Mathematical Geology

, Volume 37, Issue 8, pp 915–927 | Cite as

Multifractal Distribution of Eigenvalues and Eigenvectors from 2D Multiplicative Cascade Multifractal Fields

  • Qiuming Cheng


Two-dimensional fields (maps) generated by isotropic and anisotropic multiplicative cascade multifractal processes are common in many fields including oceans, atmosphere, the climate and solid earth geophysics. Modeling the anisotropic scaling property and heterogeneity of these types of fields are essential for understanding the underlying processes. The paper explicitly derives the eigenvalues and eigenvectors from these types of fields and proves that the eigenvalues and eigenvectors are described by non-conservative multifractal distributions. This results in a new multifractal model implemented in eigen domain to characterize 2D fields not only with respect to overall heterogeneity and singularity as characterized by the ordinary multifractal model applied to the field itself, but also with respect to orientational heterogeneity of the field. It may also result in a new way to characterize the distribution of extreme large eigenvalues involved in other studies such as principal component analysis. A newly defined operator and its properties as derived in this paper may be useful for studying other types of multifractal cascade processes.

Key Words

non-conservative multifractal eigen domain eigenvalues eigenvectors multiplicative cascade process 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Badii, R., and Politi, A., 1984, Hausdorff dimension and uniformity of strange attractors: Phys. Rev. Lett., v. 52, p. 1661–1664.CrossRefGoogle Scholar
  2. Badii, R., and Politi, A., 1985, Statistical description of chaotic attractors: The dimension function: J. Stat. Phys., v. 40, p. 725–750.Google Scholar
  3. Cheng, Q., 2004, A new model for quantifying anisotropic scale invariance and for decomposition of mixing patterns: Math. Geol., v. 36, no. 3, p. 345–360.CrossRefGoogle Scholar
  4. Cheng, Q., 1997, Discrete multifractals: Math. Geol., v. 29, no. 2, p. 245–266.Google Scholar
  5. Cheng, Q., 1999a, Multifractality and spatial statistics: Computers & Geosciences, v. 25, no. 10, p. 949–961.Google Scholar
  6. Cheng, Q., 1999b, The gliding box method for multifractal modeling: Computers & Geosciences, v. 25, no. 10, p. 1073–1079.Google Scholar
  7. Cheng, Q., Xu, Y., and Grunsky, E., 2001, Multifractal power spectrum-area method for geochemical anomaly separation: Nat. Resour. Res., v. 9, no.1, p. 43–51.Google Scholar
  8. Cheng, Q., Xu, Y., and Grunsky, E., 1999, Integrated spatial and spectrum analysis for geochemical anomaly separation, in Lippard, J. L., Naess, A., and Sinding-Larsen, R., eds., Proceedings of International Association for Mathematical Geology Meeting, Tapir, Trondheim, Norway I, p. 87–92.Google Scholar
  9. Cheng, Q., Agterberg, F. P., and Ballantyne, S. B., 1994, The separation of geochemical anomalies from background by fractal methods: J. Geochem. Explor. v. 51, no. 2, p. 109–130.CrossRefGoogle Scholar
  10. Chhabra, A. B., and Sreenivasan, K. R., 1991, Negative dimensions: theory, computation and experiment: Phys. Rev. A, v. 43, no. 2, p. 1114–1117.CrossRefGoogle Scholar
  11. Evertsz, C. J. G., and Mandelbrot, B. B., 1992, Multifrtactal measures, in Peitgen, H.-O., Jürgens, H., Saupe, D., eds., Chaos and Fractals, Springer-Verlag, New York, p. 922–953.Google Scholar
  12. Feder, J., 1988, Fractals, Plenum Press, New York, 283 p.Google Scholar
  13. Frisch, U., and Parisi, G., 1985, On the singularity structure of fully developed turbulence, in Ghil, M., Benzi, R., and Parisi, G., eds., Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, North-Holland, New York, p. 84–88.Google Scholar
  14. Grassberger, P., 1983, Generalized dimensions of strange attractors: Phys. Lett. A, v. 97, p. 227–230.Google Scholar
  15. Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., and Shraiman, B. I., 1986, Fractal measures and their singularities: The characterization of strange sets: Phys. Rev. A, v. 33, no. 2, p. 1141–1151.CrossRefGoogle Scholar
  16. Hentschel, H. G. E., and Procaccia, I., 1983, The infinite number of generalized dimensions of fractals and strange attractors: Physica, v. 8, p. 435–444.Google Scholar
  17. Li, Q., and Cheng, Q., 2004, Fractal singular-value (eigen-value) decomposition method for geophysical and geochemical anomaly reconstruction, Earth Science, a Journal of China University of Geosciences, v. 29, no. 1, p. 109–118 (in Chinese with English Abstract).Google Scholar
  18. Mandelbrot, B. B., 1972, Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, in Rosenblatt, M., Van Atta, C., eds., Statistical Models and Turbulence, Lecture Notes in Physics 12, Springer, New York, p. 333–351.Google Scholar
  19. Mandelbrot, B. B., 1974, Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier: J. Fluid Mech., v. 62, p. 331–358.Google Scholar
  20. Paladin, G., and Vulpiani, A., 1987, Anomalous scaling laws in multifractal objects: Phys. Rep., v. 156, no. 4, p. 147–225.CrossRefGoogle Scholar
  21. Schertzer, D., and Lovejoy, S., eds., 1991: Nonlinear Variability in Geophysics: Kluwer Academic Publisher, Dordrecht, The Netherlands, 318 p.Google Scholar

Copyright information

© International Association for Mathematical Geology 2005

Authors and Affiliations

  1. 1.State Key Lab of Geological Processes and Mineral ResourcesChina University of GeosciencesChina
  2. 2.Department of Earth and Space Science and Engineering, Department of GeographyYork UniversityTorontoCanada

Personalised recommendations