Journal of the International Association for Mathematical Geology

, Volume 16, Issue 7, pp 671-683

First online:

Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping

  • David M. MarkAffiliated withDepartment of Geography, State University of New York at Buffalo
  • , Peter B. AronsonAffiliated withEnvironmental Systems Research Institute

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Fractional Brownian surfaces have been widely discussed as an appropriate model for the statistical behavior of topographic surfaces. The fractals model proposes that topographic surfaces are statistically self-similar, and that a single parameter, the fractal dimension, applies at all scales. This paper presents the results of empirical examinations of 17 topographic samples. Only one of these samples shows the statistical behavior predicted by the fractals model; however, in 15 of the 17 samples, the surfaces' variograms could be adequately described by ranges of scales having constant fractal dimension, separated by distinct scale breaks. For scale ranges between adjacent breaks, surface behavior should be that predicted by the fractals model; the breaks represent characteristic horizontal scales, at which surface behavior changes substantially. These scale breaks are especially important for cartographic representations and digital elevation models, since they represent scales at which there is a distinct change in the relation between sampling interval and the associated error.

Key words

Fractals variograms topography stochastic models