Uncertainty quantification of full-waveform inversion with adaptive MCMC method

Uncertainty quantification (UQ) is crucial for seismic full waveform inversion (FWI), which is a highly ill-posed inverse problem. In the framework of Bayesian inference, we propose to use an adaptive Markov chain Monte Carlo (MCMC) sampling algorithm to quantify the uncertainties of FWI. The gradient-based MCMC algorithm for FWI is facing with challenges in parameter tuning and computation of Hessian, both are known to be non-trivial and non-efficient for large-scale FWI problem. To solve these issues, we introduce the adaptive gradient-based MCMC algorithm, in which the step length is automatically tuned to optimize the sampling efficiency. Furthermore, other than Hessian information, the posterior covariance matrix approximated using previous samples are used to propose a sample in the adaptive algorithm, which further improves the efficiency. The method is implemented with acoustic FWI with frequency domain finite difference (FDFD) solver. Numerical results using synthetic Marmousi model demonstrate the correctness of our method. By introducing the adaptive covariance matrix and step length in the proposal distribution, the algorithm is able to draw samples efficiently from posterior probability distribution (PPD) of FWI problem. The statistic features of posterior samples can be used to analyse the uncertainties for FWI problem.