J i

35

Than, the angle observation equation for the angle ct- is

v. = Az. .- Az., - a.

i i j i k i

= arc tan -r?—zr~ - arc tan -r^—rr1- ~ °^

J (3.2 1)

Ji k i

Equation (3.21) is nonlinear in the parameters.

Write this equation in function focm:

v. = f. (E- , N. , E; , N. , E,, N. )

- d.i i

vi

r i* j'

j' k k '

i

Thus, the linearized form of the angle observation

equation is

v. = f; (E], V], E). *). E°, N°k

) * 3f. Se, *|JjSh;

^f. SB- *3f;6N; +3f.8E, BfrSlT - d-3 E; ?N] ^E

k-3N

k

The observation equation for the distance d« between

two points i and j is

v. = [ (E1

- Ej)« + (N, - H.) 2]i/2 - d;

or in the functional form

V; = f; (Ej. K;, Ej, Bj ) - d-

Thus, the linearized form of the distance

observation equation is

V; = f;(E;, N°;

, Z), B) ) * ^ffSE; + Al ;5 H

j

i

INi

3 f; Se- ^ f; 5N. " d;

3E°; ^N°J

J J

For the adjustment of the traverse which was

conducted at Moss Landing with the technique of adjustment

of indirect observations, four angle and three distance

36

observation equations need to be written. The seven equa-

tions would include as unknown parameters the four coordi-

nates of stations 2 (Mossback) and 3 (Dune Temp) - Four

normal equations are formed and solved for corrections to

the approximate values (precalculated values) of the parame-

ters. The corrections are added to the approximations to

update their values, and the solution is repeated until the

last set of corrections is insignificantly small.

The linearized equations of this problems are

V1

= a n 5E2

+ a!25N

2 *dl3

6E3

+ d l4 5N3

" b1

v2

= a2i

6E2

a22

SN2

a23

5E3

+ a^S^ - b2

v?

= ayiSE

2a?25N

2ay 3

5e3

* a ?4 6n3

- b?

The zero order terms of the angle and distance

functions are

E°o-Ef

1 = arC tan N%1

f_ = arc tan tttN3 - N^

- AZ,

- arc tan El - E2Nt - N°2

f3

= arc tan N _r

f4

= AzL

- arc tanN?

*

E4~ E°3 E2- E3- arc tan

N°2- N°

3

f . = [ (E. - E2 )2 (N. - N? ) 2]211/2

f = [ (e 2- e\ ) 2 + (n; - n; )

2]1 /2

f7

= [ (E3

- E4 )2 + (N

3- N4 , 2] 1 /2

By using the data of the known coordinates (Table II) and

the approximate values (Table VII)

-2x, -11

3e°N2 - N

1

^N°- N^ 2+- (E

2- E

1)

2j

= 0. 68246 x 10-3

37

I

12

•1 3

•1 4

•21

= 11,^3

=

=

= ^2 = -

E 9 - E

lCn'-N^ 2-!- (Ej- e

1)

2

0. 16550 x 10-3

^E°-

N 3 - N 2

l(n;-n;)2

+. (e3-e°

2)

2

»1 " »2

i N 1- N°2 )

2 + (E1- £°

2 )2

-

'22

3.06868 x 10-3

o o

E 3 - E 2=U 2=

liN3~ N °2

)2 + (E °3~ E°2 )2

Ei - E-

L(Nf N°2)2 +, (El - E°

2)

:

= 1.16926 x 10-3

l 23 ll 2= Nt - N^ 1 = 2.38621 x 10-3

- M° ^2 o ,2^3- Nu

2)^ 4- (EX- E°„)

'24

31

= 3f2

= "

= 1 f3

= -

^E°2

E3 - Epi^2(N

3-N

2)^ 4- (E°

3- E

2)2

N 2 - Nii°_m° ^2[(N2-N

3)^ +- (E°

2-E

3)'

= - 1.33476 x 10-3

= 2.38621 x 10-3

32 =ll3 = Ep - E

(N°2-N°

3)2 +(E °-E

3)2

= - 1.33476 x 10-3

'33 =AI3= -

*E%Ua ~ N

(N - N° )2 ME - E° )

2u 4 j 4 j

"2 - S3

(N°2-N

3)2

f- (E2-E°

3)

2

= - 2.50541 x 10-3

34 =if3= Ea - E-

"4 "3

= 1.43291 x 10-3

n;) 2 f (E4- E

3)

2

E 2 - E 3

JN°2-N°

3)2 +-(E°

2-E

3)

2

38

a . = dt A =41

dE°

4 23n°

2

=

a . =?1 = -43

^E-

N, - N.

iJnJ-n )2+<e'-e 4 )

2

= 0.11920 x 10-3

4 4-*1

4=

3N?E-, - E, = - 0.09816 x 10-3

l

(N3-N

4)

2+-(E

3- E 4 )

2_

•51 Ej - E 2

_[(Er E°2

)

; :N1- N °2 )2]

1/2

= - 235.67466 x 10-3

a_ = 3f. = -52 3N C

Nt - N2

£(E,- E°2 )

2-f-(N

1-N 8

2 )2]V2

= 971.83201 x 10-3

53= df - =

a. .= 3f e =

54dhi°

a., = 3f =61

'62

3e°

= 3f. =

go - E 3

o .

