Adaptive anchored inversion for Gaussian random fields using nonlinear data

Abstract

In a broad and fundamental type of ‘inverse problems’ in science, one infers a spatially distributed physical attribute based on observations of processes that are controlled by the spatial attribute in question. The data-generating field processes, known as ‘forward processes’, are usually nonlinear with respect to the spatial attribute, and are often defined non-analytically by a numerical model. The data often contain a large number of elements with significant inter-correlation. We propose a general statistical method to tackle this problem. The method is centered on a parameterization device called ‘anchors’ and an iterative algorithm for deriving the distribution of anchors conditional on the observed data. The algorithm draws upon techniques of importance sampling and multivariate kernel density estimation with weighted samples. Anchors are selected automatically; the selection evolves in iterations in a way that is tailored to important features of the attribute field. The method and the algorithm are general with respect to the scientific nature and technical details of the forward processes. Conceptual and technical components render the method in contrast to standard approaches that are based on regularization or optimization. Some important features of the proposed method are demonstrated by examples from the earth sciences, including groundwater flow, rainfall-runoff and seismic tomography.