Self-affinity of landform and its measurement

Fractal geometry is expected to provide a new quantitative way to express a certain property of landform. Transect profiles of landform are considered to be self-affine because vertical and horizontal coordinates should be scaled differently, while contour lines are isotropic and self-similar. The method developed to analyze the self-affinity of curves in two-dimensional space can well express these fractal character-istics of both transect profiles and contour lines. This method is extended to analyze the three-dimensional land surfaces. The variance of elevation change Z^2, surface area S and bottom area A are measured in a number of scaling unit areas of various sizes to see whether Z^2 and A are scaled as Z^2～S^＜VZ＞ and A～S^＜VA＞. The three dimensional land surface is appeared to be self-affine with VA≅1 and 0＜VZ＜1, and the value of VZ is equivalent to the value of scaling factor H of fractional Brownian motion traces. This method of measurement well reproduces the initially introduced H value of the computer-generated fractional Brownian surfaces, and it is confirmed to work also on the real topograph