THE METHOD OF SIMULATED ANNEALING FOR THE OPTIMAL ...

iii

THE METHOD OF SIMULATED ANNEALING FOR THE OPTIMAL ADJUSTMENT

OF THE NIGERIAN HORIZONTAL GEODETIC NETWORK

BY

O. G. OMOGUNLOYE

MATRICULATION NUMBER: 840405039

Ph. D THESIS SUBMITTED TO THE

SCHOOL OF POST GRADUATE STUDIES

DEPARTMENT OF SURVEYING AND GEOINFORMATICS

FACULTY OF ENGINEERING

UNIVERSITY OF LAGOS

AKOKA, LAGOS, NIGERIA

October, 2010

iv

SCHOOL OF POSGRADUATE STUDIES UNIVERSITY OF LAGOS

CERTIFICATION

THIS IS TO CERTIFY THAT THE THESIS:

“THE METHOD OF SIMULATED ANNEALING FOR THE OPTIMAL ADJUSTMENT

OF THE NIGERIAN HORIZONTAL GEODETIC NETWORK”

SUBMITTED TO THE

SCHOOL OF POSTGRADUATE STUDIES

UNIVERSITY OF LAGOS

FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY (Ph. D) IS A RECORD OF ORIGINAL RESEARCH CARRIED OUT

BY

OMOGUNLOYE, OLUSOLA GABRIEL IN THE DEPARTMENT OF SURVEYING AND GEOINFORMATICS

----------------------------------------- ------------------------- -----

----------

AUTHOR‟S NAME SIGNATURE

DATE

---------------------------------------- ------------------------- ----

-----------

1ST

SUPERVISOR NAME SIGNATURE

DATE

----------------------------------------- ------------------------- -----

----------

2ND


DATE

----------------------------------------- ------------------------- -----

----------

v

3RD


DATE

----------------------------------------- ------------------------- -----

----------

1ST

INTERNAL SUPERVISOR SIGNATURE

DATE

----------------------------------------- ------------------------- -----

----------

2ND

INTERNAL SUPERVISOR SIGNATURE

DATE

----------------------------------------- ------------------------- -----

----------

EXTERNAL EXAMINAR SIGNATURE

DATE

----------------------------------------- ------------------------- -----

----------

SPGS REPRESENTENTIVE SIGNATURE

DATE

vi

ABSTRACT

The Horizontal Geodetic Network of Nigeria is made up of terrestrially arranged chains of

triangles augmented by precise traverses. The work on the network began early 19th century

but by 1930 the work was discarded and a re-observation of the network was carried out to

the highest possible accuracy then, enhanced with high order geodimeter traverses which

linked to other neighboring African networks.

The full network consists of 515 stations, with 2411 observations which comprise 2197 angular

observations, 40 Laplace azimuths and 174 measured distances, part of which substituted for

the sparse triangulation observations especially in the southern part of the country. The added

observations contributed to strengthening of the network in the 1977 adjustment which

however was not a holistic optimized adjustment, but rather, a phase adjustment. Based on the

1977 state of adjustment of the network, no meaningful distortion monitoring exercise can take

place until the network is adjusted by an optimized simultaneous technique in order to

ascertain the state and consistency of the network.

The use of the simulated annealing method, which has been successfully applied in other fields,

is presented for the classical geodetic problem of simultaneous adjustment of the entire

triangulation net using the least squares observation equation method. This method is an

iterative heuristic technique (a method of solving problems by learning from past experience

and investigating practical ways of finding a solution) in operations research. It uses a thermo

dynamic analogy (Cooling theory) to adjust a network of unstable stations (changes to gaseous

state) through fairly stable station coordinates (liquid state) to a stable station coordinates

(solid state) so as to offer a solution that converges in a probabilistic sense (statistically based)

to the global optimum. The simulated annealing method of optimization serves to help

determine the position of all triangulation stations by means of minimizing the volume of the

error hyper ellipsoid inherent in the solution to give an optimal configuration of the geodetic

network. Computer programs were developed using Matlab Software and run on an adequately

configured Pentium IV computer. Creation of an intelligent database was achieved through the

vii

interactive network of the data storage, processing, manipulation, analysis and retrieval of

results of the adjustment.

The result of the new adjustment produced a generally consistent trend of changes in the

distances and azimuths compared to the previous adjustments. Error analysis of all lines were

carried out and the respective standard errors in distances and azimuths were determined.

Relative and absolute error ellipses of all stations were determined and plotted. Statistical plots

and analysis of the error ellipses of the network stations were also determined. The absolute

and relative weakness/strength of the network stations coordinates after adjustment were

shown and confirmed by the error plots to have the following geometry error distributions.

That is, 90.5% of the 515 Network Stations fell within Network Standard deviation of 1- Sigma,

94.2% within 2-Sigma, while 98.3% fell within 3- Sigma

The distributions confirmed the high reliability of the Nigerian Horizontal Geodetic Network and

its data quality. Re-strengthening exercise would be necessary using either the 1-Sigma or 2-

Sigma region of network standard deviation

A data structure for the entire network was developed and necessary conclusions and

recommendations are made for further action to update/upgrade the precision of the Nigerian

horizontal geodetic network for future study.

Keywords: Geodetic network, Optimum adjustment, Simulated annealing technique, Least

squares technique.

viii

DEDICATION

This work is dedicated to The Living Word of God that made all things (John 1:1-5), Who is the

True Light that gives Light to Every man coming into the world (John 1:9), Who also is the

Alpha and Omega, the First and the Last (Revelations 1:11a), Who became flesh and dwelt

among us (John 1:14), Who is the Lamb of God Who took away the sins of the world (John

1:29), Who died for our sins (Romans 5:8), Who has a name above all names (Philippians 2:9),

Who has all authority in Heaven, on Earth and Under the Earth (Matthew 28: 18), Who is the

soon coming King, with rewards (Acts 1:11, Revelations 22:12), Who is therefore the KING OF

KINGS AND THE LORD OF LORDS (Revelations 19:16) – HIS NAME IS JESUS CHRIST.

THANK YOU LORD JESUS CHRIST - MY SAVIOR AND MY LORD

KEEP ME SAVED TILL YOU COME.

ix

ACKNOWLEDGEMENT

My greatest thanks go to the FATHER, SON AND THE HOLY SPIRIT for giving me the grace to

begin and complete this research work, which without doubt is a milestone accomplishment in

my life and in the nation (Nigeria) at large. The journey so far has been possible because His

presence has been continuously abundant in my life. The required knowledge, wisdom and

understanding for the fulfillment of this work had been solely from Him.

I will not be justified if at this state, I fail to acknowledge my spiritual and academic father as

well as my supervisor, Professor O.O. Ayeni and his wife (mummy). You have fully guided,

assisted, supported and encouraged me from the beginning of this research to the end, indeed

you are my God-sent angel, may the good Lord continue to uphold you and your family, thanks

uncountable times.

I am highly indebted to Professor J.B. Olaleye who by the abundant grace of God upon his life

became not only a leader but a father to everyone of us in the department. Your constructive

criticism and intellectual eagle eyes had contributed and added to the beauty and the eventual

accomplishment of this research work. May the good Lord always remember your labor of love

in Jesus name.

The technical aspect of this research work had been enhanced by my supervisor, Professor P.C.

Nwilo, who has stood his ground to criticize, correct and perfect the technical inputs in this

work. You always have riddles to pose that would assist one to determine to rise up in life.

Thank you very much sir.

My profound thanks go to Professor F.A. Fajemirokun, Professor Ezeigbo and late Professor

F.O. Egberongbe who during the period of my academic pursuit had contributed immensely to

this study. Thank you very much sirs.

My sincere thanks go to my academic fathers – Prof V.O.S. Olunloyo, Prof. O. Ibidapo - Obe,

Professor Olu Ogboja (late), Professor A. B. Sofoluwe (The Vice Chancellor), Professor M.A.

Salau (Dean of Faculty of Engineering), Professor O. T. Ogundipe (Dean of PG School). May the

good Lord continually bless and make you a blessing on earth.

x

Dr. J.O. Olusina (H. O. D., Department of Surveying and Geoinformatics) and Dr. O.T. Badejo

(Department of Surveying and Geoinformatics PG Coordinator), you are among the very few

that showed interest in my work by correcting and suggesting constructive ideas that brought

about the completeness of this work. Thanks a million times.

The accomplishment of this work has majorly been realized as a result of the cooperation of my

beautiful wife, MRS. HANNAH GOODNESS, ENIOLA OMOGUNLOYE, who had paid all the

utmost prize of a mother to my children through the period of this research, especially when I

had to pitch my tent in my office to be able to complete the work. She particularly showed a

rare asset of weathering all the storms of life with me especially during our rough financial

days. I pray I would be gracefully placed to reward your labor of love, thank you very much

I will, at this time, acknowledge MY BEAUTIFUL CHILDREN LIGHT OREOLUWA (LIGHTOO),

TRUTH GBOLAHAN (TRUTHEE), PEACE OTARU (PEACEE) AND PRAISE JESUS (PRAISOO), You

have all been wonderful, beautiful, great and glorious children. You are all blessed in Jesus

name.

Big thanks to Surveyor Lola (you assisted me and the Nation Nigeria to recover the lost data

for this research), Surveyor J.T. Ajayi, Mr. A.O. Adebisi, Mr. H. Mosaku, Mr. E.E. Epuh, Mr. O.E.

Abiodun, Mr. E.G. Ayodele, Mrs. A.M. Ayeni (Pastor), Mr. A. Alademomi, Surveyor M. Jegede,

Surveyor R. Adekola, Surveyor O.A. Babatunde, Ms. G.I. Inyang, Mrs. C.A. Sokenu, Mr. D.J.

Ikechukwu, Mrs. M.A. Adeyinka, Mr. O. Omojowo, Mrs. K. Sulaimon, Mrs. A. Ekanem, Mr

Adebayo, Mr Iluyemi, Brother Oshode Joseph Olusola, Mr. R. Adeyeye, Mr. Lawal, Alhaji

Jimoh, Alhaja Okonu, Mr. Oluga, Mr. Thomas, Mr. Durojaiye (late) and Mr. Ojo (late), I

appriaciate all your contributions

Associate Professor G.O. Oyekan (Daddy), and Professor D. E. Esezobor - I acknowledge your

concern for people’s progress and welfare. Dr. Adeosun, Dr. Ladi Ogunwolu, Dr. Fashanu, Dr.

Akanmu, Associate Professor Ayesimoju, Dr Kamiyo, Dr. S. Ojolo, Brother Joel, and Mummy

A. Sholiyi, thanks for all your supports and encouragements at all times. Pastor Taiye Adeoye,

Pastor F.A. Festus, thanks for your support.

xi

I say thank you to Pastor S.O. Adefarakan, Dr. (Mrs.) T.F. Ipaye, Mrs Ogunlewe, Miss Bimbo

Komolafe and Mr. Kingsley Okhiria, all of PG school, for all your persistent readiness to help,

assist, encourage, and attend to researchers, the Lord will crown all your efforts with success.

I appreciate my God Given Spiritual Children – Evangelist Yinka Akinsulire, Evangelist Elijah

Omogunloye, Evangelist Alaba Omogunloye, Brother Seun Adebayo (late – rest in peace),

Brother Wale Arowoiya, Brother Anuoluwa Ojemuyiwa, Brother Opeyemi Ayorinde, Pastor

Sunday Omogunloye, Pastor Idowu Omogunloye, Evangelist Emily Omogunloye, Deaconness

Lawal and the members of Jesus The Light, Word Outreach Ministries – You are all blessed in

Jesus Name. Thanks to Engineer and Mrs. B. R. Owolabi for all your support.

My elder sister (Mrs. Bukonla Emuleomo), her husband (Rev. Bayo Emuleomo) and Children;

Mr. Banji Omogunloye, his wife (Mummy Tops) and Children; Pastor Gbenga Omogunloye, his

wife (Mummy Dan) and children; my late brother Mr. Segun Omogunloye (rest in the bosom

of our Lord Jesus); Mrs. Kemi Adeoye and her wonderful family; Sister Funke Omogunloye;

Brother Dami; Brother Femi; Brother Deji; Sister Seye; and Brother Tobi - You’ve all been life

giver to me and my family. The good Lord will perfect all that concerns you in Jesus name.

I must not forget my in-laws - late Mr. Enesi (daddy), Mummy Sango (My mother in-law)

Reverend Dr. Peter Enesi, Sister Victoria, Reverend Abraham, Brother Godwin, Sister Bose,

Brother Lucky, Brother Johnson and your respective nuclear families – I love you all; you’ve

been so dear to me.

Finally to my Dear father, Pastor J.A. Omogunlye and my precious mother, Mrs. B.I.

Omogunloye, you have been there for me from pregnancy throughout my study days till date.

You have been an example of a good parent on the earth. The good news is that, Daddy, though

you are a carpenter and mummy, a trader, but you have a Ph.D holder as a son –

CONGRATULATIONS.

In conclusion, LORD JESUS, I return all the glory, honor, and majesty to you, for without you, we

can do nothing and by strength shall no one prevail.

xii

TABLE OF CONTENTS PAGE

CONTENTS

Title Page i

Certification ii

Abstract iii

Dedication v

Acknowledgement vi

List of Figures xiii

List of Tables xv

List of Appendices xvii

Glossary of Notations and Abbreviations xix

`

CHAPTER ONE: INTRODUCTION

1.1 Background of the Study 1

1.1.1 Previous Adjustments 2

1.2 Statement of the Problem 5

1.3 Aim and Objectives of the Research 6

1.4 Scope and Limitations of the Research 6

1.5 Significance of the Research 7

1.6 Research Questions 7

1.7 Definition of Operational Terms 8

xiii

CHAPTER TWO: GEODETIC NETWORK

2.0 Literature Review 11

2.1 Theoretical Framework 25

2.1.1 Computations of Geodetic Coordinates on the Ellipsoid 25

2.1.1.1 Direct Problem Equations 26

2.1.1.2 Indirect Problem Equations (Inverse Problem) 28

2.1.2 Observation Equations on the Ellipsoid 30

2.1.3 The Least Squares Method 32

2.1.3.1 Simultaneous Method 33

2.1.3.2 Sequential Method 35

2.1.3.3 Phase Method 37

2.1.3.4 Combined (Phase and Sequential) Method 40

2.1.4 Network Geometry Assessment 42

2.1.5 FGCC Standard and Specification for Geodetic Control Networks 43

2.1.5.1 Standards 43

2.1.5.1.1 Horizontal Control Network Standards 44

2.1.5.1.2 Monuments 45

2.1.5.2 Specifications 45

2.1.5.2.1 Triangulation 45

2.1.5.2.2 Instrumentation 46

2.1.6 Definition of Best Geometric Configuration 48

2.1.6.1 W hat is Optimization 50

2.1.6.2 Optimization Techniques 52

xiv

CHAPTER THREE: METHODOLOGY

3.1 General 55

3.1.1 Data Acquisition 55

3.1.2 Data Pre – processing and Quality Control 57

3.1.2.1 Instruments used/Date of Observation 57

3.1.3 Data Processing 57

3.2 Simulated Annealing Algorithm 58

3.2.1 Least Squares Equations Part 58

3.2.2 Equations used for the Optimization Part 60

3.3 Error Ellipse 66

CHAPTER FOUR: RESULTS AND ANALYSIS

4.1 Result 72

4.1.1 The Recovered Network Data Format 73

4.1.1.1 Network New Data Structure Format 73

4.1.1.2 Instruments used/Date of Observation of Network 74

4.1.2 Residuals Vector (V) after Adjustment 77

4.1.3 Stations Positional Corrections 80

4.1.4 Error Ellipse (Geometry) Computation 83

4.1.5 Relative Error Ellipse (Relative Geometry) computation 86

4.1.5.1 Standard Error in Azimuths (Orientation) computation 88

4.1.5.2 Standard Error in Distances (Scale) computation 91

4.2 Analysis of Results 94

4.2.1 Analysis of the Network residuals of Observations (V) after the Adjustment 94

4.2.2 Analysis of the Network Stations Position Correction (x) after the adjustment 95

xv

4.2.3 Analysis of the Network Error Ellipse (Geometry) after the Adjustment 96

4.2..4 Analysis of the Network Relative Error Ellipse (Relative Geometry) 98

4.2.4.1 Analysis of the Network Standard Error in Azimuths (Orientation) 98

4.2.4.2 Analysis of the Network Standard Error in Distances (Scale) 98

4.2.4.3 Analysis of Network Standard Deviation 99

4.2.5 Statistical Paired Sample Test analysis of the error ellipse values of the 33

stations in 1977 and their corresponding values in 2009 adjustment. 100

4.2.6 Comparison of the 515 Stations Coordinates in the 1977 102

and 2009 Adjustments.

4.2.7 The summary of the Results in 1977 and 2009 Adjustments 105

CHAPTER FIVE: CONTRIBUTIONS TO KNOWLEDGE, CONCLUSIONS AND

RECOMMENDATIONS

5.1 Conclusion 106

5.2 Contribution to Knowledge 110

5.3 Recommendation/Further research Work 110

REFERENCES 112

xvi

LIST OF FIGURES PAGE

Figure 1.0 Graphical plot of Nigerian Horizontal Geodetic Network 4

Figure 2.1.1 Ellipsoidal Polar Triangles 26

Figure 2.1.2 Sample Triangle 30

Figure 2.1.3a Network View for the Simultaneous Method 34

Figure 2.1.3b Network view for the Sequential Method 36

Figure 2.1.3c Network view for the Phase Method 39

Figure 2.1.3d Network view for the combined (Phase and Sequential) Method 41

Figure 3.1 Plot of Cooling Scheme 63

(Gradual Stabilization of Network Coordinates)

Figure 3.2 Plot of Function of Free Movement Structure 64

Figure 3.3 Plot of Standard Deviation of Network Correction Vector 65

Figure 3.4 Graph of Standard Ellipse 69

Figure 4.1a Sample Triangular Structure 74

Figure 4.1.2a Plot of V matrix vector in the Network 79

Figure 4.1.2b Plot of V matrix of some CFL lines/type of Observation 80

Figure 4.1.2c Plot of V matrix of the of the first 7 Network Triangles 80

Figure 4.1.3a Plot of Positional corrections of 515 stations in the Network 82

Figure 4.1.3b Plot of the XL and ML Secondary Stations with Large 82

Positional Corrections

Figure 4.1.4a Plot of Network Stations with Larger Absolute Error Ellipse Sizes 85

Figure 4.1.4b Plot of Network CFL Stations with Large Absolute Error Ellipse Sizes 85

Figure 4.1.4c Plot of Network ML Stations with Larger Absolute Error 86

Ellipse Size

xvii

Figure 4.1.5a Plot of Network Stations with Larger Relative Error Ellipse Sizes 88

Figure 4.1.5b Plot of Network Standard Error Vector (S.e) in 90

Azimuhs (orientation)

Figure 4.1.5c Plot of the Network CFL Stations Cross Sectional view Standard 90

Error (S.e) in Azimuhs.

Figure 4.1.5d Plot of the Network XL and ML Stations Cross Sectional 91

View Standard Error (S.e) in Azimuths.

Figure 4.1.5e Plot of Network Standard Error Vector (S.e) in Distances (Scale) 93

Figure 4.1.5f Plot of the Network CFL Stations Cross Sectional view Standard 93

Error (S.e) in Distances

Figure 4.1.5g Plot of the Network XL and ML Stations Cross Sectional 94

View Standard Error (S.e) in Distances

Figure 4.2.5a Plot of the Paired Sample Test Results of the Absolute Error 102

Ellipses of 33 stations in the 1977 and 2009 at 95% Confidence Level

Figure 4.2.6a Plot of the Optimized Adjusted Horizontal Geodetic 104

Network of Nigeria

xviii

LIST OF TABLES PAGE

Table 2.0 (a) Invar Taped Baselines 12

Table 2.0 (b) Old Azimuths (Pre-1945) 13

Table 2.0 (c) The Nigerian Triangulation Network 15

Table 2.0 (d) Coordinates of Minna Datum L-40 computed from astromical

Observation stations in widely separated areas. 19

Table 2.0 (e) Comparison of GNSS Systems 25

Table 2.1.5.1.1 Distance Accuracy Standards 44

Table2.1.5.1.2 Network Geometry 46

Table 2.1.5.1.3 Instrument Order and Class 47

Table 2.1.5.1.4 Theodolite Observation 47

Table2.1.6.1 Summary of Optimization Solution Method 53

Table 3.1a Sample Angular Data 56

Table 3.1b Sample Azimuths Data 56

Table 3.1c Sample Scale check Data 56

Table 4.1a Sample Triangular Arranged Stations ID of the Network 73

Table 4.1b Sample Triangular Arranged Lines of the Network 73

Table 4.1c Sample Triangular Arranged Angles of the Network 74

Table 4.1d Instrument used for the Nigerian Horizontal Geodetic Network 75

Table 4.1e Sample Network Assessment based on Instruments and year 76

of Observation

Table 4.1.2a Results of the Sample Network Residual Vector (V) 78

after adjustment

Table 4.1.3a Results of the Sample Network stations Positional Corrections 81

xix

Table 4.1.4a Result of Sample Network Error Ellipse Computation 84

Table 4.1.5a Results of Sample Network Relative Error Ellipse Computation 87

Table 4.1.5b Results of Sample Network Standard Error in Azimuths 89

Computation

Table 4.1.5c Results of Sample Network Standard Error in Distances 92

Computation

Table 4.2.4.3a Classification/Assessment of Network A-Posteriori Variances 99

of unit weight

Table 4.2.5a Comparison of Error Ellipse Sizes for 33 Stations used 101

in 1977 and 2009 Adjustment

Table 4.2.5b Statistical Paired Sample-Test on the Extracted Absolute 102

Error Sizes of 33 stations in 1977 and 2009 Adjustment

Table 4.2.6a Comparison of the Sample of Final Adjusted coordinates in 103

1977 and 2009

Table 4.2.7a Comparison of the Summary of 1977 and 2009 Adjustments 105

xx

LIST OF APPENDICES PAGE

Appendix (I a) Print of the Network Observed Angles (Now in soft copy) 123

Appendix (I b) Print of the Network Observed Azimuths (Now in soft copy) 129

Appendix (I c) Print of the Network Observed Distances (Now in soft copy) 130

Appendix (I d) The Programs written for the entire Research in Matlab 134

Appendix (II a) 1054 Triangular Arranged Network Stations Identity (ID) 148

Appendix (IIb) 1054 Triangular Arranged Network Lines/Stations Identity (ID) 155

Appendix (IIc) 1054 Triangular Arranged Network Observed Angles 162

Appendix (III) 3162 Network Residual Matrices Vector 169

Appendix (IV) 515 Network Stations Positional Corrections 189

Appendix (V) 515 Network Stations Error Ellipses Values (Network Geometry) 193

Appendix (VI) 3162 Network Lines/Stations Relative Error Ellipses Values 196

(Relative Geometry)

Appendix (VII) 3162 Network Standard Error in Azimuths of Lines (Network Orientation) 216

Appendix (VIII) 3162 Network Standard Error in Distances of Lines (Network Scale) 236

Appendix (IX) Comparison of the Final Coordinates in 1977 and 2009 Adjustment 256

(Stations Correction)

Appendix (X) Assessment of Instruments and Date of Observation of the Network 275

Appendix (XI a) Plot of Error Ellipses for all Chains/Stations in the Network 278

Appendix (XI b) Plot of Error Ellipses for A, some ML and XL Chains/Stations 279

Appendix (XI c) Plot of Error Ellipses showing the Weakness of the CFL Chain/Stations 280

Appendix (XI d) Plot of Error Ellipses showing the Western end of the CFL Chain/Stations 281

Appendix (XI e) Plot of Error Ellipses showing the Central part of the CFL Chain/Stations 282

Appendix (XI f) Plot of Error Ellipses showing the Eastern end of the CFL Chain/Stations 283

xxi

Appendix (XI g) Plot of Error Ellipses for D, and L Chains/Stations 284

Appendix (XI h) Plot of Error Ellipses for A, B, C, E, F, H, M, N, X, XL, ML and MR 285

Chains/Stations

Appendix (XI i) Plot of Error Ellipses for B, E, H, K, N, and P Chains/Stations 286

Appendix (XI j) Plot of Error Ellipses for A, E, F and G Chains/Stations 287

Appendix (XI k) Plot of Error Ellipses for K, and A part of the CFL Chains/Stations 288

Appendix (XI l) Plot of Error Ellipses for D, L and a part of U Chains/Stations 289

Appendix (XI m) Plot of Error Ellipses for C, F, H and M Chains/Stations 290

Appendix (XI n) Plot of Error Ellipses for A, E, F, G, XL, and ML CFL Chains/Stations 291

Appendix (XI o) Plot of Error Ellipses for B, K, N and A part of the CFL Chains/Stations 292

Appendix (XI p) Plot of Error Ellipses for C, H, M, P and U Chains/Stations 293

Appendix (XI q) Plot of Error Ellipses for R, CFL , XL Chains/Stations 294

Appendix (XI r) Plot of Error Ellipses for P and U Chains/Stations 295

Appendix (XI k) Plot of Error Ellipses for P and U Chains/Stations 296

Appendix (XII a) 49 Stations with Standard Error > 1 sigma that needs re-observation 297

Appendix (XII b) 30 Stations with Standard Error > 2 sigma that needs re-observation 298

Appendix (XII c) 9 Stations with Standard Error > 2 sigma that needs re-observation 299

xxii

GLOSSARY OF NOTATIONS AND ABBREVIATIONS

IAUE

Iterated Almost Unbiased Estimator

(LA-LG) Laplace correction

(C ) The computed value of an angle, azimuth, or distance

(O) The observed value of an angle, azimuth, or distance

A12 Azimuth of station 1 to 2

ACO Ant Colony Optimization

Ai j Azimuth of geodesic from i to j

ATPA Normal Matrix (N)

AZo The normal defined by the actual gravity vector

BA Bacteriologic Algorithms

CE Cross-entropy

The Stochastical covariance matrix model

D.F Degree of freedom

Correction to the direction from a normal section to geodesic.

