Hierarchical Bayesian Inversion of Global Variables and Large-Scale Spatial Fields

Abstract

Bayesian inversion is commonly applied to quantify uncertainty of hydrological variables. However, Bayesian inversion is usually focused on spatial hydrological properties instead of hyperparameters or non-gridded physical global variables. In this paper, we present a hierarchical Bayesian framework to quantify uncertainty of both global and spatial variables. We estimate first the posterior of global variables and then hierarchically estimate the posterior of the spatial field. We propose a machine learning-based inversion method to estimate the joint distribution of data and global variables directly without introducing a statistical likelihood. We also propose a new local dimension reduction method: local principal component analysis (local PCA) to update large-scale spatial fields with local data more efficiently. We illustrate the hierarchical Bayesian formulation with three case studies: one with a linear forward model (volume averaging inversion) and two with non-linear forward models (pumping tests and hydraulic head measurements), including a 3D case. Results show that quantifying global variables uncertainty is critical for assessing uncertainty on predictions. We show how the local PCA approach accelerates the inversion process. Furthermore, we provide an open-source Python package (https://github.com/lijingwang/hierarchicalBayes.git) on the hierarchical Bayesian framework including three case studies.