On the Non-uniqueness of Gravitational and Magnetic Field Data Inversion (Survey Article)

Abstract.

The gravitational and the magnetic field of the Earth represent some of the most important observables of the geosystem. The inversion of these fields reveals hidden structures and dynamics at the surface or in the interior of the Earth (or other celestial bodies). However, the inversions of both fields suffer from a severe non-uniqueness of the solutions. In this paper, we present a generalized approach which includes the inversion of gravitational and magnetic field data. Amongst others, uniqueness constraints are proposed and compared. This includes the surface density ansatz (also known as the thin layer assumption). We characterize the null space of the considered class of inverse problems via an appropriate orthonormal basis system. Further, we expand the reconstructable part of the solution by means of orthonormal bases and reproducing kernels. One result is that information on the radial dependence of the solution is lost in the observables. As an illustration of the non-uniqueness, we show examples of anomalies which cannot be disclosed from the inversion of gravitational data. This paper is intended to be a theoretical reference work on the inversion of gravitational but also magnetic field data of the Earth.