Probabilistic description of topographic slope and aspect

Local topographic features such as slope and aspect play a crucial role in a number of morphological, ecological, and hydrological processes. We propose a simple yet realistic probabilistic description of local slope and aspect as a function of properties of the field of elevation changes. We consider different classes of models of elevation changes and obtain the theoretical distribution of slope and aspect. We relate the features of the obtained distributions to large-scale landscape structures, such as regional trends and anisotropy. We find that the theoretical distribution of slope is strongly impacted by the parameters used to represent the distribution of elevation changes, while large-scale features play a secondary role. Conversely, the distribution of aspect is also controlled by regional trends and anisotopy, even when they are weak. The proposed statistical description of slope and aspect is applied to assess the effects of topographic features on direct solar radiation mean and standard deviation. The main control on direct solar radiation is exerted by the partial derivative variance. We consider four different landscapes across the continental United States and compare the proposed theoretical description of slope and aspect distributions to the observed histograms.