ON THE FRACTAL STRUCTURE AND STATISTICS OF CONTOUR LINES ON 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 zeta-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).