1. Random stress and Omori's law Yan Y. Kagan Department of Earth and Space Sciences, University of California Los Angeles Abstract We consider two statistical regularities that were used to explain Omori's law of the aftershock rate decay: Levy and Inverse Gaussian (IGD) distributions. These distributions are thought to describe stress behavior under the influence of various random factors, i.e., post-earthquake stress time history is described by a Brownian motion. Both distributions decay to zero for time intervals close to zero. This feature contradicts the high immediate aftershock level according to Omori's law. We propose that these distributions are influenced by the power-law stress distribution in the neighborhood of the earthquake focal zone and we derive new distributions as a mixture of power-law stress with the exponent psi and Levy as well as IGD distributions. These new distributions describe the resulting inter-earthquake time intervals and they closely resemble Omori's law. The new Levy distribution has a pure power-law form with the exponent -(1+psi/2) and the mixed IGD has two exponents, same as Levy for small time intervals and -(1+psi) for longer times. This power-law behavior should be replaced by even longer time intervals by a uniform seismicity rate corresponding to the long-term tectonic deformation. These background rates are computed based on our former analysis of the earthquake size distribution and its connection to the plate tectonics. We analyze several earthquake catalogs to confirm and illustrate our theoretical results. Finally we discuss how the parameters of random stress dynamics can be determined either through a more detailed statistical analysis of earthquake occurrence or by new laboratory experiments. As an illustration of our results we estimate long-term earthquake rates in the area of 1811-12 New Madrid events as well as aftershock rates during the last 50 years. URL: http://eq.ess.ucla.edu/~kagan/igd_index.html http://eq.ess.ucla.edu/~kagan/igd_index.html References Bird, P., and Y. Y. Kagan, 2004. Plate-tectonic analysis of shallow seismicity: apparent boundary width, beta, corner magnitude, coupled lithosphere thickness, and coupling in seven tectonic settings, Bull. Seismol. Soc. Amer., 94(6), 2380-2399 (plus electronic supplement). Bird, P., C. Kreemer, and W. E. Holt, 2010. A long-term forecast of shallow seismicity based on the Global Strain Rate Map, Seismol. Res. Lett., 81(2), 184-194 (plus electronic supplement). Enescu, B., J. Mori, M. Miyazawa, and Y. Kano, 2009. Omori-Utsu law c-values associated with recent moderate earthquakes in Japan, Bull. Seismol. Soc. Amer., 99(2A), 884-891. Helmstetter, A., Y. Y. Kagan, and D. D. Jackson, 2005. Importance of small earthquakes for stress transfers and earthquake triggering, J. Geophys. Res., 110(5), B05S08, doi:10.1029/2004JB003286, pp. 1-13. Kagan, Y. Y., 1991. Likelihood analysis of earthquake catalogues, Geophys. J. Int., 106(1), 135-148. Kagan, Y. Y., 1994. Distribution of incremental static stress caused by earthquakes, Nonlinear Processes in Geophysics, 1, 172-181.. Kagan, Y. Y., 2002a. Aftershock zone scaling, Bull. Seismol. Soc. Amer., 92, 641-655. Kagan, Y. Y., 2002b. Seismic moment distribution revisited: II. Moment conservation principle, Geophys. J. Int., 149, 731-754. Kagan, Y. Y., 2004. Short-term properties of earthquake catalogs and models of earthquake source, Bull. Seismol. Soc. Amer., 94(4), 1207-1228. Kagan, Y. Y., P. Bird, and D. D. Jackson, 2010. Earthquake Patterns in Diverse Tectonic Zones of the Globe, Pure Appl. Geoph., The Frank Evison Volume167(6/7), 721-741, doi: 10.1007/s00024-010-0075-3. Lavallee, D., 2008. On the random nature of earthquake sources and ground motions: a unified theory, Advances in Geophysics, 50, 427- 461. Matthews, M. V., W. L. Ellsworth, and P. A. Reasenberg, 2002. A Brownian model for recurrent earthquakes, Bull. Seismol. Soc. Amer., 92, 2233-2250. Powers, P. M., and Jordan, T. H., 2010. Distribution of Seismicity Across Strike-Slip Faults in California, J. Geophys. Res., 115(B05), Article Number: B05305. Vere-Jones, D., 2010. Foundations of Statistical Seismology, Pure Appl. Geophys., 167(6/7), 645-653, doi: 10.1007/s00024-010-0079-z. Distribution curves in Figs. 10-13 consist of two parts: for small time intervals they follow a power-law and for larger intervals the distribution is parallel to the Poisson rate. The difference between regression parameters in this (active continents) and upper (all earthquakes) figure is small, therefore the size of earthquake focal zone either does not change in various tectonic zones or modifies slightly. 7-th International Workshop on Statistical Seismology: 25 - 27 May 2011, Greece - Thera (Santorini) Example: New Madrid earthquake sequence of 1811-12 As an illustration of our results we estimate long-term earthquake rates in the area of 1811-12 New Madrid events as well as aftershock rates during the last 50 years. The size of the focal zone can be evaluated by using regression equations in the left Figure – log aftershock zone size vs magnitude for active continental zones. The Cottonwood fault (110-120 km length) would correspond to ~M7.3 event, total zone length corresponds to about M8 event. For strain rate 10^-9 1/yr we obtain, using Eq.17, background annual rate 0.001 M5 earthquakes, i.e. M8 event once per 2,000,000 years. The aftershock annual rate after 175 years would be ~0.25 M5 events, close to the observed ~0.5 per year (Calais et al. (2006). S. Hough (2004)