Unique reconstruction of simple magnetizations from their magnetic potential

TL;DR

This paper characterizes the conditions under which inverse magnetization problems become uniquely solvable, identifying specific subspaces that ensure uniqueness in applications like geo-sciences.

Contribution

It provides a detailed analysis of the subspaces causing non-uniqueness and identifies conditions for unique reconstruction of simple magnetizations.

Findings

Identification of subspace of harmonic gradients for unique inversion

Characterization of non-uniqueness caused by certain vector fields

Dense subspace of piecewise constant fields enabling unique solutions

Abstract

Inverse problems arising in (geo)magnetism are typically ill-posed, in particular {they exhibit non-uniqueness}. Nevertheless, there exist nontrivial model spaces on which the problem is uniquely solvable. Our goal is here to describe such spaces that accommodate constraints suited for applications. In this paper we treat the inverse magnetization problem on a Lipschitz domain with fairly general topology. We characterize the subspace of $L^{2}$-vector fields that causes non-uniqueness, and identify a subspace of harmonic gradients on which the inversion becomes unique. This classification has consequences for applications and we present some of them in the context of geo-sciences. In the second part of the paper, we discuss the space of piecewise constant vector fields. This vector space is too large to make the inversion unique. But as we show, it contains a dense subspace in $L^2$ on which the problem becomes uniquely solvable, i.e., magnetizations from this subspace are uniquely determined by their magnetic potential.