Toward Practical and Reliable Electromagnetic Inversion: A Semi-Stochastic Lévy Gradient Descent Approach With Trajectory-Based Model Assessment

Abstract

Quantitative assessment of inverted models has long been a major challenge in geophysical inversion, particularly in 3-D electromagnetic (EM) inversion. Traditional gradient-based inversion methods are computationally efficient, but their results strongly depend on the initial model and may converge to local minima. Moreover, they generally provide only a global measure of data misfit, such as the rms, rather than spatially resolved information on the reliability of the recovered model. In contrast, fully stochastic methods can deliver uncertainty estimates through posterior probability distributions, yet their computational cost remains prohibitive for 3-D EM inversion. To bridge this gap, we propose a semi-stochastic inversion algorithm based on the Lévy flight process—Lévy Gradient Descent (LGD). This method utilizes gradient information to guide the search direction and employs the Lévy flight process to generate stochastic step sizes, combining gradient-guided local search with broader stochastic exploration of the parameter space. This enables a more comprehensive and efficient exploration of the solution space. More importantly, the statistical information provided by LGD effectively offers assessments for the inversion results, serving as a crucial complementary metric beyond data misfit for interpreting inverted models. This allows for a relative assessment of the structural stability of individual model parameters in the model. We tested the LGD algorithm on both synthetic and real data. The results demonstrate that LGD produces results comparable or superior to nonlinear conjugate gradient (NLCG), while the trajectory-based structural stability indicator offers valuable insights for interpretation. Our new resistivity model of the Yellowstone hotspot region reveals a deep mantle plume head that appears to connect to shallow structures only through weak, moderately conductive channels—suggesting a waning hotspot.