Characterizing kernels of operators related to thin-plate magnetizations via generalizations of Hodge decompositions

Abstract

Abstract. Recently developed scanning magnetic microscopes measure the magnetic field in a plane above a thin-plate magnetization distribution. These instruments have broad applications in geoscience and materials science, but are limited by the requirement that the sample magnetization must be retrieved from measured field data, which is a generically nonunique inverse problem. This problem leads to an analysis of the kernel of the related magnetization operators, which also has relevance to the "equivalent source problem" in the case of measurements taken from just one side of the magnetization. We characterize the kernel of the operator relating planar magnetization distributions to planar magnetic field maps in various function and distribution spaces (e.g., sums of derivatives of L p (Lebesgue spaces) or bounded mean oscillation (BMO) functions). For this purpose, we present a generalization of the Hodge decomposition in terms of Riesz transforms and utilize it to characterize sources that do not produce magnetic field either above or below the sample, or that are magnetically silent (i.e., no magnetic field anywhere outside the sample). For example, we show that a thin-plate magnetization is silent (i.e., in the kernel) when its normal component is zero and its tangential component is divergence-free. In addition, we show that compactly supported magnetizations (i.e., magnetizations that are zero outside of a bounded set in the source plane) that do not produce magnetic fields either above or below the sample are necessarily silent. In particular, neither a nontrivial planar magnetization with fixed direction (unidimensional) compact support nor a bidimensional planar magnetization (i.e., a sum of two unidimensional magnetizations) that is nontangential can be silent. We prove that any planar magnetization distribution is equivalent to a unidimensional one. We also discuss the advantages of mapping the field on both sides of a magnetization, whenever experimentally feasible. Examples of source recovery are given along with a brief discussion of the Fourier-based inversion techniques that are utilized.

Fast inversion of magnetic field maps of unidirectional planar geological magnetization

“…This suggests that, in general, arbitrary two-dimensional magnetization distributions cannot be uniquely recovered from magnetic field maps, even if all three components of the field are known everywhere on a plane. (In some cases, sophisticated spectral/spatial unmixing techniques can be employed to tackle nonuniqueness when attempting to invert multidirectional magnetization distributions [Baratchart et al, 2013], but such an approach is outside of the scope of this paper.) In addition, it shows that, for low or moderate noise levels, a single field component carries all the information about the magnetization distribution [Lima and Weiss, 2009].…”

“…For k = 0, g is the zero matrix, which implies that the uniform (constant) component of the magnetization distribution cannot be directly recovered from magnetic field measurements. This is not surprising, since uniform planar magnetizations with support in the entire plane constitute a classic example of magnetically silent source or annihilator [Baratchart et al, 2013;Parker, 1977], generating no external magnetic field. Thus, a constant magnetization can always be added to the solution without changing the field produced by the overall distribution.…”

“…The most interesting practical case consists of a unidirectional magnetization distribution with known fixed direction but variable strength [Baratchart et al, 2013;Weiss et al, 2007a]. Retrieving such a magnetization from scanning magnetic microscopy data measured on a plane above the sample is the goal of this paper.…”

“…They must rely on solving a nonunique inverse problem to retrieve the magnetization distribution that gives rise to the measured field data. The nonuniqueness arises because there are an infinite number of solutions that produce the same observed magnetic field [Baratchart et al, 2013]. However, incorporating as much information as possible regarding the specificities of the sample under analysis and particularities of the experimental setup can in many cases narrow the space of solutions and help select physically meaningful magnetization distributions.…”

“…However, incorporating as much information as possible regarding the specificities of the sample under analysis and particularities of the experimental setup can in many cases narrow the space of solutions and help select physically meaningful magnetization distributions. In fact, we show in Baratchart et al [2013] that, under specific conditions (e.g., unidirectional magnetization), it is even possible to ensure uniqueness for the inverse problem in scanning magnetic microscopy (SMM) given that, in this application, there are no sources outside a well-defined region in space.…”