70-19 , 34-8 OBENSON, G abriel Francis Tambe, 1934— DIRECT EVALUATION OF THE EARTH'S GRAVITY ANOMALY FIELD FROM ORBITAL ANALYSIS OF ARTIFICIAL EARTH SATELLITES. The Ohio S ta te U n iv e rsity , Ph.D. , 1970 Geophysics University Microfilms, A XEROX Company, Ann Arbor, Michigan i THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED DIRECT EVALUATION OF THE EARTH'S GRAVITY ANOMALY FIELD FROM ORBITAL ANALYSIS OF ARTIFICIAL EARTH SATELLITES DISSERTATION Presented in Partial Fulfillment of the Requirements, for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Gabriel Francis Tambe' Obenson, B.Sc., M. Sc., A.R.I.C.S. The Ohio State University 1970 A pproved b Department of Geodetic Science Obenenkongho, Obenafa, Nzealu, Orang, Obenafa & Besem. AC KNOW LEDGEM EN TS I wish to thank Dr. R. H. Rapp for providing valuable guidance during the preparation of this dissertation and Dr. U. A. Uotila for the many pertinent discussions 1 had with him. My thanks also go to the Instruction and Research Computer Center for providing free computer time for the extensive program­ ming that was involved in this study. This work was supported, in part, through an Ohio State Uni­ versity Research Foundation project sponsored by the Air Force Cambridge Research Laboratories, Bedford, Massachusetts. VITA October 28, 1934 ......... Born —Mamfe', Cameroon, West Africa 1 9 6 1 .................................... B.Sc., Photogrammetric Engineering, ITC Delft, Holland 1 9 6 3 .................................... Diploma, Comprehensive Planning, Institute of Social Studies, The Hague, Holland 196 5 .................................... A. R. I.C. S., Professional Associate of the Royal Institution of Chartered Surveyors, B ritain 1968 .................................... ... M. Sc., Geodetic Science, The Ohio State University, Columbus, Ohio 1961-1965 ......................... Senior Surveyor, Survey Department, Kaduna — Nigeria, West Africa 196 6 .................................... Senior Surveyor, Lands & Surveys Depart­ ment, Buea, West Cameroon, West Africa 196 8 .................................... Teaching and Research Associate, The Ohio State University, Columbus, Ohio 196 9 .................................... Research Associate, The Ohio State Uni­ versity, Columbus, Ohio PUBLICATIONS "Production of 1/50,000 Scale Maps of Northern Nigeria. 11 C hartered Surveyor (A. R. I.C. S.) Thesis, 1965. "Photogrammetric Mapping — Practical Hints. " Northern Nigeria Survey, 1965. "Prediction of Mean Gravity Anomalies of Large Blocks from Sub-Bloc Means." M.Sc. Thesis, 1968. "Prediction Accuracies of 5°x5° Mean Anomalies from 1° xl° Means at Different Latitudes. " Reports of the Department of Geodetic Science, No. 117, The Ohio State University, Nov., 1968. FIELDS OF STUDY Major Field: Geodesy Geodetic Astronomy Dr. I. I. M ueller Physical Geodesy Dr. I. I. M ueller Satellite Geodesy Dr. I. I. M ueller Geometric Geodesy Dr. R. H. Rapp Mathematical Projections in Geodesy Dr. R. H. Rapp Adjustment Computations Dr. U. A . Uotila Minor Field: Photogrammetry Dr. S. K. Ghosh Mathematical Statistics Dr. T. A. Willke TABLE OF CONTENTS P age ACKNOWLEDGEMENTS....................................... m VITA ............................................................................................................................. iv LIST OF TABLES ................................................................................................... v iii LIST OF FIG URES ............................................................................................ x INTRODUCTION ................................................................................................... 1 Chapter 1. REPRESENTATION OF THE EARTH'S GRAVITATIONAL POTENTIAL F IE L D ........................................................................ 6 1. 1 General 1. 2 P o ten tia l of a L ev e l E llip so id 1. 3 Total Disturbing Potential from Gravity Anomalies 1.4 Total Disturbing Potential from Potential Coefficients 1. 5 Numerical Check 1. 6 Su m m ary 2. GENERATION OF SATELLITE ORBITS ................................. 13 2. 1 General 2. 2 Cowell's Method of Orbit Generation 2. 3 Method of Variation of Parameters for Orbit G en eration 2.31 Evaluation of S, T, W 2. 4 Components of the Disturbing Force from Potential Coefficients 2. 5 Check Computations v i TABLE OF CONTENTS (Continued) P age 3. PARTIAL DIFFERENTIAL EQUATIONS AND ADJUSTMENT M O DELS................................................................... 37 3. 1 General 3. 2 Evaluation of the Partial Derivatives 3. 2. 1 Numerical or Variant Method 3.2.2 Variational Partial Derivatives 3. 3 Adjustment Models 3.3.1 General 3.3.2 Minimization of Quadratic Sums 3.3.3 Variance-Covariance Matrix of the Parameters and Observables 3.3.4 Special Cases 3. 