A Measure Theoretic Approach for the Recovery of Remanent Magnetizations

Abstract

This work is motivated by the problem of recovering the magnetization M of a rock sample from a given set of measurements for the magnetic field it generates. Modeling the magnetization by an R 3 -valued measure, we focus on the study of inverse problems for the Poisson equation with source term the divergence of M; that is, ∆Φ = divM, where Φ denotes the Magnetic Scalar Potential whose gradient is assumed to be known on a set disjoint from the support of the measure M. We develop methods for recovering M based on total variation regularization of measures. We provide sufficient conditions for the unique recovery of a magnetization in cases where it is uni-directional or when the magnetization has a support which is sparse in the sense that it is purely 1-unrectifiable. In the last chapter we work on the ideal case where the magnetized sample is contained in a subset of the horizontal plane. For this case we show that all magnetizations which do not generate a magnetic field can be decomposed as a superposition of loops. The findings presented in this chapter rely on the theory of functions of Bounded Variation and sets of finite perimeter and give a characterization for magnetizations that do not generate a magnetic field. Numerical examples are provided to illustrate the main theoretical results.