Some New Random Field Tools for Spatial Analysis

This is a brief review, in relatively nontechnical terms, of recent rather technical advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric characteristics of excursion sets of random fields, such as their volume, surface area and Euler characteristics. What is particularly important in these formulae is that whereas the previous theory covered only stationary, Gaussian random fields, the new theory requires neither stationarity nor, a fortiori, isotropy. Furthermore, it covers a wide class of non-Gaussian random fields. The formulae provided by these advances have a wide range of potential applications, including new techniques of parameter estimation, model testing, and thresholding for spatial and spacetime functions. The paper reviews this theory, and provides brief descriptions of some of the applications.