Bayesian full waveform inversion with sequential surrogate model refinement

Bayesian formulations of inverse problems are attractive for their ability to incorporate prior knowledge and update probabilistic models as new data become available. Markov chain Monte Carlo (MCMC) methods sample posterior probability density functions (pdfs) but require accurate prior models and many likelihood evaluations. Dimensionality-reduction methods, such as principal component analysis (PCA), can help define the prior and train surrogate models that efficiently approximate costly forward solvers. However, for problems like full waveform inversion, the complex input/output relations often cannot be captured well by surrogate models trained only on prior samples, leading to biased results. Including samples from high-posterior-probability regions can improve accuracy, but these regions are hard to identify in advance. We propose an iterative method that progressively refines the surrogate model. Starting with low-frequency data, we train an initial surrogate and perform an MCMC inversion. The resulting posterior samples are then used to retrain the surrogate, allowing us to expand the frequency bandwidth in the next inversion step. Repeating this process reduces model errors and improves the surrogate's accuracy over the relevant input domain. Ultimately, we obtain a highly accurate surrogate across the full bandwidth, enabling a final MCMC inversion. Numerical results from 2D synthetic crosshole Ground Penetrating Radar (GPR) examples show that our method outperforms ray-based approaches and those relying solely on prior sampling. The overall computational cost is reduced by about two orders of magnitude compared to full finite-difference time-domain modeling.

Efficient Bayesian travel-time tomography with geologically-complex priors using sensitivity-informed polynomial chaos expansion and deep generative networks

Monte Carlo Markov Chain (MCMC) methods commonly confront two fundamental challenges: the accurate characterization of the prior distribution and the efficient evaluation of the likelihood. In the context of Bayesian studies on tomography, principal component analysis (PCA) can in some cases facilitate the straightforward definition of the prior distribution, while simultaneously enabling the implementation of accurate surrogate models based on polynomial chaos expansion (PCE) to replace computationally intensive full-physics forward solvers. When faced with scenarios where PCA does not offer a direct means of easily defining the prior distribution alternative methods like deep generative models (e.g., variational autoencoders (VAEs)), can be employed as viable options. However, accurately producing a surrogate capable of capturing the intricate non-linear relationship between the latent parameters of a VAE and the outputs of forward modeling presents a notable challenge. Indeed, while PCE models provide high accuracy when the input-output relationship can be effectively approximated by relatively low-degree multivariate polynomials, this condition is typically unmet when utilizing latent variables derived from deep generative models. In this contribution, we present a strategy that combines the excellent reconstruction performances of VAE in terms of prio representation with the accuracy of PCA-PCE surrogate modeling in the context of Bayesian ground penetrating radar (GPR) travel-time tomography. Within the MCMC process, the parametrization of the VAE is leveraged for prior exploration and sample proposal. Concurrently, modeling is conducted using PCE, which operates on either globally or locally defined principal components of the VAE samples under examination.