Scaledependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping
 David M. Mark,
 Peter B. Aronson
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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 selfsimilar, 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.
 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 ComputerAssisted Catography (AutoCarto 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 DAERO59173g0040, 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 selfsimilarity 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 numberarea 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”: GeoProcessing, 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 ComputerAssisted 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 EP124, Natick, Massachusetts, 20pp.
 Title
 Scaledependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping
 Journal

Journal of the International Association for Mathematical Geology
Volume 16, Issue 7 , pp 671683
 Cover Date
 19841001
 DOI
 10.1007/BF01033029
 Print ISSN
 00205958
 Online ISSN
 15738868
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Fractals
 variograms
 topography
 stochastic models
 Industry Sectors
 Authors

 David M. Mark ^{(1)}
 Peter B. Aronson ^{(2)}
 Author Affiliations

 1. Department of Geography, State University of New York at Buffalo, 14260, Buffalo, New York, USA
 2. Environmental Systems Research Institute, 380 New York Street, 92373, Redlands, California, USA