Statistical seismicity in view of complex systems

In this chapter, we investigate the statistical seismicity in view of complex systems, namely, interoccurrence time statistics and intermittency of earthquakes. To show statistical properties of earthquakes, we analyze three different kinds of data; i) natural earthquake data, ii) synthetic data, and iii) time sequence analysis based on a one-dimensional map. For i), Japan Meteorological Agency (JMA), Southern California Earthquake Data Center (SCEDC), and Taiwan Central Weather Bureau (TCWB) are analyzed. For ii), we investigate synthetic data produced by the two-dimensional (2D) spring-block model. For iii), intermittency and correlation of earthquakes are studied using JMA data and the constructed map. First, we focus on the interoccurrence time statistics, statistical properties of time intervals between successive earthquakes by analyzing both natural data and the synthetic data. We show that the probability distribution of interoccurrence time depends on the threshold of magnitude, mc and is described by the superposition of theWeibull distribution and log-Weibull distribution for all earthquake data. As mc increases, the ratio of the Weibull distribution increases. In other words, the distribution changes from the Weibull to the log-Weibull distribution according to an increase of mc, i.e., theWeibull-log-Weibull transition. This result reinforces the view that the interoccurrence time statistics contain theWeibull statistics and the log-Weibull statistics. In synthetic data, the distribution changes from the pure log-Weibull regime to the Weibull regime via the superposition regime, i.e., a sharp Weibull-log-Weibull transition. For natural data, the distribution presents a smooth Weibull-log-Weibull transition which means that the distribution varies from the superposition regime to the pure Weibull regime. For large earthquakes, the Weibull distribution with the exponent a1 < 1 is preferred, which indicates that the sequence of earthquakes is not a Possion process. We give a geophysical interpretation of theWeibull-log-Weibull transition. Second, to characterize an intermittency and the correlation of interoccurrence times in earthquakes, we analyze the JMA catalog. We demonstrate that interoccurrence times are not independent and identically distributed random variables. However, we find that the tails of all conditional probability distribution functions of interoccurrence times are universal. The tail characterizes intermittency of earthquakes. It is shown that forecasting of the occurrence of large earthquakes is more difficult than that of small earthquakes.