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University of Nottingham Department of Civil UNIVERSITY OF NOTTINGHAM DEPARTMENT OF CIVIL ENGINEERING DETERMINATION OF SATELLITE ORBITS AND THE GLOBAL POSITIONING SYSTEM by Loukis George Agrotis~ B.Sc.~ M.Inst. C.E.S. ; • i Thesis submitted to the University of Nottingham for the degree of Doctor of Philosophy October 1984 ii TABLE OF CONTENTS ABSTRACT ix ACKNOWLEDGEMENTS xi LIST OF FIGURES xiii CHAPTER 1 INTRODUCTION 1 CHAPTER 2 SATELLITE ORBIT DETERMINATION 2.1 Basic Concepts 7 2.2 COORDINATE REFERENCE FRAMES AND TIME SCALES 2.2.1 Reference Frames and Time 10 2.2.2 Precession and Nutation 13 2.2.3 Earth Rotation and Polar Motion 21 2.2.4 Coordinate Transformations 28 2.3 FORCE MODEL COMPONENTS 2.3. 1 Introduction 28 2.3.2 Earth Gravitational Attraction 29 ..... ,.. ," .. , ., . , Moon, Sun and Planetary Attractions, .::>. 32 2.3.3 t.. 2.3.4 Solid Earth Tides 35 2.3.5 Ocean Tides 43 2.3.6 Empirical Accelerations 45 , .. iii Page 2.3.7 Air Drag 46 2.3.8 Solar and Albedo Radiation 49 2.3.9 Satellite Thrust 55 2.4 NUMERICAL INTEGRATION OF THE EQUATIONS OF MJTION 2.4.1 Equations of Motion 56 2.4.2 Numerical Integration of Differential Equations 57 2.5 LEAST SQUARES ADJUSTMENT AND PARTIAL DERIVATIVES 2.5.1 The Satellite Observations 65 2.5.2 Least Squares Adjustment 67 2.5.3 The Orbit Determination Observation Equations 73 2.5.4 Orbit Determination Adjustment Requirements 82 CHAPTER 3 UNIVERSITY OF NOTTINGHAM ORBIT DETERMINATION SOFTWARE 3.1 Introduction 86 3.2 CHEBYSHEV POLYNOMIAL PROGRAM (CHEBPOL) 3.2. 1 Data Input 90 3.2.2 Program Description and Output 90 iv 3.3 THE SATELLITE ORBIT INTEGRATION PROGRAM (ORBIT) 3.3.1 Numerical Integration and Force Model Definition 94 3.3.2 Input Requirements 100 3.3.3 General OUtline 104 3.3.4 Program Output 111 3.3.5 Software Debugging 112 3.4 THE SATELLITE ORBIT ANALYSIS PROGRAM (SOAP) 3.4.1 Program Input and Output 114 3.4.2 General Description 118 CHAPTER 4 ANALYSIS OF SATELLITE TRACKING DATA AND APPLICATIONS OF ORBIT DETERMINATION 4.1 Introduction 122 4.2 RESULTS OF ANALYSIS 4.2.1 LAGEOS Tracking Data 123 4.2.2 Effect of Force Model Parameters and 127 Geopotential Models 4.2.3 Solutions for Tracking Station Coordinates 138 and Satellite Starting Elements 4.2.4 Polar Motion Solutions 143 v Page 4.2.5 Effect of Earth Rotation 149 4.3 Applications of Precise Orbit Determination 153 CHAPTER 5 THE GLOBAL POSITIONING SYSTEM 5. 1 Introduction 158 5.2 GENERAL SYSTEM DESCRIPTION 5.2.1 Space Segment 160 5.2.2 Control Segment 163 5.2.3 User Segment 166 5.3 GPS SATELLITE SIGNALS 5.3.1 Signal Structure and Receiver Measurement 167 Sequence 5.3.2 The CIA Code 172 5.3.3 The P Code 176 5.3.4 Satellite Data Message 177 5.4 MODES OF OBSERVATION AND ADJUSTMENT FOR PRECISE POSITIONING 5.4.1 Instantaneous Navigation Principles 180 5.4.2 Pseudo-Range Measurements 185 5.4.3 Phase (Doppler) Observations 189 5.4.4 Interferometric Techniques 196 vi Page 5.4.5 Reduction of Observations 204 5.4.5.1 Ionospheric Corrections 205 5.4.5.2 tropospheric Corrections 209 5.4.5.3 Relativity 212 5.5 GPS Geodetic User Equipment 212 CHAPTER 6 GPS SOFTWARE DEVELOPMENT AND ,ANALYSIS OF SAMPLE DATA 6.1 Introduction 217 6.2 THE GPS DATA 6.2.1 The NAVSTAR Geodetic Receiver System (NGRS) 218 6.2.2 Data Specification 219 6.3 NGRS DATA PRE-PROCESSING SOFTWARE 6.3.1 Aims of Pre-Processing 224 6.3.2 General Outline 226 6.3.2.1 Smoothed Pseudo-Range Mode 227 6.3.2.2 Biased Range (Phase) Mode 230 6.4 GPS LEAST SQUARES ADJUSTMENT PROGRAM (GPSPROG) 6.4.1 Introduction and Data Input 237 6.4.2 Program Description 239 vii Page 6.4.3 Future Development 245 6.5 ANALYSIS OF THE NGRS-2 DATA 6.5.1 Introduction 248 6.5.2 Biased Range Solutions 250 6.5.3 Smoothed Pseudo-Range Solutions 264 6.5.4 Single-Frequency Solutions 269 CHAPTER 7 CONCLUSIONS AND SUGGESTIONS FOR FURTHER WORK 7.1 CONCLUSIONS 7. 