The fractal properties of topography: A comparison of methods

Abstract In this study the fractal characteristics of fifty‐five digital elevation models from seven different United States physiographic provinces are determined using seven methods. The self‐similar fractal model tested in this analysis is found to provide a very good fit for some landscapes, but an imperfect fit for others. Thus, outright rejection of this model does not appear to be warranted, but neither does a blind application. The three implementations of the dividers methods considered in this study consistently produce lower dimensions than those produced by the other methods, and those dimensions consistently do not vary much between surfaces. Although the dimensions produced by the cell counting method (applied to the digital elevation model itself) display consistent intersurface variation, the dimensions are generally lower than those produced by the variogram‐based methods. Among the variogram‐based methods, the dimensions of the quarter‐sections of the digital elevation models are generally greater than the dimensions obtained from the other variogram‐based methods. The dimensions produced by the variogram method which considered the surfaces on a directional basis are very similar, on average, to the dimensions produced by the entire‐surface variogram.