Uncertainties and errors in algorithms for elevation gradients

Elevation gradients are primary components of slope and aspect. Significant concerns remain when computing gradients if noise (perturbing non-DEM data) is present. There is still a need to find ways to balance accuracy of the gradient and stability to noise for specific types of DEM. In this study, six algorithms are compared using four DEMs and analyzed for stability to base level DEM noise and added random noise. Theoretical stability and accuracy of the formulae are analyzed using harmonic (frequency or spatial scale) response. The results provide a basis to determine the most appropriate algorithm for different situations. They show that: (1) the set (Evans-Young (EY), Sharpnack (Sp), Sobel (Sb)) has a better stability to noise ratio than the set (Zevenbergen (Z), Florinsky (F), Horn (H)). EY has the smoothest surface and the highest stability to noise ratio. If stability is the primary measure in mid-frequencies, EY is a good choice. (2) Sb is good because of its accuracy in mid to high frequencies. Out to the highest frequencies, Sb is the best. (3) F has potential but should not be used with very high-frequency noise. (4) H and Z should not be used when there is substantial noise.