Parameter estimation in differential equations: Mathematical foundations for satellite gravimetry, review and perspectives
Abstract
Satellite gravimetry has become essential in many areas of earth science. However, the resolution of satellite gravitational models remains low at scales of a few hundreds km and no gravity recovery methods can take full advantages of unprecedented high accuracy of satellite tracking measurements. We first provide a unified theoretical framework of parameter estimation in differential equations for satellite gravimetry and then briefly review the mathematical methods to compute the gravity field of the Earth from satellite tracking. We focus on the collocation method, Kaula linear perturbations, two-point boundary value problems and orbit-energy-based methods. The numerical integration method is also included in this review, though it has been proved to be mathematically incorrect and physically not permitted. The reason is that it has become the standard method to routinely produce global gravitational models from satellite tracking data, which have been widely applied in many different areas of earth science. Because it is not clear how the incorrect foundation would affect gravity products from satellite tracking, we do not review any applications of these products. We then present a measurement-based perturbation theory to estimate the gravity field of the Earth, which can fully utilize both precise satellite orbits of arbitrary length and unprecedented high accuracy of satellite and inter-satellite tracking. The method is theoretically free of modeling errors, is capable of extracting any small forces from satellite and inter-satellite tracking data and provides a guarantee for high-precision and high-resolution global gravity models. Finally, we assume a reference gravity model and derive local solutions to the Newton's nonlinear governing differential equations of satellite motion for scattered tracking data that can still be important in some applications.
Related
- On the foundations of anthropic reasoning: Implications for life across the universe
- B-fields In STar-forming Region Observations (BISTRO): Technical Foundations for Quantitative 450/850 μm JCMT SCUBA-2/POL-2 Polarimetry
- Curvature–Matter Feedback in Semiclassical Cosmology: A Reduced Mathematical Framework for Geometry-Induced Quantum Matter Production
- X-reconstruction: 3D Tomographic Reconstruction in Astrophysics. I. Mathematical Formulation for Systems in the Optically Thin Regime
- Mathematical Framework for the Effects of Sample Absorption on Raman Intensity for Quantification and Detection Thresholds of Minerals in Two-Component Geological Mixtures.
BibTeX
@misc{xu2026parameter,
title = {{Parameter estimation in differential equations: Mathematical foundations for satellite gravimetry, review and perspectives}},
author = {Peiliang Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15148},
eprint = {2608.15148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15148},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Parameter estimation in differential equations: Mathematical foundations for satellite gravimetry, review and perspectives
AU - Peiliang Xu
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2608.15148
UR - https://arxiv.org/abs/2608.15148
AB - Satellite gravimetry has become essential in many areas of earth science. However, the resolution of satellite gravitational models remains low at scales of a few hundreds km and no gravity recovery methods can take full advantages of unprecedented high accuracy of satellite tracking measurements. We first provide a unified theoretical framework of parameter estimation in differential equations for satellite gravimetry and then briefly review the mathematical methods to compute the gravity field of the Earth from satellite tracking. We focus on the collocation method, Kaula linear perturbations, two-point boundary value problems and orbit-energy-based methods. The numerical integration method is also included in this review, though it has been proved to be mathematically incorrect and physically not permitted. The reason is that it has become the standard method to routinely produce global gravitational models from satellite tracking data, which have been widely applied in many different areas of earth science. Because it is not clear how the incorrect foundation would affect gravity products from satellite tracking, we do not review any applications of these products. We then present a measurement-based perturbation theory to estimate the gravity field of the Earth, which can fully utilize both precise satellite orbits of arbitrary length and unprecedented high accuracy of satellite and inter-satellite tracking. The method is theoretically free of modeling errors, is capable of extracting any small forces from satellite and inter-satellite tracking data and provides a guarantee for high-precision and high-resolution global gravity models. Finally, we assume a reference gravity model and derive local solutions to the Newton's nonlinear governing differential equations of satellite motion for scattered tracking data that can still be important in some applications.
ER -