Bayesian full waveform inversion using the shifted ordinary differential equation method with underdamped Langevin Markov chain Monte Carlo

Abstract

Full waveform inversion (FWI) is a powerful tool for constructing high-resolution subsurface models but remains fundamentally ill-posed due to sparse and noisy data, modeling errors, and the severe nonlinearity of the forward modeling. Because this inherent nonlinearity creates complex uncertainty structures that traditional deterministic optimization cannot easily resolve. To fully quantify this inversion uncertainty and explore all plausible solutions, Bayesian FWI seeks to sample directly from the posterior probability density function using Markov chain Monte Carlo (MCMC) algorithms. However, traditional MCMC methods suffer from inefficient exploration (poor mixing) in high-dimensional model spaces. In this study, we adopt an efficient sampler based on underdamped Langevin diffusion (ULD). The stochastic differential equation (SDE) governing ULD is approximated using the shifted ordinary differential equation (ODE) method, which converts the stochastic dynamics into a deterministic ODE that is easier to solve numerically. Building on this formulation, we introduce the SORT (Shifted ODE with Runge–Kutta Three) sampling method. The SORT method employs a third-order Runge-Kutta scheme to discretize the ODE, which effectively reduces discretization errors and enables highly efficient sampling. We validate the performance of SORT through synthetic 2D acoustic FWI experiments in both low- and high-dimensional model spaces. The numerical results show that SORT efficiently explores high-dimensional posterior distributions and produces stable and reliable mean and uncertainty estimates for complex subsurface structures.