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A Bell & Howell Information Company 300 North Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313-'761-4700 800/521-0600 ANALYSIS OF GPS DATA PROCESSING TECHNIQUES: IN SEARCH OF OPTIMIZED STRATEGY OF ORBIT AND EARTH ROTATION PARAMETER RECOVERY
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Graduate School of The Ohio State University
By
Dorota A. Grejner-Brzezinska, M.S., M.S.
*****
The Ohio State University 1995
Dissertation Committee: Approved by
John D. Bossier Clyde C. Goad Ivan L Mueller Adviser Department of Geodetic Science and Surveying UMI N um ber: 9612187
OMI Microform 9612187 Copyright 1996, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 To my daughter Bogna ...who always keeps my spirit up... ACKNOWLEDGMENTS
First and foremost, I wish to express my deepest appreciation and thanks to my husband Robert and my daughter Bogna for their patience, constant support and unshakable faith in me during the difficult and long but certainly fruitful years of my graduate studies. To my daughter, Bogna, I thank you for understanding my frequent absences. Very special thanks go to my mother, my sister and my mother-in-law in Poland, for their encouragement, support and belief in me. I wish to express my deep gratitude to my adviser, Dr. Clyde C. Goad, for his professional guidance and assistance, and the valuable help throughout my graduate studies at the Department of Geodetic Science and Surveying, and during the course of my doctoral research. His support, understanding and encouragement are greatly appreciated. I am also very thankful and truly indebted to Dr. Ivan I. Mueller, for his valuable guidance, expert advice and constant encouragement. I want to thank him for his support and understanding, for many stimulating discussions, and also for permission to use his invaluable and unique personal library. My gratitude and appreciation go to Dr. John D. Bossier, for the stimulating and constructive discussions and valuable suggestions which greatly enhanced the quality of this dissertation. In addition, I wish to acknowledge the invaluable contribution of all the faculty of the Department of Geodetic Science and Surveying to my education. Special thanks are also due to my fellow students, past and present: Dr. David Chadwell (now at Scripps Institution of Oceanography, La Jolla), Dr. Ming Yang (now at National Cheng Kung University in Tainan, Taiwan) and Jarir Saleh (now at the University of the West Indies, Trinidad), for their help, valuable comments and stimulating discussions during all the years we spent together in Cockins Hall. My deep appreciation goes to Mrs. Irene Tesfai, “expert grammarian and dialectician,” for her help and advice during the typing of this manuscript. VITA
March 10,1963 ...... Bom, Poland 1986 ...... M.S., Agricultural and Technical University of Olsztyn, Poland 1986-1990 ...... Agricultural and Technical University of Olsztyn, Poland 1995 ...... M.S., The Ohio State University, Columbus, Ohio 1990-present ...... Graduate Teaching Associate & Graduate Research Associate, The Ohio State University, Columbus, Ohio
PUBLICATIONS
Goad, C. C., Grejner-Brzezinska, D. A , Yang, M., Determination of High-Precision GPS Orbits Using Triple Differencing Technique, submitted to Bulletin Geodesique/manuscripta geodaetica/Journal o f Geodesy, June 1995.
Grejner, G., Grejner-Brzezinska, D. A , New Methods in Land Information Data Gathering, Proceedings of International Conference of Young Scholars, Olsztyn, Poland, September 1990.
Grejner, D. A , Successors of the Transit System. GPS - Future of the Space Methods for Positioning, Przeglad Geodezyjny 61(1), pp. 3-9, 1989.
Grejner, D. A, Precise Relative Positioning with GPS Phases - Accuracy Analysis, Acta Academiae AC Technicae Olstenesis 19, pp. 19-29,1989. FIELDS OF STUDY
Major Field: Geodetic Science and Surveying
• Studies in Geodetic Astronomy, Satellite Geodesy, Geodynamics: Prof. Ivan I. Mueller • Studies in Physical Geodesy, Geometric Geodesy: Prof. Richard H. Rapp • Studies in Global Positioning System, Advanced Satellite Geodesy, Geometric Geodesy: Prof. Clyde C. Goad • Studies in Adjustment Computations: Prof Burkhard Schaflrin TABLE OF CONTENTS
DEDICATION...... ii
ACKNOWLEDGMENTS...... iii
VITA...... v
LIST OF FIGURES...... xi
LIST OF TABLES...... xv
CHAPTER PAGE
I. INTRODUCTION...... 1 n . GLOBAL POSITIONING SYSTEM. GENERAL OVERVIEW AND DATA MODELING...... 8 2.1. GPS System and Signal Description ...... 8 2.2. GPS Data Modeling ...... 15 2.2.1. Types of Observables: Pseudoranges, Carrier Phases and Their Combinations ...... 15 2.2.2. Correlation Between Differenced Observations ...... 21 2.3. GPS Observation Error Sources and Corrections ...... 23 2.3.1. Ionospheric Refraction ...... 28 2.3.2. Tropospheric Refraction ...... 32 m . REFERENCE FRAMES AND TIME SYSTEMS...... 39
3.1. Conventional Celestial Reference System ...... 40 3.2. Conventional Terrestrial Reference System ...... 41 3.3. Transformation Between Terrestrial and Celestial Reference Systems.. 44 3.4. Relativistic Effects on Reference Frames and Time ...... 47 3.5. Time Systems...... 50 3.5.1. Sidereal and Solar Time Systems ...... 51 3.5.2. Atomic Time Scale...... 53 3.5.3. Dynamical Time Scale ...... 53 vii IV. MODELING OF THE MOTION OF THE EARTH ORBITING SPACECRAFT; TERRESTRIAL SITE DYNAMICS...... 56
4.1. Equations of Motion and Variational Equations of the Earth Orbiting Satellite ...... 56 4.2. Numerical Integration of Equations of Motion and Variational Equations ...... 59 4.2.1. Adams Type Predictor/Corrector Formulae ...... 60 4.3. Total Perturbation Model ...... 61 4.3.1. Noncentral Gravitational Potential of The Earth ...... 62 4.3.2. Third-Body Attraction and the Indirect Oblation Effect 66 4.3.3. Direct Solar Radiation Pressure ...... 67 4.3.3.1. Solar Radiation Pressure Model for GPS Satellites: ROCK4 and ROCK42 ...... 68 4.3.3.2. Acceleration due to the 7-bias ...... 70 4.3.3.3. Effect of the Eclipse on the Modeling of the Motion of GPS satellites ...... 71 4.3.3.4. Spacecraft Attitude Model ...... 77 4.3.4. Earth Radiation (Albedo) ...... 79 4.3.5. Solid Earth Tides ...... 79 4.3.6. Ocean and Atmospheric Tides ...... 81 4.3.7. Relativistic Acceleration ...... 83 4.3.8. Resonant Acceleration...... 83 4.3.9. Orbital Maneuvers ...... 85 4.4. Terrestrial Site Dynamics...... 86 4.4.1. Station Displacement Due to the Solid Earth Tides ...... 86 4.4.2. Station Displacement Due to the Pole Tide ...... 88 4.4.3. Station Displacement Due to the Ocean and Atmospheric Tides...... 89
V. ORBIT AND RELATED PARAMETER ESTIMATION USING THE BATCH LEAST-SQUARES TECHNIQUE WITH TRIPLE-DIFFERENCED PHASES...... 90
5.1. Introduction ...... 90 5.2. The Application of the Triple-Difference to GPS Orbit Determination ...... 91 5.3. Cholesky Decomposition of the Covariance Matrix ...... 96 5.4. Recursive Decorrelation Scheme and the Final Solution ...... 98 5.5. Estimation of the GPS Trajectory and Other Geodetic Parameters with Triple Difference Phases ...... 100 5.5.1. Linearization of the Triple-Difference Observation Equation 100 5.5.2. Estimation of the Satellite State Vector ...... 108 viii 5.5.3. Estimation of the Coordinates of the Terrestrial Station Ill 5.5.4. Estimation of the Tropospheric Refraction Scaling Factor... 114 5.5.5. Estimation ofthe Earth Rotation Parameters ...... 115 5.6. The Current Status of IGS Activity in GPS Orbit and Related Parameter Estimation ...... 121
VI. VARIABLE ROTATION OF THE EARTH...... 125
6.1. Variability of the Earth’s orientation ...... 125 6.1.1. Secular Variations in the Earth Rotation Parameters: An Overview ...... 129 6.1.2. Long-Periodic Variations in the Earth Rotation Parameters: An Overview ...... 130 6.1.3. Annual, Seasonal and Short-Periodic Variations in the Earth Rotation Parameters: An Overview ...... 131 6.2. Variations in the Rotational Speed ...... 132 6.2.1. Lunisolar Effects on the Rotational Velocity ...... 134 6.2.2. Effects of Ocean Tides on the Rotational Velocity ...... 135 6.2.3. Atmospheric Effects on the Rotational Velocity ...... 135 6.3. Variations in the Location of the Pole ...... 136 6.3.1. Effects of the Solid Earth Tides on Polar Motion 137 6.3.2. Effects of the Ocean Tides on Polar Motion ...... 137 6.3.3. Atmospheric Effects on Polar Motion ...... 138 6.4. Estimability of the ERPs from Satellite Techniques ...... 138 6.5. Estimation of ERPs by Means of GPS Observations ...... 142
VII. GPS TRAJECTORY AND RELATED PARAMETER ESTIMATION USING A BATCH LEAST-SQUARES ESTIMATOR: EXPERIMENTS DESCRIPTION AND ANALYSIS OF THE RESULTS...... 149
7.1. OSU Standard Solution: Description ...... 149 7.2. Experiment 1: Standard OSU Orbit Determination ...... 151 7.2.1. OSU Solution Versus IGS Solution ...... 151 7.2.2. Baseline Repeatability ...... 155 7.3. Experiment 2: Influence of the Nutation Model on the Orbits and ERP Estimates ...... 157 7.4. Experiment 3: Influence ofthe Adopted Troposphere Model on the Final Estimates of the Trajectories and ERPs ...... 159 7.5. Experiment 4: Influence of the ^4 priori ERP Update From the Absolute Technique on the Final Estimates of the Trajectories and ERPs...... 165 7.6. Experiment 5: Short-Periodic UT1 Determination with Triple Differences ...... 173 ix 7.6.1. Triple Difference Solution Versus Herring Daily and Subdaily UT1 Tidal Model...... 175 7.6.2. Application of AAM Functions and the Tidal Model in the Recovery of Diurnal and Semidiurnal UT1 Signature from the Total GPS Signal ...... 179 7.6.3. Application of Vondrak Smoother to the Satellite-derived Short-Periodic UT1 ...... 184
VIE. CONCLUSIONS AND RECOMMENDATIONS...... 190
REFERENCES...... 194 LIST OF FIGURES
FIGURE PAGES
1. Arrangement of GPS satellites in the full constellation ...... 9
2. Schematic view of a Block II GPS satellite ...... 10
3. Epoch-by-epoch widelane ambiguity N 1-N2 combination under A S 13
4. Epoch-by-epoch AS-free widelane ambiguity N1-N2 combination ...... 14
5. Estimates of the xp as a function of the number of tropospheric scaling factors estimated per day ...... 36
6. The average a posteriori standard deviation in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Matera for GPS week 784 ...... 37
7. Global distribution of GPS tracking stations ...... 43
8. Polar motion ...... 46
9. Satellite body-fixed reference system ...... 69
10. 3DRMS for 6-hour orbit overlap for PRN 1 and 28 ...... 72
11. 3DRMS for 6-hour orbit overlap for PRN 16 and 22 ...... 73
12. Cylindrical model for the Earth’s shadow ...... 74
13. Conical model for the Earth’s shadow ...... 75
14. The recursive Cholesky decomposition and decorrelation scheme operating on the lower triangular part ofthe covariance matrix and the vectorized A matrix augmented by the column Y ...... 99 15. Mean RMS values for the OSU orbit comparison to the IGS orbit for different measurement generation procedure ...... 108
16. Schematic illustration of the forces that perturb the Earth’s rotation (beetles represent the continental drift ...... 127
17. The difference between the xp derived by CODE and final Bulletin B estimation between October 1992 and December 1994 ...... 143
18. The difference between th e ^ derived by CODE and final Bulletin B estimation between October 1992 and December 1994 ...... 144
19. (UT1-UTC) derived by CODE and final Bulletin B estimates between October 1992 and December 1994 ...... 145
20. The difference between (UT1-UTC) derived by CODE and final Bulletin B estimation between October 1992 and December 1994 ...... 146
21. The difference between the xp derived by CODE and final Bulletin B estimates between January 1 and June 1, 1995 ...... 147
22. The difference between the yp derived by CODE and final Bulletin B estimates between January 1 and June 1, 1995 ...... 148
23. Baseline repeatability RMS (east component) versus baseline length for GPS weeks 784-787. All stations are used except for Pamatai and Herstmonceux due to possible receiver problems ...... 155
24. Baseline repeatability RMS (north component) versus baseline length for GPS weeks 784-787. All stations are used except for Pamatai and Herstmonceux due to possible receiver problems ...... 156
25. Baseline repeatability RMS (height component) versus baseline length for GPS weeks 784-787. All stations are used except for Pamatai and Herstmonceux due to possible receiver problems ...... 156
26. Estimates of the xp as a function of the number of tropospheric scaling factors estimated per day ...... 160
27. Estimates of the yp as a function of the number of tropospheric scaling factors estimated per day ...... 160 28. Estimates of the rate of (UT1-TAI) as a function of the number of tropospheric scaling factors estimated per day ...... 161 xii 29. The average a posteriori standard deviations in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Matera for GPS week 784 .... 164
30. The average a posteriori standard deviations in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Hobart for GPS week 784 ...... 164
31. xp from triple difference solution and the IERS solution for GPS week 784-787 ...... 165
32. yp from triple difference solution and the IERS solution for GPS week 784-787 ...... 166
33. d(UTl-TAI) from triple-difference solution and the IERS solution for GPS weeks 784-787 ...... 168
34. The differences between the standard xp solution and the final IERS estimate (routine solution), and between the xp test solution and the final IERS estimate (test solution) for GPS weeks 784-787 ...... 170
35. The differences between the standard^ solution and the final IERS estimate (routine solution), and between the test solution and the final IERS estimate (test solution) for GPS weeks 784-787 ...... 171
36. The differences between the standard d(U Tl) solution and the final IERS estimate (routine solution), and between the d(U Tl) test solution and the final IERS estimate (test solution) for GPS weeks 784-787 ...... 172
37. Geodetically-derived corrections to the Bulletin B nominal series of UT1 with 30-minute resolution for GPS week 784 ...... 176
3 8. Geodetically-derived corrections to the Bulletin B nominal series of UT1 with 30-minute resolution for GPS week 785 ...... 176
39. Geodetically-derived diurnal and semidiurnal UT1 and Herring’s model with 30-minute resolution for GPS week 784 ...... 178
40. Geodetically-derived diurnal and semidiurnal UT1 and Herring’s model with 30-minute resolution for GPS week 785 ...... 178
41. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution and Herring’s model for GPS week 785 (best fitting linear function to the GPS UT1 corrections was removed from the signal presented in Figure 38) ...... 179
42. Geodetically-derived UT1 with tides up to 35 days and the atmospheric effect removed; 30-minute resolution, GPS weeks 784 and 785 ...... 182
43. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution and Herring’s model for GPS week 784 ...... 183
44. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution and Herring’s model for GPS week 785 ...... 183
45. The difference between Herring model and GPS-derived UT1 for GPS week 784 ...... 184
46. The difference between Herring model and GPS-derived UT1 for GPS week 785 ...... 184
47. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution smoothed with Vondrak smoother (equal weights) and Herring’s model for GPS week 784 ...... 187
48. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution smoothed with Vondrak smoother (equal weights) and Herring’s model for GPS week 785 ...... 187
49. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution smoothed with Vondrak smoother (unequal weights) and Herring’s model for GPS week 784 ...... 188
50. The difference between geodetically-derived diurnal and semidiurnal UT1 series smoothed with Vondrak smoother (equal weights) and the original signal for GPS week 784 ...... 189
51. The difference between geodetically-derived diurnal and semidiurnal UT1 series smoothed with Vondrak smoother (equal weights) and the original signal for GPS week 785 ...... 189
xiv LIST OF TABLES
TABLE PAGE
1. Main sources of the errors and their contribution to the single range observation equation ...... 24
2. Relativistic effects on a clock ...... 27
3. Code measurement characteristics in a high multipath environment 28
4. Maximum vertical ionospheric range error [m] ...... 29
5. Estimated ionospheric group delay for GPS signal ...... 30
6. Sources, magnitudes and uncertainties of propagation delay ...... 35
7. ITRF93 site’s positions adopted by IGS [m] ...... 42
8. ITRF 93 site’s velocities adopted by IGS [m/y] ...... 44
9. Effect of the perturbing forces on GPS satellites ...... 62
10. Comparison of the solution characteristics between solutions with and without C„ and £* for January 15, 1995 ...... 65
11. Selected characteristics of the solutions with and without the tidal correction to the terrestrial sites’ coordinates for January 15, 1995 ...... 88
12. Double-differences and the respective set of equivalent observations for the integer ambiguity solution ...... 91
13. Double-differences and the respective set of equivalent triple-differences for the ambiguity float solution ...... 92
14. Measurement model ...... 101
XV 15. Solution parameters ...... 104
16. Selected characteristics of the solution with a full dynamic model and the solution with a fixed solar radiation pressure scaling factor in the Z-body-fixed direction, for January 15, 1995 ...... I l l
17. Selected characteristics of the solution with a full dynamic model versus the solution with a velocity field neglected, for July 3, 1995 ...... 114
18. Selected characteristics of the processing strategies of individual analysis centers including OSU (as o f December 1994) ...... 122
19. The influence of the oceans and the atmosphere on Universal Time: amplitudes of periodical changes...... 133
20. Tidal variations in rotation rate induced by the zonal tidal potential; columns 2 and 3 represent amplitudes ...... 134
21. Summary report for orbit comparison for January 15, 1995 ...... 152
22. The daily RMS of fit of the Analysis Center orbits (including OSU) to the IGS solution for GPS weeks 784-787 [m ] ...... 153
23. The Mean RMS of fit, after transformation, of orbit comparison among the IGS centers including OSU for GPS weeks 784-787 [cm] 154
24. The a posteriori standard deviations of a unit weight for the orbit estimation with 1980 nutation and Herring’s nutation model for GPS week 784 ...... 158
25. ERP estimates for GPS week 784 given at 12:00 TAI from the 1980 and Herring’s nutation models, based on the routine data processing 158
26. The a posteriori standard deviations of a unit weight for the test solutions with different tropospheric model ...... 162
27. The a posteriori coordinate standard deviations for PRN 2, 5 and 20 as a function of different number of tropospheric estimates per day [cm].. 163
28. The a posteriori standard deviations of a unit weight for the solution with Bulletin B applied only once and for the standard solution ...... 169
xvi Number ofthe eclipsing satellites per day during the GPS weeks 784-787 ...... CHAPTER I INTRODUCTION
Satellite technology, without question, has transformed the world. While its applications like the communication and information technologies, revolutionary developments by themselves, are now a part of everyday living, only a few realize the myriad of complexities involved in their proper functioning. To make certain that all the applications of satellite techniques that require a high degree of accuracy endure with an expected reliability is the purpose of research of many scientists worldwide; it forms a significant part of the overall exciting puzzle about our physical Earth and its interactions. This dissertation is a dedicated part of the overall scope of this complex issue. The focus of the research presented here is on the application of the triple- difference technique in the recovery of the NAVSTAR Global Positioning System (GPS) trajectories together with the Earth Rotation Parameters (ERPs), using the Batch Least-Squares Adjustment. In addition, a brief analysis of the state-of-the-art methods used in GPS orbit and ERP estimation is presented. The utilization of triple differences in the orbit determination scenario, as presented here, is a new application of a well-known idea. To date, all GPS orbital software packages available worldwide use undifferenced, single- or double-differenced GPS phases. The implementation of triple differences into the orbit and related parameter estimation requires special consideration of the correlation that is introduced into the data set via subsequent differencing. On one hand, triple differencing reduces significantly the number of parameters that need to be estimated, removing all the nuisance parameters such as
1 2 ambiguities and clock errors. On the other hand, the size of the observation covariance matrix is very large and finding its inverse could be very troublesome or even impossible. Thus, a convenient, fast and flexible -whitening or decorrelation scheme was developed for the purpose of decorrelating the data in order to avoid the computation of the inverse. It is an integral part of the software package that has been developed as a basis for my research. The GPS Orbit Determination Including Various Adjustments - GODIVA, is designed in a module-like form where every constituent can be easily replaced by an updated component. GODIVA, created by my colleague Ming Yang and myself consists of three major parts: Data Base created by Ming and Predictor/Corrector and Estimator developed mainly by myself. Both Ming Yang and I extensively tested the program (see Goad et al., 1995). Dr. Chris Rizos from the University of New South Wales provided us with some subprograms that were modified and included in Predictor/Corrector. The integrator used in GODIVA is DVDQ coded by Dr. Fred Krogh from the Jet Propulsion Laboratory (1969) (see section 4.2.1). Ming Yang used GODIVA to investigate the measurement modeling techniques (Yang, 1995); especially, to explore the optimal approach to generating the independent set of triple-differenced measurements. Chapter n, following this section, presents the general overview of the GPS system and data modeling. In Chapter m the reference frames and time systems applied in the orbit determination process are described. Chapter IV explains the problem of the modeling of the motion of the Earth’s orbiting spacecraft and the terrestrial site dynamics. In Chapter V the Batch Least-Squares Adjustment and the parametrization of the triple difference observation equation are presented. In Chapter VI the variability of the Earth’s rotation is discussed together with the major problems related to the ERP determination with the satellite technique. Since one of the main goals of this research was to elucidate the GPS-based determination of time and pole coordinates under different conditions such as different lengths of the arc processed, influence of the type of tropospheric refraction model used to correct the collected 3 tracking data, impact of the ERP model itself, and the influence of the dynamic model and parametrization of the satellite orbit, Chapter VU presents the number of experiments that were run for that purpose. In particular, determination of diurnal and semidiurnal Universal Time with a satellite technique is discussed in Chapter VU. The first satellite of the NAVSTAR GPS System was launched in 1978 (Rothacher, 1992, p. 1). It opened a new era in satellite positioning, specially for civilian applications. Space Geodesy has become essential for the study of Earth’s dynamics such as global/regional crustal motion, high resolution monitoring of the motion of terrestrial sites after major earthquakes or mapping of postglacial rebound of the Earth’s crust. Accurate GPS baselines with a precision of one to a few parts in 109 are essential to the study of geodynamics. Such an accuracy can be achieved only with precise satellite trajectories. Thus, high-precision orbits of the GPS satellites are the fundamental constituents of GPS-based Space Geodesy. As a byproduct of the orbit recovery from GPS observables, collected by the globally distributed network of tracking stations, high resolution ERPs can also be estimated. A dynamic response of the solid Earth to the variety of forces acting upon it is the polar motion, which is defined as a displacement of the Earth’s rotation axis with respect to a conveniently defined reference point. Also, the rotation of the Earth is not uniform. The precision of the prediction of both phenomena is limited and decreases with the period of extrapolation. For example, the accuracy of the daily ERP evaluation from the International Earth Rotation Service (IERS) Bulletin A, is SxlO*4 arcsec for the pole coordinates, and 5xl0‘s s for UT1 (UT1 is the Universal Time that is directly related to the orientation of the rotating Earth); whereas, the 10-day prediction shows an accuracy of only 3.9xl0‘3 arcsec and 16X10"4 s, respectively. The 40-day prediction is even weaker: llxlO '3 arcsec for the pole and 68x1 O'4 s for the Earth rotational time (IERS Explanatory Supplement to Bulletins A and B). In every case, the prediction of UT1 degrades faster than the extrapolation of the pole coordinates. Thus, both polar motion and rotational speed must be monitored continuously. ‘The scientific need for 4 monitoring changes in earth rotation relates to understanding a number of physical processes involving the planet’s interior structure, as well as its enveloping oceans and atmosphere. Precise knowledge ofthe earth’s orientation is also important for purposes of navigation, especially in the tracking of interplanetary spacecraft” (Salstein et al., 1993). As mentioned above, the analysis of satellite tracking data can provide precise knowledge of the Earth’s position and orientation in space, both of which are very important for diverse purposes in several branches of science and engineering. In recent years, many space-based methods have been developed at the highest level of accuracy for determining the Earth Rotation Parameters. These include Very Long Baseline Interferometry (VLBI), Lunar Laser Ranging (LLR), and Satellite Laser Ranging (SLR). These methods are now considered classical techniques for measuring ERPs, i.e., drift of the Earth’s pole (polar motion) and/or time, referred to as UT1 minus UTC (LJTC is the Coordinated Universal Time measured with atomic clocks, thus ideally having a constant rate and being a basis for most radio time signals and civil time systems). Since both time systems must naturally be connected, UT1-UTC is treated as a parameter that is estimated by space methods. The most recently developed technique applied to the determination of ERPs is the analysis of tracking data from GPS satellites. GPS currently employs a full constellation of 25 satellites and numerous ground receivers. With its geometrical strength and rigidity resulting from the capacity of providing simultaneous observations, it has already demonstrated tremendous potential in precise point and relative positioning. This system also allows obtaining accuracy similar to the classical methods of ERP determination, in some aspects even better. GPS manifests a potential for higher resolution and faster availability of the final product. Li comparison with other techniques, GPS-based applications are much more economical. It is an important factor in today’s price competitive environment. In the standard processing of GPS data observed by terrestrial stations, the possibility exists for simultaneous solution of the station coordinates defined in the 5 network rotating with the Earth along with the satellite orbits that furnish the celestial frame. Since ERPs provide the link between celestial (inertial) and terrestrial (attached to the rotating Earth) frames, polar motion and UT1-UTC can ultimately be recovered. But GPS, similar to other nonabsolute methods of ERP retrieval, suffers from an inherent defect in UT1 determination. This results from the almost perfect collinearity between the variations in the right ascension of a satellite’s orbital node and UT1 that limits the capacity of GPS to resolve UT1 using satellite observations alone. However, the rate of UT1-UTC can be recovered by the satellite technique due to the fact that the partial derivatives of the GPS observable with respect to the change in UT1 and with respect to the right ascension of the node are not linearly dependent. Theoretically, the rate of the node and the rate of UT1 are linearly dependent, but “...because we assume that the force model is (more or less) known, there is no necessity to solve for a first derivative of the node” (Beutler et al., 199S). Having the variations in UT1 resolved, one needs to connect it to the starting value of UT1-UTC based on an absolute, independent and stable reference which must be obtained by some other mechanism. For example, VLBI provides direct connection to some suitable quasars (extremely distant celestial objects emitting radio waves, offering the absolute reference orientation in space). But VLBI is an expensive technique that typically provides the solution at five-day intervals with a delay of about a month (frequency of Bulletin B). GPS, on the other hand, is much cheaper and gives an access to ERPs in almost real time. Thus, the combination of both absolute VLBI and relative GPS signals should provide an optimal solution. Since one of the main problems presented in this dissertation is GPS orbit determination, the short overview of the trajectory estimation deserves some special attention in this introductory section. In the early 1980’s, when GPS first started finding its place in geodesy and surveying, the accuracy of the point positioning as well as the baseline determination was limited to some extent by the relatively poor quality of GPS orbits (more than 10 m uncertainty in a single coordinate). But scientists 6 quickly learned that the solar radiation pressure component in the orbit dynamics representation requires special attention. Consequently, the solar radiation pressure models, ROCK4 and ROCK42 were developed. Further investigations led to the introduction of the additional solar radiation force component, Y-bias acceleration (Fliegel e l al., 1985, 1992). With these improvements, as well as with the updated tropospheric models and densified network of permanent tracking stations, the geometry has been significantly strengthened. GPS trajectories were routinely determined with an accuracy of about 5 meters in the late 1980’s and 1-2 meters in the early 1990’s in the postprocessing mode. For real-time applications, however, the broadcast ephemeris is used. But, the application of Selective Availability (see Chapter II for details) can lead to the degradation of broadcast orbits even up to the 30-50 meter range (Hoffinan-Wellenhof e t al., 1992, p. 37). It should also be mentioned that the activation of Selective Availability has not diminished the accuracy of the precise post-fit orbits, since the double- or triple-differenced phases that are commonly used are not explicitly sensitive to the erroneous satellite clock due to Selective Availability. Even if single phases are used, the satellite clock parameters enter the observation equation as unknowns and are estimated together with the orbit and other parameters of the model. In the early 1990’s, preliminary results from global GPS experiments (at least for the pole position determination) were so promising that they prompted the scientific community to develop the capability for routine and accurate monitoring of ERP changes on a daily basis. This led to the establishment of the International GPS Service for Geodynamics (IGS) with the primary goals of providing the scientific community with high quality GPS orbits, high resolution ERPs, and densification of the current International Terrestrial Reference Frame (ITRF) allowing for the monitoring of global deformations of the Earth's crust (Mueller, 1993). Currently, IGS is capable of providing highly accurate GPS trajectories at the level of 10-15 cm per coordinate (IGS Reports). The IGS Analysis Centers apply undifferenced or double-differenced GPS 7 phases for the purpose of the orbit and ERP recovery. Thus, the implementation of the triple-difference technique represents a new idea in the area of the satellite trajectory determination. In order to confirm the validity of the triple-difference solution for GPS orbits and ERPs, consistency and repeatability tests for orbits and long baselines were conducted to show that the adopted technique is capable of achieving a high quality product. However, there are still a few shortcomings of the dynamic model applied in GODIVA, as reported in Chapter VUI. The model itself has to be enhanced, especially if arcs longer than 1 day are to be determined. For that purpose, small perturbing forces such as Earth albedo and the indirect oblation effect might be considered as an extension of the force model. The order of magnitude of these accelerations is about 0.4x1 O’9 m/s2 for the Earth radiation and 2.0x1 O'9 m/s2 for the indirect oblation, which might be important for the determination of arcs longer than a day. Also, including more perturbing objects beyond the Sun and the Moon in the third-body problem should be accomplished. However, comparison of the results from GODIVA with the IGS orbits and ERP estimates has proved that the approach to the problem of precise trajectory (over a day) and related parameter of the dynamic model determination presented in this dissertation represents a high-standard, optimal and contemporary approach. CHAPTER II
GLOBAL POSITIONING SYSTEM: GENERAL OVERVIEW AND DATA MODELING
2.1. GPS System and Signal Description The NAVSTAR Global Positioning System (GPS) program was initiated in 1973 through the combined efforts of the US Army, the US Navy, and the US Air Force. The new system, designed as an all-weather, passive, continuous, global radionavigation system was developed to replace the old satellite navigation system, TRANSIT, which was not capable of providing continuous navigation data in real time on a global basis. Currently, GPS is fully operating and provides global coverage with four to eight simultaneously observable satellites above the cut-off angle of 15° (Hoffman-Wellenhof et a l, 1992, p. 17). One of the two GPS operational modes, Precise Positioning Service (PPS) is only for authorized users, whereas the second mode, Standard Positioning Service (SPS) is available for numerous civilian applications. GPS provides suitably-equipped users with three-dimensional position, velocity and precise time information. The system consists of three segments:
• space segment • control segment • user segment
8 The space segment is responsible for satellite development, manufacturing and launching. The current constellation of GPS satellites comprises 25 high-altitude (-22200 km) space vehicles on six near-circular, 12-sidereal-hour orbits inclined at 55° to the Earth’s equatorial plane. Since the period of GPS satellites is measured in sidereal time units, the repeated configuration appears every day 3 min 56 s earlier with respect to universal time. The final arrangement of satellites in the full constellation is schematically presented in Figure 1.
160'
120'
0°/ Equator I 325.7 25.7 85.7 145.7 205.7 15.7 Right ascension of the asc&nding node 320
280
240
200
Figure 1. Arrangement of GPS satellites in the full constellation (Seeber, 1993, p. 212)
There are three basic categories of GPS satellites: Block I (development satellites), Block II (production satellites) and Block IIR (replenishment satellites). Block I is the first generation of GPS satellites launched between 1978 and 1985. Their orbital plane inclination at 63° differs from the inclination for the Block II that is at 55°, as mentioned above. Currently, only one Block I satellite (PRN 12), launched in 10 September 1984, remains in operation. It has more than doubled its 4.S year design life! Block IIR satellites, which are planned to replace Block II satellites, will have a mean mission duration of ten years (Hoffinan-Wellenhof et a t , 1992, p. 15). Block I satellites were fully available to the civilian users whereas the new generation availability is restricted, as noted below. The schematic view of a Block n GPS satellite is shown in Figure 2.
Figure 2 . Schematic view of a Block n GPS satellite
The fundamental frequency of GPS signal generated by the satellite’s oscillator is 10.23 MHz. Two signals, Li and L 2, are coherently derived from the basic frequency by multiplying it by 154 and 120, respectively, yielding: 1 1 Lr = 1575.42 MHz (~ 19.05 cm) Lz = 1227.60 MHz (~ 24.45 cm).
