Skip to main content
SpringerLink
Log in

Characterization of the Structure of River-Bed Gravels Using Two-Dimensional Fractal Analysis

  • Published:
Mathematical Geology Aims and scope Submit manuscript

Abstract

This paper is concerned with the application of fractal analysis to understand the structure of water-worked gravel-bed river surfaces. High resolution digital elevation models, acquired using digital photogrammetric methods, allowed the application of two-dimensional fractal methods. Previous gravel-bed river studies have been based upon sampled profiles and hence one-dimensional fractal characterisation. After basic testing that bed elevation increments are Gaussian, the paper uses two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension. The results identify mixed fractal behavior with two characteristic fractal bands, one associated with the subgrain scale and one associated with the grain scale. The subgrain scale characteristics were isotropic and sensitive to decisions made during the data collection process. Thus, it was difficult to differentiate whether these characteristics were real facets of the surfaces studied. The second band was anisotropic and not sensitive to data collection issues. Fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation, by increasing surface irregularity and hence roughness. This is an important observation as it means that water-worked surfaces may have a distinct anisotropic signal, revealed when using a fractal type analysis.

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

  • Andrle, R., and Abrahams, A. D., 1989, Fractal techniques and the surface roughness of talus slopes: Earth Surface Processes and Landforms, v. 14, p. 197–209.

    Google Scholar 

  • Armstrong, M., 1986, Common problems seen in variograms: Math. Geology, v. 16, p. 305–313.

    Google Scholar 

  • Avnir, D., Biham, O., Lidar, D., and Malcai, O., 1998, Is the geometry of nature fractal?: Science, v. 279, p. 39–40.

    Google Scholar 

  • Biham, O., Malcai, O., Lidar, D., and Avnir, D., 1998, Reply to the discussion of ‘Is the geometry of nature fractal?’: Science, v. 279, p. 785–786.

    Google Scholar 

  • Bray, D. I., 1982, Flow resistance in gravel-bed rivers, in Hey, R. D., Bathurst, J. C., and Thorne, C. R., eds., Gravel-bed rivers: Chichester, Wiley, p. 109–133.

    Google Scholar 

  • Burrough, P., 1981, Fractal dimensions of landscapes and other environmental data: Nature, v. 294, p. 240–242.

    Google Scholar 

  • Burrough, P. A., 1983a, Multi-scale sources of spatial variation in soil. I. The application of fractal concepts to nested levels of soil variation: Journal of Soil Science, v. 34, p. 577–597.

    Google Scholar 

  • Burrough, P. A., 1983b, Multi-scale sources of spatial variation in soil. I. A non-Brownian fractal model and its application in soil survey: Journal of Soil Science, v. 34, p. 599–620.

    Google Scholar 

  • Butler, J. B., 1999, High resolution photogrammetric monitoring and analysis of the structure of gravel-bed river surfaces: unpubl. soctoral dissertation, University of Cambridge, 311 p.

  • Butler, J. B., Lane, S. N., and Chandler, J. H., 1998, Assessment of DEM quality for characterising surface roughness using close range digital photogrammetry: Photogrammetric Record, v. 16, p. 271–291.

    Google Scholar 

  • Chase, C. G., 1992, Fluvial landsculpting and the fractal dimension of topography: Geomorphology, v. 5, p. 39–57.

    Google Scholar 

  • Clifford, N. J., Robert, A., and Richards, K. S., 1992, Estimation of flow resistance in gravel-bedded rivers: A physical explanation of the multiplier of roughness length: Earth Surface Processes and Landforms, v. 17, p. 111–126.

    Google Scholar 

  • Cressie, N., and Hawkins, D. M., 1980, Robust estimation of the variogram: I: Math. Geology, v. 12, p. 115–124.

    Google Scholar 

  • Culling, W. E. H., 1986, On Hurst phenomena in the landscape: Transactions of the Japanese Geomorphological Union, v. 7, p. 221–243.

    Google Scholar 

  • Culling, W. E. H., and Datko, M. 1987, The fractal geometry of the soil-covered landscape: Earth Surface Processes and Landforms, v. 12, p. 369–385.

    Google Scholar 

  • David, M., 1977, Geostatistical ore reserve estimation: Elsevier, Amsterdam, 364 p.

    Google Scholar 

  • Furbish, D. J., 1987, Conditions for geometric similarity of coarse stream-bed roughness: Math. Geology, v. 19, p. 291–307.

    Google Scholar 

  • Gomez, B., 1993, Roughness of stable, armoured gravel beds: Water Resources Research, v. 29, p. 3631–3642.

