Efficient geostatistical inverse methods for structured and unstructured grids

Abstract

The geostatistical inverse approach of Kitanidis (1995) requires the covariance matrix of a discretized spatial random variable and matrix multiplications of the covariance matrix. The explicit computation of the full covariance matrix is prohibitive for large‐scale problems. Discrete spectral methods, which help to reduce computational costs dramatically, are restricted to regular grids. To make the geostatistical inverse approach applicable to unstructured grids, which may be of great interest for practical applications, we use the functional parameterization of the spatial random field by the Karhunen‐Loève (KL) expansion, in which the spatial variable is parameterized by weighted base functions that are derived from the covariance function. In the inverse approach, the estimation of the spatial variable becomes estimating the weights of base functions. The prior covariance matrix of the weights is the identity matrix. The base functions are continuous in space and can be discretized in any fashion. For the derivation of the base functions, we embed the domain into a larger unit cell of a periodic domain. Then, the base functions are sinusoidal and can be derived by spectral methods which are independent of the discretization used for forward and inverse modeling. For illustration, we present case studies using synthetic data to estimate hydraulic conductivities on regular and unstructured grids. The results show that we can efficiently obtain a good estimate using the inverse method in conjunction with the KL expansion. We also demonstrate how to estimate structural parameters and to generate conditional realizations in the framework of our method.