Bayesian full-waveform inversion using adaptive Markov chain Monte Carlo methods

Uncertainty quantification is crucial for seismic full-waveform inversion (FWI), which is a highly ill-posed inverse problem. In the framework of Bayesian inference, Markov chain Monte Carlo (MCMC) sampling algorithms can be used to quantify the uncertainties of FWI. However, the MCMC algorithm for FWI is faced with challenges in parameter tuning and low efficiency, which can be nontrivial for large-scale FWI problems. To solve this issue, we introduce adaptive MCMC algorithms for FWI, in which the step length is automatically tuned to optimize the sampling efficiency. Furthermore, we compare different preconditioning matrices for the proposal distribution, including an adaptive posterior covariance matrix estimated using previous samples and a local Hessian matrix. The method is implemented with an acoustic FWI with a frequency-domain finite-difference solver. The synthetic Marmousi and the 2004 BP velocity benchmark models are used to verify the effectiveness of our method. Numerical results suggest that a local Hessian matrix is preferred compared with the online estimated sample covariance matrix to be used as the preconditioning matrix for the proposal distribution. Numerical results using different starting models, data noise, and dimensionalities demonstrate the effectiveness of our method. By introducing the adaptive step length and the appropriate preconditioning matrix in the proposal distribution, the algorithm is able to draw samples efficiently from the posterior probability distribution of the FWI problem. The statistical features of the posterior samples are used to analyze uncertainties for the FWI problem.