On the Fractal Structure and Statistics of Contour Lineson a Self-Affine Surface

Contour lines of a self-affine surface with a specified value of Hurst exponent H show certain special characteristics in fractal structure and statistics. The first argument is that the fractal dimension D e of an entire pattern formed by contour lines of the same altitude, similar to coastlines, should be distinguished from the fractal dimension D c of single contour lines. It is then confirmed that D e =2- H . The exponent ζ characterizing a power-law form of the size distribution of closed contour lines, similar to islands, is found to be equal to D e . Self-avoiding fractional Brownian motion is newly introduced to derive a new scaling law D c =2/(1+ H ).