We have numerically investigated statistical properties of the so-called interoccurrence time or the waiting time, i.e., the time interval between successive earthquakes, based on the two-dimensional (2-D) spring-block (Burridge-Knopoff) model, selecting the velocity-weakening property as the constitutive friction law. The statistical properties of frequency distribution and the cumulative distribution of the interoccurrence time are discussed by tuning the dynamical parameters, namely, a stiffness and frictional property of a fault. We optimize these model parameters to reproduce the interoccurrence time statistics in nature; the frequency and cumulative distribution can be described by the power law and Zipf-Mandelbrot type power law, respectively. In an optimal case, the b-value of the Gutenberg-Richter law and the ratio of wave propagation velocity are in agreement with those derived from real earthquakes. As the threshold of magnitude is increased, the interoccurrence time distribution tends to follow an exponential distribution. Hence it is suggested that a temporal sequence of earthquakes, aside from small-magnitude events, is a Poisson process, which is observed in nature. We found that the interoccurrence time statistics derived from the 2-D BK (original) model can efficiently reproduce that of real earthquakes, so that the model can be recognized as a realistic one in view of interoccurrence time statistics.

By analyzing the Japan Meteorological Agency (JMA) seismic catalog for different tectonic settings, we have found that the probability distributions of time intervals between successive earthquakes -interoccurrence times-can be described by the superposition of the Weibull distribution and the log-Weibull distribution. In particular, the distribution of large earthquakes obeys the Weibull distribution with the exponent α1 < 1, indicating the fact that the sequence of large earthquakes is not a Poisson process. It is found that the ratio of the Weibull distribution to the probability distribution of the interoccurrence time gradually increases with increase in the threshold of magnitude. Our results infer that Weibull statistics and log-Weibull statistics coexist in the interoccurrence time statistics, and that the change of the distribution is considered as the change of the dominant distribution. In this case, the dominant distribution changes from the log-Weibull distribution to the Weibull distribution, allowing us to reinforce the view that the interoccurrence time exhibits the transition from the Weibull regime to the log-Weibull regime. PACS numbers: 91.30.Dk, 91.30.Px, 05.65.+b, 05.45.TpRecently, a unified scaling law of interoccurrence times was reported using the Southern California [3] and worldwide earthquake catalogs [4], where the interoccurrence times were analyzed for the events with the magnitude m above a certain threshold m c under the following two conditions: (a) earthquakes can be considered as a point process in space and time; (b) there is no distinction between foreshocks, mainshocks, and aftershocks. It has been demonstrated that the probability distribution of the interoccurrence time is well-fitted by the generalized gamma distribution. This scaling law is obtained by analyzing the aftershock data [5] and is derived approximately from a theoretical framework proposed by Saichev and Sornette [6]. Abe and Suzuki showed that the survivor function of the interoccurrence time can be described by a power law [7]. It has been reported that the sequence of aftershocks and successive independent earthquakes is a Poisson process [8,9]. Recent works on interoccurrence time statistics are focused on the effect of "long-term memory" [10,11,12] as well as on the determination of the distribution function. However, the effect of * Electronic address: t-hasumi.1981@toki.waseda.jp

[1] We analyze the Hurst exponent H and a power-law exponent B obtained from frequency-size distributions of avalanche events in the long-range connective sandpile (LRCS) model and study the relation between those two exponents. The LRCS model is introduced by considering the random distant connection between two separated cells. We find that the B-values typically reduce prior to large avalanches while the H-values increase. Both parameters appear precursory phenomena prior to large avalanche events. Most importantly, we show that the LRCS model can demonstrate an interesting negative correlation between the B-and H-values, which has been frequently implied in observations of seismicity and firstly verified in our present simulations.

We construct a one-dimensional piecewise linear intermittent map from the interevent time distribution for a given renewal process. Then, we characterize intermittency by the asymptotic behavior near the indifferent fixed point in the piecewise linear intermittent map. Thus, we provide a framework to understand a unified characterization of intermittency and also present the Lyapunov exponent for renewal processes. This method is applied to the occurrence of earthquakes using the Japan Meteorological Agency and the National Earthquake Information Center catalog. By analyzing the return map of interevent times, we find that interevent times are not independent and identically distributed random variables but that the conditional probability distribution functions in the tail obey the Weibull distribution.

In analyzing synthetic earthquake catalogs created by a two-dimensional Burridge-Knopoff model, we have found that a probability distribution of the interoccurrence times, the time intervals between successive events, can be described clearly by the superposition of the Weibull distribution and the log-Weibull distribution. In addition, the interoccurrence time statistics depend on frictional properties and stiffness of a fault and exhibit the Weibull -log Weibull transition, which states that the distribution function changes from the log-Weibull regime to the Weibull regime when the threshold of magnitude is increased. We reinforce a new insight into this model; the model can be recognized as a mechanical model providing a framework of the Weibull -log Weibull transition.