Statistical properties of the one-dimensional Burridge-Knopoff model of earthquakes obeying the rate- and state-dependent friction law

Statistical properties of the one-dimensional spring-block (Burridge-Knopoff) model of earthquakes obeying the rate- and state-dependent friction law are studied by extensive computer simulations. The quantities computed include the magnitude distribution, the rupture-length distribution, the main shock recurrence-time distribution, the seismic-time correlations before and after the main shock, the mean slip amount, and the mean stress drop at the main shock, etc. Events of the model can be classified into two distinct categories. One tends to be unilateral with its epicenter located at the rim of the rupture zone of the preceding event, while the other tends to be bilateral with enhanced “characteristic” features resembling the so-called “asperity.” For both types of events, the distribution of the rupture length $L_r$ exhibits an exponential behavior at larger sizes, $\approx exp[-L_r/L_0]$ with a characteristic “seismic correlation length” $L_0$. The mean slip as well as the mean stress drop tends to be rupture-length independent for larger events. The continuum limit of the model is examined, where the model is found to exhibit pronounced characteristic features. In the continuum limit, the characteristic rupture length $L_0$ is estimated to be $\sim 100$ [km]. This means that, even in a hypothetical homogenous infinite fault, events cannot be indefinitely large in the exponential sense, the upper limit being of order $\sim {10}^3$ kilometers. Implications to real seismicity are discussed.

Figure 1

The magnitude distribution of earthquake events of the 1D BK model for various frictional-parameter $b$ values (a) in the strong frictional instability regime of $b>b_c=19$ and (b) in the weak frictional instability regime of $b<b_c$. The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The system size is $N=800$.

Figure 2

The magnitude distribution of earthquake events of the 1D BK model for various $a$ values (a) in the strong frictional instability regime of $b=30>b_c=19$ and (b) in the weak frictional instability regime of $b=8<b_c$. The other parameter values are $l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The system size is $N=800$.

Figure 3

The magnitude distribution of earthquake events of the 1D BK model for various $v^{*}$ values (a) in the strong frictional instability regime of $b=30>b_c=19$ and (b) in the weak frictional instability regime of $b=8<b_c$. The other parameter values are $a=1, l=3, c=1000$, and $\nu ={10}^{-8}$. The system size is $N=800$.

Figure 4

The rupture-length $L_r$ distribution of earthquake events of the 1D BK model on a semilogarithmic plot for various frictional-parameter $b$ values (a) in the strong frictional instability regime of $b>b_c=19$ and (b) in the weak frictional instability regime of $b<b_c$. The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The system size is $N=800$. The inverse slope of the tail of the distribution yields a characteristic rupture-length scale $L_0$ as indicated in the figure. The $b$-dependence of $L_0$ is shown in panel (c).

Figure 5

The event frequency before and after the main shock occurring at the time $t=0$ (a) in the strong frictional instability regime of $b=30>b_c=19$ and (b) in the weak frictional instability regime of $b=8<b_c$. The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The system size is $N=800$. The number of events occurring within the distance $\mathrm{\Delta }r$ from the epicenter site of the main shock is counted irrespective of their magnitude. Note that the abscissa is scaled with the plate velocity $\nu$.

Figure 6

The distribution of the main-shock recurrence time of the 1D BK model. The parameter values are $a=1, b=30, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The system size is $N=800$. The dashed lines represent the recurrence-time distribution of the type I events (green) and of the type II events (blue). For the details, see the main text. Note that the abscissa is scaled with the plate velocity $\nu$.

Figure 7

Schematic spatiotemporal pattern of the type I and the type II events. Bars represent the rupture zone of the event, and crosses represent its epicenter site.

Figure 8

The stress value plotted versus the block position just before and after a typical type II event. The parameter values are $a=1, b=30, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The epicenter block is indicated by the arrow.

Figure 9

The $b$-dependence of the rate of the type I event. The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The borderline value of $b$ separating the strong and weak frictional instability regimes, $b_c=19$, is indicated by the vertical dotted line. The inset shows the $a$-dependence of the rate of the type I event at $b=b_c=19$.

Figure 10

The rupture-length $L_r$ dependence of (a) the mean slip amount $\mathrm{\Delta }u$ and of (b) the mean stress drop $\mathrm{\Delta }\tau$. The frctional-parameter $b$ value is either $b=30>b_c=19$ in the strong frictional instability regime (red), and $b=8<b_c$ in the weak frictional instability regime (blue). The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$.

Figure 11

The mean slip amount during the rupture propagation of a main shock is plotted versus the block position measured from the epicenter block. The frictional-parameter $b$ value is either (a) $b=30>b_c=19$ in the strong frictional instability regime or (b) $b=8<b_c$ in the weak frictional instability regime. The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. Events that propagate with exactly the distance $L_r=100,200$, and 300 in their longer direction are collected, and the data are averaged over the events satisfying this condition.

Figure 12

The mean stress drop during the rupture propagation of a main shock is plotted versus the block position measured from the epicenter block. The frictional-parameter $b$ value is either (a) $b=30>b_c=19$ in the strong frictional instability regime or (b) $b=8<b_c$ in the weak frictional instability regime. The other parameter values are $a=1, l=3, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. Events which propagate with exactly the distance $L_r=100,200$, and 300 in their longer direction are collected, and the data are averaged over the events satisfying this condition.

Figure 13

(a) The block-size $d$ dependence of the magnitude distribution. The parameters are $a=1, b=6, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$. The $d\to 0$ limit corresponds to the continuum limit where $l=1/d$. The system size (the total number of blocks) is $N/d$ with $N=800$. (b) The magnitude distribution near the continuum limit ($d=1/16$) for various frictional-parameter $b$ values. The other parameters are the same as in (a).

Figure 14

The rupture-length ${\tilde{L}}_r=L_{r}d=L_r/l$ distribution near the continuum limit ($d=1/16$) for various frictional-parameter $b$ values. The other parameter values are $a=1, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$.

Figure 15

The main shock recurrence-time distribution near the continuum limit ($d=1/16$) for various frictional-parameter $b$ values. The other parameter values are $a=1, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$.

Figure 16

The $b$-dependence of the rate of the type I event near the continuum limit ($d=1/16$) for $a=1$ and 0.1. The other parameter values are $c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$.

Figure 17

The rupture-length dependence of (a) the mean slip amount and (b) the mean stress drop, which represents the continuum limit of Fig. 10. The frictional-parameter $b$ value is $b=5,7,10$, while the other parameters are $a=1, c=1000, v^{*}={10}^{-2}$, and $\nu ={10}^{-8}$.