The adaptation of signals from two frequencies is a fundamental issue in the reduction of the errors due to the propagation media, mainly, ionospheric refraction. This problem is addressed in section 2.3.1. Every GPS spacecraft carries the high performance rubidium or cesium frequency standard as a precise time base with a proportional accuracy of 10 ' 12 to 10'13. The Li carrier frequency is modulated by a C/A code (clear/acquisition) and a P-code (precision or protected) or Y-code, while L 2 is modulated by a P-code or Y-code only. Both frequencies carry the broadcast message providing the user with information about the satellite clock, position and velocity. In general, C/A and P-codes are available to civilian users, while the Y-code is encrypted and is accessible only to authorized users, mainly military. The GPS time system is a continuous atomic time system defined by the Master Control Station (MCS), characterized by a GPS week number and the number of seconds elapsed since the beginning of the current week. The initial GPS epoch is
January 6 , 1980, at Oh UTC (Universal Coordinated Time). This corresponds to a Julian Date of 2444244.5 or a Modified Julian Date of 44244. The GPS time does not coincide with UTC, since every one-half year UTC might absorb a leap second which can be introduced in order to maintain the admissible difference of less than 0.7 s between UTC and UT1 (Universal Time is the fundamental time system defining the actual angular rotation of the Earth). The problem of time system and reference frames is addressed in detail in Chapter m. The operational control system (OCS) of the control segment consists of three ground antennas (GAs), five monitor stations, the prelaunch compatibility station (PCS) and the master control station (MCS) (Lamons, 1990). The primary tasks of the control segment are continuous monitoring and controlling the system, determining 1 2 GPS time, prediction of satellite ephemeris and the clock behavior, as well as updating the navigation message for every satellite. The user segment consists of numerous types of GPS receivers, providing navigators, surveyors, geodesists and other users with precise positioning and timing data. The basic types of observables provided by GPS receivers are pseudoranges (geometric range between the transmitter and the receiver, distorted by the lack of synchronization between satellite and receiver clocks), and carrier phase (a difference between the phases of a carrier signal received from a spacecraft and a reference signal generated by the receiver’s internal oscillator). This type of measurement introduces the unknown integer ambiguity, N, i.e., the number of phase cycles at the starting epoch. Both types of measurements reflect the distance between a satellite and a receiver. The Global Positioning System is exploited widely in a broad range of space- based applications in many branches of geosciences, navigation and surveying. However, the civilian user’s access to PPS, as addressed above, is limited. The U.S. Department of Defense (DoD) has established a policy for the implementation of Selective Availability (SA) and the Anti-Spoofing (AS) for the GPS signal to limit the number of unauthorized users and the level of accuracy for nonmilitaiy applications. This results in the degradation in the positioning performance and, in general, complicates the solution strategy. SA, by altering the broadcast ephemeris and/or the satellite’s clock, can reduce the user’s point positioning accuracy from 30-40 m to approximately 100 m (Georgiadou and Doucet, 1990). This accuracy can be achieved about 95% of the time; for the remaining 5%, errors of the order of 300 m or less should be expected. The corresponding accuracy level for heights is 150 m. Considering the horizontal and vertical components together as a three-dimensional position, the respective accuracy is 120 m. The velocity determination errors are of the order of 0.3 m/s. Manipulation of the spacecraft’s clock affects both the C/A and P- code ranges, as well as carrier phase measurements in the same fashion since the same atomic clock onboard a satellite controls the timing and frequency for the two carriers and both codes. Falsifying the navigation broadcast data is called the epsilon process, the dithering or systematic destabilizing of the satellite clock frequency is referred to as the delta process. But it should be emphasized that the SA effects can be removed or at least significantly diminished using differencing techniques. This aspect is explained in detail in section two of this chapter. AS, on the other hand, is another feature of PPS that modifies the GPS coded signal in a way that causes degradation in the quality of the range measurement (Davis, 1990). In essence, Anti-Spoofing introduces an additional code (the W code) that is impressed on the P-code in Li and L 2 in order to obtain the encrypted Y-code. Thus in order to recover the precise range from the P- code, the receiver must be able to generate the Y-code. Since unauthorized geodetic receivers have no a priori knowledge of the W code, they are unable to decode the Y- code. Thus, as a result, some alternative tracking scenario is required. Measuring the less precise C/A pseudorange on Li frequency and the difference (or the equivalent) between the Li and L 2 pseudoranges replaces the need for the inaccessible P-code range. Another important consequence of AS is the difficulty in recovering the integer ambiguities and their increased number with respect to a non-AS scenario (Rothacher etal., 1994). For example, the implementation of Anti-Spoofing has made the retrieval of widelane (Lw = Li - L 2 ) integers using pseudoranges more troublesome, although not impossible. In general, to compensate for the lack of the pseudorange that was normally used to identify integer ambiguities, a longer time of averaging or filtering is generally required during the data processing or an alternative scheme, where only phases are implemented must be applied. Figure 3 illustrates the behavior of the widelane ambiguity estimates, N 1-N2, on the epoch-by-epoch basis from the four- measurement combination ( double-differencedL.\ and La phases and ranges; see section 2.2.1 for more details) (Euler and Goad, 1991), collected by the TurboRogue receiver, under the impact of AS. The noisy signature shows the effect of AS as compared to Figure 4, representing the same phenomenon, but based on the Rogue receiver data 14 free from AS influence. It should be mentioned that before AS became fully operational, the widelane integers were very easily recovered in just a few epochs using the filtering technique presented by Euler and Goad (1991) (Ni and N2 are integer ambiguities onLi and L 2 phases, respectively; for details see section 2 .2 ).
a N1-N2 ffi Elevation 0 .0 ao « o c
o . - - q> "Ea 0 .0 20 ^ CM v * * a ** ;*.* as oC w 2 .0 G 2 .0 tOO 200 000 400 600 Number of Epochs
Figure 3. Epoch-by-epoch widelane ambiguity JVrJVj combination under AS
— N1-N2 -B-Elovatlon o.o 00
2.0 t.o o V.o c o.o 40 ^ SO 5 % 1.0
2.0 % ui 0 .0 0 100 200 Numbor of Epooho
Figure 4. Epoch-by-epoch AS-free widelane ambiguity JVrAfc combination As an effect of the permanent AS implementation on January 31, 1994, IGS has reported temporal increase in orbit RMS due to the poor clock solution (Kouba, 1995). Initially, the problem was pronounced mostly by hardware problems. However, GPS hardware and software upgrading improved the clock solutions that have approached again the subnanosecond level for most of the analysis centers. Also, the data processing and dynamic modeling improvements made the orbital solution RMS reach the level of 10-20 cm by the end of 1994.
2.2. GPS Data Modeling 2.2.1. Types o f Observables: Pseudoranges, Carrier Phases and Their Combinations GPS utilizes the concept of ‘bne way ranges,” thus it uses information about both the transmitter and receiver clocks. The ranges are recovered from the measured time difference between the instant of transmission (in transmitter’s time) and the epoch of reception (in the receiver’s time). The pseudorange observation equations on both
Li and L 2 frequencies are described as follows:
p,1= p ‘ + £ +t ; +c(<*, -<**)+»„+«» Ji (2 .1) % = p ‘,+ ^ + T: +c(dt,-dl‘)+ b,^el J 2
The notation follows that of (Goad and Yang, 1995; Yang, 1995). Pseudoranges on both Li and L 2 frequencies are used here in the point positioning mode for the purpose of the station clock correction determination. The carrier phase observable consists of the measured fractional part and the unknown integer count ( ambiguity ) Nf of phase cycles at the starting epoch. The dual-frequency, one-way phase observation equations in length units, are formulated in the following way: 16 =p?-4-+r+ajv* +c(dt,-dt‘)+x,(ri -«> ,)+ < , /l (2-2) K =/>: +c(dt, -<**)+»,., +*,(?; -*>„)+<. Jl where
P(*, P, * - pseudoranges measured between station / and satellite k on Li and L 2, respectively
$>*!, ® *2 - phase ranges measured between station i and satellite k on Li and L 2, respectively N* i» - ambiguities associated with®*, and 0 *2, respectively, Xi« 19 cm and « 24 cm are wavelengths of Li and L 2 phases, p k - geometric distance between the satellite k and receiver 1, /* I k -7 7 ,-7 5 - " ionospheric refraction on L| and L 2, respectively, J\ J 2 Tk - the tropospheric refraction term, dtt - the i-th receiver clock error, d t - the k-lh transmitter (satellite) clock error, fi, f i - both carrier frequencies, c - the vacuum speed of light, e*t, e *2 • measurement noise for pseudoranges on Li and L 2, , ek2 - measurement noise for phases on Li and L 2, bu, ba, bl3- interchannel bias terms for receiver i that represent the possible time non synchronization of the four measurements presented above; bui represents the interchannel bias between +&„, + st Ji2 (2.3) I pt.i = Pv +-7T + c'dtv +bv l+ e*, J\ .* /* PV2 — p V + -T £ + C’d tlt + and (2.4) Further differencing of the equations of type (2.3) leads to the removal (or at least significant reduction) of other common errors, such as receiver clock errors as well as interchannel biases. Again, the differential effect is still present but its influence is much less significant than the absolute error. The interchannel biases in equation (2.3) are highly correlated with the ionosphere and the ambiguity unknowns, which disables their effective separation in an estimation procedure (Goad and Yang, 1995). By differencing one-way observables from two receivers, i and j, observing two satellites, k and /, or simply by differencing two single differences to satellites k and /, one arrives at the double-differenced measurement, equation (2.5). M K = p “- IM J i (2.5) 19 'U _ , rptd P V ~ Py +ly (2.6) In (2.6)p'y comprises the difference between geometric distances from the two underlying receivers to satellites k and /, and the combined double difference tropospheric refraction term. As referred to above, the interchannel biases are strongly correlated with the ionospheric term and the ambiguity unknowns for single differences. However, after performing double differencing that removes the bias terms, a decoupling of the ionosphere and the phase ambiguities in (2.5) can be accomplished (ibid.). The next step in the process of the common error elimination is the differencing of two double differences, separated by the time interval dt, that effectively cancels the phase ambiguity biases, Ni and N2. As explained, Ni and N2 biases will remain constant as long as the tracking of carrier phase is continuous. The occurrence of a phase cycle slip introduces a new ambiguity unknown. Consequently, the triple-differenced measurement obtained from two double differences with cycle slips, contains the difference between the old and the new bias that itself is arbitrary and unknown. Tu y,\jt ~ P V'* j -2 lU,* J\ •/" II J, f i (2.7) nU . *y,JtJti . u Py.l.J, ~ P »•* + ~7T + J I Tu P W - ■ V.* ■ -U y,2,dt p V.* + -2 u.2.tb Jl U . irU PV-* =Py,*+Ty,* (2.8) The triple differenced p '* ,* represents the difference in the geometric distances pertaining to two underlying double differences, and the combined tropospheric propagation terms. Two different combinations of GPS phase observations exist that are particularly useful in a case when elimination or at least significant decrease of the influence of the ionospheric refraction is desired, or, conversely, when behavior of the ionosphere is to be monitored. Forming a linear combination of two double differences of the form (2.5) one arrives at ionosphere-free measurement. <&w = + d 2®2 (2.9) where coefficients cry, c& can be defined somewhat arbitrarily but should meet the following condition: a, • A*,, + a 1 • = 0 , where A*,, and A^ 2 denote the ionospheric term on Li and L 2 respectively. Examples for ay and as for the ion-free phase combination is presented by equation ( 2 .1 0 ). (2.10) Notice that a> and ay above are different from those originally presented by Goad (1985), since equation (2.9) is in length units. Similarly, the ion-free pseudorange can be formed. In that case, ay and 0 2 can be defined as follows, according to Hofinann-Wellenhof et al. 2 1 The linear combination of the form (2.9) is free from the first-order ionospheric refraction effect; however, due to the noninteger nature of the coefficients and a2t the integer characteristics of the unknown ambiguities is lost. The ionosphere-free triple difference is the basic observable applied in GODIVA. The observation equation of an ionosphere-free triple-differenced measurement between two receivers i and j, two satellites k and /, and over a time interval dt from epoch tj to epoch t2, using coefficients given in (2.9), is written in the following way: ,iono-jn*.dt = PiJ.dt + + a i S ij.\,d t + a 2S iJ.2.it ( 2 - 1 2 ) The ionospheric-only combination is, on the other hand, obtained by differencing two phase ranges belonging to the frequencies Li and Lq. The important characteristics of such a linear combination is the absence of the geometric range as shown below in equation (2.13). =«£. -® £-ii& i fry+ t.K (2.i3) J\J2 The use of the features referred to above is twofold. Since the ionospheric effect is a phenomenon that usually changes very smoothly with time, any abrupt discontinuity in the linear combination (2.13) could indicate a phase cycle slip. Secondly, since the geometric term is not present in (2.13), the ionospheric-only measurement can be used to monitor the behavior of ionospheric refraction provided that there are no losses of lock. 2.2.2. Correlation Between Differenced Observations The original data set of one-way GPS phases is assumed to be uncorrelated. Thus, the covariance matrix is simply diagonal. However, forming the single, double or triple phase differences introduces statistical correlations among such formed 2 2 observables. Therefore, the proper accounting for the correlation among differenced data is necessary in order to make the new, differenced model equivalent to the original one, based on the undifferenced, uncorrelated data (Schaffrin and Grafarend, 1986). As already discussed, the linear combinations of the single phases forming double or triple differences are no longer independent. Even for the optimal arrangement of the combined phases, the resulting covariance matrix could only be of a block-diagonal form (double differences) or a banded matrix (triple differences). Double differences at the same epoch are correlated to one another. Also, assuming that successive time differences is the chosen scheme, the correlation exists between triple- differenced observables at the same epoch interval and between consecutive epoch intervals because they share the same single difference or even many of them. Consequently, in using linear combinations of phases, one has to determine properly their covariance matrix according to the Law of Error Propagation. In its general form, the law states that given the covariance matrix of measurements, 2 J* and the respective differencing operator in matrix form, D, the covariance matrix of the linear combination of the original observables reads as follows: (2.14) However, the vectorial approach to the problem is much easier to accomplish in computer code since the sizes of matrices in (2.14) are usually very substantial. For an nx7 vector y of the original undifferenced observations, the covariance matrix is defined as Ey= cr\ - I., whereof is the a priori variance component of the raw phase observable and Im is the nxn identity matrix. Let us define cj and c; as linear combinations of the original undifferenced observation vector .y in the following way: c, =a,y (2.15) c2 = a,y where the two 7xn vectors ai and a 2 are the differencing operators whose elements are 0, -1 and 1. The number and location of the elements in vector a< depends on the type of linear combination formed, the total number of the single phases under consideration and the differencing scheme. For example, forming the double difference, comprising four one-way phases out of a set of n, can be represented as follows: a, =[0 0 0 1 -1 -1 1 0 0 0 . . . 0 0]n (2.16) The triple difference, combining eight single phases out of n in a data set, can be formulated as: a, =[0 0 1 -1 -1 1 -1 1 1 -1 0 . . . 0 0 l (2.17) The correlation between any linear combinations, cj and c;, defined above, is then obtained according to the Law of Error Propagation as follows: (2.18) This way, using every pair of vectors ak and ai, representing linearly dependent combinations based on the same set of n original observables, the resulting covariance (2.18) can be evaluated very efficiently as a dot product of two vectors multiplied by a scaling factor cr*. 2.3. GPS Observation Error Sources and Corrections In general, there are three major groups of sources introducing errors into GPS observation equations: - satellite orbital errors (imperfect orbit modeling), - station position errors 24 - propagation media errors and receiver (hardware) errors. The first two groups affect the residual vector Y in the system of observation equations, whereas the third one alters the observable directly. Corrections (due to the systematic errors from the third group) to the measurements are usually computed based on the empirical models developed from the characteristics of the observation and the phenomenon under consideration, or, alternatively, on the theoretical assumptions. They are applied to the raw measurement. Another way of systematic effects elimination is the appropriate combination of observables, as explained in the previous section. In general, differencing between receivers eliminates satellite-related biases, differencing between satellites cancels the receiver-attributed systematic effects. The impact of errors and their propagation depend substantially on the geometric configuration of the observed satellites. Table 1 presents the average error contribution from the individual sources, grouped in the sections listed above. Table 1. Main sources of the errors and their contribution to the single range observation equation (Seeber, 1993, p. 290) P-code C/A-code Source SA off SA on SA off SA on Satellite - Orbit 5 m 10 - 40 m 5 m 10 - 40 m - Clock 1 m 10 - 50 m 1 m 10 - 50 m Signal propagation - Ionosphere (2 frequencies) cm - dm cm - dm cm - dm cm - dm -Ionosphere (model) -- 2-100m 2 -100 m - Troposphere (model) dm dm dm dm - Multipath Effects 1 m 1 m 5 m 5 m - Relativistic Propagation ~ 2 cm ~ 2 cm ~ 2 cm ~ 2 cm Receiver - Observation Noise 0.1 - 1 m 0.1 -1 m 1 - 10 m 1 - 10 m - Hardware Delays dm -m dm - m m m -Antenna Phase Center mm - cm mm - cm mm - cm m m -cm 25 If the observation equation is expressed in the terrestrial reference frame, ITRF, (for details about reference frames see Chapter m), then the Earth rotation correction must be applied to the GPS observable. During the signal propagation from the * transmitter to the terrestrial antenna, the ITRF frame rotates with the Earth with respect to the satellite. As a consequence, the position of the transmitter’s antenna at the time of signal transmission has changed in the ITRF frame. Thus, the spacecraft’s coordinates at the transmission time must be rotated about the third axis of the ITRF frame by the amount equal to the propagation time dt multiplied by the Earth’s rotational velocity, a>». The angle of rotation is expressed as follows: a= eotdt (2.19) The relativistic effect in GPS is twofold. First, the Earth’s gravitation field exerts a direct relativistic effect on the orbital motion of the satellite (relativistic perturbation, discussed in Chapter IV) and also affects the phase observable. The combined effect amounts to 0.001 ppm in positioning. Second, Earth gravitation and the fact that the satellite moves, affect the satellite clock’s frequency at the order of 1 0 -to Thg dynamic and propagation effects strongly depend on the geometry between station, satellite and geocenter. The maximum effect in the range units (c5t) for the single phase measurement is 19 mm. This effect is significantly reduced (to 0.001 ppm) for the relative positioning. The phase measurement relativistic propagation correction reads as follows (Zhu and Groten 1988): St = (2GM / c’)ln[(r +R + p) / (r + R - p)J ( 2 .2 0 ) where r,R- geocentric distances to the satellite and station, respectively, p - range distance between satellite and the receiver, c - speed of light in a vacuum, 26 GM - gravitational constant multiplied by the mass of the Earth. According to the theory of general relativity, a clock’s rate depends upon the clock’s state of motion and also the gravitational potential acting on it. Since both the spacecraft clock and the receiver clock are moving in quasi-inertial frames with different velocities and are affected differently by the gravitational attraction, their rates must be different. The relativistic time difference between the receiver and the satellite is given by Schwarze etal. (1993): where z - rate of the receiver clock, tt - rate of the satellite clock, UI. - geopotential at the geoid, h - height of the station above the geoid, g( The constant drift which is a part of the total correction in (2.21) is compensated for before launch time by reducing the frequency of the satellite clock by 0.00455 Hz from its nominal value of 10.23 MHz. However, the periodic term that depends upon E cannot be removed; it has to be modeled. For GPS altitude, it has the maximum amplitude of ~30 ns in time or 10 m in distance. The periodic part can be canceled by performing between-stations differencing, but for point positioning is still harmful if not properly accommodated. Further discussion on the relativistic effect on 27 the time and reference frame definitions can be found in Chapter m. Table 2 summarizes the relativistic clock effect (Zhu and Groten, 1988): Table 2. Relativistic effects on a clock Secular Drift - Nominal Drift -4.45xlO'10, calculated from the nominal oibit parameters, corrected prior to the launch - Residual Drift less than lxlO'12, due to an off-nominal semimajor axis of the orbit: constant time and frequency offset, correctable Periodic Time Offset -Initial Value (Keplerian) ksinE(tee), k=46 ns, for e=0.02; constant time offset, correctable; too is an epoch of time at which time and frequency -Time Variable (Keplerian) offsets for satellite are redetermined with respect to GPS time - Non-Keplerian ksinE(t), can be corrected using GPS Navigation Message less than 1.2 ns (36 cm) for e=0.02 Periodic Frequency Offset for e=0.02, average for range rate: 0.6 mm/s, maximum: 2 mm/s Multipath is an effect of the interaction of reflective objects in the antenna’s surrounding with the signal coming from the satellite. This causes multiple reflection and diffraction that result in the interference of both direct and reflected signals. Thus, the signal arrives at the receiver antenna via direct and indirect paths. The magnitude of multipath tends to be random and unpredictable. It ranges up to 5 cm (maximum) for the phases, being usually smaller than 1 cm, and can be as large as 1 0 - 2 0 m for the code pseudoranges (Hofinann-Wellenhof et ai., 1992, p. 113). Multipath effects can be largely reduced by appropriate antenna design (proper signal polarization, wideband antennas, choke ring antennas) and the selection of sites that are far enough from reflective objects (Hofinann-Wellenhof et al., 1992, p. 112; Seeber, 1993, p.230). In fact, multipath and the receiver noise cannot be effectively separated, so their combined error should usually be presented as the performance characteristics for the receiver in particular conditions. Table 3 presents an estimate of noise level and noise combined 28 with the multipath effect produced by some geodetic receivers in a highly reflective environment (Lachapelle et a l, 1993). Antenna Phase Center is the point to which the received signal is referred. It usually does not coincide with the physical center of the antenna, and for GPS receivers both the Li and L 2 phase centers are generally different. The magnitudes of these offsets are provided by the manufacturer; however, the location of the phase center can vary with time. Generally, this variation should not exceed 1 - 2 cm; for modem microstrip antennas it reaches only a few millimeters. Antenna phase center offset depends on the azimuth and the elevation of the satellite as well as on the intensity of the incoming signal. Similarly, the satellite phase center does not coincide with the spacecraft’s center of mass. Suggested satellite center of mass corrections for Block I and Block II GPS satellites can be found in IERS Technical Note No. 13 (McCarthy, 1992, p. 9). Discussions of the ionospheric and tropospheric refraction are given in detail in subsequent sections of this chapter. Table 3. Code measurement characteristics in a high multipath environment (Lachapelle et al., 1993) Receiver and Observable Type Measuring Noise Noise Plus Multipath GPS Card1*1 10 cm 70 cm GPS Card1*1 with choke ring antenna 30 cm P-XII C/A-code 100 cm 300 cm P-XIl C/A-code with choke ring antenna 200 cm P-XII P-code 10 cm 70 cm P-XII P-code with choke ring antenna 30 cm 2.3.1. Ionospheric Refraction The ionosphere is the upper part of the atmosphere, ranging between SO to 1000 km in altitude above the Earth’s surface. The gas molecules in the ionosphere are ionized by the ultraviolet radiation from the Sun, and as a result, the free electrons are released. The presence of free electrons in the geomagnetic field causes a nonlinear dispersion of electromagnetic waves traveling through the ionized medium. In general, the propagation delay depends on the total electron content (TEC) along the signal’s path and on the frequency of the signal itself as well as on the geographic location and time. Carrier frequencies below 30 MHz are reflected by the ionosphere; only higher frequencies are able to penetrate it. The GPS signals are affected by the ionosphere in two different ways. The ionospheric group delay causes a delay of the satellite code signal recorded by the receiver. On the other hand, phase measurements are subject to the ionospheric phase advance. The range measurement obtained from one of the codes is measured longer than the geometric distance, and the phase-range is measured shorter with respect to the geometric distance. The total ionospheric effect on the measured range varies from less than 1 m to more than 150 m at the maximum activity of sunspots for the satellites near the horizon (Wells etal., 1986, p. 145). The maximum vertical range errors in meters are shown in the Table 4 below (Seeber, 1993, p. 304). Table 4. Maximum vertical ionospheric range error [m] Frequency lst-Order Effect 2nd-Order Effect 3rd-Order Effect ( l/f J) (1/f1) t l / f 4) u 32.5 0.036 0.002 l2 53.5 0.076 0.007 Li / L2 0.0 0.026 0.006 Brunner and Gu (1991) present an updated model that calculates the ionospheric effect along the ray path rather than along the geometric distance. In addition, they take into account the coupling between the Earth’s magnetic field and the electron density in order to evaluate properly the higher-order terms. In this model, the estimated ionospheric delay is presented in Table 5 (Brunner and Gu, 1991; Bassiri and Hajj, 1993). Table 5. Estimated ionospheric group delay for GPS signal Lt l2 Residual Range Error First Order, lit* 16.2 m 26.7 m 0.0 Second Order, lit3 ~ 1.6 cm ~ 3.3 cm ~ -1.1 cm Third Order, lit* ~ 0.86 mm ~ 2.4 mm ~ -0.66 mm Calibrated lit 3 Term Based on a Thin Layer ~ 1-2 mm Ionospheric Model The phase advance can be obtained from the above table by multiplying each number by -1, -0.5 and -1/3 for the 1/f2, l/f3and l/f 4 term, respectively. A comparison of Tables 4 and 5 proves the superiority of the model developed by Brunner and Gu over the former solution by King et a l (1985). For a dual frequency signal the method of ionospheric effect elimination (or at least significant reduction) is efficiently applied. It is based on the fact that Li and L 2 frequencies are subject to different ionospheric delays since the effect itself is a function of the carrier frequency. Therefore, the combination of both L| and L 2 ranges (Pi, P2) provides the range ionospheric correction for the code measurement on Li, bPi,™, as follows (Seeber, 1993, p. 304): (2.22) 31 This correction removes the first-order ionospheric effect from the Li pseudorange. On the other hand, the ionospheric-free combination of pseudoranges is defined by equation (2.23): Pu =P,-^-P, (2-23) Ji where the coefficients from the above linear combination are defined by equation (2.11). For the carrier phase f l («>.-AT,)} (2.24) SI -A 2 where Ni and TVz denote the respective ambiguity terms, f i and are both carrier frequencies and = -j& p*' 'jfh ;9' ( 2 ' 2 5 ) where the coefficients from the above linear combination are defined by equation (2.10). The higher-order terms of the ionospheric refraction effect are still present in the ionospheric-free linear combinations. However, they can be almost entirely canceled through the relative observations over a short baseline since both paths of the upcoming signal are propagated through nearly the same atmosphere. The longer 32 baselines still are affected by the higher-order terms that can reach 1.6 cm for Li and 3.3 cm for L 2, but the residual error term should not exceed 1.1 cm (Table 5). During periods of high solar activity, these numbers are significantly larger. Also, the geographic location of the observing site is very important since the Earth’s ionosphere can be divided into three major geographic regions: mid-latitude (with the lowest level of disturbances, where TEC can be fairly well predicted), the auroral and polar regions (significant disturbances such as phase scintillations at the diurnal frequency, correlated with the magnetic activity), and the equatorial anomaly zone, with the strongest disturbances in the ionosphere with clear diurnal fluctuations. 2.3.2. Tropospheric Refraction Tropospheric refraction (tropospheric delay) is caused by the nonionized part of the atmosphere in the region up to 40-50 km in altitude above the Earth’s surface comprising three layers: the troposphere, the tropopause, and the stratosphere. Unlike the ionosphere, the troposphere is not dispersive for frequencies below 15 GHz, so both group and phase delay are the same. As a consequence, the elimination or at least significant reduction of the tropospheric delay effect by the linear combination of dual frequency observations is not applicable. Atmospheric refraction affects relative as well as point positioning. The total effect on the GPS phases and pseudoranges reaches 2.5 m in the zenith direction and increases with the cosecant of the elevation angle up to 20 - 28 m at 5° elevation. The measured range is longer than the geometric distance from the observer to the satellite. The tropospheric propagation is usually represented as a function of temperature, pressure and relative humidity. It is separated into two components: dry and wet. The dry component, which is proportional to the density of the gas molecules in the atmosphere and changes with their distribution, represents about 90% of the total tropospheric refraction. It can be modeled with an accuracy of about 2% that corresponds to 4 cm in the zenith direction using surface measurement of pressure and 33 temperature (Seeber, 1993, p. 48). The wet refractivity is due to the polar nature of the water molecules and the electron cloud displacement (Solheim, 1993). Since the water vapor is less uniform both spatially and temporally, it cannot be modeled easily or predicted from the surface measurements. As a phenomenon highly dependent on the turbulences in the lower atmosphere, the wet component is modeled less accurately than the dry. The influence of the wet tropospheric zenith delay is about 5-30 cm that can be modeled with an accuracy of 2-5 cm according to Leick (1995, p.308). The tropospheric refraction as a function of the satellite’s zenith distance is usually expressed as a product of a zenith delay and a mapping function. A generic mapping function represents the relation between zenith effects at the observation site and at the spacecraft’s elevation. Several mapping functions have been developed (e.g., by Saastamoinen, Goad and Goodman, Chao, Lanyi), which are equivalent as long as the cutoff angle for the observations is at least 20° (Rothacher, 1992, p. 83). The tropospheric range correction can be written as follows: A „ = /,(*) A',+/„(*) A". (2.26) where fd(z), fj(z) - mapping functions for dry and wet components, respectively, A0,, A#w - verticaldry and wet components, respectively (see Seeber, 1993, p.47). Tropospheric refraction accommodates only the systematic part of the effect, and some small unmodeled effects remain. Moreover, errors are introduced into the tropospheric correction via inappropriate meteorological data (if applied) as well as via errors in the zenith mapping function. These errors are propagated into station coordinates in the point positioning and into base components in the relative positioning. For example, the relative tropospheric refraction errors affects mainly a baseline’s vertical component as reported by Santerre (1991). Error in the relative tropospheric delay at the level of 1 0 cm implies errors of a few millimeters in the horizontal components, and more than 20 cm in the vertical direction. If the zenith 34 delay error is 1 cm, the effect on the horizontal coordinates will be less than 1 mm but the effect on the vertical component will be significant, about 2.2 cm. The effect of the tropospheric refraction error increases with the latitude of the observing station and reaches its maximum for the polar sites. It is a natural consequence of a diluted observability at high latitudes where satellites are visible only at low elevation angles. The scale of a baseline derived from observations that are not corrected for the effect of the tropospheric delay is distorted; the baseline is measured too long. The usual approach in handling tropospheric refraction problems of GPS orbit determination is twofold: first, one of the refraction models is used to compute the correction that is further applied to the raw observation. Second, a tropospheric zenith delay scaling factor is introduced to the model as an additional unknown to be estimated in the least-squares adjustment at specified time intervals. This way, the systematic effect of troposphere mismodeling is absorbed by the new parameter. Some researchers estimate only the scaling factor to the wet part of the tropospheric model assuming that the dry part is sufficiently modeled. GOD1VA implements the modified Hopfield model according to Goad and Goodman (1974), and estimates the scaling factor to the total tropospheric correction at 4-hour intervals, for every tracking station. Here, the reasonable assumption that the short periodic changes in the atmosphere average out over a few hours is introduced. The wet component mapping function developed by Goad and Goodman is reported by Janes and Langley (1990) as providing the best overall agreement with ray tracing results, among all the mapping functions tested for representative atmospheric profiles. Generally, the model’s departure from the observation results falls within 3-5 mm for the zenith distance up to 85°. When the highest accuracy is required, the water vapor and water liquid content can be measured along the signal propagation path with water vapor radiometers , providing the signal path delay with accuracy of 1 to 3 cm (Seeber, 1993, p. 49). As an alternative, radiosondes can be used to measure pressure, temperature and relative humidity along the vertical path. An interesting analysis of the problem of improvement of GPS positioning accuracy by the use of water vapor radiometers is presented by Solheim (1993). He examines several different methods of accounting for the atmospheric effect during the baseline determination, applying the precise GPS trajectory. His first method implements Saastamoinen’s model alone. In his second method, Saastamoinen’s model with the scaling factor estimated at different time intervals is used. The third method is to apply Saastamoinen’s model with Kalman filtering updating. In addition, several different solutions based on the water vapor observations with a radiometer, combined with the estimation of the tropospheric offsets, are presented. The repeatability of the test baseline, specially in the vertical component, is reported to be the best for one of the solutions where the water radiometer was used. However, the application of the water vapor radiometers is quite expensive. Moreover, GPS solutions were found capable of measuring the difference of water vapor between GPS sites and thus, the tropospheric delay, in the least-squares sense. Also, the results reported by Solheim showed that the approach currently used by some of the IGS Processing Centers and implemented in GODIVA (i.e., estimation of the tropospheric scaling factor at constant intervals, for example, four hours), is satisfactory for GPS relative positioning combined with the precise orbit determination. Summarizing the topic related to the influence of the propagation media on the GPS signal, Table 6 shows the typical ranges of the propagation delay by sources together with the uncertainties of the present models (Solheim, 1993). 