Space-Time Metric Determination in Environmental Modeling

Abstract

Physical space-time metrics are used in environmental modeling to define “distance†between points in the space-time domain of a physical attribute (contaminant concentration, exposure, temperature etc.). Assessing a space-time metric is often a considerably more complicated affair than assessing a purely spatial metric. This is because the physical space-time metric suggests a certain concept of distance that blends space and time to make space-time, but at the same time, it views time as a dissimilar quantity. In this work, the determination of space-time metrics takes advantage of the strong links between the physical characteristics of the real-world attribute and the geometrical features of the composite space-time domain within which the attribute occurs. Via physical law an explicit connection is established between attribute’s space-time dependence structure (represented by the covariance function) and attribute’s domain geometry (expressed by the metric coefficients). The derived physical geometry equation can be solved for the metric coefficients. The solution depends not only on the form of the physical law, but also on the boundary/initial conditions and the randomness sources. The proposed approach turns metric coefficients into physically meaningful parameters, allowing better understanding of the space-time characteristics than the ad hoc and arbitrary metric selection in purely technical terms.

Keywords: environment, modeling, space-time, metric, stochastics, covariance, physical law