    Google Scholar 

  • Goodchild, M. F., 1980, Fractals and the accuracy of geographical measures: Math. Geology, v. 12, p. 85–98.

    Google Scholar 

  • Goodchild, M. F., and Mark, D. M., 1987, The fractal nature of geomorphic phenomenon: Annals of the Association of American Geographers, v. 77, p. 265–278.

    Google Scholar 

  • Hassan, M. A., and Reid, I., 1990, The influence of microform bed roughness elements on flow and sediment transport in gravel-bed rivers: Earth Surface Processes and Landforms, v. 8, p. 39–50.

    Google Scholar 

  • Huang, J., and Turcotte, D., 1990, Fractal mapping of digitised images: Application to the topography of Arizona and comparisons with synthetic images: Journal of Geophysical Research, v. 94, p. 7491–7495.

    Google Scholar 

  • Jakeman, A. J., Beck, M. B., and McAleer, M. J., eds., 1993, Modelling change in environmental systems: Wiley & Sons, Chichester, p. 584.

    Google Scholar 

  • Journel, A. G., and Huijbregts, C. J., 1978, Mining geostatistics: Academic Press, San Diego, 600 p.

    Google Scholar 

  • Kent, C., and Wong, J., 1982, An index of littoral zone complexity and its measurement: Canadian Journal of Fisheries and Aquatic Science, v. 39, p. 847–853.

    Google Scholar 

  • Kirchner, J. W., Dietrich, W. E., Iseya, F., and Ikeda, H., 1990, The variability of critical shear stress, friction angle, and grain protrusion in water-worked sediments: Sedimentology, v. 37, p. 647–672.

    Google Scholar 

  • Klinkenberg, B., 1992, Fractals and morphometric measures: Is there a relationship?: Geomorphology, v. 5, p. 5–20.

    Google Scholar 

  • Klinkenberg, B., 1994, A review of methods used to determine the fractal dimension of linear features: Math. Geology, v. 26, p. 23–46.

    Google Scholar 

  • Klinkenberg, B., and Clarke, K., 1992, Exploring the fractal mountains, in Palaz, I., and Sengupta, S., eds., Automated pattern analysis in petroleum exploration: Springer-Verlag, New York, p. 201–212.

    Google Scholar 

  • Klinkenberg, B., and Goodchild, M. F., 1992,The fractal properties of topography: A comparison of methods: Earth Surface Processes and Landforms, v. 17, p. 217–234.

    Google Scholar 

  • Leopold, L. B., Wolman, M. G., and Miller, J. P., 1964, Fluvial processes in geomorphology: W. H. Freeman, San Francisco, 522 p.

    Google Scholar 

  • Malinverno, A., 1989, Testing linear models of sea-floor topography: Pure and Applied Geophysics, v. 131, p. 139–155.

    Google Scholar 

  • Mandelbrot, B., 1977, Fractals: Form, chance and dimension: W. H. Freeman and Co., San Francisco, 365 p.

    Google Scholar 

  • Mandelbrot, B., 1982, The fractal geometry of nature: Freeman, New York, 460 p.

    Google Scholar 

  • Mandelbrot, B., 1998, A discussion of ‘Is the geometry of nature fractal?’: Science, v. 279, p. 783–785.

    Google Scholar 

  • Mandlebrot, B. B., and Van Ness, J. W., 1968, Fractional Brownian motions, fractional noises and applications: SIAM Review, v. 10, p. 422–437.

    Google Scholar 

  • Mark, D. M., and Aronson, P. B., 1984, Scale dependent fractal dimensions of topographic surfaces: An empirical investigation with applications in geomorphology and computer mapping: Math. Geology, v. 16, p. 671–683.

    Google Scholar 

  • Matheron, G., 1965, Les variables regionalisees et leur estimation: Masson et Cie, Paris.

    Google Scholar 

  • McBratney, A. B., and Webster, R., 1986, Choosing functions for semi-variograms of soil properties and fitting them to sampling estimates: Jour. Soil Science, v. 37, p. 617–639.

    Google Scholar 

  • Milne, B. T., 1991, Lessons from applying fractal models to landscape patterns, in Turner, M. G., and Gardner, R. H., eds., Quantitative methods in landscape ecology: The analysis and interpretation of landscape heterogeneity: Springer, New York, p. 199–235.

    Google Scholar 

  • Nakano, T., 1983, A “fractal” study of some rias coastlines in Japan: Annual Report of the Institute of Geosciences, University of Tsukuba, v. 9, p. 75–80.

