Abstract
Multifractal properties of the epicenter and hypocenter distribution and also of the energy distribution of earthquakes are studied for California, Japan, and Greece. The calculatedD q-q curves (the generalized dimension) indicate that the earthquake process is multifractal or heterogeneous in the fractal dimension. Japanese earthquakes are the most heterogeneous and Californian earthquakes are the least. Since the earthquake process is multifractal, a single value of the so-called fractal dimension is not sufficient to characterize the earthquake process. Studies of multifractal models of earthquakes are recommended. Temporal changes of theD q-q curve are also obtained for Californian and Japanese earthquakes. TheD q-q curve shows two distinctly different types in each region; the gentle type and the steep type. The steeptype corresponds to a strongly heterogeneous multifractal, which appears during seismically active periods when large earthquakes occur.D q for smallq or negativeq is considerably more sensitive to the change in fractal structure of earthquakes thanD q forq≥2. We recommend use ofD q at smallq to detect the seismicity change in a local area.
Similar content being viewed by others
References
Aki, K.,A probabilistic synthesis of precursory phenomena. InEarthquake Prediction, An International Rev., M. Ewing Series 4 (eds. Simpson, D. W., and Richard, P. G.) (Am. Geophys. Union, Washington D.C. 1981) pp. 566–574.
Badii, R., andBroggi, G. (1988),Measurement of the Dimension Spectrum f(α): Fixed-mass Approach, Phys. Lett.A131, 339–343.
Benzi, R., Paladin, G., Parisi, G., andVulpiani, A. (1984),On the Multifractal Nature of Fully Developed Turbulence and Chaotic Systems, J. Phys. Am. Math. Gen.17, 3521–3531.
Chhabra, A. B., Meneveau, C., Jensen, R. V., andSreenivasan, K. R. (1989),Direct Determination of the f(α) Singularity Spectrum and its Application to Fully Developed Turbulence, Phys. Rev.A40, 5284–5294.
Frisch, U., Sulem, P., andNelkin, M. (1978),A Simple Dynamical Model of Intermittent Fully Developed Turbulence, J. Fluid Mech.87, 719–736.
Grassberger, P., Badii, R., andPoliti, A. (1988),Scaling Laws for Invariant Measures on Hyperbolic Attractors, J. Stat. Phys.51, 135–178.
Greenside, H. S., Wolf, A., Swift, J., andPignataro, T. (1982),Impracticality of a Boxcounting Algorithm for Calculating the Dimensionality of Strange Attractors, Phys. Rev.A25, 3453–3459.
Gutenberg, B., andRichter, C. F. (1956),Earthquake Magnitude, Intensity, Energy, and Acceleration, Bull. Seismol. Soc. Am.46, 105–145.
Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., andShraiman, B. I. (1986),Fractal Measures and Their Singularities: The Characterization of Strange Sets, Phys. Rev.A33, 1141–1151.
Hentschel, H. G. E., andProcaccia, I. (1983),The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors, Physica8D, 435–444.
Hirata, T. (1989),A Correlation between the b Value and the Fractal Dimension of Earthquakes, J. Geophys. Res.94, 7507–7514.
Hirata, T., andImoto, M. (1991),Multi-fractal Analysis of Spatial Distribution of Microearthquakes in the Kanto Region, Geophys. J. Int.107, 155–162.
Ito, K., andMatsuzaki, M. (1990),Earthquakes as Self-organized Critical Phenomena, J. Geophys. Res.95, 6853–6860.
Jensen, M. H., Kadanoff, L. P., Libchaber, A., Procaccia, I., andStavans, J. (1985),Global Universality at the Onset of Chaos: Results of a Forced Rayleigh-Benard Experiment, Phys. Rev. Lett.55, 2798–2801.
Kagan, Y. Y. (1981),Spatial Distribution of Earthquakes: The Three-point Function, Geophys. J. R. Astr. Soc.67, 69–717.
Kagan, Y. Y., andKnopoff, L. (1980),Spatial Distribution of Earthquakes: The Two-point Correlation Function, Geophys. J. R. Astr. Soc.62, 303–320.
Kanamori, H.,The nature of seismicity patterns before large earthquakes. InEarthquake Prediction, M. Ewing Series 4 (eds. Simpson, D. W., and Richards, P. G.) (Am. Geophys. Union, Washington, D.C. 1981) pp. 1–19.
King, G. (1983),The Accommodation of Large Strains in the Upper Lithosphere of the Earth and Other Solids by Self-similar Fault Systems: The Geometrical Origin of b-value, Pure and Appl. Geophys.121, 761–815.
Mandelbrot, B. B.,Fractals; Form Chance, and Dimension (Freeman, W. H. and Company, San Francisco 1977).
Mandelbrot, B. B. (1989),Multifractal Measures, Especially for the Geophysicist, Pure and Appl. Geophys.131, 5–42.
Meneveau, C., andSreenivasan, K. R.,The multifractal spectrum of the dissipation field in turbulent flow. InPhysics of Chaos and Systems Far from Equilibrium (eds. Van, M. D., and Nicolis, B.) (North-Holland, Amsterdam 1987).
Ouchi, T., andUekawa, T. (1986),Statistical Analysis of the Spatial Distribution of Earthquakes—Variation of the Spatial Distribution of Earthquakes before and after Large Earthquakes, Phys. Earth. Planet. Interiors44, 211–225.
Sadvskiy, M. A., Golubeva, T. V., Pisarenko, V. F., andShnirman, M. G. (1984),Characteristic Dimensions of Rock and Hierarchical Properties of Seismicity, Izv. Acad. Sci. USSR Phys. Solid. Earth, Engl. Trans.20, 87–96.
Schertzer, D., andLovejoy, S. (1978),Physical Model and Analysis of Rain and Clouds by Anisotropic Scaling Multiplicative Processes, J. Geophys. Res.92, 9693–9714.
Stanley, H. E., andMeakin, P. (1988),Multifractal Phenomena in Physics and Chemistry, Nature335, 405–409.
Takayasu, H.,Fractals in the Physical Sciences (Manchester Univ. Press, Manchester 1990).
Takayasu, M., andTakayasu, H. (1989),Apparent Independency of an Aggregation System with Injection, Phys. Rev.A39, 4345–4347.
Turcotte, D. L. (1986),A Fractal Model for Crustal Deformation, Tectonophysics132, 261–269.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hirabayashi, T., Ito, K. & Yoshii, T. Multifractal analysis of earthquakes. PAGEOPH 138, 591–610 (1992). https://doi.org/10.1007/BF00876340
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00876340