In the field of geophysical exploration, the application of deep learning techniques has garnered significant attention. This paper proposes a new deep learning model for three-dimensional magnetotelluric inversion, named 3DInception-U. In this model, we integrate the inception module into the network architecture and combine the concatenation layer with a U-Net structure.This model has two advantages: Firstly, the inception module, along with the deep concatenation layer, enhances the network’s capability for feature extraction and representation; Secondly, the skip connections in the U-Net facilitate information propagation, enabling the design of a network with fewer parameters but better performance. We produced 10,000 3D complex samples for training by Gaussian Random Fields (GRF) and compared 3DInception-U with existing 3D MT inversion models and applied it to real geological interpretation. The results demonstrate that this network architecture achieves good inversion accuracy and robustness.

The conventional magnetotelluric inversion method is subject to the influence of the initial model, which leads to an unstable inversion process and a tendency to get trapped at local optimal solutions. In contrast, deep learning technology relies on its powerful non-linear fitting capability and can construct complex non-linear mappings directly from observation data (input) to model (output). In recent years, it has received extensive attention from researchers. Due to the difficulties in creating a sufficiently large dataset and performing extensive neural network training, most current magnetotelluric inversion methods for geophysical exploration remain limited to one-dimensional (1D) or two-dimensional (2D) scenarios. To the best of our knowledge, for deep learning-based three-dimensional (3D) magnetotelluric inversion, currently there is no reported work in the literature. In this work, we propose a 3D magnetotelluric inversion method based on deep learning technology. By designing a neural network architecture for 3D structures (MT3D-Net), we achieve an end-to-end mapping from the network input to output. To alleviate the excessive dependence of the network on the training set, we introduce a joint weighted loss function based on data-driven and physics-driven method, allowing the network to follow the physical constraints of magnetotelluric data during the training process and thus more reasonably guide the update of network parameters. Numerical experiments show that this method combines the advantages of traditional and data-driven inversions, significantly improving the stability and accuracy of magnetotelluric inversion. The proposed method has been successfully applied to synthetic models and measured field data, and has good application prospects.

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.

Multigrid (MG) solvers, typically increasing the computational time linearly with the grid size, are suitable for large-scale forward modeling problems. However, for electromagnetic (EM) problems as frequency decreases and grid is increasingly stretched, MG solvers for EM modeling can converge slowly or even diverge. We propose an efficient four-color Line Gauss-Seidel (GS) MG for finite difference (FD) frequency-domain EM solution. In this algorithm, the edge components attached to nodes on each line in one particular direction are updated simultaneously, leading to that the solution in each local region satisfies divergence free condition. Due to the fact that each local linear system of equations is completely uncoupled with that formed for its disjoint lines, we can group all lines of grid nodes into four colors with the requirement that all local systems formed for lines with the same color are disjoint. This can be utilized to parallelize or vectorize our algorithm. The correctness is verified by comparing with the analytical solution based on a three-layered model. The numerical performance is examined by comparing with other commonly used state-of-art solvers based on three increasingly more complex models, indicating the efficiency dominance, good parallelization and excellent ability on handling grid stretching for our algorithm.

Multigrid (MG) methods are among the best choice for stable and efficient three-dimensional (3D) forward modeling of electromagnetic (EM) fields over large area due to their linear dependence of computational time on grid size. However, as the frequency decreases to near zero and/or the grid is increasingly stretched, MG solvers for EM modeling based on the curl-curl equations converge slowly or even diverge. In this letter, we develop an efficient MG algorithm combined with a two-color plane Gauss-Seidel (GS) smoother for finite difference forward modeling of EM fields particularly at low frequencies. In this algorithm, we group different planes of the grid nodes into two colors. In each color, the components attached to different planes are totally decoupled and can be solved simultaneously, which can be distributed to different processors. The Dublin Test Model 1 is used to verify the accuracy of our algorithm and examine the numerical performance of our method against MG algorithms based on a four-color cell-block GS smoother and the Bi-Conjugate Gradient stabilized (BICGstab) smoother (as four-color cell-block GS MG and BICGstab-MG, respectively), and BICGstab and Quasi-Minimal Residual (QMR) both preconditioned with block incomplete lower-upper (blockILU) decomposition (as blockILU-BICGstab and blockILU-QMR, respectively). The numerical test based on OpenMP shows the good parallelization of our algorithm. Grids stretched to different degrees are designed to examine its ability to handle grid-stretching. The numerical performance comparison indicates its remarkable dominance in efficiency and stability.

Krylov subspace solvers for frequency-domain electromagnetic forward modeling problems converge remarkably more slowly as the period increases. In this article, we present an efficient four-color cellblock Gauss Seidel (GS) preconditioner for finite-difference (FD) electromagnetic modeling in geophysical applications. Rather than updating the FD electromagnetic (EM) equation edge by edge, as in a traditional GS scheme, we renew six edge components attached to one node simultaneously (i.e., in cellblock manner) effectively enforcing a local divergence free condition for currents. To improve implementation efficiency, we reorder the nodes on the FD grid into four colors so that nodes in each color are uncoupled, allowing the use of highly parallel vectorized algorithms. The four-color cellblock GS preconditioner is implemented in the MATLAB code, in conjunction with a BiCGstab solver. It is compared, in terms of iteration number and computing time, with other three commonly used preconditioners [GS, symmetric successive overrelaxation (SSOR) and incomplete lower and upper triangular matrix decomposition (ILU)] on three models-two synthetic and one modified from the version of real data inversion. The comparison indicates that the proposed algorithm is extremely stable and efficient compared with the other three preconditioners tested, over a range of periods (1-1000 s). Especially at long periods, the improvement of our proposed algorithm is substantial. In addition, a parallel implementation of the cellblock GS preconditioner is straightforward due to the independence of nodes in each color.

A label of an equation in the Four-Color Cellblock Gauss-Seidel Preconditioner section of the title article contains a writing mistake, so we are modifying it by: 1) changing “violation of (8)” to “violation of (6)” and 2) changing “free condition in (8)” to “free condition in (6).” This error does not affect the text or results presented in the article.