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：贝叶斯方法是在高维空间求解反问题的主要方法论之一。此类反问题出现在水文学中，用于根据多孔介质中的流动数据推断渗透率场。通常的做法是将未知场分解为某种基函数，并推断分解参数，而非直接推断未知量。鉴于渗透率场的多尺度特性，小波是参数化该场的自然选择。本研究采用贝叶斯方法，引入离散小波系数所具有的统计稀疏性。首先，我们通过尺度依赖的超参数，施加一种包含小波系数层级结构与重建平滑性的先验分布。接着，采用序列蒙特卡洛方法自适应地探索不同尺度上的后验密度，并基于贝叶斯因子进行模型选择。最后，利用基于第二代小波的多分辨率方法，从系数重建渗透率场。此处，压力传感器网格网络的观测值通过多水平自适应小波配点法计算。结果凸显了在反问题中先验建模对参数估计的重要性。