Seismic statistics: Universality and interim report on the 3. 11 earthquake (2011) in Fukushima-Miyagi area

We mainly study three empirical laws in earthquake statistics; the Omori formula in the aftershock frequency, the Gutenberg-Richter formula for the magnitude, and the interoccurrence time distribution, and we show a possibility to refine and to unify them by carrying out with the data analysis for natural earthquakes in Japan, Taiwan, and South California. Especially in the analysis of the 3.11 EQ (2011) in Fukushima-Miyagi area, firstly it is emphasized that the Omori formula is generalized to express the magnitude-dependent form, and that the Omori coefficient reveals the same behavior as the Gutenberg-Richter(GR) law in the nonstationary regime after the big shock. Next, the multi-fractal diagram, that connects the GR law and the Weibull distribution for interoccurrence times, is studied to characterize the shock sequences before and after the main shock. A universal relation is well confirmed in the long stationary regime before the main shock, but in the nonstationary regime such as the foreshock region and the aftershock region, the multi-fractal diagram reveals big deviations from the stationary case. In spite of those deviations, it is shown that the empirical laws are approximately confirmed even in the nonstationary regime. Lastly, the moving ensembles are used to describe the temporal change of the multi-fractal diagram. We could not find out any signals to suggest the occurrence of the big shock in statistical parameters, but the multi-fractal diagram obeys the time-dependent universal relation in the nonstationary regime. These results imply that the universality is also existing even in the nonequilibrium moving ensemble. In relation to these remarkable points, we will discuss some theoretical conjectures to seismic statistics, which enable us to understand the origin of statistical laws of earthquakes beyond the traditional ergodic-theoretical interpretation.