Correction to the direction for the height of the observed point

D2 The corrected azimuth (Total direction correction)

The correction for the deflection of the vertical

Di j Length of geodesic from i to j

DOS. Directorate of Overseas Surveys

dφ, dλ, The change in latitude, longitude between two points

E1, E2 Easting of the ends of the line

EDM. Electromagnetic Distance Measurement

EO Extremal Optimization

EP Evolutionary Programming

ES Evolution Strategies

GGA Grouping Genetic Algorithm

GP Genetic Programming

http://en.wikipedia.org/wiki/Ant_colony_optimization

http://en.wikipedia.org/w/index.php?title=Bacteriologic_Algorithms&action=edit

http://en.wikipedia.org/wiki/Cross-entropy_method

http://en.wikipedia.org/wiki/Extremal_optimization

http://en.wikipedia.org/wiki/Evolutionary_programming

http://en.wikipedia.org/wiki/Evolution_strategies

http://en.wikipedia.org/w/index.php?title=Grouping_Genetic_Algorithm&action=edit

http://en.wikipedia.org/wiki/Genetic_programming

xxiii

The prime vertical deflection of the vertical

h2 Height of target station above spheroid in metres.

HS Harmony search

IAUE Iterated Almost Unbiased Estimator

IEA Interactive Evolutionary Algorithms

S The chord distance measured between the two points A and B

L Vector of observation

L1 Vector of the First set of observation

L2 Vector of the second set of observation

L40 Minna Datum Station Symbol

λi Provisional longitude of station i

Øi Provisional latitude of station i

M Number of observations

MA Memetic Algorithm ( hybrid genetic algorithm)

MPV Most Probable Value

n Number of unknown Parameters.

N The normal matrix

Q The weight coefficient matrix

N1, N2 Northings of the ends of the line

NA Gaussian Adaptation (Normal or Natural Adaptation)

Ø Latitude of target

P Estimated standard error of observation

pi Meridional radius of curvature of spheroid at i

Deflection of the vertical (difference in directions of two gravity vectors)

R The radius of curvature in the azimuth of the line

S.e Standard error/deviation for measured angle, distance and azimuth

a-posteriori variance of unit weight of adjusted observation

o

a-priori variance of unit weight of observation

SA Simulated Annealing

http://en.wikipedia.org/wiki/Harmony_search

http://en.wikipedia.org/wiki/Interactive_evolutionary_algorithm

http://en.wikipedia.org/wiki/Memetic_algorithm

http://en.wikipedia.org/wiki/Gaussian_adaptation

http://en.wikipedia.org/wiki/Simulated_annealing

1

SO Stochastic Optimization

The unbiased estimate of the convariance matrix of adjusted parameters

Trace (Ʃʟа) Trace of the variance covariance matrice of the adjusted observation

Trace (Ʃxа) Trace of the variance covariance matrice of the adjusted parameters

TS Tabu search

V The vector of residuals

vi Radius of curvature at right angles to meridian at i

Vi j k Residual of the observation at station ijk.

vTpv Sum of the squares of the residual

W Weight matrix

Xa Vector of the unknown adjusted parameter

X Vector of the correction to unknown Parameter

The meridian deflection of the vertical

Xo The approximate values of the unknown parameters

Zo The astronomic zenith

Ϭx Error Ellipse Parameter (Semi Major Axis)

Ϭy Error Ellipse Parameter (Semi Minor Axis)

Ψ Error Ellipse Parameter (Orientation of Ellipse)

La Adjusted observation (Angles, Distances and Azimuth)

F(La) Function of the adjusted observation (Angles, Distances and Azimuth)

Lb Vector of unadjusted observation

F(Xa) Function of adjusted Parameter

Xo1 1st set of approximate solution vector of Parameter

Xo2 2nd set of approximate solution vector of additional Parameter

Xa2 2nd set of Adjusted solution vector of additional Parameters

Xo Approximate Vector of the Parameter.

http://en.wikipedia.org/wiki/Stochastic_optimization

http://en.wikipedia.org/wiki/Tabu_search

2

CHAPTER ONE

INTRODUCTION

1.1 BACKGROUND OF THE STUDY

A Geodetic Control network is a collection of identifiable stable points on the surface of the

earth tied together by observations of high accuracy. From these observations, the positional

coordinates of points are computed and published. This framework of coordinated point provides

a common basis for all surveying and mapping operations in a suitable reference system (Anon,

1971; Alliman and Hoar, 1973).

There are three National Geodetic Control frameworks, namely:

Horizontal Control framework, which is the focus of this study;

Vertical Control framework and;

Gravity Control framework;

The Nigerian Horizontal Geodetic Network is a network of terrestrial points made up of

triangulation, trilateration and traversing sub-networks, a larger part of which was observed

between 1930 and 1960. The thickly vegetated terrain of the southern part of Nigeria made the

use of triangulation method a difficult task, hence a system of Primary Traverses formed the

position control between 1923 and 1940 with later addition of microwave EDM traverses in the

south east. From 1960 to 1968, series of primary traverses were used to extend the triangulation

in the northern part of Nigeria, such as: the Trans-Africa Twelfth Parallel Geodimeter traverse

carried out by the U.S. Corps of Engineers which connected the triangulation at eight points; a

number of mapping projects were also carried out shortly after independence; controls provided

by aero triangulation were integrated into the network to fill many gaps between the main

triangulation chains; and additional scale and azimuth measurements were made to strengthen

the entire triangulation network (Anon, (1936 and 1961); Close, (1933); De Normann, (1933);

Dept. of the Army, (1953)).

The 1977 adjustment of the network, which integrated all the stations was a phase adjustment,

hence inadequate for a holistic optimal solution of the Nigerian Horizontal Geodetic Network.

3

Consequently, there is a need to implement a holistic optimal adjustment that will minimize the

volume of the error hyper ellipsoid inherent in the solution to give an optimal configuration of

the network.

This study shall comprise:

brief review of literatures on the Geodetic Network of some countries,

models used in adjustment of horizontal geodetic networks.

methods of holistic optimization using the Simulated Annealing Method.

results and analysis of results.

conclusions, contribution to knowledge and recommendations based on the outcome of the

research or study.

The end products of these work would assist in giving an optimal set of coordinates of stations in

the network which are often used directly or indirectly for:

Planning and carrying out national and local projects.

Development delineation of state and international boundaries.

Utilization of natural resources.

National defense, land management and monitoring of crustal motion.

Supporting the conduct of public business at all levels of government.

General basis of nationwide surveys, maps, and charts of various kinds (Chedtham, 1965;

Clark, 1965; Alliman and Hoar, 1973; Clark, 1965; Charles, 1942; Dare, 1995; Choi, 1998).

1. 1.1 Previous Adjustments

The network had been adjusted by some researchers and agencies. The first adjustment carried

out between 1930 and 1940 by the Directorate of Overseas Surveys was not completely

satisfactory due to misclosures between base and azimuth checks (Field, 1977). Another

adjustment by the U.S. Topocom used 440 existing stations and some observations made up to

1968. It excluded the 12th

Parallel Traverse (Field, 1977). Other adjustments between 1968 and

1977 were made on individual parts of the network, such as the Primary Traverses, which were

adjusted to connect the triangulation at eight points. The 12th

Parallel Traverse which was earlier

4

adjusted to a different origin but same spheroid (Clarke 1880), was re-adjusted to be consistent

with the triangulation network in the 1977 phase adjustment (Field, 1977). Error analysis carried

out on selected lines of the network, showed areas of strength and weakness in scale and

orientation. The standard errors indicated a weakness in the southern area.

Since none of these adjustments was able to provide a simultaneous optimal solution of network

stations coordinates, this research seeks to provide an independent simultaneous optimal

adjustment of the Nigerian Horizontal Geodetic Network .

6

1.2 STATEMENT OF THE PROBLEM

The following problems were peculiar to the past adjustments:

adjustments comprises misclosures between base lines and azimuth checks.

adjustments done individually for some sections of the network.

none of the adjustments integrated all network stations except that of 1977 which also did not

integrate all stations simultaneously.

none of the past adjustment was an optimal adjustment, hence adjustments did not converge

with a constraint probability to the global optimum in terms of:

Station coordinates;

Variance of unit weight;

Traces of variance-covariance matrices of the adjusted parameters and

observations;

Past adjustment only analyzed error on selected lines in the network;

The data was not properly structured;

There has always been the need to achieve an optimum adjustment in order to ascertain the

reliability of the network, its geometry and subsequent stations to be re-observed, so as to

strengthen the network in the optimal sense.

In the process, it is necessary to carry out new observations on the network using modern

satellite techniques. These provide a faster and easier method of data acquisition and assist in

improving the accuracyand strengthening the network stations geometry as well as providing a

platform for distortion study of the network (Field, 1977).

This research seeks to provides a simultaneous optimal adjustment of the network thereby

providing indices of the network strength and weakness at all stations coordinates, distances, and

azimuths for the determination of the network reliability and subsequent program for network

stations upgrades.

7

1.3 AIM AND OBJECTIVES OF THE RESEARCH

The aim of this research is to carry out a holistic and optimal adjustment of the Nigerian

horizontal Geodetic Network. This can be met through the achievement of the following

objectives:

(i) Carrying out a simultaneous adjustment of the Nigerian horizontal geodetic network

using the Simulated Annealing Optimization method and determination of the network

reliability.

(ii) Determining the Network stations geometry distributions in-terms of the network

standard deviation (a posteriori variance of unit weight value) after the adjustment.

(iii) Identifying areas of strength and weakness in the network stations geometry for stations

upgrade (re-observation) in order to strengthen the network.

(iv) Developing a generalized network through the creation of a comprehensive intelligent

database for the whole network which can search, query and perform calculations of any

desired parameters of the network.

(v) Recovering the lost raw data for the Nigerian Horizontal Geodetic Network before the

adjustment by searching necessary libraries within and outside Nigeria.

1.4 SCOPE AND LIMITATIONS OF THE RESEARCH

The scope of this research includes:

(i) Adjusting the Nigerian Horizontal Geodetic Network optimally using the 1977

adjustment data which are to be recovered.

(ii) Determining network reliability.

(iii) Determining the network stations geometry.

(iv) Plotting the network views, chains and stations error geometry.

(v) Creating an intelligent database for the network.

(vi) Comparing results of adjustment with the 1977 adjustment.

(vii) Determining area of strength and weakness within the network.

(viii) Providing a program for the network upgrade.

8

This research is limited to the existing data, whose quality checks had been determined in the

previous adjustments (Field, 1977) and are briefly discussed later in Section 3.1.2. This research,

however, does not include fresh field observation.

1.5 SIGNIFICANCE OF THE RESEARCH

This research would provide for the first time in the history of Nigeria, an optimal holistic

adjustment and insight into the geometrical strength of the Nigerian Horizontal Geodetic

Network. It will assist in the right choice of the network stations for upgrading and provide a real

platform for assessing the past, present and future state of network distortion as well as the

creation of an intelligent data structure for the network.

1.6 RESEARCH QUESTIONS

Ayeni, et. al. (2005), confirmed that the simultaneous least square mathematical technique has

the best suitability criteria over the phase, sequential and combine techniques [Section 2.0 (C)].

The Research Questions were formulated as follows:

(i) What method would be appropriate for optimal simultaneous adjustment of the Nigerian

horizontal geodetic network and its reliability?

(ii) How can the Network stations geometry distributions be determined?

(iii) How can one identify areas of strength and weakness in the network stations geometry

for stations upgrade (re-observation) in order to strengthen the network.

(iv) How can one develop a generalized network through the creation of a comprehensive

intelligent database for the whole network which can search, query and perform

calculations of any desired parameters of the network?

9

1.7 DEFINITION OF ACRONYMS OPERATIONAL TERMS AND SYMBOLS

FGCC: Federal Geodetic Control Committee in charge of standards and specifications.

ZOD: Zero-Order Design problem (ZOD): aims at datum definition.. Hence, in the ZOD, datum

points are the variables.

FOD: The First-Order Design (FOD) optimizes station positions and the observations to be

made. The variable in this problem is the observations‟ design matrix.

SOD: The Second- Order Design problem (SOD) aims at designing the observation weights so

that the solution is able to accomplish prescribed precision. The variable in this problem is the

observation weight matrix.

TOD: The Third-Order Design problem (TOD) deals with optimal network densification Its

design variables are the observations‟ design matrix and the observations weight matrix.

(SA): Simulated Annealing is a global optimization technique that seeks the lowest energy

instead of the maximum fitness and can also be used within a standard GA algorithm by starting

with a relatively high rate of mutation and decreasing it over time along a given schedule.

(GA): Genetic Algorithm is used to find exact or approximate solutions to search problems.

They are categorized as global search heuristics. Genetic algorithms are a class of evolutionary

algorithms which uses Evolutionary Biology, such as inheritance, mutation, selection, and

crossover or recombination.

(NA): Gaussian Adaptation (NA) is a normal or natural adaptation. NA maximizes mean fitness

rather than the fitness of the individual and is also good at climbing sharp crests.

(IAUE) Iterated Almost Unbiased Estimator is the computed variance factor ratio for a

new survey combined with network data. It is used in survey standards and specifications.

(LA-LG): Laplace correction stands for the difference between the astronomic azimuth and

Geodetic azimuths at any station.

U.S. Topocom: The United State Topography survey arm in the early 19th

century

12th

Parallel Traverse: Traverse that ran across the 12th

Parallel latitude of the country

10

Triangulation: A precise method of survey of stations widely separated from each other and

conveniently located on top of hills and mountains, in which angles within triangular formations

of stations are measured to a high precision and accuracy.

Trilateration: A precise method of survey of stations widely separated from each other and

conveniently located on top of hills and Mountains, in which lengths of triangular formations of

stations are measured to a high precision and accuracy.

Traversing: A precise method of survey of stations widely separated from each other and

conveniently located in which lengths and angles of triangular formations of stations are

measured to a high precision and accuracy.

Ratio of the a-posteriori variance covariance after adjustment to the ratio of a-priori

Variance of unit weight at the beginning of the adjustment.

GPS: Global Positioning System equipment that uses satellite system to determine position of

any point of interest.

G. T.: Great Trignometrical Triangulation Network of India

3-D: Three Dimensional view positions (Latitude, Longitude and Height).

TRF: Terrestrial Reference Framework.

ETRS89: European Terrestrial Reference System/datum 1989.

OSGB36: Ordnance Survey Great Britain 1936

ODN: Ordinance Datum Newlyn

DOS. Directorate of Overseas Surveys.

EDM. Electromagnetic Distance Measurement.

G P Genetic Programming

L40 Minna Datum Station which is the origin of the Nigerian horizontal Geodetic Network.

http://en.wikipedia.org/wiki/Genetic_programming

11

MPV Most Probable Value

S.e Standard error for measured angle, distance and azimuth

: A-posteriori variance of unit weight after adjustment

: A-posteriori variance of unit weight before adjustment

The unbiased estimates of the covariance matrix of adjusted parameters X

12

CHAPTER TWO

GEODETIC NETWORK

2.0 LITERATURE REVIEW

The literature review is discussed under the following subheading in order to justify as well as

provide a comprehensive view of relevant issues on the subject of this research work.

Nigerian Triangulation Network.

Typical network adjustment programmes already carried out.

Survey Requirement.

Application of Optimization Technique to Triangulation network.

Application of the Global Navigation Satellite Systems (GNSS) in geodetic work.

The above issues can be discussed as follows:

(A) Nigerian Triangulation Network: Field, (1977) stated that the Primary Triangulation

Network of Nigeria consists of 441 stations distributed in a series of chains over most of the

country. In the extreme North East of Nigeria, the land is flat. In the South the relief is subdued,

and the ground is covered with high tropical forests. Triangulation is difficult to practice in these

two areas and the network did not therefore extend into them. The stations were formed into a

series of 18 chains; running roughly North-South and East-West across the country, meeting at

sixteen junction points, and leaving extensive lacunae between them.

By early 1932, inspection of the individual angular measurements, and examination of the chain

misclosures made it obvious that most of the work accomplished up to 1930 was not sufficiently

accurate. Between 1931 and 1939, therefore, the bulk of the main triangulation network as it

exists today was beaconed and observed. Re-observation in most cases produced a significant

improvement in apparent precision. Bradley (1939) notes that new angular measurements in the

Ilorin-Eruwa chain reduced the average triangular misclosure from 1.70” to 0.59” for 35

triangles. The angular observations of the main network were completed by 1939 except for the

UDI-CAMEROON (C), YOLA- NKAMBE (F), LAFIA-OGOJA (H), NKAMBE-CAMEROON

–AFIKPO (M), and MAKURDI- LOKOJA (P) chains. The F and M chains were established to

provide primary control along the eastern borders of the country.

13

Further observational work on the network started in the mid-1950‟s and was almost completed

in 1961 when the then Southern Cameroons separated from Nigeria and joined the Republic of

Cameroon. The P chain was established in the late 1950‟s to provide primary control along the

lower Benue Valley. In 1963 certain stations in the Lokoja area were re-observed to eliminate a

5'' error. This was eventually found as expected at U67, and with this exercise, the main

triangulation network was completed (Field, 19977). The following attributes of the network are:

(I) Base Lines

There were originally nine taped baselines, of which the first four were measured very early in

the history of the triangulation, which includes those at Eruwa and Naraguta, measured between

1910 -1912 by a party of Royal Engineers and were also used in the 1977 adjustment. Calder

(1936) confirmed their acceptability in general terms and later measurements gave supporting

evidence as to their reliability. Other base lines were remeasured in the1930‟s as new work on

the triangulation enabled adjustments to be made to them. By the 1930‟s the original Udi base

was considered inaccurate and discarded due to reason of poor misclosure on the base of the

chain coming from Minna and Ilorin. It was thought that errors were caused by poor base

extension figures and the station U80A. A traversed base was observed in 1939 and 1950

between U12 and U34 in the same area, but its final computed length and reliability were not

easy to establish. In 1937, the first Udi base was measured to replace an earlier base in the same

area. Here again, preliminary computations revealed a large misclosure through the chains and

suspicion centred on the base extension observations. It was found possible to measure a side of

Table 2.0 (a): INVAR TAPED BASELINES

Location Line Date Length (m) Linear* Misc.

Eruwa L1-L2 1910-11 16 064.6832 1:2.2m

Naraguta L1-N2 1912 4 148.9797 1:2.1m

Minna L40-L41 1928 5 664.31069 1:13.3m

Chafe R50-R51 1935 9 875.563023 1:4.4m

Ilorin L52-L53 1936 10 410.93092 1:8.9m

Yola A1-A2 1936 10 647.39748 1:8.9m

Rijau D28-D30 1937 32 845.42362 1:5.8m

Udi U1-U2 1931 6 935. 64416 Not

established

U12-U34 1945-50 26 577.621 Not

established

(Field, 1977)

14

the triangulation scheme with the aid of a subsidiary triangle at each end (Field, 1977). The

invar taped baselines data are shown in Table 2.0 (a).

(II) Astronomical Observations

Astronomical observations associated with the base line measurements were made, though not

necessarily completed at the same time for azimuth, but usually along one side of the base

extension figures close to the base itself. Longitudes were observed at three places. The use of

telegraph must have been adopted for the time check since the observations took place before the

1920‟s. Close, (1933) confirms this and quotes an accuracy of only 15” of arc.

Another azimuth observation was made between 1920 and 1930 before the 1st World War at

some of the oldest bases observed. The reason was presumably to eliminate some of the

misclosures which had emerged in the adjustment computations. The new azimuth line did not

always coincide with the old, for instance at Naraguta (Jos) the original azimuth was observed

along the baseline from N1 to N2 in 1912. By the time re-observation was thought necessary in

1928, the town of Jos had developed in the valley across which the base had been measured. The

new azimuth was then observed from N1 to N3, a station in the first base extension figure.

At Ilorin the base and azimuth had been measured at a late stage (1936) to try and localize the

apparent errors in the Minna-Ilorin-Rijau chains. The base succeeded in splitting up the length

Table 2.0 (b): Old Azimuths (Pre-1945)

Location Date At R. O. Value Quoted S.e

Eruwa 1910 L1 L1 - -

1928 L1 L1 2130 01'37.21'' 0.55''

Naraguta 1912 N1 N1 1850 35' 09.62'' -

1928 N1 N1 2550 35' 03.01'' 0.87''

Minna 1928 L40 L40 190014' 56.37'' 0.47''

1938 L39 L39 244035' 49.28'' 0.50''

Chafe 1935 R51 R51 176049' 51.98'' 0.87''

Ilorin 1936 L53 L53 334022'07. 16'' 0.36''

1937 L19 L19 316034' 32.26'' 0.39''

Yola 1936 A1 A1 770 32' 28. 71'' 0.42''

Rijau 1937 D28 D28 690 03' 19.14'' 0.44''

Udi 1913 U2 U2 193038'43. 57'' 0.65''

1945 U12 U12 010 17' 44.72'' 0.62''

NOTE: U2 - U1 not in present network, (Field, 1977).

15

error between Minna and Rijau, but large azimuth misclosures between Ilorin and the other base

figures were now found. In 1937, another azimuth was observed along an early line, the external

side of the Ilorin base which not L19-L21 to check the original azimuth. This confirmed the

earlier observations from L52-L53. The azimuth misclosures were eventually improved by

selecting angle observations in the Minna base extension net. The Table 2.0 (b) shows the

azimuths observed before 1945 (Field, 1977).

(III) The First Adjustment

The first adjustment was aborted by the second World War, and no further computation was

possible until the late 1950‟s when a review was made of the work done up to that time. This is

summarized by Cooper, (1974) and by Stamers and Sonola, (1955). As a result, angular re-

observations were made in the Lokoja area; D.O.S. was approached to do a completely new

adjustment of the network, and additional requirements of scale and orientation were specified

before such an adjustment could usefully be made.

(IV) Azimuth and Scale Check Programme:

Before the start of the Civil War, only 9 out of 20 Laplace stations and 16 out of 20 Scale check

triangles were completed, using Wild T4 theodolite, accurate timing piece, and Tellurometer

MRA2 and MRA3 instruments.

(V) Twelfth Parallel Survey

The Twelfth Parallel Survey was observed by the United States Corps of Engineers, with Federal

Surveys between 1967 and 1971 and covered a total of 81 points including Connections at 8

points to the Nigerian Primary Triangulation network, and 6 lower order stations.

(VI) Topocom Adjustment

In 1968, the so called Topocom Adjustment was done by the U.S Corps of Engineers, using data

available to them by then, an adjustment was made of most of the main triangulation network

stations. However details of the adjustment are not available.

16

(VII) Directorate of Overseas Survey

Two kinds of measurements done by the D.O.S are supplements to the primary network. They

are: (a) Tellurometer measurements of lengths of primary triangulation lines. (b) EDM traverses

running between stations of the primary chains.

(VIII) Primary Traverses in the South

Calder (1936) describes the measurement and procedure in some detail. Zeiss traverse equipment

and 500 foot steel tapes were used, and a linear misclosure of more than 1/60,000 was obtained.

These traverses were linked to the primary triangulation.

Table 2.0 (c): The Nigerian Triangulation Network

Chain Date No Mean Remarks

Observed Stations Side(Km)

Minna base net 1932 13 23.5

Karaguta base net 1932 11 8.1

Chafe base net 1935 8 23.8

Ilorin base net 1936 10 29.6

Yola base net 1936 6 12.4

Minna-Chafe (N) 1935 16 32.2

Naraguta-Chafe (N/K) 1929-35 56 18.4 Rearranged „37

Minna-Naraguta (N) 1933 23 40.4

Birnin Gwari-Naraguta (B) 1932 23 34.6

Kwongoma-Rijau (D) 1934-35 19 35.4

Rijau-Chafe ( R) 1935 44 36

Minna-Udi (L/U) 1930-37 45 34.8 Rearranged‟ 63

Ilorin-Eruwa (L) 1937 19 37.5

Ilorin-Rijau (D) 1934 33 42.6

Bauchi-Yola (A) 1933-36 42 36.2 Includes Yola bases net

Ropp,Yola (E) 1937 20 47.5

Udi-Cameroons (C ) 1945-54

1955-57

37 37.7

Lafia-Ogoja (U) 1945 14 45.5

Makurdi-Lokoja (P) 1957 16 45.4

Yola Nkambe (F) 1958-59 21 -

Nkambe-Takum (M) 1958-59 8 4.5 Part only of N chain

Biu. Madagali (C ) 1952-50 14 3.3

(Field, 1977).

