Spatial distributions of islands in fractal surfaces and natural surfaces

The size and spatial distributions of islands in random fractal surfaces and natural islands are investigated. Multi-scale island distributions from a fractal surface are analyzed to determine the size distribution of islands, such as the number of islands greater than a particular size, and this was found to exhibit a Korcak-type empirical relation. In the same multi-scale analysis, the spatial distribution of islands, which is the number of islands less than or equal to a particular distance between a pair of islands, is also investigated to determine the effects of differences in scale. The spatial distribution of distance between uniformly distributed islands is analytically calculated and compared with the island distributions of the fractal surface. We seek to determine the degree of resemblance between simulated spatial distributions and the natural island distribution of French Polynesia. We also estimate the size and spatial distributions of islands according to different sea levels.