On the Self-Affinity of Various Curves

A numerical method proposed here enables us to scale-independently analyze the self-affinity of various curves such as noise and real topographical curves. The length N and variances for two appropriately chosen coordinates, X 2 and Z 2 , of a curve are measured between two arbitrary points on the curve by using the smallest fixed scale. The self-affinity is confirmed by testing whether X and Z are scaled as X ∼ N ν x and Z ∼ N ν z . The method is applied to transect profiles of real mountain topography, which is found to be self-affine with ν x =1 for the horizontal and ν z <1 for the vertical coordinate variations. The consistency of a scaling relation ( D =2- H ) betwee a self-affine profile with H =ν z and a self-similar contour with the fractal dimension D in the same area is also discussed.