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.
Highlights
The goal of magnetic inverse problems is to recover the magnetization of an object from the magnetic field it creates
The magnetic inverse problem is more challenging due to its vectorial setup, meaning that combinations of the vector components of magnetizations can lead to cancellations that cannot occur for scalar densities in gravimetry
By Hs(∂Ω) (0 ≤ s ≤ 1), we indicate the standard scale of Hilbertian Sobolev spaces on the boundary of a Lipschitz domain; we will only need s = 1/2
Summary
The goal of magnetic inverse problems is to recover the magnetization of an object from the magnetic field it creates. We assume magnetizations to be of L2-class on a Lipschitz domain Ω with fairly general topology (see Section 2), and discuss two subspaces of L2-vector fields that yield a unique inversion. The main result is Theorem 3.3, that parametrizes those magnetizations creating the zero magnetic potential in Θ, and shows in particular that only the harmonic gradient component of the Hodge decomposition of a magnetization can produce a non-vanishing magnetic potential Based on this characterization, we derive corollaries that address some of the geophysical constraints mentioned earlier. Harmonic functions cannot vanish on an open domain unless they are identically zero, whereas realistic magnetizations of rock samples can For another thing, many geophysical applications assume the magnetization to be piecewise constant. Throughout the paper, we provide some basic examples and try to motivate connections to geophysically relevant situations
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