Gravity Recovery Using COSMIC GPS Data: Application of Orbital Perturbation Theory by Cheinway Hwang Report No. 463 Geodetic and GeoInformation Science Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University Columbus, Ohio 43210-1275 October 1998 Gravity recovery using COSMIC GPS data: application of orbital perturbation theory by Cheinway Hwang Department of Civil Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan I Abstract COSMIC is a joint Taiwan-US mission to study atmosphere using GPS occultation. Its GPS data for precise orbit determination can be used for gravity recovery. In this report a kinematic approach was employed which assumes the positional data can be derived from the GPS data of COSMIC in the operational phase. Using the geometric relationship between the positional variations of orbit and the variations in the six Keplerian elements, improved formulae for the radial, along-track and cross-track perturbations were derived. Based on a comparison with true perturbations from numerical integrations, these formulae are more accurate than the commonly used order-zero formulae. The improved formulae were used to simulate gravity recovery using the COSMIC data. In one simulation with the OSU91A model to degree 50 as the a priori geopotential model, it is demonstrated that the EGM96 model can be improved up to degree 26 using one year of COSMIC data. A significant effort was devoted to the recovery of temporal gravity variation using COSMIC data. Sea level anomaly (SLA) was first generated using the Cycle 196 TOPEX/POSEIDON altimeter data. The steric anomaly due to thermal expansion was created using temperature data at 14 oceanic layers. The steric anomaly-corrected SLA was used to generate harmonic coefficients of temporal gravity variation. With a 3-cm noise at a one-minute sampling interval in the COSMIC data, the gravity variation cannot be perfectly reproduced, but the recovered field clearly shows the gravity signature due to mass movement in an El Niño. With a 0.1-cm noise, the temporal gravity variation up to harmonic degree 10 is almost exactly recovered and this prompts the need of a better processing technique and a sophisticated GPS receiver technology. II Acknowledgments The idea described in this report was initiated during my visit to the Ohio State University in the summer 1998, hosted by Prof. C.K. Shum. I am very grateful for his support and inspiring discussions. This research was partly supported by the National Science Council of ROC. Thanks also go to Dr. G. Balmino for providing me the program of inclination and eccentricity functions, and to Mr. S. -A. Chen for helping me to compute the steric anomaly. III TABLE OF CONTENTS Abstract Acknowledgments 1. Introduction………………………………………………………………………. 1 2. Methods and data type for gravity recovery from COSMIC…………………….. 3 3. Orbital perturbations due to the geopotential…………………………………… . 6 3.1 Radial, along-track and cross-track perturbations……………………... 6 3.2 Higher order and resonance effects…………………………………… 11 3.3 Model errors of the perturbation formulae……………………………. 12 4. Use of orbital perturbation formulae in gravity recovery……………………….. 17 5. Simulations of gravity recovery from COSMIC data…………………………… 18 5.1 Improving current gravity model……………………………………… 18 5.2 Recovering temporal gravity variation………………………………… 22 5.2.1 Generating gravity variation due to oceanic mass variation………..22 5.2.2 Recovery…………………………………………………………….27 5.2.3 The effect of data noise on recovering temporal gravity……………30 6. Conclusion and recommendation…………………………………………………37 Appendix A: Order-zero perturbations……………………………………………... 38 Appendix B: Computation of steric anomaly due to thermal expansion……………40 Appendix C: Harmonic coefficients and geoid variation from corrected sea level anomaly………………………………………….. 41 Appendix D: Important FORTRAN programs……………………………………...43 IV References…………………………………………………………………………..49 TABLES Table 1: Orbital characteristics of COSMIC mission in the operational phase……...2 Table 2: Model errors and statistics of the difference between the true and predicted perturbations…………………………………………………….. 13 Table 3: Statistics of sea level anomaly, steric anomaly and corrected sea level anomaly (in cm) over oceans with depth greater than 500 m………… 24 V Figures Fig. 1: Three methods of gravity recovery from COSMIC GPS data and applications of gravity. …………………………………………………..4 Fig. 2: Geometry showing the effects of the perturbations in argument of perigee (top), right ascension of the ascending node (center) and inclination (bottom) on the radial, along-track and cross-track perturbations of satellite position……………………………………………..7 Fig. 3: Differences between the true perturbation, and three predicted perturbations computed with three perturbation models (order- zero, Q =1 and Q = 2). Day is the elapsed number of days since the first data point…………………………………………………………...14 Fig. 4: Difference between the true and predicted along-track perturbations using 10 and 15 coefficients in the empirical model. The perturbation model uses Q = 2. Day is the elapsed number of days since the first data point……………………………………………………………...