Measurement of self-affinity on surfaces as a trial application of fractal geometry to landform analysis

Abstract 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 Z 2 , surface area, S , and basal area, A , measured in a number of scaling unit areas of various sizes, can be scaled as Z 2 ∼ S νz and A ∼ S νA , indicating the self-affinity of these surfaces. Z 2 and A then are scaled to each other as Z 2 ∼ A H ′ , and H′ = ν z ν A . In the of land surfaces, H′ ⋟ v z , because ν A is always close to 1. This means that H ′ is equivalent to the scaling (Hurst) exponent, H (0 H 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.