Inversion of VLF Data Using a Non-Linear Smoothing Operator

Abstract

Summary Inversion of electromagnetic induction data, including VLF, is generally realized using smooth inversion methods. The smoothness of the recovered models and the regularization of the ill-conditioned problem is ensured with smoothing matrices. Smoothing matrices are simple linear derivative matrices penalizing the resistivity differences between adjacent cells. Since these matrices are linear operators, they are calculated once at the beginning of the inversion process. Considering its structure, smoothing matrices can be considered similar to low-pass Gaussian filters. Similarly, it’s possible to define a non-linear smoothing operator based on rank order filtering. We have defined a non-linear smoothing constraint based on these filters and penalized the differences from the cells corresponding to the desired rank value. Since the defined constraint is non-linear it is re-calculated as the model parameters change. The defined constraint is tested on synthetic data and its results are compared to the results obtained with a traditional smoothing matrix. Accordingly, the defined non-linear rank order smoothing constraint can provide relatively focused, amplified structures, and can increase blockiness.