Tomohiro Hasumi

The Weibull–log Weibull distribution for interoccurrence times of earthquakes

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.

Hypocenter interval statistics between successive earthquakes in the two-dimensional Burridge–Knopoff model

We study statistical properties of spatial distances between successive earthquakes, the so-called hypocenter intervals, produced by a two-dimensional (2D) Burridge–Knopoff model involving stick-slip behavior. It is found that cumulative distributions of hypocenter intervals can be described by the q-exponential distributions with q<1, which is also observed in nature. The statistics depend on a friction and stiffness parameters characterizing the model and a threshold of magnitude. The conjecture which states that qt+qr∼2, where qt and qr are an entropy index of time intervals and spatial intervals, respectively, can be reproduced semi-quantitatively. It is concluded that we provide a new perspective on the Burridge–Knopoff model which addresses that the model can be recognized as a realistic one in view of the reproduction of the spatio-temporal interval statistics of earthquakes on the basis of nonextensive statistical mechanics.

The Weibull-log Weibull transition of the interoccurrence time statistics in the two-dimensional Burridge-Knopoff Earthquake model

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.

Interoccurrence time statistics in the two-dimensional Burridge-Knopoff earthquake model.

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 (2D) 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 2D 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.

Characterization of intermittency in renewal processes: application to earthquakes.

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.

The Weibull – Log-Weibull Transition of Interoccurrence Time of Earthquakes

The time intervals between successive earthquakes can be classified into two types: interoccurrence times and recurrence times (4). Interoccurrence times are the interval times between earthquakes on all faults in a region, and recurrence times are the time intervals between earthquakes in a single fault or fault segment. For seismology, recurrence times mean the interval times of characteristic earthquakes that occur quasi-periodically in a single fault. Recently, a unified scaling law of interoccurrence times was found using the Southern California earthquake catalogue (5) and worldwide earthquake catalogues (6). In Corral’s paper (6), the probability distribution of interoccurrence time, P(τ), can be written as

STATISTICAL PROPERTIES OF THE INTER-OCCURRENCE TIMES IN THE TWO-DIMENSIONAL STICK-SLIP MODEL OF EARTHQUAKES

We study earthquake interval time statistics, paying special attention to inter-occurrence times in the two-dimensional (2D) stick-slip (block-slider) model. Inter-occurrence times are the time interval between successive earthquakes on all faults in a region. We select stiffness and friction parameters as tunable parameters because these physical quantities are considered as essential factors in describing fault dynamics. It is found that inter-occurrence time statistics depend on the parameters. Varying stiffness and friction parameters systematically, we optimize these parameters so as to reproduce the inter-occurrence time statistics in natural seismicity. For an optimal case, earthquakes produced by the model obey the Gutenberg-Richter law, which states that the magnitude-frequency distribution exhibits the power law with an exponent approximately unity.

Precursory measure of interoccurrence time associated with large earthquakes in the Burridge-Knopoff model

We studied the statistical properties of interoccurrence time i.e., time intervals between successive earthquakes in the two‐dimensional (2D) Burridge‐Knopoff (BK) model, and have found that these statistics can be classified into three types: the subcritical state, the critical state, and the supercritical state. The survivor function of interoccurrence time is well fitted by the Zipf‐Mandelbrot type power law in the subcritical regime. However, the fitting accuracy of this distribution tends to be worse as the system changes from the subcritical state to the supercritical state. Because the critical phase of a fault system in nature changes from the subcritical state to the supercritical state prior to a forthcoming large earthquake, we suggest that the fitting accuracy of the survivor distribution can be another precursory measure associated with large earthquakes.

Model of Earthquake Generation Exhibiting Self-Organized Criticality with Self-Affine Fault Surfaces

Generation of earthquakes has not been still understood. However, some statistical properties of seismicity are established such as magnitude-frequency distribution indicating power law. In this study, we numerically investigate one-dimensional spring-block earthquake model, considering self-affine fault surfaces, in order to understand such statistical properties.