Seismic cycle
Consistent with previous studies (e.g., Skarbek et al. 2012; Dublanchet et al. 2013; Luo and Ampuero 2017; Yabe and Ide 2017), we observed several different types of slip behavior in our parameter studies. The first is the “total seismic” regime, in which a mainshock event ruptures the entire seismogenic zone on the fault (parameter sets A-C in Fig. 2). Here, the slip behavior between mainshocks depends on the frictional parameter. In most cases, smaller stick-slip events occur, comprising seismic ruptures of one or more cells, but not all cells (parameter sets A and B in Fig. 2). The frequency of smaller events also varies with the frictional parameters. In some cases, seismic events do not occur between mainshocks except for slow slip events (parameter set C in Fig. 2). The second regime is the “partial seismic” regime, in which each cell shows stick-slip behavior but simultaneous slip does not occur across the entire seismogenic zone (parameter set D in Fig. 2). We also observe a “slow slip” regime where the entire seismogenic zone shows stick-slip behavior, though peak slip velocity does not reach seismic slip velocity, which is defined as 1 mm/s in this study (parameter set E in Fig. 2). The last regime is the “stable slip” regime, where stick-slip events are never initiated (parameter set F in Fig. 2).
We present the detailed slip behavior of frictionally heterogeneous faults on shorter timescales in Fig. 3. In the total seismic regime, slip velocity distributions on the fault are shown in a 20 s window around the mainshock (parameter sets A–C in Fig. 3). Slip velocity decelerates monotonically after the mainshock, whereas the preseismic behavior is less uniform. Although part of the fault is accelerated before the mainshock (i.e., the nucleation), its width depends on the frictional parameters. Furthermore, in Fig. 3 for parameter sets B and C, part of the fault accelerates to seismic slip velocity for a short period before the mainshock, which represents foreshocks. During the mainshock, the entire seismogenic zone simultaneously accelerates to seismic slip velocity. In the partial seismic regime (parameter set D in Fig. 3), individual cells are accelerated to seismic slip velocity but the accelerations themselves are not simultaneous; rather, we observe a migrating cell rupture with variable time delays between adjacent ruptures. In the slow slip regime (parameter set E in Fig. 3), cells are never accelerated to seismic slip velocity; instead, the rupture of the fault propagates slowly from the center of the seismogenic zone to the edge.
To assess the dependence of the four types of slip behavior on the frictional parameters, we measure the peak value of slip velocity Vave averaged across the entire seismogenic zone (Fig. 4). The total seismic regime has a high peak slip velocity (~ 0.1 m/s) because the entire seismogenic zone simultaneously reaches seismic slip velocity. The total seismic regime is observed when (b − a)σ in the VWZ (ξw) is large and (a − b)σ in the VSZ (ψs) is small. On the other hand, when both ξw and ψs are large, the partial seismic regime has a lower peak average-slip velocity (~ 1 mm/s) because only a small part of the seismogenic zone slips seismically at one time. When ξw is small and ψs is large, stick-slip events are never initiated (i.e., the fault is in a stable slip regime). The slow slip regime is observed in a narrow parameter space between the total seismic regime and the stable slip regime, with smaller ξw.
The transition in slip behavior from the total seismic regime to the partial seismic regime corresponds to the slip behavior transitions documented by Dublanchet et al. (2013) and Yabe and Ide (2017). The transition from the total seismic regime to the slow slip regime corresponds to the slip behavior transitions documented by Skarbek et al. (2012). This transition is controlled by the spatially averaged values of frictional parameters on an infinite fault subjected to constant external stress (Yabe and Ide 2017), though the conditions of the transition vary in the finite fault system or with increasing external stress (Skarbek et al. 2012; Dublanchet et al. 2013; Luo and Ampuero 2017; Yabe and Ide 2017). In this study, we conduct parameter studies only for the a value. However, other parameters, such as cell size and the ratio of VWZ size to VSZ size, also affect the changes in the conditions of the transition. Detailed parameter studies of these changes were conducted by Luo and Ampuero (2017). Hereafter, we focus on precursory slip behavior in the total seismic regime.
Precursory behavior
Comparing the precursory slips of three examples in the total seismic regime (parameter sets A–C in Fig. 3), there are wide varieties. In parameter set A, where ξw is large and ψs is small, precursory slip is negligible and occurs only in one cell (#1). On the other hand, in parameter sets B and C, which are closer to the stability boundary between the total seismic regime and other regimes, intense precursory aseismic and seismic slip occurs across a wide area of the fault. To quantify these variations, we define the precursory period and foreshocks below, then report the relevant results for each precursory slip behavior.
