An Alternating Direction Method of Multipliers Algorithm for 1-D Magnetotelluric Anisotropic Inversion Using Fourier Series Expansion

In this study, we present a novel approach for 1-D magnetotelluric (MT) anisotropy inversion that aims to improve the reliability and efficiency of the inversion process. First, to reduce the number of inversion variables, we used the Fourier series (FS) expansion to approximate the resistivity distribution of the underground media. Consequently, the inversion variables are identified as the coefficients of the FS expansion, which are typically fewer in number than the layers. In addition, the boundary restriction of the recovered resistivity model is applied to further reduce the solution space and obtain a physically meaningful result. The objective function is formulated on the basis of the Tikhonov regularization method with an
$L_{1}$
norm, which facilitates accurate imaging for models with abrupt changes in resistivity. Then, we employ a new alternating direction method of multipliers (ADMM) algorithm to minimize the augmented Lagrangian function of the original objective function, thus deriving a set of resistivity parameters that can effectively illustrate the observation data. This process involves alternating updates of the inversion variables and two auxiliary variables during each iteration. Finally, our FS-based ADMM inversion method with the
$L_{1}$
norm, when subjected to extensive tests with synthetic and field datasets, has proved its superiority in terms of robustness, reliability, and efficiency in reconstructing subsurface resistivity distributions, compared to the conventional Gauss-Newton (GN) algorithm with the
$L_{2}$
norm. This offers the geoelectromagnetic community a novel and effective tool for the interpretation of MT data observed in anisotropic media.