The Quantification of Rill Patterns that Develop on Bare Hillslopes

Rill networks which developed along bare hillslopes because of rainfall have very complex patterns. It is impossible to know the rate of erosion with time until we express quantitatively the figure of the rill patterns. As the rill patterns are usually similar to the river networks, application of Horton'law to the rill networks has been tried. In this study, the degree of complication of rill patterns that develop on hillslopes is described as entropy and fractal dimension which has generally non-integer values, by applying the two geometrical concepts of Spectrum and that of Fractal geometry. The entropy of rill patterns can be calculated from the power spectrum which can be obtained by decomposing the rill patterns into the several single harmonic function. On the other hand, fractal dimension can be easily computed by computer graphics. In addition to this analysis, the applicability of Horton's law to rill patterns was also considered. As a result of applying this analysis to the rill patterns which develop on bare hillslopes, the entropy and the fractal dimension of rill pattrns were closely related to the number of bifurcations of the rill. And applying Horton's first and second laws to the rill patterns, it becomes clear that only Horton's first law can be applied to the rill patterns.