17

The work was completed in 1939 and because of the vulnerable nature of traverses along the

main roads, many of the stations can no longer be recovered. A further series of primary

traverses, using EDM equipment, was established in the Niger Delta area by Shell B.P. These

connect to the primary triangulation near Idah (U97) and Arochukwu (C88). The summary of the

triangulation network is shown in Table 2.0 (c).

(IX) Controls used for Adjustment

The Controls used for the adjustment can be conveniently grouped into five categories:

1. The main triangulation network

2. The Twelfth parallel traverses

3. D.O.S. topographical traverses

4. The standard traverses

5. The EDM traverses by Shell B.P.

The configuration of the present Nigerian Horizontal Geodetic Network comprises of 515

stations. This includes all the 441 main triangulation network stations, the 39 stations of the

Twelfth Parallel Traverse and the 35 D.O.S. traverse stations.

In the Station numbering, each triangulation or traverse station bears a unique number, and is

also identified by the name of the hill feature on which it is established. The primary

triangulation points were originally numbered sequentially and prefixed by an initial letter

Provisional Coordinate Values: The observation (parametric) equation method of least squares

adjustment technique, used in the exercise, computes the corrections to be applied to a set of

provisional positional values for each station in the network. The result of the computation

generally were such that the input provisional values were sufficiently close to the final figures

so that no direction changes by more than one minute of arc, and no length changes by more than

1:400 exist (Bomford, 1971).

Provisional latitude and longitude were obtained for the stations in the networks from the

following sources:

Main Triangulation Network: For those stations included in the 1968 U.S. Topocom

adjustment, the final values were used as provisional coordinates. For certain stations not

18

included a provisional latitude and longitude were taken from either the 1930‟s first adjustment

or from the D.O.S. preliminary computations (M chain).

Twelfth Parallel Traverse: Final adjusted values for this traverse are shown in the 1977

adjustment by Field, (1977), including values for the eight points common with the Nigerian

triangulation network. As may be seen in Field, (1977), there is a difference of almost exactly 3”

are in latitude and longitude between origin of the survey. The final values have therefore been

corrected by 3” in arriving at the provisional coordinates for this adjustment.

D.O.S Traverse Stations : Approximate values for the 35 stations were supplied by D.O.S.

They have not been adjusted in sympathy with the triangulation, so cannot be expected to be as

precise as values for the other points. For this reason it can be anticipated that these set of

stations when incorporated would require several adjustment iterations before they are

acceptable.

(X) Spheroid, Origin and Unit of Length

Measurements are taken on the surface of the earth, which in this area lies between 0 and 2400

metres above the mean sea level (equipotential surface or geoid), (Field, 1977), a surface to

which all observations in the field are referred. The geoid is nearly spheroidal but is not exactly

regular in shape. For this reason, computations are made on a mathematical surface chosen to

approximate to the geoid shape in the Survey area. This surface is the ellipsoid of revolution

known as the spheroid.

Following Bomford (1971), there are eight independent constants to define in a spheroidal

reference system. They are:

1. The relation of the axis of revolution to the earth‟s

2. Mean polar axis.

3. Length of the major axis in the plane of the equator.

4. Relation of the lengths of the major and minor axis.

5. Latitude of survey origin.

6. Longitude of survey origin.

7. Spheroidal height at origin.

19

8. Reference longitude.

The Minor Axis of the Spheroid which lies along the polar axis of the earth varies in position in

relation to the body of the earth from year to year. The movement is in part periodic, due to

meteorological and dynamic causes, and in part constant. The total effect is a circular movement

about a mean pole of about 0.3” arc. Referring the movement to x and y axes along meridians 00

and 900 West, corrections can be made to convert astronomically observed values to the

Conventional International Origin (CIO) of 1930.

For observed azimuths the correction is - (x sin+ y cos ) cosφ (Field, 1977).

Lengths of the Major and Minor axes: The figure of the earth adopted in Nigeria, in common

with most other countries in Africa, South of the Sahara is the Clarke 1880 Spheroid. The

fundamental data are quoted by Calder (1936) and agree with MCCaw (1939) and Bomford

(1971)

a (semi – major axis) = 20 926 202 ft = 6 378 249.145m

f (flattening) = 1/293. 465

b (semi - minor axis) = a (1-f)

The conversion factor from international metres to geodetic feet is the Clarke foot legal meter

relation, (1 international meter = 3.2808 6933 geodetic feets).

Station L40, in Minna, is the origin of the Nigerian triangulation framework. The position of this

station is defined by Morley (1938) as:

Latitude 090 38 09”.000 N

Longitude 060 30‟ 59”.000 E

The height of L41 with reference to the mean sea level at Lagos is given by Calder (1936) as

follows:

Spirit Level height of L41 = 768.72ft.= 767.410 ft = 233.96 meters

20

The assumed latitude and longitude of L40 were obtained by combining values computed

through the triangulation network from astronomical observation stations in widely separated

areas. These are shown in Table 2.0 (d).

Table 2.0 (d): Coordinates of Minna Datum (L40) computed from astronomical

observation stations in widely separated areas.

Location Latitude Longitude

Enugu U12 90 38' 06.26'' -

Ilorin L53 - 60 30' 59.24''

Kaduna REW 1 90 38' 10.0'' 6

0 30' 54.4''

Kano K2 90 38' 08.8'' 6

0 30' 59.55''

Lagos Observatory 90 37' 58.95'' * 6

0 31' 0.6''

Lafia N26 90 38' 04.60'' 6

0 30' 56.9''

Minna L40 90 38' 11.7'' 6

0 30' 53.85'' **

Naraguta 90 38' 13.4'' 6

0 31' 0.5''

Olokomeji L3 90 38' 07.3'' 6

0 30' 57.65''

Onitsha U21 90 38' 07.36'' -

Zaria N144 90 38' 10.1'' 6

0 31' 0.95''

(Field, 1977).

* The latitude from Lagos Observatory was considered to be incorrect and hence omitted in

the computation of the origin values.

** It will be noticed there is an laplace correction (A-G) of –6E-15 x sin 90 38'' to the base

azimuth at Minna.

Reference Longitude is defined as the Greenwich Meridian.

(B) Typical network adjustment programmes already carried out around the world

These include the following:

(I) Los Angeles

Horizontal Geodetic Network measurements of six years of Global Positioning System (GPS)

data, 20 years of trilateration data and a century of triangulation, taped distance, and astronomic

azimuth measurements which were combined to provide the Horizontal Geodetic network of Los

Angeles (Land Information New Zealand, 2000).

21

(II) India

A huge amount of geodetic data and accurate maps on different scales exist. The Indian geodetic

control network is noted for its high precision in the world. The geodetic data, collected through

centuries of dedicated efforts, consists of the Great Trignometrical (G.T.) Triangulation Network

of India, the Satellite Survey Control Network, the High Precision, Precision and Secondary

Levelling Network, the Laplace Stations Network, the Gravity Stations Network, the Tidal

Stations Network and the Geomagnetic Stations Network. The topographical maps by

Government of India are another such database. A wealth of information is contained in this

databank.

(III) Korean peninsula

The first Nationwide Geodetic Network in the Korean peninsula was established in 1910-1915

by the Bureau of Land Survey. The Government-General of Korea in cooperation with the

Japanese Military Land Survey. The major network of the old triangulation consisted of thirteen

baselines, primary and secondary networks, and were connected to the Tokyo Datum with the

triangulation through Tsushima Islands. After the World War II, the network over the Korean

(Tsushima straight) was resurveyed in 1954 by US Army Map Service Far East in cooperation

with Geographical Survey Institute of Japan, in order to strengthen the connection between

Korea and Japan.

To keep consistency with the old coordinates system, the Primary Precise Geodetic Network

(PPGN) was adjusted in the way that its official coordinates are same as the old ones.

Unfortunately, original records of the old survey were lost during the Korean War, and we only

have a set of coordinates of triangulation points now.

The establishment of PPGN was carried out in 1975-1994 by National Geography Institute of

Korea. The PPGN consists of 1155 points including 175 (~15 %) old first- and second-order

triangulation points (normal points) not damaged by the war, and its mean side-length is about 11

km. The coordinates of PPGN derived in 1995 have been held fixed since. During the 1980's and

1990's the increased use of satellite based geodetic measuring systems - such as the GPS - began

to impact on the utility of the national geodetic datum in Korea. Under this impact, new

geocentric datum, new Korean Geodetic Datum 2000 (KGD2000), designed and built during

22

1998, is realized through ITRF97 and used the GRS80 ellipsoid (Choi, 1998: Grant, Blick,

Pearse, Beavan and Morgan, 1999; Stephen, Sever, Bertiger, Heflin, Hurst, Muellerschoen, Wu,

Yunk, 1996).

(IV) Great Britain

In Great Britain Geodetic Network, three coordinate systems are considered:The National GPS

Network, a modern 3-D TRF using the ETRS89 datum. This coordinate system is the basis of

modern Ordnance Survey control survey (the surveyor's jargon for adding local points to a TRF

for mapping purposes), and will become the basis of definition of all Ordnance Survey

coordinates over the next few years. A subset of the Active Layer of the National GPS Network

has been ratified as the official densification of ETRF89 in Great Britain.

The National Grid, a traditional horizontal coordinate system, which consists of: a traditional

geodetic datum using the Airy ellipsoid; a TRF called OSGB36® (Ordnance Survey Great Britain

1936) which was observed by theodolite triangulation of trig pillars; and a Transverse Mercator

map projection, allowing the use of easting and northing coordinates. This coordinate system is

important because it is used to describe the horizontal positions of features on British maps.

However, its historical origins and observation methods are not of interest to most users.

National Grid coordinates are these days determined by GPS rather than theodolite triangulation.

Ordinance Datum Newlyn (ODN), a 'traditional' vertical coordinate system, consisting of a tide

gauge datum with initial point at Newlyn (Cornwall) and a TRF observed by spirit levelling

between 200 fundamental bench marks (FBMs) across Britain. The TRF is densified by more

than half a million lower-accuracy bench marks. Each bench mark has an orthometric height (not

ellipsoid height or accurate horizontal position). This coordinate system is important because it is

used to describe vertical positions of features on British maps (for example, spot heights and

contours) in terms of height above mean sea level. Again, its historical origins and observation

methods are not of interest to most users. The word Datum in the title refers strictly speaking, to

the tide gauge initial point only, not to the national TRF of levelled bench marks.

The National GPS Network provides a single 3-D TRF which unifies ODN and OSGB36 via

transformation software Using transformation techniques, precise positions can be determined

by GPS in ETRS89 using the National GPS Network and then converted to National Grid and

23

ODN coordinates. This is the approach used today by Ordnance Survey (Paul and Jackson, 1999;

Land Information New Zealand, 2000).

(V) Uganda

The Uganda Triangulation Network was established by the British colonial administration in the

20th

century. This was done in Phases, starting with the primary, followed by the secondary and

finally the tertiary network. A total of 1730 stations were established throughout the country. The

computation of the network was based on the Clark 1858 ellipsoid and the curvilinear horizontal

positions were projected onto the UTM projection which was used to map the country. Later, the

re- computation of the country‟s network was based on Clark 1880 modified ellipsoid. There is a

program in place to re-observe the existing network and other new points with GPS satellite

method (Rainsford, 1948; Okia and Kitaka, 2000).

(VI) Kenya

In Kenya, the Directorate of overseas surveys began the establishment of the primary, secondary

and tertiary triangulation and traverse networks around 1950. The mapping of the country was

also done by them until they departed from the country in 1983. The reference ellipsoid and the

projection used for mapping the country are Clark 1880 and UTM respectively (Kenya Institute

of Surveying and mapping Nairobi, 2000).

(VII) Egyptian

The existing Egyptian geodetic network, which dates back to the first decade of the twentieth

century, has been studied and adjusted in two and three-dimensions by several researchers.

All previous trials showed that there is a problem of some kind of distortion due to the inaccurate

adjustment and lack of geoidal information. GPS is used extensively in the last decade. The first

order geodetic horizontal control network of Egypt contains two main networks, Network (1) and

Network (2), [Cole, 1944] and these networks were extended to other African nations. The

networks comprise 402 stations and were established between 1907 and 1968 (Saad and Elsayed,

2007).

(C) Survey Requirement

Like any other field of study, most especially in Geodesy, the need and importance of the

bearable limit of any process is an essential consideration in the design, construction and

24

measuring stages in order to meet some given specifications. Surveying constitutes a unique

field where this consideration operates with much emphasis laid on the accuracy achievable in a

given process such as Geodetic Networks. The various practical requirements for accuracy can

be attained with the minimum effort and time through a suitable choice of most appropriate

instrument and an efficient and cost saving measuring procedure.

Observational precision in contemporary surveying practice is characterized by the standard

deviation or variance of individual observations. In order that useful statistical propagation of

this error can occur, these variances (variance covariance) are assumed to have a multivariate

normal distribution with vector of zero mean. This implies that the variances must be composed

of random errors and that any error or inaccuracy which is systematic in nature has already been

accounted for and removed, either by solving for the systematic component through an

adjustment process, eliminating it through appropriate observation procedures, or eliminating it

by other empirical techniques.

The determination of the geodetic coordinates of points on the ellipsoid can be expressed as

Direct and Indirect Problems (Ashkenazi, 1967 and 1970). The least squares method of

observation equation which take into consideration the curvilinear shape of the earth expresses

the adjusted observation such as angles, distances, azimuths as a function of the unknown

stations coordinates (Bomford, 1971 and Ashkenazi, 1972).

Ayeni (1980, 2002 and 2003) and Ayeni, et. al, (2005) from their work – “Determination of the

most appropriate least squares method for position determination in a triangulation

network” at an International workshop on Geodesy & Geodynamics, in Toro, Bauchi State of

Nigeria showed that, of the four mathematical techniques (simultaneous; sequential; phase and

combined) used in a sample geodetic adjustment, the simultaneous technique yielded the best

suitability criteria in terms of network standard deviation, trace of the variance covariance matrix

of the adjusted parameters and observations, and least time of computation.

(D) Application of Optimization Technique to Triangulation network

Of the three successful optimization algorithms (Simulated Annealing (SA); Genetic

Algorithms (GA); and Gaussian adaptation (NA);) used in geodetic network problem, Goldberg,

(1989 and 2002), Berne and Baselga (2003) and Donald (1970) confirmed that the simulated

annealing gave the best suitability criteria because it seeks the lowest energy state of geodetic

coordinates (stable state or final adjusted coordinates of station) of individual stations instead of

25

the maximum fitness of the network. Other benefits are less time of computation, lower cost, less

storage, high reliability for global solution. The Simulated Annealing was used for the First-

order design of sample geodetic networks by Berne and Baselga (2003).

Ayeni, et.al, (2006) in their work on “Application of Optimization Technique (Simulated

Annealing Method) to Triangulation network.” showed better suitability criteria for the method

of Simulated Annealing over the Simultaneous method in terms of network standard deviation,

trace of the variance covariance matrix of the adjusted parameters and observations, and high

reliability for global solution because it seeks the lowest energy state of geodetic coordinates of

individual stations instead of the maximum fitness of the network. Beside the additional

optimization algorithm in the SA, it incorporate simultaneous iterative scheme which ensures a

holistic adjustment of the network.

(E) Application of Global Navigation Satellite Systems (GNSS)

At present the Global Navigation Satellite Systems (GNSS) are used for geodetic work. GNSS is

the standard generic term for satellite navigation systems ("sat nav"). It provide autonomous geo-

spatial positioning with global coverage. It allows small electronic receivers to determine their

location (longitude, latitude, altitude) to within a few meters using time signals transmitted

along a line of sight by radio from satellites. Receivers calculate the precise time as well as

position, which can be used as a reference for scientific experiments.

As at the year 2010, the United States Navstar Global Positioning System (GPS) is the only fully

operational system. Today, the Russian Glonass (GNSS) and its in the process of being restored

to full operation (21 of 24 satellites are operational). The European Union‟s Galileo Positioning

System is a GNSS in initial deployment phase, scheduled to be operational in 2014. The People‟s

Republic of China has indicated it will expand its regional Beidou navigational system into the

global Compass navigation system by 2020.

The global coverage for each system is generally achieved by a constellation of 20–30 Medium

Earth Orbit (MEO) satellites spread between several orbital planes. The actual systems vary, but

use orbit inclinations of >50° and orbital periods of roughly twelve hours (height 20,000 km /

12,500 miles). Table 2.0 (e) shows the comparison of GNSS systems.

26

Table 2.0 (e):Comparison of GNSS Systems

System Country Coding Orbital height & period

Number of satellites Frequency Status

GPS United States CDMA 20,200km,

12.0h ≥ 24

1.57542 GHz (L1 signal) 1.2276 GHz (L2 signal)

operational

GLONASS Russia FDMA/CDMA 19,100km, 11.3h

24 (30 when CDMA signal launches)

Around 1.602 GHz (SP) Around 1.246 GHz (SP)

operational with restrictions, CDMA in preparation

Galileo European Union CDMA 23,222km,

14.1h

2 test bed satellites in orbit 22 operational satellites budgeted

1.164-1.215 GHz (E5a and E5b) 1.215-1.300 GHz (E6) 1.559-1.592 GHz (E2-L1-E11)

in preparation

COMPASS China CDMA 21,150km, 12.6h 35

[5]

B1: 1,561098 Ghz B1-2: 1.589742 Ghz B2: 1.207.14 Ghz B3: 1.26852 Ghz

5 satellites operational, additional 30 satellites planned

(Wikipedia, 2010).

2.1 THEORETICAL FRAMEWORK

Generally the methods for the adjustment of Geodetic networks are based on the following

considerations:

(i) Computations of geodetic coordinates on the ellipsoid.

(ii) The observation equations on the ellipsoid.

(iii) Least Squares method.

(iv) Error ellipse equations.

(v) Federal Geodetic Control Committee (FGCC) standards and specifications.

(vi) Optimization technique suitability for large geodetic network.

2.1.1 COMPUTATIONS OF GEODETIC COORDINATES ON THE ELLIPSOID.

The problems of the geodetic coordinates of points on the ellipsoid can be expressed as Direct

and Indirect Problems, (Ashkenazi, 1967 and 1970).

http://en.wikipedia.org/wiki/Global_navigation_satellite_system#cite_note-4

27

Figure 2.1.1: ELLIPSOIDAL POLAR TRIANGLES (Fajemirokun lecture notes, 1988)

2.1.1.1 Direct Problem Equations

The geodetic coordinates of points on the surface of the earth are usually specified as latitude

(φ), and longitude (λ). Given the coordinates of a starting point 1 represented by (φ1, λ1), a

distance (S) and azimuth ( A12) to a second point 2, we desire to compute the coordinates of the

second point 2 represented by (φ2,λ2) as well as the azimuth (A21) from the second point to the

first, (Ashkenazi and Cross, 1972). The direct problem is given by equations 2.1, 2.2 and 2.3:

φ2 = f1(φ1,λ1,A12,S) 2.1

λ2 = f2(φ1,λ1,A12,S) 2.2

A21= f3 (φ1,λ1,A12,S) 2.3

A convenient formulation of the extended equations is given by Jordan (1941). Equation relating

the latitude from point 1 – point 2 is (Ashkenazi, 1973) and (Chang et al, 1996),

2.4

)15152(30

)45301(120

2

4)29

1364(12

)931(24

)1(2

)931(6

2

3

2

1

42

4

3242

4

4

3

3

2

22

3

222222

3

4

22

2

32222

2

2

222

2

12

tte

uvtt

e

uv

te

ut

tte

uvttt

e

v

te

utt

e

uv

tue

tve

uV

27

Equation relating the Longitudes from point 1 to point 2

2.5

Equation relating the forward azimuth and backward azimuth between points 1 and point 2

2.6

where in our notation:

2.7

where

t = λ2 – λ1 2.8

)30201(15

)15152(15

)31(15

)32(3

)31(3

)31(33

1cos) - (

42

4

2342

4

422

4

5

22

3

332

3

3

22

2

22

2

3

12

tte

uvtt

e

vutt

e

v

tte

vutt

e

uv

te

vut

e

v

vute

v

)12018061(120

)24028058(120

)24201(120

)8624

285(24

)8224201(24

)465(6

)21(6

)21(2

'

42

4

23

42

4

2342

4

5

2224

2

3

322242

3

3

422

2

222

2

3

22

ttte

uv

ttte

uvttt

e

v

tt

te

vuttt

e

uv

tte

vutt

e

v

te

vuvt

28

a= length of semi major axis

b=length of semi minor axis

e = 1st eccentricity

e' = 2nd

eccentricity

For angular units in radians set to 1. Bagratuni (1962 and 1967) indicated accuracy up to

130km; While Grushinsky (1963) indicated accuracy up to 600-800km.

2.1.1.2 Indirect Problem (Inverse Solution)

The indirect problem can be expressed as shown in equations 2.14, 2.15 and 2.16

Given the coordinates of the end points of the line, we desire to find the azimuths ( A12), (A21)

and distance S.

S = f4(φ1,λ1, φ2 , λ2) 2.14

A12 = f5(φ1,λ1, φ2 , λ2) 2.15

A21= f6 (φ1,λ1, φ2 , λ2) 2.16

The solution of the inverse problem can be solved using series expansions and it is done through

an iterative procedure, (Baarda, 1977), (Bannister and Raymond, 1975)

29

where A and B are functions of s, A12, and 1. We now solve equations 2.17 and 2.18 assuming

A and B are known, (Bradley, 1939 and Butler, 1966).

Letting =2- 1 and = 2 - 1

We have:

V12=1+1

2 ` 2.21a

12=e‟

2cos

21 2.21b

Dividing equation (2.19) by equation (2.20) and rearranging terms:

[

] 22

In addition S can be found from either of the equations (2.17 and 2.18) as

Knowing , , and 1, and setting A and B to zero as a first approximation to the azimuth

(A12(1)

) in equation (2.22), we have from equation (2.17 and 2.18), (Bomford, 1971 and

Richard, 1980):

[

]

Setting A equal to zero in equation (2.23), and using the azimuth from (2.24) we compute the

first approximation to the distance as:

30

Using the now known values of A(t)

12 and S(1)

we can compute values for A and B which can

then be used in equations (2.17) and (2.18) to find better values for A12 and S. The process is

iterated until the values of A12 and S do not change beyond a specified amount (Stamers and

Sonola, 1955; U.S. Department of Commerce, 1935; and Gazdzicki, 1976).

In this study both the direct method and indirect methods are used in the adjustment.

2.1.2 THE OBSERVATION EQUATIONS ON THE ELLIPSOID

These are equations used in geodetic adjustment which take into consideration the curvilinear

shape of the earth. Bomford (1971) and Ashkenazi (1972) gave equivalent formula linking dA12,

dφ, dλ, and dD where the first three are in seconds of arc, the last is in linear units as defined

below:

Ai j is azimuth of geodesic from i to j;

Di j is length of geodesic from i to j;

φi is provisional latitude of station I

λi is provisional longitude of station I

pi is meridional radius of curvature of spheroid at i

vi radius of curvature at right angles to meridian at i

The Figure 2.1.2 is a sample of triangulation net with nodes/station I, J and K.

31

The following defined terms (equations 2.26 – 2.33) can computed for the angular, distance and

azimuth observation equations for the network (equations 2.36 – 2.38).

From these basic equations, observations equations may be formed, and grouped into a series of

simultaneous linear equations dAij and dDij are the small changes in the observations needed to

convert from observed (O) to computed (C) values) (Bomford, 1971 and Richard, 1980).

Thus

Vij in equations (2.36 – 2.38) is the small correction to the observed value in the solution. The

observed values in this network are of three kinds, clockwise angles at pont j indicated by i j k,

distance i j, and Laplace azimuths i j.

The three types of observation equation according to Bomford (1971) and Ashkenazi (1972)

are:

32

(i) Clockwise angles i j k

(Ki k –Ki j) dφ i + (Li k – Li j) dλi – Mi j dφ j – Ni j dλj + Mi k dφ k + Ni k dλk =

(O – C)i j k + Vi j k 2.36

where: (O – C) i j k is the small difference in seconds between observed (O) and computed (C)

angles; Vi j k is the correction to the observed angle.

(ii) Distance i j

Pi j dφ i + Qi j dλi + Ri j dφ j + Si j dλi = (O – C)i j + Vi 2.37

(O – C) is in linear units.

(iii) Laplace azimuths

Ki j dφ i + (Li j – sinφ i) dλi + Mi j dφ j + Ni j dλj = (O – C)i j + Vi j 2.38

(Bomford, 1971 and Richard, 1980)

In this equation the expression (Li j – sinφ i) allows for the Laplace correction (LA – LG) sinφ i

which changes the geodetic azimuth of the line in accordance with the deviation of the vertical at

the point of observation. Successive approximations to the correct geodetic azimuth are thus not

necessary (Even-Tzur and Papo, 1996; Even-Tzur, 2001; Grant, Blick, Pearse, Beavan and

Morgan, 1999).

2.1.3 LEAST SQUARES MATHEMATICAL METHOD:

It is generally accepted that the precision of a measurement may be improved by increasing the

number of observations. The redundant observations arising therefore, however creates a number

of problems. For example discrepancies may occur between repeated observations since each

observation has a certain amount of insecurity (precision) attached to it. Such discrepancies

therefore have to be reconciled (adjusted) so as to obtain the most satisfactory (most probable or

adjusted) values of unknown quantities. Also, redundant observations may lead to redundant but

consistent equations in which there are more equations than unknown quantities. There is

therefore the need to obtain not only the most probable values of unknown quantities but also to

find a unique solution for these quantities.

