Why is topography fractal ?

The power spectrum S of linear transects of the earth's topography is often observed to be a power-law function of wave number k with exponent close to ?2: S(k) / k ?2. In addition, river networks are fractal trees that satisfy several power-law relationships between their morphologic components. A model equation for the evolution of the earth's topography by erosional processes which produces fractal topography and fractal river networks is presented and its solutions compared in detail to real topography. The model is the diiusion equation for sediment transport on hillslopes and channels with the diiusivity constant on hillslopes and proportional to the square root of discharge in channels. The dependence of diiusivity on discharge follows from fundamental equations of sediment transport. We study the model in two ways. In the rst analysis the diiusivity is parameterized as a function of relief and a Taylor expansion procedure is carried out to obtain a diierential equation for the landform elevation which includes the spatially-variable diiusivity to rst order in the elevation. The solution to this equation is a self-aane or fractal surface with linear transects that have power spectra S(k) / k ?1:8 , independent of the age of the topography, consistent with observations of real topography. The hypsometry produced by the model equation is skewed such that lowlands make up a larger fraction of the total area than highlands as observed in real topography. In the second analysis we include river networks explicitly in a numerical simulation by calculating the discharge at every point. We characterize the morphology of real river basins with ve independent scaling relations between six morphometric variables. Scaling exponents are calculated for seven river networks from a variety of tectonic environments using high-quality digital elevation models. River networks formed in our model match the observed scaling laws and satisfy Tokunaga side-branching statistics.