Uncertainty quantification in Bayesian full waveform inversion based on underdamped Langevin Markov Chain Monte Carlo

Abstract

Full waveform inversion (FWI) is a powerful technique for building high-resolution subsurface models, but it is fundamentally ill-posed. Traditional FWI cannot quantify uncertainties arising from noise, modeling errors, the intrinsic high nonlinearity, and other sources. In Bayesian inference, Markov chain Monte Carlo (MCMC) methods facilitate direct sampling from the posterior distribution to resolve these concerns. However, MCMC approaches are slow to converge and inefficient for exploration in high-dimensional spaces due to the complex posterior distribution in FWI. In this study, we develop an uncertainty quantification framework for Bayesian FWI based on underdamped Langevin dynamics, which adds momentum and inertia to sample trajectories, allowing for more efficient posterior exploration. We implement two splitting schemes, Strang splitting and OBABO splitting (named after the sequence of operator applications in underdamped Langevin dynamics), to discretize the stochastic differential equations (SDE). These splitting methods divide the Langevin dynamics into simpler components, each of which can be solved more accurately, and can then be recombined. This reduces discretization errors and improves stability, allowing more efficient sampling. We validate the proposed approach through numerical experiments on two examples. The results show that our methods can effectively explore high-probability regions of the posterior distribution, enabling reliable estimation of posterior means, variances, and marginal probability densities for uncertainty quantification. The Strang splitting scheme exhibits a slightly faster convergence rate and lower computational overhead. Overall, at an acceptable computational cost, our methods achieve rapid convergence, provide robust uncertainty quantification, and yield posterior statistics that are largely insensitive to the choice of initial model.