Neighborhood size and spatial scale in raster-based slope calculations

Raster-based slope estimation is routine in GIS. Like many other terrain attributes, the slope at a location is determined from elevations of surrounding cells. This spatial extent – ‘neighborhood size’ – is often treated as the ‘spatial scale’ of the calculation. In fact, neighborhood size and spatial scale are two connected yet different concepts, but few studies have investigated the relationship between them. The distinction is important because neighborhood size is under user control whereas spatial scale is merely implicit in the computational method. This article attempts to clarify and provide a more precise meaning of the two terms by considering slope operators from the standpoint of the frequency (or wavenumber) domain. This article derives analytical expressions for the amplitude response functions of four popular slope estimators. These are used to characterize the individual methods and also to show that the neighborhood size and spatial scale of a slope calculation are not numerically the same. In fact, because there is no single spatial scale that can be unambiguously associated with a given neighborhood size, neighborhood size cannot be an adequate indicator of spatial scale. Furthermore, this article shows that different indices of ‘scale’ yield different impressions about the action of a slope estimator and its response to changing neighborhood size. Therefore, it is necessary to examine the amplitude response function when investigating the spatial scale. The article also provides guidance for GIS practitioners when selecting a slope estimation method.