Stochastic models for the Earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands.
The degree of irregularity in oceanic coastlines and in vertical sections of the Earth, the distribution of the numbers of islands according to area, and the commonality of global shape between continents and islands, all suggest that the Earth's surface is statistically self-similar. The preferred parameter, one which increases with the degree of irregularity, is the fractal dimension, D, of the coastline; it is a fraction between 1 (limit of a smooth curve) and 2 (limit of a plane-filling curve). A rough Poisson-Brown stochastic model gives a good first approximation account of the relief, by assuming it to be created by superposing very many, very small cliffs, placed along straight faults and statistically independent. However, the relative area predicted for the largest islands is too small, and the irregularity predicted for the relief is excessive for most applications; so is indeed the value of the dimension, which is D = 1.5. Several higher approximation self-similar models are described. Any can be matched to the empirically observed D, and can link all the observations together, but the required self-similarity cannot yet be fully explained.
General Sciences Department, IBM T.J. Watson Research Center, Yorktown Heights, New York 10598.
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Mandelbrot