Analytical basis for determining slope lines in grid digital elevation models

An analytical basis for the determination of slope lines in grid digital elevation models is provided by using the D8-LTD method (eight slope directions, least transverse deviation). The D8-LTD method's capability to predict consistently exact slope lines as the grid cell size goes to zero is shown analytically by applying mathematical analysis methods. The use of cumulative, least transverse deviations is found to be the key factor allowing for globally unbiased approximations of slope lines. The D8-LTD method's properties are also demonstrated numerically by using digital elevation models of a synthetic sloping surface obtained from the Himmelblau function. It is shown that slope lines obtained from the D8-LTD method can approximate the exact slope lines as close as desired by selecting a grid cell size that is small enough. In contrast, the standard D8 method is found to produce significantly biased results even when small grid cells are used. The D8-LTD method outperforms the D8 method over a wide range of grid cell sizes (up to 20 m in this application), beyond which grid cell size becomes too large to validly represent the underlying sloping surface. It is therefore concluded that the D8-LTD method should be used in preference to the standard D8 method in order to obtain slope lines that are only limited in reliability by the detail of topographic data, and not by the accuracy of the slope direction method applied.