Magnetic inversion to recover the subsurface block structures based onL1 norm and total variation regularization

Abstract

This paper presents a new sparse inversion method based on L₁ norm regularization for 3-D magnetic data. In isolation, L₁ norm regularization yields model elements which are unconstrained by the input data to be exactly zero, leading to a sparse model with compact and focused structure. Here, we complement the L₁ norm with a penalty minimizing total variation, the L₁ norm of the model gradients; it is expected that the sharp boundaries of the subsurface structure are not compromised by incorporating this penalty. Although this penalty is widely used in the geophysical inversion studies, it is often replaced by an alternative quadratic penalty to ease solution of the penalized inversion problem; in this study, the original definition of the total variation, that is form of the L₁ norm of the model gradients, is used. To solve the problem with this combined penalty of L₁ norm and total variation, this study introduces alternative direction method of multipliers, which is a primal-dual optimization algorithm that solves convex penalized problems based on the optimization of an augmented Lagrange function. To improve the computational efficiency of the algorithm to make this method applicable to large-scale magnetic inverse problems, this study applies matrix compression using the wavelet transform and the preconditioned conjugate gradient method. The inversion method is applied to both synthetic tests and real data, the synthetic tests demonstrate that, when subsurface structure is blocky, it can be reproduced almost perfectly.