3n

[(E'-E^)2 -MN

2-N

3;

o o

N 2 - N3

2 11/2

2 [(E'-El) 2 ^(N'-Nl) 2]

1 ^2 3

= - 488. 17948 x 10-3

= - 872.74326 x 10-3

a_ = 3f c = -63

9e°

E 2 " E °3

I(E 2 -E°3)2 +(N

2-N

3 )2]1/

= 488.17948 x 10"3

643n°

N 2 - N3

5Z^ e;)2 +(n;-n o

•2 -3' '2 3'

= 872.74326 x 1-3

a_ = af-, =71

3E°o

"72= _3f 7 =

39

0E°3

74 =f*7

"

E 3 - E

L[(E3-E 4)

2 ^(N3-N4

)

N°3 ' N4

2 11/2

[(E3-E

4) 4-(N

3-N

4)


2J1/2

= - 635.68254 x 10-3

= - 771. 95059 x 10-3

A = 10-3

0.68246

-3.06868

2.38621

0.00000

-235.67466

0.16550

1. 16926

-1.33476

0.00000

971.83201

-448. 17948 -872.74326

0.00000

2.38621

-2.50541

0. 11920

0.00000

488.17948

0.00000 0.00000 -635.68254

0. 00000

-1. 33476

1.43291

-0.09816

0.00000

872.74326

-771.95059

The computation of B matrix, by using the data

from Tables II, III, IV, and VII

= d1

- f1

=

d2" f2

=

= C*3- f

3=

= CL - f =4 L

4

- f5 "= d

= d. - fs

=

0.00000 radians

0.00000 radians

0.00002 radians

0.00002 radians

0.00000 meters

0.00000 meters

= d3

- f 7= -0.01426 meters


BT

= [ 0.00002 0.00002 -0.01426 ]

The weight matrix can be computed by using the vari-

ance of the observed angles and distances from Tables III

and IV by applying to Equation (3.13). For uncorr elated

observations of unequal precision, the weight matrix is

defined as the diagonal ? matrix.

40

'1

1

•22

333D'44

D' 55D

'66

77

1 / sins (1.984")

1 / sin* (1.405")

1 / sin2 (1.203")

1 / sin2 (1.614")

1 / (0.001)2

1 / (0.001)2

1 / (0.003)2

10,808,537,426.79957

21,552,498,219.02474

29,398,082,997.43487

16,332,144,194.08730

1,000,000.00000

1,000,300. 00000

111,111.11111


the values of A, B, and P matrices are applied to Equation

(3.18). The correction vector is found to be

5e2

= 0.01284

8n2

= 0.00336

5e3

= 0.01079

6n3

= 0.00495

The approximation values were improved by adding the

corrections to the first approximation values. The improved

approximation values after the first iteration are

E 2= 607,943.44238 + 0.01284 = 607,943.45522 m.

N2

= 4,073,939.74473 + 0.00336 = 4,073,939.74809 a.

E3

= 608,121.99110 + 0.01079 = 608,122.0 0189 m.

N3

= 4,074,258.94534 + 0.00495 = 4,074,2 58.95029 m.

Using these values, the solution is iterated. After the

second iteration, the correction vector is zero to six

decimal places, so the improved approximation values are

the final estimates of the coordinates.

For uncorrelated observations of equal precision,

the correction vector is found by solving Equation (3.19).

The correction vector after the first iteration is

X =

§ E.

5e.

5il

= 0.01756

= 0.00426

= 0.01660

= 0.00479

41

The improved approximation values after the first iteration

are

607,943.44238 + 0.01756 607,943.45994 m.E2 =

N2

= 4,073,939.74473 0.00426 = 4,073,939.74899 a.

E3

= 608,121.99110 + 0.01660 = 608,122.0 0770 m.

N3

= 4,074,258.94534 + 0.00479 = 4,074,258.95013 m.

The solution is iterated by using these adjusted

values. The correction vector is zero to six decimal places

after the second iteration. These values are the final

estimates of the coordinates.

The standard deviation of an observation which has

unit weight can be found by the following equations, for

uncorrelated observations of unequal precision,

& = vT pyn _ m

1/2

for uncorrelated observations of equal precision,

0" = VT7

n -m1 /2

(3.2 2)

(3.23)

where n is the number of observation equations and m is the

number of unknowns [Eef. 7, p. 249]. After calculating the

best estimate values of the unknowns, the V matrix can be

computed from Equation (3.20).

The standard deviations of the best estimate values

foe the unknowns are then given by the following equations:

6.= 6. 1/2 (3. 24)

where 6- is the standard deviation of the ith adjusted

quantity. The quantity in the ith row of the X matrix, S--

is an element of the (AT PA) - 1 matrix for uncorrelated

observations of unequal precision. For uncorrelated

observations of equal precision, Sjj is an element of the

(AT A)-i matrix [ Ref - 7, p. 250].

42

(AT PAJ~ 1 or {AT A)

- 1 =

21

'ml

'12

'22

'm2

'1m

7m

mm

The standard deviations of adjusted angles and

distances can be computed by the following equation:

6; = 6. Qi1 /2 (3.2 5)

where 6. is the standard deviations of the ith adjusted

quantity. The quantity in the ith row of the V matrix, Q-j

is an element of A (A PA) -1 AT matrix for uncorrelated

observations of unequal precision. For uncorrelated

observations of equal precision, Q-- is an element of the

A(AT A)-iAT matrix [Ref. 3, p. 912].

