Directional rates of change under spatial process models

Spatial process models are now widely used for inference in many areas of application. In such contexts interest is often in the rate of change of a spatial 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. Because the spatial surface is viewed as a random realization, all such rates of change are random as well. We formalize the notions of directional finite difference processes and directional derivative processes building upon the concept of mean square differentiability as developed by Stein and Banerjee and Gelfand. We obtain complete distribution theory results under the assumptions of a stationary Gaussian process model either for the data or for spatial random effects. We present inference under a Bayesian framework which, in this setting, presents several advantages. Finally, we illustrate our methodology with a simulated dataset and also with a real estate dataset consisting of selling prices of individual homes.