Self-Affine Scaling of Earth's Islands
Earth's relief is approximately self-affine, meaning a zoom-in on a small region looks statistically similar to a large region upon rescaling. Fractional Brownian surfaces give an idealized self-affine model of Earth's relief with one parameter, the Hurst exponent $H$, characterizing the roughness of the surface. We compile a large dataset of topographic profiles of islands (N=131,063 with the range of areas covering 8+ orders of magnitude) and obtain four estimates for the Hurst exponent of Earth's surface by fitting four statistical laws from the theory of self-affine surfaces concerning islands: (i) distribution of areas, (ii) volume-area relationship, (iii) perimeter-area relationship, and (iv) maximum height-area relationship. The estimated Hurst exponents indicate different fractal scaling behavior for different geometric features, and are sorted in order of increasing expected influence of coastal processes. This sheds light on the impact of coastal erosion and sedimentation on island geomorphology.