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.

Figure 1

(a) 2D spring-block model. In our calculation, the system size is $(100\times 25)$ on the $(x,y)$ plane. $k_c^x$, $k_c^y$, and $k_p$ are spring constants. The friction force acts on the surface between the block and the bottom plate. (b) $\alpha$-dependence of the nonlinear dynamical friction function. $\sigma =0.01$ throughout the simulation.

Figure 2

(Color online) The frequency distributions of the interoccurrence time for different $\alpha$ (▵: $\alpha =1.5$, ◻: $\alpha =2.5$, 엯: $\alpha =3.5$, and +: $\alpha =4.5$). $\overline{\tau }$ is the characteristic interoccurrence time, with fixing $l_x=1$ and $l_y=\sqrt{3}$. $b^{\prime }$ is calculated from the slope of the dashed line. All plots except for in the case of $\alpha =4.5$ are shifted vertically for clarity.

Figure 3

(Color online) The cumulative distributions of the interoccurrence time for $\alpha =2.5$ and 3.5 with fixing $l_x=1$ and $l_y=\sqrt{3}$. In this figure, the dotted, solid, and broken lines correspond, respectively, to the simulation data, the Zipf-Mandelbrot type power law defined in Eq. (5), and the exponential distribution.

Figure 4

(Color online) The frequency distributions of the interoccurrence time for different $l_x$ and $l_y$ (▵: $l_x=1$ and $l_y=\sqrt{10}$, ◻: $l_x=\sqrt{2}$ and $l_y=\sqrt{5}$, 엯: $l_x=l_y=\sqrt{3}$, and +: $l_x=\sqrt{3}$ and $l_y=3$), with $\alpha =3.5$. All plots except for in the case of $l_x=1$ and $l_y=\sqrt{10}$ are shifted vertically for clarity.

Figure 5

(Color online) The cumulative distributions of the interoccurrence time for $l_x=l_y=\sqrt{3}$ in (a) and $l_x=\sqrt{3}$ and $l_y=3$ in (b), while $\alpha$ is fixed to be 3.5. In this figure, the dotted, solid, and broken lines correspond, respectively, to the simulation data, the Zipf-Mandelbrot type power law defined in Eq. (5), and the exponential distribution.

Figure 6

(Color online) The interoccurrence time statistics increasing the threshold of magnitude $M_c$, for $l_x=1$, $l_y=\sqrt{3}$, and $\alpha =3.5$ (엯: $M_c=-0.70$ and ▵: $M_c=-0.60$). The solid line corresponds to the exponential distribution. The frequency and cumulative distributions are in (a) and in (b), respectively.

Figure 7

(Color online) The interoccurrence time statistics obtained from the model and real earthquake data. The frequency distribution and the cumulative distribution are shown in (a) and (b), respectively. In (b) the fitting parameter of the cumulative distribution yields $q=1.13$ and ${\tau }_0=1.72$, $q=1.05$ and ${\tau }_0=1.58$, and $q=1.08$ and ${\tau }_0=2.05$ from the Southern California data, and the Japan data, and our simulation, respectively.

Figure 8

(Color online) Size dependence of the interoccurrence time statistics for the case of $l_x=1$, $l_y=\sqrt{3}$, and $\alpha =3.5$. We vary the system size $N$ from 625 to 22 500 (▵: $N=625$, ◻: $N=2500$, 엯: $N=10000$, and +: $N=22500$).