36 Table 6. Sources, magnitudes and uncertainties of propagation delay Delaying Medium Typical Range of Magnitude at Typical Uncertainty of Zenith Present Modelsfor Elevation Angle > 15* Ionosphere 10 m (night) to 50 m (day) 2 to 20 cm for single path to one satellite Troposphere: - Dry component 2.35 to 2.45 m at the sea level predictable, 1 mb = 2.4 mm - Water vapor 6 to 30 cm stochastic, at zenith: 2 cm anisotropies: 2 cm - Hydrometeors (cloud, rain, 0 to several cm 0 to several cm fog, haze) Propagation Multipath: -At the Receiver Antenna 0 to several cm removed by averaging -At the Spacecraft Antenna unknown but small not removed An extensive discussion of the tropospheric effect is presented here due to its significance in the problem of the ERP and orbit recoveiy from the GPS observations. The results presented in Chapter VH show that proper modeling of the troposphere is critical for determination of the pole and rotational time from the satellite signal. For example, the impact of the number of tropospheric scaling factors estimated per day on the x component of the pole, Xp, is presented in Figure 5. 37 -a-4-hour aolutlon -■-8-hour aolutlon -o-18-hour aolutlon -♦-no tropoaphara —O— Bullatln B m E IS IS 17 is ia so si day ol yaar Figure S. Estimates of the xp as a function of the number of tropospheric scaling factors estimated per day The satellite trajectory and terrestrial station coordinates are also affected by the number of tropospheric scaling factors estimated per day. The a posteriori standard deviation per satellite coordinate can grow from 9.5 cm for the solution with tropospheric scaling factors estimated at four-hour intervals to 44.5 cm (see Table 27 in Chapter VII) when no tropospheric scaling factors are estimated! Figure 6 represents an example of the influence of the adopted tropospheric model on the terrestrial station coordinates, especially on the height component. For more detail see Section 7.4. 38 ■ 4-hour solution b 8-hour solution E3 16-hour solution m no troposphsrs S.00 A7C 2.SS E a t.oo 0.00 Figure 6 . The average a posteriori standard deviation in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Matera for GPS week 784 CHAPTER m REFERENCE FRAMES AND THE TIME SYSTEMS The coordinate system is most commonly referred to as three mutually perpendicular axes and a specifically defined origin. But the term “reference system ” has much broader meaning; it comprises the definition and description of the physical environment, as well as the theories and constants used in the definition of the coordinates (e.g., the theory of precession and nutation involved in the definition of the celestial reference system) (Moritz and Mueller, 1987, p. 526). Such a defined system must be realized by a set of conveniently chosen reference points that form a reference fram e (e.g., coordinates of some quasars form the celestial reference frame for the VLBI system), giving the user physical access to the system. The most common way of representing a position is with a set of three Cartesian coordinates. Since in general, and especially in astronomy and space geodesy, the objects are moving with respect to the reference frames, and frames are also moving in space, the time information related to the epoch of observation and the reference time at which the frame is defined, must to be specified. Two most significant and commonly used reference frames in space geodesy are the celestial reference frame (CRF), defined by the barycentric coordinates of selected extragalactic objects and the terrestrial reference frame (TRF) rotating with the Earth. Since any terrestrial frame changes its orientation in space, establishment of such a frame requires linking to the underlying celestial frame in order to define and monitor what is generally known as Earth Orientation. Earth orientation defines the 39 40 transformation between the celestial and the terrestrial systems, usually expressed in terms of precession, nutation, polar motion and rotation about the Earth’s polar axis in terms of the Greenwich Apparent Sidereal Time. 3.1. Conventional Celestial Reference System The ideal celestial reference system is an inertial system such that the equations of motion in that system are described rigorously without any rotational components. However, such a system cannot be realized in practice; to some extent it can only be approximated (Seidelmann, 1992, p. 95). Also, the theory of relativity requires some additional consideration as noted in section 3.4. As an example, Moritz (1980) shows that in theory, the practical materialization of the inertial system is done best by means of the VLBI-determined coordinates of a set of quasars. But quasars are a typical phenomenon of an expanding universe and as such must be described by the theory of general relativity for which no rigorous inertial system exists! Thus, some approximations of the rigorous theoretical approach are necessary to realize practically the celestial system that is satisfactorily close to an inertial system. “... all our practical inertial systems are nonrigorous in the sense of general relativity but still perfectly useful. For the region of our galaxy, we may easily consider space-time to be essentially flat, apart from local gravitational irregularities” (Moritz, 1980). The conventional celestial reference system, as any other reference system, must comprise a specification of the reference frame at a given date (time component), all the necessary procedures and constants that are required for the proper transformation of that frame in time (i.e., precession and nutation), and also transformation between the underlying frame and any other reference frame (i.e., terrestrial reference frame). A detailed description of a transformation problem is presented in section 3.3. The International Earth Rotation Service (IERS) Celestial Reference Frame (ICRF) is a barycentric frame, defined by coordinates of extragalactic objects (quasars) determined by VLBI. It “... is a realization of a system of directions which are 41 consistent with those of the FK5” (McCarthy, 1992, p. 12). The polar axis points to the direction of the mean pole at the standard epoch J2000.0. The origin of right ascension (X-axis) is close to the dynamical vernal equinox at J2000.0,” (McCarthy, 1995), the Z-axis completes the right-handed system. The inertial frame adopted in GODIVA is Mean of Reference Date (MoD). Its Z-axis coincides with the mean pole and X-axis points to the mean equinox of the reference date, i.e., 0 h 0 min 0 s of International Atomic Time (TAI) of the day at which starting elements are given. The 7-axis completes the right-handed system. 3.2. Conventional Terrestrial Reference System The important requirement that must be met by the conventional terrestrial reference system is that geophysical phenomena can be related to it. One example is the Earth’s gravity field, represented by a set of spherical harmonic coefficients in the reference frame tied to the conventional terrestrial system. The origin of the conventional terrestrial system is defined by the coefficients C io , C n , and Sn of the gravity field. For the gravity model GEM-T3 those coefficients are zero, thus imposing the constraint that the conventional terrestrial frame is geocentric. The orientation of the conventional terrestrial frame is primarily defined by the potential coefficients C 21, S21 (orientation of the third axis) and C 22, S22 (orientation of the first axis). C 21 and S 21 are zero only when the third axis of the reference system coincides with the Earth’s axis of maximum inertia. C 22 and S 22 would vanish only if the Earth had complete rotational symmetry or if a principal axis of inertia fell on the zero reference meridian (Heiskanen and Moritz, 1979, p.62). Since C 21 and S21 for GEM-T3 are constrained to zero but in fact the third axis of the terrestrial reference system does not coincide with the Earth’s axis of maximum inertia, the IERS recommends the use of the averaged values of C 21 and S 21 that define the location of the mean figure axis (i.e., the axis of maximum inertia) of the Earth. The recommended values are: C 21 - - 0.17x10'® and S 21 = 1.19x10-®. 42 The conventional terrestrial reference frame currently applied for GPS orbit determination, as recommended by the IERS, is the International Terrestrial Reference Frame 1993 (TTRF93), defined by the coordinates and velocity model for 13 core stations, listed in Tables 7 and 8 (Boucher et al., 1994, IGSMA1L #819). Figure 7 represents the world-wide geometry of the GPS tracking stations, including the 13 reference core stations. The consistency of the IERS series of the Earth Rotation Parameters (defining a transformation between celestial and terrestrial frames) with the given realization of the ICRF and ITRF is at the level of 0 . 0 0 1 arcsec (McCarthy, 1992, p. 13). The origin of the current ITRF is located at the center of mass of the Earth with an uncertainty of 5 cm according to the IERS Explanatory Supplement to Bulletins A and B. This accuracy is reported by SLR solutions that provide ITRF's access to the geocenter (Boucher and Altamimi, 1992). For the future ITRF ‘The goal ... is to have a set of sites permanently or repeatedly monitored by at least two IERS techniques [i.e., VLBI, LLR, SLR, GPS and Doris] including at least four sites on each of the major tectonic plates, and two on the smaller ones” (IERS: Missions and Goals for 2000, IERS publication, 1995). Table 7. TTRF93 Site Positions adopted by IGS [m] Site Name XYZ Tromso 2102940.360 721569.398 5958192.092 Madrid 4849202.459 -360329.148 4114913.089 Kootwyk 3899225.260 396731.803 5015078.324 WettzeU 4075578.593 931852.662 4801570.020 Hartebeesthoek 5084625.431 2670366.543 -2768493.990 Algonquin 918129.510 -4346071.228 4561977.846 Yellowknife -1224452.487 -2689216.070 5633638.283 Goldstone -2353614.169 -4641385.389 3676976.474 Fairbanks -2281621.422 -1453595.760 5756961.945 Kokee-Park -5543838.126 -2054587.365 2387809.642 Santiago 1769693.278 -5044574.137 -3468321.048 TidbmbMa -4460996.070 2682557.105 -3674443.836 Yarragadee -2389025.427 5043316.850 -3078530.871 O IERS Fiducial Stations of ITRF93, Epoch95.0 O Estimated Stations Ny-Alesund Jo h n s . . Kitob Mi „ Taiwan Kokee Park ©•.Pamatai 0 - Hartebeesthoek Tidbin^ja e Kerquelen Hobart Davis Casey lurdo Figure 7. Global Distribution of GPS Tracking Stations 44 Table 8. ITRF 93 Site Velocities adopted by IGS [m/y] Site Name Vx VY Vz Vm vN Vv Tromso -0.0252 0.0162 0.0065 0.0232 0.0194 -0.0008 Madrid -0.0141 0.0222 0.0201 0.0209 0.0253 0.0010 Kootwijk •0.0218 0.0212 0.0122 0.0231 0.0230 -0.0027 Wettzell -0.0252 0.0191 0.0123 0.0241 0.0231 -0.0042 Hartebeesth oek -0.0054 0.0176 0.0216 0.0183 0.0215 -0.0060 Algonquin -0.0217 -0.0021 0.0066 -0.0219 0.0067 0.0032 Yellowknife -0.0289 0.0006 -0.0025 -0.0268 -0.0112 0.0025 Goldstone -0.0191 0.0061 -0.0047 -0.0197 -0.0060 -0.0002 Fairbanks -0.0285 -0.0019 -0.0101 -0.0139 -0.0274 0.0018 Kokee-Park -0.0129 0.0614 0.0292 -0.0617 0.0303 0.0025 Santiago 0.0228 -0.0063 0.0256 0.0197 0.0290 -0.0030 TidbinbiUa -0.0354 -0.0017 0.0412 0.0198 0.0502 0.0000 Yarragadee -0.0459 0.0090 0.0403 0.0377 0.0485 0.0050 In Table 8 subscripts X , Y, Z denote the geocentric Cartesian components, and E, N, V denote the local topocentric components in East, North and Vertical directions, respectively. 3.3. Transformation Between Terrestrial and Celestial Reference Systems Transformation between the Terrestrial Reference System (TRS) and the Celestial Reference System (CRS) is defined by the set of rotations described by equation (3.1) below: [CRS] = PN{f) x R(t) x W(t) x [7KS] (3.1) where PN(t) - transformation matrices arising from the motion of the Celestial Ephemeris Pole (CEP) is space, i.e., precession and nutation (Moritz and Mueller, 1987, pp. 189, 394; Seidelmann, 1992, p. Ill), R(t) = R 3(- GAST) - diurnal rotation of the Earth about the CEP axis, 45 W(t) = Rifyp) R 2(xJ - combined polar motion rotation about the X and Y axes of the Earth-fixed frame shown in Figure 8 , where xp and yp are the coordinates of the CEP relative to the reference point that is the ITRF reference pole. Precession and nutation define the motion of the CEP in space. Lunisolar precession is a smooth, long periodic motion of the mean celestial pole of the Earth equator about the pole of the ecliptic caused by torques of the Sun, and the Moon exerted on the dynamical figure of the Earth. Its period is about 26000 years. Planetary precession describes the motion of the ecliptic due to the gravitational attraction of the planets. It results in a westerly motion of the vernal equinox of about 12 arcsec/century and a decrease in the obliquity of about 47 arcsec/century. Planetary and lunisolar precession combine into general precession. Nutation is the short-periodic motion of the true celestial pole around the mean celestial pole caused mainly by the ellipticity of the Earth’s orbit, the elliptic character of the Moon’s orbit, and the noncoincidence of the Moon’s orbit with the ecliptic in conjunction with the retrograde motion of the lunar nodes. The principal period for nutation is 18.6 years, with an amplitude of about 9 arcsec. Precession modeling used in GODIVA is in accordance with IERS Standards that follow the theory of Lieske et al. (1977). The nutation model follows recent determination according to Herring (McCarthy, 1995, p. 25), based on the analysis of VLBI and LLR data. 46 y=A=270° ITRF reference pole CEP x= A = (f Figure 8. Polar motion Polar motion is the motion of the true celestial pole (CEP) that defines the instantaneous rotation axis of the Earth with respect to a reference point, i.e., the conventional terrestrial pole (ITRF reference pole). The ITRF reference pole is defined by a set of adopted coordinates and velocities of the terrestrial sites that furnish the conventional terrestrial reference frame. The x-axis is in the direction of the IERS reference meridian (IRM), the y-axis is in the direction of 90° west longitude (see Figure 8 ). Diurnal rotation of the Earth about the CEP axis is described by the angle called Greenwich Apparent Sidereal Time (GAST), that is the Greenwich hour angle of the true (thus affected by nutation) vernal equinox. Greenwich Apparent Sidereal Time is computed as a sum of Greenwich Mean Sidereal Time (GMSTo h u t i ) at 0 hours of the Universal Time (UTI; for the definition see section 3.5 below), the interval of GMST from 0 hours UTI to the epoch of observation, and the Equation of Equinoxes (nutation in right ascension), as described by eq. (3.2), (3.3) and (3.5) below (McCarthy, 1992). GMST = GMST0UJTi + r[(OTl - UTC) + UTC] (3.2) and 47 GMSTokuti = 6*41"50'.54841+8640184'.812866 x Tu +0'.093104 x T* (3.3) —6 ' .2 x 1 0 " 6 x f where /•[(OT1 - C/7U) + f/7U] - the interval of GMST from 0 hours UTI to the epoch of observation converted to sidereal units, UTI, UTC - universal (solar) and coordinated times, respectively, for definition see Section 3.5, r - ratio of the universal time to sidereal time: r = 1.002737909350795 + 5.9006x 10"n xT-5.9x 10 15 x 2? (3.4) Tu=d 136525 where d is the number of Julian days elapsed since 2000 January 1, 1 2 h UTI. Finally: GAST = GM ST+A'Fcos* (3.5) where A\|/cosf is the Equation of Equinoxes and A\|/ represents nutation in longitude; e is the obliquity of the ecliptic. The problems associated with the Earth’s orientation, its determination, predictability and reasons for variations are provided in Chapter VI. 3.4. Relativistic Effects in Reference Frames and Time The theory of special relativity is an extension of classical mechanics when objects under consideration move with very high speed. This theory holds in the absence of the gravitational field. On the other hand, the theory of general relativity provides the refinement of Newtonian theory of gravitation for the case of very strong gravitational fields (Moritz, 1980). In Newtonian mechanics, the inertial frame is a frame that is either at rest or moves with respect to absolute space with uniform translational velocity and no rotational component. Both special and general relativity theories require some changes in that concept. Classical mechanics defines also quasi- 48 inertial frames that are frames moving in space without any rotational component but their origins may have acceleration (Moritz and Mueller, 1987, p. 530). A nonrotating geocentric Cartesian coordinate system is an example of a quasi-inertial system, since its origin moves on an elliptic orbit around the Sun in a relatively weak gravitational field. Thus, in classical mechanics, and also in the special relativity as explained below, inertial and quasi-inertial frames are in some sense ‘privileged” frames. However, they lose this status according to the theory of general relativity. According to Moritz (1980), “... there are no inertial systems in general relativity. All possible coordinate systems are, in principle, equivalent; there are no privileged systems.” But the relativistic treatment of reference systems allows for some practical approximations to the inertial systems. In the theory of general relativity, Einstein defines the inertial system as a freely falling coordinate system in accordance with the local gravitational field which is due to all the matter in the universe. The special theory of relativity, on the other hand, defines the inertial frame not in absolute space; instead, it is defined in a space-continuum in the absence of the gravitational field (Seidelmann, 1992, p. 96). The local inertial systems are introduced for ‘&mall” regions (for instance our solar system) and global, nearly-inertial systems (harmonic coordinate systems) that are considered as a natural generalization of a classical inertial system to a curved space time manifold. Within the solar system, the harmonic coordinate system approximates an inertial system, and at infinity it reduces to an inertial system in the sense of the theory of special relativity (Moritz, 1980). The use of Euclidean geometry and classical mechanics in geodesy, geodynamics, or satellite dynamics is valid if an accuracy of about 1 part in 1 0 8 is required. For higher precision the special and general theories of relativity must be considered by applying relativistic corrections to the systems moving in space. Terms vV2 c and V/(? represent a special-relativistic correction and general-relativistic correction, respectively, where V is the classical Newtonian gravitational potential, c is the vacuum speed of light, and v is the velocity of the object (atomic clock). For 49 objects moving with the Earth around the Sun the effect of special relativity is of the order of 1 C8, similarly, the general relativistic correction due to the Sun’s gravitational potential effect at the Earth’s orbit reaches the order of 10 ‘8 (Moritz, 1980; Seidelmann, 1992, p. 96). The general line element is defined in general relativity as follows: (3.6) where V, c are as defined above, x,y,z - rectangular spatial coordinates, t - fourth coordinate - time (Coordinate time”, defined as time measured by a synchronized set of portable clocks or electromagnetic signals in the local inertial frame (Ashby and Allan, 1979) ). In the absence of the gravitational field, eq. (3.6) reduces to the special relativity form, represented by eq. (3.7) below (Moritz, 1980). ds2 ={dx2 +dy2 +dz2)-c 2dt2 (3.7) As noted, for the objects moving in the gravitational field, the time measured by their atomic clocks onboard will be affected by both gravitational red shift due to a constantly changing gravity potential (or gravitational potential if the attracting body is also rotating), and the Doppler effect due to the motion of the clock itself. According to Ashby and Allan (1979), relativistic time corrections associated with the spinning Earth may be as large as 200 ns. The reduction of the atomic time x, measured by a clock in motion, affected by a gravity field to coordinate time t, is given by (ibid.) : (3.8) 50 This reduction provides a uniform time scale that is not affected by motion or any gravitational irregularities. If the centrifugal force is present, i.e., the gravitation- producing body rotates, potential V must be augmented with the centrifugal potential in every equation presented above. The relativistic effect on the GPS observable and the time measured by a satellite’s clock was addressed in detail in Chapter n. In addition, the relativistic correction to the equations of motion of the GPS satellite is explained in Chapter IV. 3.5. Tim e System s There are three basic time systems that can be defined as follows: - rotational time {sidereal and universal (solar) times based on the diurnal rotation of the Earth that is not uniform), - dynamical time, defined by the motion of the celestial bodies in the Solar System; it is the independent variable in the equations of motion of dynamics, - atomic time, based on the electromagnetic oscillations produced by the quantum transitions of an atom with the basic unit being an atomic second, defined as the duration of9192631770 cycles of radiation corresponding to the transition between two hyperfine levels of the ground state of cesium 133 (Seidelmann, 1992, p. 40). A summary of the notation for time scales and related quantities is presented below: UT = UTI - universal time, counted from 0 hours at midnight; basic unit - mean solar day (interval between two consecutive transits of the mean sun over a meridian); not affected by polar motion, UTO - UTI not corrected for polar motion, UT2 - XJTl corrected for periodic variations in the rotational velocity of the Earth; still affected by the secular and irregular variations, 51 GMST - Greenwich Mean Sidereal Time (GMST) = Greenwich hour angle of mean vernal equinox; basic unit is the mean sidereal day, defined as the interval between two successive upper transits of the mean vernal equinox across some meridian, GAST - Greenwich Apparent Sidereal Time = Greenwich hour angle of true vernal equinox, TAI - International Atomic Time; unit is the SI second at mean sea level. The origin of TAI is defined by UTI-TAI = 0 on 1958 January 1. The instability of TAI is about six orders of magnitude smaller then that of UTI (Explanatory Supplement to IERS Bulletin A and B). UTC - Coordinated Universal Time; differs from TAI by an integral number of seconds, and is the basis for most radio time signals and legal time systems; unit is SI second, TDT - Terrestrial Dynamical Time, used as a time scale for ephemerides for observations from the Earth’s surface; TDT=TAI+32*.184, TDB - Barycentric Dynamical Time, used as time scale of ephemerides referred to the barycenter of the solar system, TCB - Barycentric Coordinate Time, dynamical time that replaced TDB, TCG - Geocentric Coordinate Time, dynamical time that replaced TDT. 3.5.1. Sidereal and Solar Time Systems Solar (universal) and sidereal time systems are directly related to the diurnal rotation of the Earth. Universal Time (UT) is given by equation (3.9) (Moritz and Mueller, 1987, p.402). UT = 12* + ht (3.9) where h, is the Greenwich hour angle of the fictitious (mean) sun. The solar and sidereal times are equivalent forms of time, related by the rigorous formula: 52 GMST = UT + a - Ylhours (3.10) where a, is the right ascension (RA) of the fictitious sun, and is expressed as follows: a t - GMSTfjWIX + Ylhours (3.11) GMST oh u t i is described by eq. (3.3). In some applications, it is convenient to express the epoch as an integer number of days and the fraction of day that elapsed since some reference epoch, usually in the past. The Julian day number is the number that denotes the day in such a continuous system in solar time. The Julian date (JD) commences with 0 Julian day number at 12 hours UT on January 1, 4713 BC. In addition, m odified Julian date (MJD) is usually defined as follows: MJD = J D - 2400000.5 (3.12) Both solar and sidereal times, as the equivalent time systems based on the diurnal rotation of the Earth about its axis, are equally affected by the irregularities in the rotation (see Chapter VI). Due to all variations in the rotational velocity the distinction should be made between universal time as observed by the terrestrial station (UTO) and universal time corrected for those irregularities. UTI is obtained from UTO by adding the correction for polar motion (since the meridian of observation is subject to the changes due to the pole variations) (Mueller, 1969, p. 164). And UT2 is UTI with the seasonal (mostly atmospheric in origin) variations removed. However, UT2 is still affected by secular variations; as yet it is the best approximation to the uniform time scale (see also Chapter VI). 53 3.5.2. Atomic Time System Atomic time is based on the electromagnetic oscillations produced by the quantum transition of a cesium 133 atom, as noted earlier. The unit of TAI is the SI second defined at mean sea level, and the main objective of TAI is to achieve and maintain the definition of a second at sea level. Since TAI is independent of the Earth’s rotation, the concept of Coordinated Universal Time (UTC), that is in some prescribed way connected to the rotational time, was introduced in 1961, taking advantage of the stability, predictability and almost immediate accessibility of TAI. UTC is based on the atomic second, thus its rate is uniform. Also, its epoch is manipulated accordingly so that the difference between UTI and UTC is maintained on a level less than or equal to 0.7 s. For that purpose UTC is modified by introducing a leap second, when required, e.g., on December 31 and/or June 30. As a result, UTC and TAI always differ by an integer number of seconds that can change only every year or one-half year. In January 1995 TAI- UTC = 29 seconds. The difference between TDT and TAI equals exactly 32.184 seconds (TDT=TAI+32\184). GPS uses its own time system that is based on the atomic time scale. The initial GPS epoch (week 0 ) is Oh UTC of January 6 , 1980. This corresponds to a Julian Date of 2444244.5 or a Modified Julian Date of 44244. Since UTC is altered as explained, the difference between UTC and GPS time grows. On the other hand, the difference between GPS time and TAI is constant and equals the difference between UTC and TAI at the GPS starting epoch, i.e., GPS time = TAI -19 seconds. 3.5.3. Dynamical Time Systems The unit of dynamical time is the atomic second. It is the time system used in celestial mechanics as the independent variable in the equations of motion for celestial objects. Due to the eccentricity of the Earth’s orbit, there exists the periodic variation between the time scales related to the geocenter and the barycenter (relativistic effect). Thus, the Barycentric Dynamical Time (TDB) and Terrestrial Dynamical Time (TDT) are used in the equations of motion with respect to the barycenter of the solar system and with respect to the geocenter, respectively. TDB = ZDr+0.001658sin£+0.000014 sin2^ (3.13) where g is the mean anomaly of the Earth around the Sun (McCarthy, 1992) and g = 357.° 53 + 0.° 98560028(JD- 2451545.0). (3.14) The strictly periodic difference between TDT and TDB introduces a scale difference between the geocentric and the barycentric frames in order to keep the speed of light unchanged in both frames. According to McCarthy (1992, p. 124; 1995, p. 72) the scale difference vanishes if Coordinate Time is used. The difference between TCB and TCG is defined by the four-dimensional space transformation as follows (ibid.): (3.15) where xt, v. - barycentric position and velocity of the Earth’s center of mass, x - barycentric position of the observer, - Newtonian potential of all the solar system bodies apart from the Earth evaluated at the geocenter, c - speed of light, to - 0 h 0 m 0 s TAI on January 1, 1977, t - TCB. The difference between TCG and TT (Terrestrial Time, considered equivalent to TDT) and TCB and TDB is given in seconds as follows: 55 TCG -TT= 6.96929019 x 10~* (M/D - 43144.0) • 86400 (3.16) TCB - TDB = 1.5505019748 x 10"* (M/D - 43144.0) • 86400 (3.17) CHAPTER IV MODELING OF THE MOTION OF THE EARTH ORBITING SPACECRAFT TERRESTRIAL SITE DYNAMICS 4.1. Equations o f Motion and Variational Equations o f the Earth Orbiting Satellite Trajectory generation is most commonly accomplished through the numerical integration of the orbital equations of motion defined by Newtonian mechanics and other forces acting on the satellite which can be modeled mathematically provided that the starting elements at an initial epoch are known. Equations of motion are usually expressed in terms of the rectangular components of the acceleration vector of the space vehicle and are referred to a geocentric inertial reference system. The general form of the equations of motion at the time epoch t in the geocentric inertial reference system takes the following form (Cappellari, 1976, p. 4-2): H t) = f(r,t,t,Ptolal) (4.1) where r{t) - the vector of accelerations of the satellite in the inertial coordinate system at the time t, r - column vector of vehicle position in the inertial coordinate system at the time t, r - column vector of vehicle velocity in the inertial coordinate system at the time t, P mat " vector of size n=m+l containing position and velocity of the spacecraft at epoch t0, and dynamic parameters of the model (i.e., gravitational harmonic 56 57 coefficients, aerodynamic drag coefficient, solar radiation pressure constant, 0 O - vector of dynamical parameters that are not estimated; size m, 0 - vector of size I of dynamic parameters, such as 0 = (r(f0), r (t0), 0 *); 1 corresponds to the number of model parameters that are estimated, r(f0) and t(t0)- position and velocity of the spacecraft at epoch t0, i.e., state at t0, 0*- in GODIVA, comprises three scaling factors to the solar radiation pressure: in X , Y (7-bias) and Z satellite body-fixed directions. Equation (4.1) can be rewritten as follows: r(0 = ~ 4 r(0 + A(0 (4.2) r where p. - the GM, where G is the gravitational constant and M is the mass of the Earth, and 4 r ( 0 represents the acceleration due to the spherical Earth potential. A(r) is a vector of accelerations induced by the following forces at time V. - nonspherical part of the Earth’s gravity field - the luni-solar and planetary gravitational attraction - solar radiation pressure - Earth radiation (albedo) - indirect oblation - solid earth, ocean and atmospheric tides - atmospheric drag (negligible for GPS satellites due to their high elevation, about 2 2 0 0 0 km.) - relativistic effect - pole tide - resonant acceleration 58 The sum of all the accelerations is numerically integrated in order to recover the position and the velocity of the satellite at subsequent epochs within the required time span. In addition, if the orbit is to be improved based on observations, the partial derivatives of the instantaneous orbital elements with respect to the initial state vector and any other unknown parameters are required in the differential correction process. These are known as variational partial derivatives and constitute a state transition matrix that is used in the least-squares adjustment called an orbit improvement process. Variational partial derivatives are obtained by the integration of the differential equations, called variational equations, of the form: d 2 d t1 d p d fi d p y «•’> where P is the vector containing the parameters defined above. Variational equations (4.3) can be represented as a system of the linear differential equations of the form: Yit) = A (0 • Y + B(r) • Y(0 + C(t) (4.4) where drjf) A (r) = £ L dr d r 3*3 B(f) = dr(t) d r dr 3*3 dr(t) dr(t) dr(t) c(0 = df~ f dp\ _ dr, ’ drt ’ dp * _ > w 3d explicit Y(t) = ^ Y (0= ^ 59 The variational equations are numerically integrated to obtain Y(t) and Y(t). The initial conditions for the variational equations are the following: A(/o) = [0]3x3 B(f0) = [0 ]3x3 (4.5) dir C(t0) = ®3*3>®3*3> ~df* where 0 3 x 3 is a 3x3 null matrix, d r columns of (1-6) explicit partial derivatives of acceleration with respect to the d p ' model parameters; I is the size of p vector, as defined before. 4.2. Numerical Integration o f Equations o f Motion and Variational Equations There are two basic methods for the numerical integration process: single-step and multistep techniques. The class of single-step integrators (e.g., Runge-Kutta) is very efficiently used as the starting scheme for the multistep integrators. The principal assumption is that the solution can be represented as a converging series, thus, in that sense it is similar to the Taylor expansion of the given function f(x,y) (Kelly, 1967). Within the class of multistep methods there are some degrees of freedom of choosing the one which fits the underlying purpose best. One can pick the technique which integrates the second- or higher-order equations directly or reduces them to a set of several first-order differential equations. The order of the process can also be suitably chosen depending on the application, accuracy and stability requirements. Generally, the higher-order formulae are more accurate but less stable (Krogh, 1969). Efficiency of the integration process dictates the use of the integration step size control for highly eccentric orbits which can be done analytically by changing the independent variable in the equation of motion (time regularization) or by numerical monitoring of local errors during the integration (Cappellari etal., 1976, p. 6-22). 4.2.1. Adams Type Predictor/Corrector Formulae The method employed in GODWA is the multistep predictor-corrector of the Adams type (Krogh, 1969). It integrates directly the second-order differential equations of the type given in (4.1). The major features of this integrator are reliability, efficiency, flexibility, low integration overhead, and small storage requirements (Krogh, 1969). The integrator is self-starting. The basic formulae resulting from that method are, generally speaking, of the Newtonian type derivable from the standard difference operator techniques. They are as follows: (4.6) where Is„ and Hsn are first and second inverse sums of X(t), denoted as V"1, V-2, respectively, which satisfy the following relationships: V' 1 X(t) - V" 1 X(t ~h) = X(t) (4.7) V-J X(t) -V~2X(t-h) = V 'l X(t) and the backward difference operator is defined as: (4.8) where a/, Pi - are coefficients given by recursive formulae, h - integration step size, 61 k - order of integration. The corrector formulae take the following form: (4.9) where the notation is analogous to (4.6). Predictor and corrector formulae can be used in several modes which differ slightly but the most efficient one seems to be the predict/evaluate/correct/pseudo correct scheme that is also very stable. 4.3. Total Perturbation Model The orbit improvement technique requires a prediction of a trajectory based on the dynamical model and the initial orbital elements, as already explained. Thus, the underlying force model is the important issue in the problem of orbit determination since the quality of the prediction is the limiting factor for the proper fit to the real data. The selection of the force model depends on the type of spacecraft and its orbital characteristics. For example, high-altitude satellites such as GPS, are not directly affected by the Earth’s atmosphere, thus this part of the total force can be omitted in the modeling of their motion. Similarly, the high-altitude satellites do not require the modeling of the gravity field beyond degree 8-12. The total acceleration including the central part (-GMZr3) of the Earth gravity field can be symbolically represented by equation (4.10) as follows: (4.10) 62 where notation is explained in Table 9 below (Rothacher, 1992, p. 48; Zhu and Groten, 1988). Table 9. Effect of the perturbing forces on GPS satellites Perturbing Force Notation in Eq. Acceleration in Effect on the (4.10) [m/P] Orbit After One Day [ml Repletion Term (Central Part ofthe Earth Gravity Field) K 0.59 00 Earth’s Oblateness (J£ Y 5.0xl0‘5 10000 NSB The Geopotential Harmonics (all the coefficients S,„* and C**, excluding JJ Y 3.0xl0"7 200 **NSB Third Body (Sun) 2.0X10"6 800 Third Body (Moon) X ^ 5.0X10-6 3000 Direct Solar Radiation it 6.0x10-® 200 SotRad Y-bias Y 0.8xl0*9 1.4 T-bUu Albedo (Earth’s Radiation) 4.0xl010 0.03 X jk Indirect Oblation K 2.0xl0'9 ? Earth Tides K l.OxlO"9 0.3 Ocean Tides %or 5.0xl0'10 0.04 Relativistic Effect 3.0xl0'10 ? 