Abstract
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.
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Allder, W. R., Caruso, V. M., Pearsall, R. A., and Troup, M. I., 1982, An overview of digital elevation model production at the United States Geological Survey: Proceedings, Fifth International Symposium on Computer-Assisted Catography (Auto-Carto 5), p. 23–32.
Church, M. and Mark, D. M., 1980, On size and scale in geomorphology: Prog. Phys. Geog., v. 4, p. 342–390.
Dutton, G. H., 1981, Fractal enhancement of cartographic line detail: Amer. Cartog., v. 8, p. 23–40.
Evans, I. S., 1972, General geomorphometry, derivatives of altitude, and description statistics,in, Chorley, R. J. (Ed.), Spatial analysis in geomorphology: Methuen & Co., London, p. 17–90.
Evans, I. S., 1979, An integrated system of terrain analysis and slope mapping: Final Report on U.S. Army Grant DA-ERO-591-73-g0040, Department of Geography, University of Durham, England, 192 pp.
Fournier, A., Fussell, D., and Carpenter, L., 1982a, Computer rendering of stochastic models: Comm. ACM, v. 25, p. 371–384.
Fournier, A., Fussell, D., and Carpenter, L., 1982b, Authors' reply to: Comment on computer rendering of fractal stochastic models (by B. B. Mandelbrot): Commun. ACM, v. 25, p. 583–584.
Freiberger, W., and Grenander, U., 1977, Surface patterns in theoretical geography: Comput. Geosci., v. 3, p. 547–578.
Goodchild, M. F., 1980, Fractals and the accuracy of geographical measures: Math. Geol., v. 12, p. 85–98.
Goodchild, M. F., 1982, The fractional Brownian process as a terrain simulation model: Proceedings, Thirteenth Annual Pittsburg Conference on Modelling and Simulation, v. 13, p. 1133–1137.
Mandelbrot, B. B., 1967, How long is the coast of Britain? Statistical self-similarity and fractional dimension: Science, v. 156, p. 636–638.
Mandelbrot, B. B., 1975, Stochastic models of the Earth's relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands: Proceedings of the National Academy of Sciences, v. 72, p. 3825–3828.
Mandelbrot, B. B., 1977, Fractals: form, chance and dimension: Freeman, San Francisco, 365 p.
Mandelbrot, B. B., 1982a, Comment on computer rendering of fractal stochastic models: Comm. ACM, v. 25, p. 581–583.
Mandelbrot, B. B., 1982b, The fractal geometry of nature: Freeman, San Francisco.
Mark, D. M., 1975, Geomorphometric parameters: A review and evaluation: Geografiska Annaler, series A, v. 3, p. 165–177.
Mark, D. M., 1978, Comments on Freiberger and Grenander's “Surface patterns in theoretical geography”: Comput. Geosci., v. 4, p. 371–372.
Mark, D. M., 1979, Review of B. B. Mandelbrot's “Fractals: Form, Chance and Dimension”: Geo-Processing, v. 1, p. 202–204.
Mark, D. M., 1980, On scales of investigation in geomorphology: Can. Geog., v. 24, p. 81–82.
McEwan, R. B., 1980, USGS digital cartographic applications program: Jour. Surv. Mapp. Div., ASCE, November 1980, p. 13–22.
Moultrie, W., 1970, Systems, computer simulation, and drainage basins: Bulletin of the Illinois Geographical Society, v. 12, p. 29–35.
Peucker, T. K., Fowler, R. J., Little, J. J., and Mark, D. M., 1978, The triangulated irregular network, Proceedings, Digital Terrain Models (DTM) Symposium, American Society of Photogrammetry, May 9–11, 1978, St. Louis, Missouri, p. 516–540.
Seginer, I., 1969, Random walk and roughness models of drainage networks: Water Resour. Res., v. 5, p. 591–607.
Shelberg, M. C., Moellering, H., and Lam, N., 1982, Calculating the fractal dimensions of empirical cartographic curves: Proceedings, Fifth International Symposium on Computer-Assisted Cartography (Auto Carto V), p. 481–490.
Shreve, R. L., 1979, Models for prediction in fluvial geomorphology. Math. Geol., v. 11, p. 165–174.
Sprunt, B., 1972, Digital simulation of drainage basin development,in Chorley, R. J. (Ed.), Spatial analysis in geomorphology: Methuen & Co., London, p. 371–389.
Tausworthe, R. C., 1965, Random numbers generated by linear recurrence modulo two, Math. Comput., v. 19, p. 201–209.
Wood, W. F. and Snell, J. B., 1960, A quantitative system for classifying landforms: Quartermaster Research and Engineering Center, Technical Report EP-124, Natick, Massachusetts, 20pp.
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Mark, D.M., Aronson, P.B. Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping. Mathematical Geology 16, 671–683 (1984). https://doi.org/10.1007/BF01033029
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DOI: https://doi.org/10.1007/BF01033029