  Implementations of Markov chain Monte Carlo (MCMC) methods need to confront
two fundamental challenges: accurate representation of prior information and
efficient evaluation of likelihoods. Principal component analysis (PCA) and
related techniques can in some cases facilitate the definition and sampling of
the prior distribution, as well as the training of accurate surrogate models,
using for instance, polynomial chaos expansion (PCE). However, complex
geological priors with sharp contrasts necessitate more complex
dimensionality-reduction techniques, such as, deep generative models (DGMs). By
sampling a low-dimensional prior probability distribution defined in the
low-dimensional latent space of such a model, it becomes possible to
efficiently sample the physical domain at the price of a generator that is
typically highly non-linear. Training a surrogate that is capable of capturing
intricate non-linear relationships between latent parameters and 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 not met when employing latent variables derived from DGMs. In this
contribution, we present a strategy combining the excellent reconstruction
performances of a variational autoencoder (VAE) with the accuracy of PCA-PCE
surrogate modeling in the context of Bayesian ground penetrating radar (GPR)
traveltime tomography. Within the MCMC process, the parametrization of the VAE
is leveraged for prior exploration and sample proposals. Concurrently,
surrogate modeling is conducted using PCE, which operates on either globally or
locally defined principal components of the VAE samples under examination.
