Geostatistical reduced-order models in underdetermined inverse problems

Abstract

[1] Reduced-order models (ROMs) approximate the high-dimensional state of a dynamic system with a low-dimensional approximation in a subspace of the state space. Properly constructed, they are used to significantly reduce the computational cost associated with the simulation of complex dynamic systems such as flow and transport in the subsurface. A key component in model reduction is to construct the subspace where we look for approximate solutions. In this work, we apply model reduction in inverse modeling and use the solution parameter space of underdetermined geostatistical inverse problems to construct the subspace in which we seek approximate solutions for any given parameters needed in the inversion process. The subspace is constructed by collecting state variable (e.g., pressure) distributions in the flow domain. Each of the distributions, called snapshots, contains the result of full forward model simulation for a given test with a basis vector in the solution parameter space as input parameters. We then use linear combinations of the snapshots to approximate the forward model solution for any parameters needed in inverse modeling. In geostatistical inverse modeling, the solution parameter space is spanned by the cross-covariance of measurements and parameters; hence, we name the ROM as the geostatistical reduced-order model (GROM). We also show that with minor loss of accuracy in the forward model, the accuracy in parameter estimation is still high, and the saving in computational cost is significant, especially for large-scale inverse problems where the number of unknowns is enormous.