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Real earthquake fault systems are not composed of identical, homogenous materials. The variety of materials with different physical properties can cause different spatial and temporal behaviors. In order to study a system with particular aspects of spatial heterogeneities, an earthquake fault model based on Olami-Feder-Christensen (OFC) and Rundle-Jackson-Brown (RJB) cellular automata models is created. In this model, some percentage of stronger sites called 'asperity cells' are added to the system, which are significantly stronger than surrounding lattice sites but do rupture when the applied stress reaches their greater threshold stress. We found some periods of activity which start with the gradually increasing number of bigger events (foreshocks) and ends with a tail of decreasing activity (aftershocks). This recurring activity produces characteristic-type events with apparent accelerating seismic moment release and foreshocks, where the ratio of the number of foreshocks to aftershocks is dependent on a limited number of internal parameters.
Understanding the dynamics of seismic activity is a fundamental key for investigation of the earthquake process. To avoid all the complications in the dynamics of earthquakes, some simple models of statistical fracture have been used to test some of the typical assumptions and effective parameters and their possible outcomes (Burridge and Knopoff, 1967; Otsuka, 1972; Rundle and Jackson, 1977; Rundle, 1988; Carlson and Langer, 1989; Nakanishi, 1990, Rundle and Brown, 1991; Olami et al., 1992; Alava et al. 2006). Most of these models assumed a spatially homogeneous earthquake fault and since inhomogeneity plays an important role in the seismicity of an earthquake fault, the following step of modeling can be the expansion of the regular homogeneous models to inhomogeneous models where some parameters might vary from site to site. There are several previous studies on the inhomogeneous OFC model (Janosi and Kertesz, 1993; Torvund and Froyland, 1995; Ceva, 1995; Mousseau, 1996; Ramos et al. 2006; Bach et al., 2008; Jagla, 2010). These models considered different ways to impose inhomogeneity in system and most of them considered a system with short stress transfer range.
In this work, the focus is on the spatial heterogeneity in the system with long stress transfer range. The presented model is a cellular automaton version of earthquake faults based on the Olami-Feder-Christensen (Olami et al., 1992) and Rundle-Jackson-Brown (Rundle and Jackson, 1977; Rundle and Brown, 1991) models with some small variations. Here, inhomogeneities are imposed in the model by converting a percentage of randomly selected sites to stronger sites that can support much higher stress. We found some periods of activity which start with the gradually increasing number of bigger events (foreshocks) and ends with a tail of decreasing activity (aftershocks). This period begins with the failure of the first asperity cell, which propagates a higher amount of stress to the entire system and causes the failure of the rest of asperity cells, triggering larger events. Recurring activity produces characteristic-type events with apparent accelerating seismic moment release and foreshocks, where the ratio of the number of foreshocks to aftershocks is dependent on a limited number of internal parameters.
Our model is a two-dimensional cellular automaton model with periodic boundary condition. In this model every site in the lattice is connected to z number of neighbors which are defined as sites within a certain distance or stress interaction range, R. A homogeneous residual stress Ïƒr is also assigned to all the sites in the lattice. To impose spatial inhomogeneity into the lattice, two sets of failure thresholds are introduced; 'regular sites' with a constant failure threshold of ÏƒF and 'asperity sites' with a much higher failure threshold (ÏƒF(asperity)=ÏƒF+Î”ÏƒF) in order to initiate some stronger sites which are able to bear higher stress before failure. These asperity sites do not break as fast as the regular sites, but they accumulate stress and fail with higher amount of stress.
At the start, an internal stress variable, Ïƒj(t), as a function of time is randomly distributed to each site in a way that the stress on all sites sets between the residual stress and failure stress thresholds (Ïƒr<Ïƒi(t=0)<ÏƒF). Given the initializing procedure, it is clear that at t=0 no sites will have Ïƒi > ÏƒF and hence the procedure for inducing failures must start. There are several ways to do this but we used the so-called zero velocity limit (Olami et al., 1992). In this procedure the entire lattice is searched for the site that minimizes (ÏƒF - Ïƒi) and that equal amount of stress adds to each site such that the stress on at least one site is now equal to its failure threshold. Therefore one site fails and some fraction of its stress given by Î± [ÏƒF-(ÏƒrÂ±Î·)] is dissipated from the system, where Î± is a dissipation parameter (0<Î±â‰¤1) which describe the portion of stress dissipated from the failed site and Î· is a randomly distributed noise. The failed site's stress is lowered to (ÏƒrÂ±Î·) and the remaining stress is distributed to its predefined neighbors. After the first site failure, all neighbors are searched to see if the stress added to the neighbors of the failed site caused any of them to fail. If so, the described procedure repeats for those neighbors and if not, the time step (known as plate update) increases by unity and the lattice is searched again for next site which minimizes (ÏƒF - Ïƒi). The size of the event is considered as the number of failures that expand from the first failed site.