A(AT PA) -iAT

or

A(AT A) -1AT

Oil

Q 21

n1

«1S

Q 22

n2

•1n

2n

nn

A closed traverse was conducted at Moss Landing.

After the second iteration, the V matrix can be found by

applying the values of A, B, and X matrices to Equation

(3.20).

43

For uncorr elated observations of unequal precision,

ths V matrix is found to be

V =

ai

Va2

Va3

7a4Vdl

Vd2

d3

0.93149 x 10-s radians or 1.92134"

-1.63120 x 10-s radians or -3.36459"

-1.28363 x 10-5 radians or -2.64768"

-2.30436 x 10-s radians or -4.75308"

0.00024 meters

0.00039 meters

0.00358 meters

The standard deviation of unit weight is found by solving

Equation (3.22), the result is

6 = ± 2.69685a

The values of (AT PA)- 1 are

10-s

1.25 178

0.22019

1.25575

0. 10788

0. 22019

0. 13819

0. 21810

0. 1 1127

1.25575

0.21810

1.46408

0. 03315

0.10788

0. 11127

0.03815

0.22515

The standard deviations of the best estimate values for

positions can be found by solving Equation (3.24), the

results are

E2~

>N2 =

'E3 =

N3

± 0.00954 meters

± 0.0031

7

meters

± 0.01032 meters

± 0.00405 meters

44

The values of A(AT PA)- 1 AT are

^0.064 -0.052 -0.021 0.010 -4.165 -6.963 -63.38

-0.052 0.225-0.175 0.002 3.415 4.690 41.59

-0.021 -0.175 0.211 -0.014 2.563 3.574 31.79

0.010 0.002 -0.014 0.002 -1.813 -1.301 -9.50 1

-4.165 3.415 2.563 -1.313 9917 -117.4 -1047

-6.963 4.690 3.574 -1.301 -117.4 9830 -1525

-63.38 41.59 31.79 -9.501 -1047 -1525 76323

The standard deviations of adjusted angles and distances are

found by solving Equation (3.25) , the results are

;ai = ± 1.40346

da2 = ± 2-64103

<3a3 = ± 2.55260

<5a4 = ± 0.26137

d,, = ± 0.00269

662

= ± 0.00267

6d3

= ± 0.00745

seconds

seconds

seconds

seconds

meters

meters

meters


the V matrix is found to be

V =

ai

7a2

Va3

7d1

7d27d3

= 1.26896 x 10-s radians or 2.61742"

= -1.56893 x 10-s radians or -3.23615"

= -1.75414 x 10-s radians or -3.61817"

= -2.23358 x 10- 5 radians or -4.60710"

= 0.14037 x 10~ 7 meters

= 0.23844 x 10- 7 meters

= 0.24365 x 10-7 meters

The standard deviation of unit weight is found by

solving Equation (3.23), the result is

tf = ± 2.01 144 x 10-s

45

The values of (AT A)- 1 are

0.28193 0.06827 -0.85449 0.70380

0.06827 0.01664 -0.20731 0.17075

-0.85449 -0.20731 2.59561 -2.13724

0.70380 0.17075 -2.13724 1.75999

Ths standard deviations of the best estimate values for

positions can be found by solving Equation (3.24), the

results are

Ê2 = ± °* 00 107 meters

tfN2 = ± 0.00026 meters

C5E3 = ± 0.00324 meters

= ± 0.00267 metersN3

T _i*TThe values of A (A A) _1 A are

1.472 -27.197 26.960 -1.235 -0.518 -0.343 -0.356

-27.197 503.1 -498.7 22.855 0.348 0.480 0.473

26.960 -498.7 494.4 -22.66 0.370 0.523 0.518

-1.235 22.855 -22.66 1.038 -0.201 -0.159 -0.134

-0.518 0.348 0.370 -0.201 10000 -0.001 -0.001

-0.843 0.480 0.523 -0.159 -0.001 10000 -0.001

-0.856 0.473 0.518 -0.134 -0.001 -0.001 10000


found by solving Equation (3.25) , the results are

â i - ± 0.15917 seconds

â2 = ± 2.94270 seconds

â3 = ± 2.91732 seconds

a4 = ± 0.13370 seconds

^di = ± 0.00002 meters

'd2

5d3

= ± 0.00002 meters

= ± 0.00002 meters

46

The computations in this subsection were performed

by the NPS IBM 3033 computer using 16 decimal places and

were rounded off to 5 decimal places prior to output.

3 . Least Squares Adjust ment by_ the Con dit ion Squat ion

Method

The general principle of least squares adjustment by

the condition equation method in surveying is

to minimize a function consisting of the sum of thesquares of the corrections to the observations plus thenecessary mathematical conditions involving some or allthe corrections; each condition by itself is made equalto zero bv adding the corrections to the discrepancydetermined from a pread justed calculation (i.e. calculatedfrom the observed values, rather than from adjustedvalues); thus, the magnitude of the sum of tha squares isnot changed by adding conditions [Ref. 3].


the principle condition of least squares function may be

expressed in matrix form as follows:

= 7T P7

= 7T P7

= 7T PV

= 7T PV

= 7T P7

2K[ B7 + ff ]