4.3.1. Noncentral Gravitational Potential o f The Earth The major portion of the perturbing force acting on the Earth-orbiting spacecraft is the nonspherical part of the Earth gravity field. In general, the geopotential is represented as a series of spherical harmonics of the form (McCarthy et al., 1991): GM ntntx n U 1 + I Z (4 .11) Jf=2 m- j • P f (sin cos mk + sinw/l] 63 where G,M,r as defined in Section 4.1, a , - the Earth’s mean equatorial radius, P„"(sini>) - associated Legendre functions of degree n and order m, C„m, Sm, - gravitational coefficients, A, The geopotential model applied currently in GODIVA is the degree 8 and order 8 portion of normalized GEM-T3 as recommended by IERS, where nominal values for C21 and S 2l coefficients are admitted according to IERS standards (McCarthy, 1992, p. 43). Their values are given in Chapter HI. The GM value reported with GEM-T3 is 398600.436 km 3/s2, and the semimajor axis of the reference ellipsoid, which equals 6378137 m, is used as a scale-defining parameter for the geopotential coefficients. _ 6 4 The nominal values of C2, and S n coefficients change due to the polar motion. In order to account for the offset of the conventional reference pole and the current location of the pole, the C 21 and S 2l should be computed as follows: C21 = —1.3 x lO"* ■ (xp) (4.13) S2l=-Uxl0*.(-y,) where xp=ml - x and yp=m2-y \ mi and m2 represent the instantaneous polar motion elements with units of arc seconds, x and y are the six-year mean values of the polar motion. The importance of proper handling of C2I and S2l coefficients is shown in Table 10 which displays some characteristics of the two solutions: the standard one with the full dynamic model, and the solution where C 21 and S 2l were set to zero (Test Solution in Table 10). The a posteriori standard deviation for the orbit solution increased less than 1% by ignoring the nonzero C2I and S21 but still can be observed. The a posteriori standard deviations per coordinate for satellites remained basically unchanged as shown in Table 10. However, the major effect is observed in the ERPs. They absorb most of the effect of the difference between the terrestrial frames in which the geopotential and the station coordinates are given if C 21 and S u are neglected. The difference between the respective xp and yp components from both solutions reaches 0.15 mas and 0.098 mas, respectively; the xp variation exceeds the formal uncertainty claimed by both solutions. 65 Table 10. Comparison of the solution characteristics between solutions with and without Cjt and S21 for January 15,1995 Solution Standard Solution Test Solution Difference Between Characteristics Solutions 1.203 1.207 0.004 < To Xp 1 2 9 .9 2 9 130.079 0.150 yB 4 5 3 .0 1 0 4 5 2 .9 1 2 -0 .0 9 8 0 .1 1 0 0 .1 1 0 0.000 0 .1 2 0 0 .1 2 0 0.000 C T c for PEN 1 0 .1 3 0 0 .1 3 0 0.000 tre for PEN 2 0 .0 9 0 0 .0 9 0 0.000 tXc for PEN 4 0 .1 1 0 0 .1 1 0 0.000 Notation in Table 10 is as follows: Another important issue that should be addressed here is the resonance effect that is caused by the fact that GPS satellites have a period that is equal almost exactly to a half of a sidereal day. That leads to the resonance effect caused by the second order sectorial and tesseral harmonics of the gravity field. The resonance effect that results in the magnification of perturbation in some of the orbital elements (i.e., orbit semimajor axis) can be utilized in the process of the resonant harmonic coefficient determination. In fact, Beutler e t al. (1994) report some preliminary results of C 32 and S 32 estimation during the GPS orbit improvement. The results are promising and indicate that GPS might also be a powerful tool contributing to the gravity field estimation. 4.3.2. Third-Bocty Attraction and the Indirect Oblation Effect Currently, GODIVA accommodates the perturbation coming from the point mass Sun and the Moon. According to Seeber (1993, p. 8 8 ) and Mlani et al. (1987, p. 17) planetary attraction is negligible. The largest contribution comes from the Venus (the acceleration equals roughly 1.2x1 O ' 8 m/s2). However, for orbit accuracy reaching the decimeter level it might become significant, and therefore it is applied by the majority of IGS Analysis Centers (IGS Questionnaires). The planetary attraction should be considered as a future enhancement in GODIVA. Geocentric positions of the Sun and the Moon in J2000 mean inertial frame are obtained from JPL Development Ephemeris DE200 and then transformed to the MoD inertial frame described in Chapter m. The acceleration arising from the point mass Sun and the Moon is expressed by the following equation: (4.14) where Mm and Ms - masses of the Moon and the Sun, respectively, r, i*m rs - geocentric position vectors of the spacecraft, the Moon and the Sun in the inertial frame, r, rM, rg - geocentric distances to the spacecraft, the Moon and the Sun. The assumption that the attracting bodies are point masses is the first-order approximation. In fact, the perturbing acceleration arises also from the nonspherical 67 portion of the gravitational potential of the attracting bodies. The resulting indirect acceleration of the spacecraft is equal and opposite to the acceleration experienced by the reference body, i.e., the Earth. It is represented by the following formula: (4.15) where Mb - mass of the Earth, X „(£) - acceleration of the point mass Moon due to the nonspherical Earth, Xs (A/) - acceleration of the point mass Earth due to the nonspherical Moon. A similar equation represents the indirect acceleration due to the mutual attraction between the Sun and the Earth. The indirect oblation effect is usually evaluated based on the second zonal harmonic for the Earth, the Sun and the Moon (Cappellari, 1976, pp. 4-18). The magnitude of the indirect oblation acceleration due to the Moon’s oblateness is of the order of 2.0 x 10 "9 m/s2 and 1.0 x 10-1J m/s 2 due to the Earth’s oblateness. These are not modeled in GODIVA due to their small effect on the satellite motion and also due to the fact that indirect oblation is not a part of IERS standards that GODIVA closely follows. 4.3.3. Direct Solar Radiation Pressure Solar radiation pressure is one of the perturbing forces that is very difficult to model for GPS due to the complexity of the shape of the spacecraft. The total acceleration due to the direct solar radiation effect can be expressed in the following form (Rothacher, 1992, p. 55): (4.16) where 68 v - eclipse factor: v = 0 if satellite is in the shadow (umbra), v = 1 if satellite is in the sunlight, 0 X y-m« - 7-bias acceleration. To account for all the complexities in the spacecraft’s shape and the constantly changing orientation of the body-fixed reference frame with respect to the inertial one, more sophisticated solar radiation models were developed for the GPS satellites. These are ROCK4 and ROCK42 models that are based on the conventional Rockwell International satellite model and accommodate the direct radiation for Block I and Block Il/HA satellites, respectively, according to Fliegel et al. (1992), Gallini and Fliegel (1995). Since the original models are rather complex, the simplified formulae for direct radiation including the thermal reradiation effect in Z and X directions are recommended by IERS and applied in GODIVA, as shown in the following section. 4.3.3.1. Solar Radiation Pressure Model for GPS Satellites: ROCK4 and ROCK42 The original ROCK4 and ROCK42 models are expressed in the satellite body frame where the Z axis points towards the geocenter along the antenna. The 7 component, pointing along one of the solar panel center beams, is defined by the normalized cross product of the Z vector and the unit vector from the spacecraft to the Sun. The positive X direction is located in the half plane that contains the Sun and completes a right-handed system (Figure 9). The spacecraft is maneuvered in such a way that the Sun is always in the spacecraft’s plane of symmetry and the angle between 69 the Sun and the spacecraft’s antenna falls between 0°and 180°. If that angle, called 27, is less than 14°, the satellite is eclipsed (Feltens, 1989). Figure 9. Satellite body-fixed reference frame Currently recommended models of the acceleration components in X and Z body-fixed directions are T10 for the Block I and T20 for the Block n. They are computed as a function of the angle B in the following way (McCarthy, 1995): X = -4.55 sini? + 0.08sin(2B + 0.9) - 0.06 cos(4B + 0.08) + 0.08 Z = -434 cos B + 020 sin(2£ - 03) - 0.03 sin(4£) (4‘ for Block I, and X = -856 sin 5 + 0.16 sin 32? + 0.1 Osin 52? - 0.07 sin IB (4.18) Z = -8.43 cos 5 70 for Block n. Units in the above equations are 10's N, and both X and Z components are given at the AU distance, so they have to be rescaled by multiplying by the ratio [AU/|r-r. | ] 2 and dividing by the mass of the spacecraft that is given in Fliegel et al. (1992). Angle B is in radians. The formulae (4.17) and (4.18) represent the approximation of the output from the original ROCK4 and ROCK42 with an uncertainty of 1.5%, whereas the original thermal model accounts for about 97-98% of the total observed solar radiation pressure effect. Since errors in the solar radiation manifest themselves as a scale problem, the scaling factors to the a priori accelerations are usually estimated in the orbit improvement process. In GODIVA, scaling factors are evaluated for both X and Z directions. The variations in these scaling factors are correlated with the eclipse season but in general they change very slowly, about 7% over a year (Rothacher, 1992, p. 57), thus they are usually computed once per day. The solar radiation force model for GPS Block HR is currently under development by The Aerospace Corporation. It will incorporate the Earthshine effect and the thermal and visible light effects providing an error of 1% in the total solar radiation effect evaluation (Gallini and Fliegel, 1995). 4.3.3.2. Acceleration Due to the 7-bias The along-track acceleration that is different for all the GPS satellites is produced by the force acting along the solar panel beams, i.e., in the body-fixed +7 or -7 direction. There are several possible reasons for the 7-bias. First, the solar panel center beams are neither perfectly straight nor exactly normal to the satellite’s body median plane. Next, the solar sensors are not perfectly aligned with the Z axis as defined above. Fliegel (1985) reports that misalignment of about 0.5° to 1° might count for that along-track acceleration component. A third possible reason for the 7-bias is the fact that heat generated in the spacecraft’s body is radiated mostly from louvers that are located on the 7-side for Block I satellites. Since the effect of all of these factors cannot be computed before launch time, the 7-bias has to be estimated as a part of the 71 model in the orbit improvement mode. GODIVA estimates the scaling factor to the a priori 7-bias equal to 10 "9 m/s2. 4.3.3.3. Effect of the Eclipse on the Modeling of the Motion of GPS Satellites The configuration Earth-Sun-satellite changes constantly with time due to the spacecraft’s revolution around the Earth and the Earth’s revolution around the Sun. For the GPS satellites the Sun crosses the orbital plane twice per year during its apparent revolution around the Earth (Feltens, 1989). This is called the eclipse season and lasts for 30-40 days depending on the orbital plane (Fliegel et al., 1992). The time of a single shadow event ranges from 0 to 57 minutes. During the eclipse season, the spacecraft is periodically in the Earth shadow, thus the solar radiation pressure changes dramatically when the satellite enters and exits the shadow. As was reported by Colombo (1986), the accuracy of the orbit determination is significantly degraded during the eclipse season. Some special effects associated with the shadow events are responsible for the deterioration of the accuracy of the orbit determination. These effects are: solar panel misalignment, solar panel radiation and Earth albedo. The total effect of the panel misalignment causes a velocity increment of 2 . 0 xlO -5 m/s that can result in large along track residuals. A second effect, solar panel radiation, is practically set to zero by ROCK4 and ROCK42 during the eclipse. In fact, the satellite "... continues to radiate heat after eclipse begins, and is abnormally cool after the eclipse ends” (Fliegel e ta l, 1992). This produces a small impulse away from the Sun upon the shadow entry and a small impulse toward the Sun after the end of the shadow period. In practice, since both effects are almost equal, but with opposite signs, they cancel each other and thus the thermal effect is considered negligible for the eclipse period. The third effect, the Earth albedo is reported to have the greatest impact on a spacecraft’s orbit during the eclipse seasons (ibid.). One of the most useful measures of the orbit quality is the difference between two overlapping orbits. During the eclipse, the three-dimensional RMS of fit ( 3DRMS), 72 defined by equation (4.19), shows significant increase not only for the eclipsing satellites, but also for those that are not in the shadow, since they are combined with the eclipsing satellites in a triple-differenced observables which introduces correlation among the satellites. 3 DBMS = JEMS2 +RMS2r + RMS\ (4.19) Notation in equation (4.19) is the following: RMSx, RMST, RMSZ denote the RMS for X, Y and Z coordinates, respectively, computed for the six-hour orbit overlap. Figure 10 shows 3DRMS for a six-hour overlap for PRN 1 and PRN 29 during GPS week 785. The number of the eclipsing satellites increased from four at the beginning to eight by the end of the week. PRNs 1 and 29 did not themselves eclipse, but their 3DRMS show dramatic growth when the number of eclipsing satellites increased. -^ 3 D R M S for PRN 1 -*-3D RM S for PRN 29 2 .1 22 21 2425 26 27 25 day< In January 1695 Figure 10.3DRMS for 6 -hour orbit overlap for PRN 1 and 29 Figure 11 shows the behavior of 3DRMS for the satellites that experienced the shadow event during that week. PRN 16 entered the eclipse season on January 26, 1995, and consequently the quality of the overlap between days 25 and 26 declined 73 with respect to the former days. PRN 22 was in the eclipse season for the entire week, and its 3DRMS increased together with the number of spacecraft experiencing the shadow event. — 3DRMS for PRN 16 — 3DRMS for PRN 22 22 22 24 2S 24 27 21 d*y* In January 1605 Figure 11.3 D R M S for 6-hour orbit overlap for PRN 16 and 22 In order to properly account for the eclipse effect on the force acting on the spacecraft, respective scaling of the solar radiation pressure based on the eclipse model must be implemented. The two most commonly used models for the Earth’s shadow are cylindrical and conical. They are illustrated in Figures 12 and 13, respectively. For the conical model the condition for the eclipse is the following: r * r v = 0 if cos a = — L < 0 and A = r • v l-co s 2 a < aE r r , v = 1 otherwise, (420) where as is the equatorial radius of the Earth and v is the eclipse factor that scales the solar radiation pressure and 7-bias accelerations. The conical model is shown in Figure 13, where the penumbra (partial shadow) extends outward behind the terminator plane 74 and the umbra (total shadow) is a conical region collapsing inward behind the spherical Earth. EARTH SUN ...... Figure 12. Cylindrical model for the Earth’s shadow The condition for the eclipse in that case is the following (Casali, 1992): for the umbra: 0 M < abs(aE-a t) (4.21) for the penumbra: abs(aB - a s) < < (a E + a s) (4.22) where 0s e is the angle between the r and rs vectors. Angles as and as are as shown in Figure 13. These conditions are applied in GODIVA to detect if a satellite is in sunlight, full or partial shadow. 75 •v. •s”' Tyvs« Vr >, > > < v >»»>»»■>» »»»»i»i»SS5 SATELLITE Figure 13. Conical model for the Earth’s shadow The proper sampling of the shadow is a very important factor for accurate orbit integration by numerical methods. The integration step size must be small enough to sample the penumbra adequately. A typical integration step size equals 1/100 of the orbital revolution or about 6-7 minutes for GPS satellites. If a variable mesh integrator is used, the step size is usually reduced automatically upon the penumbra boundary crossing. However, the penumbra time can be too short to allow the integrator to account for the partial shadowing effect. This is most likely to happen during the first and the last day of the eclipse season when the penumbra time may be shorter than the typical integration step size. Also, the shadow boundary crossing introduces a discontinuity in the force model, and thus violates the assumption about the characteristics of the force that has to change smoothly and continuously with time. The solar radiation pressure force varies as a function of the amount of the shadowing. If the step size is small enough, the total force can be assumed as changing smoothly with time provided that the conical shadow model is used. However, it may happen that the integrator will not determine the starting epoch of the apparent discontinuity with submillisecond accuracy that is required for the precise orbit determination (Beutler, 1995, private communication, see also Beutler et al., 1994). In order to avoid such a mistiming and to accommodate properly for the shadow event, the times of the shadow boundary crossing should be computed and passed to the integrator as the starting and ending points for the ‘shadowed’ and ‘sunlight’ arcs. These times are used to divide the entire arc into segments enclosing the ‘shadow’ and ‘sunlight’ parts that are integrated separately with different step sizes. In essence, the integrator is restarted at every boundary crossing using the last orbit determination from the former segment as starting elements for the subsequent arc. This method is implemented in GODIVA. First, the angle B, whose magnitude indicates the eclipse season, is computed based on the initial elements for every orbit and the geocentric position of the Sun. It changes roughly 1° per day. As a first approximation, the Keplerian orbit representation is used to predict the position of the spacecraft and compute the eclipse factor v at one-minute intervals for the orbits with B less than 14°. The adjacent eclipse factors are compared in order to determine the sunlight/shadow crossing. The quadratic approximation to the orbital arc where the shadow crossing occurs is applied as described below. The information about the shadow entry and exit times is passed to the numerical integrator. Orbit prediction is initiated with the step size of 900 s. At entry to the penumbra, known from the preprocessing step described above, the integration is restarted with the smaller step size. This step size is maintained during the eclipse event. After it ends the integration is restarted again with the original step size. The Keplerian approximation does not, however, provide times that are accurate enough. It has been found that the typical mistiming of the boundary crossing ranges between 0 - 2 0 seconds. Consequently, in order to obtain submillisecond accuracy, the times of the shadow events must be updated during the orbit improvement. Thus, a timetable containing the information about shadow boundary crossing times is updated every iteration together with the improved orbit. The determination of the location of the eclipse boundaries is 77 done according to Anderle (1973) as described by Casali (1992). It is assumed that three equally-spaced-in-time position vectors of the satellite are known at times tj, t2 and t3. The boundary crossing occurs between ti and t3. In addition, geocentric positions of the satellite and the Sun can be computed at any instant between tj and t3. For the conical model, the boundary crossing occurs when the function described by equation (4.23) equals zero. F=Qa - a B-yaa (4.23) where / = 1 if a sunlight/penumbra boundary is sought, and / = - 1 for an umbra/penumbra crossing. Notation in (4.23) is the same as for equation (4.22). In order to estimate when F=0, a parabolic approximation to the function (4.23) is formed using the values of F at times ti, t2 and t3. The crossing time tc is found by solving the quadratic equation; tc = f, - St where St is the root of the quadratic approximating function F, having smaller absolute value. This procedure is iterated with the smaller separation between tj, t2 and t3 until convergence is reached. 4.3.3.4. Spacecraft Attitude Model The attitude control system is used to establish and maintain a stable orientation of the spacecraft during flight. In order to keep the proper orientation (i.e., keep the Z axis pointing to the geocenter) the satellite rotates once per revolution about an axis that is perpendicular to the orbital plane (Rothacher, 1992, p. 61). However, due to the torques resulting from the perturbing forces this orientation cannot be maintained without an active control mechanism. Mismodeling of the spacecraft’s attitude can lead to decimeter-level error in precise geodetic applications (Bar-Sever, 199S). Also, during the shadow event, the satellite can rotate randomly even up to a full turn as a consequence of the design of the GPS attitude control system (ACS) that uses Sun sensors to determine the yaw attitude. During the eclipse, the signal from the Sun sensors vanishes and .. the output from the sensors is essentially zero and the ACS is driven in an open loop mode by the noise in the system,” thus the yaw becomes undetermined (Bar-Sever et al., 1995). Therefore, the control over the satellite’s orientation is very important since the random yaw attitude, especially during the eclipse, introduces significant kinematic and dynamic mismodeling (Wu et al., 1993; Bar-Sever, 1994). From the dynamical point of view, the solar radiation pressure effect is strongly correlated with the satellite’s orientation in terms of magnitude and direction. Kinematically, it is important for two reasons: first, the unmodeled yaw angle (i.e., the angle between the spacecraft-fixed X axis and the orbital plane along the general direction of the spacecraft’s velocity vector, restricted to be between -180° and 180°) affects the location of the transmitter antenna phase center, since it is located about 2 0 cm from the satellite’s rotation axis. It results in errors in both pseudorange and phaserange that can reach a level of 8 cm. The second effect which is a function of the relative orientation of a transmitter-receiver pair is the mismodeling of the phase wind-up. In general, any mismodeled rotation of a satellite’s antenna will be interpreted by a receiver as a change in a tracking phase and consequently the measured p haserange will be erratic. As a result of the mismodeled phase wind-up, the error in measured phase-range can be as large as 10 cm for some receivers (Bar-Sever et al., 1995). Since the change in the observed phase is a function of the orientation of the transmitter and receiver antennas and also a direction of the line of sight, the effect of the mismodeled location of the transmitter’s antenna does not cancel by differencing of the phases, especially over the long baselines (Wu et al., 1993). On the other hand, the wind-up mismodeling does cancel when single differencing (between stations) is performed, since both stations should observe the same wind-up effect. The solution to the attitude problem of the eclipsing satellites as proposed by J. Anselmi (Bar-Sever et al., 1995), is to bias the yaw angle by the small but known amount of 0.5°, so that during the shadow event this bias will dominate the noise and the yaw angle will be known. More details about the new yaw-attitude model implementation and its effect on the precise orbit determination are given by Bar-Sever et al. (1995) and Bar-Sever 79 (1994,1995). As reported by Bar-Sever et al. (1995) the application of the new attitude model improved the solution for all eclipsing and noneclipsing satellites. The implementation of the new ACS should be considered as further improvement to GODIVA. 4.3.4. Earth Radiation (Albedo) A portion of the solar radiation is reflected back into space by the Earth’s surface and its atmosphere. The ratio of reflected radiation to the incoming solar flux is called albedo. The force resulting form the reflected light is called Earth albedo force or Eartkshine and can reach the level of 1.6% of the nominal solar radiation pressure (Gallini and Fliegel, 1995). At every epoch, satellite acceleration due to albedo is computed as a sum over all the Earth surface elements “seen” by the satellite that reflect solar radiation (Tapley, 1989). According to King et al. (1985), the magnitude of the perturbing acceleration due to the Earth albedo at GPS altitude is approximately equal to 4.0 x 1 0"19 m/s2. Since the orbital error arising from its omission is of the order of 3 cm after one day, it can be important when decimeter level accuracy is approached. The albedo radiation pressure modeling for GPS satellites is described in detail by Beutler et al. (1994). However, these authors did not report any significant improvement in the solution after implementation of the albedo model due to the strong correlation of the albedo model parameters and the solar radiation model parameters. The correlation is so strong that it makes the separation among those parameters practically impossible. 4.3.5. Solid,Earth Tides The gravitational attraction of the Sun and the Moon, as described in section 4.3.2., represents a direct component of the total effect on the motion of the spacecraft due to that source. The indirect effect is caused by the tidal deformation of the nonrigid Earth induced by the gravitational attraction of the Sun and the Moon that results in the change in the geopotential. Variations in the geopotential are most easily modeled by corrections to the standard coefficients Cm and S™ (McCarthy, 1992, Chapter VII). The two-step procedure, as recommended by IERS, is implemented in GODIVA. The first step uses the frequency-independent Love number k2; in the second step the corrections due to the variable Love number are computed. In the first step, second- degree normalized coefficient corrections are computed as follows: (4.24) (4.25) (4.26) where k2 - nominal Love number, aB - equatorial radius of the Earth, GMe, GMj - gravitational constant multiplied by the mass of the Earth, the Moon (j=2) and the Sim (j=3), respectively, rj - distance form geocenter to the Sun or Moon, body-fixed geocentric latitude and longitude of Moon or Sun. The second step corrections to the normalized coefficients are: fl+ja f 1 \ M R (4.27) where 81 5k, - difference between the frequency-dependent Love number from Wahr’s model and the nominal Love number, H, - amplitude of the term at frequency s of the harmonic expansion of the tide generating potential, O,=n-fi=in,0, (4.29) J -l n - vector of sue multipliers of the Doodson variables, 0, - the Doodson variables (i.e., mean lunar time, Moon’s mean longitude, Sun’s mean longitude, longitude of the lunar mean perigee, negative longitude of the mean lunar ascending node, longitude of solar mean perigee). IERS standards recommend the use of six diurnal terms to modify C21 and S2l and two semidiurnal terms modifying Cu and S21. If the zero frequency term (the permanent tide) is included in the adopted C20, it should be removed from (4.24) before applying it to the nominal coefficient. The zero frequency tidal correction to C20 can be computed from the following equation: (AC2B) = A0H0k2 = -1.39119 x Kr**2 (4.30) where the notation is as explained above. Since GEM-T3 value of C20 does not comprise the permanent tidal deformation, it should be restored together with the variable part. 4.3.6. Ocean and Atmospheric Tides The effect of the ocean tides on the geopotential is implemented as a periodic variation to the coefficients of the gravity field. The model that is recommended by 82 IERS and applied in GODIVA is from Schwiderski (1983a, 1983b) and is formulated in the following way (McCarthy, 1992, Chapter VII): AC - /AS. (4.31) where \ + kn (4.32) g - 9.798261 m/s2, G - the gravitational constant, Pv, - density of sea water = 1025 kg m'3, k'n - load deformation coefficients of degree n, (ibid.); they allow for the deformation of the Earth under the variable ocean load, C l , N2, M2, S2, K2 . These tides are expressed at any point in the world’s ocean by an amplitude and a phase. Thus, the tidal effect can be computed as the sum of all eleven tidal frequencies for which the amplitudes and the phases are given on 1° x 1° grid intervals in latitude and longitude. The summation symbol implies addition of the expression using the top signs (the prograde waves C l , 5 1 ) to that using bottom signs (the retrograte waves C l , 5 l) . For most precise applications, the coefficient 83 with n=2 and m=2 for the S 2 tidal argument can be modified in order to account for the atmospheric tide as follows: C£= -0.537 cm and S22 =0.321 cm (4.33) GODJVA currently has the ability to accommodate this correction. 4.3.7. Relativistic Acceleration The relativistic acceleration of the Earth-orbiting satellite due to Earth’s gravity field is described as follows according to the theory of general relativity (McCarthy, 1992, p. 123): (4.34) where r and r are the geocentric position and velocity vectors of the satellite, G and M are the gravitational constant and the mass of the Earth, respectively, and c is the speed of light in vacuum. The above model, following the 1ERS standards is implemented in GODIVA. In equation (4.34) only the point mass effect is included. At GPS altitude the J2 and the higher-order terms effect can be omitted. For the effect of the quadruple (J 2) effect see (Soffel et al. , 1988; Soffel, 1989, p. 96). 4.3.8. Resonant Acceleration Colombo (1989) recognizes that errors in the GPS ephemerides are mostly due to the resonance that occurs between the orbit and the perturbing forces. The assumption is that the unmodeled small forces acting on the GPS spacecraft are periodic in nature with the major period equaling the orbital revolution. For example, the resonant effect arising from the gravitational attraction comes from the harmonics 84 whose orders are multipliers of the number of orbital revolutions per day. Similarly, the solar radiation or 7-bias can resonate with the orbit as a consequence of the orientation of the solar panels and the spacecraft’s body with respect to the Earth and the Sun that is approximately repeated in every revolution. Since only the small constituents of the total perturbing forces are considered, first-order perturbation theory is used to study their influence on the orbit. The linearized form of Hill’s equations is used (equation 4.35). Their solution provides the errors in the acceleration in the radial, across- and along-track directions. Ax = Ax + Aax Ay = -2w0 Ai + Aay (4.35) Az = 3w#2 At+2n0Ay + Aa, where n0 - the mean orbital frequency of the spacecraft, x,y,z- designate the across-track, along-track and radial directions, respectively, Aax, Aay, Aaz - errors in the acceleration in x, y and z directions, respectively. Solution of the equations (4.35) is developed by Colombo (1989) and presented in the form of Fourier transforms of the acceleration errors Aax, Aay, Aat. The sums of the Fourier components of the across-track, along-track and radial resonant accelerations is written as follows: Aaxt = AXj cosnct + BXf sin n0i + CH (4.36) where i = 1,2,3 for x, y, z and AXt, BXi and CXi are parameters that are estimated during the orbit adjustment process. Currently, GODIVA does not implement Colombo’s model but it could be considered as a further improvement of the orbital dynamic modeling. 4.3.9. Orbital Maneuvers Orbital maneuvers represent one of the operational modes of the satellite’s attitude control system established to maintain a stable orientation of the satellite (see Section 4.3.3.3). The orbital parameters of the satellites change constantly due to the perturbing forces, so during the midcourse correcting maneuvers the attitude control system changes the orientation (and also position) of the satellite in order to attain predetermined (nominal) conditions. The attitude control mechanisms generate low- thrust forces that accelerate the satellite by performing multiple propulsive maneuvers to bring it to the desired orientation and orbital placement. The maneuver, called a momentum dump, can occur even once per week, if necessary, and its time is given in the broadcast message in the TLM word (Rothacher, 1992, p. 62). Momentum dumps are needed to decelerate the momentum wheels that are a part of the satellite’s control mechanism and are done by firing the thrust boosters. The station keeping maneuvers are performed from time to time and are announced by the U.S. Coast Guard. Delta-V maneuvers are the most significant and the largest maneuvers. They are related to the relocation of the space vehicles within a particular orbit. It is obvious that every change in the satellite’s location and/or orientation should be properly accommodated in the modeling of its motion. One of the options is to introduce the new velocity (energy) components at the time of the maneuver that represent the maneuver components as described by Lichten and Bertiger (1989). The currently practiced procedure is the resetting of the satellite state vector, including the solar radiation pressure parameters, at the times of orbital maneuvers or any modeling problems that may occur (Beutler et al ., 1995; Kouba, 1995). The solar radiation pressure parameters have to be reset since the area-to-mass ratio changes for the satellite after the maneuver due to the loss of the thruster’s fuel mass. The example below shows the significance of proper modeling. The modeling problems for PRN 23 occurred on July 4, 0 h UT, 1995, due to the maneuver. The solution provided by GODIVA for that day, including PRN 23, shows an RMS of fit to the IGS orbit of 25 cm. After the removal of this satellite from the data set, the RMS of fit dropped to 17 cm! GODIVA does not currently have a capability to properly handle maneuvers (i.e., resetting of the state vector). It should be considered as a future enhancement to the software. 4.4. Terrestrial Site Dynamics 4.4.1. Station Displacement Due to the Solid Earth Tides The effect of the solid Earth tides on the terrestrial site is computed in GODIVA according to McCarthy (1992, p. 56) applying a two-step procedure. For a precision of 1 cm, only the second-degree tides are accommodated. The first step correction, based on the nominal Love and Shida numbers, is given as follows: 3 4 GM, -r 2 A r = 2 J (4.37) PiGMb ‘Rj 2 where GMj, GMb - gravitational parameters for the Moon (j=2), for the Sun (j=3) and for the Earth, Rj - unit geocentric vector from the Sun or the Moon, r - unit geocentric vector to the station, h2 = 0.609 and l2 - 0.0852 - nominal second-degree Love and Shida numbers. In the second step, only the Ki tidal frequency, with the corresponding Love number equal to 0.5203, needs to be used. And for sufficient accuracy only the radial displacement of the station should be corrected as follows: j 3 si n p cos p sin(G4Sr + x + X) (4.38) 87 Using the appropriate values for the amplitude HXl of the Ki term and the difference between the frequency dependent and nominal Love numbers, ShKx, equation (4.38) can be rewritten in the following way: dhm = -0.0253 sin^?cos 0 > sin (G ^ J+ X) (4.39) where q> and X denote the geocentric latitude and longitude of the station. According to Lambeck (1988, p.259) the variation in the terrestrial site position is predominantly radial with the maximum displacement of about 16 cm every 1 2 hours (for the dominant M 2 tide). The tangential component (orthogonal to the meridian plane) is only 4 cm. Equation (4.37), describing the first step correction, contains the zero frequency term. Thus, if the given coordinates of the station contain the permanent tide, it should be subtracted from the radial and north components of the terrestrial station location. The permanent tide in the radial and north directions in the local topocentric system are described by the following equations: Ar • r = -0.12083^ sin 2 Ar • er = -0.05071cos^ sin Table 11. Selected characteristics of the solutions with and without the tidal correction to the terrestrial sites’ coordinates for January 15,1995 Solution Characteristics Standard Solution Test Solution Difference Between Solutions (T o 1.203 1.542 0.339 X. 129.929 130.235 0.306 Vo 453.010 453.116 0.106 0.110 0.140 0.030 0.120 0.150 0.030 CTe fo r PRN 1 0.130 0.160 0.030 Notation in Table 11 follows the notation in Table 10. 