A Comparison of Two Approaches to Ranking Algorithms Used to Compute Hill Slopes

The calculation of slope (downhill gradient) for a point in a digital elevation model (DEM) is a common procedure in the hydrological, environmental and remote sensing sciences. The most commonly used slope calculation algorithms employed on DEM topography data make use of a three-by-three search window or kernel centered on the grid point (grid cell) in question in order to calculate the slope at that point. A comparison of eight such slope calculation algorithms has been carried out using an arti®cial DEM, consisting of a smooth synthetic test surface with various amounts of added Gaussian noise. Morrison's Surface III, a trigonometrically de®ned surface, was used as the synthetic test surface. Residual slope grids were calculated by subtracting the slope grids produced by the algorithms on test from true/reference slope grids derived by analytic partial differentiation of the synthetic surface. The resulting residual slope grids were used to calculate root-mean-square (RMS) residual error estimates that were used to rank the slope algorithms from ``best'' (lowest value of RMS residual error) to ``worst'' (largest value of RMS residual error). Fleming and Hoffer's method gave the ``best'' results for slope estimation when values of added Gaussian noise were very small compared to the mean smooth elevation difference (MSED) measured within three-by-three elevation point windows on the synthetic surface. Horn's method (used in ArcInfo GRID) performed better than Fleming and Hoffer's as a slope estimator when the noise amplitude was very much larger than the MSED. For the large noise amplitude situation the ``best'' overall performing method was that of Sharpnack and Akin. The popular Maximum Downward Gradient Method (MDG) performed poorly coming close to last in the rankings, for both situations, as did the Simple Method. A nonogram was produced in terms of standard deviation of the Gaussian noise and MSED values that gave the locus of the trade-off point between Fleming and Hoffer's and Horn's methods.