Methods for generating non-separable spatiotemporal covariance models with potential environmental applications

Environmental processes (e.g., groundwater contaminants, air pollution patterns, air–water and air–soil energy exchanges) are characterized by variability and uncertainty. Spatiotemporal random fields are used to represent correlations between fluctuations in the composite space–time domain. Modelling the effects of fluctuations with suitable covariance functions can improve our ability to characterize and predict space–time variations in various natural systems (e.g., environmental media, long-term climatic evolutions on local/global scales, and human exposure to pollutants). The goal of this work is to present the reader with various methods for constructing space–time covariance models. In this context, we provide a mathematical exposition and visual representations of several theoretical covariance models. These include non-separable (in space and time) covariance models derived from physical laws (i.e., differential equations and dynamic rules), spectral functions, and generalized random fields. It is also shown that non-separability is often a direct result of the physical laws that govern the process. The proposed methods can generate covariance models for homogeneous/stationary as well as for non-homogeneous/non-stationary environmental processes across space and time. We investigate several properties (short-range and asymptotic behavior, shape of the covariance function etc.) of these models and present plots of the space–time dependence for various parameter values.