Two-dimensional Magnetotelluric Inversion using Optimal Transport and Automatic Differentiation

Magnetotelluric (MT) data inverse problem is inherently characterized by strong non-linearity. Consequently its solution using the conventional gradient-based algorithm is highly dependent on the initial model. Usually, conventional inversion schemes (e.g. Non-linear Conjugate Gradient, NLCG) employ L₂ norm to form the objective function, measuring the data misfit. However, this metric often fails to capture the phase shifts and global structural features of complex response curves, causing the inversion iteration to become easily trapped in local minima. To address this challenge, we propose a novel 2D MT inversion framework based on Optimal Transport (OT) theory, introducing the Wasserstein distance (W₂) as a robust misfit measure.

We represent the MT dataset as a six-dimensional point cloud within a joint space-feature domain. In this framework, frequencies and station coordinates constitute the spatial dimensions, while the multi-modal responses—including apparent resistivity and phase for both TE and TM modes—form the feature dimensions. By incorporating coordinates and frequencies into the distance computation, the W₂ metric effectively constrains the overall morphological evolution of the response curves across both spatial and spectral domains, providing stronger geometric and topological constraints than the L₂ norm. We implemented the algorithm using the GeomLoss library and PyTorch, leveraging entropy-regularized Sinkhorn distances and Automatic Differentiation (AD) for efficient and precise gradient computation.

Numerical experiments on 2D synthetic models demonstrate that the OT-based inversion exhibits superior convergence stability compared to traditional methods, particularly under demanding conditions where the initial model significantly deviates from the ground truth. Furthermore, the proposed method maintains high resolution for deep conductive anomalies, even under significant noise levels. These results indicate that treating MT data as high-dimensional point clouds within an Optimal Transport framework provides a robust, geometry-sensitive, and innovative technical path for geophysical imaging.

This research was supported by grants from National Major Science and Technology Projects of China: Local Funds for the "Double First-Class" Initiative (924041), Deep Earth Probe and Mineral Resources Exploration (2024ZD1000200) and National Natural Science Foundation of China General Program (42474103).