Simulation of Stationary Gaussian Random Fields with a Gneiting Spatio-Temporal Covariance

Abstract

The nonseparable Gneiting covariance has become a standard to model spatio-temporal random fields. Its definition relies on a completely monotone function associated with the spatial structure and a conditionally negative semidefinite function associated with the temporal structure. This work addresses the problem of simulating stationary Gaussian random fields with a Gneiting-type covariance. Two algorithms, in which the simulated field is obtained through a combination of cosine waves are presented and illustrated with synthetic examples. In the first algorithm, the temporal frequency is defined on the basis of a temporal random field with stationary Gaussian increments, whereas in the second algorithm the temporal frequency is drawn from the spectral measure of the covariance conditioned to the spatial frequency. Both algorithms perfectly reproduce the correlation structure with minimal computational cost and memory footprint.