Summary
Statistical topography involves the study of the geometrical properties of the iso-sets (contour lines or surfaces) of a random potential ψ(r). Previous work [1,2] has addressed coastlines on a random relief ψ(x, y) that possess a single characteristic spatial scale λ with topography belonging to the universality class of the random percolation problem. In the present paper this previous analytical approach is extended to the case of a multiscale random function with a power spectrum of scales, ψλ ∝ λH, in a wide range of wavelengths, λ0 < λ < λ m . It is found that the pattern of the coastline differs significantly from that of a monoscale landscape provided that −3/4 <H < 1, with the case −3/4 <H < 0 corresponding to the long-range correlated percolation and 0 <H < 1 to the fractional Brownian relief. The expression for the fractal dimension of an individual coastline is derived,D h = (10 − 3H)/7, the maximum value of whichD h = 7/4, corresponds to the monoscale relief. The distribution functionF(a) of level lines over their sizea is calculated:F(a) ∝a −4(1-H)/7, for λ0 ≪a ≪ λ m . A comparison of the theoretical results with geographical data is presented.
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Isichenko, M.B., Kalda, J. Statistical topography. I. Fractal dimension of coastlines and number-area rule for Islands. J Nonlinear Sci 1, 255–277 (1991). https://doi.org/10.1007/BF01238814
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DOI: https://doi.org/10.1007/BF01238814