A multiple level set method for three-dimensional inversion of magnetic data

Abstract

We propose a multiple level set method for inverting three-dimensional magnetic data induced by magnetization only. To alleviate inherent non-uniqueness of the inverse magnetic problem, we assume that the subsurface geological structure consists of several uniform magnetic mass distributions surrounded by homogeneous non-magnetic background such as soil, where each magnetic mass distribution has a known constant susceptibility and is supported on an unknown sub-domain. This assumption enables us to reformulate the original inverse magnetic problem into a domain inverse problem for those unknown domains defining the supports of those magnetic mass distributions. Since each uniform mass distribution may take a variety of shapes, we use multiple level-set functions to parameterize these domains so that the domain inverse problem can be further reduced to an optimization problem for multiple level-set functions. To compute rapidly gradients of the nonlinear functional arising in the multiple-level-set formulation, we utilize the fact that the kernel function in the field-susceptibility relation decays rapidly off the diagonal so that matrix-vector multiplications for evaluating the gradients can be speeded up significantly. Numerical experiments are carried out to illustrate the effectiveness of the new method.