4.4.2. Station Displacement Due to the Pole Tide Pole tide is caused by the perturbing centrifugal potential that changes periodically due to the variations in the pole location with the period of more than a year (Chandler period) (Moritz and Mueller, 1987, pp. 285, 521; see also Wahr, 1987). Variable centrifugal acceleration is a consequence of the permanent variability in the location of the Earth’s principal axis of inertia. The accommodation for the variation in the station coordinates caused by the pole tide is recommended by IERS standards in the following form: Sr = -32 sin20 • ( xp cos A - y p sin A) SB = -9cos20 • (x, cosA - y p sinA) (4.42) SA= 9 cosO • sin X+yp cos A) The units in the equation above are in millimeters, 0 is a colatitude, xp and yp are pole coordinates in seconds of arc. Sr defines the radial displacement, S& and Sx are the horizontal displacements defined in the local horizon system, where positive directions are upwards, south and east, respectively. These corrections are currently applied in GODIVA 4.4.3. Station Displacement Due to the Ocean and Atmospheric Tides The local site displacement due to ocean loading can be computed at time t in radial, East-West and North-South directions as the sum of the contributions of individual ocean tide components, according to IERS standards (Seidelmann, 1992, p. 244). Application of this correction is necessary when millimeter level accuracy is required, especially for the stations close to the coast line. The effect on the VLBI baseline determination could reach several millimeters of seasonal variations according to Seidelmann (1992, p. 248). The atmospheric loading effect can be computed in the vertical direction as a function of the local pressure anomaly (McCarthy, 1995, p. 109). These corrections are currently not available in GODIVA. CHAPTER V ORBIT AND RELATED PARAMETER ESTIMATION USING THE BATCH LEAST-SQUARES TECHNIQUE WITH TRIPLE-DIFFERENCED PHASES S.l. Introduction In this study, the batch least-squares estimator is applied to determine the best estimate of all the parameters. Since the triple-difference observation equation that represents the functional relationship between the observed quantity and the unknown parameters of the model is nonlinear, its linearization and the application of an iterative scheme to the related system of the normal equations is required. In this chapter, the observation equation for the ion-free triple-difference is presented, together with the related partial derivatives that constitute its linear form. Also, the theoretical justification of the triple-difference implementation in the orbit determination is addressed. The decorrelation scheme of the triple-differenced phases is explained in detail as an essential component of the technique presented here. In addition, the estimation process of the GPS trajectories and other geodetic parameters is described. Finally, the state-of-the-art of the orbit determination problem is discussed based on the results and the activity reports presented by IGS and participating centers. 90 91 5.2. The Application o f the Triple-Difference to GPS Orbit Determination The concept of triple differencing itself is not new but it has never been applied to the problem of GPS orbit estimation, as presented here. The triple-differenced phase observable was used for the first time by Goad and Remondi (1984) in the baseline determination problem in order to obtain good starting conditions for the single- and double-difference approach. Also, Eren (1986, 1987) reports a successful implementation of the triple difference phases in the terrestrial network adjustment, but he provides no reference to the problem of observation decorrelation. Since the classical approach to orbit determination utilizes only the undifferenced or double differenced phases, the technique presented here shows departure from previous practice. However, as explained below, special attention has to be given to appropriate modeling of the triple-difference covariance matrix to accommodate properly the mathematical correlation resulting from the differencing scheme when switching from double- to triple-differenced phases. Given a set of double differences over N epochs, one can always create a set of equivalent observations comprising the double differenced phases at epoch one and all the triple differences created by the subsequent differencing of the double differences over the time epochs, as explained in Chapter n and presented in Table 12. Table 12. Double-differences and the respective set of equivalent observations for the integer ambiguity solution Double Differences Over N Epochs Equivalent Set of Observations Y, Y, y 2 y 2- y , y 3 y 3- y 2 • y 4- y 3 • y n., yn yn - y n_, In general, the phase ambiguity is estimated either as a real number in a float solution, or as an integer value that must be recovered using an integer resolution technique. The integer solution provides the highest accuracy since the precise double difference phase ranges are characterized by only a millimeter level noise. In general, the ambiguities associated with baselines shorter than a few hundred kilometers in length are easier to resolve as integers than for the longer baselines. However, in orbit determination, very long baselines of at least several thousand kilometers contribute primarily to the establishment of the orbital control that makes the problem of integer resolution very difficult or virtually impossible as reported by Dong and Bock (1989). As a result, all the IGS analysis centers apply a float solution to the estimation of the ambiguity biases (IGS Questionnaires in IGS Electronic Mail 654-657 and 695). But in the approach presented here, the main goal was to avoid entirely the estimation of the ambiguities in order to reduce both the processing time related to data editing and the cycle slip fixing, and the size of the normal matrix. In such a case the double differences Yi, containing the information about ambiguities should be removed from the data set presented in the second column of Table 12. The new equivalent data sets are presented in Table 13. Table 13. Double-differences and the respective set of equivalent triple-differences for the ambiguity float solution Double Differences With Float Equivalent Set of Triple Difference Ambiguity Over N Epochs Observations Yt Y2-Y! y 2 y 3- y 2 y 3 y <-y 3 • Y n -i yn Y n -Y k., 93 As Schaffrin and Grafarend (1986) have proved, elimination of the nuisance parameter vector rj from the partitioned linear Gauss-Markov model described as follows: £{Y} = A£ + B7 , D{Y} = P 1cr2 (5.1) leads to the system that provides the least-squares solution for £ identical to the estimates obtained from the original system, under the condition that the covariance matrix for the reduced system is modeled properly. A and B are design matrices, such that rk(A) + rk(B) = rk[A,B], thus column spaces for A and B are complimentary, so that separability of both groups of nonstochastic unknown vectors £ and rj is assured. The reduced system is obtained by finding a «x(n-rk(B)) matrix R of maximum column rank such that: R rB = 0 and r*(R)+r*(B) = n (5.2) where n is the number of rows in A and B and 0 denotes a ‘zero’ matrix. Thus the new, R -transformed Gauss-Markov model can be characterized as follows: £ { r t y} = R rA£ andD[RT Y} = R 'P 'R a ’ (5.3) The bias vector is not present in the system, and the dispersion matrix of the new model is also R-transformed according to the law of error propagation. Schaffrin and Grafarend analyze the example of GPS phase observation equations that are further differenced, providing the system of double and then triple differences, so that the nuisance parameters vanish. They proved that the subsequent use of a differencing operator of the form: 94 '-1 1 0 . . . 0 o' 0 -1 1 . . . 0 0 (5.4) 0 0 0 . . . -1 1 provides the required transformation matrix R , such that condition (5.2) holds and system (5.1) is reduced to (5.3). The structure of the matrix A describes the scheme of differencing among epochs, transmitters and receivers, applied consecutively to obtain the triple-difference observation equation. The same transformation matrix is applied to the (diagonal) covariance matrix of the one-way phases to determine the covariance matrix for triple differences in order to fulfill (5.3) and preserve the equivalence of the original and transformed models. The approach explained above is valid only when the ambiguity bias is estimated as a real-valued quantity in the float solution of the original, non-differenced system. The reduced systems, i.e., single-, double- and triple-differenced will generate the same least-squares estimates of the vector £ if the mathematical correlations among measurements at all differencing levels are properly accommodated. However, it should be emphasized that when the double-difference solution provides the integer estimates of the ambiguity biases, the system no longer represents the linear Gauss-Markov model. Instead, it is a type of constrained Gauss-Markov model and the equivalence between the original and the subsequently differenced systems vanishes. The approach presented here meets all of the equivalency requirements, thus the triple difference solution preserves the parity with the undifferenced or double-differenced results provided by the IGS centers. The obvious advantage of the triple difference application is having no need for human interaction on cycle slip detection and correction. Rather, in triple differencing the cycle slips are treated as data outliers and are detected, and consequently eliminated from the data set during the adjustment. Every cycle slip or bad quality phase is seen in the triple difference data as a large residual in the system of the observation equations. 95 The rejection level currently applied in GODIVA is 20 cm for triple difference; the sampling interval equals 900 s. Typically observed is an average rejection level between 600 and 800 measurements for a 32-hour data set, collected at 36 globally distributed stations above the cutoff angle of 16°. This approach assures no risk related to the undetected or erratically corrected cycle slips that may introduce systematic offsets into the phase measurements. Ultimately, all the cycle slips and other significant variations or biases in the observations that are atmospheric in origin or multipath- or clock- related (receiver and satellite) are removed from the data set at the level of the attained convergence. The average number of unknown ambiguity biases associated with a one-day data set collected by about 40 stations ranges between 260 and 500 (G. Gendt, private communication, 1995). These numbers were obtained from the analysis of one week of data (days of year: 57-63) from the IGS center GeoforschungsZentrum (GFZ), located in Potsdam, Germany. The upper limits are reached since the implementation of Anti-Spoofing that has been activated since January 1994. The dramatic increase in the number of cycle slips contained in GPS phase data is also reported by Rothacher et al. (1994) as the result of the influence of AS on the tracking loop of civilian GPS receivers, which results in a degraded quality of the pseudoranges. Based on the numbers presented above, it can be concluded that roughly only about 50% of our rejections are associated with outliers. The rest are cycle slips or losses of lock that normally introduce the new unknown ambiguity. Since the technique presented here eliminates a class of nuisance parameters, the number of columns in the triple-difference normal matrix will be on average reduced by 300 to 500 per day with respect to the solution based on undifferenced or double-differenced ionospheric free combination of phases. On the other hand, the price one has to pay when the triple differences are utilized is the more complicated form of observation covariance matrix than for undifferenced or double-differenced observables. Therefore, the most numerically challenging task is the proper handling of the correlation between two 96 consecutive triple differences that share a common epoch. This problem is addressed in sections 5.3 and 5.4 and follows the notation of Goad etal. (1995). 5.3. Cholesky Decomposition o f the Covariance Matrix Since triple-differenced observations belonging to consecutive epochs are correlated by sharing the same epoch via differencing, the covariance matrix for this method is not diagonal or even block-diagonal as for the undifferenced, single- or double-differenced phases. It is, however, a banded matrix with a profile that depends on the manner in which the triple differences are built. Its inverse could possibly be a full matrix. Due to numerous observations involved in the problem of orbit determination with triple differences, the size of the covariance matrix for observations is very large. Thus, its size and form make it difficult or even impossible to handle by standard matrix inversion procedures. Therefore, a convenient decorrelation scheme, employing the Cholesky factorization of the covariance matrix is applied instead. It replaces the need for the inverse of the covariance matrix, utilizing a simple forward substitution scheme for the solution of the system of the linear equations of the form (5.12) as given below. In that case, the maximum size of the matrix that needs to be loaded in the computer memory is equivalent to the sum of the number of observations for two neighboring epochs. Solution of the system (5.12) provides uncorrelated observations, with the covariance matrix transformed to an identity. Denoting the covariance matrix of triple differences by 2, its Cholesky decomposition is given as follows: 2=LLr (5.5) where L is the lower triangular matrix with positive diagonal elements (Cholesky factor of 2 ), thus: E_1 = (LLr )_l = ( l-1)T L 1 (5.6) Writing the system of normal equations for a Gauss-Markov model we have: 97 (A7 ! : 1 A )| = Ar r 1Y (5.7) where A - design matrix, Y - vector of measurements, £ - vector of unknown parameters. Substituting (5.7) in (5.6) above one gets: [a t (L-1)7 L-1a]^ = JjL^Aj^L'AjJ^ = Ar (L~1)JL"1Y (5.8) and finally: {[L‘‘A]r[L‘'A] ^ = [L-'A]rL-‘V (5.9) The new, decorrelated system of normal equations is written as follows: ArA£ = ArY (5.10) where A = L-1A and Y = L 1 Y (5.11) Even though L '1 is present in the equation above, all one needs to obtain is the Cholesky factor L to find whitened (decorrelated) A and Y matrices. Equations (5.11) can be rewritten as follows: LA = A and LY = Y, where (5.12) L, A and Yare known matrices. Thus A and Y can be successfully recovered by forward substitution. In practice, the columnwise vectorized augmented matrix [ A | Y] is used, together with similarly vectorized Cholesky factor, L. 98 5.4. Recursive Decorrelation Scheme and the Final Solution The process of both Cholesky decomposition and the decorrelation is done recursively. In general, in order to decorrelate observations at epoch i+1, all data pertaining to epochs i and i+1 must be present in memory at the same time. What is needed are two subblocks A, and Ai+I of the design matrix, augmented by the Y, and Y/+/ columns, and the lower triangular block of the covariance matrix for epoch i+1, that is 2/+/, together with a cross-correlation 2/+;,/ between both epochs. Also, the Cholesky factor of the covariance matrix for epoch / that was obtained during the previous decorrelation step is required (Figure 14). Respective parts of A , Y and 2 in the vectorized form reside in the memory and are used to decorrelate observations of the two current consecutive epochs. During the decorrelation of the epoch i, the [A |Y ] matrix belonging to this section of data is overwritten by its whitened counterparts. Also, an epoch’s covariance matrix is replaced successively by its Cholesky factor. First, the Cholesky factor of the subblock 2,+j,,- of 2 is obtained by calling another subroutine, and then, forward substitution is performed using [A/,;+j|Y/,/+;] to obtain [A t/+j|Yt/+j], where i and i+1 stand for epochs / and i+1, respectively. For the next step, the current epoch one is discarded and epoch two is relocated so that it occupies the same place in memory as the former block one. Then, the next subblock is read and located as epoch two. The correlation block between adjacent epochs and the covariance for the epoch two, together with its design matrix, are also loaded into memory. Each step described above is repeated until all the data are decorrelated. The whitening process is accompanied by the successive accumulation of the normal matrix that is eventually used to obtain the final solution. This convenient, flexible and efficient way of recursive whitening and accumulation of the normal equations makes it possible to save computer time and memory. A and Y need not be stored since all one needs is the normal system to find the final solution that can be, again, obtained by the Cholesky factorization. 99 Am Yj Am r 2 A 3 r s A, Y, Am Ym Am Ym Ah-1 r-i A m r« i Figure 14. The recursive Cholesky decomposition and decorrelation scheme operating on the lower triangular part of the covariance matrix and the vectorized A matrix augmented by the column Y The scheme described above is successfully implemented as an integral part of GODIVA in a PC environment, under WindowP* NT. Its major characteristics are reliability, efficiency, low computation overhead and small storage requirements. The process of triple difference whitening is done much faster than the datg screening and cycle slip detecting/fixing required by single- or double-differenced phases. Fully automated procedures for all phases of data processing are employed in the generation of orbit estimates along with other parameters of the model so the human interaction is minimal. The combined processing time needed for downloading of the observational data, creation of the database, selection of the complete set of independent measurements, and determination of parameters is about 5.5 hours on a P-90 personal computer. Furthermore, the use of a personal computer has added enormous flexibility to the orbital and geodetic parameter estimation problem, as the seven IGS analysis 100 centers rely on the processing power available from at least a workstation which, in turn, has limited the accessibility of the problem for many researchers around the world. For most of the applications to date, triple difference solutions have been treated as tools to detect outliers in the data and to obtain preliminary solutions before the main one, usually based on double differences, is obtained. However, in our case, the triple difference is the only solution generated. The mechanism that detects outliers in the data is built into the software and assures the automatic elimination of the cycle- slip-affected or bad quality observations, as explained in the preceding sections. The saving in time is substantial with respect to the cycle-slip fixing procedure. 5.5. Estimation o f the GPS Trajectory and Other Geodetic Parameters with Triple Difference Phases 5.5.1. Linearization o f the Triple-Difference Observation Equation In the current section, the linear form of the triple-difference observation equation is developed by the first-order Taylor expansion of the nonlinear function (5.13). In addition, the characteristics of the observation model are summarized in Table 14. As noted earlier, the basic observable applied in GODIVA is the ion-free triple-difference phase represented by equation (2.12) in Chapter n. For clarity of showing the differencing between epochs h and t2 and in order to make the form of the partial derivatives more explicit, it can be rewritten here as follows: ®ijjono-/re*, Sampling rate: 15 minutes Weighting: uniform, with 1 cm standard deviation for the single phase Elevation angle cutoff: 16 degrees Ground Antenna Phase Offset • applied Center Elevation-dependent phase center correction - not applied Troposphere Modified Hopfteld with mapping junction developed by Goad and Goodman Ionosphere Not modeled, ion-free combination used Plate Motion TTRF93 Station velocities, fixed Station Tidal Displacement Solid Earth tides, according to IERS Standards Station Displacement Due to According to IERS Standards thePole Tide Satellite Center of Mass Block!. 0.211 m, 0.000 m, 0.854 m Correction Block 1I/1IA: 0.279 m, 0.000 m, 1.023 m The double-differenced geometric range and the tropospheric term are defined by the equations (5.14) and (5.15), respectively. Pit = (pi ~Pkj)~{pli ~Plj) = >/(** -*i)2 +(yk-y/)2 +(** ~zt) 2 - yj(xl - xt)2 +(yl - y tf +{zl- z , ) 2 - (5.14) >/(** - * /) 2 + 0 * - ys)2 +(** -Zj)2 + T^S^-Th-SjiTf-Tj) (5.15) 102 The tropospheric terms Ttl ,T,k , T j , l f in (5.15) are referred to as single phase corrections, St and Sj are the scaling factors to the tropospheric mapping function at stations i and j with a priori value of one. It should be emphasized that the station coordinates are given in the ITRF93 terrestrial reference frame, whereas the satellite coordinates are determined in the MoD inertial reference frame, as explained in Chapter m. Therefore, the coordinates of the observing stations are transformed to the adopted inertial reference frame following the formulae defined in Chapter m. The tropospheric refraction correction is evaluated according to the modified Hopfield model, as given in Chapter m. The general form of the nonlinear observation equation can be written as follows, where the time dependency is indicated by the presence of the independent variable 1: 0 (0 - C(t, r (/), F(0, ~P, r„A, trop, x,, ^ , (UTl - TAI)) = d C d r d C dr d C dr - d C /C i — — dr* +— — dr, +— ^=d/J + — -d r„A + (5.16) dr dr0 0 dr dr0 0 d r d P df^“STA dtrop where —— denotes the partial derivative with respect to the unknown parameter, and, d ,— =? represent the variational partial derivatives as explained in d r0 d r 0 d p Chapter IV, 0(t) - the actual observation, i C- the computed observation based on the approximated values of the unknown parameters, t - time in the TAI scale, 103 r0(0 and f a( t ) - (3x1) satellite position and velocity vectors in the inertial frame at the initial time epoch to, r ( t ) and r ( t ) - same as above, but at the time epoch t, P - satellite state vector such as P = (r(tt), r(tt), P*) where P* denotes model parameters that are estimated together with the orbit: solar radiation pressure scaling factors in the and Z satellite-body-fixed directions and the scaling factor to the 7-bias. All the unknown parameters including the satellite state vector are listed in the Table IS together with the sources of their a priori values and a priori standard deviations. rnA - vector of the earth-fixed coordinates of the tracking stations, trop - vector of the tropospheric scaling factors for the tracking stations of the form: Si+1>h ... Sl>ti Sl+l>li ...SU3 Sl+Ui...], where subscript i, i+1, etc. denote the stations and t i, t2, etc., denote the consecutive time intervals where the tropospheric scaling factors are assumed constant. xp - the x component of the polar motion, yp - the.y component of the polar motion, (UTl-TAI) - the change of UTl, usually estimated as a rate of (UTl-TAI) due to the collinearity of the Earth rotation and the motion of the orbital node, as addressed in section 5.5.4., e - the observation noise. 104 Table IS. Solution parameters Product A priori Value A priori Constraint Satellite Position former solution 1.0 m Satellite Velocity former solution 10“* m Solar Radiation Pressure In order to evaluate all the terms of the form ——, i.e., the partial derivatives of o the computed triple difference observable with respect to the model parameters, the computed range must be defined first: +3?]*-U *+#]*)„.= (5.17) C+Mp?]* ~[dp»\ + [ < ] f2 - [ < % where C = l^p^ + Ttf ]fj -[p$ +T^]t^ and symbol “ 0” denotes the approximation. Ctrut is the theoretical, computed observation, that is never accessible, and as such, has to be approximated by the Taylor expansion. C is evaluated as a difference between two pf} and Tjj1 belonging to the neighboring epochs ti and t2, based on the prior information about the unknown parameters, and dp and dT denote the first-order Taylor expansion differential corrections of the function Cm* and are defined in the following way: 105 dx, Xk - X , . X, - X i yk-yt . y'-yt z k - z t . z l - z , d p » = k t k I k I dxk dxk ' yk - y t *k -h x -Xj y -yj Z~ - Z , k k k dyk dyk L Pi Pi Pi . i p ) Pi Pi J dzk dzk dxl dxl x'-Xj yl - y t -Zj 1 X -Xj y -yj Z - Z j i i i dyl I I I dyl . Pt Pt Pi J Pi Pi Pi dz 1 dz 1 (5.18) and K = t t * -(If - T j ) d S j (5.19) where dSt and dSj in (5.19) are the differential corrections to the tropospheric scaling factors for stations i and j, and cbc, efy, dz with the superscripts k and j in (5.18) are the differential corrections to the a priori satellite coordinates. Terms dx, efy, dz with the subscripts i and j represent the differential corrections to the a priori station coordinates. Tropospheric scaling factors are currently evaluated every four hours, thus, assumed constant for every station within the four-hour interval. When tj and fe both fall in the same four-hour time interval, the partial derivatives for dSt and dSj, respectively, are combined for both epochs. If the triple difference is created over the boundary of the neighboring sections, the partial derivatives for dS{ and dSj are computed separately for both intervals. Term C (the computed measurement) is evaluated based on the stochastic prior information about the unknown parameters that is known up to a certain accuracy, defined by the a priori standard deviation. In GODIVA, the prior information is introduced into the solution as a pseudo observation of the form: b 0 - » ®o ~ (®») (5.20) where § - the (mxl) vector of the unknown parameters with the prior information, b« - the stochastic prior information vector associated with the unknown vector e# - the random error vector of the stochastic prior information, with a corresponding positive definite covariance matrix, Zo. Combination of the stochastic prior information, bo, associated with the unknown vector with the original observation vector, y, leads to the Extended Gauss-Markov Model of the following form (Koch, 1988, p. 206): y _ a , (5.21) b0J L 1 where £r and & represent the unknown parameter vectors with and without prior information, respectively. Ai and A 2 are the respective design matrices related to and £ vectors. Symbol I denotes the (mxm) identity matrix. Solving (5.21) under the least- squares principle provides the following system of normal equations: 'a J 'S - ^ + E ; 1 AjZ^A,! pi] rAjZ-V + S^bo (5.22) A jZ ^ A j A J E ^ a J _ A j z _1y that is formed in GODIVA. This system is solved by Cholesky factorization and reduction and provides the least-squares estimates of the unknown model parameters. Another critical factor of the approach presented here is the way the baseline connection is furnished. The technique widely practiced by the IGS centers is to have a fixed set of baselines connecting stations whose data are used in the adjustment process. In such a case, some information can be repeated since linear dependence among the baselines should be expected unless only independent baselines are used. But the common practice is to have redundant baselines which provide stronger geometry in the solution. The higher the number of baselines connecting more and more stations, the better and stronger the geometry. But having a predetermined set of baselines may lead to some loss of information over the baselines that are not used. In addition, it requires human interaction to control the data availability and quality over the defined station connections. This approach was first implemented in GODIVA. However, it was found that the automated procedure for generating an optimum set of linearly independent ionosphere-free triple differences according to Goad and Mueller (1988) works much better. This approach allows access to 100% of linearly independent information. The linear dependency between the measurements is revealed by displaying a zero diagonal element on the corresponding position in the Cholesky factor of the measurement covariance matrix. Single precision arithmetic, less precise than the double ( 8 -byte operation) but much faster, is applied to operate on the covariance matrix of all the possible triple differences. It provides a sufficient number of significant figures for our purpose. The selection of the independent observations could be combined ideally with the decorrelation process. However, the whitening algorithm involves double precision arithmetic which is more precise but much slower. That can be computationally too expensive for the data selection process where the total number of triple-differences per epoch can be as large as 500, whereas only about 30% of all the observations are linearly independent. The process is again automated so it is fast and flexible in the sense that no separate checking for data availability is necessary. In essence, the baseline configuration is variable from epoch to epoch and the optimum information with no redundancy is acquired. Figure 15 presents the RMS 108 of fit of the OSU orbits to the IGS solutions obtained with the fixed set of 61 baselines connecting 36 stations and with the new approach (Goad et a l, 1995; Yang, 1995). Clearly, the dramatic decrease in the RMS value can be observed for the solution obtained from the linearly independent data. 18 17 18 15 14 Optimal 13 Baseline Set 12 11 15 18 17 18 IB 20 21 Days of Year Figure 15. Mean RMS values for the OSU orbit comparison to the IGS orbit for different measurement generation procedure 5.5.2. Estimation o f the Satellite State Vector The satellite state vector consists of three initial elements of the satellite position and three initial elements of its velocity, denoted as r0(/) and ro (t), respectively. In addition, two scaling factors, Sx and S2, to the solar radiation pressure components in X and Z satellite-body-fixed directions are determined, together with the scaling factor, Sr-bia*, to the a priori 7-bias acceleration. Since the triple difference is not the explicit function of the satellite state vector at the starting epoch, the respective 109 d C partial derivatives are formed by the combination of the geometrical partials, ——, of dr the computed phase range with respect to the satellite coordinates at the given time epoch t, and the corresponding variational partial derivatives that constitute a (3x9) state transition matrix, providing the mapping of the current state to the epoch to. The variational partial derivatives, (see equation (5.16)), are obtained by dT%'d? d c k' Xk -X j y k - y j z k -Zj' £ 1 z £ l 1 _ \i£*A - I * * - * ' & k k k dyk L * * L ti ti ti 1 . P> P> Pj . dzk dzk d c k ' ' d x k ' x k - X j y k- y } z k - Z j dCjr"] yk~y> zk~z>] _ . d y k k k k d y k ** ml I ti ti ti J„ . Pj Pj Pj 11 d z k d z k *2 (5.24) for the time epoch fc. Similarly, for the satellite /: 110 d x ‘ 'd x r 2 -Z. ■yj z’ -z. l £ j r ‘] \x' - x> y'-y< dyl • d y ' ** L p‘ p\ Pi P i n d z ' d z‘ (5.25) for the time epoch t j , and d x ‘ d_ X -x, y'-y, z'-z, x‘-Xj y‘ —y} z '- z , * d y l + i l l dy1 H ■ - . Pj Pj Pj . n 0 - J'2 dz‘ L " " ' J J,t dz' & L J<* (2 L Jii (5.26) for the time epoch t2. These geometrical partials are then combined with the respective variational partial derivatives as shown in (5.16) and create the portion of the linearized observation equation that corresponds to the unknown state vector. The proper parametrization of the state vector is very important for the quality of the total solution. For example, removing one of the solar radiation scaling factors from the state vector (Test Solution in Table 16) causes some deterioration in the quality of the solution, as shown in Table 16. The a posteriori standard deviation shows a 1.5% increase with respect to the solution with a full dynamic model (called Standard Solution in Table 16). The missing scaling factor Sz (that was fixed to its nominal value 1) introduces an error in the remaining elements that absorb the change in the body-fixed Z component of the solar radiation that is not modeled. ERPs, especially yp for which the difference with respect to the standard solution is 26 pas, seem to absorb a part of the orbital mismodeling. Also the scaling factor in jf-body- fixed direction is affected by different parametrizations as can be observed in Table 16. Both solutions are very similar, yet with the current accuracy and precision of the IGS solution, even an improvement of 1.5% is important. Ill Table 16. Selected characteristics of the solution with a full dynamic model and the solution with a fixed solar radiation pressure scaling factor in the Z-body-fixed direction, for January 15,1995 Solution characteristics Standard Solution Test Solution Difference Between Solutions Oi 1.203 1.218 0.015 129.929 129.931 0.002 y. 453.010 452.984 -0.026 0.110 0.110 0.000 0.120 0.120 0.000 O r for PUN 1 0.130 0.130 0.000 O r for pm 2 0.090 0.090 0.000 O r forPSN9 0.100 0.100 0.000 S xforPRNl 0.900 0.859 -0.041 S xfor Pm 2 0.988 0.991 0.003 S xfor Pm 9 0.952 0.998 0.046 Notation in Table 16 is as follows: o' -a posteriori standard deviation for xp in mas, o yf - a posteriori standard deviation for yp in mas, o0-a posteriori standard deviation of a unit weight, d - a posteriori standard deviation per coordinate in m, Sx- solar radiation scaling factor in X-body-fixed direction, xp and yp are given in mas. 5.5.3. Estimation o f the Coordinates o f the Terrestrial Station The geometrical partial derivatives of the double difference measurement with respect to the station position in the celestial reference system (CRS) are shown explicitly in the equation (5.18) and can be written as follows for two time epochs involved in the triple difference measurement: 112 d C x k - X, , x l - xt zk -z, i y k- y t h . y l -y, . *l -*t g pCRS _* 1 _/ Jc Pi Pi I h Pi pli . (5.27) d C xk -Xf xl - xt zk -zt . zl -z, i yk - y t . y l - y t d -CRS _k ' I H i h Pi Pi Pi p lt . for station i at both receive time epochs, and r d C Xk-Xj Xl-Xj yk-yj yl -y} zk-zj zl-2j d Y c*s P) Plj P) Plj Pj P\ L j Jh (5.28) d C X -Xj Xl-Xj y-yj yl -y } z * -2j z‘-Zj d -CRS k „k I Pj p 'j Pj Pj Pj L j Jh for station j at both receive time epochs. The partial derivative of the vector of the inertial coordinates with respect to the coordinates in the terrestrial reference system (TRS) can be obtained from equation (3.1), that is presented here in more explicit form as follows: rCRS = GAST)-R±(yp)'R%(xp)- i^g (5.29) where the notation follows that of Chapter m. The following equation represents a (3x3) matrix of the partial derivatives of the station coordinates in the celestial (inertial) frame with respect to the terrestrial coordinates. ^ - = P N R 3(rGAST)-Rl(yp)R,(xr) (5.30) u r.TRS Finally, the combined partial derivative can be written in the following way: 113 r e c ' p c ' ' PC 0 r C R S \ P r S T A i P r™ drf** P r T S S _ Xk -Xf x l - x ( y k - y t . y l - y t z k - z { _ z l - z ( P r C R S 1 - k 1 I „ k A A c 1 I Pi Pi Pi Pi Pi Pi h - - xk -X, xl - xt y k - y t . y l ~ y t zk -zt zl-z, VCRS 1 k 1 I A A k 1 I Pi Pi Pi Pi Pi Pi . 'i*- VTRS J,, (5.31) for station i, and for station j: ' d C ' PC PC P r CRS -7 -TRS A =CRS P t , p r TSS 0 * S T A _ i -d x i . Xk - Xj z l - Z j X ~ X J yk-yj yl-y, p *CRS A A c P r TRS_ Pj plJ P i P) P i p li . *2 X - X t X - X z l — Zj j y -ys y -y} P r CRS k „k A c P i _ P r TSS . P) P i P) P i p lJ . h (5.32) It should be emphasized that the station coordinates that are used in the evaluation of the computed measurement and the partial derivatives are first corrected for all the systematic effects listed in Chapter IV that are related to the terrestrial site dynamics (see Sections 4.3.9-4.3.11). The impact of the improper handling of the tidal correction to the station coordinates is presented in Section 4.4.1. The table below illustrates the importance of the correct application of the velocity field, as defined in Table 8 , to the quality of the final solution, especially ERPs. The standard solution, with the full dynamic model and the solution where the velocity field was not applied, (Test Solution in Table 17) are presented. As can be observed, the a posteriori standard deviations are very similar in both solutions. The same can be concluded for 114 the trajectories. However, the major impact can be observed in the ERP estimates. The lack of the velocity corrections, especially for the 13 fiducial stations, introduces a misorientation of the 1'1'RF which in turn causes errors in the ERP estimates. It should be emphasized that the velocity components are defined at 0 h UT, January 1, 1995. Thus for the epoch presented in Table 17 (July 3, 1995) it is very important to account for the motion of the stations. However, for epochs close to January 1, 1995 the effect of station velocity may not be that harmful due to the relatively small displacements. But the motion accumulates with time and the correction should certainly be applied rigorously during the course of a year. Table 17. Selected characteristics of the solution with a full dynamic model and the solution with a velocity field neglected, for July 3,1995 Solution Characteristics Standard Solution Test Solution Difference Between Solutions (To 1.499 1.506 0.007 X p 282.506 282.439 -0.067 y B 373.254 373.082 -0.172 0.110 0.110 0.000 0.130 0.130 0.000 Oc for PR1V1 0.110 0.110 0.000 CFc for p m 5 0.090 0.090 0.000 tXc for pm 6 0.100 0.100 0.000 Notation in Table 17 follows that of Table 16. 