33

The method of least squares is a method which makes use of redundant observations in the

mathematical modeling of a given problem with a view to minimizing the sum of the squares of

discrepancies (residuals) between the observations and their most probable (adjusted ) values

subject to the prevailing mathematical model.

There are two types of models used in least square problems, (Ayeni, 2002 & 2003).

(i) Condition equation, which can be expressed as:

where is the adjusted observable, e.g. angles and distances

(ii) Observation Equation, which can be expressed as:

where: is the adjusted parameter.

In this study, we wish to determine the parameter ( , hence equation 2.40 shall be considered.

There are four common mathematical techniques used in the least squares observation equations

method, (Ayeni, 1980, 2002 and 2003 and Cross, 1974). They are:

2.1.3.1 Simultaneous Method.

Observation Equation method of Simultaneous approach is shown in Figure 2.1.3a and given by

equation (2.40). Each of the network triangle observations are combined in a single process of

adjustment to achieve a unique and best estimate values of the network stations coordinates.

Linearised form is given by equation (2.41)

( )

V= Ax + L 2.41a

where La = L

b + V,

Lb = Vector of observation,

34

Xª = adjusted parameters

V = the vector of residuals,

X = the solution vectors,

x = the correction vector = Xa - X

o

Xº = the approximate parameters,

The resulting normal equation is shown in equation (2.42)

Figure 2.1.3a: SIMULTANEOUS METHOD

35

P = weight matrix design as the inverse of the square of the standard error of observations.

Iterative solution is necessary to compensate for linearization in equation (2.41) giving rise to

equation - (2.43)

Xª (i+1) = X1º + xi 2.43

At convergence 2.43a

where i represent the ith

iteration

2.1.3.2 Sequential Method.

Observation Equation method which is used for Sequential approach is shown in Figure (2.1.3b)

and defined by equation (2.44) and (2.45) (Ayeni, 2001). This method allows the effect of

additional observations and new parameters on observation equation model of a previously

adjusted observation and parameter (coordinates) of some triangles within the network. Hence

triangles in Figure 2.1.3b are chosen on sequential bases to allow for the addition of a new

triangle observation and parameter in any subsequent adjustment.

V = the vector of residuals,

X = the solution vectors,

x = the correction vectors,

Xº = the approximate parameters,

The resulting normal equation is shown in equation (2.42)

V1 = A1x + L1 2.46

36

V 2 = A2x + L2 2.47

Where L1a = L1 + V1; L2

a = L2 + V2;

V1 and V2 are the respective vectors of residuals from the observables Lb

1 and Lb

2; X1 and X2

are the solution vectors (adjusted parameters); A1, A2 and A3 are the respective design matrices;

while X1o and X2

o are the respective approximate values of the parameters at which L1 and L2 are

computed.

where Δx shows the effect of the new observation in equation (2,45), given by:

(Ayeni, (2001))

Figure 2.1.3b: SEQUENTIAL METHOD

37

where ∆X is the effect of the additional observation and new parameters on observation equation

model. Iterative solution is necessary to compensate for linearization

where i represent ith iteration which converges as xi = 0 and Xai+1 is the final adjusted

parameters (coordinates) of the network stations.

N1= A1TP1A1

N2 =A2TP2A2

U1 = A1TP1L1

U2 = A2TP2L2

Xa1 and X

a2 are adjusted parameters; L

a1 and L

a2 are adjusted observations;

L1 = 1(Xo1) – L

b 1,

L2 = 2(Xo1) – L

b 2

V1 and V2 are the vectors of residuals; X1 and X2 are the solution vectors; A1 and A2 are the

respective design matrices; X1o and are the approximate values of the parameters.

P2 = weight matrix of L2

2.1.3.3: Phase Method.

Observation Equation method which is used for Phase approach is shown in Figure (2.1.3c) and

defined by equations (2.52) and (2.53), (Ayeni, 1985 and 2003), (Biacs, Krakiwsky and

Lapucha, 1990). This method allows the network triangle observation and parameter to be

38

adjusted in phases. Adjoining phases incorporates any result of previously adjusted parameter

reappearing in it for subsequent adjustment.

La1 = f1 (X

a1) 2.52

La2 = f2 (X

a2) 2.53

The constraint model from equation (2.52) after adjustment of the first phase is:

and their linearized forms are given by equations (2.54, 2.55 and 2.55a)

V1 = A1x1 + L1 2.54

V2 = A2x2 + L2 2.55

Vx = x1 + L3 2.55a

Note that the adjusted parameters in the first Phase, equation (2.52) are used as approximate

values in the second Phase, equation (2.53).

where La1 = L1 + V1,

La2 = L2 + V2

39

Figure 2.1.3c: PHASE METHOD

V1 and V2 are the vectors of residuals; X1 and X2 are the solution vectors; A1 and A2 are the

respective design matrices; X1o and X2

o are the approximate parameters; Px is the weight

constraint derived from the adjusted coordinate from the first phase which appears in the second

phase.

The resulting normal equations are shown in equations (2.56, 2.57).

x1 = - (A1TP1A1)ˉ¹A1

TP1L1 2.56

x2 = - (A2TP2A2 + Px)ˉ¹(A2

TP2L2 + PxLx) 2.57

where x1 and x2 are correction vectors for parameters in the first and second phases respectively.

Iterative solution is necessary to compensate for linearization in equation (2.56) and equation

(2.57) giving rise to equations (2.58) and (2.59)

X1a(i+1) = X

oI + x1i 2.58

X2a( i +1) = X2

o+ x2i 2.59

where i represent ith iteration which converges as x1i = 0 and x2i

40

2.1.3.4 Combined (Phase & Sequential) Method.

Observation Equation method which is used for Combined approach is shown in Figure (2.3d)

and defined by equations (2.60) and (2.61), (Ayeni, 2001 and 2003; Cooper, 1974; and Cross,

1974). Here the network triangles are adjusted sequentially in phases. Additional phases

observation and parameter are added to all the result of the previous adjustment, sequentially.

La1 = 1 (X

a1) 2.60

La2 = 2 (X

a1, X

a2) 2.61

STEP 1

The equations (2.60 and 2.61) can be considered for phase adjustment, in which case equation

(2.60) is adjusted first to obtain:

x*1 = - (A1TPA1)ˉ¹ A1

TPL1 2.62

Using X1* as observations in the adjustment of equation (2.60) along with weight matrix QX1

QX1 = - (A1TPA1 )ˉ¹

STEP 2

LaX1 = X

a1 2.63

La2 = 2 (X

a1, X

a2) 2.64

Linearising equation (2.63) and (2.64)

Vx1 = x1 + Lx1 2.65

V2= A2x1 + A3x2 + L2 2.66

where Lax1 = Lx1 + Vx1

41

Figure 2.1.3d : COMBINED METHOD

La2 = L2 + V2

Vx1 & V2 are the vectors of residuals; x1 and x2 are the correction vectors of the parameters; A1,

A2 & A3 are the respective design matrices; X1o and X2

o are the approximate parameters.

The resulting normal equations are shown in the equations below:

X1= x1* - Pˉ¹x1A2TK2 2.67

X2 = (A3T(A2Pˉ

¹X1A2

T+Pˉ

¹2)ˉ

¹A3

T(A2Pˉ

¹X1A2

T+Pˉ

¹2 )ˉ¹(A2x1* +L2) 2.68

Where

K2 = (A2 Pˉ¹X1 A2

T+Pˉ

¹2)ˉ

¹(A2X2+A2x1* +L2)

PX1 = σ2

o Σˉ¹Χª1

42

X1 = Xª1

L2 = Lº2 – L2

Lº2 = 2(Xº1 ,Xº2 )

Xª2 = Xº2 + X2

P2 = weight matrix of L2;

Iterative solution is necessary to compensate for linearization in equations (2.59) and (2.63)

giving rise to equation - (2.66), (2.67 and 2.68).

Xª1( i+1) = XºI + XI 2.69

Xª2 (i+1) = Xº 2 + X2 2.70

Where i represent ith iteration which converges as: XI = 0 & X2 = 0

2.1.4 NETWORK GEOMETRY ASSESSMENT

The least squares estimate of the adjusted parameters Xa is given for the linear case by equation

(2.71):

X = N -1

U 2.71

Where N = ATPA; U = A

TPL

b;

N is called normal coefficient matrix.

The unbiased estimate of the covariance matrix of adjusted parameters is given by equation

(3.21) as:

(

)

N-1

= weight coefficient matrix and denoted by Q i.e Q = N-1

.

Where ∑x is an error matrix since standard errors associated with the adjusted parameters may be

estimated from its diagonal elements.

43

The covariance matrix for the unknown parameter before adjustment is given as equation (3.1)

as:

Where: σ²o is the a-priori variance of unit weight of the observations and P is the weights of the

observations.

The details of these equations are discussed under methodology in chapter three.

2.1.5 FEDERAL GEODETIC CONTROL COMMITTEE (FGCC) STANDARDS AND

SPECIFICATIONS.

FGCC is a committee set up by world survey body to control: Specifications to Support

Classification; Standards of Accuracy; and General Specifications of Geodetic Control Surveys;

The Nigerian Institution of Surveyors standards and specifications (NIS, 2008) also agrees with

(FGCC, 2006). These are necessary due to the rapid changes in the requirements and methods for

acquisition of geodetic control (Hotline, 1935; Holland, 1975; Ingham, 1975; Hongsic Yun

2001).

2.1.5.1 Standards

The classification standards of the National Geodetic Control Networks are based on accuracy.

This means that when control points in a particular survey are classified, they are certified as

having datum values consistent with all other points in the network, not merely those within that

particular survey. It is not observation closures within a survey which are used to classify control

points, but the ability of that survey to duplicate already established control values. This

comparison takes into account models of crustal motion, refraction and any other systematic

effects known to influence the survey measurements (Baarda, 1977).

A variance factor ratio for the new survey combined with network data is computed by the

Iterated Almost Unbiased Estimator (IAUE) method and (FGCC, 2006). If the variance factor

ratio is reasonably close to 1.0 (typically less than 1.5), then the survey is considered to check

44

with the network, it is classified with the provisional (or intended) accuracy. If the variance

factor ratio is much greater than 1.0 (typically 1.5 or greater), it is considered not to check with

the network, and both the survey and network measurements will be scrutinized for the source of

the problem (Ralph, 1985 and Roeloss, 1950).

.

2.1.5.1.1 Horizontal Control Network Standards

The general horizontal control standards for all network is shown in Table 2.1.5.1.1

Distance accuracy (a) is computed from a minimally constrained, correctly weighted, least

squares adjustment by:

a = d/s

where: a = distance accuracy denominator

s = propagated standard deviation of distance between survey points obtained from the least

squares adjustment

d = distance between survey points (FGCC, 2006).

Table 2.1.5.1.1 Distance accuracy standards

Classification Minimum distance accuracy

First-order 1:100,000

Second-order, class I 1: 50,000

Second-order, class II 1: 20,000

Third-order, class I 1: 10,000

Third-order, class II 1: 5,000

(FGCC, 2006 and NIS, 2008)

45

2.1.5.1.2 Monumentation

Control points should be part of the National Geodetic Horizontal Network only if they possess

permanence, horizontal stability with respect to the Earth's crust and a horizontal location which

can be defined as a point. First-order and second-order class I control points should have an

underground mark, at least two monumental reference marks at right angles to one another, and

at least one monumental azimuth mark no less than 400 m from the control point. Replacement

of a temporary mark by a more permanent mark is not acceptable unless the two marks are

connected in timely fashion by survey observations of sufficient accuracy (FGCC, 2006).

2.1.5.2 Specifications

All measurement systems regardless of their nature have certain common qualities. The

measurement system specifications follow a prescribed structure as outlined in Section 2.1.5.2.1.

These specifications describe the important components and state permissible tolerances used in

a general context of accurate surveying methods. The user is cautioned that these specifications

are not substitutes for manuals that detail recommended field operations and procedures

(McCaw, 1935; Morley, 1938; Munsey, 1949; Lindlohr, and. Wells 1985; Atilola, 1985 and

1986; Omogunloye, 1988 and 1990; and FGCC, 2006).

2.1.5.2.1 Triangulation

Triangulation is a control establishment system comprising of joined or overlapping triangles of

angular observations supported by occasional distance and astronomic observations (at Laplace

Stations). Triangulation can also be used to extend horizontal control (Richardus, 1966; Ralph,

1985; and FGCC, 2006). The Network geometry (Table 2.1.5.1.2) shows the configuration

usually used in the specifying of different orders and classes of network.

The new survey is required to tie to at least four well spaced network control points. These

Network points must have datum values equivalent to or better than the intended order and class

of the new survey. For example, in an arc of triangulation, at least two network control points

should be occupied at each end of the arc. Whenever the distance between two new unconnected

survey points is less than 20 percent of the distance between those points traced along existing or

new connections, a direct connection should be made between those two survey points. In

addition, the survey should tie into any sufficiently accurate network control points within the

46

station spacing distance of the survey. These network stations should be occupied and sufficient

observations taken to make these stations integral parts of the survey. Non-redundant geodetic

connections to the network stations are not considered sufficient ties. Control stations should not

be determined by intersection or resection methods. Simultaneous reciprocal vertical angles or

geodetic levelling are observed along base lines. A base line need not be observed if other base

lines of sufficient accuracy were observed within the base line spacing specification in the

network, and similarly for astronomic azimuths (Bannister, 1975; Baarda, 1977; Ordnance

Survey, 1967; Nickerson, 1978; and FGCC, 2006).

Table 2.1.5.1.2: Network Geometry

Order First Second Second Third Third

Class I II I II

Station spacing not less than (km) 15 10 5 0.5 0.5

Average minimum distance angle† of figures not less than 40° 35° 30° 30° 25°

Minimum distance angle† of all figures not less than 30° 25° 25° 20° 20°

Base line spacing not more than (triangles) 5 10 12 15 15

Astronomic azimuth spacing not more than (triangles) 8 10 10 12 15

† Distant angle is angle opposite the side through which distance is propagated.

(FGCC, 2006 and NIS, 2008)

2.1.5.2.2 Instrumentation

In Triangulation, only properly maintained theodolites are adequate for observing directions and

azimuths. Precisely marked targets, mounted stably on tripods or supported towers, should be

employed. The target should have a clearly defined centre, resolvable at the minimum control

spacing. Optical plummets or collimators are required to ensure that the theodolites and targets

47

are centred over the marks. Microwave-type electronic distance measurement (EDM) equipment

is not sufficiently accurate for measuring higher-order base lines (FGCC, 2006). Tables 2.1.5.1.3

and 2.1.5.1.4 show the least counts, standard deviations and rejection limits applicable in

geodetic network.

Table 2.1.5.1.3: Instrument Order and Class


Class I II I II

Theodolite, least

count 0.2" 0.2" 1.0" 1.0" 1.0"

Source: FGCC, (2006) and NIS, (2008)

Table 2.1.5.1.4: Theodolites Observations


Class I II I II

Directions

Number of positions 16 16 8 or 12† 4 2

Standard deviation of mean not to exceed 0.4" 0.5" 0.8" 1.2" 2.0"

Rejection limit from the mean.

4"

4"

5"

5"<

Source: FGCC, (2006) and NIS, (2008)

48

2.1.6 DEFINITION OF BEST GEOMETRIC CONFIGURATION

One of the objectives of this research is to provide an insight into the true geometry of the

Nigerian Horizontal Geodetic Network. Therefore, there is a need to examine the best geometric

configuration of geodetic networks. The best geometric configuration of a new geodetic network

can be defined, using the range of situations which can vary, as follows:

(i) The case where the possible location of the stations is so constrained by exterior

conditions such as visibility, natural features, private properties, etc.

(ii) A case where there is almost no choice for the most adequate location of stations

because there is no margin of movement.

(iii) A case where any possible location of stations within an area is acceptable

(Barricelli, 1954 and 1963; Berne and Baselga, 2003).

(iv) Beyond the simplest statements that merely recommend certain geometric principles

(considering angles over 30 degrees, similar distances and so on), some mathematical

methods are required to search for the best design.

As the margin of choice grows, there is an increasing need for reliable criteria to determine the

most appropriate network design. Criteria for appropriateness should rely on the minimum

indetermination (error) at the defined points, considering both the type and the number of

observations to be done.

The optimum geometric design problem for a geodetic network is classified into four main

groups (Schwefel, 1974; Syswerda, 1989; Whitley, 1994; Schmitt, 2001, 2004). These are:

The Zero-Order Design problem (ZOD).

The First-Order Design problem (FOD).

The Second- Order Design problem (SOD).

The Third-Order Design problem (TOD).

(i) The Zero-Order Design problem (ZOD): ZOD aims at datum definition. The constraints are

imposed on the model to find a solution that is free from the influence of fixed coordinate

inaccuracies. Hence, in the ZOD, datum points are the variables (Shepherd, 1999 and Vose,

1999).

http://en.wikipedia.org/w/index.php?title=Nils_Aall_Barricelli&action=edit

49

(ii) The First-Order Design (FOD): FOD optimizes station positions and the observations to be

made. The variable in this problem is the observations‟ design matrix.

(iii) The Second- Order Design problem (SOD): It aims to design the observation weights so that

the solution is able to accomplish prescribed precision. The variable in this problem is the

observation weight matrix.

(iv) The Third-Order Design problem (TOD): This deals with optimal network densification. It

can be considered to a certain extent as a mixture of the first and second order problems. Because

its design variables are the observations‟ design matrix and the observations weight matrix (Wolf

and Ghilani, 1997; Wright et al. 2003).

Historically, the design problems were defined and initially dealt with in the 1970s and the

1980s. The first study pioneered by Baarda, 1973 and Grafarend, 1974 were collected in the

classic book by Grafarend and Sanso (1986); Walter and Wells (1986); Wells and Grafarend

(1987);

Later notable contributions were made by Jagar (1988) and Jagar and Kaltenbach (1990) on the

introduction of spectral analysis in relation to network stiffness. Xu and Grafarend (1995)

examined the introduction of multi-objective optimality for SOD while an overview of recent

works was done by Kuang (1996).

The application of the FOD has recently proved to be essential in geodynamics geodetic

networks as shown by Johnson and Wyatt (1994), Gerasimenko (1997) and Gerasimenko et.al.

(2000) in GPS network design by Dare and Saleh (2000) and in general survey networks by

Chang et al, (1996).

One of the most rigorous approaches is to determine the position of all the stations to be located

by means of minimizing the hyper volume of the error ellipsoid inherent to the solution (Berne,

& Baselga, 2003).This criterion is developed in Section (2.1.6.1).

50

2.1.6.1 WHAT IS OPTIMIZATION ?

G.W. Leibniz, derived the notion “Optimization from the Latin word optimus, which means

more or less the best.

We need optimization for the following reasons:

To ensure the most economic field campaign.

To help in identifying, eliminating, or minimizing the effects of the gross and systematic

errors existing in the observation data.

To avoid misinterpreting measuring errors as deformation phenomena.

A global optimization problem can be formulated as:

Minimizing an objective function f(x)

minimum f(x)

Subject to the constraint function Q given by:

[ ]

Where x Є Q

Function Q in this study would help to check the integrity of an already observed network.

The three different parts that can be distinguished in this type of problem are:

(a) The objective function: The function to be minimized is known as the objective function and

is defined under the following terms.

A-optimality where the trace in the covariance matrix of the parameters is minimized,

thereby minimizing the average variances of the parameter estimates.

D-optimality where the determinant of the covariance matrix of the parameters is minimized.

It has the statistical significance of minimizing the volume of the hyperellipsoid inherent to

the solution. The D-Optimality shall be employed in this study, since it focus on the variance

covariance matrix of the parameter from which the network geometry can be determined.

51

E-optimality where it minimizes the largest eigen value of the covariance matrix for the

parameter estimates (Graferand and Sanso 1986).

Spectral optimization related to network stiffness

Criterion matrices.

(b) Optimization Variables: Optimization variables are those related to the optimization design

problem under consideration and are classified as follows:

Zero Order Design (ZOD): The variables are the datum points (fixed coordinates in the

network).

First Order Design (FOD): , the variables are the A-matrix (the design matrix dF(X)/dXo)

which represents the geometry of the network where Xº is the approximate of the

parameters.

Second Order Design (SOD): The variables are the P-matrix of observation weights

Third Order Design (TOD): The variables are A-matrix (the design matrix dF(X)/dXo) and

the P-matrix of observation weights.

This study is an FOD problem, since the heart of this work is to determine the geometry of the

Nigerian horizontal geodetic network.

( c ) The Solution Method: The solution methods are classified under Local optimization and

Global optimization.

Local optimization technique: Searches for an optimum in the neighborhood of a starting

point.

Global optimization technique: The starting point is sufficiently close to an optimum. The

nature of the objective function must permit it (Pardalos and Romeijn 2002).

Since this study covers stations (coordinates) covering the entire country, it is therefore a global

optimization problem.

52

2.1.6.2 Optimization Techniques

Optimization ensures the most economic field campaign, and helps to identify, eliminate, or

minimize the effects of the gross and systematic errors existing in the observation data prior to

the estimation of the deformation parameters in order to avoid misinterpreting measuring errors

as deformation phenomena, (Rechenberg, 1971; Legault, 1985; Michalewicz, 1996; Mitchell,

1996 and Wikipedia, 2007).

Optimization problems can be divided into two categories: local and global optimization. Local

optimization techniques search for an optimum in the neighborhood of a starting point.

Unfortunately, the achieved solution is not the global optimum unless the starting point is

sufficiently close to it and the nature of the function permits it. So for an objective function with

a global optimum hidden among many other local optima, such a technique is likely to fail in

determining the global optimum (Schaffrin, 1985; Schmitt, Lothar, Nehaniv and Fujii, 1998).

The incapacity of local optimization procedures to always obtain the global solution has directed

efforts in recent decades to global optimization methods ( Pardalos and Romeijn, 2002)

Following Xu (2002), successful algorithms on geodetic network can be divided into three main

categories:

(i) Simulated Annealing (SA).

(ii) Genetic Algorithms (GA).

(iii) Gaussian adaptation (NA).

(i) Simulated Annealing (SA)

It is a global optimization technique that traverses the search space by testing random mutations

on an individual solution. A mutation that increases fitness is always accepted while a mutation

that lowers fitness is accepted at a probability based on the difference in fitness and a decreasing

temperature parameter. SA seeks the lowest energy state of individual station coordinates in the

network and hence the maximum fitness of the network and can also be used within a standard

GA algorithm by starting with a relatively high rate of mutation and decreasing it over time

along a given schedule (Baudry et al, 2005).

http://en.wikipedia.org/wiki/Simulated_annealing

53

(ii) Genetic Algorithm (GA)

It is used to find exact or approximate solutions to search problems. They are categorized as

global search heuristics. Genetic algorithms are a class of evolutionary algorithms which use

Evolutionary Biology, such as inheritance, mutation, selection, and crossover or recombination

(Gwenn, Sale and Mark, 2006; Wikipedia, 2007).

Table 2.1.6.1 SUMMARY OF SOLUTION METHODS

CRITERIA GENERIC

ALGORITHM

GAUSSIAN

ADAPTATION

SIMULATED ANNEALING

DEFINITION

Uses evolutionary

biology, such as

Inheritance,

mutation,

selection, and

crossover or

recombination to

find exact or

approximate

solutions to search

problems.

Uses normal or

natural

adaptation (NA)

It relies on a

certain theorem

valid for all

regions of

acceptability and

all Gaussian

distributions.

A technique involving heating and

controlled cooling of a material to

increase the size of its crystals and

reduce their crystallographic defect.

The heat causes the atom(s) to become

unstuck from their initial positions (a

local minimum of the internal energy)

and wander randomly through states of

higher energy; the slow cooling gives

them more chances of finding

configurations with lower internal

energy than the initial one.

SUITABILITY

FOR GEODETIC

NETWORK

(1) Though

suitable, It does

not reveal the

lowest energy state

of geodetic

coordinates of

stations.

(2) More time,

(3) More storage

(4) Expensive

(1) Less Suitable.

(2)Maximizes

mean fitness

rather than the

fitness of the

individual

(3) Less time,

(4) Less storage

(5)Less expensive

(1) More Suitable because it seeks the

lowest energy state of geodetic

coordinates of individual station

coordinates and hence its maximum

fitness.

(2) Less time required.

(3) Less expensive.

(4) Required less Storage.

(5) High Reliability for global solution.

(6) It was used for the First-order

design of sample geodetic networks.

(Berne and Baselga, 2003; Wikipedia, 2007)

54

(iii) Gaussian Adaptation or Normal Adaptation (NA)

Gaussian Adaptation or Normal Adaptation (normal or natural adaptation NA), maximizes

manufacturing yield of signal processing systems and may also be used for ordinary parametric

optimization. The efficiency of NA relies on information theory and a certain theorem of

efficiency which is defined as information divided by the work needed to get the information.

NA maximizes mean fitness rather than the fitness of the individual and is also good at climbing

sharp crests by adaptation of the moment matrix (FFOX, 1964; Kjellström, 1970, 1991 and 1996;

Koza, 1992; Crosby, 1973; Konak, 1994 and Fentress, 2005).

