3D Magnetotelluric Inversion Using Levy Gradient Descent Scheme

Geophysical inversion algorithms can be categorized into gradient-based quasi-linear inversion and stochastic-search-based fully nonlinear inversion. Quasi-linear methods, such as gradient descent and conjugate gradient, rely on the gradient information of the objective function with respect to the model parameters for updates. These methods are prone to getting trapped in local minima and lack quantitative methods for evaluating inversion results. Fully nonlinear inversion methods, such as genetic algorithms and Monte Carlo methods, search the solution space through various stochastic processes, offering the advantage of avoiding local minima. However, they incur excessively high computational costs in 3D scenarios, making them difficult to apply to field data at present. We propose the Levy Gradient Descent (L-GD) method based on the Levy flight process and explore its performance in 3D magnetotelluric (MT) inversion.

The Levy flight process, derived from the Levy distribution introduced by the French mathematician Paul Lévy, is a random walk process that combines high-frequency small steps with low-frequency large steps. The heavy-tailed characteristic of Levy flights demonstrates unique advantages in escaping local minima and extensively exploring the solution space, making it highly suitable for solving large-scale complex optimization problems. This makes the L-GD algorithm a highly promising semi-stochastic search-driven algorithm. We have implemented the 3D MT L-GD inversion algorithm based on the open-source ModEM framework. Building upon this existing codebase, we integrated an inversion module based on the L-GD algorithm. Furthermore, we accelerated the forward modeling process using the cuBiCG solver developed by Dong et al. (2024) on GPU, transforming it into a practical 3D MT inversion method. Additionally, traditional methods typically rely solely on the RMS data misfit to evaluate the final inversion result. To provide model evaluation criteria beyond data misfit, we calculate statistical information for models along the search path, obtaining the mean and standard deviation of all models during the inversion iterations. This statistical information is essentially a weighted combination of gradients, reflecting the characteristics of the marginal distribution for each parameter in the high-dimensional solution space, thereby providing crucial indirect information for the reliability analysis of the optimal model.

We conducted a series of synthetic and field data tests on the L-GD algorithm. The results indicate that this algorithm can achieve better data misfit compared to the NLCG algorithm, and the model statistical information provides intuitive reference for evaluating the optimal model. Furthermore, our experiments demonstrate that for semi-stochastic inversion algorithms like L-GD, the traditional cooling method—which gradually reduces the regularization factor of the model constraint term during iterations—is not conducive to obtaining better inversion results. Instead, fixing the regularization factor at a small, low value proves to be a superior strategy.

This research was supported by grants from National Major Science and Technology Projects of China: Deep Earth Probe and Mineral Resources Exploration (2024ZD1000200) and National Natural Science Foundation of China General Program (42474103).