5.5.4. Estimation o f the Tropospheric Refraction Scaling Factor Since the atmospheric conditions are different at every station, the tropospheric correction is evaluated for the single phase between every satellite and station at both epochs, and then combined to form the triple-difference tropospheric correction applied to the phase range observable. However, the tropospheric models and mapping functions provide only the approximation to the true tropospheric correction, thus, the scaling factor associated with the mapping function must be evaluated for each station in order to obtain the best estimate of the tropospheric effect. Normally, one scaling factor is determined every three or four hours for eveiy station. For two stations i and j involved in the triple-difference measurement, the partial derivatives of the computed observation with respect to the tropospheric scaling factor St and Sj are formulated in the following way: (5.33) and (5.34) according to equations (S.IS) and (S. 19). The nominal values of the tropospheric refraction terms T(k - Tfl and Tk - Tj are evaluated from the modified Hopfield model, as explained in Section 5.5.1. 5.5.5. Estimation o f the Earth Rotation Parameters The standard processing of GPS data observed by the terrestrial stations enables the simultaneous estimation of station coordinates defined in the network rotating with the Earth along with the satellite orbits that furnish a dynamical connection to the celestial frame as addressed in the preceding sections. Since ERPs provide the link between celestial (inertial) and terrestrial (attached to the rotating Earth) frames, polar motion and UT1-UTC can ultimately be recovered. But GPS, similar to other nonabsolute methods of ERP retrieval, suffers from an inherent defect in UT1 determination. This is the result of the collinearity between the variations in the right ascension of a satellite’s orbital node and UT1 which limits the capacity of GPS to resolve UT1 using satellite observations alone. Due to the correlation among the model parameters, any small mismodeling of the motion of the orbital node will affect the estimates of UT1. The dynamic model of a spacecraft’s orbital motion accommodates for the major part of the nodal drift in the UT1 estimate, but the accuracy limitations of the model parameters restrict the level of resolution of the motion of the orbital node. However, the rate of change in UT1-UTC (that is, a negative of the change in length- of-day (l.o.d.)) can be recovered, usually over a short period of time such as 1-3 days. The rate of UT1-UTC can be recovered by the satellite technique due to the fact that the partial derivatives of the GPS observable with respect to the change in UT1 and with respect to the right ascension of the node are not linearly dependent. Theoretically, the rate of the node and the rate of UT1 are linearly dependent, but “... because we assume that the force model is (more or less) known, there is no necessity to solve for a first derivative of the node” (Beutler et al., 1995). Having the variations in UT1 resolved over a short interval, one needs to connect this sequence to the starting value of UT1-UTC based on an absolute, independent and stable reference which must be obtained by another mechanism, for example, VLBI which provides direct connection to some suitable quasars that offer the absolute reference orientation in space. This way both VLBI and GPS signals should be optimally combined as addressed in more detail in Chapter VI. Since observation equations are expressed in the inertial frame (MoD), the partial derivatives of the computed measurement with respect to ERPs are computed using a chain rule. The geometrical partial derivatives with respect to the station coordinates in the Celestial Reference System (CRS) are computed first, as shown in equations (5.27) and (5.28). Next, the partial derivatives of the station coordinates in CRS with respect to the polar motion components and (UT1-TAI) are computed based on equation (5.29). 117 d r CRS & [P-N-R3 (-GAST) • R \iyp ) • -^(x )] rriis d x p d x p (5.35) = P -N R i(-GAST)R1(yp) dx. ‘TBS for the xp component, and dJcRS___e__ [P - N 'R^i-GAST)'Ri(yp)'R2 (x»)l *W &yP # y P (5.36) #Ri(yP) = p .N R 3(-GAST)—^ ^ R 2(xp) fTRS a yp for th e ^ component, where cos(xp) 0 -sin(Xp) * 2 0 0 = o 1 o (5.37) sin(xp) 0 cos(Xp) j 1 0 0 * iO O = 0 cosCyp) sin (yp) (5.38) 0 -sinO^p) cos(yp) and the respective derivatives of the above matrices are as follows: -sin(xp) 0 -cos(Xp) [*zOO] P'J _ 0 0 0 (5.39) cos(xp) 0 -sin(X p) 0 0 0 e [*iOO]pji - 0 -sinC yp) cosCVp) (5.40) 0 -cos(>p) -sin(>p) 118 The rotation by GAST is expressed by the following matrix: cos(-GAST) sin(-G4iS'7) o' cos(GAST) -sin(GAST) O' R3(-GAST) = -sin (-GAST) cos (-GAST) 0 = sin(GAST) cos(GAST) 0 (5.41) 0 0 1 0 0 1 and the corresponding partial derivative: - sm(GAST) - cos(GAST) 0 0 [R3(-GAST)] cos (GAST) -sin (GAST) 0 (5.42) d GAST 0 0 0 In order to obtain the partial derivative with respect to UT1 or equivalently, (UT1-UTC) or (UT1-TAI), the relationship between GAST and UT must be applied, as shown below. GAST = GMST + equinox = GMSTohun + ri^UTl - UTC) + £/7U] +equinox (5.43) where equinox denotes the equation of equinoxes, and approximated r= 1.002737909350795, as explained in Chapter m. Thus the partial derivative of GAST with respect to UT1 is the scaling factor r that provides the conversion from solar increments to sidereal time increments. Finally, the combined partial derivatives of the computed measurement with respect to the pole offset and the rotational velocity can be written as follows: 119 d C _ d C dtjgs # xp #*ms d*p d C d C dxms (5.44) dyp d *m dyP d C _ dC drm dGAST _ dC dxm d(UTl- TAI) ~ d fm dGAST d(UT\-TAI) ~ dxm dG ASTT Equations (5.44) are written for stations i and j that are involved in the triple-difference observable and the respective partial derivatives are summed since the receive time is common for both stations. As discussed above, estimations of either change in UT1, change in (UT1-UTC), or variation in (UT1-TAI) are equivalent since both UTC and TAI are uniform time scales that are not dependent on the rotational velocity of the Earth and are based on the atomic second. Thus, only the rate of UT1 is variable and estimated together with the orbit and other parameters of the model. Also, the pole solution is very sensitive to the adopted terrestrial reference frame (Zhu et al.y 1993). If the daily estimates of the terrestrial stations are not sufficiently accurate or the antenna height corrections are not properly applied, as a consequence, the final estimates of the pole coordinates will be distorted (see Table 17). The quality of the pole coordinates depends largely on the number of tracking stations and their relative geometry. Pole coordinate estimation with GPS is presented in more detail in Chapter VI. Currently, GODIVA offers three models for Earth Rotation Parameters: step function, linear function, and piecewise continuous linear function. Representation of ERPs as a linear function is the routine approach, similar to the majority of the IGS processing centers. Two additional options are treated as an extension from the common approach of ERP modeling. The piecewise continuous linear function is not very much in use currently since the routine approach requires only a 32-hour arc. This model could be particularly useful for the longer arc (more than two days) applications that are not presented here. However, the piecewise continuous linear model was tested 120 for the 32-hour arc that first was divided into two 16-hour bins and then for four 8 - hour bins. In both cases, the results did not improve with respect to the routine solution. However, as reported by Rothacher et al. (1994), the modeling of the polar motion as a piecewise continuous linear function over a 72-hour arc improved the pole estimates with respect to the case when the pole was treated as a discontinuous linear function within every 24-hour arc participating in a three-day solution. The step function approach was intensively tested and the results are presented and discussed in Chapter VI. To complete the partial derivatives required for the ERP estimation, the following linear representation of the pole coordinates and (UT1-TAI) should be considered for the routine approach: ERP = a+ b-At (5.45) where a represents (UT1-TAI), xp or yp at the reference epoch, i.e., middle of the arc (12:00 TAI of the middle day), that are obtained from the IERS Bulletin B or A prediction. Coefficient b represents the rate of change of the pole coordinates or negative rate of l.o.d. Finally, the partial derivative of the underlying ERP with respect to the linear model parameters are: dERP , . dERP . ~z>— = 1 and — = A/ (5.46) d a db where At is the time in seconds that elapsed between the reference epoch and the current epoch. Equations (S.46) are combined with (5.44) to provide the partial derivatives of the ERP model parameters a and b. The a priori standard deviations for ERPs that are routinely used are as follows: xp and yp offsets - 0.097 arcsec xp and yp rates - 0.097 x 10 -8 arcsec/day (UT1-UTC) - 6.47x 10 ~3 s 121 • d(UTl-UTC) - 6.47x l Clearly, the routine solution tightly constrains (UT1-TAI) offset and pole rates. In fact, it was found that the final trajectories and also pole coordinates present better quality if only pole offsets are estimated. For that reason the a priori pole rates are set to zero and are tightly constrained, as shown above. For the step function approach only the a coefficients remain in (S.4S) and the final partial derivatives are represented by (5.44). 5.6. The Current Status o f IGS Activity in GPS Orbit and Related Parameter Estimation The state-of-the-art of GPS orbit and other geodetic parameter estimation strategy applied at the IGS analysis centers (AC) is outlined in this section. Also, the method of unification of the participating orbits into the combined solution is addressed. The following analysis centers are currently involved in IGS activities: Center for Orbit Determination in Europe (COD), Bern, Switzerland; Natural Resources, Canada (NRCan), formerly Energy, Mines and Resources (EMR), Ottawa, Canada; European Space Agency (ESA), Darmstadt, Germany; GeoforschungsZentrum (GFZ), Potsdam, Germany; Jet Propulsion Laboratory (JPL), Pasadena, California, USA; National Geodetic Survey (NGS), Silver Spring, Maryland, USA; and Scripps Institute of Oceanography (SIO), San Diego, California, USA Table 18 summarizes the activity reports provided to IGS by the participating centers (IGS Questionnaires in IGS Electronic Mail 654-657 and 695; Bock et a l, 1993; Kouba, 1995). Contrary to the triple-difference (TDF) approach presented here, the ACs utilize the undifferenced (UDF) or the double-differenced (DDF) phase as a basic observable. In Chapter VII, where the results of the test data are presented, the comparison of the triple-difference approach with the IGS products is also given to show that the newly adopted strategy provides highly competitive results. In order to make the comparison of different 122 approaches valid, all the constituents of the different procedures should be considered and cross-examined. For that purpose, Table 18 is presented to show the similarities and disparities in the strategies applied by AC and our modus operandi described in the preceding chapters. Table 18. Selected characteristics of the processing strategies of individual analysis centers including OSU (as of December 1994) COD EMR ESA GFZ JPL NGS SIO OSU Basic DDF UDF DDF UDF UDF DDF DDF TDF Observable # of Stations 47 22 23 38 32 33 32 36 Used #of Fixed 12 12 12 18 12 23 16 13 Stations Arc Length in 72 24 48 32 30 31 24 32 Hours Observation 3 7.5 6 6 5 0.5 2 15 Interval fntinl Gravity Model GEMT3 GEMT3 GEMT3 JGM2 JGM3 GEMT3 GEMT3 GEMT3 (8,8) (8,8) (8,8) (8,8) (12,12) (8,8) (8,8) (8,8) # ofRadiation Pressure Parameters 2 3 2 2 2.5 2 3 3 Including Y-bias Estimation weighted square- weighted Helmert square- weighted weighted weighted Strategy least- root least- blocking root least- least- least- squares informa squares informa squares squares+ squares tion tion Kalman filter filter filtering Besides the data editing technique applied and data validation, the other critical factors that determine the precision of the global orbit solution is the number of fixed stations and the length of the arc processed (Kouba, 1995). Additionally, the number of estimated solar radiation scaling factors is very important due to the complicated nature of this perturbing force, as explained in Chapter IV. For example, the difference in RMS between two solutions with a different number of scaling factors can be as large as 10 cm (Kouba, 1995). The number 2.5 for JPL is related to the stochastic f representation of the scaling factors in x and z body fixed directions, both starting with the same a priori value. In contrast, no significant difference due to the application of different gravity models is reported. The maximum difference between GPS oibits based on IERS recommended GEMT2 and JGM2 is well below 2 cm, as reported by Kouba (from private communication with J. Klokocnik and J. Kostelecky, 1995). More information about the numerical analysis of the problem of the gravity field influence on the high orbiting satellites can be found in Klokocnik and Kostelecky (1987). The IGS orbit is created as a weighted mean of the orbit estimates from the seven analysis centers as detailed by Springer and Beutler (1993), Beutler et al. (1993), Kouba (1995) and Beutler et aJ. (1995). The orbit weight for eveiy center is computed from the corresponding absolute deviation from the equally weighted mean orbits. Since every angle solution is given in the terrestrial reference frame that has individually determined orientation with respect to the inertial frame, every trajectory that contributes to the combined solution must be rotated to a common frame whose orientation is determined by IERS Bulletin A or B before the actual averaging takes place. Furthermore, a seven- parameter similarity transformation is applied to remove remaining systematic reference frame differences between any two orbits. Details about the averaging procedure are given in Springer and Beutler (1993), Beutler et al. (1993) and Kouba (1995). According to the IGS processing principle, all the information submitted by an AC should be used in the combined solution. However, solutions from equally-weighted averaging with orbit RMS of lm and larger, or satellite clock solution with errors exceeding a few tens of ns are usually excluded (Kouba, 1995). This is most likely to happen when the rapid IGS orbit is generated. Solutions formerly excluded are contained in the final IGS solution if the source of the error was detected by the individual AC and the problem removed from the updated solution submitted to IGS. The remarkable agreement among the IGS centers is a consequence of continuous cooperation, product evaluation, standardization of the activities, and the terrestrial frame improvement. The remaining differences are most likely due to the lack of temporal correspondence between the AC orbits and ERPs, as pointed out by Beutler e t al. (1993). The significant improvement of the terrestrial reference frame ITKF93 with respect to ITRF92 is clearly seen from the coordinate standard deviations that are about 50% of the corresponding standard deviations for ITRF92. Improvement of ITRF93 coordinates and site velocities has made the reference system more consistent with the IERS ERP series, although some bias with respect to the NNR NUVELA velocity model was introduced. As should be expected, replacing ITRF92 by its 93 counterpart introduced some discontinuities in all the IGS series, but they are not significant for most applications. For more precise geodynamical applications, continuity can be recovered since the relationship between the 1994 and 1995 IGS series is known (Kouba, 1995). The current quality of the IGS product is given by the RMS of fit of the participating solutions to the combined orbit that ranges from slightly less than 10 to about 28 cm. The accuracy of satellites and station receivers’ clock solutions is reported at the subnanosecond level, after some temporal deterioration due to the AS implementation at the beginning of 1994, and shows a remarkably good quality for global precise time transfers. CHAPTER VI VARIABLE ROTATION OP THE EARTH "Earth orientation data provide a means of obtaining information about irregular motions offluid on a global scale, which provides most of their geophysical interest, as this information is typically poorly determined by other means. At the same time, rotational variations can generally not be understood in isolation, but only using theory and data from other areas of geophysics. The study of the rotation of the Earth is thus probably the most interdisciplinary of geosciences, as it interacts closely with research in meteorology, oceanography, geomagnetism, hydrology and other fields" (Eubanks, 1993) 6.1. Variability o f the E arth’s O rientation In this chapter, a short overview of the problem of the variability of Earth orientation, i.e., rotational speed and location of the instantaneous axis of rotation, is presented (for a more detail discussion see Moritz and Mueller, 1987; McCarthy and Carter, 1990; Lambeck, 1980 and 1988). In the most common approach to the representation of the Earth’s orientation, it is conveniently separated into three elements: precession and nutation that describe the orientation of the Earth in space, polar motion (wobble), representing the orientation of the rotational axis with respect to the Earth crust, and changes in the rotational speed (changes in length-of-day) that describe the rotational variations about the instantaneous pole. Change in length-of-day represents the excess from 24 solar hours - 86400 s and is directly related to the change in the rate ofU Tl according to equation (6.1) (Lambeck, 1980, p. 63). 125 U=-(UTJ-TAI)=(TAI-UTJ) and is considered positive if the Earth rotation rate is slow relative to UTC, a)3 and denote the instantaneous and mean rotational velocity, respectively. Precession, nutation and polar motion describe the variability in the orientation of the axis of rotation, whereas changes in l.o.d. represent the variation in magnitude of the rotational speed. Precession, nutation, forced polar motion, Earth tides and partially the change in l.o.d. have the same origin - lunar, solar and planetary gravitational attraction. In this study, only polar motion and the variations in the rotational speed are presented. Both components, rotational velocity and the polar motion that form the Earth Rotation Parameters, change periodically and nonuniformly with time. These changes can be predicted only to some extent limited to the quality of the models, like for example, the solid Earth and ocean tides models. The free part of the motion must be monitored in order to properly determine the Earth orientation. The major reasons for the variability in ERPs are the tidal effects (solid Earth and ocean), atmospheric influence, and the variable dynamics of the Earth itself. Fluctuations in the Earth’s dynamics are related to the solid mass and the ground water redistribution within the crust and the interaction between the layers that constitute the planet. They are schematically presented in Figure 16 (Lambeck, 1988, p. 547). 127 Viscous torques currents / Plate tectonics Figure 1 6 . Schematic illustration of the forces that perturb the Earth’s rotation (beetles represent the continental drift) These phenomena contribute to a diversity of frequencies that describe the spectrum of the rotational variation of our planet. The discussion presented here recognizes different sources and the associated frequencies, -with special emphasis placed on the short-periodic (daily and subdaily) variations in the rotational speed and the pole location. Duly and subdaily resolution of the ERPs, especially from satellite techniques, is currently an area of very extensive research, whereas the longer period terms are fairly well known and understood (some exceptions like uncertainty about the reasons for the Markowitz wobble or excitation/dumping mechanism for the Chandlerian motion are given in the following sections). The high-resolution ERPs with 128 expected accuracy of 0.1 mas and 0.01 ms for the pole and UT1, respectively (Montag et al., 1992), should allow better understanding of the interaction of the inner and outer forces that change the Earth’s rotation. In particular, precise knowledge of the Earth’s rotation in the time scale of hours to months will allow further investigation of interactions between the solid Earth, oceans, and the atmosphere. However, the problem of insufficient orbit modeling that is reflected in the degraded quality of the high-frequency ERPs is the major issue in ERP determination from satellite techniques. For example, the long-period mismodeling in the orbital dynamics will contaminate the solution of the diurnal pole signature. In addition, the diurnal retrograde polar motion cannot be decomposed from the nutation series (Beutler et al. 1995; Ibanez-Meier et a l., 1994; Watkins e t al., 1994; Montag e t al., 1992). Moreover, the satellite tracking data have to be of very high quality and should be evenly distributed. Thus, the crucial factors are the precise orbit modeling combined with the proper data selection and the parametrization of the observation model. The tidal potential, as noted earlier, is one of the driving forces that induce the periodic variation in the Earth orientation. The solid lunisolar tides modify the Earth’s gravitational potential making it time dependent, with frequencies that are functions of the orbital motion of the Sun, the Moon and the Earth. As shown by Moritz and Mueller (1987, p. 10), the genuine deformation of the Earth is caused by the tidal potential of degree £ 2 , where the second-degree tides have been found dominant. Tesseral (diurnal) tidal potential is responsible for precession, nutation and forced polar motion. Zonal (long-periodic) tides generate the rotational variations, sectorial (semidiurnal) tides are used in evaluation of the tidal friction that causes the secular retardation of the Earth’s rotation (Mueller and Moritz, 1987, p. 16). According to (Mueller and Moritz, 1987, p. 192), polar motion occurs for a rigid earth as well as for an elastic earth, but variation of rotational velocity is a typical phenomenon of nonrigidity.” In essence, the tidally driven redistribution of masses inside the Earth produces coherent changes in the inertia tensor (matrix containing the principal 129 moments of inertia on the main diagonal and off-diagonal products of inertia) that is reflected in the respective motion of the pole and the changes in the rotational rate. In addition, the center of mass will also change its location with respect to its equilibrium position (Pavlis, 1994). The influence of the solid Earth tides on the change of the inertia tensor is the largest and most important effect. However, ocean and atmospheric tides having the same frequency as the lunisolar tides contribute to the changes in the inertia tensor (for the major tidal frequencies see Moritz and Mueller, 1987, Table 1.1, p. 28 or McCarthy, 199S, Tables S.l and 5.2, pp. 33-34). For most of the tidal frequencies the ratio of the changes in the inertia tensor due to the elastic Earth and due to the ocean tides is about 10-15. The respective ratio of the solid Earth and the atmospheric tide is about 100, so the dominance of the solid Earth is evident (Lambeck, 1980, p. 107). The current accuracy of the daily and subdaily ERP determination is also limited by the inaccuracies in the models of the solid Earth and ocean tides, ocean currents, and limited accuracy of the atmospheric excitation determination, as well as the relatively weak knowledge of the ground water distribution. The sources of the rotational variability that can be modeled must be removed from the total observed signal in order to analyze the residual signature. The remaining part of the spectrum should, ideally, be of the frequency that is not a part of the effect removed, plus the observation noise. 6.1.1. Secular Variations in the Earth Rotation Parameters: An Overview Both polar motion and the rate of rotation undergo the secular variability detected by decades of observations of astronomical latitude, LLR, SLR, and VLBI (pole) and analysis of the discrepancies of the recorded and computed eclipses (rotational speed). The pole moves about 0.002-0.004 arcsec/year in the direction of 280° longitude. The reason for this motion is not clear but the major ‘Suspects” are the mass exchange between the Greenland ice sheet and the oceans, and continental drift 130 (Moritz and Mueller, 1987, p. 509). Based on ancient records of solar and lunar eclipses, lunar occultations and more precise recent LLR, SLR and VLBI observations, Stephenson and Morrison (1995) reported the secular increase in l.o.d. of the order of 1.7 ms/cy. That is reflected in a secular deceleration of the Earth. However, the increase in l.o.d. of 2.3 ms/cy is expected from the analysis oftidally driven acceleration of the Moon of -26 arcsec/cy2, caused mainly by the dissipation of the energy of the ocean tides. Thus, there has to be a nontidal, accelerative component that shortens the l.o.d. by about 0.6 ms/cy. ‘The nontidal acceleration may be associated with the rate of change in the Earth’s oblateness, attributed to viscous rebound of the solid Earth from the decrease in load following the last deglaciation” {ibid.). The currently adopted rate of change in the Earth’s zonal coefficient h of -2.5 x 10 ' 11 per year implies a rotational acceleration equivalent to the rate of l.o.d. of -0.44 ms/cy, which can explain most of the 0 . 6 ms/cy shortening in l.o.d. 6.1.2. Long-Periodic Variations in the Earth Rotation Parameters: An Overview Long period polar motion within a range of periods of 25-30 years displays an amplitude of about 0.02-0.03 arcsec in each component. This is called the ‘Markowitz wobble” and the reasons for it are still unclear (Moritz and Mueller, 1987, p. 509). According to Eubanks (1993) the long period and decadal polar motion can be explained to some extent by the mass transfer of ground water, glacial ice and sea level. Very long-periodic rotational variability can be explained to some extent by the electromagnetic and topographic coupling between the core and the mantle. It has a period of about 1500 years with semiamplitude of about 4 ms (Stephenson and Morrison, 1995). In addition, there exists some “... enigmatic fluctuations in length-of- day on time scales of a few decades. These changes are so large that their existence has been known for the best part of a century, yet their explanation remains obscure” (Lambeck, 1988, p. 633). L.o.d. exhibits also interannual fluctuations, i.e., “... variations on time scales between 1 and 1 0 years with peak to peak amplitude of about 131 0.5 ms” (Eubanks, 1993). These changes are associated with periodic global oscillations in the oceans and the atmosphere. There are also long-periodic tidal variations in l.o.d., with the major period of 18.6 years (principal period for nutation) and amplitude of 1720498xl0'7 s. 6.1.3. Annual, Seasonal and Short-Periodic Variations in the Earth Rotation Parameters: An Overview Annual and seasonal variations in the pole position, and the rotational speed are mostly atmospheric in origin. In addition, all the fluctuations within the range of a few days to a year result from a mass redistribution in the atmosphere, as well as from changing wind patterns (Moritz and Mueller, 1987, p. 512). Short-periodic tidal variations with periods ranging from about a year up to one day or half a day are also present. For the major tidal frequencies and associated amplitudes affecting UT1, see Moritz and Mueller (1987, Table 8.10, p. 519). Diurnal and semidiurnal variations in l.o.d. and polar motion were analyzed by Herring and Dong (1994) based on eight years of VLBI data. ERP variability at these frequencies is largely tidally driven. Respective periods correspond to the largest lunar and solar tides, and the amplitudes range from 8 . 6 pis to 25.82 pis for the change in Universal Time. The amplitudes of the tidal changes in the pole coordinates are between 39 pias and 256 pias. Influence of the Atmospheric Angular Momentum on l.o.d. with diurnal and semidiurnal frequencies is estimated at the level of 1 pis. 132 6.2. Variations in the Rotational Speed The solid Earth, oceans, and atmosphere constitute a large mechanical entity whose parts interact in a process of angular momentum conservation within a system. Thus, any change in the effective angular momentum within one or more components of the system is reflected in the variability of the remaining counterparts. However, the dominant role of the solid Earth in this three-component entity has to be recognized here. Due to the predominance of the most stable and solid part within the system, the major portion of Earth rotation has been highly regular over the planet’s long history. The mean rotational velocity of the Earth equals 7292115xl0'n s'1, with the rate of rotation of 15.041067 arcsec per SI second, that corresponds to the period of rotation of 23h 56m 4.1s (1 sidereal day) (Mueller and Moritz, 1987, p. 5). The major problem of variable rotation determination is the estimation of the short-periodic (daily and subdaily) changes that are superimposed on the long-periodic and secular parts. The secular effect due to tidal friction (transfer of the angular momentum between the Earth and the lunar orbit) has been accumulated in l.o.d. in terms of hours during the Earth’s history (Brosche, 1994). The dominant role of the solid Earth is the main contributor to the regularity of Earth rotation. ‘The main causes of variability, detected by ever refined clocks and angular measurements, are connected with the two outer nonsolid parts [i.e., oceans and atmosphere] despite of their small mass fractions” (ibid.). Moreover, the influence of the atmosphere on Earth rotation is stronger than the oceans, despite its smaller mass with respect to the total mass of the fluid component (see Table 19). ‘The key for this lies in the occurrence of much greater relative motion in the atmosphere as compared with the oceans” (ibid.). The average values describing the solid Earth-ocean-atmosphere system, as well as its variations are displayed in Table 19 (Brosche, 1994). 133 Table 19. The influence of the oceans and the atmosphere on Universal Time: amplitudes of periodical changes Moment of Inertia Angular Mass Pig] Momentum Ikgnts1] Whole Earth sSolid 6.0x1024 8.1xl037 5.9xl033 Earth Oceans 1.4xl021 3.9x 10m 2.8X1030 Atmosphere 5.1xl0lg 1.4xl032 l.OxlO2* Corresponding Torque [kgnts*] Angular Momentum Change in UT of pig m2 s'1] the Solid Earth [ms] Atmosphere Seasonal 5.0x10** 2.4x10“ 20-30 Oceans Seasonal 3.0xl017 1.5X1024 1 Tidal (MJ 3.4x 102‘ 2.4x10“ 0.03 Secular Retardation of 5.0xl0lfi 1.5x10“ 2.3 ms/cyforl.o.d. the Whole Earth by (constant) (loss per year) change Tidal Friction Another important factor that has already been noted here is the secular deceleration of the Earth’s rotation due to the tidal friction. Tidal friction is one of the ‘fcide” effects of the solid Earth and ocean tides. It is responsible for the dissipation of energy that slows the Earth down and delays (in time) the tidal bulge that consequently does not stay in phase with the tide-generating body. The effect of tidal friction is more significant for the ocean tides and is two times larger than the tidal friction due to the solid Earth tide. The resultant secular increase of the length of day is given in Section 6 .1.1., together with the respective tidally-driven secular acceleration of the Moon. In addition to the tidal and atmospheric effects on the rotational speed that are mainly addressed in this chapter, irregular changes in l.o.d. up to about 0 .0 0 S s and periods from five years to several decades exist. They are most probably caused by the internal coupling between core and mantle due to the roughness of the core-mantle interface (Lambeck, 1975). In addition, small changes in the Earth inertia tensor can be related to some tectonic displacements caused mainly by large earthquakes. That leads 134 to some small variations in the Earth’s orientation. For example, two large earthquakes of this century, Chile, 1960 and Alaska, 1964, produced a shift of the mean pole of 0.01-0.02 arcsec (Lambeck, 1988, p. 566). 6.2.1. Lunisolar Effects on the Rotational Velocity Since the lunisolar potential changes the Earth’s inertia tensor through the tidal attraction, and the change caused mainly by the zonal part of the tidal potential is responsible for the change in l.o.d., thus we speak of the tidal variations of rotational velocity. The major tidal variations in the rotational rate of the elastic Earth with the liquid core (column 2) and for the solid elastic Earth (column 3) induced by the zonal tidal potential are displayed in Table 20 (Mueller and Moritz, 1987, p. 199). It can be observed form this table, that the presence of the liquid core introduces the difference in l.o.d. by about 11% with respect to the solid elastic Earth model. Table 20. Tidal variations in rotation rate induced by the zonal tidal potential; columns 2 and 3 represent amplitudes Period [days] A(Lo.d.) [ms] AfLo.d.) [ms] Elastic Earth with Solid Earth the Liquid Core 6798 -0.1270 -0.1420 3399 0.0013 0.0014 365.3 0.0224 0.0251 365.2 -0.0012 -0.0013 182.6 0.1410 0.1580 177.8 -0.0035 -0.0039 121.7 0.0083 0.0092 31.8 0.0307 0.0343 27.6 0.1600 0.1790 14.8 0.0266 0.0297 13.7 0.3040 0.3400 13.6 0.1260 0.1410 9.1 0.0582 0.0651 135 6.2.2. Effects o f the Ocean Tides on the Rotational Velocity The layer of water contained between the mean sea level and the variable tidal surface produces a twofold attraction on the solid Earth: a direct gravitational attraction of the layer itself and the loading effect due to the time-varying load on the solid Earth. This load deforms the elastic Earth and causes the respective change in its gravitational potential. The gravitational attraction of the oceanic water layer is expressed in terms of the spherical harmonics and "... serves as ‘driving potential’ for ocean loading effects, in almost exactly the same way as the lunisolar tidal potential... serves as a ‘driving potential’ for lunisolar effects” (Moritz and Mueller, 1987, p. 297). This potential affects both Earth’s rotational speed and the solid Earth tides (elastic, lunisolar deformations). Again, the second-degree zonal part is responsible for changes in l.o.d. The order of magnitude of the UT1 changes due to the ocean loading effect is about 0.02 to 0.07 ms according to Yoder e t al. (1981), which is smaller by about two orders of magnitude than the corresponding lunisolar effect (see Section 6 .2 .1 ). The major frequencies are monthly, fortnightly, daily and subdaily — similar to the frequencies of the major solid Earth tides. 6.2.3. Atmospheric Effects on the Rotational Velocity The total angular momentum of the atmosphere consists of two major terms: relative angular momentum due to wind velocity (motion or wind term) and the so-called matter or pressure term that is the function of the tensor of inertia of the atmosphere. Thus, the second term is mainly a function of the atmospheric mass distribution. The pressure term can be considered as a load term that changes the Earth inertia tensor, and thus, the Earth rotation. Any motion of the masses of the atmospheric layer changes its angular momentum and causes a change in the solid Earth’s tensor in order to preserve the total angular momentum of the system. The effect of the variations in the atmosphere on the ERPs are computed from the equatorial and axial Atmospheric Angular Momentum (AAM) functions (Moritz and Mueller, 1987, p. 300). There are equatorial components, responsible mainly for the change in the pole location and the axial component, causing the variations in the rotational speed. The axial component is mainly wind dependent. The major atmospheric effect on the rotational speed has a period of about a year, displaying strong seasonal character, and changes l.o.d. by about 1 ms (Moritz and Mueller, 1987, p. 321). Eubanks (1993) gives the amplitude of the annual cycle of approximately 0.35 ms and for the semiannual cycle about 0.3 ms. According to Salstein et al. (1993), the seasonal change in l.o.d. due to the atmospheric excitation can even reach 2 ms. The zonal Atmospheric Angular Momentum variations dominate the nontidal variations of l.o.d. at periods of one year and about 40 days (Eubanks, 1993). There are also atmospheric/oceanic oscillations in l.o.d. of a period between 40-50 days that can reach 0.5 ms (ibid.). Periods as small as 8-14 days show some influence from the oceans and the core as well. 6.3. Variations in the Location o f the Pole Polar motion is defined as the rotation of the true celestial pole (celestial ephemeris pole, CEP) around the reference pole, that is, the pole of the selected Conventional Terrestrial Frame. The major components of the motion of the pole is the Chandlerian motion, with a period of about 434 ± 2 days and variable amplitude, and the annual motion. Both are counterclockwise rotations as viewed from the North Pole. The amplitude of the Chandlerian wobble varies from 3 to 11 m (about 0.2-0.3 arcsec) indicating that there must be a source of excitation and also a dumping mechanism. However, the primary reasons for both dumping and excitations are still unclear. According to Lambeck (1988, pp. 562-569) the source of the dissipation of the Chandlerian wobble energy is in the mantle, whereas the source of the excitation energy comes from the atmosphere, large earthquakes, ground water redistribution and the electromagnetic torques on the base of the mantle. Eubanks (1993) suggested that the discrepancy between the predicted (theoretical) and the observed Chandlerian 137 period arises from the influence of the ‘honequilibrium Chandler pole tide,” and from the dispersion associated with mantle nonelasticity. The pole tide is due to the variability in the centrifugal part of the gravity potential caused by the change in the orientation of the rotational axis. It is driven by the Chandlerian frequency and by the annual pole variations. Its magnitude is only a few millimeters and can be observed by very stable superconducting gravimeters (Moritz and Mueller, 1987, pp. 285, 521; Lambeck, 1988, p.559). Chandlerian wobble as well as nearly diurnal free wobble (still remains largely unobserved) represent the free rotational mode. In addition, continuous mass redistribution in the oceans and the atmosphere causes some excitation that results in the variations in the pole location. The motion of the CEP averages out over a period of six years. 