1.1 Conclusions on Orbit Determination 215 7.1.2 Conclusions on the Global Positioning System 216 7.2 SUGGESTIONS FOR FURTHER WORK 7.2.1 Suggestions for Further Work on Orbit 218 Determination 7.2.2 Suggestions for Further Work on GPS 219 APPENDICES A ROTATION AND REFLECTION MATRICES 281 B COORDINATE REPRESENTATIONS 283 c ORBIT DETERMINATION PARTIAL DERIVATIVES 291 viii Page D METHODS OF NUMERICAL INTEGRATION FOR ORBITS 301 E POLYNOMIAL REPRESENTATION AND INTERPOLATION 307 F WEIGHTED LEAST SQUARES 310 G COMPUTATION OF GPS TIME AND SATELLITE 317 COORDINATES REFERENCES AND BIBLIOGRAPHY 331 ix DETERMINATION OF SATELLITE ORBITS AND THE GLOBAL POSITIONING SYSTEM ABSTRACT An artificial satellite orbit determination (OD) computer program is the most essential tool in satellite geodesy. Such a program has been developed at Nottingham as part of this research and was tested with Satellite Laser Ranging (SLR) observations of the Laser Geodynamics Satellite (LAGEOS). This thesis describes the basic theory behind orbit determination and the software development at Nottingham. It includes details of the adopted force model, coordinate reference frames, and numerical integration and interpolation techniques. It is also explained how several geodetic parameters can be determined. The thesis discusses the results of two separate determinations of the LAGEOS orbit with an emphasis on the solutions for station coordinates and for earth rotation and polar motion. The NAVSTAR Global Positioning System (GPS) is on schedule to replace Transit as the most important satellite navigation system. When fully operational, in 1988, it will consist of 18 satellites which will provide continuous global coverage. This thesis describes the Global Positioning System and outlines the theory behind the most accurate techniques of adjustment of the CPS observables. It derives the equations for interferometric techniques and shows that, by differencing the observations, several undesirable unknowns can be eliminated. x GPS data from the NAVSTAR Geodetic Receiver System (NGRS) have been provided for Nottingham by the US Defence Mapping Agency (DMA). The thesis describes the software development to analyse these data and gives the results of several solution schemes to derive the absolute coordinates of the NGRS antenna. It is also shown how the software can be modified to incorporate interfero- metric techniques. Significant improvements over the NGRS solutions can be expected when GPS is fully operational, with refinements in both receiver hardware and software. xi ACKNOWLEDGEMENTS This thesis is the result of three years of fulfilling research in the Department of Civil Engineering at the University of Nottingham. The work was carried out with the support of the Heads of Department, Professor R. C. Coates and Professor P. S. Pell, and was funded by a University of Nottingham Postgraduate Studentship