It is important to study the responses of the magnetotelluric (MT) method in anisotropic media. However, MT anisotropy research has focused mainly on one-dimensional (1D) and two-dimensional (2D) solutions. Therefore, we developed a three-dimensional (3D) finite element (FE) algorithm for MT modelling in anisotropic media. This approach is based on the weak formulation of the governing Maxwell equations using Coulomb-gauged potentials. The node-based FE method is adopted here, and the values of the coefficient matrixes are obtained with hexahedral meshes. To validate the correctness and accuracy of this method, its results are compared with previous solutions for a 2D anisotropic model and a 3D arbitrary anisotropic model, respectively. Different solvers with different preconditioners are tested, and the results show that the quasi-minimum residual method with the incomplete LU preconditioner is more stable and faster compared with the other schemes. We then studied a 3D anisotropic model in three different conditions, and analysed the results in detail. Finally, three main conclusions are obtained: the xy- and yy-mode apparent resistivities remain almost unchanged if a principal conductivity is in the x-direction; the yx- and xx-mode apparent resistivities remain almost the same if a principal conductivity is in the y-direction; a principal conductivity in the z-direction has almost no influence on apparent resistivities.

Controlled-source audio-frequency magnetotellurics (CSAMT) is an important geophysical technique. Numerous studies have shown that the electrical anisotropy in the Earth cannot be ignored because it would probably lead to misinterpretations of electromagnetic data. It is necessary and meaningful to study CSAMT responses in three-dimensional (3D) electrical anisotropic media; therefore, we have developed an edge-based finite element method for 3D CSAMT forward modelling in generalised anisotropic media. The total electric field in this approach is decomposed into a primary electric field and a secondary electric field, and the Galerkin weighted residuals method is adopted to obtain the variational equation. The accuracy of this algorithm was initially validated by comparing solutions with those obtained in previous work on a 3D arbitrary anisotropic model. We then studied the responses of an oblique source as well as equatorial and axial configurations and the tensor source for anomalies with different Euler’s angles. Several meaningful conclusions can be derived from this work and a synthetic model is developed; the results also confirm the validity of previous conclusions. This study shows that a tensor source is necessary for a CSAMT survey in 3D anisotropic media, and the anisotropic parameters of anomalies have complex and significant influences on CSAMT responses.

As a popular population based heuristic evolutionary algorithm, differential evolution (DE) has been widely applied in various science and engineering problems. Similar to other global nonlinear algorithms, such as genetic algorithm, simulated annealing, particle swarm optimization, etc., the DE algorithm is mostly applied to resolve the parametric inverse problem, but has few applications in physical property inversion. According to our knowledge, this is the first time DE has been applied in obtaining the physical property distribution of gravity data due to causative sources embedded in the subsurface. In this work, the search direction of DE is guided by better vectors, enhancing the exploration efficiency of the mutation strategy. Besides, to reduce the over-stochastic of the DE algorithm, the perturbation directions in mutation operations are smoothed by using a weighted moving average smoothing technique, and the Lp-norm regularization term is implemented to sharpen the boundary of density distribution. Meanwhile, in the search process of DE, the effect of Lp-norm regularization term is controlled in an adaptive manner, which can always have an impact on the data misfit function. In the synthetic anomaly case, both noise-free and noisy data sets are considered. For the field case, gravity anomalies originating from the Shihe iron ore deposit in China were inverted and interpreted. The reconstructed density distribution is in good agreement with the one obtained by drill-hole information. Based on the tests in the present study, one can conclude that the Lp-norm inversion using DE is a useful tool for physical property distribution using gravity anomalies.

This study introduces a computational method aimed at accelerating multifrequency 3D magnetotelluric (MT) forward modeling in 3D conductivity structures with general anisotropy. The approach combines an edge-based finite-element (FE) method with a rational Krylov subspace method. In this method, the MT source term is expressed as a planar current source, and the frequency-dependent electric field response is the product of a transfer function and a constant vector of the current source. Consequently, the rapid approximate electric field for multifrequency calculations can be obtained by constructing the orthogonal basis of the rational Krylov subspace. By applying this technique, the large sparse matrix resulting from FE discretization can be projected onto a much lower-order matrix, significantly reducing its size to tens or hundreds. Incorporating the direct solver PARDISO to construct the orthogonal basis enables a considerably faster solution compared with the previous FE method. Moreover, the algorithm is implemented using Julia, a high-level programming language known for its readability, maintainability, and extensibility. The validity of the algorithm is demonstrated through a 1D anisotropic model, and two additional 3D models are designed to further evaluate the applicability and effectiveness of our algorithm. Numerical experiments illustrate a notable increase in modeling speed compared with the traditional edge-based FE method.