Two-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 surfaces are colored by height using the "Topography" gradient mapping and level curves.
The code for generating the data is from Roman E. Maeder, The Mathematica Programmer II, New York: Academic Press, 1996.