ABSTRACT
The granite of these experiments is a self-affine surface, so that means a magnification α in the lateral xy-plane corresponds to a magnification αH in vertical z-direction. The resulting Hurst coefficient H gives the fractal dimension D = 3 − H. This self-affinity is true only below the macroscopic scale denoted by the lateral cut-off length ξ|| and its corresponding vertical length ξ⊥. They can be calculated with the height-difference correlation function:
C xz ( ) ( ( ) (z ))λ λxz > 2
It describes the height difference of two points separated laterally by the distance λ. Below these cut-off length ξ|| and ξ⊥ the curve of self-affine surfaces shows a slope H in the correlation function, whereas above no correlation between the analyzed points is found. The linear relationship between Cz and the surface descriptors ξ⊥, ξ|| and D is given by the following expression:
C forZ
ξ λ ξ
λ ξ( )λ ⎛⎝ ⎞ ⎠⊥
If more scaling ranges should be necessary, these formulas can be expanded to any number of multifractality.
