Process modeling for slope and aspect with application to elevation data maps

Learning about the behavior of land surface gradients and, in particular, slope and aspect over a region from a dataset of levels obtained at a set of (possibly) irregularly spaced locations assumes importance in a variety of applications. A primary example considers digital terrain models for exploring roughness of land surfaces. In a geographic information system software package, gradient information is typically extracted from a digital elevation/terrain model (DEM/DTM), which usually presents the topography of the surface in terms of a set of pre-specified regular grid points, each with an assigned elevation value. That is, the DEM arises from preprocessing of an originally irregularly spaced set of elevation observations. Slope in one dimension is defined as “rise over run”. However, in two dimensions, at a given location, there is a rise over run in every direction. Then, the slope at the location is customarily taken as the maximum slope over all directions. It can be expressed as an angle whose tangent is the ratio of the rise to the run at the maximum. In practice, at each point of the grid, rise/run is obtained through comparison of the elevation at the point to that of a set of neighboring grid points, usually the eight compass neighbors, to find the maximum. Aspect is defined as the angular direction of maximum slope over the compass neighbors. We present a fully model-based approach for inference regarding slope and aspect. In particular, we define process versions of the slope and aspect over a continuous spatial domain. Modeling slopes takes us to directional derivative processes; modeling angles takes us to spatial processes for angular data. Using a stationary Gaussian process model for the elevation data, we obtain distribution theory for slope and associated aspect as well as covariance structure. Hierarchical models emerge; fitting in a Bayesian framework enables attachment of uncertainty. We illustrate with both a simulation example and a real data example using elevations from a collection of monitoring station locations in South Africa.