The above methods have been applied in the following studies: Artificial Creativity; Code-

breaking using the GA to search large solution spaces of ciphers for the one correct description;

Design of water distribution systems; Timetabling problems, such as designing a non-conflicting

class timetable for a large university; Traveling Salesman Problem; as well as other fields of

engineering studies (Fraser, 1957; Fogel, 1998; Fogel, 2000 and 2006; Berne and Baselga, 2003).

In Table 2.1.6.1, Goldberg (1989 and 2002), Berne and Baselga (2003) and Donald (1970)

ascertained the suitability criteria of the Simulated Annealing method over others in geodetic

network problems.

55

CHAPTER THREE

METHODOLOGY

3.1 GENERAL

There are a number of ways to achieve a global solution in a geodetic optimization problem. The

global solution method applied in this research is the Simulated Annealing described by Berne

and Baselga (2003), which have been found for solving geodetic problems in that it seeks the

lowest energy state (state of stability) of geodetic coordinates of individual stations and hence the

maximum fitness of the network. It requires less time; lower costs; less Storage and gives high

reliability for global solution. It has been used for the first-order design of sample geodetic

networks ( Berne and Baselga, 2003 and Wikipedia, 2007).

The simultaneous observation equation methods of least squares, discussed in Sections (2.1.2

and 2.1.3.1) as well as the methods of computation of coordinates on the ellipsoid (direct and

indirect method) discussed in Section (2.1.1), were applied with the optimization method

(Section 2.1.6) to achieve consistent global estimates of the network stations coordinates as

discussed in Section (2.1.5). The error ellipse method (Sections 2.1.4 and 3.3) was applied to

determine the network geometry. The geometry comprises the absolute/relative numerical values

and graphical plots of the strength/weakness of the network in terms of positions, scales and

orientations. The results of this study will be compared with the 1977 adjustment and analyzed

statistically.

3.1.1 Data Acquisition

The study has recovered the lost network data by searching for possible libraries within and

outside the country where the data could be found. The data used for this study were obtained

from the report on the 1977 adjustment by Field (1977). The original data were collected in 1977

by Field from the Federal Surveys of Nigeria. The data format of the network consists of 441

primary stations, 74 secondary and tertiary stations, 2197 reduced angles, 39 Laplace azimuths,

174 scale checks, stations provisional coordinates, and standard error for all observations

(Section 1.4). The sample hard copy formats of the Network observations for angles, azimuths

and scale checks, which now have soft copy versions are shown in Tables 3.1 a – c. Details are

given respectively in Appendices (Id, Ie and If).

56

Table 3.1a: SAMPLE ANGULAR DATA

ANGLES

AT LEFT RIGHT DEG MINUTE SECOND ESTIMATED

S E (sec)

S/N

K18 K29 K30 95 17 28.68 0.72 1

K18 K19 K29 38 34 37.64 0.72 2197

Table 3.1b: SAMPLE AZIMUTH DATA

AZIMUTHS

AT RO DEG MINUTE SECOND ESTIMATED S E S/N

R26 R23 271 0 42.61 1.00 1

U12 U34 1 17 44.72 3.00 40

Table 3.1c: SAMPLE SCALE CHECKS DATA

DISTANCES

AT TO DISTANCE (M) ESTIMATED SE PPM S/N

L1 L2 16064.083 1.00 1

U12 U34 26577.621 3.00 174

3.1.2 Data Pre-Processing and Quality Control

The data quality checks and reductions, were carried out in the previous adjustment (Field, 1977)

and were confirmed to possess high precision and reliability, based on their assigned individual

standard error, which were made use of in this study. The data were converted to a soft copy

57

format and three independent checks of the transcribed soft copy data were done manually by

different competent survey personnel to ascertain the correctness of the transcription and to

eliminate both large and small errors of transcription. Then the data set was arranged into their

corresponding terrestrial triangular formations serially and a total of 1054 triangles were

obtained for the entire network (Appendix Ia - c). The deviations from 180o of sum of the angles

in each of the 1054 triangles were computed to check for any undetected wrong data entry using

program written in Matlab (Appendix Ib).

3.1.2.1 Instruments used/Date of Observation

The various instruments used from 1929 to the time of adjustment in 1977, were related to their

respective chains and network stations to ascertain if there is any relationship between year of

observation, least count reading of instrument and the stations error ellipse values using the

intelligent data format created in this study.

3.1.3 Data Processing

The data processing stage includes written programs in Matlab for all the computational steps in

Chapters 2 and 3 to achieve the following: (Appendices I -XI):

(i) Computation of the approximate coordinates of all the 515 stations in the network using

the supplied controls coordinates, the reduced 2197 observed angles, the 174 given

scale and 39 azimuth checks of some lines in the network using the model discussed

Section (2.1.1).

(ii) Computation of the L matrix (differences between the observed and computed angles,

bearings and distances) using the least squares observation equation method discussed in

Sections (2.1.2, 2.1.3.1 and 3.2).

(iii) Computation of the lengths of all lines and their respective azimuths using both the direct

and indirect methods discussed in Sections (2.1.1.1, 2.1.1.2 and 3.2).

(iv) Computation of the design matrices, correction vectors for the parameters, residual vector

of the observations, variance covariance matrices of both the adjusted parameters and the

adjusted observations and their respective traces using the models discussed in Sections

(2.1.1, 2.1.2, 2.1.31, and 3.2).

58

(v) Computation of the error matrix and error ellipse radial distance (resultant of the semi-

major and semi-minor error ellipse values) of all stations in the network using the method

discussed in Section (3.3).

(vi) Plots and annotation of all graphs/maps including absolute/relative graphs to show trends

of the strength/weakness of the network geometry in terms of residuals of the adjusted

observations, positional corrections, standard deviations in azimuths and distances, as

well as error ellipses of network using programs written in Matlab (Appendix Id.)

(vii) Creation of an intelligent data structure/data base of the network which would make

possible, access to information on any required distance, angle, azimuth, and other salient

queries on any of the network triangles using computational models discussed in Sections

(2.1.1, 2.1.2, 2.1.3, 3.1, 3.2 and 3.3) as well as Appendices (XI a - r).

3.2 SIMULATED ANNEALING ALGORITHM

The optimization solution method used in this study is the Simulated Annealing (SA) approach

and the method is discussed fully in this section.

SA approach consists of two parts:

(1) The Least Squares Equations and

(2) The Optimization Equations.

3.2.1 Least Squares Equations

The linearised form of the Least Squares (LS) system of observation equation is given by

Equation (2.41), (Ayeni, 2001; Berne and Baselga, 2003).

The stochastic model given by the covariance matrix is:

where: σ²o is the a-priori variance of unit weight of the observations and P is the weights of the

observations.

59

The least squares solution is given by Equation (2.42) in Section 2.1.3.1 with the covariance

matrix for the unknowns parameter given by

(Ayeni, 1980 and 2001; Berne and Baselga, 2003)

In an adjustment, the reference factor

(a-posteriori variance of unit weight) obtained from n

observation residuals, which provides the final m unknowns, is estimated as:

(Ayeni, 1980; Berne and Baselga, 2003)

Without having actually observed the network, since

We can consider σ²o = 1 before the adjustment.

Hence, the covariance matrix of the unknown parameters before adjustment yields

The network coordinates are given by Equation (2.42)

where is approximate and a constant vector

The covariance matrix of the network coordinates during adjustment after the first estimates of

- which represents the m-dimensional error figure of the defined network – is given by Equation

(3.6)

As the aim is to minimize the determinant of the covariance matrix of the adjusted parameter

( ), which is the volume of the hyper ellipsoid.

60

This involves minimizing its determinant (Equation 3.7).

min det ( ) = min det 3.7

or the equivalent problem of maximizing the determinant of its inverse (Equation 3.8)

max det( ) = max det 3.8

3.2.2 Equations used for the Optimization Part

The least square part in Section (3.2.2) is introduce into this part. Once Equations (3.7) or (3.8) is

true, the correction matrix for the parameters in the Section (3.2.1) is accepted and introduced

into Equation (3.9), or else accepted and introduced at a an empirically given probability, or

otherwise rejected, keeping the last correction vector and introducing it into Equation (3.9) as

discussed below.

The optimization part is sub-divided into four parts:

(A)The iterative process structure.

(B) The cooling scheme structure (conversion from unstable random coordinates to a stable one).

(C) Functions of free movement structure.

(D) Objective function and acceptance criteria structure.

(A) The iterative process structure:

1. Take an arbitrary initial value of correction matrix (xi) of size (515 by 2) and its

corresponding objective function f(xi )

2. Generate an increment ∆xi by means of a multivariate normal random distribution called free

movement function (Equation 3.9).

∑

∑

61

with zero mean (vector of zero) and variance-covariance matrix Σ

making sure that :

xi + ∆ xi Є Q

where Q is

[ ]

Otherwise the new point is rejected and another one is generated.

3.With the new solution

xi + 1 = xi + ∆xi 3.11

∆xi is the new correction vector of the parameter generated as the least square estimate during

each iteration while xi is from the last iteration.

or xi + 1 = xi 3.12

Depending on the acceptance criteria

4. Decrease temperature (stations coordinates being less error prone or being more stable than its

value in the last iteration) following the corresponding cooling scheme and return to step 2 until

the finish criteria is fulfilled. The number of iterations may be in the range of several hundreds.

(B) Cooling Scheme This is a scheme that ensures the gradual change of approximate coordinate

to an optimal stable one. Given a sufficiently high initial temperature To of the unknown

parameter, some successful cooling functions (Equations 3.13, 3.14, and 3.15) are as follows:

3.13

3.14

3.15

(Van laarhoven and Aarts, 1987).

62

where t = the time (or equivalently the iteration number),

To = the initial temperature, determined empirically as 10000

β = scale factor of gradual cooling (cooling rate) which is 0.999

With a high enough initial temperature, for instance, some hundred thousands, and a slow

enough rate of cooling. σ(t) in Equation (3.16 or 3.17) defines the amplitude of possible

movement of the free function in Equation (3.9). It can be demonstrated that the simulated

annealing algorithm converges the standard deviation to the optimal solution at a given

probability as shown in Figures (3.1, 3.2, 3.3) by Van Laarhoven and Aarts, (1987).

The final unchanging (stable) coordinates state which must occur at the least temperature in the

adjustment is the temperature at which to stop the iterations and can be stated as a fraction of the

initial one, say 10²־ To, or as the moment when the differences between two consecutive

correction vectors of the parameters solutions become negligible for our purposes. For this study

iteration was stopped when the determinant of the covariance matrix of the parameter is minimal

(value remain constant).

The process of stabilizing the coordinates optimally (cooling scheme) is given in Figure 3.1

which gives a three in one graphical view of the plots of Equations (3.13, 3.14, and 3.15).

Equation 3.15 was chosen for this study, because it gave the best gradual stabilization process

(cooling scheme) of the network stations coordinates state of stability as shown by the plot of

Equations (3.13, 3.14, and 3.15) in Figure (3.1).

( C ) Functions of Free Movement Structure

Another critical point, as mentioned above, is the correct modelling of the thermal agitation of

particles. This agitation is a random movement whose amplitude depends on the present

temperature. One of the most suitable functions for defining this random movement is the

multivariate normal distribution, whose density function is given by Equation (3.9).

Using the function of free movement in Equation (3.9), with zero mean (vector of zero) and

variance-covariance matrix Σ σ – 1

.

63

It is also possible to use the Cauchy distribution. The aim is to accurately emulate the movement

of particles at a certain temperature.

The standard deviation, defining the amplitude of possible movement, has to be related to this

temperature. Thus, initial values for Σ σ – 1

have to be set depending on the particular problem.

Note that one standard deviation provides the limit displacement for a 68% probability.

Fig. 3.1: COOLING SCHEME (PROCESS OF STABILIZING THE NETWORK

STATIONS COORDINATES)

Secondly, a criterion for decreasing σ, as temperature descends has to be adopted. Two of the

simplest but also highly efficient possibilities are given by Equations (3.16 and 3.17) and plotted

in Figure (3.3).

64

For an initial value σo, a factor β, a temperature T and time t (or equivalently iteration number)

are involved to determine the successive standard deviation of the amplitude of the free

movement function. Consequently, ∆xi displacement is a random one derived from Equation

(3.9) using equation (3.15) together with Equations (3.16) or (3.17) if the new point is located

inside the acceptable domain define by the objective function and the acceptance criteria below,

otherwise ∆xi is zero.

Fig. 3.2: GRAPH OF FUNCTION OF FREE MOVEMENTS OF THE NETWORK

STATIONS COORDINATES

(D). Objective Function and Acceptance Criteria.

The aim of the algorithm (least squares part) mentioned in Section 3.2.1 is to find the global

optimum of a function that defines the goodness of the network design. This function f(xi ) stated

65

in Equation (3.7) or equivalently Equation (3.8) for our purposes – is known as the objective

function.

At first, we can think of admitting every new point xi that betters the previous function value and

rejecting every one that makes it worse. This was the initial approach of the simulated annealing

method (Metropolis et al, 1953). Nevertheless, in some cases the algorithm may fall into local

optimum and not be able to escape if only aided by successive standard deviations subject to the

acceptance criterion in Equation (3.18) only.

Fig. 3.3: PLOT OF THE NETWORK STANDARD DEVIATION SCHEME

66

Here in equation (3.19), the application of a modified criterion can be of more interest; if the new

point provides a better objective function then it is accepted, otherwise the new point (with a

worse solution) is accepted with certain probability P, in order to be able to escape from local

minima. Otherwise we can admit every previous point xi that betters the function value and reject

every one that makes it worse. Expression for the acceptance probability (p) of worse solutions

can be as simple as a given fixed value; for instance values such as 0.2, 0.1, 0.05, 0.01

probability level can be used. Another much more sophisticated possibility, continuing with

thermo dynamical analogies is Boltzman‟s distribution given by equation (3.20).

p takes into consideration the amount of worsening (correction matrix) and the temperature

(stations energy state). This introduces a new problem since the variation range of the objective

function (f (xi + 1 ) - f(xi )) may not correspond to the order of the temperature, and so the

increment f (xi + 1 ) - f(xi ) has to be properly scaled, hence p is chosen empirically, (Pardalos

and Romeijn, 2002).

Equations (3.1 to 3.20) are introduced into the main program of the adjustment to minimize the

determinant of the covariance matrix of the solution matrix thereby minimizing the volume of

the error hyper ellipsoid in the solution.

3.3 ERROR ELLIPSE

It has been shown in equation 2.42 that the least squares estimate of the adjusted parameters X is

given for the general case as:

X = -N -1

U;

where: N=ATPA; U= A

TPL

b;

N = normal coefficient matrix. The unbiased estimate of the covariance matrix of adjusted

parameters is given by:

67

3.21

N-1

= weight coefficient matrix and is denoted by Q i.e. Q = N-1

.

x is an error matrix since standard errors associated with the adjusted parameters may be

estimated from its diagonal elements. The geometries interpretation of the error matrix may be

illustrated by the quadratic form

(Rainsford, 1949; Rainsford, 1957 and Paul, 1999).

which represents the equation of a hyper ellipsoid (multiaxial ellipsoid) centre in

m-dimensional space, where m is the number of parameters (Hamilton, 1964).

When β = 1 we have an ellipsoid of standard deviation which may be projected on the (x,y) plane

to obtain a standard ellipse whose equation is given by Richardus (1966) as:

o2 is a-posteriori variance of unit weight

The weight coefficient matrix Q(2x2) is given by

3.24

x (λ) and y(φ) are the two adjusted vectors of parameters contained in the vector X

68

The equation of the ellipse may be written in terms of the elements of the N coefficient matrix; N

is given for 2 parameters by:

3.27

By making the following substitution in Equation 3.23;

We have equations (3.28 and 3.29)

Graphically the standard ellipse is defined as the ellipse to which the sides of a rectangle given

by Equations (3.28 and 3.29) are tangents (Figure 3.4).

There are several possible ellipses of different orientations (and centered at x,y) which can be

drawn tangentially to the rectangle defined by Equations (3.28 and 3.29). It is therefore

important to derive formulas which will assist us in defining the semi-major and minor axes as

well as the orientation of the axes of the ellipse. Let us consider an orthogonal transformation of

x,y system into a U, V system as follows.

U = xcos + sin 3.30

V = -xsin + cos 3.31

(Biddle, 1958; Baarda, 1968 and Ayeni, 2003)

where is the angle between u- and x- axes. According to the law of propagation of errors

(Ayeni, 2001); given:

oxxxxyyxyxyxx

yqxyqxqqqq

222

2ˆ)2(

)(

1

NNDD

nq

D

nq

D

nq yyxyxx )(det,,, 221222

cossin

sincos,G

v

uZ

69

3.32

where:

3.33

Fig. 3.4: GRAPH OF STANDARD ELLIPSE

By using the following Trigonometric identities

Sin2 = 2sin cos 3.34

Cos2 = cos2 + sin

2 3.35

1 = cos2 + sin

2

Hirvonen (1965) has shown that:

quu=½(qxx+qyy)+½(qxx+qyy)cos2+ qxysin2

yyxy

xyxx

x

vvuv

uvuu

zqq

qqQ

qq

qqQ ˆ,

70

3.36

Therefore the parameter for ellipse orientation is given by Equation (3.37) as:

3.37

3.38

3.39

where H =

quu=½(qxx+ qyy – H) 3.40

(for minimum)

Similarly from:

qvv = qvv cos2 Ψ–2qxysinΨcosΨ + qxxsin

2 Ψ

it can be shown that:

qvv=½(qxx+qyy– H) 3.41

(for maximum)

The semi-major and minor axes of the error ellipse (parameters for the size of the ellipse) are

therefore defined by Equations (3.42 and 3.43)

3.42

3.43

02sin)(2cos2

yyxxxy

uu qqqq

yyxx

xy

qq

q

22tan

H

qxy22sin

H

qq yyxx 2cos

uuou q ˆ

vvov q ˆ

xyyyxx qqq 22 4)(

71

The resultant value (σuv ) of Equations (3.42) and (3.43), which represent the radial distance error

of each station coordinates can be derived from Equations (3.42 and 3.43) as given by equation

(3.44) .

σuv = (u2 + v

2)½ 3.44

These axes and their orientation may also be computed directly from the covariance matrix. Two

characteristics of error ellipse must be noted. x2

+ y2 are invariant under the orthogonal

transformation so also is x2y

2 - xy

2

i.e. x2

+ y2 = u

2 + v

2

x2y

2 - xy

2 = u

2v

2 - uv

2

If xx= yy , then xy =0, and therefore the error ellipse becomes a circle.

72

CHAPTER FOUR

RESULTS AND ANALYSIS

4.1 RESULTS

All results in this study had passed through statistical testing during the course of adjustment.

The simulated annealing method which is based on the least square approach had also

incorporated statistical algorithm that ensures the following (Section 3.2):

(i) Functions of free movement structure which is a multivariate normal distribution function.

(ii) Objective function and acceptance criteria structure which was incorporated into the function

of free movement to compute the correction vectors of the parameters at a given probability level

(95% confidence level) and hence, the residuals, positional corrections, the error ellipses values,

the standard error in azimuths and distances of all lines in the network. The ATPV matrix after

adjustment is equal to zeroes, this denotes that the residual matrix of the adjusted observation

only contain white random noise.

The a-posteriori variance of unit weight computed from the adjustment, at an acceptance criteria

of 95% probability level was found to be equal to 1meter. This represents the statistical network

standard error of 1 sigma ( and shall be used for the analysis of the distribution of the

network stations geometry.

The use of the simulated annealing method had in the adjustment had help to minimize the

determinant of the covariance matrix of the solution matrix, thereby minimizing the volume of

the error hyper ellipsoid in the solution.

Paired sample statistical test was carried out with the use of SPSS software on the extracted

absolute error values of the 33 stations used in 1977 and their corresponding values in this study

at 95% confidence level. The result is shown in Table (4.2.5b) and Figure (4.2.5a).

The results of the network adjustment and plots are discussed under the following sub –sections:

73

4.1.1 Newly Arranged old Network Data and Result Formats

Intelligently arranged old network data, results from computations and plots were achieved in

this study and discussed in section (4.1).

4.1.1.1 Newly Arranged old Network Data Format

The structure of the newly arranged old network data comprises 1, 054 Triangles, with 515

Stations Identity (Tables 4.1a – c). 3162 lines identity and angles values (Table 4.1b), and 3162

network triangle angles in Tables ( 4.1c). Samples of the newly structured network data format

with details are given respectively in Appendices (IIa, IIb, and IIc). This triangular arrangement

applies to all forms of information on lines, angles, azimuths, station coordinates, residual

values, station correction vectors, and so on (Section 3.1.3 ). The sample graphical view of the

arrangement of the stations is shown in Figure 4.1a

Table 4.1a: SAMPLE TRIANGULARLY ARRANGED STATIONS ID

OF THE NETWORK

TRIANGLE S/N STATION 1 STATION 2 STATION 3

1 K52 K44 A50

1054 N138 N135 N133

Table 4.1b: SAMPLE TRIANGULARLY ARRANGED STATION - STATION/LINES

OF THE NETWORK

TRIANGLE S/N LINE 1 LINE 2 LINE 3

1 K52 - K44 K44 - A50 A50 - K52

1054 N138 - N135 N135 - N133 N133 - N138

74

Table 4.1c SAMPLE TRIANGULARLY ARRANGED ANGLES OF

THE NETWORK ( deg min sec)

TRIANGLE

SR NO ANGLE 1(deg min sec) ANGLE 2(deg min sec) ANGLE 3(deg min sec)

1 59º 52' 57.3623'' 60º 02' 20.8878'' 60º 04' 41.7500''

1054 60º 44' 40.393'' 65º 53' 24.3873'' 53º 21' 55.2200''

Figure 4.1a show a sample of the network plot for any required section of the network, such as

lines, error ellipse plot, residual plot, station correction plot, and so on.

Fig. 4.1a: PLOT OF NETWORK SAMPLE TRIANGULAR STRUCTURE

4.1.1.2 Assessment of the Instruments used/Date of Observation of the Nigerian Horizontal

Geodetic Network

Table 4.1d shows the different instruments used from 1929 to the time of adjustment in 1977.

They are relatively close in least count reading and the deficiency of any of these instrument

were compensated for by the number of rounds of measurements. This consequently created no

75

significant difference between the type of instrument and the date of observation as can be seen

in sample Table (4.1e) with details in Appendices (X). Here the absolute error ellipse values of

the network stations were tabulated in ascending order with type of instrument/date of

observation.

Table 4.1d: INSTRUMENTS USED FOR THE NIGERIAN

HORIZONTAL GEODETIC NETWORK.

TYPE OF SURVEY OBSERVED

ANGLES

INSTRUMENT

SCALE CHECKS

INSTRUMENT

AZIMUTHS INSTRUMENT

DOS TRAVERSE 1‟‟ THEODOLITE

READING ON 8

ZEROES

S.E. = 1‟‟

TELLUROMETER

DISTANCES

51 USABLE LINES OF

TRIANGULATION NET

S.E. = 3 PPM

12TH

PARALLEL

TRAVERSE 2‟‟ THEODOLITE

(WILD T4)

S.E. = 0.70‟‟

GEODIMETER

DISTANCES

46 LINES OF 12TH

PARALLEL

TRAVERSE

S.E. = 2 PPM

2‟‟ THEODOLITE (T4)

23 AZIMUTHS

TRIANGULATION

NETWORK 4‟‟ THEODOLITE

(WILD T3)

READING ON (10 -

20) ZEROES

S.E. = 0.72‟‟

INVAR TAPED BASES

(INVAR WIRES)

USED FOR THE

TRIANGULATION NET

S.E. = 1 PPM

8 OLD AZIMUTHS

REMEASURED

8‟‟ MICROMETER &

4‟‟ THEODOLITE (T3)

8 NEW AZIMUTHS :

8‟‟ MICROMETER and 4‟‟

THEODOLITE (T3)

76

Table 4.1e: NETWORK ASSESSMENT BASED ON INSTRUMENTS STANDARD

DEVIATIONS/YEAR OF OBSERVATION (where applicable)

S/N

ERROR ELLIPSE

VALUE IN

ASCENDING ORDER

(meter)

YEAR

OBSERVED

STANDARD

ERROR OF

ANGLES

(second)

CHAIN/ STANDARD ERROR

IN SCALE CHECKS (PPM)

Chain ID/

S.e.( Azimuth in Second)

Date observed

1 6.04899210414509E-02 '1929-1935' '0.75 -0.77' 'N5', 0.67 'N5', 1.00, 1912'

2 7.85708063404021E-02 '1929-1935' '0.75 -0.77' 'N3', 0.67 'N3', 1.00, 1912'

5 8.57794637219177E-02 '1929-1935' '0.75 -0.77' 'N1', 0.67 'N1-N2, 1.00, 1912'

8 1.08049128289698E-01 '1929-1935' '0.75 -0.77' 'N2', 0.67 'N2'

9 1.08180564119912E-01 '1930-1937' '0.92' 'L39' 'L39-L41, 1.00, 1938'

10 1.09854275072479E-01 '1937' '0.92' 'L37' 'L37'

11 1.13371500413491E-01 '1929-1935' '0.75 -0.77' 'N10-N12, 0.96, N10-N14, 0.67 'N10-N12, 1.00, 1963'

15 1.29346292880717E-01 '1935' '0.75' 'R47' R47-N127,1.00, 1969'

158 3.45171339803536E-01 '1967-1971' '1.00' 'ML52, 3.00' 'ML52'

160 3.47261829144649E-01 '1967-1971' '1.00' 'ML308, 3.00' 'ML308'

506 2.96097513511009E+00 '1967-1971' '0.70' 'CFL11, 2.00'

'CFL11-CFL10, 1.00,

1968'

507 3.01507821965987E+00 '1967-1971' '0.70' 'CFL17, 2.00'

'CFL17-CFL16, 1.00,

1969'

514 3.41360888872390E+00 '1967-1971' '0.70' 'CFL24, 2.00'

'CFL24-CFL23,

1.00,1969'

515 4.82062729063526E+00 '1967-1971' '0.70' 'CFL37, 2.00' 'CFL37'

77

4.1.2 RESULTS OF THE RESIDUAL VECTOR (V) AFTER ADJUSTMENT

The Residual Vector (V) of all observations (angles, distances, azimuths) in the entire network

were computed and plotted as discussed in Sections (2.1.1, 2.1.2 and 2.1.3) to provide a

graphical view of the magnitude of the corrections applied to each observation during the

adjustment. The numerical values whose units are in radian are arranged in ascending order with

their corresponding line identities for all the 3162 lines in the 1054 triangles that made up the

network.

