Mathematical Geology

, Volume 29, Issue 7, pp 919–932 | Cite as

Multifractal Modeling and Lacunarity Analysis

  • Qiuming Cheng
Article

Abstract

The so-called “gliding box method” of lacunarity analysis has been investigated for implementing multifractal modeling in comparison with the ordinary box-counting method. Newly derived results show that the lacunarity index is associated with the dimension (codimension) of fractal, multifractal and some types of nonfractals in power-law relations involving box size; the exponent of the lacunarity function corresponds to the fractal codimension (E – D) for fractals and nonfractals, and to the correlation codimension (E – τlpar;2)) for multifractals. These results are illustrated with two case studies: De Wijs's zinc concentration values from the Pulacayo sphalerite-quartz vein in Bolivia and Cochran's tree seedlings example. Both yield low lacunarities and slightly depart from translational invariance.

multifractals fractal lacunarity gliding box box-counting moments 

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REFERENCES

  1. Agterberg, F. P., 1993, Calculation of the variance of mean values for blocks in regional resource evaluation studies: Nonrenewable Resources, v. 2,no. 4, p. 312–324.Google Scholar
  2. Agterberg, F. P., 1994, Fractals, multifractals, and change of support, in Dimitrakopoulus, R., ed., Geostatistics for the next century: Kluwer Acad. Publ., Dordrecht, p. 223–234.Google Scholar
  3. Allain, C., and Cloitre, M., 1991, Characterizing the lacunarity of random and deterministic fractal sets: Physical Review A, v. 44,no. 6, p. 3552–3558.Google Scholar
  4. Cheng, Q., 1994, Multifractal modeling and spatial analysis with GIS: gold potential estimation in the Mitchell-Sulphurets area, northwestern British Columbia: unpubl. doctoral dissertation, Univ. Ottawa (Canada), 268 p.Google Scholar
  5. Cheng, Q., 1997, Discrete multifractals: Math. Geology, v. 29,no. 2, p. 245–266.Google Scholar
  6. Cheng, Q., and Agterberg, F. P., 1995, Multifractal modeling and spatial point processes: Math. Geology, v. 27,no. 7, p. 831–845.Google Scholar
  7. Cheng, Q., and Agterberg, F. P., 1996a, Multifractal modeling and spatial statistics: Math. Geology, v. 28,no. 1, p. 1–16.Google Scholar
  8. Cheng, Q., and Agterberg, F. P., 1996b, Comparison between two types of multifractal modeling: Math. Geology, v. 28,no. 8, p. 1001–1015.Google Scholar
  9. Cheng, Q., Agterberg, F. P., and Bonham-Carter, G. P., 1995, GIS treatment of multifractality of spatial objects, in Proc. Geomatics '95 on CDRom available from Surveys and Mapping Branch, NRCan, 615 Booth Street, Ottawa, Canada.Google Scholar
  10. Cheng, Q., Agterberg, F. P., and Bonham-Carter, G. P., 1996, Fractal pattern integration for mineral potential mapping: Nonrenewable Resources, v. 5,no. 2, p. 117–130.Google Scholar
  11. Cochran, W. G., 1963, Sampling techniques (2nd ed.): John Wiley & Sons, New York, 413 p.Google Scholar
  12. De Wijs, H. J., 1951, Statistics of ore distribution: Geologie en Mijnbouw, v. 13,no. 8, p. 365–375.Google Scholar
  13. Evertsz, C. J. G., and Mandelbrot, B. B., 1992, Multifractal measures, in Peitgen, H.-O., Jürgens, H., and Saupe, D., eds., Chaos and Fractals: Springer-Verlag, New York, p. 922–953.Google Scholar
  14. Feder, J., 1988, Fractals: Plenum Press, New York, 283 p.Google Scholar
  15. Gefen, Y., Meir, Y., and Aharony, A., 1983, Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices: Physical Review Letters, v. 50,no. 3, p. 145–148.Google Scholar
  16. Gefen, Y., Aharony, Y., and Mandelbrot, B. B., 1984, Phase transitions on fractals: III. Infinitely ramified lattices: Physics A: Mathematical and General, v. 17,no. 6, p. 1277–1289.Google Scholar
  17. Henebry, G. M., and Kux, H. J. H., 1995, Lacunarity as a texture measure for SAR imagery: Intern. Jour. Remote Sensing, v. 16,no. 3, p. 565–571.Google Scholar
  18. Lin, B., and Yang, Z. R. 1986, A suggested lacunarity expression for Sierpinski carpets: Jour. Physics A: Mathematical and General, v. 19,no. 2, L49–L52.Google Scholar
  19. Mandelbrot, B. B., 1983, The fractal geometry of nature (updated and augmented edition): W. H. Freeman and Company, New York, 468 p.Google Scholar
  20. Plotnick, R. E., Gardner, R. H., and O'Neill, R. V., 1993, Lacunarity indices as measures of landscape texture: Landscape Ecology, v. 8,no. 3, p. 201–211.Google Scholar
  21. Plotnick, R. E., Gardner, R. H., Hargrove, W. W., Prestegaard, K., and Perlmutter, M., 1996, Lacunarity analysis: a general technique for the analysis of spatial patterns: Physical Review E, v. 53,no. 5, p. 1–8.Google Scholar
  22. Schertzer, D., and Lovejoy, S., eds., 1991a, Non-linear variability in geophysics: Kluwer Acad. Publ., Dordrecht, 318 p.Google Scholar
  23. Schertzer, D., and Lovejoy, S., 1991b, Nonlinear geodynamical variability: Multiple singularities, universality and observations, in Schertzer, D., and Lovejoy, S., eds., Non-linear Variability in Geophysics: Kluwer Acad. Publ., Dordrecht, p. 41–82.Google Scholar
  24. Stanley, H., and Meakin, P., 1988, Multifractal phenomena in physics and chemistry: Nature, v. 335,no. 6189, p. 405–409.Google Scholar

Copyright information

© International Association for Mathematical Geology 1997

Authors and Affiliations

  • Qiuming Cheng
    • 1
  1. 1.Department of Earth and Atmospheric Science, Department of GeographyYork UniversityNorth YorkCanada

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