One-Dimensional Fractional Brownian Motion

method

random additions

Fourier synthesis

points

400

Hurst exponent h

0.5

randomize

Two methods for generating a fractional Brownian motion to simulate a natural surface are demonstrated here. The Hurst exponent

h

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

V(t)-V(t+dt)∝

h

dt

. Ordinary Brownian motion has

h=

1

2

.