ON THE FRACTAL DIMENSION OF SELF-AFFINE PROFILES

One-dimensional profiles f(x) can be characterized by a Minkowski-Boul igand dimension D and by a scale-dependent generalized roughness W(f, epsilon). This roughness can be defined as the dispersion around a cho sen fit to f(x) in an epsilon-scale. It is shown that D = lim(epsilon- ->0)[2 - ln W(f, epsilon)/ln epsilon] holds for profiles nowhere diffe rentiable. This establishes a close connection between the roughness a nd the fractal dimension and proves that D = 2 - H for self-affine pro files (H is the roughness or Hurst exponent). Two numerical algorithms based on the roughness, one around the local average (f(x))(epsilon) (usual roughness) and the other around the local RMS Straight line (a generalized roughness), are discussed. The estimates of D for standard self-affine profiles are reliable and robust, especially for the last method.