Table (4.1.2a) shows a sample of the residual matrix while the details are given in the

Appendices (III). The graphical plot reveals the magnitude of this residuals (Figure 4.1.2a).

Figure 4.1.2a reveal the size of residuals shown as bars, which represent the amount of

corrections effected to the original observables such as angles, azimuths and scale checks

throughout the network. It can also be seen that numerical values of the residual and their

corresponding sizes of bars for the CFL‟s stations are relatively smaller than others (Table 4.1.2a

and (Figure 4.1.2a). This is because the residual vector formed from the linearised observation

equations, contain more of observations from the triangulation network (an observation

appearing in two or more triangles) than those from the traverse lines. Consequently these

residual corrections (V) from the CFL‟s observations yielded a corresponding position correction

values. The number of time a line appears as bar is the number of time such a line was used in

the network adjustment.

78

Table 4.1.2a ASSESSMENT OF THE SAMPLE NETWORK RESIDUAL MATRIX

S/N OF

LINES/OBSERVATIONS

STATION ID OF

LINES

V MATRIX OF OBSERVATIONS

(radian)

1 ML453XL451 -1.32727861241487E-04

2 R48R47 -9.12734779968862E-05

3 ML451ML403 -8.87450817614267E-05

4 ML54ML53 -7.93193635187504E-05

5 B12B9 -7.91263456432537E-05

6 ML201XL202 -7.58267630321441E-05

7 ML656ML403 -7.18591405053288E-05

1599 L45L27 -8.49110319592170E-09

1600 R42R3 -6.85095221256699E-09

1601 CFL24CFL23 2.52732832055954E-08

1602 CFL14CFL13 3.62725573499121E-08

3155 N42L56 6.47364477112486E-05

3156 A46A42 6.60515085080113E-05

3157 ML502XL453 6.72911136471098E-05

3158 L1L3 6.82468014508886E-05

3159 N128R48 7.03178221369891E-05

3160 L23D2 7.34825231312190E-05

3161 ML705XL453 7.81416569281016E-05

3162 ML456ML453 1.67093443878418E-04

79

Figure 4.1.2a: PLOT OF V (RESIDUALS) MATRIX OF ADJUSTED OBSERVATIONS

OF ALL STATIONS/LINES (3162 lines) IN THE NETWORK

(Charts could not show all stations identity due to scale)

Figure 4.1.2b shows the size of residual (V) represented as bars along the Twelfth Parallel

Traverse (CFL) stations of the network. These bars reveal the number of observation at that point

and their corresponding correction. For the CFL, the observations comprise scale checks,

azimuths and angles.

Figure 4.1.2c shows the contribution of redundancy in triangulation method of survey. More

residual (bars) are revealed for the triangulation stations (Figure 4.1.2c) than the traverse

Network (CFL‟s) in Figure (4.1.2b), thereby providing better estimates of residual corrections of

the observables. Most of the redundant observations are angles.

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

0.00004

0.00006

0.00008

0.0001M

L45

3X

L45

1

R4

7N

12

8

E15

E13

C3

9C

40

D2

2D

20

ML6

51

E1

A4

9K

30

CFL

26

CFL

27

XB

15

2X

B1

53

R4

6R

51

CFL

27

CFL

28

D1

0D

12

N1

39

N2

5

F23

F22

F8M

R5

52

V matrix(radian)

V matrix(rad)

80

Figure 4.1.2b: V (RESIDUALS) MATRIX OF ADJUSTED OBSERVATIONS OF

SAMPLE 11 CFL STATIONS/LINES IN THE NETWORK.

Figure 4.1.2c: PLOT OF V (RESIDUALS) MATRIX OF ADJUSTED OBSERVATIONS

OF 11 TRIANGULATION STATIONS/LINES IN THE NETWORK.

4.1.3 RESULTS OF STATIONS POSITIONAL CORRECTIONS

The correction Vector (x) to all the stations in the network as discussed in Sections 3.1, 3.2 and

3.3 were computed as shown in Table 4.1.3a and plotted to provide a graphical view of the

-1.2E-06

-0.000001

-8E-07

-6E-07

-4E-07

-2E-07

0

0.0000002

0.0000004

0.0000006

CFL

10

CFL

11

CFL

10

CFL

11

CFL

10

XB

15

3

CFL

11

CFL

12

CFL

11

CFL

12

CFL

11

XB

15

3

CFL

12

CFL

10

CFL

12

CFL

13

CFL

12

CFL

13

CFL

13

CFL

11

CFL

13

CFL

14

V matrix(radian)

V matrix(rad)

-0.00001

-0.000005

0

0.000005

0.00001

0.000015

0.00002

0.000025

0.00003

0.000035

0.00004

K1

9K

29

K1

9K

29

K1

9K

30

K1

9K

52

K2

9K

18

K2

9K

30

K2

9K

30

K3

0A

49

K3

0A

50

K3

0K

18

K3

0K

18

V matrix(rad)

V matrix(rad)

81

magnitude of the corrections applied to each station during the adjustment as shown in Figure

(4.1.3a). The numerical values whose units are both in seconds of arc and meter are arranged in

ascending order with their corresponding station identity for all the 515 stations that make up the

network as shown in the sample Table (4.3a) with its details in Appendix (IV). Similarly, each

bar size in the graphical plot reveals the magnitude of the corrections at each station of the

network (Figure 4.1.3a). It should be noted that all computations were done in curvilinear

coordinates. Conversion of these result to rectangular units were done for simplification purpose.

Table 4.1.3a: ASSESSMENT OF NETWORK STATIONS POSITIONAL

CORRECTIONS VECTOR X

S/N STATION ID POSITION ERROR (Second)

POSITION ERROR

(Meter)

1 XB153 8.13339175790117E-03 2.51219972192022E-01

2 B1 8.70975118276397E-03 2.69022262183277E-01

3 N141 8.74078524556394E-03 2.69980826165645E-01

4 P14 9.42009629528608E-03 2.90963032371948E-01

5 XB152 9.94963742110609E-03 3.07319223105513E-01

6 CFL9 9.96749799870022E-03 3.07870891332015E-01

7 CFL37 1.00623597219236E-02 3.10800930875127E-01

8 XB106 1.06731628334801E-02 3.29667099537293E-01

9 CFL8 1.08534605331535E-02 3.35236041062116E-01

10 CFL26 1.21229268245101E-02 3.74446655269099E-01

503 U84 1.46825513922256E-01 4.53507006948193E+00

504 U70 1.85456126516067E-01 5.72827232881613E+00

503 U84 1.46825513922256E-01 4.53507006948193E+00

504 U70 1.85456126516182E-01 5.72827232881971E+00

505 U75 1.86021781099743E-01 5.74574397324335E+00

506 ML255 1.92503526543460E-01 5.94594875356083E+00

507 XL202 2.45672400962927E-01 7.58820127879591E+00

508 XL201 2.52086910742817E-01 7.78632931891695E+00

509 ML201 2.52269014217547E-01 7.79195403628204E+00

510 ML403 3.84265201498420E-01 1.18689835812981E+01

511 ML656 4.37056438612328E-01 1.34995718419527E+01

512 ML601 4.40288000686979E-01 1.35993866496856E+01

513 ML453 4.40697469115091E-01 1.36120340974153E+01

514 ML452 4.45482576972756E-01 1.37598340188651E+01

515 ML451 4.48473229133684E-01 1.38522077265467E+01

82

Figure 4.1.3a: BAR CHART OF ALL POSITIONAL CORRECTIONS APPLIED TO

THE 515 STATIONS IN THE NETWORK

(Charts could not show all stations identity due to scale of plot).

Figure 4.1.3b: SOME OF THE XL AND ML SECONDARY STATIONS OF THE

NETWORK WITH LARGER POSITIONAL CORRECTIONS,

0.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

1.40E+01

XL4

53

MR

55

0

ML3

08

XL4

51

XL2

51

ML7

05

ML2

55

XL2

02

XL2

01

ML2

01

ML4

03

ML6

56

ML6

01

ML4

53

ML4

52

ML4

51

POSITION CORRECTION (m)

POSITION ERROR (m)

0.00E+00

2.00E+00

4.00E+00

6.00E+00

8.00E+00

1.00E+01

1.20E+01

1.40E+01X

B1

53

R1

4

N1

25

F26

K4

4

C1

8

A2

9

L22

C2

5

B9

A1

7

L4 R4

3

POSITION CORRECTION (m)

POSITION ERROR (m)

83

From the Figures (4.1.2a, 4.1.2b, 4.1.3a and 4.1.3b), size of bar reveal the corrections carried out

on the observations through the computed values of the residuals (V) and thus, subsequently on

the approximate coordinates of all stations in the network.

Figure (4.1.3b) reveals the magnitude (bar size) of the corrections applied to the approximate

coordinates of some of the secondary stations of the network during the adjustment.

4.1.4 RESULTS OF ERROR ELLIPSE COMPUTATION ON THE NETWORK

The computed resultant values (semi-major, semi-minor, orientation) of the error ellipse

discussed in Section (3.3) after the adjustment, are meant to give the magnitude of displacement

of the network from its true geometry. The better the estimates of the residuals to the observables

and the corresponding correction vectors to the approximate coordinates of stations, the smaller

the value and size of the error ellipse, which consequently depicts closeness to the true geometry

of the network.

The resultant absolute error ellipse values for all the 515 stations of the network were computed

as discussed in section (3.3) and their numerical values shown in ascending order in sample

Table (4.1.4a) with its details in Appendices (V) and (XIb). Figure (4.1.4a) displays a graphical

view of the magnitude of these error ellipses of the adjusted network from the true geometry.

The numerical values are tabulated in units of seconds of arc and meter, together with their

corresponding station identity (Table 4.1.4a).

Figure (4.1.4a) displays a graphical view of the magnitude of these error ellipses of the adjusted

network from the true geometry. The numerical values are tabulated in units of seconds of arc

and meter, together with their corresponding station identity (Table 4.1.4a). The size of bar

reveals the size of absolute error ellipse and the weakness or strength of the network stations.

From Figure (4.1.4b), size of bar reveals the size of absolute error ellipse and weakness/strength

of the stations along the 12th

Parallel Traverses (CFL‟s).

Figures (4.1.2a and 4.1.3a) displays smaller estimates of the residuals of the observables and the

corresponding correction vector for the CFL stations and thereby making their error ellipse

values bigger (Table 4.1.4a, Figure 4.1.4a and 4.1.4b).

84

Figures (4.1.2a and 4.1.3a) display larger estimates of the residuals of the observables and the

corresponding correction vector for the secondary and tertiary stations of the network. However

some of these stations still have error ellipse of larger values (Table 4.1.4a and Figures 4.1.4a-c).

Table 4.1.4a: ERROR ELLIPSE VALUES OF THE NETWORK STATIONS

SR N0 STATION ID

ERROR ELLIPSE VALUE

(Second)

ERROR ELLIPSE VALUE

(Meter)

1 N5 1.95839614558421E-03 6.04899210414509E-02

2 N3 2.54377525450968E-03 7.85708063404021E-02

466 B5 3.10949519384057E-02 9.60444694390861E-01

467 XL202 3.25930730159967E-02 1.00671787864847E+00

484 CFL13 6.17928980018689E-02 1.90862687790903E+00

485 XD456 6.23319152426084E-02 1.92527576194908E+00

486 CFL8 6.47906554094047E-02 2.00122004875045E+00

487 U70 6.50004537434476E-02 2.00770019051824E+00

488 H14 6.61063886742350E-02 2.04185973315785E+00

504 CFL10 9.46459254435395E-02 2.92337409358354E+00

505 ML54 9.51937849541689E-02 2.94029609305464E+00

506 CFL11 9.58632808893351E-02 2.96097513511009E+00

507 CFL17 9.76148995131038E-02 3.01507821965987E+00

508 CFL16 9.79878427877285E-02 3.02659749745556E+00

509 XB152 9.90863528421629E-02 3.06052770437810E+00

510 ML452 1.02486426694956E-01 3.16554741622460E+00

511 CFL15 1.08362979529948E-01 3.34705932217180E+00

512 CFL33 1.08845689582055E-01 3.36196901907028E+00

513 CFL5 1.08981998948910E-01 3.36617927186152E+00

514 CFL24 1.10517560200283E-01 3.41360888872390E+00

515 CFL37 1.56070593955791E-01 4.82062729063526E+00

85

Figure 4.1.4a: PLOT OF ABSOLUTE ERROR ELLIPSE VALUES FOR 515 STATIONS

IN THE NETWORK (Charts not showing all stations (515 stations) identity due to scale)

Figure 4.1.4b: PLOT OF ABSOLUTE ERROR ELLIPSE VALUES FOR SOME CFL

STATIONS IN THE NETWORK

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

4.00E+00

4.50E+00

5.00E+00N

5

G2

L14

M1 E3 H4

P5

ML6

01

F23

H5

N2

2

CFL

6

ERROR ELLIPSE VALUES (m)

ERROR ELLIPSE VALUES(m)

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

4.00E+00

4.50E+00

5.00E+00

CFL

22

CFL

31

CFL

10

CFL

11

CFL

17

CFL

16

CFL

15

CFL

33

CFL

5

CFL

24

CFL

37



86

Figure 4.1.4c: CHART SHOWING SOME ML, XL, XD and XB SECONDARY

STATIONS WITH THEIR ABSOLUTE ERROR ELLIPSE VALUES IN THE

NETWORK

4.1.5 RESULTS OF RELATIVE ERROR ELLIPSE (RELATIVE GEOMETRY)

COMPUTATION

Table 4.1.5a shows the sample network relative error ellipse values in Section (3.3) of all

lines/stations in their ascending order with its details in Appendix (VI). It shows a very close

agreement with the absolute error ellipse values of stations in the network. The size of ellipse

reveals the relative weakness/strength along each line or station to station as well as the relative

geometry of all stations in the network. From Table 4.1.5a, it can be seen that the CFL‟s and

some of the secondary ML‟s and XL‟s stations have weaker geometry closure. The reason for

this might be due to the lower precisions of observations within the affected chains.

Figure 4.1.5a displays the network relative error ellipse plot of all lines/stations. It shows a very

close agreement with the absolute error ellipse plot (Figure 4.1.4a) of stations in the network.

The size of ellipse reveals the relative weakness/strength along each line or station to station as

well as the relative geometry of all stations in the network. From Figure 4.1.5a, it can be seen

that the CFL‟s and some of the secondary ML‟s, XL‟s stations have weaker geometry closure as

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

XL2

02

ML2

ML3

52

ML7

05

XD

45

6

XB

15

3

XD

70

2

YC2

02

XB

10

6

ML5

4

XB

15

2

ML4

52



87

Table 4.1.5a ASSESSMENT OF RELATIVE ERROR ELLIPSE

COMPUTATION ON NETWORK

S/N

LINES/STATIONS

ID

LINE RELATIVE ERROR

ELLIPSE (Second)

LINE RELATIVE ERROR

ELLIPSE (Meter)

1 N4N1 0.0000409984303 0.0012663381819281

2 N1N4 0.0000409984303 0.0012663381819281

2854 U71U74 0.0101589097639 0.3137831183282140

2855 N106L36 0.0101811772762 0.3144709056612820

3037 M7M9 0.0202911928071 0.6267438043652180

3038 CFL28CFL26 0.0205822941602 0.6357351914773390

3081 ML705XL453 0.0297613775723 0.9192539433314840

3082 XD456CFL31 0.0307290912899 0.9491441810651530

3090 U70U66 0.0407925753614 1.2599798402646900

3091 CFL12CFL10 0.0438826575083 1.3554246896134100

3090 U70U66 0.0407925753614 1.2599798402646900

3091 CFL12CFL10 0.0438826575083 1.3554246896134100

3155 ML452ML453 0.0911816511538 2.8163713920500300

3156 R47CFL17 0.0939583885540 2.9021378119185100

3157 CFL33CFL35 0.0947975895201 2.9280586146578300

3158 CFL5R16 0.0953597823286 2.9454233335737800

3159 K12CFL24 0.1007079695166 3.1106153563661700

3160 N131CFL24 0.1032314779836 3.1885601727209500

3161 CFL35CFL37 0.1418910742963 4.3826576660826800

3162 CFL37A39 0.1424702537777 4.4005470605122800

88

Figure 4.1.5a: STATIONS WITH LARGER RELATIVE ERROR ELLIPSE VALUES IN

THE NETWORK (Chart could not show all line (3162 lines) identities due to scale).

4.1.5.1 Results of Standard Error in Azimuths

The Network Standard Error in Azimuths of all lines are shown in ascending order along with

their station to station identity in the sample result in Table 4.1.5b with details in Appendices

(VII) and Figures (4.1.5b, 4.1.5c, and 4.1.5d). The network strength/weakness of azimuths

(orientation) of all lines can be deduced in ascending order in this table.

The 12th

parallel traverse (CFL‟s stations) and the secondary/tertiary stations (ML‟s, XB‟s,

XD‟s, XL‟s) are weaker in orientation compared to the other stations.

0.00E+005.00E-011.00E+001.50E+002.00E+002.50E+003.00E+003.50E+004.00E+004.50E+00

N4

N1

C2

3C

25

K4

0K

38

C8

C9

N3

4N

66

M2

M1

E4E6

N1

0N

14

L13

L15

G2

G4

D1

9D

22

N1

40

N3

5

RELATIVE ERROR ELLIPSE SIZES OF LINES (m)

RELATIVE ERROR ELLIPSESIZEOF LINES (m)

89

Table 4.1.5b SAMPLE NETWORK STANDARD ERROR OF AZIMUTHS OF LINES

S/N

STATIONS/LINES

ID STANDARD ERROR OF AZIMUTH (Second)

1 R37R35 1.04273651424594E-04

2 R37R35 1.04273651424594E-04

3 R35R37 1.04273651424594E-04

4 E15L38 1.31406589778790E-04

450 N136N131 1.56261808528142E-03

451 N131N136 1.56261808528142E-03

3114 CFL35CFL34 4.60676384754674E-02

3115 CFL35CFL34 4.60676384754674E-02

3118 CFL11CFL12 4.78302222671061E-02

3119 ML54XL301 4.86920209597843E-02

3121 ML54ML53 5.31106006786564E-02

3122 ML54ML53 5.31106006786564E-02

3123 ML452XL202 5.32611317019209E-02

3124 XL202ML452 5.32611317019209E-02

3125 A36ML54 5.41316652479105E-02

3153 CFL17N127 1.17710866557099E-01

3154 CFL6XB106 1.20992821839288E-01

3155 CFL32CFL33 1.24009632199593E-01

3156 CFL33XD702 1.41806320987866E-01

3157 XD702CFL33 1.41806320987866E-01

3158 CFL32CFL30 1.44612575096434E-01

3159 CFL34XD702 1.45666908976288E-01

3160 XD456CFL31 1.46459880605601E-01

3161 CFL31CFL30 1.47615953546203E-01

3162 CFL30CFL31 1.47615953546203E-01

90

Figure 4.1.5b: CHART OF STANDARD ERRORS IN AZIMUTHS OF THE NETWORK

(Chart could not show all lines (3162 line)) identities due to scale).

Figure 4.1.5c: CHART SHOWING SOME CFL STATIONS/LINES WITH LARGER

STANDARD ERROR IN AZIMUTHS VALUES IN THE NETWORK.

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

R3

7R

35

N4

N2

E11

E12

L54

L52

K4

7B

17

A7

A3

N1

14

N1

10

L26

L25

N4

0N

42

R3

5R

36

H1

2N

26

N1

08

N1

04

P1

5P

16

AZIMUTH STANDARD ERROR(Sec)

AZIMUTH STANDARDERROR(Sec)

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

1.40E-01

1.60E-01

CFL

8C

FL7

R4

7C

FL1

7

N1

27

CFL

17

CFL

17

N1

27

CFL

6X

B1

06

CFL

32

CFL

33

CFL

33

XD

70

2

XD

70

2C

FL3

3

CFL

32

CFL

30

CFL

34

XD

70

2

XD

45

6C

FL3

1

CFL

31

CFL

30

CFL

30

CFL

31

STANDARD ERROR IN AZIMUTH (Sec)

STANDARD ERROR INAZIMUTH (Sec)

91

Figure 4.1.5d: CHART SHOWING SOME NETWORK SECONDARY

STATIONS/LINES AND THEIR CORRESPONDING STANDARD ERROR IN

AZIMUTHS VALUES

4.1.5.2 Results of Standard Error in Distances

The Network Standard Errors in distances of all lines are shown in ascending order along with

their station to station identity in the sample Table (4.1.5c) with details in Appendix (VIII) and

Figures (4.1.5e, 4.1.5f, and 4.1.5g). The network strength/weakness of distances (scale) of all

lines can be deduced in ascending order from this table. The 12th

parallel traverse (CFL‟s

stations) and the secondary/tertiary stations (ML‟s, XB‟s, XD‟s, XL‟s) are weaker in scale

compared to the other stations.

0.00E+002.00E-024.00E-026.00E-028.00E-021.00E-011.20E-011.40E-011.60E-01

ML4

53

XL2

02

ML5

4X

L30

1

ML5

4M

L53

ML5

4M

L53

ML4

52

XL2

02

XL2

02

ML4

52

XL2

01

ML4

52

XD

70

2C

FL3

1

XD

70

2C

FL3

2

ML4

52

ML4

53

XB

15

3C

FL1

0

XB

15

2C

FL1

0

YC2

02

CFL

15

XB

10

6C

FL7

XD

70

2C

FL3

3

XD

45

6C

FL3

1



92

Table 4.1.5c SAMPLE NETWORK STANDARD ERROR OF DISTANCES OF LINES

S/N STATIONS/LINES STANDARD ERROR OF DISTANCE(m) 1

N4N3 3.69374907436008E-11

2 N3N4 3.69374907436008E-11

3 N3N1 1.82970257816117E-10

4 N1N3 1.82970257816117E-10

450 A10E18 1.70886118939992E-09

451 F2F1 1.71099756278415E-09

3114 XD702CFL31 5.85573434284588E-07

3115 ML54ML53 6.02206891875569E-07

3116 ML54ML53 6.02206891875569E-07

3117 A36ML54 6.15973692027048E-07

3122 CFL14CFL12 7.52493998813762E-07

3123 ML452ML453 7.56546576160243E-07

3124 CFL32XD702 8.11472732421762E-07

3125 XD702CFL32 8.11472732421762E-07

3153 CFL17N127 2.11317241160263E-06

3154 CFL6XB106 2.19806869672546E-06

3155 CFL32CFL33 2.30671225593517E-06

3156 CFL33XD702 3.02269071611815E-06

3157 XD702CFL33 3.02269071611815E-06

3158 CFL32CFL30 3.16104201450632E-06

3159 CFL34XD702 3.17961574121035E-06

3160 XD456CFL31 3.21444886672260E-06

3161 CFL31CFL30 3.26582106323275E-06

3162 CFL30CFL31 3.26582106323275E-06

93

Figure 4.1.5e: CHART SHOWING DISTANCE STANDARD ERROR VALUES OF ALL

STATIONS/LINES (3162 lines) IN THE NETWORK.

(Chart could not show all stations/lines due to scale)

`

Figure 4.1.5f: CHART SHOWING DISTANCE STANDARD ERROR VALUES OF

SOME CFL STATIONS/LINES (19 lines) IN THE NETWORK.