2[B7 + ff ]T K

2[ (B7)T S T ]K

2[7T BT + W T ]K

27T BT K 2HT K

= minimum

where D is the least squares function matrix, K is a matrix

of Lagrange multipliers, P is the weight matrix, V is the

correction vector, B is the constant coefficients of the

corrections, and W is the precalcula ted discrepancy

[Ref. 9], It is numerically more convenient for later

development to multiply by 2. Taking the partial

derivatives of the a matrix with respect to each of the

corrections and equating to zero leads to

U7

2_? = 2PV - 2BT K =

3Vor

7 = P-1BT K (3.26)

This equation is called a correlate equation or correlate

equation matrix. The solution of the Lagrange multipliers

matrix can be derived by multiplying Equation (3.26) by the

B matrix then adding the V matrix

BV W = BP-iBT K R

(B7 + W) is the condition equation matrix which must equal

zero. The solution of the Lagrange multiplier vector is

BP-iBT K = -H

K = (BP-iB1)-i (-17) (3. 27)

The correction vector is derived by substituting

Equation (3.27) into Equation (3.26)

V = P~iBT (BP-iB7 )-! (-W) (3.28)


the correction vector can be derived similarly to the

unequal precision case or by using the inverse of the weight

matrix which is the identity matrix. Equation (3.28)

becomes

V = BT (BBT )~i (-W) (3.29)

Closed traverse station positions may be adjusted by

using the technique of least squares adjustment by the

Condition Equation Method. There are two condition equa-

tions. They are the azimuth and coordinate condition equa-

tions. The coordinate condition equations are divided into

two parts, which are E and N coordinate condition equations.

48

The azimuth condition equation is the sum of the

corrections to the angles in a traverse plus a precalcula ted

discrepancy W1

(Equation 3.6) which must equal zero. Where

v • equals the corrections to the angles, the general equa-

tion can be written;

lv,or

+ W. =at 1

i : 1

n n-1

A*ai + ^°- Vd!

+ S1

= ° <3 - 30 >

1=1 i = 1

where v.- is the correction to the distances. However,d i

this term equals zero and does not effect the equation, but

it does reserve space in matrix notation for distance

corrections. A closed traverse which was conducted at Moss

Laading can be written with an aziauth condition of

1.v +1.v + 1 - v , + 1 - v . . v . . +0.7. + . v , .,

a1 a2 a3 a4 d1 d2 d3

+ W^ = (3.31)

where the precalcula ted discrepancy (Wj) is +3.844" or

+0.00004288 radians.

The simple linearized form of an E coordinate condi-

tion equation is the sum of the adjusted departures and must

equal zero. 3y applying Equations (3.2) and (3.7), the

general formula for E coordinate condition equation isn-1

]>(d;

+ vd

. ) sin Aza[

- (ET

- E,) = (3.32)

u '

r 1

wherei

Aza|

= Azp

^_{dj

vaj )

- (i - 1)180° (3.33)

i = 1

Aza ;

is the adjusted azimuth. 3y using Equations (3.32) and

(3.33) with the data conducted at [loss Landing, these

equations become

(d1

+ vd1 ).sin Az

a1+ (d

2+ vd2 ).sin Aza2

+ (d 3 + vd3 )-sin Aza3

- (ET

- E1

) =0 (3.34)

49

where

Azal

= AZF * d

i

+ Vai

= AZi + va i

AZa2

= AZF * d

1

+ Va1

+ *2 + Va2 " 180 °

AZ2

+ Va1

+ Va2

Az a3= Az F

+ d1

Tal

* cl2

va2

d 3+ va3- 360O

= Az3 * V

a1+ V

a2+ V

a3

Azj is the precalculated azimuth or unadjusted

azimuth (Equation 3.1).

From trigonometry

sin ( A +AA ) = sin A.cosAA cos A.sinAA

in which Aa is a very small angle in radians, let

cosAA = 1 and sinAA = AA

therefore,

sin ( A + AA ) = sin A + Aa.cos A

from the first term of Equation (3.34)

(d, + vd1 Jsin Az a1

= (d1

+ vd1

)sin (l Zl + va1 )

= (d1

+ vd1

)[sin Az1

+ va1

cos Az1 ]

= d1sin Az

1+ v . d

1cos Az-| + v_ sin AZj + v.

1v

^cos Az

1

(3.35)

(v .v ) is very small, by letting this term equal zero,

Equation (3.35) becomes

( d i+ v

d1 ) sin Azai

= d. sin Az. + v , d„ cos Az. + v.. sin Az. (3.36)i l 3ii i a i I

50

The second and third terms of Equation (3.34) can be derived

similar to Equation (3.36), so

(d2

+ vd2 )sin Aza2

= d2sin Az

2(v

a1va2

)d2cos Az

2+ v

d2sin Az

2(3.37)

(d3

+ vd3)sin Az

a3

= d3sin Az

3+ (v

a1+v

a2+v

g 3) d

3cos Az

3+ v

d3sin Az

3 (3.33)

Substituting Equations (3.36), (3.37), and (3. 38)

into Equation (3.34) and rearranging terms

va1 (d1cos Az-i + d

2cos Az

2* d

3cos Az

3 )

