Least squares adjustment computer programs for horizontal and vertical positions

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AN ABSTRACT OF THE THESIS OF
Ramesh Lal Shrestha for the degree of Master of Science
in Civil Engineering presented on August 6, 1979
Title: LEAST SQUARES ADJUSTMENT COMPUTER PROGRAMS FOR HORIZONTAL
AND VERTICAL POSITIONS
Redacted for Privacy
Abstract approved:
Professor rt J. Schultz
The application of the least squares adjustment techniques to the
area of surveying has been considered and two computer programs
called HCONTRL and VCONTRL are developed for horizontal and vertical
control respectively. The principle of least squares is discussed
in chapter II. The observation and condition equations are compared
and the observation equations technique is applied in the programs.
The general forms of the observation equations for distance and
angle (or direction) are derived in chapter III. The solution and the
precision of the solution from the least squares adjustment technique
are also discussed. A brief discussions on future systems; the satel-
lite positioning, the inertial positioning, and the non-classical
method, are included in chapter III.
In chapter IV, the development and the procedures to be used in
using the computer program HCONTRL are discussed. The least squares
application to the various techniques of the horizontal position con-
trol; traverse, intersection, resection, triangulation, trilateration,
and combined networks, are described in chapter V.
The theory of vertical position control is discussed in chapter
VI where techniques of direct leveling and trig leveling are described.
This chapter also includes a brief description on; gravimetric leveling,
barometric leveling, hydrostatic leveling, tacheometric leveling,
satellite altimetry, and steric leveling. The development and proce-
dures of the computer program VCONTRL, to adjust the direct leveling
and the trig leveling by the least squares technique are discussed in
chapter VII. The application of the least squares to the direct level-
ing and the trig leveling is given in chapter VIII. In chapter IX,
practicality of the least squares adjustment is considered.
The summary and conclusions are in chapter X. Appendix I con-
sists of the numerical examples of the least squares adjustment of
the various techniques in the horizontal position control by the com-
puter program HCONTRL. The numerical examples of direct leveling and
trig leveling by the computer program VCONTRL are given in Appendix II.
A numerical example of the gravimetric leveling computation to find
the geopotential numbers is also presented in Appendix II. Appendix
III contains the complete listings of the computer programs HCONTRL
and VCONTRL.
Least Squares Adjustment Computer Programs For
Horizontal and Vertical Positions
by
Ramesh Lal Shrestha
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed August 6, 1979
Commencement June 1980
APPROVED:
Redacted for Privacy
Professor of Civil Engineer
in charge of ma
Redacted for Privacy
Head of Wartment of Civil Engineering
Redacted for Privacy
Dean of Graduate School
Date thesis is presented August 6, 1979
Typed by Shanda L. Smith for Ramesh Lal Shrestha
ACKNOWLEDGEMENT
The author would like to take this opportunity to acknowledge
for the continuous support given by Mr. Owen G. Miller, Jr., of
Portland, Oregon, during the course of study of the author at
Oregon State University. Mr. Miller's support is truly and grate-
fully appreciated.
The author is sincerely very grateful to Professor Robert J.
Schultz for patiently editing and correcting this paper, and for the
proper guidance in the completion of the paper.
The author would like to express his gratitude to Assoc. Pro-
fessor John Seaders for the academic inspiration given to do this paper.
The computer programs developed in this paper are the result
of the assistance provided by the Computer Center of Oregon State
University. The author is sincerely very grateful and thankful for
the assistance.
TABLE OF CONTENTS
I. Introduction 1
II. The Least Squares Principle 4
Weights
9
Problem Analysis 12
Observation Equations vs. Condition Equations 15
III. Theory of the Horizontal Position Control 18
Distance Equation 18
Direction or Angle Equation 21
Least Squares Solution 25
Precision from the Least Squares 29
Future Systems 34
Satellite Positioning 35
Inertial Positioning 37
Non-Classical Technique 38
IV. Development of the Computer Program HCONTROL for the
Horizontal Position Control 40
Program HCONTRL 40
V. Application of Least Squares Solution for the Hori-
zontal Position Control Techniques 52
Traverse 52
Intersection 55
Resection 57
Triangulation 59
Trilateration
61
Combined Networks 63
VI. Theory of the Vertical Position Control 67
Direct Leveling 68
Trig (Trigonometric) Leveling 70
Atmospheric Refraction 72
Curvature of the Earth 75
Gravimetric Leveling 77
Non-Classical Techniques 81
Barometric Leveling 82
Hydrostatic Leveling 82
Stadia (Tacheometric) Leveling 83
Satellite Altimetry 84
Steric Leveling 84
VII. Development of the Computer Program VCONTROL for the
Vertical Position Control 86
Program VCONTRL 86
VIII. Application of Least Squares Solution for the Vertical
Position Control Techniques 95
Direct Leveling 95
Trig (Trigonometric) Leveling 98
IX. Practicality of the Adjustment 102
X. Summary and Conclusions 104
XI. Bibliography 106
XII Appendices
Appendix I 110
Traverse 110
Intersection 115
Resection 118
Triangulation 121
Trilateration 125
Combined Networks 128
Appendix II 134
Direct Leveling 134
Trig Leveling 136
Gravimetric Leveling 139
Appendix III 148
HCONTRL 148
VCONTRL 156
LIST OF ILLUSTRATIONS
Figure
Page
1. Sketch for Distance Observation Equation 19
2. Sketches for Direction and Angle Observation Equations 21
3. Position Fix from Artificial Satellites 36
4. Sample Coefficients Computations 41
5. Deck Set Up for HCONTRL 46
6. Flow Chart for HCONTRL 47
7. Sample Sketch for HCONTRL Input 49
8. Traverse Sketch 54
9. Intersection Sketch 56
10. Resection Sketch 58
11. Triangulation Sketch 59
12. Sketch for Combined Networks 64
13. Reference Surface 67
14. Direct Leveling 69
15. Trig Leveling 70
16. Refraction and Curvature of Earth 73
17. Equipotential Gravity Surfaces 79
18. Non-parallelism of Equipotential Gravity Surfaces 79
19. Stadia Leveling 83
20. Sample Level Net 88
21. Deck Set Up for VCONTRL 90
22. Flow Chart for VCONTRL 91
23. Level Net
97
Figure
Page
24. Trig Leveling Net
100
25. Sample Traverse Computation
111
26. Level Net and Gravity Observation Sequence
139
LIST OF ILLUSTRATIONS
Table Page
1. Techniques of Least Squares Adjustment 14
2. Effects of Refraction and Curvature 77
3. Surface Gravities, Station Elevations, and Geo-
potential Numbers
81
4. Extract from Field Book of Vertical Angle
Observations
100
5. Gravity Computation 141
6. Geopotential Numbers 143
7. Calibration Values
144
8. Earth Tide Corrections - April 20, 1979 145
9. Earth Tide Corrections - April 21, 1979
146
10. Earth Tide Corrections - May 8, 1979
147