0.00E+005.00E-011.00E+001.50E+002.00E+002.50E+003.00E+003.50E+00

N4

N3

N3

N7

M4

M5

N1

12

N1

03

K4

7K

38

L33

L32

A1

8A

21

D1

9D

17

U8

5U

13

N1

03

N1

01

H4

P1

4

U6

9U

72

A3

6M

L52

DISTANCE STANDARD ERROR (PPM meter)

DISTANCE STANDARDERROR (PPM meter)

0.00E+005.00E-011.00E+001.50E+002.00E+002.50E+003.00E+003.50E+00

A3

9C

FL3

6

CFL

11

XB

15

3

CFL

15

YC2

02

CFL

16

YC2

02

CFL

37

A3

9

CFL

36

CFL

37

CFL

37

CFL

36

CFL

35

CFL

37

CFL

16

N1

27

CFL

7X

B1

06

CFL

8C

FL7

CFL

17

N1

27

CFL

6X

B1

06

CFL

32

CFL

33

CFL

33

XD

70

2

CFL

32

CFL

30

CFL

34

XD

70

2

CFL

30

CFL

31

CFL

31

CFL

30


DISTANCE STANDARD ERROR (PPMmeter)

94

Figure 4.1.5g: CHART SHOWING DISTANCE STANDARD ERROR VALUES OF

SOME SECONDARY STATIONS/LINES (16 lines) IN THE NETWORK.

4.2 ANALYSIS OF RESULTS

The analysis of results is divided into six subsections in accordance with the objectives of this

research. A close view of each of the bar charts in section 4.1 clearly shows the tabulated range

of values against the stations/lines being considered at any given time.

It should be noted that due to the small scale of plotting used in the charting of each of the

Figures 4.1.2a, 4.1.3a, 4.1.4a, 4.1.5a, 4.1.5b, and 4.1.5e which represents the whole network

plot, most of the stations/lines identities are hidden in the individual plots, but shown in the other

figures depending of the chosen area of interest/analysis.

4.2.1 Analysis of the Network residuals of Observations (V) after the Adjustment

Before Network Adjustment

The observations used in this adjustment were eliminated of gross errors and the following

results were achieved.

Triangular closure (TC) lies in the region (-3.6 < TC < 3.7) seconds (Appendices Ia).

Mean Triangular closure is 0.9358633776118294 second (Appendices Ia).

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

XB

15

3C

FL1

0

ML4

52

XL2

02

XL2

02

ML4

52

XL2

01

ML4

52

XB

15

2C

FL1

0

ML4

53

XL2

02

YC2

02

CFL

15

ML2

01

ML2

ML2

ML2

01

XL2

01

ML2

ML1

ML2

01

XB

10

6C

FL7

R4

7C

FL1

7

N1

27

CFL

17

XD

70

2C

FL3

3

XD

45

6C

FL3

1


DISTANCE STANDARD ERROR(PPM meter)

95

After network Adjustment

Residual vector (V) after adjustment is shown in Table 4.1.2a and appendix (III), V matrix of

size (3162 by 1) ranges between {-7.8182E-05 and - 9.7688E-05} radian for both angles and

distances and this can be categorized into V < ± 1 second for all angles and V < ± 1m for all

distances.

Figure 4.1.2a shows the plot of the network residuals values in radian against their corresponding

stations/lines used in the adjustment. Its distribution gave higher and better estimates of the

residuals at the extreme ends and lower estimates of the residuals towards the middle of the

charts, thereby revealing and relating stations/lines identity and their corresponding residuals.

It seems that the triangulation stations lies at the extreme ends of the chart with higher and better

estimates of residuals (Figure 4.1.2c), the secondary stations follows, while the CFL

stations/lines occupies the middle section of the (Figure 4.1.2b) chart and thus denoting least

estimate of residuals in the network.

Triangulation stations involve interconnected lines from other triangles stations. It provides the

opportunity of using a given station/line observation as many times as it appears in a network,

thereby providing redundant observations for the adjustment of such station/line. This in turn

would give higher/better estimates of the residual V as seen on line K29K30 in Figure (4.1.2c).

Sample of the residual values is shown in Table (4.1.2a) and the Appendix (III) for all the

stations.

Other stations/lines like the CFL traverse stations configuration which contributes less redundant

observations as against the triangulation stations configuration during adjustment gave

lower/poorer estimates of V, while the secondary stations/lines gave a better estimates of the

residuals than the CFL stations/lines as observed in Figure 4.1.2a

4.2.2 Analysis of the Network Stations Position Correction (x) after the adjustment

The resultant of 515 stations position corrections of size (515 by 2) lies in the region (0.25122m

- 5.74574m) for all stations, with the exception of the 10 tertiary stations (9 ML-chains stations

and 1 XL-chain station) which lies in the region of {5.94595m and 13.85221} as shown in Table

96

(4.1.3a) and Appendix (IV). Triangulation stations/lines with large redundant observations gave

higher/better estimates of x. Other stations/lines like the CFLs with less redundant observations

gave lower/poorer estimates of x, while the secondary stations/lines with higher redundant

observations produced a larger/better estimates of x as shown in Figures (4.1.2a, b, and c.)

where same line bar is repeated, especially in triangulation where an observation can appear in

two or more triangles during adjustment.

Figure 4.1.3a shows the plot of the network positional corrections (x) due to the residuals values

against their corresponding stations/lines used in the adjustment. Its distribution gave higher and

better estimates of x towards the extreme right, lower estimates of the x towards the extreme left

of the chart, thereby revealing and relating stations/lines identity and their corresponding

positional correction values.

It can be seen that the triangulation stations lies at the extreme right of the chart with higher and

better positional corrections (Figure 4.1.3a), the secondary stations follows, while the CFL

stations/lines occupies the extreme left of chart and thus denoting least estimate of residuals in

the network.

Sample of the positional corrections values is shown in Table (4.1.3a) and the Appendix (IV) for

all the stations. Figure (4.1.3b) shows the distribution of the positional corrections in the

secondary stations/lines of the network.

4.2.3 Analysis of the Network Error Ellipse (Geometry) after the Adjustment

Stations resultant error ellipse values (radial error) lies in the region ( 0.061 – 4.821) meters

(Table 4.1.4a). Triangulation stations/lines with large redundant observations gave higher/better

estimates of V and x and hence a lower values of resultant error ellipse values (radial error) as

shown in Tables (4.1.2a, 4.1.3a, 4.1.4a), Figures (4.1.2a, 4.1.3a, 4.1.4a) and Appendix (XIb)

Other stations/lines CFLs with less redundant observations gave lower/poorer estimates of V and

positional corrections and hence higher values of error ellipse (radial eerror) (Tables 4.1.2a,

4.1.3a, 4.1.4a) and (Figures 4.1.2a, 4.1.3a, 4.1.4a and Appendix (XIb)). The absolute/relative

error ellipse values of any station/line directly denotes the relative geometry of such station/line

in the network.

97

Table (4.1.4a) and Figure (4.1.4a and Appendix (XIb)) shows the plot of the network absolute

error ellipse values against their corresponding stations used in the adjustment. Its distribution

gave higher estimates at the extreme right and lower estimates at the extreme left of the charts,

thereby revealing and relating stations identity and their corresponding absolute error values.

It can be seen that the triangulation stations lie at the extreme left of the chart with smaller and

better estimates/stronger section in geometry of the network stations (Figure 4.1.4a), while the

CFL and some of the secondary stations occupy the extreme right of the chart and thus signifying

higher estimate/weaker section in the network geometry. Details of the distribution is in

Appendix (V) and Appendix (XIb).

Figure (4.1.4b) and Figure (4.1.4c) reveals the magnitudes of the network absolute error ellipses

for the CFL and the secondary stations respectively.

Plan for strengthening the network can be achieved as follows:

The Network a-posteriori variance of unit weight after adjustment was computed to be 1 meter,

and this represents the standard deviation of the network ( ⏞ = 1 meter). Classification can thus

be made of the network error ellipse values as a factor of the standard deviation of the network to

ascertain likely possible number of stations that would need re-observation so as to strengthen

the network.

The number of stations that fall outside each class of the network standard deviation

classification thus represents the network stations that would need to be re-observed, so as to

strengthen the network. For example the last 49 stations in Table 4.1.4a, appendix (V), Appendix

(XIb and (XII a) would need a re-observation to attained a standard deviation of ≤ ( 1 ⏞ ) as

shown in Table 4.5d. 90.5% of the present network stations fall within error ellipse values of ≤

( 1 ⏞ ).

4.2.4 Analysis of the Network Relative Error Ellipse (Relative Geometry)

Stations relative error ellipse size vector (Relative E.Lx) showed that 94.2% ( error ellipse values

of stations ≤ 2 meter) of stations standard deviations lies within 2-sigma network standard

deviation. The relative error ellipse values range within (1.26634E-03 – 4.40055) meter and of

size (3162 by 1).

98

Table (4.1.5a) and Figure (4.1.5a) shows the plot of the network relative error ellipse values in

meters against their corresponding stations/lines identity used in the adjustment. Its distribution

gave higher and better estimates towards the extreme ends and lower estimates towards the

extreme left of the charts.

4.2.4.1 Analysis of the Network Standard Error in Azimuths (Orientation)

Station – Station (Line) Standard error (S.e.) in azimuth lies in the region

( 0.00010" – 0.14762") as in Table (4.1.5b).and Appendix (VI).

Table (4.1.5b) and Figure (4.1.5b) showed the plot of the network standard errors in azimuth

values against their corresponding stations/lines used in the adjustment. Its distribution gave

higher estimates at the extreme right and lower estimates at the extreme left of the charts, thereby

revealing and relating stations/lines identity and their corresponding standard error in azimuths.


better estimates/stronger section in orientation of the lines in the network (Figure 4.1.5b), while

the CFL and some of the secondary stations/lines occupy the extreme right of the chart and thus

signifying higher estimate/weaker section in orientation of the lines in the network. Details of the

distribution is in Appendix (VII).

Figure (4.1.5c) and Figure (4.1.5d) reveals the magnitude of the standard error in azimuths for

the CFL and the secondary stations/lines in the network respectively.

4.2.4.2 Analysis of the Network Standard Error in Distances (Scale)

Station – Station (Line) Standard error (S.e.) in Distances lies in the region

( 0.00037 – 3.26582) PPM in meters as in Table (4.1.5c).

Table (4.1.5c) and Figure (4.1.5e) shows the plot of the network standard errors in distance

values against their corresponding stations/lines used in the adjustment. Its distribution gave

higher estimates at the extreme right and lower estimates at the extreme left of the charts, thereby

revealing and relating stations/lines identity and their corresponding standard error in distances.

99


better estimates/stronger section in scale of the lines in the network (Figure 4.1.5e), while the

CFL and some of the secondary stations/lines occupy the extreme right of the chart and thus

signifying higher estimate/weaker section in scale of the lines in the network. Details of the

distribution are shown in Appendix (VIII).

Figure (4.1.5f) and Figure (4.1.5g) reveals the magnitude of the standard error in distances for

the CFL and the secondary stations/lines in the network respectively.

It can be deduced that the Nigerian Horizontal Geodetic Network Standard error in distances,

falls in the acceptable specification of a Primary Network of 1/100,000 (Table 2.1.5.1.1 and

Appendix VIII).

4.2.4.3 Analysis of the Network Standard Deviation

The classification of the Network Standard Deviation is given in Table 4.2.4.3a

Table 4.2.4.3a CLASSIFICATION /ASSESSMENT OF NETWORK A- POSTERIORI

VARIANCE OF UNIT WEIGHT (NETWORK STANDARD DEVIATION ⏞ = 1 meter)

NETWORK STANDARD

DEVIATION ( ⏞ = 1 meter)

CLASSIFICATION

NUMBER OF

NETWORK STATIONS

WITHIN A CLASS

NUMBER OF

NETWORK

STATIONS OUTSIDE

A CLASS

NUMBER OF NETWORK

STATIONS WITHIN A

CLASS PERCENTAGE

(%)

≤ ( 1 ⏞ ) 466 49 90.5%

≤ ( 2 ⏞ ) 485 30 94.2%

≤ ( 3 ⏞ ) 506 9 98.3%

≤ ( 4 ⏞ ) 514 1 99.8%

≤ ( 5 ⏞ ) 515 NONE 100%

It can therefore be deduced that the Nigerian Horizontal Geodetic Network Standard error in

azimuths, falls in the acceptable specification of a primary network which is 0.4seconds (Table

2.1.5.1.2).

4.2.5 Statistical Paired Sample test analysis of the error ellipse values of the 33 Stations of

the Network used in 1977 and their corresponding values in 2009 adjustment.

100

The results of the error ellipse values in the 1977 and 2009 adjustments are shown in Table

(4.2.5a). The result of the statistical paired sample test of the data in Table (4.2.5a) shown in

Table (4.2.5b) at 95% confidence level reveals that the misclosure in the Geometry of the 33

stations have been reduced by 36% (Figure 4.2.5a) in the 2009 adjustment.

101

Table 4.2.5a: COMPARISON OF ERROR ELLIPSE VALUES FOR 33 STATIONS USED

IN 1977 AND THEIR CORRESPONDING VALUES IN 2009 ADJUSTMENT

RESULT OF ERROR ELLIPSE

PARAMETER IN 1977

ADJUSTMENT

RESULT OF ERROR

ELLIPSE PARAMETER

IN 2009 ADJUSTMENT

S/N STN ID error ellipse values (m) error ellipse values (m)

1 A50 0.339705755 0.532768419

2 B9 0.502493781 0.31087458

3 C1 0.914002188 0.365302356

4 C18 0.517397333 0.446354699

5 C30 0.651920241 0.293212354

6 C32 0.498196748 0.450049134

7 C6 0.915478017 0.378443483

8 D15 0.660681466 0.40139855

9 D34 0.468614981 0.367893166

10 D5 0.714212853 0.315136174

11 E13 0.829276793 0.380398444

12 E3 0.412310563 0.359156202

13 F1 0.587962584 0.39253392

14 H1 0.686221539 0.483649711

15 H11 0.452769257 0.304410309

16 H12 0.632534584 0.300091819

17 K11 0.428018691 0.40582027

18 K17 0.322024844 0.496442103

19 L19 0.758221603 0.264783661

20 L33 0.296984848 0.199516126

21 L8 1.098362417 0.28215929

22 M1 0.735527022 0.330514462

23 N104 0.240831892 0.526722116

24 N12 0.403112887 0.210405589

25 N124 0.428018691 0.360634383

26 N38 0.304138127 0.380111136

27 P15 0.538144962 0.784287696

28 P9 0.651152824 0.341421165

29 R34 0.81467785 0.439254982

30 R5 0.702922471 0.293145068

31 U67 0.731095069 0.365039143

32 U71 0.759275971 0.381462273

33 U78 0.915478017 0.506848498

102

Table 4.2.5b: PAIRED SAMPLE TEST ON THE EXTRACTED ABSOLUTE

ERROR VALUES OF 33 STATIONS

IN THE 1977 AND 2009 ADJUSTMENTS AT 95% CONFIDENCE LEVEL

Fig. 4.2.5a: 95% CONFIDENCE LEVEL PLOT OF ABSOLUTE ERROR

ELLIPSE BAR FOR 33 STATIONS

IN THE 1977/2009 ADJUSTMENTS

4.2.6 Comparison of the 515 Stations Coordinates in the 1977 and their corresponding

values in 2009 adjustments.

The results of a sample of the 515 Network stations coordinates in 1977, the change in the

coordinates due to the 2009 adjustment, and the new coordinates are shown in sample Table

4.2.6a with details in Appendices (IX). The plot of the adjusted stations coordinates are shown in

Figure 4.2.6a.

103

APPENDIX (IX)

FINAL ADJUSTED COORDINATE 1977 CORRECTION TO STATION COODINATES FINAL ADJUSTED COORDINATE 2009

SR N0 STATION ID LATITUDE φ(deg)

LONGITUDE λ(deg) ∆φ(sec) ∆λ(sec)

LATITUDE φ(deg)

LONGITUDE λ(deg)

1 K52 10.2831915 9.604600833 -0.00741 -0.08656 10.28318944 9.604576872

2 K44 10.26249606 9.226191111 -0.00205 -0.04646 10.2624955 9.22617815

3 A50 10.59671464 9.396634167 0.01383 -0.08222 10.59671847 9.396611189

4 A49 10.68438167 9.670446667 -0.00105 -0.06442 10.68438139 9.670428689

5 K30 10.76705286 9.069961944 0.00287 -0.07963 10.76705367 9.069939825

6 K40 10.40356747 8.963409444 -0.00926 -0.05603 10.40356489 8.963393742

7 K47 10.11895292 8.701388611 0.00605 -0.07102 10.11895461 8.701368939

505 XL451 9.848630917 10.91423056 -0.02125 -0.1066 9.848625014 10.91420042

506 ML502 9.680435028 11.03806111 -0.00202 -0.08862 9.680434467 11.03803567

507 XL453 9.505554833 10.97334444 0.04652 -0.08446 9.505567756 10.97331989

508 ML705 9.379028028 11.48335556 0.01048 0.14127 9.379030939 11.48339428

509 ML751 9.199734472 11.49636667 0.03741 -0.01545 9.199744864 11.49636281

510 MR550 7.900946139 11.76091389 0.02036 -0.09479 7.900951794 11.76088739

511 U68 7.550463778 6.469128333 0.012 -0.05032 7.550467111 6.469114244

512 R51 12.05971761 6.885535278 0.00547 -0.07482 12.05971914 6.885514578

513 R50 11.97057358 6.890548611 0.00319 -0.08283 11.97057447 6.890525547

514 U70 7.807286333 6.713752222 0.04091 -0.17726 7.807297697 6.713703067

515 A45 10.67423722 10.28772778 -0.0361 0.0299 10.67422778 10.28771642

Table 4.2.6a: COMPARISON OF THE FINAL ADJUSTED COORDINATES IN 1977 ADJUSTMENT AND

2009 ADJUSTMENT

105

4.2.7 Analysis of the Summary of the Results in the 1977 and 2009 adjustments.

The Summary of both adjustments are shown in Table (4.2.7a).

Table 4.2.7a COMPARISON OF 1977 & 2009 ADJUSTMENT

S/N SUMMARY OF 1977 ADJUSTMENT SUMMARY OF 2009 ADJUSTMENT

1 Adjustment by observation equation

method.

Least square Phase adjustment was used.

Adjustment by observation equation method.

Least square Optimized Simultaneous

adjustment was used.

2 A posteriori variance of unit weight is

1.17

A posteriori variance of unit weight is 1.0

3 The least element of AT

PV is not zero

while the largest element was 1.02E-09.

All elements of AT

PV are zeroes which

denotes presence of only white random noise

in the adjusted observation.

4 Maximum standard error (S.e.) of

Azimuths after adjustment is 75'' while

that of distance is 546ppm

Maximum S.e of Azimuths after adjustment

is 0.147615953546203'' while that of

distance is 3.26582106323275ppm

5 Standard error of mean for absolute error

ellipse values for 33 lines selected was

3.677E-02

Standard error of mean for absolute error

ellipse values for 33 lines selected was

1.197E-02

6 Standard deviation for the absolute error

ellipse values for the 33 lines used in

1977 was 0.2112

Corresponding Standard deviation for the

absolute error ellipse values of the 33 lines

selected in 2009 was 0.1101

7 Mean for absolute error ellipse values

for the 33 lines selected was 0.6034

Mean for absolute error ellipse values for the

33 lines selected was 0.3833

8 Weakness in the southern part of the

network

Weakness in CFL and few secondary and

tertiary stations of the network.

9 Error analysis not carried on the entire

network.

The standard deviation of network was found

to be 1.0m and 94.2% of the network stations

with error ellipse values ≤ 2 meters represent

2-sigma statistical interval in the network

standard deviation distribution.

106

CHAPTER FIVE

CONCLUSIONS, CONTRIBUTIONS TO KNOWLEDGE AND RECOMMENDATIONS

5.1 CONCLUSIONS.

An optimized holistic adjustment of the Nigerian horizontal geodetic network has been achieved

with a- posteriori variance of unit weight = 1.0 meter, which represents the network

standard deviation (which is an indication of an optimal design and adjustment of the network)

and the product of ATPV matrix being zeroes (which confirms the presence of only white

random noise in the residuals of the adjusted observation) (Table 4.2.7a).

Comparison of the 2009 and 1977 adjustments showed that:

(a) 2009 adjustment has a maximum Standard error (S.e) in Azimuths of

0.147615953546203'' while that of 1977 was 75'' (Figures 4.1.5b –d, Tables 4.1.5b and

4.2.7a), hence the optimized holistic approach used in this study has greatly improved

and revealed the strength of the network in orientation.

(b) 2009 adjustment has maximum Standard errors (S.e) in distance of

3.26582106323275ppm. while that of 1977 was 546ppm (Figures 4.1.5e –g, Table 4.1.5c

and Table 4.2.7a), hence the optimized holistic approach used in this study has greatly

improved and revealed the strength of the network in scale.

(c) 2009 adjustment has an a-posteriori variance of unit weight of 1.0 (an indication of

optimal design of the network) while that of 1977 was 1.17 (Table 4.2.7a).

(d) 2009 adjustment has improved the geometry of the 33 selected stations/lines used for

error ellipse analysis in 1977 by 36% as confirmed by the following values (Table 4.6d):

Standard error of mean: 3.677E-02 (in 1977); 1.197E-02 (in 2009)

Standard deviation: 0.2112 ( in 1977); 0.1101 (in 2009)

Mean: 0.6034 (in 1977); 0.3833 (in 2009).

The study has recovered the lost network data by searching for possible libraries within and

outside the country where the data could be found. Also, the creation of a comprehensive

intelligent database for the whole network which can search, query and perform calculations of

any desired parameters of the network has been achieved. For example the data base can be used

to compute the variance covariance matrix of any control station or the azimuth of any line in the

107

network (Figure 4.1a and Tables 4.1a –f). The products of the database are tabulated in the

appendices. The software for the data base is shown in appendix 1d.

The following Network geometry have been determined in-terms of: the residuals of the

observations (Figures 4.1.2a –c, Table 4.1.2a); positional corrections (Figures 4.1.3a –b, Table

4.1.3a); error ellipse values (Figures 4.1.4a – c, and 4.1.5a; Tables 4.1.4a, 4.1.5a, 4.2.4.3a and

4.2.7a and Appendix XII a, b, c); standard errors in azimuths (Figures 4.1.5b –d; Tables 4.1.5b

and 4.2.7a); and standard errors in distances (Figures 4.1.5e –g; Tables 4.1.5c and 4.2.7a).

The plotting and annotation of any figure in the entire network using the intelligent data base

created have been achieved.

The identified areas of strength/weakness (geometry) in scale and orientation in the network are

as stated below:

(a) The Triangulation section of the Network was found to be relatively less error prone due

to the enormous redundant observations that connected the stations. hence these stations

were more stable than other chains (Figures 4.1.2a, 4.1.2c, 4.1.3a, 4.1.4a, 4.1.4c; Tables

4.1.2a, 4.1.3a, 4.1.4a and 4.1.5a).

(b) Triangulation stations (U70, U68, N26, and H14) which occurred towards the end of their

respective chains, and therefore were connected by few observations were identified as

relatively weaker sections of the triangulation network (Figures 4.1.2a, 4.1.2c, 4.1.3a,

4.1.4a, 4.1.4c; Tables 4.1.2a, 4.1.3a, 4.1.4a and 4.1.5a).

(c) With the exception of a few secondary and tertiary stations (XD456, XL202, XB152,

ML54 and ML452) with error ellipse values > 1m, other secondary and tertiary stations

have error ellipse values < 1 meter. This adjustment has helped to improve the accuracy

of these stations (Figures 4.1.2a, 4.1.3a-b, 4.1.4a, 4.1.4c; Tables 4.1.2a, 4.1.3a, 4.1.4a).

(d) The CFL stations incorporated few observations as compared to the triangulation stations

in the adjustment, therefore giving relatively poorer estimates of its residuals/correction

vectors, as shown in the larger values of error ellipses. The CFL section is therefore the

relatively weaker part of the network (Figures 4.1.2a, 4.1.2b, 4.1.3a, 4.1.4a-b; Tables

4.1.2a, 4.1.3a, 4.1.4a and 4.1.5a).

108

The identified network stations which require upgrade so as to achieve improved network

geometry are listed in order of urgency (Table 4.2.4.3a):

(a) All the CFL section of the network (Tables 4.1.2a, 4.1.3a, 4.1.4a and 4.1.5a);

(b) The following secondary and tertiary stations section of the network: XD456, XL202,

XB152, ML54 and ML452 (Tables 4.1.2a, 4.1.3a, 4.1.4a and 4.1.5a).

(c) The sections towards the end of the triangulation chains, such as: U70, U68, N26, and

H14.

(d) All the other stations of the network are also necessary for upgrade to improve on the

network geometry (Tables 4.1.2a, 4.1.3a, 4.1.4a and 4.1.5a

(e) The network re-strengthening/station upgrade exercise can be classified thus (Table

4.2.4.3a): 49 stations at 1-sigma network standard deviation; 30 stations at 2-sigma

network standard deviation; while at 3-Sigma, 9 stations would be re-observed.

The different computer programs created in this study, such as that of adjustment; error analysis;

and data structure; now form an efficient platform for any future adjustment and study of the

Nigerian Horizontal Geodetic Network (Sections 3.1-3.4). An example is the creation of a

coefficient matrix space' that can help compute the design matrix of any large system, (Appendix

Ic),

This platform comprises:

(a) Network program for the computation of the residual vector of the observations

(Appendix Ic, Figures 4.1.2a –c, and Table 4.1.2a).

(b) Network program for the computation of the positional corrections (Appendix Ic, Figures

4.1.3a –b, Table 4.1.3a)

(c) Network program for the computation of the error ellipse value (positional shift) of

4.81metres (Appendix Ic, Figures 4.1.4a –c, and 4.1.5a; Tables 4.1.4a, 4.1.5a and 4.2.7a).