Foreshock and Aftershocks
Considering the seismicity as an indicator of the dynamics of earthquakes provides some information about the spatial and temporal nature of earthquakes. For instance, aftershocks are one of the most noticeable patterns of clustering in the catalog of earthquakes. Aftershocks usually occur close to their triggering main shock and their rate start to decay with time thereafter. Some features of aftershock activities can lead to the theory that the likelihood of earthquake occurrence at any specified time and place is related to the seismic history of the nearby region. The famous ETAS (Epidemic Type Aftershock-Sequences) model (Ogata, 1999 and Helmstetter and Sornette, 2002a), in which every event, regardless of its size, increases the probability of upcoming events is created based on this concept. In this perspective, it is not necessary for the aftershocks to be smaller than their triggering events. In fact, in some cases the largest event is triggered by some earlier smaller events, which can be called as foreshocks. In recent years by improving the seismicity data especially for the smaller earthquakes and based on the temporal and spatial clustering behavior in seismicity data, tons of research has been done with the intention of the relatively smaller events prior to the larger earthquakes (Kanamori, 1981; Bakun et al., 1986; Haberman, 1981; Swan et al., 1980; Mogi, 1969; Yamashita and Knopoff, 1989; Dodge et al., 1996; Eneva and Ben-Zion, 1997; Press and Allen, 1995; Nanjo et al., 1998; Brodsky, 2006; Stein, 1999; Rundle, 1989; Bowman et al., 1998; Smyth et al., 2011).
Accelerating Moment Release (AMR)
In 1989, Mogi observed a regional increase of seismicity before great earthquakes. He found an increase in the overall level of seismicity in the crust surrounding the future rupture zone, together with a relative shortage of events along or near the fault. Ellsworth et al (1981) also noticed an increase in the rate of M5 events over a broad region in the years preceding the 1906 San Francisco earthquake. This precursory regional increase in the seismicity (AMR) has been documented in a variety of papers (Bowman and King, 2001, Bowman et al, 1998, Sornette and Sammis, 1995, Bufe and Varnes, 1993, and Skyes and Jaumé, 1990). It is defied by the equation
Îµ(t)=A-B(tf - t)m,
where Îµ(t) has been interpreted as the accumulated seismic moment, the energy release or the Benioff strain release within a specified region, from some origin time t0, up to time t.
Where N(t) is the number of events in the region between t0 and t and q=0, 1/2, 1. The remaining quantities, A, B, tf and m, are parameters characterizing the seismic episode under study. In 2002, Ben-Zion & Lyakhovsky analyzed the deformation preceding large earthquakes and obtained a 1-D analytical power-law time-to-failure relation for accelerating moment release before big events. They found that phases of accelerating moment release exist when the seismicity occurring immediately before a large event has broad frequency-size statistics. These and similar results of Turcotte et al. (2003) and Zoller et al. (2006) are consistent with observed seismic activation before some large earthquakes.
Results and discussions
A system with 1% of randomly spatially distributed asperity sites in a two dimensional lattice of linear size L=256 with periodic boundary conditions is studied. We considered a homogeneous failure threshold for the regular sites as ÏƒF=2.0, homogeneous residual stress for the entire lattice as Ïƒr=1.0, temporal random distribution of noise as Î·=[-0.1,+0.1]. The failure threshold for asperity sites is also considered as ÏƒF(asperity)=ÏƒF+10.
We let the system run with the above initial conditions and waited long enough to make sure the system passes its preliminary transient state to collect the statistics. We investigated the difference between our inhomeneous model and a regular homogeneous model with no asperity sites by looking at time series and also the distribution of avalanche sizes. Figure 1 shows time series (6*105 plate update steps) and distribution of avalanches (collected during 107 plate update steps) for three different values of stress dissipation parameters Î±. In this figure, first diagram (i) in each set is the time series of the avalanches for the heterogeneous model with 1% of asperity sites. The shaded areas are the time steps in which an asperity sites breaks; second diagram (ii) is the time series of avalanches for the homogeneous model with no asperity sites; Figure 1-d is the comparison between the distribution of avalanches for different values of Î± with and without asperity sites. By looking at the time series we can clearly see despite the random spatial distribution of asperity sites, they do not tend to break randomly in the time domain. The distribution of the avalanches also confirms that in the models with higher stress dissipation, the system is unable to produce big avalanches. In addition, the end tail of avalanche distribution diagram shows that the model with 1% of asperity sites generates bigger events compared to the homogeneous model.
Since in the dynamics of our model there is at least one broken site every time steps, we bin time into coarse-grained units of Î”t=500 tp.u. and count the number event bigger than a predefined threshold in each bin. Figures 2-a and 2-b show one of the activation periods from the cases where Î±=0.2 and Î±=0.4 respectively. In the first figure in each set (i) we can see the time series of avalanches and also the time steps where an asperity breaks. The second figure of each set (ii) shows the distance of each event from the biggest event in the series (main shock). In the third figure (iii) we can see the number of events at each coarse-grained time bins. Forth figure (iv) is the distribution of avalanches during the selected time period and the fifth one (v) is the accumulated number of event bigger than the chosen threshold versus coarse-grained time. Based on these observations we can clearly see time clustering of events which start with the gradually increasing number of bigger events (foreshocks) and ends with a tail of decreasing activity (aftershocks). The increasing number of large event prior to the mail shock in figures 2-a-v and 2-b-v also seem similar to Accelerating Moment Release (AMR) behavior before big events. We also investigate the length of the foreshock and aftershock activation for different stress dissipations in our model and our results show that the average rate of aftershocks is higher in areas with higher stress dissipation (Fig 3-a&b). The total activity period also seems to be related to the stress dissipation parameter. Figure 3-c shows the model with higher stress dissipation parameters tend to have a longer avtivity period.
By adding some stronger asperity sites to the lattice, we find periods of activity which start with the gradually increasing number of bigger events or foreshocks and ends with a tail of decreasing activity or aftershocks (Figure 2) . This increasing number of large events prior to the main shock seem similar to Accelerating Moment Release (AMR) behavior before big events (Figures 2 -a-v and 2-b-v) .
Our results also suggest that the length of the foreshock and aftershock activation is related to one or more controlling parameters of the model, including the stress dissipation (Figure 3).