+ v2

(d2cos Az

2+ d

3cos Az

3) va3 d

3cos Az

3

+ v sin Az. + v^.sin Az v ._ sin Az_di I a 2 d aj J

+ d1sin Az

1+ d

2sin Az

2+ d

3sin Az

3- (E T

- E.,) =

This equation can be written in the general formula asn-1 n-1 n-1 n -1

2< val 2 d

kcos Az

k>+ 2^di

sin Az. + ]>d. sin azj

I Z 1 k= i « = 1j ; 1

- (2T

- E,) =0 (3.39)

substituting Equations (3.2), (3.3), and (3.7) into Equation

(3.39)

n-1 n-1 n-1

(vai JAN.) + 3>H .sin Az. + » 9= (3.40)ai ^-

l

^->

k' .^— di i 2I = 1 k:l I = 1

For example, if the number of the observed angles (n) at

Moss Landing is four, this equation can be expanded to

*al (A»i + An 2 *An

3) + v

a2 (AN2

+ An3

)va3 AN 3

vd1

sin Az1

+ v sin Az2

« v sin Az3

+ 5?2

=

51

Substituting the numerical values from the precalculat ed of

this traverse into this equation, the result is

(6702.37612) v, + (53 18.48345)

v

n9 + (4999 . 28 284) v , + O.vv ' ai ' ' a* * a3 a 4

-(0.23567) v + (0.48818) vd2

+ (0. 63570) vd3

+ »2

= (3.41)

where the precalcula ted discrepancy (57 ) is 0.08608 m and

O.v is equal to zero, which does not effect this equation,

but space is reserved in matrix notation for tha last angle

correction.

The N coordinate condition equation is the sum of

the adjusted latitudes and must eqaal zero. By applying

Equations (3.3) and (3.7), the general condition equation is

n-1

i = 1

(d. + vdi ) cos Az - (NT

- N,) =

This equation can be derived similarly to the S coordinate

condition equation case, resulting inn-

1

n-1 n-1

"Svaf]>AE k) 2*

d;C0S Az. W

3= (3.42)

1=1 kzl ir1

From the data acquired at Moss Landing, this equation can

be expanded to

-va1 (Ae

1+Ae

2+Ae

3)

- va2 (Ae2*Ae

3) va3 Ae 3

v cos Az, v „ cos Az + v cos Az- + W-, =d 1 1 d2 2 d3 3-1

Substituting the numerical values from the calculation of

this traverse into this equation, the result is

-(3959.89460) va1

- (4 295. 49626) va 7 ( 4 1 1 6. 94755) v

a3+ 0.v

4

+ (0.97183)v + (0.87274) v + (0. 77 1 94) v ._ + W-, = (3.43)d1 d 2 Qj->

where the precalcula ted discrepancy (W3 ) is -0.08936 m.

52

Equations (3.30), (3.40), and (3.42) can be written

in the matrix form as (BV + W) , where 3 is the constant

coefficients of the corrections, V is the correction vector

which is composed of the angle and distance corrections, W

is the precalculated discrepancy. Equations (3.31), (3.4 1),

and (3.43) are written in matrix form as

BT =

1 6702.37612

1 5318.48345

1 4999.28284

1 0.00000

-0.23567

0.48818

0.63570

-3959.89460

-4295.49626

-41 16-94755

0.00300

0.97183

0.87274

0.77194

r

V =

¥a1

7a2

'a 3

"a 4

di

7d2

d3

-W =

-W1

= -0. 00004

-W2

= -0. 08608

-W, = 0.08936


the inverse of the weight matrix is the diagonal matrix,

which is equal to the observation's variance. The diagonal

elements of the inverse of the weight matrix is

p-i = sin* (1.984") = 0.92519 x 10-»o11

p-* = sin* (1.405") = 0.46398 x 10-*°2 2

p-i = sin2(1.203") = 0.34016 x 10~*°r33

p-i = sin2(1.614") = 0.61229 x 10-*°44

p-» = (0.001)2

p-i = (0.001) 2

66

p-i = (0.003)277

= 0.10000 x 10-s

= 0.10000 x 10-s

= 0.90000 x 10-s

53


the correction vector is found by solving Equation (3.28)

V =

'a1

'a2

'a 3

'a 4

dl

762

'd3

= 0.93141 x 10-5 radians or 1.92117"

= -1.63114 x 10- 5 radians or -3.36446"

= -1.28358 x 10-s radians or -2.64758"

= -2.30438 x 10-s radians or -4.75312"

= 0.00024 meters

= 0.00039 meters

= 0.00357 meters

The corrected angles are

C*1 = 246° 05' 45. 12117"

a 222° 51' 05.23554"

Ck3

= 190O 14' 59. 95242"

OL = 2770 05' 12.24688"4

The corrected distances are

d-, = 1424.00424 m.

d 2= 365.74439 m.

d 3= 6476.27457 m.

Using the corrected data to recompute the coorlinates is

similar to the initial traverse computations. The adjusted

coordinates are

E2

= 607,943.45521

= 4,073,939.74809

608,122.00139

= 4,074,258.95029

612,238.85256

= 4,079,258.31754

N,

N.

S

m

.

m.

m.

m.

m.

m.

The last adjusted coordinates are the same values as the

fixed coordinates. Hence, it is not necessary to iterate.

For uncorrelated observations with equal precision, the

correction vector is found by solving Equation (3.29)

54

V =

V

"al

7a3

'a 4

d1

7d2

7d3

1.26884 x 10-s

-1.56886 x 10- s

-1.75408 x 10-s

-2.23360 x 10-s

0. 14034 x 10" 7

0.23842 x 10" 7

0.24362 x 10- 7

radians or 2.61717"

radians or -3. 23601"

radians or -3. 61804"

radians or -4.60713"

meters

meters

meters

The corrected angles are

C^l = 246° 05 1 45. 81717"

OU = 2220 51 i 05.36399"

oc. = 190O 14' 58.98196"

C*4

= 2770 05 » 12.39287"

The corrected distances are

a, = 1424.00400 m.

d2

= 365.74400 m.

a = 6476.27100 m.

