The subsurface conductivity of geological media is often anisotropic, making three-dimensional (3D) anisotropic inversion of controlled-source electromagnetic (CSEM) essential for resolving complex geologic settings. However, compared to isotropic inversion, anisotropic inversion involves a substantially greater number of model parameters, increasing the severity of the non-uniqueness problem and enhancing interpretation complexity for large-scale field data. To address these challenges, we present an innovative anisotropic inversion approach that incorporates a Gramian-based constraint, which promotes similarity between horizontal and vertical conductivity models without relying on a prior information. We formulate the inverse problem within a Gauss-Newton framework and employ the finite element method on unstructured grids, leveraging parallel direct solvers for computational efficiency. Synthetic tests on complex anisotropic land and marine CSEM models show that the Gramian-constrained inversion significantly suppresses spurious anomalies and improves reliability compared to conventional anisotropic inversion. Application to field CSEM data from the Huaniushan Pb-Zn mining area in Gansu Province, China, demonstrates high consistency with real geology and drilling results. These findings highlight the proposed approach as a computationally efficient, robust, with direct relevance to hydrocarbon, mineral, and geothermal exploration.

Accurate estimation of the Earth’s interior temperature is essential for solving fundamental scientific and applied geothermal problems. Currently, there is no universal method for determining deep temperature fields; however, such a method may be based on resistivity, a temperature-dependent proxy parameter. We propose an electromagnetic (EM) geothermometer based on the temperature–pressure coupled resistivity model (TPCRM). This geothermometer can accurately determine the relationship between the normalized resistivity, temperature, and pressure in deep formations based on well-logging, gravity, and EM data, thus allowing to visualize the temperature distribution. The TPCRM is utilized to predict the subsurface temperature in the Xiong’an New Area and shows an accuracy of 76.35%–96.58%. Sensitivity analysis of the critical variables of the TPCRM reveals that the TPCRM relatively weakly depends on the number of constraining boreholes and that the optimization of the subdivision spacing of the well-logging data can significantly improve temperature prediction accuracy. In addition, the effect of the spacing of inverted resistivity normalization grid nodes on the temperature prediction accuracy is relatively weak because the TPCRM considers the factor of the overburden pressure. The TPCRM is a promising tool for studying thermal genetic mechanisms, as well as fine evaluation of geothermal resources for their large-scale and efficient development and utilization.

In this study, we present an efficient and memory saving 3D inversion algorithm for interpreting controlled-source electromagnetic (CSEM) data using the total electric field formulation. To tackle CSEM problems involving complex geometries, we discretize the study domain for both forward and inversion problems using unstructured tetrahedral elements. Our inversion scheme combines the parallelized finite element (FE) method with the Gauss-Newton optimal strategy. Additionally, we transform the conventional model space inversion into data space inversion, significantly reducing the computation time and Random Access Memory (RAM) requirements during the inversion process. To begin, we validate the effectiveness and stability of the developed data space inversion algorithm by utilizing a synthetic land CSEM basin model and a synthetic marine CSEM model with bathymetry. These validation experiments further demonstrate that compared with the conventional model space inversion method, the computational efficiency of the data space inversion scheme is greatly improved and the memory required is significantly reduced. Furthermore, we apply the inversion method to survey CSEM data to demonstrate the practical applicability of the new inversion scheme.

We present a regularized inversion method for three-dimensional (3D) magnetotelluric (MT) data with axial anisotropic conductivities based on the edge-based finite element (FE) method. The Gauss–Newton (GN) approach is used to minimize the inversion objective function, including data misfit and regularization penalties, considering both structural complexity and anisotropic penalties. The most time-intensive task in the 3D MT inversion process is solving the large sparse system of linear equations. To speed up the inversion calculation, a hybrid direct–iterative solver combined with a block-diagonal preconditioner that has not yet been applied in anisotropic inversion is developed to accelerate the solutions for the sparse linear system resulting from forward modeling and sensitivity computations. In each GN iteration, a preconditioned conjugate gradient (PCG) method is adopted to overcome the difficulty of the sensitivity matrix storage for the anisotropic scene and obtain a model update without explicitly calculating and storing the sensitivity matrix. Before the inversion test, we use a model to demonstrate that the hybrid solver is computationally beneficial in terms of memory usage and time spent as compared to the direct solver. The good convergence properties and efficiency of the GN–PCG inversion scheme are demonstrated by two synthetic models and USArray data. The proposed inversion scheme can be an important supplement to existing anisotropic inversion algorithms and provide technical support for MT data interpretation.

Efficient three-dimensional magnetotelluric anisotropy forward modeling is one of the key research techniques used for inversion interpretation. We propose an improved multi-level down-sampling scheme to reduce the degrees of freedom of the stiffness matrix derived from the edge-based finite element method to improve the computational efficiency, saving on memory usage and calculation time for the forward modeling. Then, to further reduce the memory requirements and speed up the solution of the discretized electric system, we develop a multiple right-hand direct–iterative hybrid solver based on a block rational Krylov preconditioner. The solver we propose can further save computational costs and time based on the multi-level down-sampling scheme. Moreover, the convergence performance of the direct–iterative solver is less affected by the frequency, which solves the problem of slow convergence of the electric field control equation at low frequencies. We also use the high-level language Julia, which is easy to load into third-party packages, to ensure the stability and efficiency of the program. Finally, the validity and advantages of the two schemes are analyzed in detail using three examples. The example results show that the multiple right-hand direct–iterative hybrid solver and improved multi-level down-sampling can significantly reduce the computational memory and save computational time.

The controlled-source electromagnetic (CSEM) method is a crucial tool for near-surface investigations and hydrocarbon exploration because of its economic benefits. Limited resolution is one of the inherent defects of the CSEM method, and to obtain high contrast results a multinary transform function was introduced to CSEM 3-D inversion. The multinary transform function was constituted by superposing several error functions, and the transform function transformed the model parameters from continuously distributed space into a semistep distributed multinary space. To deal with complex geometries, the edge-based finite element (FE) method with an irregular hexahedral grid was applied to the modeling and inverse problem. We used the Gauss–Newton optimization method to minimize the Tikhonov parametric function. The complex chessboard model and the marine CSEM model with complex geometry were used to validate the ability of the new method in improving the resolution of the CSEM method. By comparing the results of conventional, focusing, and multinary inversion methods, the effectiveness of the multinary inversion method in depicting sharp boundaries of different physical properties was proved. Additionally, the comparisons proved that multinary inversion method can, to some extent, overcome the insensitivity to low conductors of the CSEM method.