Uncertainty interval estimates for computing slope and aspect from a gridded digital elevation model

The first order derivatives of a Digital Elevation Model (DEM) defined over a regular grid are usually computed without an uncertainty estimate. The standard procedure involves a compact 3 x 3 window. Using a Taylor expansion, an uncertainty interval for each partial derivative as a function of the cell size was devised using two estimates, either of different resolution or of different order. The intervals for slope and aspect can be derived afterwards. We carried out an experiment comparing some different estimates of the slope and aspect over a synthetic surface representative of a real topography and amenable to offer an exact derivative. The partial derivatives were numerically estimated with four different procedures: the Simple procedure defined by Jones over a 2 x 2 window, the Evans-Young procedure using a centered difference over a 3 x 3 window, and using a 5 x 5 window both with an extrapolated Evans-Young procedure and the expression due to Florinsky. The results confirm that intervals for both slope and aspect always included the exact value even after drastically increasing the cell size. Finally, a real case with an integer-valued DEM was considered, illustrating the combined effect of Representation and Truncation error.