4 Relevant Matrices for Satellite Analysis 3. 5 Some Additional Notes 3. 6 Su m m ary 4. NUMERICAL RESULTS AND ANALYSIS ......................................... 81 4. 1 General 4. 1. 1 Definition of Parameters and Weighting 4. 2 Wrong Orbital Elements and Station Coordinates 4. 3 W rong A n om alies 4.4 Wrong Orbital Elements, Gravity Anomalies and Station Coordinates 4. 5 Wrong Orbital Elements and Gravity Anomalies 4. 6 Analysis of the Results 5. CONCLUSION ................................................................................................... 116 APPENDIX A. Potential C oefficients .................................................................. < . 118 B. 15°xl5° Equal Area Mean Gravity Anomalies ........................... 121 C. 15°xl5° Equal Area Terrestrial Mean Gravity Anomalies . 124 D. Set Up for Integration of Cowell's Equations of Motion. 129 REFERENCES 131 LIST OF TABLES Table Page 1 Comparison of Satellite Positions from Potential Coefficients and Gravity Anomalies ............................................... 32 2 Comparison of Disturbing Forces, S, T, W ............................ 33 3 Comparison of the Generalized Stokes' Function and Its Derivatives for Various Evaluation Points .... 35 4 Satellites First U se d ......................................................................... 82 5 Comparison of True and Predicted Differences in Right Ascension, a*, Declination, 6 *, and Range, r*, for Wrong Orbital Elements and One Station ........................... 8 6 6 Recovered Errors of Orbital Elements and Station Coordinates from One Revolution of One Satellite . 90 7 Check of Variational Equations for Wrong Gravity Anomalies Alone for One Revolution of One Satellite . 91 8 Comparison of Predicted and True Differences in Right Ascension, a*, Declination, 6 *, and Range, r*, Using Wrong Orbital Elements, Station Coordinates, and Gravity Anomalies of Three S a tellites ........................... 94 9 Recovered Errors in Orbital Elements from One Revolution Each of Three Satellites ............................................... 95 10 Recovered Errors in Station Coordinates from One Revolution Each of Three Satellites ............................................... 95 11 Recovered 15°xl5° Equal Area Mean Gravity Anomaly Errors from One Revolution Each of Three S a t e l l i t e s .................................................................................................... 96 v iii LIST OF TABLES (Continued) Table - Page 12 Second Set of Satellites Used . 101 13 Comparison of True and Predicted Differences for Right Ascension, av, and Declination, 6 V, of Vanguard 2 ............................................................................................. 103' 14 Recovered Input Errors of Orbital Elements from About 15 Revolutions Each of Five Satellites .... 104 15 Recovered Error 3 in Mean Gravity Anomalies from About 15 Revolutions Each of Five Satellites .... 105 16 Conversion Terms for Anomalies from International Gravity Formula System to GRS 1967 ........................................ 125 17 15°xl5° Equal Area Terrestrial Mean Anomalies . 127 i * LIST OF FIGURES T able Page 1 Components of the Disturbing Force at a Satellite Position, P ................................................................................................... 14 4 2 Differences Between True and Predicted Differences in Right Ascension, a , Declination, 6 , and R ange, r * ................................................................................................... 88 3 Schematic Form of the Observation Equations and Normal E quations ................................................................................ 93 4 Recovered Mean Gravity Anomaly Errors in Relation to Satellite O rb its ..............................................................................................100 5 Differences of (Predicted-True) Differences in Right & a , Ascention, a , and Declination, o , for Vanguard 2 . 104 6 Recovered Mean Gravity Anomaly Errors from Five Satellites .................................................................................................... 110 x INTRODUCTION Gravity anomalies have significant uses in geodesy, for investi­ gating the earth's gravitational potential field and for absolute position determination of points on the earth's surface. They also have signifi­ cant uses in non-geodetic fields such as geology and space mechanics. However, accurately observed gravity anomalies are not currently available on a world-wide basis because of observational and other physical problems. To obtain anomalies in those areas where there are no observed values, various techniques have been used and are