By using the corrected data to recompute the coordinates,

the adjusted coordinates are

E2

= 607,943.45994 m.

N2

= 4,073,939.74899 m.

E3

= 608,122.00770 m.

N3

= 4,074,258.95013 m.

E4

= 612,238.85256 m.

N4

= 4,079,258.31754 m.

The last adjusted coordinates are the same values as the

fixed coordinates. Hence, it is not necessary to iterate.

55

The standard deviation of an observation which has

unit weight can be found by the following equations. For

uncorrelated observations of unequal precision,

6. = VT PV 1 /2 (3.44)

and, for uncorrelated observations of equal precision,

C5. = 7TV 1 /2 (3.45)

where r is the number of condition equations. There are

three condition equations for a closed traverse (r=3). The

standard deviations of adjusted angles and distances can be

found by the following equation:

6. = 6oj_Q;Jl ' 2 (3.46)

where 6- is the standard deviation of the ith adjusted

quantity. The quantity in the ith row of the V matrix, Q--

is an element of the P- 1 - P- 1 B (BP- 1 BT

)- 1 BP- 1 matrix for

uncorrelated observations of unequal precision

[Ref. 3, p. 918]. For uncorrelated observations of equal

precision, Q;j is the element of the I - B (BB~)- X B matrix

The matrix P-i - P-i

B

T (BP-iBT )-iBP-i or the matrix

I - 3T

(BE T)-iB is equal to

Q11

Q 21

1 2

22

Qln

22n

Qnl Qn2 Qnn

56

the standard deviation of unit weight is found by solving


<5o

= ± 2.69679

The results of P- 1 - P-i3T (BP-i3T) -» BP-* are

10-io

0.064 -0.052 -0.021 0.010 -4.165 -6.963 -63.88

0.052 0.225-0.175 0.002 3.415 4.639 41.59

0.021 -0.175 0.211 -0.014 2.563 3.574 31.79

0.010 0.002 -0.014 0.002 -1.813 -1.301 -9.500

•4.165 3.415 2.563 -1.313 9917 -117.4 -1047

•6.963 4.689 3.574 -1.301 -117.4 9830 -1525

•63.88 41.59 31.79 -9.500 -1047 -1525 76323


found by solving Equation (3.46) ; the results are

<5a1

= ± 1.40340 seconds

6a2 = ± 2.64096 seconds

Oa 3 = ± 2.55253 seconds

(5a4 = ± 0.26136 seconds

<3d1= ± 0.00269 meters

662 = ± 0.00267 meters

663 = ± 0.00745 meters

Foe uncorrelated observations of equal precision, the

standard deviation of unit weight is found by solving


tf = ± 2.0 1139 x 10-s

57

The results of I - 3 T (BBT )—*B are

10-3

1.472 -27.199 26.962 -1.235 -0.518 -0.843 -0.856

-27.199 503.1 -498.7 22.855 0.348 0.480 0.473

26.962 -498.7 494.4 -22.66 0.370 0.523 0.518

-1.235 22.855 -22.66 1.038 -0.201 -0.159 -0.134

-0.513 0.348 0.370 -0.201 10000 -0.001 -0.001

-0.843 0.480 0.523 -0.159 -0.001 10000 -0.001

-0.356 0.473 0.518 -0.134 -0.001 -0.001 10000


found by solving Equation (3.46) ; the results are

tfa i= ± 0.15918 seconds

<5a2 = ± 2.94262 seconds

(5a3 = ± 2-91722 seconds

<5a 4= ± 0.13369 seconds

6dl = ± 0.00002 meters

^d2 = ± 0.00002 meters

<5d3 = ± 0.00002 meters

The computations in this subsection were also

performed on the NPS IBM 3033 computer; they used 16 decimal

places, and were rounded off to 5 decimal places in the

final solutions. These are more decimal places than are

normally used in practice because such precisions are not

generally attainable by corresponding observations.

However, in comparing two computational methods that are

theoretically equal, one method may be more sensitive to

round off error than the other. This will be commented on

in the next chapter.

58

IV. DISCUSSIONS AND ANALYSIS g_F RESULTS

Three programs were used to compute the initial traverse

and adjust tha closed traverse station positions. Programs

1 , 2, and 3 were used to compute and adjust the traverse

station positions by the Approximate Method, the Indirect

Observations Method, and the Condition Equation Method,

respectively. The programs were written in W'ATFIV language

for implementation on the NPS IBM 3033 computer. The

computer output and listings of Programs 1, 2, and 3 are

provided in Appendices A, B, and C, respectively. The

maximum number of intermediate traverse station positions

that can be computed and adjusted by these programs is 30.

The programs were tested using several fictitious data sets

to ensure their performance in handling the various

intermediate traverse station positions.

The computer storage area and CPU time of Programs 1, 2,

and 3 has been compared (Table X). The adjustment of a

closed traverse by the Approximate Method did not use a

weight matrix in the matrix computations. The computer

program for this method was written by using the variables

in only one dimension for storage of both data and results.

