We present an efficient method for uncertainty quantification in 3D magnetotelluric (MT) inversions based on variational inference principles and the variational autoencoder (VAE) framework. In this approach, we replace the VAE’s encoder and decoder respectively with a forward modeling and a 3D optimization module, where the latent variables represent updates in the constrained model space. Utilizing reparameterization, the gradient of objective function can propagate back to the mean and variance of the latent variables, and thus restores the Gaussian distribution of resistivity models. By setting appropriate prior and initial distributions, our method explores the solutions with high log-variance to fit the data. To ensure a stable convergence, we apply a smooth constraint on the log-variance of latent variables and employ a modified adaptive moment estimation (Adam) optimizer. Numerical experiments confirm that our method can accurately estimate both the resistivity model and standard deviation at a time cost only 2-3 times that of conventional inversion, which provides a highly efficient solution for uncertainty analysis. The tests with varying regularization parameters reveal that the standard deviation estimates are sensitive to both the Kullback-Leibler (KL) divergence and log-variance smoothness terms, while the adaptive smoothing strategy can well balance these effects. The application to a dataset from USArray survey in the Northwestern US demonstrates the method can achieve robust inversion and uncertainty quantification.

Magnetotelluric (MT) and teleseismic tomography are critical techniques in deep Earth exploration for geodynamic studies, internal energy circulation, plate tectonics, volcanic systems and so on. However, due to the uneven data distribution, the inherent non-uniqueness of inversion, and differing sensitivities to subsurface media, the velocity and resistivity structures derived from independent inversions often exhibit substantial discrepancies and contradict conventional geological understanding. To address these issues, we propose a novel three-dimensional (3D) joint inversion method for MT and teleseismic traveltime data, applicable to different types of grid discretization. The method first establishes a parameter mapping for different grid types. Then, a joint inversion framework based on the local Pearson correlation constraints (LPCC) is developed using a virtual grid technique. The inversion alternately updates the resistivity and velocity models by ensuring structural similarity between them. Numerical experiments demonstrate that the proposed method can effectively integrate the advantages of both techniques, enhance resolution and stability. Additionally, the tests on hyperparameter selection provides guidance for optimizing joint inversion parameters. Finally, the developed joint inversion method is applied to MT and teleseismic traveltime data in southeastern Australia and yields a constrained 3D resistivity and velocity model of the region. The results offer significant theoretical and practical insights into the lithospheric structure, the magma transport pathways, and the geodynamic processes in this area.

The conventional magnetotelluric (MT) inversions rely primarily on L2norm to quantify the misfit between the observed data and the predicted ones. However, it often suffers from issues of unreasonable weighting of inversion data, sensitivity to outliers, and poor resolution to weak anomalies. To address these limitations and achieve better data fitting across varying scales, we propose to replace the L2norm with the Wasserstein distance to measure the distributional discrepancy between the predicted and observed data. This method adopts the Wasserstein Generative Adversarial Network (WGAN) framework and utilizes the Kolmogorov-Arnold Network (KAN) as the discriminator to implement the Wasserstein metric. By substituting the neural generator in WGAN with a physics-based forward modeling, we achieve a purely physics-driven iterative process that is similar to the conventional MT inversion workflows to ensure an adequate data fitting. Numerical experiments on synthetic models demonstrate that MTGAN-KAN can better fit data distributions and yield higher-resolution inversion results than the conventional methods. Compared to WGAN that employs a fully connected neural network (FCNN) as the discriminator, the incorporation of KAN can significantly improve the convergence and stability of MTGAN. An inversion test using MT data from Newer Volcanic Province in Australia further highlights the superior characteristics of data fitting and resolution of MTGAN-KAN in the inversion of MT data.

The choice of data misfit measure has a great impact on the convergence of electromagnetic inversion. The conventional measure based onl2-norm tends to excessively amplify the weights of a larger misfit, inadvertently neglecting data with a smaller misfit during the inversion process, thereby diminishing the resolution to a certain degree. To solve this problem, we propose a robust inversion strategy based onl1-norm data misfit and adaptive moment estimation (Adam). In this scheme, we use the Ekblom-typel1-norm to simplify the derivative computation of the absolute value function. The Adam algorithm is further applied to optimize this type of non-smooth objective function, which incorporates momentum terms and adaptive steps, allowing it to better adapt to the irregularities in gradient changes. The inversion results obtained from both synthetic models and field measurements demonstrate that the Adam method performs considerably better than the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method for optimizing thel1-l2norm of objective function. Compared with the conventionall2-norm data misfit, thel1-norm data misfit can effectively avoid excessive optimization of data with large misfits and achieve high-resolution inversion results.

In a heterogeneous medium (usually called a rough medium) with fractured formations, the propagation of an electromagnetic (EM) field is a type of subdiffusion. Current mainstream geophysical EM data processing methods cannot be applied to data acquired on heterogeneous Earth, as they are not governed by the classic diffusion theory. To evaluate the influence of roughness on the transient EM (TEM) signal for a complex model and contribute to data inversion, we proposed a novel three-dimensional (3-D) forward modeling scheme for TEM in rough media. First, we derived the governing equation with a fractional-order time derivative for the subdiffusion of EM waves in rough media. Then, we proposed a novel time discretization using an unequal step length for the Caputo operator, which significantly reduces the total number of time steps. Finally, an implicit time-domain finite-element method using unstructured tetrahedron discretization was adopted to solve the 3-D forward problem. Furthermore, an efficient time segmentation strategy combined with parallel RHS construction was proposed to accelerate modeling. The numerical results prove that the proposed method is accurate and efficient, and will be a powerful numerical method for analyzing TEM wave propagation and processing TEM data in areas with multiscale fractures or porosity.