A general class of models for stationary two-dimensional random processes

A parametric family of spectral density-covariance function pairs for stationary spatial processes is introduced. The spectral densities are rational functions of elliptic forms along with factors in the numerator that may be of mixed hyperbolic, parabolic or elliptic form. The resulting covariance functions are linear combinations of modified Bessel functions of the second kind, which have been shown to be the natural basis for two-dimensional covariance functions (Whittle, 1963). Recursive computation techniques make calculation of the covariance functions feasible for even the most complicated models. Stochastic differential equations are used to provide a physical basis for the models as well as to develop methods for generation of the resultant processes. The results provide a step toward development of a general theory and methodology for the identification and estimation of two-dimensional processes to parallel the rational spectral density/autoregressive-moving average approach for one-dimensional processes.