Bayesian seismic inversion: A fast sampling Langevin dynamics Markov chain Monte Carlo method

Abstract

In this study, we aim to solve the seismic inversion in the Bayesian framework by generating samples from the posterior distribution. This distribution incorporates the uncertainties in the seismic data, forward model, and prior information about the subsurface model parameters; thus, we obtain more information through sampling than through a point estimate (e.g. maximum a posteriori method). Based on the numerical cost of solving the forward problem and the dimensions of the subsurface model parameters and observed data, sampling with Markov chain Monte Carlo (MCMC) algorithms can be prohibitively expensive. Herein, we consider the promising Langevin dynamics MCMC algorithm. However, this algorithm has two central challenges: (1) the step size requires prior tuning to achieve optimal performance and (2) the Metropolis-Hastings acceptance step is computationally demanding. We approach these challenges by proposing an adaptive step-size rule and considering the suppression of the Metropolis-Hastings acceptance step. We highlight the proposed method's potential through several numerical examples and rigorously validate it via qualitative and quantitative evaluation of the sample quality based on the kernelized Stein discrepancy (KSD) and other MCMC diagnostics such as trace and autocorrelation function plots. We conclude that, by suppressing the Metropolis-Hastings step, the proposed method provides fast sampling at efficient computational costs for large-scale seismic Bayesian inference; however, this inflates the second statistical moment (variance) due to asymptotic bias. Nevertheless, the proposed method reliably recovers important aspects of the posterior, including means, variances, skewness and 1-D and 2-D marginals. With larger computational budget, exact MCMC methods (i.e. with a Metropolis-Hastings step) should be favoured. The results thus obtained can be considered a feasibility study for promoting the approximate Langevin dynamics MCMC method for Bayesian seismic inversion on limited computational resources.