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.

翻译：蒙特卡洛马尔可夫链（MCMC）方法通常面临两个基本挑战：先验分布的精确表征与似然函数的高效计算。在贝叶斯层析成像研究中，主成分分析（PCA）在部分场景下可简化先验分布的显式定义，同时支持构建基于多项式混沌展开（PCE）的高精度替代模型，以替代计算密集型全物理正演求解器。当PCA无法直接简化先验分布定义时，可采用深度生成模型（如变分自编码器（VAE））作为替代方案。然而，精确构建能够捕捉VAE潜在参数与正演输出间复杂非线性关系的替代模型仍是一大挑战。事实上，当输入输出关系可通过低阶多元多项式有效近似时，PCE模型可提供高精度，但这一条件在利用深度生成模型导出的潜在变量时通常难以满足。本文提出一种结合VAE在先验表示方面的优异重建性能与PCA-PCE替代模型精度的策略，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的参数化方法进行先验探索与样本提议，同时基于PCE开展正演建模，该建模过程作用于所考察VAE样本的全局或局部主成分分量。