    Google Scholar 

  • Nakano, T., 1984, A systematics of “transient fractals” of rias coastline; an example of rias coast from Kamaishi to Shizugawa: Annual Report of the Institute of Geosciences, University of Tsukuba, v. 10, p. 66–68.

    Google Scholar 

  • Nikora, V. L., Goring, D. G., and Biggs, B. J. F., 1998, On gravel-bed roughness characterisation: Water Resources Research, v. 34, p. 517–527.

    Google Scholar 

  • Oliver, M. A., and Webster, R., 1986, Semi-variogram for modelling the spatial pattern of land form and soil properties: Earth Surface Processes and Landforms, v. 11, p. 491–504.

    Google Scholar 

  • Orford, J. D., and Whalley, W. B., 1983, The use of fractal dimension to quantify the morphology of irregularly-shaped particles: Sedimentology, v. 30, p. 655–668.

    Google Scholar 

  • Ouchi, S., and Matsushita, M., 1992, Measurement of self-affinity on surfaces as a trial application of fractal geometry to landform analysis: Geomorphology, v. 5, p. 115–130.

    Google Scholar 

  • Pfeifer, P., 1984, Fractal dimension as a working tool for surface-roughness problems: Applications of Surface Science, v. 18, p. 146–164.

    Google Scholar 

  • Pfeifer, P., 1998, A discussion of ‘Is the geometry of nature fractal?’: Science, v. 279, p. 785.

    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: Photogrammetric Engineering and Remote Sensing, v. 57, p. 1329–1332.

    Google Scholar 

  • Rammal, R., 1984, Random walk statistics on fractal structures: Journal of Statistical Physics, v. 36, p. 47.

    Google Scholar 

  • Robert, A., 1988, Statistical analysis of sediment bed profiles in alluvial channels: Math. Geology, v. 20, p. 205–225.

    Google Scholar 

  • Robert, A., 1991, Fractal properties of simulated bed profiles in coarse-grained channels: Math. Geology, v. 23, p. 367–382.

    Google Scholar 

  • Robert, A., and Richards, K. S., 1988, On the modelling of sand bedforms using the semi-variogram: Earth Surface Processes and Landforms, v. 13, p. 459–473.

    Google Scholar 

  • Roy, A., Gravel, G., and Gauthier, C., 1987, Measuring the dimension of surfaces: A review and appraisal of different methods: Proceedings Auto-Carto, v. 8, p. 68–77.

    Google Scholar 

  • Russ, J. C., 1994, Fractal surfaces: Plenum Press, London, p. 309.

    Google Scholar 

  • Schumm, S. A., and Lichty, R. W., 1965, Time, space and causality in geomorphology: American Journal of Science, v. 263, p. 110–119.

    Google Scholar 

  • Shelberg, M. C., Moellering, H., and Lam, N. S., 1983, Measuring the fractal dimensions of surfaces: Proceedings Auto-Carto, v. 6, p. 319–328.

    Google Scholar 

  • Strickler, D., 1923, Beiträge zur fragen der Geschwundigkeitsformel und der Rauhigkeitszahlen für Ströme, Kanale und Geschlossene Leitungen: Mitteilung der Eidgenossenschaft Amtes für Wasserwirtschaft Nr. 16, Bern, Switzerland, 16 p.

  • Suzuki, M., 1984, Finite-size scaling for transient similarity and fractals: Progress in Theoretical Physics, v. 71, p. 1397–1400.

    Google Scholar 

  • Weissel, J. K., Pratson, L. F., and Malinverno, A., 1994, The length-scaling properties of topography: Journal of Geophysical Research, v. 99, p. 13997–14012.

    Google Scholar 

  • Xia, Z., 1993, The use and limitations of fractal geometry in digital terrain modelling: Unpubl. doctonal dissertation, City University of New York.

  • Xu, T., Moore, I. D., and Gallant, J. C., 1993, Fractals, fractal dimensions and landscapes-a review: Geomorphology, v. 8, p. 245–262.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Geography, University of Cambridge, Downing Place, Cambridge, CB2 3EN, UK

    Justin B. Butler

  2. School of Geography, University of Leeds, Leeds, LS2 9JT, UK

    Stuart N. Lane

  3. Department of Civil Engineering, Loughborough University, Loughborough, LE11 3TU, UK

    Jim H. Chandler

Authors
  1. Justin B. Butler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stuart N. Lane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jim H. Chandler
    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

Butler, J.B., Lane, S.N. & Chandler, J.H. Characterization of the Structure of River-Bed Gravels Using Two-Dimensional Fractal Analysis. Mathematical Geology 33, 301–330 (2001). https://doi.org/10.1023/A:1007686206695

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1007686206695

Search

Navigation