One-Dimensional Fractional Brownian Motion
Two methods for generating a fractional Brownian motion to simulate a natural surface are demonstrated here. The Hurst exponent describes the raggedness, with higher exponents leading to smoother surfaces. Fractional Brownian motion is a generalization of ordinary Brownian motion that has been used successfully to model a variety of natural phenomena, such as terrains, coastlines, and clouds. It has the scaling property . Ordinary Brownian motion has .
Random addition refines the list of points by interpolation and adding random offsets.
Fourier synthesis generates a random spectrum such that the resulting data has the correct scaling property.
The code for generating the data is from Roman E. Maeder, The Mathematica Programmer II, New York: Academic Press, 1996.