Skip to main content
SpringerLink
Log in

Estimating fractal dimension of profiles: A comparison of methods

  • Published:
Mathematical Geology Aims and scope Submit manuscript

Abstract

This paper examines the characteristics of four different methods of estimating the fractal dimension of profiles. The semi-variogram, roughness-length, and two spectral methods are compared using synthetic 1024-point profiles generated by three methods, and using two profiles derived from a gridded DEM and two profiles from a laser-scanned soil surface. The analysis concentrates on the Hurst exponent H,which is linearly related to fractal dimension D,and considers both the accuracy and the variability of the estimates of H.The estimation methods are found to be quite consistent for Hnear 0.5, but the semivariogram method appears to be biased for Happroaching 0 and 1, and the roughness-length method for Happroaching 0. The roughness-length or the maximum entropy spectral methods are recommended as the most suitable methods for estimating the fractal dimension of topographic profiles. The fractal model fitted the soil surface data at fine scales but not at broad scales, and did not appear to fit the DEM profiles well at any scale.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Ables, J., 1974. Maximum Entropy Spectral Analysis,in Proc. Symp. on the Collection and Analyses of Astrophysical Data, Vol. 15,Astron. Astrophys. Suppl. Series: p. 383–393.

  • Armstrong, A., 1986, On the Fractal Dimensions of Some Transient Soil Properties: J. Soil Sci., v. 37, p. 641–652.

    Google Scholar 

  • Bell, T., 1979, Mesoscale Sea Floor Roughness: Deep Sea Res., v. 26A, p. 65–76.

    Google Scholar 

  • Berry, M., and Lewis, Z., 1980, On the Weierstrass-Mandelbrot Fractal Function: Proc. R. Soc. Lond. A., v. 370, p. 459–484.

    Google Scholar 

  • Brown, S., 1987, A Note on the Description of Surface Roughness Using Fractal Dimension: Geophys. Res. Lett., v. 14, n. 11, p. 1095–1098.

    Google Scholar 

  • Brown, S. and Scholz, C., 1985, Broad Bandwidth Study of the Topography of Natural Rock Surfaces: J. Geophys. Res., v. 90(B14), n. 12, p. 12,575–12,582.

    Google Scholar 

  • Burg, J., 1967, Maximum Entropy Spectral Analysis: 37th Meeting Soc. of Exploration Geophysicists, Oklahoma City, Oct. 31, 1967.

  • Burg, J., 1968, A New Analysis Technique for Time Series Data: NATO Advanced Study Institute on Signal Processing, Enschede, Netherlands, Aug. 12‥-23, 1968.

  • Carr, J., and Benzer, W., 1991, On the Practice of Estimating Fractal Dimension: Math. Geol., v. 23, n. 7, p. 945–958.

    Google Scholar 

  • Denham, C., 1975, Spectral Analysis of Paleomagnetic Time Series: J. Geophys. Res., v. 80, n. 14, p. 1897–1901.

    Google Scholar 

  • Dicke, M., and Burrough, P., 1988, Using Fractal Dimensions for Characterizing Tortuosity of Animal Trails: Phys. Entomol., v. 13, p. 393–398.

    Google Scholar 

  • Fox, C., 1989, Empirically Derived Relationships Between Fractal Dimension and Power Law Form Frequency Spectra: Pure Appl. Geophys., v. 131, n. 1/2, p. 211–239.

    Google Scholar 

  • Goodchild, M., 1980, Fractals and the Accuracy of Geographical Measures: Math. Geol., v. 12, n. 2, p. 85–98.

    Google Scholar 

  • Hakanson, L., 1978, The Length of Closed Geomorphic Lines: Math. Geol., v. 10, p. 141–167.

    Google Scholar 

  • Hough, S., 1989, On the Use of Spectral Methods for the Determination of Fractal Dimension: Geophys. Res. Lett., v. 16, n. 7, p. 673–676.

    Google Scholar 

  • Huang, C., White, I., Thwaite, E., and Bendeli, A., 1988, A Noncontact Laser System for Measuring Soil Surface Topography: Soil Sci. Soc. Am. J., v. 52, n. 2, p. 350–355.

    Google Scholar 

  • Hutchinson, M., 1989. A New Procedure for Gridding Elevation and Stream Line Data with Automatic Removal of Spurious Pits: J. Hydrol., v. 106, p. 211–232.

    Google Scholar 

  • Jones, J., Thomas, R., and Earwicker, P., 1989, Fractal Properties of Computer-Generated and Natural Geophysical Data: Comput. Geosci., v. 15, n. 2, p. 227–235.

    Google Scholar 

  • Klinkenberg, B., 1992, Fractals and Morphometric Measures: Is There a Relationship?: Geomorphology, v. 5, p. 5–20.

    Google Scholar 

  • Klinkenberg, B., and Goodchild, M., 1992, The Fractal Properties of Topography: A Comparison of Methods: Earth Surf. Proc. Landforms, v. 17, p. 217–234.

    Google Scholar 

  • Lam, N., 1990, Description and Measurement of Landsat TM Images Using Fractals: Photogr. Eng. Rem. Sensing, v. 56, n. 2, p. 187–195.

    Google Scholar 

  • Mandelbrot, B., 1967, How Long is the Coast of Britain: Statistical Self-Similarity and Fractional Dimension: Science, v. 156, p. 636–638.

    Google Scholar 

  • Mandelbrot, B., 1975, Stochastic Models for the Earth's Relief, the Shape and the Fractal Dimension of the Coastlines, and the Number-Area Rule for Islands: Proc. Nat. Acad. Sci. USA, v. 72, n. 10, p. 3825–3828.

