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.

翻译：马尔可夫链蒙特卡洛（MCMC）方法的实现需要面对两个基本挑战：先验信息的准确表示和似然函数的高效评估。主成分分析（PCA）及相关技术可在某些情况下简化先验分布的定义与采样，并支持使用多项式混沌展开（PCE）等方法训练精确的替代模型。然而，具有尖锐对比度的复杂地质先验需要更复杂的降维技术，例如深度生成模型（DGM）。通过在该模型的低维潜空间中定义低维先验概率分布并进行采样，可以在构建通常高度非线性的生成器为代价的前提下，高效地采样物理域。训练能够捕捉潜参数与正演输出之间复杂非线性关系的替代模型是一项显著挑战。事实上，当输入-输出关系能够有效地通过较低阶的多项式进行近似时，PCE模型可提供高精度，但在使用来自DGM的潜变量时，通常无法满足这一条件。本文提出了一种结合变分自编码器（VAE）卓越重建性能与PCA-PCE替代模型建模精度的策略，并将其应用于贝叶斯探地雷达（GPR）走时层析成像。在MCMC过程中，利用VAE的参数化进行先验探索和样本提议。同时，使用PCE进行替代建模，该建模基于所考察的VAE样本的全局或局部主成分。