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“Dissecting” Landscapes with Hölder Exponents to reconcile process and form
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  • Chris J Keylock,
  • Arvind Singh,
  • Paola Passalacqua,
  • Efi Foufoula-Georgiou
Chris J Keylock
Loughborough University

Corresponding Author:c.j.keylock@lboro.ac.uk

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Arvind Singh
University of Central Florida
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Paola Passalacqua
University of Texas at Austin
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Efi Foufoula-Georgiou
University of California, Irvine
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A long-standing question in geomorphology concerns the applicability of statistical models for elevation data based on fractal or multifractal representations of terrain. One difficulty with addressing this question has been the challenge of ascribing statistical significance to metrics adopted to measure landscape properties. In this paper, we use a recently developed surrogate data algorithm to generate synthetic surfaces with identical elevation values as the source dataset, while also preserving the value of the Hölder exponent at any point (the underpinning characteristic of a multifractal surface). Our primary data are from an experimental study of landscape evolution. This allows us to examine how the statistical properties of the surfaces evolve through time and the extent to which they depart from the simple (multi)fractal formalisms. We also study elevation data from Florida and Washington State. We are able to show that the properties of the experimental and actual terrains depart from the simple statistical models. Of particular note is that the number of sub-basins of a given channel order (for orders sufficiently small relative to the basin order) exhibit a clear increase in complexity after a flux steady-state is established in the experimental study. The actual number of basins is much lower than occur in the surrogates. The imprint of diffusive processes on elevation statistics means that, at the very least, a stochastic model for terrain based on a local formalism needs to consider the joint behavior of the elevations and their scaling (as measured by the pointwise Hölder exponents).