Abstract
We address a claim frequently raised by one author that determination of Earth’s gravity field by satellite observations using traditional methods is mathematically flawed. Specifically, attention is drawn to the practice of setting to zero the initial sensitivity matrix in the variational equations for force model parameters. It is asserted that this would lead to mathematical contradictions. In this paper we establish necessary and sufficient conditions for the initial sensitivity matrix to be zero—conditions that are well founded and accepted worldwide in classical satellite-based determinations of the gravity field. To claim otherwise is shown to be without basis. We inspect a proposed counterexample, and find it, too, requires zero initialization of the sensitivity matrix. In addition, we review proofs and derivations from a classic textbook that also confirm zero initialization. In a numerical exercise, perfect, synthetic data for the central force problem are processed with standard procedures, and results confirm the validity of zero initialization of the sensitivity matrix.
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Acknowledgements
Our special thanks go to Professor Jürgen Kusche, Dr.-Ing. Anno Löcher, and an anonymous reviewer; whose thoughtful comments were very helpful in improving this paper. We thank Dr. Xiaopeng Li, National Geodetic Survey, NOAA, for inviting us to a webinar on orbit determination which ignited our curiosity. A helpful comment by Prof. Artem Novozhilov crystalized the Coddington and Levinson proof regarding the initial condition of the sensitivity matrix. We thank L. F. Shampine and H. A. Watts for writing the DDEABM differential equation integration software package. Thanks to the organizers of the SLATEC mathematical library and to NETLIB for hosting the Fortran 77 version of the DEPAC collection.
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Milbert, D., Jekeli, C. On the initialization of the sensitivity matrix in variational equations. J Geod 97, 88 (2023). https://doi.org/10.1007/s00190-023-01776-4
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DOI: https://doi.org/10.1007/s00190-023-01776-4