On the Unique Reconstruction of Induced Spherical Magnetizations

TL;DR

This paper proves that the uniqueness of reconstructing induced spherical magnetizations from exterior magnetic field data can be guaranteed if the magnetization is assumed to be compactly supported on the sphere, extending planar results.

Contribution

It establishes a uniqueness result for induced spherical magnetizations under the assumption of compact support, filling a gap in geophysical magnetization recovery theory.

Findings

Uniqueness holds for compactly supported induced spherical magnetizations.

Hardy-Hodge decomposition indicates recoverable components of magnetization.

Extension of planar techniques to spherical setting.

Abstract

Recovering spherical magnetizations $m$ from magnetic field data in the exterior is a highly non-unique problem. A spherical Hardy-Hodge decomposition supplies information on what contributions of the magnetization $m$ are recoverable but it does not supply geophysically suitable constraints on $m$ that would guarantee uniqueness for the entire magnetization. In this paper, we focus on the case of induced spherical magnetizations and show that uniqueness is guaranteed if one assumes that the magnetization is compactly supported on the sphere. The results are based on ideas presented in Baratchart et al. (2013) for the planar setting.