Random fractals: Self-affinity in noise, music, mountains, and clouds

Abstract

Many of nature's seemingly complex shapes can be effectively characterized and modeled as random fractals based on generalizations of fractional Brownian motion, fBm. As a function of one dimension, t, the trace VH(t) provides a model for the "1/f{hook}" noises. Extending fBm's to higher dimensions gives VH(x,y) as landscapes and VH(x,y,z) as clouds. Although all such fBm's are statistically self-affine, as characterized by the parameter H or the spectral density exponent β, either zerosets or trails of independent fBm's are statistically self-similar and may be represented by the fractal dimension D. © 1989.