The efficiency of solving geophysical inverse problem largely relies on the efficiency of solving the corresponding forward problem. As for electromagnetic (EM) data forward modeling in frequency domain, the conventional numerical methods, e.g. finite difference method (FDM), discretize the governing equations resulting a large linear system which is usually expensive to solve. Meanwhile, for inversion iteration we normally do not need to solve the forward problem in high precision. Thus a rapid surrogate modeling approach which uses the neural network is promising for replacing the forward modeling module in the inversion scheme. Here we proposed an algorithm which uses the neural operator to solve the EM data modeling problem in the frequency domain. To develop a surrogate model for EM data forward problem, we introduce an extended Fourier neural operator (EFNO) that enables the calculation at least 100 times faster than the conventional FDM solver while maintaining good precision. Moreover, by adding a sub-network the proposed neural operator has good generalization which has the capacity of predicting solution at any site locations and frequencies. Due to the discretization-invariance of Fourier neural operator, the neural operator trained on coarse grids can easily transfer to fine grids with only retraining part of parameters, resulting in a super-resolution prediction capability. We test our proposed method with 2-D and 3-D magnetotelluric (MT) data modeling problems, demonstrating that the EFNO has great potentials for severing as a general rapid surrogate forward solver in EM data inversion scheme.

The non-linear conjugate gradient (NLCG) algorithm is one of the popular linearized methods used to solve the frequency domain electromagnetic (EM) geophysical inverse problem. During the NLCG iterations, the model gradient guides the searching direction while the line-search algorithm determines the step length of each iteration. Normally, the line search requires solving the corresponding forward problem a few times. Since line search is usually computationally inefficient, we introduce the adaptive gradient descent (AGD) algorithm to accelerate solving the frequency domain EM inverse problem within the linearized framework. The AGD algorithm is a variant of the classical gradient descent method and has been well developed and widely used in deep learning. Rather than the time-consuming line search, its core idea is to algebraically manipulate the cumulative gradients and updates of the model from previous iterations to estimate the model parameter variables at the current iteration. For the inversion of magnetotelluric (MT) data, we here designed and implemented a framework by using the AGD algorithm combined with the cool-down scheme to tune the regularization parameter. To improve the convergence performance of the AGD algorithm (specifying to Adam and RMSProp), we proposed a tolerance strategy which has been tested numerically. To optimize the global learning rate, we carried out some comparative trials in the proposed inversion framework. The inverted results of synthetic and real-world data showed that both AGD algorithms (Adam and RMSProp) can recover comparable results and save more than a third of CPU time compared to the NLCG algorithm.

SUMMARY: Effective viscosity of the upper mantle is a critical parameter for comprehending the dynamics of the lithosphere and plate tectonics. In recent years, magnetotelluric (MT) surveys have gained attention as a potential tool for determining viscosity. However, a direct physical basis for the relationship between effective viscosity and electrical resistivity still needs to be improved. To address this issue, we have established the basis that connects electrical resistivity and effective viscosity under different thermochemical conditions and the principle of electrical neutrality. The creep and electrical conductions of rocks in the lithospheric mantle are all the thermally activated atomic-scale random motion of particles in solids or melts, controlled by the slowest and fastest particles, respectively. Due to the sizeable electrostatic interaction energy, the concentrations of the two particles must always satisfy the charge-neutral condition. Hence, we derived two ideal models for anhydrous and hydrous olivine. Then, to demonstrate their applicability, we converted the resistivity cross-section constructed from long-period MT data to the effective viscosity distribution in the Tarim–Tianshan–Junggar area. We found that the results matched those obtained by the previous method well. Overall, this study provides valuable insights into the potential of MT surveys for determining the effective viscosity of the upper mantle.

SUMMARY: 3-D magnetotelluric (MT) inversion is a powerful tool for imaging the Earth’s electrical structure, yet its computational demands remain a major challenge, particularly when simulating responses across broad frequency bands. Conventional inversion schemes rely on a unified mesh for forward simulation at all frequencies, which inflates the number of model parameters and greatly extends runtime. To overcome these limitations, we present a finite-volume-based frequency-domain survey decomposition (FVFSD) method that adaptively constructs forward simulation meshes according to the skin depth of each frequency while maintaining a fixed horizontal discretization. This design decouples forward simulation meshes from the inversion mesh, striking a balance between accuracy and efficiency. To further improve the treatment of resistivity contrasts, an equivalent circuit scheme is employed to compute effective conductivities within control volumes, outperforming conventional volume-weighted averaging. We validate the proposed method through comprehensive numerical experiments, including synthetic benchmarks and real-world case studies from the Akebasitao region (China) and the Southern African MT experiment. Results demonstrate that FVFSD achieves accuracy comparable to standard finite-difference forward simulations, while significantly reducing computational time. In large-scale MT inversions, this acceleration directly translates into faster convergence and more efficient recovery of lithospheric-scale resistivity structures. The method is fully compatible with existing inversion algorithms and solvers, making it straightforward to integrate into standard workflows. Overall, FVFSD provides a scalable and accurate strategy for advancing 3-D MT inversion, with clear implications for lithospheric studies, resource exploration and tectonic investigations.