……..16 Fig. 5: Configuration of the COSMIC constellation in the operational phase at the initial epoch of orbit integration (0 hour UTC, Jan. 1, 2001). The 8 satellites are placed on 8 orbital planes with evenly spaced right ascensions and arguments of latitude……………..19 Fig. 6: Relative errors of the recovered harmonic coefficients from the degree-50 solution for (a) Cnm without constraint (b) S nm without constraint (c) Cnm with constraint and (d) S nm with constraint….20 Fig. 7: Relative errors of recovered zonal coefficients from the degree-50 VI solution with constraint……………………………………………………...21 Fig. 8: Geoid errors by degree computed from the coefficient errors of the EGM96 model, and the models from the COSMIC solutions using 7 days and 1 year of data…………………………………...23 Fig. 9: TOPEX/POSEIDON-observed raw sea level anomaly (top), temperature-derived steric anomaly (center), and corrected sea level anomaly (by steric anomaly) at Cycle 196………………………...25 Fig. 10: Degree amplitude and cumulative percentage power of geoid variation computed from the corrected sea level anomaly in Fig. 9……….26 Fig. 11: Contour maps of geoid variation up to degrees 5 (top), 15 (center) and 50 (bottom) from the corrected sea level anomaly in Fig. 9. Unit is mm………………………………………………………………….28 Fig. 12: True and modeled degree variances of gravity variation due to oceanic mass variation…………………………………………………..29 Fig. 13: Perturbations of COSMIC orbit due to the mass variation of corrected sea level anomaly up to degree 50. Day is the elapsed number of days since the first data point………………………….31 Fig. 14: Relative errors of the recovered harmonic coefficients of gravity variation for (a) Jnm and (b) K nm using one week of COSMIC data and degree-50 solution……………………………32 Fig. 15: Contour maps of recovered geoid variation up to degrees 5 (top), 15 (center) and 50 (bottom) using one week of VII COSMIC data and degree-50 solution. Unit is mm………………………..33 Fig. 16: Relative errors of the recovered harmonic coefficients of gravity variation using one week of COSMIC data and degree-15 solutions for (a) Jnm (3-cm noise) (b) K nm (3-cm noise) (c) Jnm (1-cm noise) (d) K nm (1-cm noise) (e) Jnm (0.1-cm noise) (f) K nm (0.1-cm noise)……………………………………………………35 Fig. 17: Contour maps of recovered geoid variation up to degrees 5 (top), 10 (center) and 15 (bottom) using one week of COSMIC data and degree-15 solution with noise = 0.1 cm. Unit is mm…………………………….………….….36 VIII 1. Introduction The Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) is a joint Taiwan (ROC)-US satellite mission to study atmosphere using the GPS occultation technique. This mission is to be launched in 2003 and will deploy a constellation of 8 micro- satellites, each equipped with one GPS receiver and two antennas. One antenna is to receive occulted signals for atmospheric study, and the other to receive un-occulted signals for precise orbit determination (POD) (Kuo and Lee, 1999). Although the COSMIC mission is primarily for atmospheric research, its POD GPS data can be used for geodetic research. In the early, geodetic phase of COSMIC, selected satellites will fly in the tandem mode at altitudes ranging from 300 km to 700 km. When using the GPS data in the tandem mode the effect of the non-conservative forces can be reduced because the non-conservative forces acting on a satellite pair are highly correlated. In a simulation study Chao et al. (2000) show that the COSMIC GPS data from the geodetic phase can improve the accuracy of the EGM96 model (Lemoine et al., 1998) for harmonic coefficients up to degree 40 and for selected terms of higher degrees and orders. The geodetic phase of COSMIC will last for less than one year and the mission will be soon shifted to the operational phase. Table 1 lists orbital characteristics of COMIC satellites in the operational phase, which are largely based on Kuo and Lee (1999). The 80 degree inclination of COSMIC is not officially determined, but any inclination less than 80 degree will introduce significant polar gaps in the data coverage and will not be ideal for global gravity recovery. In comparison, the GRACE and GOCE missions have polar and nearly polar obits that are ideal for global gravity recovery (Balmino et al., 1998). At the altitude of 800 km, the gravity content sensible to