MEASUREMENT OF SELF-AFFINITY ON SURFACES AS A TRIAL APPLICATION OF FRACTAL GEOMETRY TO LANDFORM ANALYSIS

Landforms, which are often referred to as good examples of fractal geometry, are considered to be self-affine, because they appear flatter as the viewing point is moved farther away. The line-scaling method, which can express the self-affinity of various curves (including self-similar curves as a limiting case) by separately scaling two coordinates, was expanded and applied to surfaces developing in three-dimensional space (area-scaling method), for the purpose of deriving a parameter which might be useful in landform analysis. The variance of height change Z2, surface area, S, and basal area, A, measured in a number of scaling unit areas of various sizes, can be scaled as Z2 approximately S(vz) and A approximately S(vA), indicating the self-affinity of these surfaces. Z2 and A then are scaled to each other as Z2 approximately A(H), and H'= v(Z)/v(A). In the case of land surfaces, H' congruent-to v(Z), because v(A) is always close to 1. This means that H' is equivalent to the scaling (Hurst) exponent, H (0 < H < 1), of fractional Brownian motion records. Values of H', measured by the area-scaling method, closely reproduced the initially introduced H values of computer-generated, fractional Brownian surfaces. Although vertical exaggeration can make a surface look very different from the original appearance, the measured values of H' do not change significantly, owing to vertical exaggeration. The H' value seems to express a "relief texture" essential for the surface. A surface with minor H' has a greater local roughness relative to the total relief than surfaces with large H'. The area-scaling method then was applied to actual landforms in two areas of well-dissected, low mountains using equally scaled topographic maps, with the same grid spacing (25 m grid on 1/5000 maps). The obtained values of H' are considered reasonable for the difference in "relief texture" shown by the block diagrams of these areas. H' values, which were evaluated for four different landscapes on 1/25,000 topographic maps with a 125 m grid, moreover, also seem to indicate their relief texture expressed by the 125 m grid spacing. H' values obtained by the area-scaling method may become a useful parameter to express the degree of self-affinity or "relief texture" of landform surfaces, although more research is necessary to establish the reliability of the H' parameter for landform analysis.