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On the gravitational inverse problem. (English) Zbl 1246.70012

Summary: 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.

MSC:

70F99 Dynamics of a system of particles, including celestial mechanics
65R32 Numerical methods for inverse problems for integral equations
86A20 Potentials, prospecting
35R30 Inverse problems for PDEs
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