FLUVIAL LANDSCULPTING AND THE FRACTAL DIMENSION OF TOPOGRAPHY

Quantitative models of landform development can help us to understand the evolution of mountains and regional topography, and the effects of tectonic motions and climate on landscape, including its fractal geometry. This paper presents a general and powerful three-dimensional model of fluvial erosion and deposition at hill- to mountain-range scale. The model works by accumulating the effects of randomly seeded storms or floods (precipitons) that cause diffusional smoothing then move downslope on digital topography grids, that erode portions of elevation differences, that transport a slope-limited amount of eroded material, and that deposit alluvium when their sediment-carrying capacity is exceeded.The iteration of these simple and almost linear rules produce very complicated simulated landscapes, demonstrating that complex landscapes do not require complex laws. Each process implemented in the model is affected differently by changes in horizontal scale. Erosion, a scale-free process, roughens topography at all wavelengths. This roughening is balanced by diffusive processes (scaling as 1/L2) at short wavelengths and deposition (scaling as 1/L2) at long wavelengths. Such a mixture of scale-free and scale-dependent processes can produce multifractal behavior in the models. The fractal dimension of the model topography is much more sensitive to climatic variables than to tectonic uplift. Landscape evolution may be fractal, but it does not seem to be chaotic. Analysis of topography of areas in southern Arizona using variograms shows approximately fractal behavior, with mean fractal dimension around 2.2-2.3. Departures from an exact fractal relationship imply that the topography is in detail multifractal. The fractal dimension at short wavelengths is less than that at long wavelengths. This variation could either be caused by the relative strengths of diffusive and erosional processes shaping the topography, or a result of changes in climatic or tectonic conditions still preserved in the landscape.