Bayesian approaches are one of the primary methodologies to tackle an inverse problem in high dimensions. Such an inverse problem arises in hydrology to infer the permeability field given flow data in a porous media. It is common practice to decompose the unknown field into some basis and infer the decomposition parameters instead of directly inferring the unknown. Given the multiscale nature of permeability fields, wavelets are a natural choice for parameterizing them. This study uses a Bayesian approach to incorporate the statistical sparsity that characterizes discrete wavelet coefficients. First, we impose a prior distribution incorporating the hierarchical structure of the wavelet coefficient and smoothness of reconstruction via scale-dependent hyperparameters. Then, Sequential Monte Carlo (SMC) method adaptively explores the posterior density on different scales, followed by model selection based on Bayes Factors. Finally, the permeability field is reconstructed from the coefficients using a multiresolution approach based on second-generation wavelets. Here, observations from the pressure sensor grid network are computed via Multilevel Adaptive Wavelet Collocation Method (AWCM). Results highlight the importance of prior modeling on parameter estimation in the inverse problem.

翻译：贝叶斯方法是解决高维反问题的主要方法之一。在水文地质中，此类反问题通过多孔介质中的流动数据推断渗透率场。通常，将未知场分解到某种基函数上，通过推断分解参数而非直接推断未知量。鉴于渗透率场的多尺度特性，小波是参数化的自然选择。本研究采用贝叶斯方法，纳入离散小波系数的统计稀疏性。首先，我们通过尺度相关超参数引入先验分布，该分布包含小波系数的层次结构及重建的光滑性。随后，利用序贯蒙特卡洛（SMC）方法自适应地探索不同尺度上的后验密度，并通过贝叶斯因子进行模型选择。最后，基于第二代小波的多分辨率方法，从小波系数重建渗透率场。其中，压力传感器网格观测数据通过多层自适应小波配置法（AWCM）计算。研究结果强调了先验建模对反问题参数估计的重要性。