Unique reconstruction of simple magnetizations from their magnetic potential

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