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.
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.
Bell, T., 1979, Mesoscale Sea Floor Roughness: Deep Sea Res., v. 26A, p. 65–76.
Berry, M., and Lewis, Z., 1980, On the Weierstrass-Mandelbrot Fractal Function: Proc. R. Soc. Lond. A., v. 370, p. 459–484.
Brown, S., 1987, A Note on the Description of Surface Roughness Using Fractal Dimension: Geophys. Res. Lett., v. 14, n. 11, p. 1095–1098.
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.
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.
Denham, C., 1975, Spectral Analysis of Paleomagnetic Time Series: J. Geophys. Res., v. 80, n. 14, p. 1897–1901.
Dicke, M., and Burrough, P., 1988, Using Fractal Dimensions for Characterizing Tortuosity of Animal Trails: Phys. Entomol., v. 13, p. 393–398.
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.
Goodchild, M., 1980, Fractals and the Accuracy of Geographical Measures: Math. Geol., v. 12, n. 2, p. 85–98.
Hakanson, L., 1978, The Length of Closed Geomorphic Lines: Math. Geol., v. 10, p. 141–167.
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.
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.
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.
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.
Klinkenberg, B., 1992, Fractals and Morphometric Measures: Is There a Relationship?: Geomorphology, v. 5, p. 5–20.
Klinkenberg, B., and Goodchild, M., 1992, The Fractal Properties of Topography: A Comparison of Methods: Earth Surf. Proc. Landforms, v. 17, p. 217–234.
Lam, N., 1990, Description and Measurement of Landsat TM Images Using Fractals: Photogr. Eng. Rem. Sensing, v. 56, n. 2, p. 187–195.
Mandelbrot, B., 1967, How Long is the Coast of Britain: Statistical Self-Similarity and Fractional Dimension: Science, v. 156, p. 636–638.
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.
Mandelbrot, B., 1983.The Fractal Geometry of Nature: Freeman, New York, 468 p.
Mandelbrot, B., 1985, Self-Affine Fractals and Fractal Dimension: Phys. Scripta, v. 32, p. 257–260.
Mandelbrot, B., and van Ness, J., 1968, Fractional Brownian Motions, Fractional Noises and Applications: SIAM Rev., v. 10, n. 4, p. 422–437.
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.
Matheron, G., 1963, Principles of Geostatistics Econ. Geol., v. 58, p. 1246–1266.
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.
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.
Palmer, M., 1988, Fractal Geometry: A Tool for Describing Spatial Patterns of Plant Communities: Vegetatio, v. 75, p. 91–102.
Pietgen, H., and Saupe, D. (Eds.), 1988,The Science of Fractal Images: Springer Verlag, New York.
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.
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.
Press, W., Flannery, B., Teukolsky, S., and Vetterling, W. T., 1989.Numerical Recipes. The Art of Scientific Computing: Cambridge University Press, Cambridge, 702 p.
Pryor, L., and Brewer, R., 1954, The Physical Environment,in H. White (Ed),Canberra—A Nation's Capital: Angus and Robertson, London.
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.
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.
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.
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.
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.
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.
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.
Xu, T., Moore, I., and Gallant, J., 1993, Fractals, Fractal Dimensions and Landscapes: A Review: Geomorphology, v. 8, p. 245–262.
Author information
Authors and Affiliations
Rights 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
Issue Date:
DOI: https://doi.org/10.1007/BF02083489