Fractional Brownian Motion adds correlated memory to white noise integration, controlled by the Hurst exponent H

Brownian Motion (BM) is the integral of white noise: each position increment is independent. Fractional Brownian Motion (fBm) introduces memory — past increments influence future ones. Positive memory (H > 0.5) means changes tend to continue in the same direction, producing smoother paths. Negative memory (H < 0.5) makes changes reverse rapidly, producing rougher paths. The Hurst exponent H ∈ (0,1) encodes this, where H=0.5 is standard BM. In computer graphics, fBm is approximated by summing octaves of noise rather than performing a true fractional integral, but understanding H’s effect on smoothness/roughness guides parameter choices for terrain, clouds, and texture generation.

Examples

H=1 (G=0.5) produces smooth, isotropic terrain shapes matching natural mountain profiles. H=0 (G=1) produces pink noise — rougher, more random. H=0.5 is the stock-market-curve-like standard BM.

Assessment

Explain why H=1 (not H=0.5) is the most common fBm setting in graphics implementations. What property of natural terrain does it match?