Self-Affine Scaling of Earth's Islands

TL;DR

The paper tests whether Earth's islands follow self-affine fractal scaling by assembling a global island dataset and fitting four scaling laws derived from a fractional Brownian surface to estimate the Hurst exponent $H$ from area, volume, perimeter, and maximum height. Three relations (area distribution, volume-area, and perimeter-area) yield distinct but plausible $H$ values, while the maximum-height relation poorly matches the one-dimensional theory, revealing bimodality in large-island volumes. The resulting $H$ estimates vary by geometric feature (roughly $H\approx0.57$ from volume, $H\approx0.95$ from perimeter, $H\approx0.32$ from max height), indicating that a single-parameter self-affine model cannot capture all observed scaling, likely due to erosion and geomorphological processes. The study provides a rich, public dataset and motivates erosion-aware, multi-parameter modeling to better understand the scaling of Earth's topography and its implications for coastal geomorphology.

Abstract

Earth's relief is approximately self-affine, meaning a zoom-in on a small region looks statistically similar to a large region upon a suitable 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. To quantitatively assess agreement with Earth elevation data, 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 differ greatly, indicating different fractal scaling behavior for different geometric features, but are sorted in order of increasing expected influence of erosion at the shorelines.

Figures (3)

• Figure 1: (right) Eckert IV (equal-area) projection of all islands in our constructed dataset. Pixels considered part of an island are plotted with uniformly sized circles to make small islands easily visible, circles are randomly colored according to connected component (island). Red square indicates location of the zoom-in in the left panel. (left) Mercator projection of $[1.6^\circ S, 1.8^\circ N]$$\times$$[4.3^\circ E, 8.3^\circ E]$, with yellow indicating islands and blue representing ocean. Inlays show top-down view and 3-dimensional surface plot of São Tomé Island, with axis units in kilometers. In the surface plot, the height is exaggerated by a factor of 5 to show detail.
• Figure 2: (i) Empirical complementary cumulative distribution function (equal to one minus the cumulative distribution function) of area in square kilometers. (ii) Scatterplot of volume in cubic kilometers vs area in square kilometers. (iii) Scatterplot of discretized perimeter in kilometers vs area in square kilometers. (iv) Histogram of normalized maximum (maximum height divided by area to the power $H/2$). Plots (i), (ii), and (iii), show the best fit line. Plot (iv) shows the best fit analytical probability density function $f_H$ for plot (iv), though we note in this case the fit is poor.
• Figure 3: (left) Probability density functions for the maximum of a collection of $N$ landmasses assuming independently distributed areas from a true power-law distribution with the exponent fit from the empirical complementary cumulative distribution function of area. Sizes of notable landmasses (New Guinea (the largest island in our dataset), Greenland, the continents, and Pangaea) marked along with the value of the theoretical CDF at these points. (right) Three different fits of the CCDF for area with a range of fit slopes. $N$ is the number of landmasses in our dataset above the minimum area considered in each fit.