This method did not require any iterations. Tha least

squares adjustment of a closed traverse characteristically

is used to simultaneously eliminate closing errors in

azimuths and distances, and Programs 2 and 3 both utilized

such an adjustment. The computation by either Least Squares

Method used matrices in two dimensions to compute the

correction vector. Likewise the computer programs were

written using variables in two dimensions. Included are all

subroutines from Program 1 and extra subroutines for each

individual program. Therefore, both Programs 2 and 3 used

more computer storage area and CPU time than Program 1.

59

w

CD

Eh

4-1

O

03

e•HEh

E=>

a.

U

C

n

cumenit1t3

U CO *4->

CO »«N

P0) *Pi-

CuO)s sO fd

en0) oJ3 P4JCu

c03

*+J

CD

CQ

WcowHP<d

a.

Ou0)

-CEH

1

1

c3

03

-P

CO

u03

CO

1

3- CO ,_ O coCO i T— CN « T— r— • T—

m =r r- t . r*< m t • f r- f 1

vO^ cn

CN

T—r~

-f-"

cd CTl m ViJ LDu sr I • in t m vO in 3-

en CN r» *— r— CO T— • LT t

CO .. .. * T— CN t CN t—u r— »— =r I • .. ^" • • • •

Dm enCN

T— T— «—

1 ^D1 © r— m ^m CN »—

1

1

1

1

1

r~ insr r—

1 IT)

inCN

t • 1

• »-1

1

1 0)

1 c CN CN1 O • • • •

1 w 0) •p r" rn1 s e rrl fll -u1 in td 0) 03 (Tt U) 01

1 u P P P -P a s1 en en cd cd cd 01 cd ' 03

1 O o CI) u u1 p u P 0) O) o tn en

a, Oi rd encd

encd -p

op

op

«w 4-4 03 U p a. cu

o O cn o T3en a p 4-1 03 4-1 <w«u w 01 P 03 rn 01 4-> O oC 4-1 4-> o .. 03

cu c P p P T— p 3 03 03

B 03 O) 03 03 0) r—

i

£ £0/ e e -P T3 p H3 H •P4J 03 1) P 3 c 3 cd 4J +J(d +J M 03 Ou cd Cm+j td id -P S S p O s0) -p +j 3 o CN o CN o Cu cu

03 01 Q. u .. u • • m u U1 4-4 S r— PO

o 4H 4-1 O 4-1 4H i) ip 4h-

1 O m O u o 03 O 01 £ o O1 P • • e E •H1 0) o T— o i-H o rrt O rrl 4-> O O1 .fl •H •H (O •H P •H P •P •H1 B +J T3 +J +J P en 4J en 3 4-1 m 4-1

3 id C id o 03 O 03 O CU 03 • • 03

C p (d p p P PCm

p pCu

u P r" P

03 0) rg 03 cn 03 0) 03 03 03 ae! 03

-^ ^ • • -C • • -C J5 M-l -G m -C JH c -Cj

EH H r~ Eh m i-i H O Eh o Eh E^^ 03 H

1

1

,_ CN m rr LO vO r- CO•

ON1

1

1

1

1

60

The Indirect Observations Method requires that the

number of observation equations be equal to the sum of

observation equations of both observed angles and distances.

The number of observation equations is not constant. It

changes depending on the number of the traverse stations--

this means that the row dimension of each matrix will change

depending on the number of observation equations.

Therefore, the subroutines affected by the number of trav-

erse stations are difficult to write in the general program.

The coefficients in the A matrix are computed by taking- the

partial derivatives of the functions with respect to each

unknown variable, where the number of unknown variables is

not constant. The number of unknown variables changes

depending on the number of the traverse stations.

Therefore, the subroutine for the computed A matrix is also

difficult to write in the general program. The Condition

Equations Method has only three equations which makes it

easier to derive the general form and write the computer

program. The adjustment using the Indirect Observations

Method does not apply the corrections to the observed values

directly; therefore, it requires the approximate values of

the unknown coordinates be computed for the correction

vector of the unknown coordinates. This method requires at

least two iterations to check the insignificance of the

correction vector when it is compared with an arbitrarily

selected small number. The adjusted coordinates from the

Condition Equations Method can be checked at the first iter-

ation. Thus, Program 3 is more economical as it uses less

computer storage area and CPU time than Program 2 (Table X) .

The closed traverse at Moss Landing originated and

terminated at control points with known positions which were

determined by third-order methods. The azimuth closure was

2.211" per station and the position closure was 1:66,617.

The classification, standards of accuracy, and general

61

specifications for a third-order class I traverse (Table I)

indicate the azimuth closure is not to exceed 3" per station

and position closure must be better than 1:10,000. This

traverse met the specifications and standards of accuracy

for a third-order class I traverse.

Using Programs 1, 2, and 3, the traverse was computed

and adjusted in UTH grid coordinates. The final adjusted

coordinates were transformed from tJTM grid coordinates to

geographic coordinates [Ref. 4, pp. 319-321]. The differ-

ence between the coordinates for each method and NOS

[Ref. 10] have been computed (Table XI). The technique of

Least Squares for both methods yields identical computa-

tional results to at least five decimal places (Table XI,

Methods 3.1 and 4.1; 3.2 and 4.2). Least Squares provides

the best estimates for positions of all traverse stations.

Program 2 performs a statistical test, yielding the standard

deviations of both adjusted positions and observed values.

