A first approach to learning a best basis for gravitational field modelling

Abstract

Gravitational field modelling is an important tool for inferring past and present dynamic processes of the Earth. Functions on the sphere such as the gravitational potential are usually expanded in terms of either spherical harmonics or radial basis functions (RBFs). The (Regularized) Functional Matching Pursuit ((R)FMP) and its variants use an overcomplete dictionary of diverse trial functions to build a best basis as a sparse subset of the dictionary and compute a model, for instance, of the gravity field, in this best basis. Thus, one advantage is that the dictionary may contain spherical harmonics and RBFs. Moreover, these methods represent a possibility to obtain an approximative and stable solution of an ill-posed inverse problem, such as the downward continuation of gravitational data from the satellite orbit to the Earth's surface, but also other inverse problems in geomathematics and medical imaging. A remaining drawback is that in practice, the dictionary has to be finite and, so far, could only be chosen by rule of thumb or trial-and-error. In this paper, we develop a strategy for automatically choosing a dictionary by a novel learning approach. We utilize a nonlinear constrained optimization problem to determine best-fitting RBFs (Abel-Poisson kernels). For this, we use the Ipopt software package with an HSL subroutine. Details of the algorithm are explained and first numerical results are shown.