Summary Bayesian formulations of inverse problems are attractive due to their ability to incorporate prior knowledge, account for various sources of uncertainties, and update probabilistic models as new information becomes available. Markov chain Monte Carlo (MCMC) methods sample posterior probability density functions (pdfs) provided accurate representations of prior information and many evaluations of likelihood functions. Dimensionality-reduction techniques such as principal component analysis (PCA) can assist in defining the prior pdf and the input bases can be used to train surrogate models. Surrogate models offer efficient approximations of likelihood functions that can replace traditional and costly forward solvers in MCMC inversions. Many problem classes in geophysics involve intricate input/output relationships that conventional surrogate models, constructed using samples drawn from the prior pdf fail to capture, leading to biased inversion results and poor uncertainty quantification. Incorporating samples from regions of high posterior probability in the training may increase accuracy, but identifying these regions is challenging. In the context of full waveform inversion, we identify and explore high-probability posterior regions using a series of successively-trained surrogate models covering progressively expanding wave bandwidths. The initial surrogate model is used to invert low-frequency data only as the input/output relationship of high-frequency data are too complex to be described across the full prior pdf with a single surrogate model. After a first MCMC inversion, we retrain the surrogate model on samples from the resulting posterior pdf and repeat the process. By focusing on progressively narrower input domain regions, it is possible to progressively increase the frequency bandwidth of the data to be modeled while also decreasing model errors. Through this iterative scheme, we eventually obtain a surrogate model that is of high accuracy for model realizations exhibiting significant posterior probabilities across the full bandwidth of interest. This surrogate model is then used to perform an MCMC inversion yielding the final estimation of the posterior pdf. Numerical results from 2D synthetic crosshole Ground Penetrating Radar (GPR) examples demonstrate that our method outperforms ray-based approaches, as well as results obtained when only training the surrogate model using samples from the prior pdf. Our methodology reduces the overall computational cost by approximately two orders of magnitude compared to using a classical finite-difference time-domain forward scheme.
Meles et al. (Thu,) studied this question.