One-Dimensional Fractional Brownian Motion

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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 .

Contributed by: Roman E. Maeder (March 2011)
Open content licensed under CC BY-NC-SA



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.

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