3-D inversion of magnetic data based on the L1\u2013L2 norm regularization

Mitsuru Utsugi

doi:10.1186/s40623-019-1052-4

Abstract

Magnetic inversion is one of the popular methods to obtain information about the subsurface structure. However, many of the conventional methods have a serious problem, that is, the linear equations to be solved become ill-posed, under-determined, and thus, the uniqueness of the solution is not guaranteed. As a result, several different models fit the observed magnetic data with the same accuracy. To reduce the non-uniqueness of the model, conventional studies introduced regularization method based on the quadratic solution norm. However, these regularization methods impose a certain level of smoothness, and as the result, the resultant model is likely to be blurred. To obtain a focused magnetic model, I introduce L1 norm regularization. As is widely known, L1 norm regularization promotes sparseness of the model. So, it is expected that, the resulting model is constructed only with the features truly required to reconstruct data and, as a result, a simple and focused model is obtained. However, by using L1 norm regularization solely, an excessively concentrated model is obtained due to the nature of the L1 norm regularization and a lack of linear independence of the magnetic equations. To overcome this problem, I use a combination of L1 and L2 norm regularization. To choose a feasible regularization parameter, I introduce a regularization parameter selection method based on the L-curve criterion with fixing the mixing ratio of L1 and L2 norm regularization. This inversion method is applied to a real magnetic anomaly data observed on Hokkaido Island, northern Japan and reveals the subsurface magnetic structure on this area.

Highlights

The inversion of geomagnetic field data has been considered by many studies that aim to determine the property and geometry of subsurface magnetic structures
This paper proposes a regularization method with an L1 and L2 norm combined penalty, which is the same as the “Elastic Net” proposed by Zou and Hastie (2005)
We can see that, the magnetization of the causative bodies was estimated larger (≃ 4 A/m), and the size of blocks was estimated to be smaller. These results suggest that, in the case of the L1 norm regularization with bound constraint, the value of βmax is critical for the shape and magnetization of the derived model

Summary

Introduction

The inversion of geomagnetic field data has been considered by many studies that aim to determine the property and geometry of subsurface magnetic structures. In this calculation, a conventional weighting function wS1 of Eq (9) was used. These results show that, by using wS2 as the weighting function, we can obtain an appropriate model robustly regardless of the noise amplitude From these results, we can see that, wS2 seems to outperform wS1 , and wS2 is suitable for the proposed L1–L2 norm regularized magnetic inversion. From the results described it is suggested that, in the proposed inversion framework, that is, L1–L2 norm combined regularized inversion with CDA, the intensity of the weights of conventional wS1 seems to be not enough in the deep part to reproduce the true model, and the weighting function wS2 seems to outperform wS1. If geomagnetic intensity is 51,000 nT, the induced magnetization is about 1 A/m, which is 10 times smaller than the estimated

Findings

Discussion

Conclusions