A new MCMC algorithm for seismic waveform inversion and corresponding uncertainty analysis

It is superior to formulate an inverse problem in a Bayesian framework and fully solve it by stochastically constructing the posterior probability density (PPD) distribution using Markov chain Monte Carlo (MCMC) algorithms. The estimated PPD can also be used to compute several measures of dispersion in the model space. However, for realistic application, MCMC methods can be computationally expensive and may lead to inaccurate PPD estimation as well as uncertainty analysis due to the strong non-linearity and high dimensionality. In this paper, to address the fundamental issues of efficiency and accuracy in parameter estimation and PPD sampling, we incorporate some new developments into a standard genetic algorithm (GA) to design more powerful algorithms for the practical geophysical inverse problems such as a non-linear pre-stack seismic waveform inversion. First, a multiscale real-coded hybrid GA is developed to facilitate exploitation of the model space for optimal parameters at a fine scale. It is demonstrated that, by using real-coding and especially multiscaling to trade information between the model vectors defined at different resolutions, we attain a substantial speed-up in computation and obtain accurate parameter estimations. This new optimization method is further adapted to a new multiscale GA based MCMC method, in which multiple MCMC chains defined at different scales are run simultaneously in parallel. To gain the benefits of both the faster convergence of coarse scales and the greater detail of fine scales, realizations of chains at different scales are combined for intelligent proposals that facilitate exploration of the model space at the fine scale. In this study, the new MCMC is justified using an analytical example and its performance on PPD estimation, and uncertainty quantification is demonstrated using a non-linear seismic inverse problem. We find that incorporation of multiscaling in the Bayesian approach shows a great promise in solving inverse problems that involve computationally intensive forward modelling. We also find that the multiscaling is particularly attractive in addressing the model parametrization issue since the job of 'model selection' could be done based on the results of 'parameter estimation' at multiple scales, which circumvents separately solving a model selection problem in a Bayesian framework.