pure and applied geophysics

, Volume 131, Issue 1–2, pp 5–42 | Cite as

Multifractal measures, especially for the geophysicist

  • Benoît B. Mandelbrot


This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach inMandelbrot (1974). The generalization from fractalsets to multifractalmeasures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function ϱ(α), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity α is called Hölder exponent. In terms of the alternative functionf(α) used in the approach of Frisch-Parisi and of Halseyet al., one has ϱ(α)=f(α)−E for measures supported by the Euclidean space of dimensionE. Whenf(α)≥0,f(α) is a fractal dimension. However, one may havef(α)<0, in which case α is called “latent.” One may even have α<0, in which case α is called “virtual.” These anomalies' implications are explored, and experiments are suggested. Of central concern in this paper is the study of low-dimensional cuts through high-dimensional multifractals. This introduces a quantityDq, which is shown forq>1 to be a critical dimension for the cuts. An “enhanced multifractal diagram” is drawn, includingf(α), a function called τ(q) andDq.

Key words

Fractal multifractal measure Hölder limit theorem 


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  1. Azencott, R., Guivarc'h, Y., andGundy, R. F.,Ecole de Saint-Flour 1978, Lecture Notes in Mathematics (Saint Flour, 1978) Vol.774 (Springer, New York 1980).Google Scholar
  2. Billingsley, P.,Ergodic Theory and Information (J. Wiley, New York 1967 p. 139.Google Scholar
  3. Book, S. A. (1984),Large deviations and applications, InEncyclopedia of Statistical Sciences 4, 476.Google Scholar
  4. Cates, M. E., andDeutsch, J. M. (1987),Spatial Correlations in Multifractals, Phys. Rev.A 35, 4907.Google Scholar
  5. Chernoff, H. (1952),A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations, Ann. Math. Stat.23, 493.Google Scholar
  6. Chhabra, A., andJensen, R. V. (1989),Direct Determination of the f(α) Singularity Spectrum, Phys. Rev. Lett.62, 1327.Google Scholar
  7. Dacunha-Castelle, D.,Grandes Déviations et Applications Statistiques (Astérisque 68) (Societé Mathématique de France, Paris 1979).Google Scholar
  8. Daniels, H. E. (1954),Saddlepoint Approximations in Statistics, Ann. Math. Stat.25, 631.Google Scholar
  9. Daniels, H. E. (1987),Tail Probability Approximations, International Statistical Review55, 37.Google Scholar
  10. Ellis, R. S. (1984),Large Deviations for a General Class of Random Vectors, The Annals of Probability12, 1.Google Scholar
  11. Feder, J.,Fractals (Plenum, New York 1988).Google Scholar
  12. Fourcade, B., Breton, P., andTremblay, A.-M. S. (1987),Multifractals and Critical Phenomena in Percolating Networks: Fixed Point, Gap Scaling and Universality, Phys. Rev.B36, 8925.Google Scholar
  13. Fourcade, B., andTremblay, A.-M. S. (1987),Anomalies in the Multifractal Analysis of Self-similar Resistor Networks, Phys. Rev.A36, 2352.Google Scholar
  14. Frisch, U., andParisi, G.,Fully develped turbulence and intermittency, InTurbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (International School of Physics “Enrico Fermi”, Course 88) (eds. Ghil, M.) (North-Holland, Amsterdam, 1985) p. 84.Google Scholar
  15. Grassberger, P. (1983),Generalized Dimensions of Strange Attractors, Phys. Lett.97A, 227.Google Scholar
  16. Gutzwiller, M. C., andMandelbrot, B. B., (1988),Invariant Multifractal Measures in Chaotic Hamiltonian Systems, and Related Structures, Phys. Rev. Lett.60, 673.Google Scholar
