Models for Spatial-Temporal Covariances

A good model for the covariance function of a stationary process in space and time should accurately describe the variances and correlations of all linear combinations of the process. In particular, it does not suffice to find a model that describes the purely temporal covariances and the purely spatial covariances accurately. Rather, it is critical to capture the spatial-temporal interactions as well. We consider a number of properties of spatial-temporal covariance functions and how these relate to the spatial-temporal interactions of the process. First, we examine how the smoothness away from the origin of a spatial-temporal covariance function affects, for example, temporal correlations of spatial differences. Second, we examine the implications of a Markov assumption in time on spatial-temporal covariance functions. Third, we consider models that are asymmetric in space-time: the correlation between site A at time t and site B at time s is different than the correlation between site A at time s and site B at time t. We examine some of these issues for sulfate concentrations as measured by monitors, as produced by an air quality model (CMAQ) and for the difference between observed and modeled concentrations.