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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