During the interseismic period, the precursory period (Fig. 5a) begins at the last time when the slip velocity averaged over the seismogenic zone exceeds the plate velocity before the mainshock. The end of the precursory period (or equivalently, the beginning of the mainshock) is defined as the last time at which the average slip velocity exceeds 1 mm/s before the peak averaged slip velocity during the mainshock. Foreshock events are defined as precursory seismic events, during which the maximum average slip velocity exceeds 1 mm/s (Fig. 5b).
The first measure of the activity level of the precursory slip is the amount of aseismic slip during the nucleation process. In the case of a frictionally homogeneous fault governed by a rate- and state-dependent friction law, fault slip velocity is expected to increase proportionally to the inverse of time remaining before the mainshock (Dieterich 1992). In the case of a heterogeneous fault, the accelerated aseismic slip is expected to drive foreshocks, and the occurrence of foreshocks perturbs this simple relationship. However, it still holds true that aseismic slip velocity outside of the foreshock period accelerates in proportion to the inverse of the time before the rupture (Noda et al. 2013). Therefore, the background aseismic slip velocity Vb could be expressed as Vb = D/tr, where D is a constant and tr is the time remaining before the mainshock. The constant D (hereafter called the nucleation level) is a proxy for the amount of aseismic slip during the nucleation process. The average slip velocity is plotted against tr as in Fig. 6a. To define the background aseismic slip velocity Vb, we need to define the slip velocity, which is not perturbed by the occurrence of foreshocks. For this purpose, we stacked the slip velocity evolutions of several precursory periods for mainshocks in Fig. 2 and measured the bottom 10th percentile value of the average slip velocity in each time bin, divided equally in log space, from 1 s before the mainshock to the beginning of the nucleation phase. Picked values of Vb were then fitted using the function shape of D/tr. This procedure was repeated for all parameter sets in the total seismic regime, and the results are summarized in Fig. 6b.
In Fig. 6a, the bottom envelopes of the slip velocity are roughly consistent with a slope of − 1, which supports our hypothesis that the background aseismic slip velocity and time to rupture are inversely proportional, even in a frictionally heterogeneous fault. The value of the nucleation level is very small for parameter set A, which indicates that the precursory slip behavior is negligible, as suggested by Fig. 3. For parameter sets B and C, the nucleation level is larger for C than for B, which indicates that aseismic slip during the nucleation is larger for C. This is also consistent with Fig. 3 because the width of the aseismic slip before the mainshock is much larger in parameter set C. The results for all parameter studies show that the nucleation level is higher around the stability boundary, though parameter sets with smaller ξw tend to have higher nucleation levels.
The other measurement of the activity of the precursory slip behavior is the amount of seismic slip during the precursory period. The seismic slip of foreshocks is driven by the background aseismic slip, which is quantified in Fig. 6. Such dynamic behavior is quantified by calculating the energy consumed by the radiation damping term, which mimics the energy lost by seismic wave radiation (Rice 1993). We calculate the following values using the slip velocity Vave and slip xave averaged across the seismogenic zone:
$$ {E}_{\mathrm{ave}}=\int \frac{\mu }{2\beta }{V}_{\mathrm{ave}}d{x}_{\mathrm{ave}}. $$
(5)
The cumulative energy during the precursory period is plotted against the time remaining before the mainshock in Fig. 7a. This represents the activity of seismic slip during the precursory slip period because the energy consumption due to the radiation damping term shows a greater increase when the slip velocity is higher. The cumulative energies at 1 s before the mainshock, averaged over several precursory periods, are plotted for all parameter sets in Fig. 7b.
Because foreshocks are driven by the background aseismic slip of the nucleation process, their seismic slip should be active around the stability boundary of the fault, where the background aseismic slip is most active in Fig. 6b. The cumulative energy is actually negligible for parameter set A, where the background aseismic slip is very small, but larger around the stability boundary of the fault where the background aseismic slip is also larger. Comparing parameter sets B and C, the former has larger values of cumulative energy, which indicates that seismic slip is more active during the precursory period of parameter set B. However, the nucleation level is higher in parameter set C (Fig. 6). This inconsistency between aseismic and seismic slip during the precursory period around the stability boundary of the fault should be related to the frictional parameters. The parameter set B has larger values of both ξw and ψs. A large value of ξw makes the nucleation size smaller and better facilitates seismic slip of individual VWZs. In contrast, parameter set C has smaller values of both ξw and ψs, which facilitates simultaneous slip across a larger region of the fault. This difference is also reflected in the amount of seismicity between mainshocks (Fig. 2), i.e., many smaller events occur between mainshocks for parameter set B, whereas no seismic events occur for parameter set C. These same tendencies should also exist in precursory slip.