On the Gravitational Inverse Problem

TL;DR

This paper explores the mathematical challenges of inverting gravitational data to determine mass distributions, focusing on the non-uniqueness caused by the infinite-dimensional kernel of the forward map.

Contribution

It characterizes the kernels of two key gravitational forward maps and discusses conditions for unique inversion, advancing understanding of the inverse gravitational problem.

Findings

Kernel of the Newtonian potential map characterized

Kernel of the gravity gradient tensor map characterized

Conditions for unique inversion under constraints identified

Abstract

We discuss some mathematical aspects of the problem of inverting gravitational field data to extract the underlying mass distribution. While the forward problem of computing the gravity field from a given mass distribution is mathematically straightforward, the inverse of this forward map has some interesting features that make inversion a difficult problem. In particular, the forward map has an infinite-dimensional kernel which makes the inversion fundamentally non-unique. We characterize completely the kernels of two gravitational forward maps, one mapping mass density to the Newtonian scalar potential, and the other mapping mass density to the gravity gradient tensor, which is the quantity most commonly measured in field observations. In addition, we present some results on unique inversion under constrained conditions, and comment on the roles the kernel of the forward map and non-uniqueness play in discretized approaches to the continuum inverse problem.