Efficient simulation of multivariate three-dimensional cross-correlated random fields conditioning on non-lattice measurement data

Abstract

It is challenging to simulate large-scale or fine-resolution multivariate three-dimensional (3D) cross-correlated conditional random fields because of computational issues such as inverting, storing or Cholesky decomposition of large correlation matrices. Recently, an efficient univariate 3D conditional random field simulation method was developed based on the separability assumption of the autocorrelation functions in the vertical and horizontal directions. The developed simulation method allows for Kronecker-product derivations of the large correlation matrices and thus does not need to invert and store large matrices. Moreover, it can handle univariate non-lattice data (e.g., all soundings measure the data of one soil property and there exists missing data at some depths at some soundings). It may be more common to see multivariate non-lattice data (e.g., all soundings measure the data of multiple soil properties and there exists missing data of some properties at some depths at some soundings) in practical site investigations. However, the proposed method is not applicable to multivariate non-lattice data because it cannot directly account for the cross-correlation among different variables The purpose of the current paper is to extend the previous method to accommodate the multivariate non-lattice data. The extended method still takes advantage of the Kronecker-product derivations to avoid the mathematical operation of the large correlation matrices. A simulated example is adopted to illustrate the effectiveness of the extended method.