Earthquake size distribution: power-law with exponent ?

Yan Y. Kagan Department of Earth and Space Sciences, University of California Los Angeles

AbstractWe propose that the widely observed and universal Gutenberg-Richterrelation is a mathematical consequence of the critical branching natureof earthquake process in a brittle fracture environment. Thesearguments, though preliminary, are confirmed by recent investigationsof the seismic moment distribution in global earthquake catalogs and bythe results on the distribution in crystals of dislocation avalanchesizes. We consider possible systematic and random errors in determiningearthquake size, especially its seismic moment. These effects increasethe estimate of the parameter beta of the power-law distribution ofearthquake sizes. In particular we find that the decrease in relativemoment uncertainties with earthquake size causes inflation in thebeta-value by about 1-3%. Moreover, earthquake clustering greatlyinfluences the beta-parameter. If clusters (aftershock sequences) aretaken as the entity to be studied, then the exponent value for theirsize distribution would decrease by 5-10%. The complexity of anyearthquake source also inflates the estimated beta-value by at least3-7%. Taking all these effects into account, we propose that therecently obtained beta-value of 0.63 could be reduced to about0.52-0.56: near the universal constant value (1/2) predicted bytheoretical arguments. We also consider possible consequences of theuniversal beta-value and its relevance for theoretical and practicalunderstanding of earthquake occurrence in various tectonic and Earthstructure environments. Using comparative crystal deformation resultsmay help us understand the generation of seismic tremors and slowearthquakes and illuminate the transition from brittle fracture toplastic flow. http://arxiv.org/abs/0908.1207

URL: http://eq.ess.ucla.edu/~kagan/beta_index.html

Conclusion: is a universal constant, equal to 0.5, its variations in earthquake catalogs are caused by systematic and random factors.

References• Bird, P., and Y. Y. Kagan, 2004. Plate-tectonic analysis of shallowseismicity: apparent boundary width, beta, corner magnitude, coupledlithosphere thickness, and coupling in seven tectonic settings, Bull.Seismol. Soc. Amer., 94(6), 2380-2399 (plus electronic supplement).• Csikor, F. F., C. Motz, D. Weygand, M. Zaiser, S. Zapperi, 2007.Dislocation Avalanches, Strain Bursts, and the Problem of PlasticForming at the Micrometer Scale, Science, 318(5848), 251-254.• Dahmen, K. A., Y. Ben-Zion, and J. T. Uhl, 2009. Micromechanical Model for Deformation in Solids with Universal Predictions for Stress-Strain Curves and Slip Avalanches, Phys. Rev. Lett., 102(17), Article Number: 175501.• Kagan, Y. Y., 1991. Seismic moment distribution, Geophys. J. Int.,106(1), 123-134.• Kagan, Y. Y., 1991. Likelihood analysis of earthquake catalogues,Geophys. J. Int., 106(1), 135-148.• Kagan, Y. Y., 1999. Universality of the seismic moment-frequencyrelation, Pure Appl. Geoph., 155(2-4), 537-573. • Kagan, Y. Y., P. Bird, and D. D. Jackson, 2009. Earthquake Patterns in Diverse Tectonic Zones of the Globe, accepted by Pure Appl. Geoph., The Frank Evison Volume.• Kagan, Y. Y., and L. Knopoff, 1981. Stochastic synthesis of earthquake catalogs, J. Geophys. Res., 86(B4), 2853-2862.• Miguel, M.-C., A. Vespignani, S. Zapperi, J. Weiss and J.-R. Grasso,2001. Intermittent dislocation flow in viscoplastic deformation,Nature, 410, 667-671, doi:10.1038/35070524.• Schwartz, S.Y., and J.M. Rokosky, 2007. Slow slip events and seismic tremor at circum-pacific subduction zones, Reviews Geophysics, 45(3), Art. No. RG3004• Vere-Jones, D., 1976. A branching model for crack propagation, PureAppl. Geophys., 114, 711-725.• Zaiser, M., 2006. Scale invariance in plastic flow of crystallinesolids, Advances Physics, 55(1-2), 185-245.

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Contour maps of relative log-likelihood, showing preferred value (at peak, +3) and 95%-confidence interval (within zero contour) of the parameters β and Mc (or mc ) of tapered Gutenberg-Richter frequency/moment distributions

Zaiser, Advances in Physics, 2006. Compressive deformation of microsamples of pure Ni (top); shown are stress vs. time and displacement vs. time curves as recorded in microtesting experiments.

Miguel et al., Nature, 2001

Csikor et al. Science, 2007

Magnitude errors

Aftershocks

Focal mechanism complexity

Tremors (Schwartz & Rokosky, 2007)