6.3.1. Effects o f the Solid Earth Tides on Polar Motion The second-degree tesseral part of the tidal potential induces changes in the Earth’s inertia tensor that are responsible for the variations in the pole location. The effect of the solid Earth tides on polar motion is of the order of 60 cm that corresponds to 0.02 arcsec (Moritz and Mueller, 1987, pp. 140, 300). 6.3.2. Effects o f the Ocean Tides on the Polar Motion The tesseral part of the ocean tide gravitational potential is responsible for the ocean-driven changes in polar motion, precession and nutation. According to Yoder et al. (1981), the effect of ocean loading on polar motion is of the order of 0.0001 to 0.0003 arcsec, that corresponds to about 1 cm in the shift of the pole. The maximum effect due to water redistribution is about 50 cm for the tidal period of 18.6 years, 37 cm for the annual and 32 cm for the semiannual terms. Displacement of ocean water with periods of one day and shorter contributes to polar motion at the order of 0 . 0 1 cm. 138 6.3.3. Atmospheric Effects on the Polar Motion The angular momentum functions that describe the change in the pole location due to the atmosphere are mainly pressure-term dependent. They are called the equatorial components of the Atmospheric Angular Momentum functions (see Section 6.2.3). Variations in the pole location due to the atmospheric effect are mainly annual and seasonal. However, practically continuous mass redistribution in the atmosphere results in high frequency changes in the pole location, showing even diurnal and semidiurnal variability. The annual period has an amplitude of about 0.05-0.1 arcsec (Torge, 1980, p. 38). The motion of the pole is generally more sensitive to variations in the atmosphere than l.o.d. For example, the variation of 4.10 Pascals in the surface pressure causes the change in the pressure term of the atmospheric excitation functions of 1 mas. Similar variation in the wind term is caused by the 2.59 mm/s change in the atmospheric winds. In order to obtain 1 ms variation in the axial angular excitation, the surface pressure or the wind have to change by 3900 Pascals or 2.69 m/s, respectively (Eubanks, 1993). 6.4. Estim ability ofERPs from Satellite Techniques Earth Rotation Parameters cannot be observed directly by any means. They can be estimated, for example, based on the change in the location of some reference points (telescopes, VLBI antennas) with respect to the adopted inertial reference frame. In case of satellite techniques like GPS, the range (or phase range) between the spacecraft and the terrestrial site is observed. The ERPs become a part of the GPS observation equation providing the transformation between the terrestrial frame, where the station coordinates are given, and the inertial frame, in which the satellite positions are determined, as explained in Chapter V. Since satellite techniques are dynamical by nature, they determine the Earth’s orientation with respect to the motion of the spacecraft with limitations set up by errors in the orbital dynamics. So without 139 additional information, satellite techniques are not capable of separating the changes in the Earth’s orientation from the variations in the orbital elements. As noted many times throughout this volume, satellite techniques are not sensitive to the origin of UT1 simply because the motion of the orbital node is correlated with the rotation of the Earth. Due mainly to the mismodeling in the even zonal harmonics of the Earth gravity field, ocean tides, and atmospheric effects, the right ascension of the orbital ascending node is affected by the long-term errors, preventing the resolution of UT1 (Gambis et al., 1993). Thus, only the rate of rotation (i.e., the first derivative of UT1) as explained in Section 5.5.5 can be determined by means of satellite observations, since ‘GPS measurements ... are sensitive to UT1-UTC time derivative” (Zumberge et a l., 1993), provided that the rate of the orbital node is not estimated (since the rate of UT1-UTC and the rate of the node are correlated). However, as addressed in Section 5.5.5, there is no need for the estimation of the drift in the right ascension of the node with the currently applied dynamical model. Another limitation is the data distribution, addressed already in this chapter. With the current density of GPS permanent tracking array this requirement should be fulfilled. In order to provide the absolute rotational orientation with long time stability, VLBI estimates ofUTl are introduced in addition to the GPS-derived time series. GPS-derived rates of UT1 are linked to the VLBI offsets of UT1. The question of how often GPS needs to be updated by the VLBI-based time estimates is addressed in Chapter VII. Currently, GPS is capable of providing high quality daily estimates of dUTl, i.e., the rate ofU Tl (see Chapter VQ). UT1 determination with resolution from a few hours up to 30 minutes is currently an area of intensive research. Chapter VII presents some results related to the determination of daily and subdaily part of the spectrum of l.o.d. changes with triple differences. The diurnal and semidiurnal variability in l.o.d. are attributed to the tidal effects of the oceans but the influence of the atmospheric tides in the diurnal signal cannot be excluded (Freedman et al., 1993; Eubanks, 1993). Since the variability of AAM over a day is very small, it cannot be effectively separated from the ocean tidal 140 effect at the same frequency. Herring and Dong (1994) also listed effects of triaxiality of the Earth and the equatorial second-degree harmonic of the core-mantle boundary as contributors to the subdaily l.o.d. and daily prograde (eastward) polar motion variability. The offset of the pole is well observed with once per day resolution from the GPS observations. Applying longer arcs, for example three days, allows for the solution of the pole offset and the rate (Rothacher et a l , 1994). However, a major problem appears when one attempts to solve for the pole in the diurnal and semidiurnal band. “There is degeneracy between the polar motion parameters and the nutation and precession parameters, in that it is not possible to distinguish between a periodic variation in [nutation in longitude] and 5e [nutation in obliquity] and a corresponding diumally modulated variations in xp and yp” (Eubanks, 1993). Thus it is possible to observe only the linear combination of nutation and polar motion parameters. In particular, the retrograde (westward) diurnal polar motion cannot be separated from the nutation term. The coupling between the polar motion and the nutation can be easily concluded from the analysis of the changes in the total rotation matrix, describing the transformation between the terrestrial and inertial frames (see Chapter m ), as presented by Eubanks (1993). For more involved discussion see Montag et a l (1992) and Eubanks (1993). Moritz and Mueller (1987, p. 567) show a very straightforward and elegant explanation of these phenomena; the influence of any change in the nutation on the ERPs is described as follows: A xp = m = n e 'B (6.3) Clearly, the nutation error looks like a diurnal polar motion with sidereal frequency 0 . Another problem, as briefly outlined above, arises from the mismodeling of the orbital elements that manifests itself as an error in the transformation matrix between the celestial inertial frame and the dynamically defined, satellite-based quasi- inertial frame. These long-periodic orbital errors, as viewed from the inertial frame (Eubanks, 1993), cannot be distinguished from the Earth rotation. Using a complex .notation to represent the variations in the orbital elements: h = A I -lA fltan/, where AH and A0 denote variations in 0 and £2 (orbital node) and AT represents variations in the orbital inclination, the following can be written (Watkins e ta l., 1994): h = im e,ae~19 = ineiQ (6.4) Clearly, any constant nutation error looks like a long period orbit error, with frequency of the orbital node, and a constant polar motion error looks like a nearly diurnal orbit error. Conversely, any orbital error can be observed as nearlydiurnal polar motion, as shown in eq. (6.5). m = -ihe~,£Ve (6.5) According to Lambeck (1988, p. 321) the variations in the orbital inclination I and node £ 2 due to the erratic polar motion and sidereal time reads as follows: 142 A / = m sin(n - © - A) (6.6) AQ=A0 + m cos( 0 - 0 - A)co t/ where A = tan describes the direction of the rotation axis. Thus, the \ y , orientation of the orbit is subject to the diurnal perturbations (frequency 0 ) with amplitudes corresponding to the errors in the polar motion components, as already shown in eq. (6.4). Perturbations of the node absorb directly any errors in the sidereal time. However, if the corrections to nutation are also estimated in the process of orbit determination, the diurnal retrograde polar motion should be absorbed by the nutation (Herring e ta l., 1994). 6.5. Estimation o f ERPs by Means o f GPS Observations The GPS-derived ERPs have been routinely provided to IERS since October 1992 (Bulletin B No. 58) by the IGS centers. More than a two-and-a-half-year long history of GPS ERP estimates calls for a presentation of these results and the combined IERS solution in order to show the gradual improvement gained by a satellite technique during the course of the IGS development. Figures 17, 18 and 2 0 show the differences between ERP estimates derived by the Center of Orbit Determination in Europe (CODE) and the final Bulletin B solution that is dominated by VLBI. Figures 17 and 18 show that the scatter between GPS-derived and IERS pole components decreases with time, making the satellite technique more and more compatible with VLBI. However, the difference still exists. In general, the satellite-derived yp coordinate agrees with the IERS combined solution better than the xp component, as can be observed in Figures 17 and 18. 143 — xp(Bull«tfn B)-xp(CODE) #.0 # - 2.00 0 100 200 Tima opocho ovary 5 dayo Figure 17. The difference between x p derived by CODE and final Bulletin B estimates between October 1992 and December 1994 For both pole components the initial agreement with Bulletin B final estimates was much weaker at the beginning of IGS but keeps improving during the course of the IGS development. This improvement is due to the significant progress in the dynamical modeling, continuous cooperation among IGS analysis centers, product evaluation, standardization of the activities, and the terrestrial frame improvement. A constantly increasing number of permanent tracking stations and better quality of GPS receivers are also important contributors to quality enhancement. 144 -*-yp(Bullatln B)-yp(CODE) 1.00 • .1.00 •4.00 0 100 200 Tlmo opocha ovary 5 daya Figure 18. The difference between y p derived by CODE and final Bulletin B estimates between October 1992 and December 1994 Comparison of GPS-derived UT1-UTC time series with VLBI-dominated IERS estimates is not very straightforward. It is due to the fact that the satellite technique uses its l.o.d. estimates and UT1-UTC value from VLBI at some epoch to reconstruct the UT1-UTC series. Thus, the agreement between both series depends not only on the quality of GPS-derived l.o.d., but also on the adopted absolute value of UT1-UTC that might cause the offset between the two series. 145 CODE (UT1-UTC) Bulletin B (UT1-UTC) 800.00 700.00 600.00 500.00 400.00 300.00 200.00 100.00 0.00 - 100.00 • 200.00 -300.00 -400.00 10« 2 0 * Tima epocha avary 6 daya Figure 19. (UT1-UTC) derived by CODE and final Bulletin B estimates between October 1992 and December 1994 Figure 19 represents the comparison of UT1-UTC series derived by GPS and the combined IERS solution, strongly dominated by VLBI. The jumps at epochs 55 and 128 refer to the introduction of a leap second on July 1, 1993, and July 1, 1994, respectively. The difference between the two time series is plotted in Figure 20. 146 — (UT1-UTC) {Bulletin B-CODE) • 1110 200 Tlmo opooho ovry S days Figure 20. The difference between (UT1-UTC) derived by CODE and final Bulletin B estimates between October 1992 and December 1994 Clearly, at epoch 85 that corresponds to November 28, 1993, there must have been a reset of the absolute UT1-UTC to which CODE estimates of Lo.d. are linked. It should be emphasized here that the high quality currently claimed for GPS-derived ERPs by the OSU and IGS centers cannot be truly observed based on the graphs above since the differences between the CODE and IERS series over the course of two years are significant and certainly exceed the formal uncertainties of the satellite technique. But it should be understood that the IERS series presented here is based on several techniques and thus represent some form of average solution and is also smoothed. Moreover, recent improvements in the IGS processing strategy led to a significant increase in the quality of satellite-derived ERPs. The most spectacular improvement in 147 IGS ERP determination was obtained by introducing die new ITRF93 reference frame on January 1, 1995. Figures 21 and 22 present the CODE xp m lyp estimates and the IERS solution during the time period between January 1, 1995, and June 1, 1995. The dramatic increase in the quality of the GPS-derived pole series is clearly seen when compared to Figures 17 and 18. The UT1-UTC series derived by CODE for the same period of time shows the fit to the IERS solution similar to the one presented in Figure 20. However, as mentioned before, the quality of fit depends on the initial UT1-UTC offset adopted in the solution. Rather, what should be compared is l.o.d., but CODE-derived l.o.d. is not published in Bulletin B. The comparison of GPS-derived l.o.d. and IERS l.o.d. is discussed and illustrated in Section 7.5 where it is shown that the GPS signal provides valid ERPs, highly compatible with the VLBI solution at the time basis of at least one month. — xp(Bulfetln B)-xp(CODE) 2.00 1.00 m•» E 0.00 0 10 20 30 40 Tima apocha avary 6 day* Figure 21. The difference between xp derived by CODE and final Bulletin B estimates between January 1 and June 1, 1995 Figure 22. The difference between y , derived by CODE and final Bulletin B B Bulletin final and CODE by derived , y between difference The 22. Figure [maa] 0.400 0 0 .0 0 0.200 o.ooo o.aoe 0.000 0.400 o.ooo o.ooo 0.700 a.ooo o.ooo 0.100 t.ioo 1.000 estimates between January 1 and June 1,1995 June and 1 January between estimates 10 20 20 10 ia pca vr 5 daya 5 avary apocha Tima y(ultn B)-yp(CODE) -yp(Bullotln CHAPTER VII GPS TRAJECTORY AND RELATED PARAMETER ESTIMATION USING A BATCH LEAST-SQUARES ESTIMATOR: EXPERIMENTS DESCRIPTION AND ANALYSIS OF THE RESULTS 7.1. OSU Standard Solution: D escription In this chapter, the results of the experiments of the application of the triple-difference to the estimation of the precise GPS trajectories and other model parameters are presented. The numerical results were obtained from the GPS data collected at 36 terrestrial sites for the continuous time period of four weeks (GPS weeks 784-787 that correspond to January 15 - February 11, 1995). The data were collected at 30-second intervals and processed every 15 minutes with an elevation angle cutoff of 16°, as noted in Chapter V. The 32-hour arc length adopted here results from some tests that were performed to determine the optimal length for the data file processed. The 24-hour arc was found to have the largest deviations from the IGS orbit at the beginning and at the end of the arc, whereas the middle portion agreed fairy well. This led to the concept of the overlapping orbits for which several different options were tested. First, a 12-hour overlap was examined. The fit to the IGS solution over the entire middle 24 hours improved dramatically, but the processing time was much longer due to the larger data set used. The next logical step was to decrease the length of the overlap preserving the middle-arc quality at the same time. This way eight-, six- and finally four-hour overlaps were tested and the shortest one was found to be the best in the sense that the fit to IGS was maintained and the smallest possible data file size was adopted, decreasing the processing time. 149 150 In the orbit determination process nine orbital parameters are estimated per satellite, including the initial position and velocity in the adopted mean of reference day (MoD) inertial reference frame, solar radiation pressure scaling factors in the satellite body-fixed X and Z directions, and the scaling factor to the a priori value of lx 10_ 9 ms' 2 of the Y-bias parameter. A priori values for the satellite state vector are obtained from the previous day’s solution with the following a priori standard deviations (assumed uncorrelated): • position (x, y, z) - 1 m • velocity (x, y, z ) - 0 . 0 0 0 1 ms' 1 • solar radiation X, Z scaling factors - 10% • Y-bias scaling factor - 15 % By constraining the positions and velocities of the 13 IERS fiducial stations according to their standard deviations as given in IGS Mail #819, the adopted terrestrial frame is realized as the International Terrestrial Reference Frame 1993 (ITRF 93) at epoch 1995.0. The coordinates of the remaining 23 stations are almost freely adjusted with a 50 m a priori standard deviation for each of the Cartesian coordinate components. The troposphere is typically estimated at four-hour intervals per station as a scaling factor to the mapping function of the modified Hopfield model. The ionosphere-free (first-order) phase is used as the fundamental measurement. Second- and higher-order ionospheric terms are ignored (Seeber, 1993, p.44). The Earth Rotation Parameters that are estimated once per day are xp and yp components of the polar motion and the rate of (UT1-TAI), referred to as negative rate of change in length of day. The IERS Bulletin B predictions are taken as a priori values with the following uncorrelated standard deviations, commonly used by the IGS analysis centers (IGS Mail, numbers 654 to 657, 695): 151 • xp and yp - 0.097 arcsec • d(UTl-TAI) - 6.47x 10 -3 s/day The results corresponding to the routine approach described in Chapter V and in this section are presented first. Additional experiments were run in order to test the capability, flexibility, and quality of the new technique to provide a valid solution under different conditions. The test results are thoroughly discussed in the following sections. The departures from the standard processing are indicated in every case together with the underlying reasons. For some of the tests described here the final combined IGS orbits for GPS weeks 784-787 were used as a reference to the OSU solution (name adopted here for the solution generated by GODIVA at The Ohio State University). Also, the final Earth Rotation Parameters from IERS Bulletin B are used as a quality standard for the OSU ERP solution. 7.2. Experiment One: Standard OSU Orbit Determination 7.2.1. OSU Solution Versus IGS Solution Orbit determination test results over four consecutive weeks (GPS weeks 784-787) are presented here. The intercomparison of OSU, IGS analysis centers and the combined IGS orbit estimates is completed by the ORBCMP software developed by the IGS analysis center at Bern, Switzerland. The daily summary report tables containing the orbit comparison statistics comprise seven-parameter similarity transformation components for every pair of analyzed trajectories. The mean RMS of the orbit difference per coordinate is also listed as an indication of the agreement of any two orbits after the removal of the systematic differences between respective reference frames (by performing a seven-parameter transformation). An example of a full summary report table is given below in Table 21. It is the output file from the ORBCMP program that performs the similarity transformation between every pair of trajectories specified in the input option file. 152 Table 21. Summary report for orbit comparison for January 15,1995 Translations and RMS - [meters], Rotations - [mas], Scale - [ppb] MODIFIED JULIAN DATE DAY MONTH YEAR 49732 15 1 1995 DX DY DZ RX RY RZ SCALE RMS -0.005--0.002 0.002 -0.4 -0.3 0.0 0.0 0.09 igs0784-->cod0784 0.001 0.021 0.028 -0.7 -0.2 -0.1 0.0 0.10 igs0784-->emr0784 0.005 0.007 0.030 -0.1 -0.2 0.4 0.0 0.16 igs0784-->esa0784 0.033 0.013 0.011 0.3 -0.5 0.0 0.6 0.12 igs0784- ->gfz0784 0.000-•0.054 0.017 0.0 -0.6 -0 .2 - 0 .3 0 .0 9 ig s0 7 8 4 - -> jp l0 7 8 4 -0.003-•0.009-0.007 0.3 -1.1 0.2 -1.1 0 .2 1 igsQ784-->ngs0784 -0 .0 1 1 0.049 0.026 -0 .9 -0.6 0.4 0.1 0.12 igs0784- ->osu0784 -0 .0 5 8 0.083-0.241 0.2 -1 .3 0.5 0.2 0.23 igs0784- -> sio0784 0.006 0.023 0.027 -0.3 0.1 -0.1 0.1 0.16 cod0784- ->emr0784 0.010 0.009 0.028 0.3 0.1 0.4 0.0 0.17 cod0784- ->esa0784 0.038 0.015 0.010 0.6 -0 .2 0.0 0.7 0.16 cod0784- ->gfz0784 0.006-•0.052 0.015 0.4 -0.3 -0.2 -0.2 0 .1 3 cod0784- -> jp l0 7 8 4 0.003-•0.007-0.008 0.7 -0.8 0.2 -1 .1 0.24 cod0784- ->ngs0784 -0.006 0.051 0.024 -0.5 -0.3 0.4 0.2 0.15 cod0784- ->osu0784 -0.053 0.086-0.243 0.6 -1.0 0.5 0.3 0.28 cod0784- -> sio0784 0.004--0.014 0.002 0 .6 0 .0 0.5 0.0 0.21 emr0784-->esa0784 0.032-■0.008-0.017 1.0 -0.3 0.1 0.6 0.18 emr0784- ->gfz0784 -0.001--0.075-0.011 0.7 -0.3 -0.1 -0.3 0.16 emr0784-->jpl0784 -0.004-0.030-0.035 1.0 -0.9 0.3 -1.1 0.24 emr0784- ->ngs0784 -0.013 0.028-0.003 -0.1 -0.4 0.5 0.1 0.14 emr0784- ->osu0784 -0.060 0.062-0.270 0.9 -1.1 0.6 0.2 0.24 emr0784- -> sio0784 0.028 0.006-0.018 0.4 -0.3 -0 .4 0 .6 0.21 esa0784- ->gfz0784 -0.005-■0.062-0.013 0.1 -0.4 - 0 .6 -0.2 0.21 esa0784- -> jp l0 7 8 4 -0.008-•0.016-0.037 0.4 -0.9 -0 .2 -1 .1 0.28 esa0784- ->ngs0784 -0.017 0.042-0.004 -0.7 -0.4 0.0 0 .1 0.21 esa0784- ->osu0784 -0 .0 6 4 0.076-0.271 0.3 - 1 .1 0.1 0.2 0.31 esa0784-->sio0784 -0.032--0.067 0.005 -0.3 -0.1 -0 .2 - 0 .9 0.16 gfz0784- -> jp l0 7 8 4 -0.036-■0.022-0.018 0.0 - 0 .6 0.2 - 1 .7 0.27 gfz0784- ->ngs0784 -0 .0 4 4 0.036 0.014 -1 .1 - 0 .1 0.4 -0 .5 0.17 gfz0784- ->osu0784 -0 .0 9 1 0.070-0.253 -0 .1 -0.8 0.5 -0.4 0.28 gfz0784-->sio0784 -0.003 0.045-0.024 0.3 -0.5 0.4 -0.9 0.26 jpl0784- ->ngs0784 -0.012 0.103 0.009 - 0 .9 0 .0 0.5 0 .4 0.16 jp l0 7 8 4 - ->osu0784 -0.059 0.138-0.258 0.2 - 0 .7 0 .7 0.5 0.25 jp l0 7 8 4 - -> sio0784 -0.009 0.058 0.032 -1 .1 0.5 0.2 1 .2 0.26 ngs0784- ->osu0784 -0.056 0.092-0.235 - 0 .1 -0 .2 0 .3 1 .3 0 .30 ngs0784- -> sio0784 -0.047 0.034-0.267 1.0 - 0 .7 0.1 0 .1 0.25 osu0784-->sio0784 In Table 21 DX, DY, DZ denote the translation components in X, Y and Z Earth-centered-fixed directions, respectively. RX> RY, RZ denote the rotation angles 153 about X , Y and 2 Earth-centered-fixed directions, respectively. SCALE and RMS denote the scale coefficient and the mean RMS of fit after the similarity transformation between every pair of orbits. In Table 22 the daily RMS of fit of every analysis center orbit (including OSU) to the IGS solution is presented. Clearly, the OSU comparisons to the IGS orbit are very consistent for all 28 days analyzed. Table 22. The daily RMS of fit of the Analysis Center orbits (including OSU) to the IGS solution for GPS weeks 784-787 [m] Day of Year IGS-* IGS—* IGS—* IGS—* IGS-* IGS—* IGS—* IGS-* COD EMR ESA GFZ JPL NGS OSU SIO 15 0.09 0.10 0.16 0.12 0.09 0.21 0.12 0.23 16 0.13 0.10 0.19 0.13 0.10 0.23 0.14 0.28 17 0.11 0.13 0.18 0.12 0.10 0.24 0.16 0.26 18 0.13 0.19 0.17 0.13 0.10 0.21 0.16 0.20 19 0.10 0.14 0.66 0.12 0.15 0.20 0.16 0.20 20 0.11 0.12 0.18 0.13 0.10 0.24 0.16 0.21 21 0.13 0.11 0.16 0.12 0.09 0.25 0.14 0.21 22 0.09 0.14 0.13 0.12 0.09 0.21 0.14 0.19 23 0.10 0.11 0.17 0.16 0.09 0.19 0.15 0.21 24 0.11 0.11 0.17 0.15 0.10 0.23 0.15 0.21 25 0.11 0.12 0.17 0.13 0.10 0.20 0.14 0.22 26 0.10 0.13 0.18 0.11 0.09 0.18 0.16 0.33 27 0.10 0.15 0.16 0.13 0.08 0.15 0.15 0.19 28 0.11 0.13 0.17 0.12 0.09 0.18 0.18 0.23 29 0.11 0.13 0.16 0.10 0.09 0.19 0.16 0.25 30 0.10 0.12 0.16 0.10 0.09 0.19 0.16 0.19 31 0.10 0.13 0.16 0.10 0.20 0.20 0.18 0.24 32 0.10 0.15 0.20 0.10 0.09 0.20 0.15 0.23 33 0.11 0.12 0.18 0.12 0.09 0.21 0.14 0.22 34 0.10 0.12 0.15 0.10 0.09 0.19 0.16 0.29 35 0.10 0.10 0.14 0.12 0.09 0.19 0.15 0.25 36 0.10 0.11 0.16 0.09 0.09 0.19 0.15 0.24 37 0.12 0.13 0.17 0.13 0.08 0.24 0.16 0.26 38 0.11 0.11 0.17 0.10 0.08 0.22 0.17 0.26 39 0.10 0.12 0.16 0.10 0.10 0.21 0.16 0.23 40 0.10 0.12 0.15 0.10 0.09 0.20 0.15 0.24 41 0.11 0.14 0.14 0.13 0.09 0.25 0.15 0.33 42 0.13 0.11 0.15 0.12 0.09 0.24 0.15 0.22 154 The summarized results of orbit comparison are displayed in Table 23. Clearly, the triple-difference approach together with the adopted force model and the iterative solution strategy implemented in this study have achieved the required accuracy of one part or less in 10* (20 cm accuracy in the satellite trajectory) for today’s precise GPS orbit determination. Table 23. The Mean RMS of fit, after the transformation, of orbit comparison among the IGS centers including OSU for GPS weeks 784-787 [cm] EMR ESA GFZ JPL NGS SIO OSU IGS COD 18.0 20.3 17.9 15.3 24.8 28.6 18.2 10.75 EMR 22.9 18.7 17.0 25.3 27.0 19.2 12.46 ESA 20.5 22.6 28.8 30.9 22.3 18.21 GFZ 16.7 26.6 28.0 18.9 11.79 JPL 25.0 26.3 18.5 9.75 NGS 29.6 27.0 20.86 SIO 28.2 23.64 OSU 15.36 Judging from the RMS values of orbit comparison between the IGS and the seven analysis centers, it appears that the fit of JPL and COD solutions to the average orbit is the best. While the RMS values between JPL and the other centers range from IS.3 to 26.3 cm, the RMS between JPL and IGS is merely 9.7S cm. The RMS values between COD and the other centers ranges from 15.3 to 28.6 cm, and the RMS between COD and IGS is 10.75 cm. These numbers suggest that those two centers gain larger weights in the averaging process than the rest of the participants. Even though OSU did not participate in the computation of the IGS orbit, the mean RMS between OSU and IGS is only 15.36 cm. The mean RMS of fit of the OSU solution to the IGS most heavily weighted solutions provided by JPL and COD is 18.5 and 18.2 cm, respectively. 155 7.2.2. Baseline Repeatability Another important indication of the precision achieved in the orbit determination process is daily baseline repeatability. As described earlier, accurate GPS baseline estimates with precision of one part or less in 1 0 8 can only be obtained with high quality satellite trajectories. For the entire four-week period all possible baselines connecting the global stations used in this study are estimated. The mean RMS values of east, north and height components of the daily baseline repeatability versus baseline distance are presented in Figures 23, 24, and 25, respectively. 0.08 — 0.05 — 0.04 — «0 - ® 0.03 — © z 0.02 — 0.01 — 0.00 — 0 2000 4000 6000 8000 10000 12000 14000 Baseline Length (km) Figure 23. Baseline repeatability RMS (east component) versus baseline length for GPS weeks 784-787. All stations are used except for Pamatai and Herstmonceux due to possible receiver problems 156 £ 0.03 4-1. + + 44. 4 4-4.# ,44 4. 4 4 4* ... + #,„ * + V + -% + ?PtiW+III! 1 ■! H I iTHl i 1i 1r 2000 4000 6000 8000 10000 12000 14000 Baseline Length (km) Figure 24. Baseline repeatability RMS (north component) versus baseline length for GPS weeks 784-787. All stations are used except for Pamatai and Herstmonceux due to possible receiver problems 0.06 du 0.05 0.04 + ++ + M 4 + + 4 4 + + 4f + + *f+ + 4 A + + * 4 - y £a> 0.03 2 . j L * + 4 + 4 ^ f 4H + + $ + .p“ - 0.02 + +++44,#- ~+ A + -T+ ^ +i# 4 4VV 4 4 4 +W4,4. + .1. 4 # 444 44 0.01 +++ ++iv +v +^ ++ ^ + ^ + ++ 44* 4H*- H+ “fc|4 + + -w- + + ++ + + + ++H4. ++ 44 £4+ + i f t t j S t 4-ltt.“ 4f4-*^^+iH4ff+4. jtl- iu . 4J- 0.00 I '---- 1---- 1---- 1---- 1---- 1--- 1 1---- 1— 2000 4000 6000 8000 10000 12000 14000 Baseline Length (km) Figure 25. Baseline repeatability RMS (height component) versus baseline length for GPS weeks 784-787. All stations are used except for Pamatai and Herstmonceux due to possible receiver problems 157 Except for some obvious outliers, the majority of the RMS values of the east and north components are below 2 cm. The RMS values appear to grow somewhat g with increasing length of baseline and the relative accuracy ranges from one part in 1 0 9 to a few parts in 10 . The height component is clearly noisier, yet still with the majority of the RMS values below 3 cm that corresponds to the relative accuracy ranging from a g 9 few parts in 1 0 to a few parts in 1 0 . 7.3. Experiment 2: Influence o f the Nutation Model on the Orbits and EBP Estimates The standard IAU 1980 nutation model without corrections (Wahr, 1981; McCarthy, 1992, Chapter V) was initially implemented in GODIVA. However, the proposed IERS Standards (McCarthy, 1995) introduce the 1995 nutation theory according to Herring that provides very accurate a priori estimates of nutation. The Herring nutation model is an empirical model based on the analysis of VLBI and LLR data. In fact, this model will not yet replace the 1980 theory of nutation, and the corrections to the nutation in longitude and obliquity for Wahr’s model will still be published by IERS. Since nutation is not estimated in applications presented here, and the correct orientation of the terrestrial frame with respect to the inertial system is extremely important, especially for the ERP estimation, the accurate a priori nutation values are crucial. For that reason, the new nutation model was implemented in GODIVA. The test results are presented here. One week of data (GPS week 784) was used as test data for the comparison of the results obtained from the old and the new nutation models. The model parameters, their a priori values and standard deviations, and the estimation strategy correspond to those presented in Chapter V and Section 7.1. The only difference is the nutation model. The a posteriori standard deviations are presented in Table 24. Table 25 contains the ERP estimates resulting from both solutions. Clearly, the new nutation model does not have a significant influence on the trajectory and ERP estimation. Since at the current stage the Herring model does not 158 require any additional corrections based on the prediction or the real data (in post processing mode) contrary to the 1980 theory of nutation, it has been permanently implemented in GODIVA. The software comprising all the subroutines that compute the lunisolar and planetary nutation in longitude and obliquity was provided by Dr. Dennis McCarthy from the U.S. Naval Observatory. Table 24. The a posteriori standard deviations of a unit weight for the orbit estimation with 1980 nutation and Herring nutation model for GPS week 784 Nutation day 1 day 2 day 3 day 4 day 5 day 6 day 7 Model 1980 1.20383 1.27453 1.04293 1.02702 1.00883 1.11422 1.08762 1995 1.20333 1.27443 1.04228 1.02736 1.00915 1.11436 1.08839 Table 25. ERP estimates for GPS week 784 given at 12:00 TAI from the 1980 and Herring nutation models, based on the routine data processing Nutation day 1 day 2 day 3 day 4 day 5 day 6 day 7 Model 1980 x, : -0.12993 -0.12926 -0.12862 -0.12825 -0.12740 -0.12620 -0.12500 yp * 0.45300 0.45520 0.45758 0.45941 0.46129 0.46271 0.46447 UTl-TAI: -28.6383 -28.6406 -28.6429 -28.6453 -28.6480 -28.6507 -28.6535 d(UTl-TAI): -0.00217 -0.00218 -0.00230 -0.00246 -0.00256 -0.00271 -0.00278 1995 Xp : -0.12992 -0.12925 -0.12861 -0.12825 0.12740 -0.12620 •0.12500 yp * 0.45301 0.45521 0.45758 0.45939 0.46129 0.46271 0.46447 UTl-TAI: -28.6383 -28.6406 -28.6429 -28.6453 -28.6480 -28.6508 -28.6535 d(UTl-TAI): -0.00217 -0.00218 -0.00230 -0.00246 -0.00256 -0.00272 -0.00279 In Table 25 xp and yp are given in [arcsec], (UT1- TAI) in seconds and d(UTl-TAJ) in [s/day]. 159 7.4. Experim ent 3: Influence o f the Adopted Troposphere M odel on the Final Estimates o f the Trajectories and ERPs In this section the results of the experiment related to the influence of the troposphere modeling during the least-squares adjustment on the ERPs and the final orbit estimates are presented (see also Fankhauser et al., 1993). Several tests were run for the data of GPS week 784 with a common processing strategy (see Chapter V and Section 7.1), except for the number of tropospheric scaling factors estimated. The length of the interval for which the tropospheric conditions were assumed constant for every terrestrial site are different for all the tests. The adopted values are: four hours, eight hours, and 16 hours. In addition, the test run was made where no tropospheric parameters were estimated. The strength of the influence of the adopted tropospheric zenith delay model on the ERP estimation is clearly seen from the results presented in Figures 26, 27 and 28 that show the ERP estimates for all the test runs and the final Bulletin B evaluation. The differences between ERP estimates stay relatively close to each other as long as the troposphere is estimated. Fixing the tropospheric conditions over the whole data span results in dramatic deterioration of the ERP estimates. The impact of the number of tropospheric scaling factors estimated per day on the trajectory estimation can be observed in Table 26 which presents the a posteriori standard deviations of a unit weight for all the tests. 160 -®-4-hour solution -■-fl-hour solution -o-18-hour solution -♦-no troposphsrs -D-Bullotin B - 121.000 •124.000 5 -127.000 121.000 • 121.000 •130.000 1S1.000 16 10 17 IS 10 20 21 dsy of yssr Figure 26. Estimates of the as a function of the number of tropospheric scaling factors estimated per day -®-4-hour solution -■-8-hour solution -o-16-hour solution —♦—no troposphsrs -Q-Bullstln B 404.000 462.000 46S.OOO 468.000 464.000 452.000 16 10 16 19 20 21 dsy of yssr Figure 27. Estimates of the yp as a function of the number of tropospheric scaling factors estimated per day 161 -a-4-hour solution -•-8 -h o u r solution -o— 16-hour solution —♦—no troposphsrs -o-Bullotln B - 2.100 - 2.200 - 2.200 •2.SQ0 -2.700 10 II 17 10 20 21 dsy of yssr Figure 28. Estimates of the rate of (UTl-TAI) as a function of the number of tropospheric scaling factors estimated per day Clearly, decreasing the number of the tropospheric zenith delay estimates per day causes the increase in the standard deviation of a unit weight that indicates the weaker estimation of the orbit and other parameters of the model. However, it should be mentioned that the solution based on four-, eight-, and 16-hour intervals for troposphere evaluation do not show significant differences in ERP estimates. The critical part in the procedure is to consider the tropospheric zenith delay factors as the estimable parameters. But in general the longer the period for which the troposphere is assumed constant, the weaker the estimation of the parameter model. It can be observed in Figures 26, 27, and 28 that the difference between the standard OSU ERP estimates and IERS solution ranges from 0.1-0.38 mas for xp and yp. For the rate of (UTl-TAI) the agreement with final Bulletin B is within 0.01-0.04 ms. These numbers are very similar to the quality of fit to IERS reported by the IGS analysis centers (see 162 IERS Technical Note 17, 1995). For the GPS week 784 the OSU pole coordinates were overestimated with respect to Bulletin B estimates, and the rate of (UTl-TAI) was underestimated. The increase of the a posteriori standard deviation of a unit weight of about 2 0 % can be observed between every subsequent model presented in Table 26. However, the most dramatic increase of about 300-500% with respect to the four-hour model in the standard deviation of a unit weight can be seen in the last column of Table 26 representing the model with fixed atmospheric conditions over the entire 32-hour data span. Table 26. The a posteriori standard deviations of a unit weight for the test solutions with different tropospheric model Bay of Year 4-Hour Solution 8-Hour Solution 16-Hour Solution No Troposphere IS 1.203 1.354 1.715 8.011 16 1.274 1.579 1.886 5.434 17 1.042 1.190 1.332 3.920 18 1.027 1.167 1.451 3.787 19 1.009 1.183 1.347 4.744 20 1.144 1.454 1.824 5.218 21 1.088 1.234 1.423 3.810 The a posteriori coordinate standard deviations for selected PRNs are presented in Table 27. The coordinate standard deviation is clearly inversely proportional to the number of the tropospheric parameters estimated per day. The largest increase in coordinate standard deviation is observed when the troposphere is held fixed over the entire time span. Again, the number of tropospheric scaling factors estimated per day seems to be one of the critical factors that determine the quality of the orbit improvement process. 