Conversely, Program 3 only yields the standard deviation of

adjusted observed values.- Comparisons of the standard devi-

ations of adjusted observed angles and distances were made

between these computed by the Indirect Observations Method

and the Condition Equations Method (Table XII) . The

computed standard deviations of the adjusted angles differed

significantly in the fourth decimal place in several cases,

which is a slight indication that the Indirect Observations

Method is more sensitive to round off error. The standard

deviation estimates are larger for the Indirect Observations

Method in every case of a significant difference in the

fourth decimal place. However, a significant difference in

the fourth decimal place is insignificant in terms of the

observational precision. The standard deviations of the

adjusted distances for both methods yield identical results

(Table XII). Both Methods differ from the NOS positions in

the fourth decimal place in seconds of arc for both latitude

62

and longitude. The horizontal control for third-order stan-

dards requires accuracy to three decimal places in seconds

of arc for both latitude and longitude--thus, these methods

can be used for computing third-order positions.

63

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65

7. CONCLUSIONS AND HECOMiiEN DATION

A. CONCLUSIONS

The computer programs developed by the author and

contained in this -thesis have a variety of useful and prac-

tical hydrographic applications. In hydrography, geodetic

field work plays a key role- Horizontal control is neces-

sary in determining positions and, even in areas which

appear to initially have adequate control, additional

control stations often need to he established after arrival

in the field. These field-generated control points must be

adjusted to fit within the existing survey net am! to mini-

mize errors in the field Jie a surevents. The errors can be

minimized by these computer programs. Direct application of

these programs would be enormously beneficial, since no

eguivalent software exist at NPS to perform such adjustments

at the present time.

In the Approximate Method, a traverse is adjusted by

computations using a hand calculator. This method is suit-

able for field computation and the adjusted coordinates meet

the specifications and standards of accuracy for a third-

order class I traverse. Although the accuracy from this

method is less than the Least Squares Metnod, it does not

require a computer and it furnishes coordinates which can be

checked in the field by the field party.

The Least Squares Method provides the best estimates for

positions of all traverse stations. Least Squares does

require more of a computational effort than the Approximate

Method. However, the accuracy of Least Squares is better

than the Approximate Method, and the Least Squares Method

should be used for the final office computations. Both

66

Least Squares Methods yield identical results, but the

Condition Equations Method is more economical aad easier to

derive than the Indirect Observations Method. Therefore,

the Condition Equations Method should be applied at HPS.

B. EECOHHENDATION

Studies should be continued at NPS to compare the

economy of adjustment methods used in this thesis with other

methods of least squares adjustment.

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LIST OF REFERENCES

1. Bomford. Brigadier G. , Geodesy., Second Edition / p. 1

,

Oxford university Press, T9"5"Z

.

2. Palikaris, Athanasios E. , Methods of HydrographicSurveying, used by Different countries, ?.?. Tnesis,ITaval Postgraduate Softool. Monterey, California, p.139, 1983.

3. Davis, Raymond E. , Foote, Francis S. , Anderson, Jair.esM. , and Mikhail, Edward M. , Surveying Theory andPractice, Sixth Edition, McGraw-rTiII, 79B"T.

4. Department of the Army Technical Manual (TM 5-237) ,

Surveying Comp ute r' s Manual, 1964.

5. U.S. Department of Commerce, Federal Geodetic ControlCommittee, Classification, Standards of Accuracy, a ndGeneral Specifications of Geodetic Control Surveys,pp7~F-7, -TT7ZT7

6. U.S. Department of Commerce, Environmental ScienceServices Administration, Coast and GeodeticSurvey,Horizontal Control Data, Quad 361214, Stations1 37, 1 0"5"87~T05~9A", "and"TU6H7~T96 3 .

7. Mikhail, Edward M. , and Gracie, Gordon, Analysis andAdjustment of Sur vey Measurements, Van NostranT"RemEoIcI Company, T98T.

8. Hardy, Holland L. , A Brief Outline and Demonstrationof the Theorv and Practice or least Sguares, p. T,TJnpuEIisEed frote.

9. Hardy, Rolland 1., Least Scuares Adjustment ofTr averse, p. 4, UnpuBIIshed ftote.

10. Pacific Marine Center, NOS, Horizontal Control Reportfor Moss Landing - Naval Postgraduate ScEgol Survey,

148

BIBLIOGRAPHY

Durgin, Casper M. and Sutcliffe, Walter D., Manual ofFirst-order Traverse, Coast and Geodetic Survey SpecialPuETIcaTiou to. T37, U.S. Department of Commerce, 1927.

Ewing, Clair E. and Mitchell, Michael M., Introduction tojgodesy, Elsevier North Holland Publishing Company, 19T9.

Graham and Neil, Introduction to Computer Science AStructure Approach", ftesl FuMisKIng Company, T982.

Hodgson, C.V., Manual of Second and Third OrderIriangulation and Traverse, CoasE and Geodetic SurveyS'peciaT'FubTica^ion Wo. TS5, U.S. Department of Commerce,1935.

149

INITIAL DISTRIBUTION LIST

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151

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ThesisA9622c.l

AumchantrA comparison of

methods of least squaresssadjustment of traverses.3.

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ThesisA9622c.l

AumchantrA comparison of

methods of least squares

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a comparison of methods of least squares adjustment of ... the nps institutional archive theses and...

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