Fractal Structure and Statistics of Computer-simulated and Real Landforms

Abstract

A vertical profile of landform looks similar to the one-dimensional Brownian motion trace. It is regarded as a self-affine curve characterized by the Hurst scaling exponent, H (0<H<1). The fractal dimension D_e for the pattern of entire contour lines including all islands, and D_c for a single contour line can be both calculated from H as D_e=2−H and D_c=2⁄(1+H). The size distribution of islands follows the power-law (the Korcak’s law) with the exponent, ζ, characterized by D_e as ζ=D_e⁄2. We confirmed these scaling laws both on computer-simulated landforms and real coastlines, although the former is dependent on the system size and the latter includes disturbing effects of coastal processes. The value of H, which expresses a relief characteristic of the three dimensional self-affine surface such as landform, can be calculated from a horizontal section by these three scaling laws.