163 Table 27. The a posteriori coordinate standard deviations for PRN 2, 5 and 20 as a function of different number of tropospheric estimates per day [cm] PRN 2 Model dov 15 day 16 day 17 day 18 day 19 doy 20 doy 21 1 9.5 10.6 10.8 10.4 9.5 9.5 10.2 2 9.8 12.7 12.1 11.7 10.4 11.2 11.2 3 12.5 14.4 12.7 13.5 11.6 13.3 12.0 4 44.5 34.6 31.4 30.3 33.0 31.4 27.4 PRN 5 1 7.3 9.0 9.8 10.4 8.0 8.7 8.4 2 8.1 11.2 10.8 11.7 8.7 10.1 9.3 3 9.8 12.8 12.0 13.5 9.7 12.4 10.3 4 37.3 32.0 28.8 30.3 27.6 29.0 23.5 PRN 20 1 8.7 10.2 10.8 11.1 8.0 8.8 9.0 2 9.4 12.1 11.6 12.0 8.8 10.2 9.5 3 11.5 13.7 12.3 14.6 9.7 12.3 10.9 4 42.5 33.5 31.4 32.4 28.3 30.0 25.1 In Table 27 day denotes day of year; models 1, 2, 3, and 4 denote 4-hour, 8 -hour, 16-hour solutions and solution without the tropospheric scaling factor, respectively. To complete the discussion about the influence of the number of tropospheric scaling factors estimated per day on the model parameters, the a posteriori standard deviations in local east (E), north (N) and vertical, i.e., “up” (U) directions for stations in Matera, Italy, and Hobart, Tasmania, are presented in Figures 29 and 30. Clearly, the major impact of the troposphere modeling can be seen in the height component, as already discussed in Chapter II (see also Rothacher, 1992, section 3.2.2 and Santerre, 1991). Four vertical bars in E, N and U directions represent the standard deviations associated with four tropospheric models presented here. 164 ■ 4-hour solution s B-hour solution h 16-hour solution d no troposphoro s.oo — ------ Figure 29. The average a posteriori standard deviation in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Matera for GPS week 784 ■ 4-hour solution u 8-hour solution s 18-hour solution U no troposphsrs s.oo 6.00 4.00 £ 3.00 2.00 1.00 0.00 Figure 30. The average a posteriori standard deviation in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Hobart for GPS week 784 165 7 .5 . Experiment 4: Influence o f the A priori ERP Update From the Absolute Technique on the Final Estimates o f the Trajectories and ERPs The purpose of this experiment is to find how often GPS ERPs need to be updated from the absolute technique. Discussion of this problem follows the presentation of the results of the ERP estimates from the OSU routine approach (see Chapter V and Section 7.1 for explanation) in comparison with the IERS final solution. In Figures 31, 32, and 33 the xp and yp offsets, and the rate of (UTl-TAI) obtained from GODIVA are plotted along with the final IERS solution, for GPS weeks 784-787. The maximum difference in the xp offset between the OSU and IERS solutions is about 0.709 mas, with the mean .RMS' of fit equal to 0.295 mas. For thej'p offset, the numbers are 0.692 mas and 0.269 mas for the difference and RMS, respectively. — xp OSU solution - ■ - x p Bullstln B -•o.ooo • 100.000 • 110.000 • 120.000 • 100.000 10 20 so 40 so day of yoar Figure 31. xp from triple difference solution and the IERS solution for GPS weeks 784-787 Two main reasons for the OSU and IERS difference can be named here. The first one is the difference between terrestrial frames in the IERS solution (where the reference frame is defined by VLBI and SLR sites, and GPS fiducial points) and the 166 terrestrial frame adopted here. The second reason is the GPS orbit mismodeling. Yet the quality of the OSU pole solution is very similar to the accuracy of the pole estimates reported by one of the top IGS centers, CODE, that is within 0.3 mas (Rothacher etal., 1994). Clearly, the triple-difference estimates of the pole location are highly compatible with the combined IERS solution that is accurate up to 0.2 mas (IERS, 1995), but further improvements especially in the orbit modeling are necessary (see Chapter VIII) to reach the VLBI accuracy level. —*-yp OSU solution - • - y p Bullstln B 610.000 600.000 • 400.000 470.000 450.000 10 20 00 40 50 day of yoar Figure 32. from triple difference solution and the IERS solution for GPS weeks 784-787 The fit of the OSU estimates of the rate of Universal Time to the final Bulletin B solution is somehow weaker. The maximum difference with respect to IERS is about 8 8 ps/day with the mean RMS of fit equaling 51.6 ps. For comparison, the fit of the l.o.d. estimate from one of the currently leading IGS ACs, CODE, to the final Bulletin B estimates is about 30 ps/day (Rothacher et al., 1994). On average GPS provides about 60 ps precision in dUTl (IERS, 1995). During the first analyzed week (784) the 167 OSU rate of (UTl-TAI) fits fairly well to the IERS estimates, whereas during week 785 and parts of weeks 786 and 787 the apparent offset with respect to the Bulletin B estimates exists, most probably due to the GPS orbit mismodeling. A commonly recognized source of the satellite-derived UT1 contamination is the mismodeling of the Earth’s gravity even zonal harmonics, ocean and atmospheric tides (Gambis etal., 1993). These are reflected in the mismodeled orbital dynamics that can be observed as erratic ERPs. Therefore, “Universal Time solution combined by IERS from the individual series is mainly based on VLBI inertial techniques ...” but “... it is still possible to combine the high-frequency fluctuations contained in the GPS UT1 series with the long-term variations in the VLBI solution to derive a mixed UT1(VLBI+GPS) solution of great interest for its accuracy, time resolution but also for its economic advantage” (Gambis etal., 1993). The a posteriori standard deviations for xp and yp are between 0.1 and 0.14 mas, and for the rotational rate 1 0 ps/day or less for all the solutions within GPS weeks 784-787. These numbers are probably too optimistic, especially for dUTl estimates, since the fit to IERS is five times weaker. The uncertainties of IERS individual series of l.o.d. range between 10 and 20 ps/day for VLBI (occasionally, up to 200 ps/day), 30-70 ps/day (SLR), and 90-270 ps/day (LLR). The respective numbers for the pole are the following: 0.1-0.2 mas (VLBI), 0.1-0.7 mas (SLR), reaching occasionally even 1.3 mas (Bulletin B, No. 8 6 , April 3, 1995). The uncertainties in ERP determination by IGS centers as reported by Bulletin B are: 0.1- 0.2 mas in the pole coordinates and 10-30 ps/day in the rotational speed. From the above discussion it can be concluded that the quality of ERP estimates obtained with the triple-difference technique as implemented in GODIVA is currently in the IGS mid ranges. Thus, further improvement, especially in the orbit modeling, is desired (see Chapter Vm). The upgrading of the Earth even zonal harmonics, ocean and atmospheric tide models are also expected to improve the orbit quality. 168 — d(UH-TAI) OSU solution —■—d(UT1-TAI) Bullotin B •2.10 • 2.20 -2.40 -2 .0 0 •2.70 10 20 SO 40 SO day of year Figure 33. d(UTl-TAI) from triple-difference solution and the IERS solution for GPS weeks 784-787 In addition to the routine solution, the results of a test that was run to determine how often GPS needs the a priori values for ERPs from the Bulletin B prediction are presented. The standard approach described in Chapter V and Section 7.1 was modified accordingly to provide the new option where the frequency of the Bulletin B prediction used as a priori ERP values is a user-defined parameter. The a priori ERP values for Bulletin B were applied at the first day of the experimental data span that covers GPS weeks 784-787. The a priori ERPs for every consecutive day were obtained from the former day’s estimates. Table 28 presents the a posteriori standard deviations of unit weight for the solution with VLBI-derived a priori ERPs applied only once (Test Solution in Table 28) and for the standard solution. 169 Table 28. The a posteriori standard deviations of a unit weight for the solution with Bulletin B applied only once and for the standard solution Day of Year Standard Solution Test Solution IS 1.203 1.203 16 1.274 1.265 17 1.042 1.091 18 1.027 1.027 19 1.009 1.023 20 1.114 1.154 21 1.088 1.095 22 1.209 1.215 23 1.077 1.102 24 1.057 1.070 25 1.073 1.061 26 1.531 1.524 27 1.344 1.321 28 1.526 1.523 29 1.496 1.471 30 1.308 1.266 31 1.308 1.349 32 1.291 1.285 33 1.336 1.341 34 1.345 1.346 35 1.161 1.163 36 1.349 1.279 37 1.355 1.337 38 1.191 1.411 39 1.444 1.360 40 1.291 1.278 41 1.147 1.141 42 1.161 1.155 The numbers in Table 28 show clearly that the quality of the solution is maintained even if VLBI estimates of ERPs are used only at the beginning of a four-week period. In many cases, the standard deviation of unit weight is even smaller for the test run than for the standard approach. The stability of the pole and UT1 rate solution is clearly seen in Figures 34-36. However, in case of the Universal Time estimation, GPS has to be linked to the more stable solution coming from VLBI, due to the colinearity of the orbital node and UT1, as explained in Chapter VI. Figures 34, 35 170 and 36 present the comparison of ERPs from the standard solution and test solution and final estimates from Bulletin B. The fit of the standard solution to the IERS final evaluation was already presented for all the ERPs in Figures 31, 32 and 33. Therefore, the differences between the standard solution and the IERS final evaluation (del(xp), del(yi>), de\(d(UTl)) routine solution in Figures 34, 35, 36) and between the test results and the final estimates from Bulletin B estimates (test solution in Figures 34, 35, 36) are plotted here. It should be mentioned that commonly accepted IGS practice is to apply VLBI a priori ERPs at the beginning of every arc processed or at the first day of a week (EMR). -*-d»l(xp) routlno solution -B-dol(xp) toot solution 0.00 10 20 10 40 B0 day of year Figure 34. The differences between the standard xp solution and the final IERS estimate (routine solution), and between thejtj, test solution and the final IERS estimate (test solution) for GPS weeks 784-787 -*-d«l(yp) routine solution -u-d»l(yp) tost solution t.oo w m l o.os E -1.00 10 20 so 40 50 day of year Figure 35. The differences between the standard y p solution and the final IERS estimate (routine solution), and between the test solution and the final IERS estimate (test solution) for GPS weeks 784-787 The stability of dUTl estimated by GPS is remarkable. Yet, the long-term stability in an absolute (i.e., UT1) sense has to be maintained by an absolute technique (i.e., VLBI). The high-frequency information coming from the satellite technique is currently combined with the IERS determination of ERPs to provide the final estimates of the Earth orientation (IERS, 1995). However, the absolute technique is needed at most once per month. In addition, almost real-time accessibility of GPS products, makes the satellite technique superior with respect to the IERS prediction, that is, good only up to 0.5 mas for the pole and 50 |is for UT1 for the five-day prediction, whereas GPS provides daily pole coordinates and l.o.d. with the uncertainty of 01-0.2 mas and 10-30 |is/day, respectively. 172 -*-dBl(dUT1) routlna solution -»-d«l(dUT1) toot solution 0.00 • 1 0 0 .0 0 10 40 so day of year Figure 36. The differences between the standard d ( U T l) solution and the final IERS estimate (routine solution), and between the d ( U T l) test solution and the final IERS estimate (test solution) for GPS weeks 784-787 Ten- and 40-day predictions are naturally weaker, and the respective accuracy they provide is as follows: 4 mas (11 mas) for the pole and 110 ps (650 ps) for UT1. Numbers in parentheses pertain to the 40-day prediction. However, the quality of satellite-derived dUTl (and pole location) is highly dependent on the quality of the trajectories. For example, any mismodeling in the orbital node might contaminate the rotational speed solution as can be seen in Figure 33 where dUTl is evidently affected by some orbital problems during most of week 785 and parts of weeks 786 and 787. It can be only partially attributed to the increased number of eclipsing satellites in the middle of the analyzed four-week period as shown in Table 29. An additional factor that has a significant influence on the quality of orbits and ERPs is the data coverage. Due to a receiver problem (IGS Mail #857), data from the Matera station were excluded from the solution on days 22, 23, 24, 25 and 27, and data from Algonquin (ITRF station) were excluded on day 24, which weakened the ERP solution for these days considerably, as can be observed in Figures 31, 32 and 33. During the third week 173 of the four-week period analyzed here the following stations were temporarily excluded from the solution: Kootwijk (ITRF station), Ny-Alesund, and Tromso on day 30; Algonquin, Greenbelt, Kootwijk, Tromso (ITRF station), and Tsukuba on day 31; Kootwijk on days 32 and 33 and Ny-Alesund, Perth, and Tromso on day 34. During day 36, stations in Yellowknife (TTRF station) and Penticton were excluded. For day 39, data from Algonquin were not used. Matera and Santiago (ITRF station) were excluded on day 41. All of these stations were excluded due to receiver problems (JPL Analysis Report for week 786, 787). In addition, during week 787, PRN 23 that was eclipsing was reported with some additional modeling problems (ESA Analysis Report for week 787). Thus, any significant data loss is reflected in a quality of the solution that deteriorated with respect to the case when all the data were used. Table 29. Number of the eclipsing satellites per day during the GPS weeks 784-787 doy 15 doy 16 dov 17 doy 18 doy 19 doy 20 doy 21 4 4 4 4 4 4 4 doy 22 doy 23 doy 24 doy 25 doy 26 doy 27 doy 28 4 4 5 6 8 8 8 doy 29 doy 30 doy 31 dov 32 doy 33 doy 34 doy 35 8 7 4 4 4 4 4 doy 36 doy 37 doy 38 doy 39 doy 40 doy 41 doy 42 4 4 4 4 4 4 4 7.6. Experiment 5: Short-Periodic VT1 Determination with Triple Differences This section presents the results of the short-periodic UT1 determination using triple-difference phases. Satellite trajectories, station coordinates and the tropospheric scaling factors were estimated according to the standard OSU solution (see Chapter V and Section 7.1). Earth Rotation Parameters were estimated every 30 minutes as a step function, with a priori standard deviations of 6.0xl0 ' 3 s for UT1 offset, and 9.7 x 10 “ 2 arcsec for the pole components. A priori values for ERPs were obtained from Bulletin B prediction. Tidal variations in UT1 with frequencies up to 35 days were restored in the interpolated values. So the total nominal signal should comprise tidal and atmospheric effects in the Earth’s rotation contaminated by model errors and prediction uncertainty. The observed signal with 30-minute resolution should be capable of providing the signature of daily and subdaily frequencies of UT1 and pole variability. However, due to the lack of separability of the orbit and the daily retrograde polar motion (see Chapter VI) the solution for pole coordinates is not presented here. The pole signature is contaminated by the orbital error and the errors in the nutation series, so the true signal cannot be fully recognized. Some authors perform the spectral analysis of the observed signal and remove the retrograde diurnal polar motion from it before making any comparison with the VLBI-derived series (Ibanez-Meier et al., 1994). Others estimate specific parts of the spectrum excluding the retrograde diurnal part (Watkins et al., 1994; Ibanez-Meier et al., 1994). In this case, this signature will be absorbed by the satellite orbit or long-periodic nutation if correction to the nutation series are included in the model parameters. On the other hand, the offset of UT1 is seen fairly well by the 30-minute solution, as presented below. The typical a posteriori standard deviation for UTl-TAI was between 8xl0's s and 9x1 O ' 3 s and is higher than in the typical solution (see Section 7.2). The major reasons are the strong correlation among the model parameters and the orbital errors (see Chapter VI). The correlation among subsequent xp estimates is as high as 50-99% and according to Reigber et al. (1994) this shows a strong dependence on the interval adopted for ERP estimation. The shorter the interval for ERPs, the stronger this correlation will be. A similar correlation can be observed among subsequent and UT1 estimates. Correlation up to 60% exists among the xp and jr, estimates, whereas the respective correlation between Universal Time and the pole estimates is less than 10%. Similarly, correlation between ERPs and the station coordinates (also the tropospheric scaling factors) is at the level of 2%. Pole coordinates show the correlation with satellite position and velocity 175 elements between 1-60%, and with the solar radiation scaling factors 1-8%. Stronger correlations, at the level of 1-80%, are observed between the time estimates and the initial orbital elements. UT1 is weakly correlated with the solar radiation scaling factors. 7.6.1. Triple Difference Solution Versus Daily and Subdaily Herring UT1 Tidal Model The Herring empirical tidal model for the short-periodic (duly and subdaily band) changes in ERPs based on eight years of VLBI data (Herring et al., 1994) is used here as a comparison with the results obtained form the satellite technique. A tidally induced signal was extracted from the GPS-derived UT1 signature and compared with the empirical model of the ocean tides. Ocean tides are considered a major reason for daily and subdaily UT1 fluctuations (see Chapter VI). However, it should be emphasized here that since the Herring model is an empirically-derived model, it may contain not only the ocean tidal part, but also some diurnal atmospheric effects. The GPS-derived UT1 series was differenced with the Bulletin B reference series with the tidal effect of periods less than 35 days restored in order to show the part of the signal that was resolved by the satellite technique, as presented in Figures 37 and 38. 176 -ffi-GPB correction to nominal UT1 -o.teo -o.**o 7 JE -0 .1*0 -0.400 0 100 200 300 400 opocha ovary 30 mln Figure 37. Geodetically-derived corrections to the Bulletin B nominal series of UT1 with 30-minute resolution for GPS week 784 -S -0 P S correction to nominal UT1 • 0.100 - 0.200 "3* g -0.300 -0.400 •0.600 ------1------1------1------1------r 0 100 200 300 400 epoch* ovory 30 mln Figure 38. Geodetically-derived corrections to the Bulletin B nominal series of UT1 with 30-minute resolution for GPS week 785 Signals seen in both figures show some long-periodic signature with the short-periodic part superimposed on it. Since the tidally-induced part of the UT1 spectrum on daily and subdaily frequencies is sought, the geodetic signal has the best- 177 fitting quadratic subtracted to remove the long-periodic fluctuations. This part of the procedure can be considered as not fully rigorous, since the frequencies removed this way cannot be truly recognized; moreover, the quadratic could also be questioned as the best fitting curve here. However, this method is commonly practiced by research groups (see for example Freedman et al., 1993). In essence, the “average combined” long-periodic fluctuations are removed by the best-fitting quadratic. A more rigorous approach where specific bands of frequencies are explicitly removed from the GPS signal is presented in Section 7.6.2. (see Figures 43 and 44). The empirical tidal part of the Herring model was computed with subroutines provided by Professor Thomas Herring from Massachusetts Institute of Technology. This signal is compared with the geodetically-derived signal on diurnal and semidiurnal bands, as shown in Figures 39 and 40. The high-frequency behavior of the GPS signal is very similar to the VLBI-developed model. Thus, most of the geodetic signal on diurnal and semidiurnal bands can be truly attributed to the oceanic tidal processes. What should be emphasized here is that no smoothing was applied to the geodetic signal. What is presented is the true signature seen by the satellite technique. Yet it shows fairly good agreement with the VLBI-derived series for most of the analyzed period. As can be seen from the figures, the difference in the amplitude between both signals is present, but there is no significant phase offset, so the diurnal variability of both signals is very similar. Thus, Herring’s model reflects properly the actual variability of UT1 at diurnal and semidiurnal frequencies. Herring's modal QP8 aignal 9.20 o.ie o 100 200 000 400 apoeha avary 30 mln Figure 39. Geodetically-derived diurnal and semidiurnal UT1 and Herring model with 30-minute resolution for GPS week 784 Herring's modal QP8 aignal 0.20 o.to o • 0.00 • 0.10 - 0.20 100 200 300 400 apoeha avary 30 mln Figure 40. Geodetically-derived diurnal and semidiurnal UT1 and Herring with 30-minute resolution model for GPS week 785 179 Since the quadratic fit to the GPS-derived corrections to UT1 can be questioned as appropriate, especially for week 784 where a rather linear trend can be observed in the signal, a linear fit to the curve presented in Figure 38 was also performed. The residual signal is plotted against the Herring model in Figure 41. Clearly, no significant difference between Figures 40 and 41 can be observed, thus either the linear or quadratic fit effectively removes the combined long-periodic frequencies that were left in the signal after differencing the GPS-derived series and the Bulletin B reference series. -■■■- Harrlng'a modal GPS aignal 0.20 o.io u m 0.00 E • 0.10 • 0.20 0 100 200 300 400 apoeha avary 30 mln Figure 41. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution and Herring model for GPS week 78S (best fitting linear function to the GPS UT1 corrections was removed from the signal presented in Figure 38) 7.6.2. Application o f AAMFunctions and the Tidal Model in the Recovery o f Diurnal and Semidiurnal UT1 Signature from the Total GPS Signal In addition, a more rigorous approach was applied to the analysis of the total GPS signal in order to recover daily and subdaily variability in ERPs. The tidal model 180 with frequencies up to 35 days and the Axial Angular Momentum function were used to remove short-periodic tidal (periods from 5.64 to 34.85 days) and atmospheric signature from the total signal observed by satellite technique. The function %3 is associated with changes in l.o.d. (see Chapter VI). It is evaluated from the meteorological data based on the following formulas (Salstein et al., 1993): p 0.70R 4 tt 3 j , j X 3 = —p,— II Ps cos tp dX dtp Cg 3 (7-1) X t = ^ ^ n i« c°s2 tp dX dtp dp where x $ and X™ denote the pressure and wind terms of the AAM (see Chapter VI) R - radius of the Earth, C - axial moment of inertia of the Earth, £2 - mean rotational rate of the Earth, g - acceleration of gravity, u - zonal wind, p t - surface pressure, tp and X - latitude and longitude. The total 3 is the unitless quantity related to the l.o.d. changes in the following way (Barnes e ta l., 1983): A (tl.o.d .) , , X 3 = . , " const (7.2) l.o .d . where const is an arbitrary constant. In this study const was assumed to be zero, thus the effect of AAM on l.o.d. in seconds equals Xt,x l.o.d. The respective effect on the UT1 is the same but has the opposite sign according to equation ( 6 .1). It follows from the definition of changes in Lo.d. and changes in the UT1 with respect to some 181 reference time scale. In essence, a change in l.o.d. introduces the same variation in UT1 series when UT1 variations are defined with respect to a reference time in the sense of TAI-UT1, however, the value UTl-TAI is used here. Therefore, the effect of AAM on l.o.d. (equals Z i x l o.d.) should be added to UT1R-TAI derived from the geodetic observations. Values of at six-hour interval were provided by Dr. David Salstein from Atmospheric and Environmental Research, Inc., Cambridge, Massachusetts, and were converted to seconds according to (7.2). They represent the combined effect due to the zonal winds and the surface pressure. The pressure term does not include the inverted barometer correction. The inverted barometer term is used to correct the pressure term of Z i for the response of the ocean to the variable atmospheric pressure distribution. “In the inverted barometer response, the ocean surface is supposed to be depressed (or raised) locally by a local increase (or decrease) in atmospheric pressure, so that, globally, there are no horizontal pressure gradients within the hydrosphere. This generally has the effect of reducing the magnitude of the pressure term in the excitation function” (Barnes e t al., 1983). However, the inverted barometer effect on Z s is small and varies insignificantly over the time period investigated here (ibid.; Salstein, private communication, 1995), therefore it was not applied. The total atmospheric effect was removed from UT1R obtained by eliminating the tidal effect (frequencies up to 35 days) from the total GPS-derived signal. The resulting time series for weeks 784-785 is plotted in Figure 42. 182 h b -UTIR-AAM 4(0.00 7> 470.00 “ 4 (0 .0 0 460.00 440.00 100 200 100 400 600 000 700 opocho ovory 30 min Figure 42. Geodetically-derived UT1 with tides up to 35 days and the atmospheric effect removed; 30-minute resolution, GPS weeks 784 and 785 In order to obtain daily and subdaily signature from the signal presented in Figure 42 the best linear fit was removed, and the resulting time series are displayed in Figures 43 and 44. Clearly, the results of this method are very similar to the results obtained by the approximated technique presented above. A comparison of Figure 39 with 43 and Figure 40 with 44 shows that there are some small differences between the respective series, but the signature is very similar. Thus, the first method, although approximate, provides valid information. In the following section, the short-periodic tidal signal in UT1 obtained with the rigorous method is used. In addition, Figures 45 and 46 present the difference between the Herring model and the GPS-derived diumal and semidiurnal UT1 for GPS weeks 784 and 785. It can be observed that the beginning and the end of week 784 and also the middle of 785 fit fairly well to the Herring model. The scatter is larger in the middle of week 784 and the beginning and the end o f785, where some outliers appeared in the GPS-signal. QP8 elgnsl Herring's model Herring's model QP8 elgnel 300 400 300 400 200 epochs every 30 mln epochs every 30 min 100 100 0 o.ioo o.ooo 0.200 0.200 o.ooo 0.200 •o.ioo - - Geodetically-derived diurnal and semidiurnal with TJT1 30-minute resolution and Herring model for GPS week784 resolution and Herring model forGPS week 785 [OOOUl] [OOOUl] Figure 44. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute 184 -*-H#rrlng'a mod#l-GP8 signal o.ioo o.ooo -o.ioo -o.aoo 100 200 400300 opocha avary 30 min Figure 45. The difference between Herring model and GPS-derived UT1 for GPS week 784 -*-Hsrring's modsl-QPS signal 0.200 0.100 o m E 0.000 -0.100 0 100 200 300 400 apocha avary 30 mln Figure 46. The difference between Herring model and GPS-derived UT1 for GPS week 785 7.6.3. Application o f Vondrak Smoother to the Satellite-Derived Short-Periodic UT1 Smoothing of a series of measurements is usually implemented to remove the high frequency noise or to free the series from short-term instability. It is usually done 185 for the purpose of interpolation of the signal between the subsequent entries in the series or to improve the representation of the measured quantity (Feissel and Lewandowski, 1984). In this study, the signal presented in Figures 43 and 44 was smoothed with a Vondrak smoother in order to remove the observation noise. The smoothing method implemented here has been used successfully by IERS to provide the smoothed values of ERPs published in IERS Bulletins A and B (Feissel and Lewandowski, 1984). The smoothing method is discussed by Vondrak (1969 and 1977) and by Feissel and Lewandowski (1984). The smoothing softwareapplied in this study was provided by Dr. Jan Vondrak from the Astronomical Institute of the Czech Republic. Given a series of unequally spaced measurements y ' with their weights p , the smoothed series y can be computed by minimizing the quantity: Q = F + e S (7.3) where F = (n-*V±P,{?,-y$ (7-4) /■I and C7.5) t, L at J e - degree of smoothness; e = 0 implements the quadratic function as the best fitting smoothed signal. For e going to infinity the smoothed curve runs through the original data points, n - number of nodal (data) points, The smoothed series, y', belongs to Herring's modsl smoothed GPS signal s. 100 o.ooo o • 0 .2 0 0 0 100 200 000 400 epochs ovary 30 mln Figure 47. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution smoothed with Vondrak smoother (equal weights) and Herring model for GPS week 784 187 Harring'a modal amoothad QP8 signal -o.too -0.200 o too 200 000 400 apocha avary 30 mln Figure 48. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution smoothed with Vondrak smoother (equal weights) and Herring model for GPS week 785 Finally, the unequally-weighted smoothing was performed for the data of week 784 in order to check the influence of the weighting on the final smoothed series. Weights were obtained from the differences between the original series and the smoothed one with e= 2, in the following way: ( 7 7 ) The smoother was run with the same value of e, but with the new weights defined by equation (7.6). The result is displayed in Figure 49. 188 Htrrlng't mod*l smoothed QP8 signal o.ooo -0.100 - 0.200 o 100 200 000 400 opocho ovary 30 min Figure 49. Geodetically-derived diurnal and semidiurnal UT1 with 30-minute resolution smoothed with Vondrak smoother (unequal weights) and Herring model for GPS week 784 Clearly, the unequally-weighted solution is very similar to the equally-weighted one but the standard deviation in that case is slightly worse and equals 0.019 (was 0.014 for week 784). Thus, the equally-weighted solution is preferred here. Finally, the difference between the original and the smoothed signals are plotted in Figures 50 and 51 to show the influence of the smoothing procedure on the original signature of the GPS-derived UT1 series. ------amoothad-original aariaa -*^amoothad>origlnal aariaa •pocha avary 30 min apocha avary 30 mln 0 0 100 200 >00 400 1 1 1 1 1 1 < 1 ' 0 0 100 200 >00 400 ) 0 . o.oi o.oi 0.04 0.04 0 0.02 a.oo 0 .0 2 •o.oi •o.oi • -e.oa The difference between geodetically-derived diurnal and semidiurnal original signal for GPS week 784 -0.04 UT1 series smoothed with Vondrak smoother (equal weights) and the original signal forGPS week 785 UT1 series smoothed with Vondraksmoother (equal weights) and the [”•“1 d [ot.uil Figure 51. The difference between geodetically-derived diurnal and semidiurnal CHAPTER Vm CONCLUSIONS AND RECOMMENDATIONS The main goal of this research was to learn more about the triple-difference approach applied to the GPS orbit and related parameter determination. In particular, satellite-based estimations of Universal Time and pole coordinates were thoroughly investigated. In addition, the influence of the type of tropospheric refraction model used to correct the tracking data, the impact of the ERP model itself and the influence of the dynamic model and parametrization of the satellite orbit on the ERP estimates were addressed. Based on the results presented in Chapter VII, the following conclusions can be drawn. • The triple difference technique, although not new as a concept for GPS data processing but certainly innovative in the field of orbit determination, has demonstrated very satisfactory processing capacity, flexibility, efficiency and reliability. • It was found that the automated procedure for generating an optimum set of linearly independent ionosphere-free triple-differences, which allows access to 100% of linearly independent information, has improved the quality of the solution with respect to the solution based on a fixed set of baselines. The average improvement is about 11% based on a comparison between the RMS of fit to IGS from both solutions with optimum data set and with fixed baselines. • The triple-difference technique does not require separate data editing since cycle slips are treated as data outliers and are rejected during the least-squares 190 191 adjustment. There are no nuisance parameters (ambiguities) in the system of observation equations, thus the size of the normal matrix is significantly reduced with respect to the normal matrix for undifferenced, single- or double-differenced observations. • The validity of the triple-difference solution for GPS orbits and ERPs was confirmed by the consistency and repeatability tests for orbits and long baselines, showing the RMS ranging from a few parts in 10s to a few parts in 109 for the East, North and Vertical baseline components. The three-dimensional RMS (3DRMS) for the orbit repeatability is currently at the level of 10-70 cm and strongly depends on the number of the eclipsing satellites participating in the solution. • A comparison between the IGS and OSU solutions clearly shows that the triple-difference technique meets all standards for the high-precision requirements for the trajectory and Earth orientation determination. The RMS of fit of the OSU solution to the combined IGS orbit was at the level of 1S.4 cm for a period of four weeks analyzed in this study (GPS weeks 784-787), showing superiority with respect to the fit presented by ESA, NGS and SIO, agencies which participate in the final IGS solution. • Application of the nutation series according to Herring is recommended for GPS orbit and ERP estimation. • VLBI estimates for the ERPs can be introduced in the GPS-derived series only once per month without any loss in the accuracy of the final product. • A strong dependency of the ERPs on the tropospheric model requires that the atmospheric scaling factor has to be carefully modeled and estimated at three to four hour intervals. • The triple-difference technique is capable of providing daily estimates of the Earth’s rotation with a quality similar to the one provided by the undifferenced, single- or double-difference techniques. The short-periodic UT1 series for GPS weeks 784-785 shows good agreement with the VLBI-derived empirical model of Herring. 192 • Due to the strong correlation between the long-periodic errors in the orbit and the daily retrograde polar motion, as well as the long-periodic nutation and daily retrograde polar motion, it is impossible to separate the true daily and subdaily signature of the polar motion from the satellite technique. However, as shown for Lageos data, the major portion of the signal can be estimated if the model explicitly contains specific parts of the spectrum, i.e., semidiurnal prograde and retrograde polar motion together with the daily prograde signal (Watkins et al., 1994). Yet, the retrograde diurnal part will still be absorbed by the orbit or by the nutation corrections, if nutation is also estimated. It should also be emphasized here that one is not able to resolve directly the nutation terms with GPS (for the reason similar to the case of UT1). However, it is still possible to determine rates for nutation in longitude and obliquity (Beutler et al., 1995). Clearly, the adopted triple-difference technique is capable of achieving a high quality product. However, there are still a few shortcomings of the dynamic model applied in GODIVA which should to be eliminated by enhancing the model itself, especially for arcs longer than two to three days. For that purpose, the small perturbing forces such as Earth albedo and indirect oblation effect should be added. The order of magnitude of these accelerations is about 0 .4 x 10~* m/s2 for the Earth radiation and 2 .0 X 1 0 " 9 m/s2 for the indirect oblation, and these might be important for the determination of arcs longer than a day. Also, including more perturbing objects beyond the Sun and the Moon in the third-body problem should be accomplished. Another important kinematic and dynamic part of the model is the proper implementation of the new GPS attitude model, as explained in Chapter IV. An additional part of the dynamic model that should be enhanced is the direct solar radiation pressure. The current two-parameter model might be, for example, replaced with a more sophisticated model similar to Colombo’s resonance model presented in Section 4 . 3 . 8 , with the major period equaling an orbital revolution. Beutler et al. 193 (1994) claim that “the essential improvement [in long arc determination] was achieved through the new model for the direct radiation pressure.” Thus, introducing a new [nine-parameter] radiation pressure model can allow for longer arc processing. That in turn might improve the pole coordinate determination, as reported by Rothacher e t al. (1994) who process three-day arcs in order to estimate the pole rate and iteratively improve xp and yp components. 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H., (1993):JET Propulsion Laboratory IGS Analysis Center Report, 1992, Proceedings of the 1993 IGS Workshop, March 25-26, Astronomical Institute University of Bern, Switzerland, pp. 154-164.> Sz Scaling former solution 0.1 Factors and Y-bias Sailing Factor 0.15 Coordinates for 23 Tracking Stations former solution 50.0 m (13 Fixed IERS Stations) IGS mail 819 3-5 mm Tropospheric Scaling Factors (at four - hour interval) 1.0 0.1 Earth Rotation Parameters: Rate o f (UT1-TAI) IERS Bulletin B 6.5 x KT* s/day xm yr prediction 9.7 x 10'2arcsec Total Arc Length: 32 hours (4+24+4)