    Google Scholar 

  • Mandelbrot, B., 1983.The Fractal Geometry of Nature: Freeman, New York, 468 p.

    Google Scholar 

  • Mandelbrot, B., 1985, Self-Affine Fractals and Fractal Dimension: Phys. Scripta, v. 32, p. 257–260.

    Google Scholar 

  • Mandelbrot, B., and van Ness, J., 1968, Fractional Brownian Motions, Fractional Noises and Applications: SIAM Rev., v. 10, n. 4, p. 422–437.

    Google Scholar 

  • Mark, D., and Aronson, P., 1984, Scale-Dependent Fractal Dimensions of Topographic Surfaces: An Empirical Investigation, with Applications in Geomorphology and Computer Mapping: Math Geol., v. 16, n. 7, p. 671–683.

    Google Scholar 

  • Matheron, G., 1963, Principles of Geostatistics Econ. Geol., v. 58, p. 1246–1266.

    Google Scholar 

  • Oliver, M., and Webster, R., 1986, Semi-Variograms for Modelling the Spatial Pattern of Landform and Soil Properties: Earth Surf. Proc. Landforms, v. 11, p. 491–504.

    Google Scholar 

  • Owen, M., and Wyborn, D., 1979, Geology and Geochemistry of the Tantangera and Brindabella Area: Bull. 204, Bureau Mineral Resources., Geol. and Geophys., AGPS, Canberra.

    Google Scholar 

  • Palmer, M., 1988, Fractal Geometry: A Tool for Describing Spatial Patterns of Plant Communities: Vegetatio, v. 75, p. 91–102.

    Google Scholar 

  • Pietgen, H., and Saupe, D. (Eds.), 1988,The Science of Fractal Images: Springer Verlag, New York.

    Google Scholar 

  • Polidori, L., Chorowicz, J., and Guillande, R., 1991, Description of Terrain as a Fractal Surface, and Application to Digital Elevation Model Quality Assessment: Photogr. Eng. Rem. Sensing, v. 57, n. 10, p. 1329–1332.

    Google Scholar 

  • Power, W., and Tullis, T., 1991. Euclidean and Fractal Models for the Description of Rock Surface Roughness: J. Geophys. Res., v. 96(B1), p. 415–424.

    Google Scholar 

  • Press, W., Flannery, B., Teukolsky, S., and Vetterling, W. T., 1989.Numerical Recipes. The Art of Scientific Computing: Cambridge University Press, Cambridge, 702 p.

    Google Scholar 

  • Pryor, L., and Brewer, R., 1954, The Physical Environment,in H. White (Ed),Canberra—A Nation's Capital: Angus and Robertson, London.

    Google Scholar 

  • Robert, A. and Roy, A., 1990, On the Fractal Interpretation of the Mainstream Length-Drainage Area Relationship: Water Res. Res., v. 26, n. 5, p. 839–842.

    Google Scholar 

  • Rosso, R., Bacchi, B., and La Barbera, P., 1991, Fractal Relation of Mainstream Length to Catchment Area in River Networks: Water Res. Res., v. 27, n. 3, p. 381–387.

    Google Scholar 

  • Roy, A., Gravel, G., and Gauthier, C., 1987, Measuring the Dimension of Surfaces: A Review and Appraisal of Different Methods,in N. Chrisman (Ed.).Proc. Auto-Carto 8: Baltimore, Maryland, March 29–April 3, 1987, p. 68–77.

  • Saupe, D., 1988, Algorithms for Random Fractals,in H. Pietgen and D. Saupe (Eds),The Science of Fractal Images: Springer Verlag, New York, p. 71–136.

    Google Scholar 

  • Taylor, J., Moran, C., White, I., Hairsine, P., and Carrigy, N., 1995, Behaviour Characterisation and Calibration of a Soil Surface Laser Scanner: Trans. Am. Soc. Agr. Engrs., in review.

  • Vignes-Adler, M., Page, A. L., and Adler, P., 1991, Fractal Analysis of Fracturing in Two African Regions, from Satellite Imagery to Ground Scale: Tectonophysics, v. 196, p. 69–86.

    Google Scholar 

  • Voss, R., 1985a, Random Fractal Forgeries,in R. A. Earnshaw (Ed.),Fundamental Algorithms for Computer Graphics, Vol. F17, NATO ASI Series: Springer-Verlag, Berlin, p. 813–829.

    Google Scholar 

  • Voss, R., 1985b. Random Fractals: Characterization and Measurement,in R. Pynn and A. Skjeltorp (Eds.),Scaling Phenomena in Disordered Systems: Plenum Press. New York, p. 1–11.

    Google Scholar 

  • Voss, R., 1988, Fractals in Nature: From Characterization to Simulation,in H. Pietgen and D. Saupe (Eds.),The Science of Fractal Images: Springer Verlag, New York, p. 21–70.

    Google Scholar 

  • Xu, T., Moore, I., and Gallant, J., 1993, Fractals, Fractal Dimensions and Landscapes: A Review: Geomorphology, v. 8, p. 245–262.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Resource and Environmental Studies, Australian National University, 0200, Canberra, ACT, Australia

    John C. Gallant, Ian D. Moore & Michael F. Hutchinson

  2. Centre for Resource and Environmental Studies, Australian National University, Australia

    Paul Gessler

  3. CSIRO Division of Soils, Canberra, ACT, Australia

    Paul Gessler

Authors
  1. John C. Gallant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ian D. Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael F. Hutchinson
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paul Gessler
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gallant, J.C., Moore, I.D., Hutchinson, M.F. et al. Estimating fractal dimension of profiles: A comparison of methods. Math Geol 26, 455–481 (1994). https://doi.org/10.1007/BF02083489

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02083489

Key words

Search

Navigation