and by an Overseas Research Students Fees Support Scheme Award. The author was also sponsored by British Petroleum. The author is indebted to his supervisor, Professor V. Ashkenazi, for his guidance, support, and encourage- ment throughout this study. The data and other documentation vital for this research have been provided by the Royal Greenwich Observatory and by the US Defence Mapping Agency. Sincere thanks are extended to the staff of both these establishments and especially to Dr. A. T. Sinclair, Dr. G. A. Wilkins, Mr. G. Appleby, Mr. M. M. ~comber , Mr. K. I. Dougherty and Dr. M. Kumar. Other scientists who have given valuable advice include Dr. P. J. Hargrave, Dr. C. C. Goad, Dr. R. J. Anderle, Dr. B. R. Hermann, Dr. G. lachapelle, Dr. C. C. Counselman, Dr. P. F. ~cDoran and Mr. J. Dow. The author acknowledges the assistance received from the other members of the Nottingham Surveying Group, notably from Dr. R. Wood, who gave advice on software development., and.from Dr. A. H. Dodson, Mr. T. Moore, Mr. P. Howard and Dr. S. Grist. xii The author expresses his gratitude to his parents for their encouragement and support and to Libby, whose patience and help have been invaluable throughout the course of this work. Finally the author wishes to thank Mrs. Beryl Greaves for typing this thesis in her neat and efficient style. xiii LIST OF FIGURES Figure No Title Page 2.1 Orbit DeterminationBlock Diagram 8 2.11 The Celestial Sphere and Precession 15 2.111 The FundamentalArguments 18 2.IV The Nutation Angles 19 2.V Earth Rotation and Polar Motion 23 2.VI Coordinate Reference Frames and Time Scales 27 2.VII Third Body GravitationalAttraction 33 2.VIII Earth Tides 37 2.IX Satellite Velocity Relative to Atmosphere 47 2.X Umbra and Penumbra 52 2.XI Shadow Test for Satellite 53 3.1 Nottingham Orbit DeterminationSoftware 87 3.11 Flow Diagram for CHEBPOL 93 3.111 TYpical Magnitudes of LAGEOS Force Mbdel 98 Constituents 3.IV Data Input and Output for ORBIT 101 3.V ORBIT Flow Chart 108 3.VI SOAP Input and Output 115 4.1 SL5 Coordinatesof LAGEOS Tracking Stations 124 4.11 LAGEOS Tracking Sites 125 4.111 Distribution of Normal Points for the Two 126 Data Sets xiv Figure No Title Page 4.IV Range Residuals From Solution for Satellite 130 State Vector and Station Coordinates 4.V Range Residuals From Solution for Satellite 131 State Vector, Station Coordinates, and GM 4.VI Range Residuals From Solution for Satellite 1~ State Vector, Station Coordinates, GM, CR, ~~ 4.VII Range Residuals From Solution for Satellite 133 State Vector, Station Coordinates, GM, CR, ~~ 4.VIII Range Residuals From Solution for Satellite 134 State Vector, Station Coordinates, GM, CR, and Ca 4.IX Range Residuals From Solution for Satellite 135 State Vector, Station Coordinates, GM, CR, ~~ 4.X Range Residuals From Solution for Satellite 136 State Vector, Station Coordinates, GM, CR, and Ca 4.XI Force Model Parameter Solution 137 4.XII Solutions for Tracking Station Coordinates 140 4.XIII Comparison Between OS1, 032, and SL5 141 Station Coordinates 4.XIV Solutions for LAGEOS Starting Elements 142 4.xv Polar Motion Solutions 144 xv Figure No Title Page 4.XVI ComparisonBetween OS1, DS2, and SL5 Station 146 CoordinatesAfter Polar Motion Solution 4.XV!I Daily Earth Rotation Solutions 151 4.XVIII Comparisonof Daily Changes in l.o.d.
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