On smoothness properties of spatial processes

For inferential analysis of spatial data, probability modelling in the form of a spatial stochastic process is often adopted. In the univariate case, a realization of the process is a surface over the region of interest. The specification of the process has implications for the smoothness of process realizations and the existence of directional derivatives. In the context of stationary processes, the work of Kent (Ann. Probab. 17 (1989) 1432) pursues the notion of a.s. continuity while the work of Stein (Interpolation of Spatial Data; Some Theory for Kriging, Springer, New York, 1999) follows the path of mean square continuity (and, more generally, mean square differentiability). Our contribution is to clarify and extend these ideas in various ways. Our presentation is self-contained and not at a deep mathematical level. It will be of primary value to the spatial modeller seeking greater insight into these smoothness issues.