  17. Guivarc'h, Y. (1987),Remarques sur les Solutions d'une Equation Fonctionnelle Non Linéaire de Benoít Mandelbrot, Comptes Rendus (Paris)3051, 139.Google Scholar
  18. Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., andShraiman, B. I. (1986),Fractal Measure and their Singularities: The Characterization of Strange Sets, Phys. Rev.A33, 1141.Google Scholar
  19. Hentschel, H. G. E., andProcaccia, I. (1983),The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors, Physica (Utrecht)8D, 435.Google Scholar
  20. Huang, K.,Statistical Mechanics (J. Wiley, New York 1966).Google Scholar
  21. Kahane, J. P., andPeyrière, J. (1976),Sur Certaines Martingales de B. Mandelbrot, Adv. in Math.22, 131.Google Scholar
  22. Mandelbrot, B. B.,Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, InStatistical Models and Turbulence (Lecture Note in Physics, Vol. 12), Proc. Symp. La Jolla, Calif. (eds. Rosenblatt, M., and Van Atta, C.) (Springer-Verlag, New York 1972) p. 333.Google Scholar
  23. Mandelbrot, B. B. (1974),Intermittent Turbulence in Self Similar Cascades; Divergence of High Moments and Dimension of the Carrier, J. Fluid Mech62, 331; also Comptes Rendus278A, 289, 355.Google Scholar
  24. Mandelbrot, B. B.,The Fractal Geometry of Nature (W. H. Freeman, New York 1982) pp. 373–381 discuss α and the “Lipshitz-Hölder heuristics”, then the “nonlacunar fractals”=multifractals.Google Scholar
  25. Mandelbrot, B. B. (1984),Fractals in Physics: Squig Clusters, Diffusiions, Fractal Measures and the Unicity of Fractal Dimension J. Stat. Phys.34, 895.Google Scholar
  26. Mandelbrot, B. B. (1986),Letter to the Editor: Multifractals and Fractals, Physics Today11.Google Scholar
  27. Mandelbrot, B. B., (1988),An introduction to multifractal distribution functions, InFluctuations and Pattern Formation (Cargèse, 1988) (eds. Stanley, H. E., and Ostrowsky, N.) (Kluwer, Dordrecht-Boston 1988) pp. 345–360.Google Scholar
  28. Mandelbrot, B. B.,Examples of multinomial multifractal measures that have negative latent values for the dimension f(α), InFractals (“Etteor Majorama” Centre for Scientific Culture, Special Seminar) (ed. Pietronero, L) (Plenum, New York 1989).Google Scholar
  29. Mandelbrot, B. B. (forthcoming),Fractals and Multifractals: Noise, Turbulence and Galaxies (Selecta, Vol. 1) (Springer, New York).Google Scholar
  30. Meakin, P. The growth of fractal aggregates and their fractal measures, InPhase Transitions and Critical Phenomena (eds. Domb, C., and Lebowitz, J. L.) (Academic Press London 1988)12, 335.Google Scholar
  31. Meneveau, C., andSreenivasan, K. R. (1987),Simple Multifractal Cascade Model for Fully Developed Turbulence, Phys. Rev. Lett.,59, 1424.Google Scholar
  32. Meneveau, C., andSreenivasan, K. R. (1989),Measurement of f(α) from Scaling of Histograms, and Applications to Dynamic Systems and Fully Developed Turbulence, Phys. Lett.A137, 103–112.Google Scholar
  33. Prasad, R. P., Meneveau, C., andSreenivasan, K. R. (1988).Multifractal Nature of the Dissipation Field of Passive Scalars in Fully Turbulent Flaws, Phys. Rev. Lett.61, 74.Google Scholar
  34. Volkmann, (1958),Ober Hausdorffsche Dimensionen Von Mengen, Die Durch Zifferneigenschaften Sind, Math. Zeitschrift68, 439.Google Scholar

Copyright information

© Birkhäuser-Verlag 1989

Authors and Affiliations

  • Benoît B. Mandelbrot
    • 1
    • 2
  1. 1.Physics DepartmentIBM T. J. Watson Research CenterYorktown HeightsUSA
  2. 2.Mathematics DepartmentYale UniversityNew HavenUSA

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