Reformulation of Bayesian Geostatistical Approach on Principal Components

Abstract

In this note, we reformulate Bayesian geostatistical inverse approach based on principal component analysis of the spatially correlated parameter field to be estimated. The unknown parameter field is described by a latent-variable model as a realization of projections on its principal component axes. The reformulated geostatistical approach (RGA) achieves substantial dimensionality reduction by estimating the latent variable of projections on truncated principal components instead of directly estimating the parameter field. We provide solutions for best estimates and posterior variances for linear and quasi-linear inverse problems. The number of normal equations to be solved is reduced to k+p, where k is the number of retained principal components and p is the number of drifts, both independent of the number of observations. To determine the Jacobian matrix for quasi-linear problems, the number of forward model runs in each iteration is reduced to k+p+1. There is no need to evaluate the Jacobian matrix in terms of the unknown parameter field. RGA unifies the problem setup and computational techniques for large-dimensional inverse problems introduced previously, which are now naturally built in the reformulated framework. RGA is more efficient and scalable for both large-dimensional inverse problems and problems with a massive volume of observations. Moreover, conditional realizations of the parameter field can be conveniently generated by generating conditional realizations of latent variables on truncated principal component axes. We also relate the new approach to the classical geostatistical approach formulas. Large-dimensional hydraulic tomography problems are used to demonstrate the application of the reformulated approach. Key Points: • Bayesian geostatistical approach is reformulated to estimate projections on principal component axes and generate conditional realizations • The new framework unifies inverse problem setup and computational techniques for large-dimensional inverse problems • Both forward model runs and normal equations are reduced to the number of retained principal components, providing high scalability Correspondence to: J. Luo,

High‐Dimensional Groundwater Flow Inverse Modeling by Upscaled Effective Model on Principal Components

“…To be complete, the following presents a brief review of the reformulated geostatistical approach (RGA) for estimating the projections or coefficients on dominant principal axes for large-scale spatial fields (Zhao & Luo, 2020). The general observation equation describing the relationship between data or dependent variables and target parameters is given by…”

A Quasi‐Newton Reformulated Geostatistical Approach on Reduced Dimensions for Large‐Dimensional Inverse Problems

“…EQUATION represents a significant dimension reduction by transforming the estimation of s to the estimation of the latent‐variables or the projection of s on k retained principal axes, where k is typically less than 100. The reformulated approach is to minimize the negative logarithm of the posterior distribution, (Zhao & Luo, 2020): …”

“…Rearranging Equation leads to: which yields the same formula as the Gauss‐Newton method for solving the nonlinear equations (Zhao & Luo, 2020). The approach was named as “Reformulated Geostatistical Approach” (RGA) because the random process assumes the typical geostatistical structure, the framework is the same as the Bayesian geostatistical approach, and Equation can be conveniently related to classic geostatistical formulas such as cokriging equations (Zhao & Luo, 2020).…”

“…We aim to examine if the quasi‐Newton updating algorithms can yield satisfactory inverse results on reduced dimensions but with much cheaper computational costs. Our approach is based on the new Reformulated Geostatistical Approach (RGA) framework (Zhao & Luo, 2020). Unlike previous methods mentioned above approximating the Hessian matrix of the objective function, our approach approximates the Jacobian of the forward model outputs with respect to the retained principal axes.…”

“…Zha et al (2018) incorporated the low-rank approximation method into the Successive Linear Estimator (SLE) for large-scale hydraulic tomography. Zhao and Luo (2020) provided a new framework that reformulated the inverse problem by estimating the projections on retained principal axes instead of the unknown parameter fields. Thus, the dimension reduction is realized at the stage of problem setup, transforming an underdetermined inverse problem to an overdetermined inverse problem.…”