The Hurst exponent directly encodes the fBm's statistical self-similarity: zooming in by U horizontally scales amplitude by U^(-H)

Self-similarity in fBm means a zoomed-in portion resembles the whole (statistically). The H parameter makes this quantitative: if you zoom in horizontally by factor U, you must scale vertically by V = U^(-H) to produce a visually identical curve. This is exactly why the gain G = 2^(-H) produces self-similarity when doubling frequency — amplitude halves by the right amount to maintain the statistical shape. For H=1 (G=0.5), horizontal and vertical zoom factors are equal (isotropic), matching natural mountain profiles where taller mountains are also proportionally wider at the base.

Examples

G=0.707 (H=0.5) yields anisotropic self-similarity like stock market curves. G=0.5 (H=1) yields isotropic self-similarity matching natural terrain. Empirical measurement of mountain-range silhouettes confirms -9dB/octave spectral decay (H=1).

Assessment

If an fBm has H=0.5 and you zoom in horizontally by factor 4, what vertical scale factor makes it look statistically the same? Show the calculation.