Fractals and the accuracy of geographical measures

  • Michael F. Goodchild
Article

Abstract

The problems of estimating line length, area, and point characteristics in the earth sciences have generated substantial but independent literatures. All three problems are of increasing concern given the current interest in digital capture, processing, and the storage of geographically referenced data. In the case of qualitative maps, all three are shown to be related to Mandelbrot's fractional dimension D (Mandelbrot, 1977) which allows the dependence of each on sampling density to be predicted. The general results are confirmed by simulation on surfaces of constant D. They also imply that certain improvements can be made in a number of previously proposed methods.

Key words

Fractals spatial distributions map analysis 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beckett, P., 1977, Cartographic generalization: Cartographic J., v. 14, p. 49–50.Google Scholar
  2. Frolov, Y. S. and Maling, D. H., 1969, The accuracy of area measurements by point counting techniques: Cartographic J., v. 6, p. 21–35.Google Scholar
  3. Goodchild, M. F. and Moy, W. S., 1977, Estimation from grid data: the map as a stochastic process,in Proceedings of the Commission on Geographical Data Sensing and Processing, Moscow, 1976, (R. F. Tomlinson, ed.), Ottawa: International Geographical Union, Commission on Geographical Data Sensing and Processing.Google Scholar
  4. Håkanson, L., 1978, The length of closed geomorphic lines: Mathematical Geol. v. 10, p. 141–167.Google Scholar
  5. Lloyd, P. R., 1976, Quantization error in area measurement: Cartographic J. v. 13, p. 22–26.Google Scholar
  6. Mandelbrot, B. B., 1967, How long is the coast of Britain? Statistical self-similarity and fractional dimension: Science v. 156, p. 636–638.Google Scholar
  7. Mandelbrot, B. B., 1975a, 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.Google Scholar
  8. Mandelbrot, B. B., 1975b, On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars: J. Fluid Mechanics v. 72, p. 401–416.Google Scholar
  9. Mandelbrot, B. B., 1977, Fractals: Form, Chance and Dimension: Freeman, San Francisco, 365 p.Google Scholar
  10. Maling, D. M., 1968, How long is a piece of string?: Cartographic J. v. 5, p. 147–156.Google Scholar
  11. Matheron, G., 1967, Éléments pour une Théorie des Mileux Poreux: Masson, Paris, 166 p.Google Scholar
  12. Moy, W. S., 1977, Estimation from grid data: the map as a stochastic process. Unpublished M.A. Dissertation, Department of Geography, University of Western Ontario.Google Scholar
  13. Muller, J. C., 1978, Map gridding and cartographic errors: a recurrent argument: Canadian Cartographer v. 14, p. 152–167.Google Scholar
  14. Orey, S., 1970, Gaussian sample functions and the Hausdorff dimension of level crossings. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete v. 15, p. 249–256.Google Scholar
  15. Richardson, L. F., 1961, The problem of contiguity: General Systems Yearbook v. 6, p. 139–187.Google Scholar
  16. Scheidegger, A. E., 1970, Theoretical geomorphology (2nd ed.): Springer Verlag, New York, 333 p.Google Scholar
  17. Switzer, P., 1975, Estimation of the accuracy of qualitative maps,in Display and analysis of spatial data, (J. C. Davis and M. J. McCullagh, eds.): Wiley, London, 378 p.Google Scholar
  18. Tobler, W. R., 1974, The accuracy of categorical maps: Department of Geography, University of Michigan, Cartographic Laboratory Report Number 4.Google Scholar
  19. Tomlinson, R. F., Calkins, H. W. and Marble, D. F., 1976,Computer Handling of Geographical Data UNESCO Press, Paris, 214 p.Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • Michael F. Goodchild
    • 1
  1. 1.Department of GeographyThe University of Western OntarioLondonCanada

Personalised recommendations