(d) Network program for the computation of the Standard error in Azimuths (orientations) of

0.147615953546203" (Appendix Ic, Figures 4.1.5b –d; Tables 4.1.5b and 4.2.7a).

(e) Network program for the computation of the Standard error in distance of

109

3.26582106323275ppm (Appendix Ic, Figures 4.1.5e –g; Tables 4.1.5c and 4.2.7a) which

include

Program for plotting and annotating the network triangles (Appendix Ic, Figure 4.2.6a);

Program for plotting and annotating the network residuals (Appendix Ic, Figures 4.1.2a-c);

Program for plotting and annotating the network positions corrections (Appendix Ic, Figures

4.1.3a-b);

Program for plotting and annotating the network absolute and relative error ellipse values

(Appendix Ic, Figures 4.1.4a-c and 4.1.5a);

Program for plotting and annotating the network standard errors in lengths(scales) and

azimuths (orientation), as shown in Appendix Ic, and Figures 4.1.5a-g;

(k) The above subsections (6a – j) would together assist to identify areas of strength/weakness

in the network and consequently allow for the upgrade of the network geometry at a minimal

cost.

There are many important parameters about the Nigerian Horizontal Geodetic Network which

before this study were unknown but now revealed: for example

(a) The network stations position correction is distributed in the range 0.25122m -

13.85221m within the network (Table 4.1.3a).

(b) The Nigerian Horizontal Geodetic Network fulfils the conditions of a primary order

network (94.2% of stations standard deviations fall within 2-sigma network standard

deviation).

The optimized adjusted network plot was not superimposed on the map of Nigeria because it is

not within the scope of this study. If the coordinates of the boundary points with other related

features are known as shown in the old Nigerian network map (Appendix XIII), the intelligent

data base can readily complete the plot accurately.

110

5.2 CONTRIBUTIONS TO KNOWLEDGE.

(1) For the first time in the history of Nigeria, an optimized holistic adjustment of the Nigerian

horizontal geodetic network has been achieved. A- Posteriori variance of unit weight of 1.0 was

achieved which is an indication of an optimal design and adjustment of the network. Also the

ATPV vectors after adjustments are zeroes which confirms the presence of only white random

noise in the residuals of the adjusted observation (Table 4.2.7a). Both confirms the network

reliability.

(2) The areas of strength and weakness in scale and orientation in the Nigerian horizontal

geodetic network have been determined.

(3) A future re-strengthening and re-observation plan of the Nigerian horizontal geodetic

network at different network standard deviations have been achieved and recommended in this

study (Table 4.2.4.3a and Appendix XII a, b, c).

(4). The study has created a generalized network through the creation of a comprehensive

intelligent database for the whole network which can search, query and perform calculations of

any desired parameters of the network. For example you can ask the data base to compute the

variance covariance matrix of any control station or the azimuth of any line in the network

(Figure 4.1a and Tables 4.1a-f).

(5).Through this study, the missing data for Nigerian Horizontal Geodetic Network have been

recovered by searching for possible libraries within and outside the country where the data could

be found (Appendix Ia).

5.3 RECOMMENDATIONS

(1) The findings in this study now provide the foundational platform for the Nigerian geodetic

network, and is therefore recommended to the Federal Government of Nigeria, Nigerian

Institution of Surveyors, Surveyors Council of Nigeria and other stake holders for an immediate

plan on the network re-observation and re-strengthening in order to know the possible trend of

changes like network distortion, that might have taken place especially since the advent of

mineral exploitations in Nigeria.

111

(2) The intelligent data base created in this study would assist accelerating the speed of any

future update/upgrade adjustment and study of the Nigerian Horizontal Geodetic Network.

(3) The transformation parameters of the Nigerian geodetic system should be determined as a

matter of urgency using the result of this adjustment where necessary.

(4) The software written for this study is recommended for use by Survey Department, Surveyors

and other relevant stake holders.

(5) If the Nigerian boundary stations coordinates and other related features are known as shown

in the Nigerian map (Appendix X111), I recommend that, the intelligent data structure created

in this study, be used to accurately complete the plot of the Nigerian map containing the adjusted

network.

The following recommendations are therefore made for further study on the Nigerian geodetic

network:

(6) The incorporation of GPS satellite observations into the network will assist in reducing the

geometric error in the network. This new set of observations can be used for the distortion study

of the Nigerian geodetic network.

(7) Re-observations of the identified weak stations must be carried out with the space satellite

technique on the network to re-strengthen the Nigerian Horizontal Geodetic network. This would

improve the network geometry at a minimal cost.

(8). New observations must be carried out on the entire network alongside other new well

selected stations all over the country, using the space satellite technique to re-strengthen the

network and it‟s distortion study to avert or minimize future disaster.

112

REFERENCES

Allman, J.S. and Hoar G.J. (1973), “Optimisation of Geodetic Networks.” Survey Review vol

XXII No. 167, pp181-207.

Anon, (1936), “Nigeria Survey Department Instructions for Primary Triangulation.” Government

Printer Lagos, pp1-45.

Anon, (1961), “Modern Computing Methods.” Notes on Applied Science No. 16. 2nd

edition.

National Physical Laboratory. H.M.S.O. pp1-170.

Anon, (1971), “Final report of the 12th

Parallel Survey in the Republic of Nigeria.” Corps of

Engineers U.S.A. pp1-217.

Ashkenazi, V. (1967), “Solution and error analysis of large geodetic net-works - 1 Direct

methods.” Survey Review Vol XD No. 146 pp166-173, No. 147 pp194-206.

Ashkenazi, V. (1970), “Adjustment of Control Networks for precise Engineering Surveying.”

Chartered Surveyor Vo1, 102 no. 7 pp314-320

Ashkenazi, V. and Cross, P.A. (1972), “Strength analysis of Block VI of the European

Triangulation.” Bulletin Geodesique pp6-23.

Ashkenazi, V., Cross P.A., Davies M.J.K. and Proctor D.W. {1972), “The Re-adjustment of the

Re-triangulation of Great Britain.” Ordnance Survey Professional papers new series no.

24. pp1-51.

Ashkenazi, V. (1974), “An assessment of the proposed re-adjustment of the North American

Geodetic Control networks, U.N.E Symposium”, Springer Berlin/Heidelberg, pp247-250.

Atilola, O. (1985-86), “Unpublished Lecture notes on Adjustment Computation I.” Department

of Surveying, University of Lagos, pp1-40.

Ayeni, O. O. (1980) “An investigation into the choice of the most appropriate Least Squares

method for position determination” Proceedings of the International Workshop on

Geodetic Positioning.University of Lagos, Nigeria, September 8- 10, 1980, pp99-132.

Ayeni, O. O. (1980) “Adjustment computation by Least Squares” unpublished lecture note,

Department of Surveying, University of Lagos, pp1-35.

113

Ayeni, O.O. (1985) , “ Comparison of least squares of adjustment methods for Geodetic

Level Network.” Proceedings of the NAG conference on Datum Determination in

Nigeria, pp89-93.

Ayeni, O.O. (2001) “ Adjustment and Analysis of Data.“, Lecture notes, Department of

Surveying and Geoinformatics, University of Lagos, pp1-300.

Ayeni, O.O. (2002) “Statistical Adjustment and Analysis of Data (With applications in Geodetic

Surveying and Photogrammetry).“, Lecture notes, Department of Surveying and

Geoinformatics, University of Lagos, pp1-49.

Ayeni, O.O. (2003) “ Adjustment computation” Unpublished Lecture notes (SVY 908)”, Dept.

of Surveying and Geoinformatic, University of Lagos, pp1-55.

Baarda, W. (1968), “A Testing Procedure for Use in Geodetic Networks”, Publication on

Geodesy, Netherlands Geodetic Commission, pp1-97.

Baarda, W., (1977), “Measures for the Accuracy of Geodetic Networks.” IAG Symposium.

Sopron, Hungary pp1-26.

Bagratuni, G. V. (1962), “Course in Spheroidal Geodesy”, translated from Russian to English

language in 1967 by the Translation Division, Foreign Technology, pp1-305.

Bannister, A. and Raymong, S (1972). “Surveying “Published by Pitman Press, London, pp1-89.

Barricelli N. A. (1954), "Esempi numerici di processi di evoluzione". Methodology,pp45-68.

Barricelli, N. A. (1963), "Numerical Testing of Evolution theories, Part II. Preliminary tests

of performance, symbiogenesis and terrestrial life". Acta Biotheoretica (16): pp99-126.

Baudry, B., Franck F., Jean-Marc J. and Yves L. T. (2005), "Automatic Test Case Optimization:

A Bacteriologic Algorithm". IEEE Software: pp76-82.

Berne, J L. and Baselga, S. (2003). “First-order design of geodetic networks using the simulated

annealing method” Springer Berlin Heidelberg, pp47-54.

Biacs, Z, Krakiwsky, E, and Lapucha, D, (1990), “Reliability Analysis of Phase Observations

in GPS Baseline Estimation”, Journal of Surveying Engineering, ASCE, pp204-224.

http://en.wikipedia.org/w/index.php?title=Nils_Aall_Barricelli&action=edit

http://en.wikipedia.org/wiki/Jean-Marc_J%C3%A9z%C3%A9quel

http://www.irisa.fr/triskell/publis/2005/Baudry05d.pdf

http://www.irisa.fr/triskell/publis/2005/Baudry05d.pdf

114

Bies, Robert R; Muldoon, Matthew F; Pollock, Bruce G; Manuck, Steven; Smith, Gwenn and

Sale, Mark E (2006). "A Genetic Algorithm-Based, Hybrid Machine Learning Approach

to Model Selection". Journal of Pharmacodynamics: pp196-221.

Biddle, C.A. (1958), “Textbook of field astronomy.” H.M.S.O pp1-247.

Bomford, G. (1971), “Geodesy.” 3rd

edition. Clarendon, pp1-731.

Bradley, H.E. (1939), “The year 1937 In Nigeria.” Empire Survey Review Vol. V no. 31,

pp50-52

Butler, T.L. (1966), “Draft specifications for the observation of latitudes, longitudes, and

azimuths using the Wild T4.” Unpublished communication, pp1-51.

Calder W. J. (1931), “The Triangulation of Nigeria.” Proceedings of the Empire Survey

Officer‟s Conference 1931. pp 4-21.

Calder W. J. (1936-37), “The survey framework of Nigeria.” Empire Survey Review Vol 111

No 21 pp 386 - 395, No.22 pp 450 – 469, and Vol. IV No. 23 pp 2-12.

Cerny V. (1985), Grafarend (1974) and Baarda (1973), “Application of optimization method

to the famous „Travelling salesman problem‟ Wikipedia 2007, pp1-166.

Cerny, V. (1985), “A thermodynamical approach to the travelling salesman problem: an efficient

simulation algorithm.” Journal of Optimization Theory and Applications 45, pp41-51.

Chang, Y. M., Chen, C. H. and Chen, C.S. (1996), “Optimal observation design of a surveying

artificial neural Geomat Res Aust 64: pp1–16

Charles B. B. (1942) “Surveying” Wiley, New York, pp1-495.

Chedtham, G. (1965), “Textbook of topographical surveying” H.M.S.O. pp1-388.

Choi J. H., (1998) " Evaluation of the Accuracy of the new Primary Precise Geodetic network in

Korea", Earth, Planets and Space, Vol. 50, No.8, pp611-708.

Clark, D. (1965), “Plane and Geodetic Surveying for Engineers” Vol 11 Higher Surveying . 5th

Edition. Constable. Pp1-385.

Close C.A. (1933), “A 50 years retrospect.” Empire Survey Review Vol. 11 No. 8,pp1-30.

Cooper, M.A.R. (1974), “Fundamentals of Survey measurement and analysis.” Crosby

Lockwood Staples pp1-107.

115

Cross, P. A. (1974), “A comparison of modern position control networks with particular

reference to Kenya.” Journal of Geodesy, Springer Berlin/Heidelberg, pp169-180.

Cross, P. A. (1985), “ Numerical methods in network design: In optimization and design of

geodetic networks .” edited by Grafarend and Sanso. Springer: Berlin etc.pp429-435.

Crosby, Jack L. (1973), “Computer Simulation in Genetics.” London John Wiley & Sons, pp1-

492.

Dare, P. (1995), “Optimal Design of GPS Networks Operational Procedures,” Phd. Thesis,

School of Surveying, University of East London.pp1-400.

Dare, P. and Saleh, H. (2000), GPS design: logistic solution optimal and near-optimal methods.

Journal of Geodsy 74(6): pp467–478.

De Normann, A.W.N, (1933), “The year 1931 in Nigeria.” Empire Survey Review XII No. 7

pp44-46.

Dept. of the Army, (1953), ”Topographic Surveying” U.S. Govt. Print off (Washington) pp1-

280.

Even-Tzur, G.,and Papo, H., (1996), “Optimization of GPS Networks by linear programming”,

Empire Survey Review pp537-545.

Even-Tzur, G. (2001) “GPS Vector Configuration Design For Monitoring Deformation

Network in the North of Israel”, The 10th

FIG International Symposium on Deformation

Measurements, Orange, California, USA, pp146-151.

Fajemirokun, F.A. (1988) “ Unpublished Lectures Notes on Geometric Geodesy” Dept. of

Surveying and Geoinformatics, University of Lagos, pp1-40.

Falkenauer E. (1997), “Genetic Algorithms and Grouping Problems.” Chichester,

England: John Wiley & Sons Ltd, pp627-645.

Fentress, S. W. (2005). “Exaptation as a means of evolving complex solutions.” MA

thesis, University of Edinburgh Springer Berlin/ Heidelberg, pp301-309.

Ffox, L. (1964), “An introduction to numerical linear algebra.” Clarendon pp1-328.

FGCC (2006), “Specification of Geodetic Network”, Wikipedia, 2007 pp1-16.

http://en.wikipedia.org/w/index.php?title=Emanuel_Falkenauer&action=edit

http://www.inf.ed.ac.uk/publications/thesis/online/IM050329.pdf

116

Field N. J. (1977), “The adjustment and Strength Analysis of the Primary Triangulation

Network of Nigeria” M.Phil Thesis, University of Nottingham, 1977 pp1-417.

Fogel, D. B. (editor) (1998). Evolutionary Computation: The Fossil Record. New

York: IEEE Press, pp8-14.

Fogel, D. B. (2000). Evolutionary Computation: Towards a New Philosophy of

Machine Intelligence. New York: IEEE Press, pp1-140.

Fogel, D. B (2006), Evolutionary Computation: Toward a New Philosophy of Machine

intelligence, IEEE Press, Piscataway, Third Edition, pp1-372.

Fraser, A. S. (1957). "Simulation of Genetic Systems by Automatic Digital Computers. I.

Introduction", Australian Journal of Biological Sciences 10, pp484-491.

Fraser, A. S. and Donald B. (1970). “Computer Models in Genetics.” New York: McGraw-Hill

pp597-611.

Gazdzicki, J., (1976), “Strength Analysis of Geodetic Control Network”, Bull Geodesique, Paris,

France.

Gerasimeko MD (1997), “First order design of the deformation with the minimal number of

geodetic points and their optimal allocation”. Far East Math Rep 4: pp86–94

Gerasimeko MD, Shestakov NV, Kato T (2000), “On optimal geodetic design for fault-

mechanics studies” .Earth Planet Space 52: pp985–987

Goldberg, David E (1989), “Genetic Algorithms in Search, Optimization and Machine

Learning, “ Kluwer Academic Publishers, Boston, MA pp1-432

Goldberg, David E (2002), “The Design of Innovation: Lessons from and for Competent Genetic

Algorithms,” Addison-Wesley, Reading, MA pp721-739.

Grafarend E (1974), “Optimization of Geodetic” Bulleting Geodetic Science Aff 33: pp351–406

Graferand, E. and Sanso, F. (1986), “Classical first Order Design Problem” Wikipedia, 2007

pp103-123.

Grant, D. B., Blick, G. H, Pearse, M. B., Beavan, R. J. and Morgan P. J.(1999). "The

Development and Implementation of New Zealand geodetic Datum 2000", Presented at

IUGG99 General Assembly, Birmingham UK, 1999, pp18-30.

http://en.wikipedia.org/wiki/Alex_Fraser_%28scientist%29

117

Grushinskiy, N. P. (1963), “The Theory of the Figure of the Earth” translated from Russian to

English in 1969, Hajila AD694748, pp17-78.

Holland J. H. (1975), Adaptation in Natural and Artificial Systems, University of Michigan

Press, pp1-28.

Hong-sic Y. (2001), New Technology for a new century, FIG International Conference,

pp415-423.

Hotline, M. (1935), “The East African Arc III Observations.” Empire Survey Review Vol 111

No. 16 pp77-80

Ingham, A.E. (1975) “Sea Surveying”, Empire Survey Review, pp1-146.

Jagar R. (1988). “Analogies between Geodetic networks and Discrete Elastomechanical

Systems” Series C, rep 342, Journal of Geodesy, Sringer Berlin/Heidelberg, pp1-308.

Jager R.,kaltenbach H. (1990) and Schmitt J. M. (1997), “Introducing Spectra analysis to

Network stiffness” Manual of Geodesy 15: pp302–311.

Johnson and Wyatt (1994), “Application of first order design in Geodynamics Geodetic

Networks” Manual of Geodesy 19: pp309–323

Jordan- Eggert, (1962), “Jordan‟s Handbook of Geodesy”, translation of 8th

edition, 1941 by

Martha W. carta, (U. S. Army Map Service, Washington, D. C.) pp1-124.

Kenya Institute of Surveying and mapping Nairobi, 2000, “Geodetic Report of Kenya” pp1-12

Kirkpatrick, S and Gelatt, C. D. and Vecchi, M. P (1983), “ Optimization by Simulated

Annealing”, Science, Vol 220, Number 4598, pp671-680.

Kjellström, G. (1970). "Optimization of electrical Networks with respect to Tolerance

Costs.". Ericsson Technics (3): pp157-175.

Kjellström, G. (Dec. 1991). "On the Efficiency of Gaussian Adaptation". Journal of

Optimization Theory and Applications. (3): pp589-597

Kjellström, G. (January 1996). "Evolution as a statistical optimization algorithm".

Evolutionary Theory (11): pp105-117.

118

Koza, John (1992), Genetic Programming: On the Programming of Computers by Means of

Natural Selection, MIT Press. ISBN 0-262-11170-5, pp1-387.

Kuang, S., (1996), “Geodetic Network Analysis and Optimal Design”, Ann Arbor Press,

Inc.pp1-368.

Kuang, S. (1996), “An overview of recent work on Network design.” Ann Arbor Press, Inc.

pp1-368.

Land Information New Zealand, (2000) "Specifications for the Installation of GPS

Permanent Tracking Stations, Version 1.0, Office of the Surveyor General, pp1-11.

Legault M. M. (1985) “Surveying” Empire Survey Review, pp1483-1485.

Lindlohr, W. and D. Wells (1985), “GPS Design Using Undifferenced Carrier Phase

Observations”, Manuscripta Geodetica pp1-69.

Mccaw, G.T. (1935), “The two metres: the story of an African foot.” Empire survey review Vol.

9 No. 52 pp96-105.

Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. andTeller, E (1953).

"Equations of State Calculations by Fast Computing Machines". Journal of Chemical

Physics, 21(6): pp1087-1092.

Michalewicz Z. (1999),”Genetic Algorithms + Data Structures = Evolution

Programs” Springer-Verlag pp1-209.

Mitchell M., (1996), An Introduction to Genetic Algorithms, MIT Press, Cambridge, MA,

pp1-270.

Morley, O.J. (1938), “History of the Minno-Udi-Ilorin series.” Unpublished records of Federal

Surveys Lagos, pp1-46.

Munsey, D.F. (1949), “Base measurement in the Anglo-Egyptian Sudan.” Empire Survey

Review Vol. X No. 74, pp1-34.

Nickerson, B.G. (1978) “A Priori Estimation of Variance for Surveying Observables”:Dept.

of Surveying Engineering, University of New Brunswick Federiction, N.B., Canada

pp1-58.

http://en.wikipedia.org/wiki/MIT_Press

http://en.wikipedia.org/w/index.php?title=Special:Booksources&isbn=0262111705

119

Olliver, J.G. and Clendinning, J. (1970) “Principles of Surveying” 4th

, Ed., vol 1: Plane

Surveying, Empire Survey Review pp1-267.

Omogunloye O.G. (1988), “Optimum number of traverse station between azimuths checks in

A traverse network” Unpublished B.Sc. Project, Dept. of Surveying, University of Lagos

pp1-61.

Omogunloye O.G. (1990) “Quality control in geophysical prospecting” Unpublished M.Sc.

Project,Department. of Surveying and Geoinformatics, University of Lagos, pp1-88.

Ordnance Survey (1967), “History of the retriangulation of Great Britain.” H.N.S.O. pp1-395.

Pardalos P. M.and Romeijn H. E. (2002). “Global Optimization problems” Wikipedia, 2007

pp1-569.

Paul, T. R. J. (1999) " The Need for Dynamic Datum‟s", Geometrics Research

Australasia No. 71, 1999, pp87-102.

Rainsford, H.F. (1949), “Long lines on the Earth: various formulae. ” Empire Survey Review

Vol. X No. 71 and 72 pp19-29.

Rainsford, H.F (1957), “Survey adjustments and least squares.” Constable & Co Ltd. London.

Pp1-326.

Ralph H. (1985) “Guide to Site Surveying” Empire Survey Review, pp1-310.

Richardus, P. (1966), “Project surveying.” Elsevier North Holland pp1-427.

Roeloss, R. (1950), “Astronomy applied to land surveying.” Argue Amsterdam pp1-259.

Saad, A. A. and Elsayed, M. S. 2007. “Simple Model for Improving the Accuracy of the

Egyptian Geodetic Triangulation Network” FIG Working Week, Hong Kong, pp1-24.

Schaffrin, B. (1985), “Aspects of Network Design, Optimization and Design of Geodetic

Networks,” Grafarend and Sanso, editions. Springer – Verlag, Berlin, pp548-597.

Schmitt, L. M., Nehaniv C. N., Fujii R. H. (1998), “Linear analysis of genetic Algorithms.”

Theoretical Computer Science (208), pp111-148.

Schmitt, L. M. (2001), Theory of Genetic Algorithms, Theoretical Computer Science (259),

pp1-61

120

Schmitt, L.M (2004), “Theory of Genetic Algorithms II: models for genetic operators over the

string-tensor representation of populations and convergence to global optima for arbitrary

fitness function under scaling,” Theoretical Computer Science (310), pp181-231.

Schwefel, H. (1974): Numerical Optimization of Computer Models (PhD thesis), Reprinted by

Birkhäuser (1977) pp1-389.

Shepherd, F.A. (1982) “Advance Engineering Surveying Problems and Solutions.” Hodder rnold

pp1-288..

Stamers-Smith H.A. and Sonola J.A (1955), “The primary triangulation of Nigeria.” Proceedings

of the Commonwealth Survey Officer‟s Conference 1955. Paper no. 23 pp67-76.

Syswerda, G. (1989). "Uniform crossover in genetic algorithms". J. D. Schaffer Proceedings of

the Third International Conference on Genetic Algorithms, Morgan Kaufmann pp2-9..

Trutmann, H.O. “The Theodolite and Its Application” translated into English by Ward, A.H.

Published. By Wild Heerbrugg Ltd.,Heerbrugg, Switzerland pp1- 110.

U.S. Dept. of Commerce (1935) “Manual of Second and Third Order Triangulation and

Traverse” Special Publication No.145, revised, pp1-35.

Vespe, F., G. Bianco, M. Fermi, C. Ferraro, A. Nardi, C. Sciarretta, "The Italian GPS Fiducial

Network: Services and Products", Journal of Geodynamics 30 (200) pp327-336.

Vose, M. D. (1999), The Simple Genetic Algorithm: Foundations and Theory, MIT Press,

Cambridge, MA pp1-220.

Wikipedia (2007), “Traveling Salesman Problem” Wikipedia, 2007, pp1-17.

Wikipedia (2010), “Global Navigation Satellite Systems (GNSS)” Wikipedia, 2010, pp1-50.

Walter, F. R. (1934) “Manual of Triangulation Computation and Adjustment”, Special

Publication Journal of Geodesy, No. 138, pp1-242.

http://en.wikipedia.org/wiki/Traveling_Salesman_Problem

121

Whitley, D. (1994). “A genetic algorithm tutorial”. Statistics and Computing 4, pp65–85.

Wolf, H., and Ghilani, C. D., (1997), Adjustment Computation: Statistics and Least Squares in

Surveying and GIS, John Wiley and Sons, Inc., ISBN 0-471-16833-5. Pp1- 564.

Wright, A.H.; et al. (2003). "Implicit Parallelism". Proceedings of the Genetic and Evolutionary

Computation Conference, pp1-15.

Xu, P. L. (2002), “A hybrid global optimization method: the one-dimensional case.” J Compute

Applied Math 147: pp301–314

Xu, P. L. and Grafarend, E. (1995), A multi-objective second order optimal design.”

physics Journal International 120: pp577–589.

THE METHOD OF SIMULATED ANNEALING FOR THE OPTIMAL ...

Documents

Transcript of THE METHOD OF SIMULATED ANNEALING FOR THE OPTIMAL ...