A multivariate variance matrix for use with emulators

Conventional emulation considers a random field as a function of position in parameter space. This document shows how that framework may be generalized naturally to accommodate multivariate output. In the univariate case, a correlation matrix A is defined and the corresponding variance matrix is 2A, where 2 is a scalar. In the multivariate generalization presented here, the variance matrix contains a spatial component and an overall covariance component. The two-variable case is given first, followed by some simple examples. The method generalizes naturally to arbitrary numbers of variables.