Bayesian inverse modeling of large-scale spatial fields on iteratively corrected principal components

Abstract

Inverse modeling of large-scale spatially variable parameters fields at fine resolution can be reduced to estimating the projections on dominant principal components of the underlying parameter fields based on principal component analysis of the spatial covariance. For unknown or biased prior structural parameters of spatial covariance models, an iterative procedure consisting of two successive steps is usually implemented, i.e., estimation of spatial covariance followed by estimation of the spatially variable parameter fields conditional on the spatial covariance and observations. In this study, we develop an iterative, computationally efficient method to update dominant principal components for nonlinear inverse problems of large-scale spatial fields that adaptively corrects the bias from the initially defined prior spatial covariance. Our algorithm involves two-layer iterations: the inner iteration is to obtain the best estimates of projections on given retained principal components, and the outer iteration implements an efficient rank-one updating to correct the retained principal components using the posterior covariance associated with the best estimates of the projections. Numerical experiments show that inversion results can be significantly improved for large-scale inverse problems with biased structural parameters for spatial covariance. The experiment results show that the iterative correction is essentially to match the distribution patterns of the spatially correlated parameter field with its most dominant principal components. We also investigate the performance of the developed method under different biased covariance model initialization, including model type bias, variance bias and correlation length bias. The correction cannot fundamentally change the smoothness defined by the covariance model type, but can still describe major distribution patterns including anisotropy. Biased variance can be corrected and yields similar best estimates and variance maps, and biased correlation length can be corrected within an applicable range.