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

摘要

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-homogeneouson-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.