Spatiotemporal correlations of the two-dimensional spring-block (Burridge-Knopoff) model of earthquakes are extensively studied by means of numerical computer simulations. The model is found to exhibit either ``subcritical'' or ``supercritical'' behavior, depending on the values of the model parameters. Transition between these regimes is either continuous or discontinuous. Seismic events in the ``subcritical'' regime and those in the ``supercritical'' regime at larger magnitudes exhibit universal scaling properties. In the ``supercritical'' regime, eminent spatiotemporal correlations, {\it e.g.}, remarkable growth of seismic activity preceding the mainshock, arise in earthquake occurrence, whereas such spatiotemporal correlations are significantly suppressed in the ``subcritical'' regime. Seismic activity is generically suppressed just before the mainshock in a close vicinity of the epicenter of the upcoming event while it remains to be active in the surroundings (the Mogi doughnut). It is also observed that, before and after the mainshock, the apparent $B$-value of the magnitude distribution decreases or increases in the ``supercritical'' or ``subcritical'' regimes, respectively. Such distinct precursory phenomena may open a way to the prediction of the upcoming large event.
An earthquake is a stick-slip dynamical instability of a pre-existing fault driven by the motion of a tectonic plate [Scholz, 2002;Scholz, 1998]. While an earthquake is a complex phenomenon, certain empirical laws such as the Gutenberg-Richter (GR) law and the Omori law concerning its statistical properties are known to hold. These are both power-laws without any characteristic energy or time scale. Such "critical" features of the statistical properties of earthquakes lead to the view that earthquake might be a phenomenon of "self-organized criticality (SOC)". This view is opposite to the other widely-spread view of "characteristic earthquakes" where an earthquake is regarded to possess its characteristic energy or time scale.
Modeling earthquakes and elucidating its statistical properties have been a fruitful strategy in earthquake studies. One of the popular model might be the so-called spring-block model originally proposed by Burridge and Knopoff (BK) [Burridge and Knopoff, 1967].
In this model, an earthquake fault is simulated by an assembly of blocks, each of which is connected via the elastic springs to the neighboring blocks and to the moving plate. All blocks are subject to the friction force, the source of the nonlinearity in the model, which eventually realizes an earthquake-like frictional instability. While the spring-block model is obviously a crude model to represent a real earthquake fault, its simplicity enables one to study its statistical properties with high precision.
Carlson, Langer and others [Carlson and Langer, 1989a;Carlson and Langer, 1989b;Carlson et al., 1991;Carlson, 1991a;Carlson, 1991b;Carlson et al., 1994] studied the statistical properties of the BK model quite extensively, paying particular attention to D R A F T November 5, 2018, 2:34pm D R A F T the magnitude distribution of earthquake events. Most of these simulations have been done for the simplest one-dimensional (1D) version of the model. It was then observed that, while smaller events persistently obeyed the GR law, i.e., staying critical or nearcritical, larger events exhibited a significant deviation from the GR law, being off-critical or “characteristic” [Carlson and Langer, 1989a;Carlson and Langer, 1989b;Carlson et al., 1991;Carlson, 1991a;Carlson, 1991b;Schmittbuhl et al., 1996]. The spring-block model has also been extended in several ways, e.g., taking account of the effect of the viscosity [Myers and Langer, 1993;Shaw, 1994;De and Ananthakrisna, 2004], modifying the form of the friction force [Myers and Langer, 1993;Shaw, 1995;Cartwright, 1997;De and Ananthakrisna, 2004], considering the long-range interactions between blocks [Xia et al., 2005[Xia et al., , 2007], driving the system only at one end of the system [Vieira, 1992], or by incorporating the rate-and state-dependent friction law [Ohmura and Kawamura, 2007].
In the previous paper, the present authors studied the statistical properties of the 1D BK model, focusing on its spatiotemporal correlations [Mori and Kawamura, 2005;2006].
This study has revealed several interesting features of the 1D BK model. For example, preceding the mainshock, the frequency of smaller events is gradually enhanced, whereas, just before the mainshock, it is suppressed in a close vicinity of the epicenter of the upcoming event (the Mogi doughnut). The time scale of the onset of the doughnut-like quiescence depends on the extent of the frictional instability. Furthermore, the apparent B-value of the magnitude distribution increases significantly preceding the mainshock under certain conditions.
It should be remembered, however, that real earthquake faults are 2D rather than 1D.
Hence, it is clearly desirable to study the 2D version of the model in order to further clarify D R A F T November 5, 2018, 2:34pm D R A F T the statistical properties of earthquakes. In the present paper, we study spatiotemporal correlations of the 2D BK model. The 2D model studied here is intended to represent a 2D fault plane itself, where the direction orthogonal to the fault plane is regarded to be rigid and not considered explicitly in the model [Carlson, 1991b]. The other possible version of the 2D model is the one where the second direction of the model is taken to be orthogonal to the fault plane [Myers et al., 1996].
Previous calculations on the 2D BK model were performed with main interest in their magnitude distribution, while there were very few systematic studies of its spatiotemporal correlations. In the present paper, we wish to fill this gap by investigating the spatiotemporal correlations of the 2D BK model, extending our previous study on the 1D BK model [Mori and Kawamura, 2005;2006]
Our model is the 2D version of the spring-block BK model, representing a “fault plane”, which is taken to be the xz plane. The plane consists of a 2D square array of blocks containing N x blocks in the x-direction and N z blocks in the z-direction. All Blocks are assumed to move only in the x
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