ABSTRACT

Spatial data are widely modeled using spatial processes that assume, for a study region D, a collection of random variables {Y (s) : s ∈ D} where s indexes the points in D. This set is viewed as a randomly realized surface over D, which, in practice, is only observed at a finite set of locations in S = {s1, s2, . . . , sn}. Once such an interpolated surface has been obtained, investigation of rapid change on the surface may be of interest. Here, interest often lies in the rate of change of the surface at a given location in a given direction. Examples include temperature or rainfall gradients in meteorology, pollution gradients for environmental data, and surface roughness assessment for digital elevation models. Since the spatial surface is viewed as a random realization, all such rates of change are random as well.