Statistical estimation of differential equation parameters: the mathematical foundation for satellite gravimetry from tracking
Authors: Xu Peiliang 
Published: March 2024

The numerical integration method has been routinely used by major institutions worldwide (for example, NASA Goddard Space Flight Center and GFZ) to produce global gravitational models from satellite tracking measurements. Such Earth&#8217;s gravitational products have found widest possible multidisciplinary applications. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown force parameters under the zero initial conditions. From the statistical point of view, satellite gravimetry from satellite tracking is essentially to estimate the unknown parameters in the Newton&#8217;s nonlinear differential equations from satellite tracking measurements --- the mathematical foundation for satellite gravimetry from tracking. From this perspective, it is rather trivial to prove that the numerical integration method, originating from Gronwall on Ann Math almost 100 years ago and currently implemented and used in mathematics/statistics, chemistry/physics, and satellite gravimetry, is groundless, even though, up to this moment, many researchers in the geoscientific community still have problems in understanding this side point of my research. In this talk, we focus on presenting three different methods to derive local solutions to the Newton&#8217;s nonlinear differential equations of motion of satellites, given unknown initial values and unknown force parameters. They are mathematically correct and can be used to estimate unknown differential equation parameters, with applications in gravitational modelling from satellite tracking measurements as a typical example in geodesy. These solution methods are generally applicable to any differential equations with unknown parameters. More precisely, we develop the measurement-based perturbation theory and construct global uniformly convergent solutions to the Newton&#8217;s nonlinear differential equations of motion of satellites, given unknown initial values and unknown force parameters. From the physical point of view, the global uniform convergence of the solutions implies that they are able to exploit the complete/full advantages of unprecedented high accuracy and continuity of satellite orbits of arbitrary length and thus will automatically guarantee theoretically the production of a high-precision high-resolution global standard gravitational models from satellite tracking measurements of any types. Finally, we develop an alternative method by reformulating the problem of estimating unknown differential equation parameters, or the mixed initial-boundary value problem of satellite gravimetry with unknown initial values and unknown force parameters as a standard condition adjustment model with unknown parameters.Xu P (2018) Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry. Commun Nonlinear Sci Numer Simulat, 59, 515-543. DOI 10.1016/j.cnsns.2017.11.021Xu P (2008) Position and velocity perturbations for the determination of geopotential from space geodetic measurements. Celest Mech Dyn Astr, 100, 231&#8211;249. Xu P (2009) Zero initial partial derivatives of satellite orbits with respect to force parameters violate the physics of motion of celestial bodies. Sci China Ser D, 52, 562&#8211;566. Xu P (2012) Mathematical challenges arising from earth-space observation: mixed integer linear models, measurement-based perturbation theory and data assimilation for ill-posed problems. Invited talk, joint mathematical meeting of American mathematical society, Boston, January 4&#8211;7.