Rupture processes in simulated earthquakes
The simulations showed rich behaviors depending on the parameters α and β (Figure 2). Noda et al. (2013a) classified the earthquakes that appeared in their simulations. Here, we use the same classification and discuss characteristics of the coseismic moment rate and moment acceleration functions for each class. Large (L) events are initiated by large nucleation (LL), by dynamic cascade-up rupture growth (cL), or by delayed cascade-up (dcL), which directly follows a preceding precursory small (pS) event. Sometimes, an LL event takes place after a pS event that has promoted the emergence of large nucleation. In addition, we obtained non-precursory small events (nS) whose afterslip decayed to the interseismic level before the occurrence of the next large earthquake. In the following subsections, we briefly review characteristics of those dynamic ruptures, which were previously reported by Noda et al. (2013a), before discussing coseismic source-time functions. In the next section, we shall discuss the effect of α and what controls the coseismic characteristics.
In Noda et al. (2013a), the earthquakes were defined by a threshold of the spatially maximum slip rate equal to 0.1 m/s. In the present paper, we used the same criterion for detecting the events so that we could use the same event catalog, but plots of moment rate and moment acceleration functions were shifted in time so that time was zero when the moment acceleration became 10 PNm/s2 (=1016 Nm/s2) at the beginning of events in order to ease comparison. The examples shown are, unless otherwise noted, from a single simulation of a sequence of earthquakes with α = β = 3 in which all the defined earthquake types appeared before the twentieth event spanning Patch L.
Non-precursory small earthquake nS
After a large earthquake, which spans Patch L, it locks, and later, a creeping motion invades into it from the rim. A small nucleation within Patch S takes place after the creep front reaches it (Figure 3a).
For α ≥ β, it is often the case that a small nucleation in Patch S (Figure 3a) initiates a rupture that spans Patch S (Figure 3b,c,d) and fails to grow further dynamically (Figure 3e). This is a small earthquake (S event) caused by rupturing of Patch S. If the afterslip of an S event settles down to the interseismic level safely, it is classified as a non-precursory S event (nS). An S event may be shortly followed by a large event before the afterslip settles down to the interseismic level. Such an S event is classified as a precursory S (pS) event and discussed later. The preseismic moment release rate follows that for acceleration of the small nucleation dictated solely by the local frictional properties (Noda et al. 2013a), which might be the case only for the aging law (2013b).
Figure 4a,b shows the moment rate and moment acceleration functions, respectively, for nS events calculated by spatial integration of the slip rate within Patch L. An nS event typically yields a rather simple single-peaked moment rate function and single-wiggled moment acceleration function. Note that heterogeneities in frictional properties at further smaller length scales are not considered in our simulations, and hence potential cascade-up from them, which may cause complexity in the initiation of earthquakes as discussed later, is not included in the results. Some of the nS events are larger than the typical ones, and these have a minor second peak in the moment rate function and unusually large amplitude of the afterslip. Those nS events managed to propagate a little outside Patch S, but did not cascade-up or prompt a following large earthquake.
Large earthquakes initiated by large nucleation LL
For α ≥ β, large earthquakes that span Patch L (L events) may be initiated by large nucleation (LL) (Figure 5). The preseismic moment release rate follows that for acceleration of the large nucleation. When the rupture front of an LL-event sweeps Patch S, it accelerates and decelerates because of the locally small fracture energy (Figure 5c,d), and it radiates a high-frequency wave ahead of the rupture front (Figure 5d,e,f, waves propagating to the upper-right direction).
LL events yield single-peaked moment rate functions (Figure 6), similarly to nS events. If we compare LL events and nS events when the moment acceleration reaches a threshold of 10 PNm/s (i.e., zero in the horizontal axis in Figure 6), LL events typically show larger moment rates and milder changes in the moment acceleration than nS events. If this happens in natural earthquakes, larger earthquakes will have a longer nucleation phase as Shibazaki and Matsu'ura (1998) argued. It should be noted that the absolute value of the moment acceleration is maximum for the stopping phase, which is more significant than that for nS events and causes non-symmetric shapes in the moment rate functions. The Patch L is the largest scale simulated, and hence is surrounded by a region of increased a-b, which acts as a stronger barrier against dynamic rupture propagation, compared with the region surrounding Patch S where only L is increased while a-b is kept constant across the boundary (See Figure 1).
There was one exception indicated by a star in Figure 6 and duplicated in Figure 7a,b in which the onset of the moment rate and moment acceleration functions were similar to those for nS events despite the fact that a large nucleation was present. This type of earthquake is initiated by interaction of ongoing large nucleation and Patch S (Figure 7c, also see Figure A5 in Noda et al. (2013a)). The dynamic rupture begins with sweeping Patch S and the large nucleation at the same time, which yields the first peak in the moment acceleration (Figure 7d,e). Then, the second peak in the moment acceleration appears as the rupture spans Patch L (Figure 7e,f,g), and this is followed by a stopping phase (Figure 7h). This example shows that an observation of short nucleation phase does not necessarily preclude operation of large nucleation. The chance of interaction between nucleation and locally fragile regions should be larger in a system with a larger number of Patches S. Investigation of such a system deserves further study.
Large earthquakes initiated by cascade-up rupture growth cL
Unless α > > β, a rupture spanning Patch S initiated by a small nucleation (Figure 8a,b,c) may dynamically expand outwards (Figure 8d,e) ending up with a large earthquake (Figure 8f,g,h). This is a cascade-up L (cL) event. If α ≥ β, a rupture spanning Patch S succeeds in cascading up only when the Patch L is ready, and otherwise, it fails; both S events and cL events are of common occurrence in our simulations (Figure 2). Notice the difference in the overall background slip rate between Figure 3a and Figure 8a.
cL events are initiated from small nucleation, and thus the initial part of the moment rate and moment acceleration functions are similar to those for nS events (Figure 9). The amount of released moment during the initial small rupture spanning Patch S is, however, larger than that in the typical nS event. During the main rupture spanning Patch L, it is sometimes the case that the moment acceleration increases with some complexity (for example, c to e in Figure 9b) because of heterogeneous rupture expansion (Figure 8c,d,e). The total rupture duration is longer than LL events on average, and there is larger variation.
If α < β, the system falls into the one limit cycle having only one cL event; eventually all the earthquakes are cL events except just after the initiation of the simulations. The example shown in Figure 10 is for α = 2.5 and β = 3. In this case, Patch L is ruptured by a strong perturbation (i.e., rupture of Patch S) before it ripens so that a large nucleation would occur shortly thereafter; the recurrence interval of large earthquakes (12.8 ± 3.3 years) is significantly shorter than the case without Patch S (17.5 ± 3.2 years) (Noda et al. 2013a). The rupture in such a cL event propagates unwillingly. In the example shown, the small rupture spanning Patch S (Figure 10a,b,c,d) is followed by bilateral (clockwise and counterclockwise) rupture propagation guided by the creeping region near the rim of Patch L (Figure 10e,f). The counterclockwise rupture stops, but the clockwise rupture front manages to propagate (Figure 10f,g), which supplies enough energy release rate for the counterclockwise rupture front to resume (Figure 10h). Those two rupture fronts finally meet at a point on the rim of Patch L, but not on the opposite side from the rupture initiation because of the non-symmetric rupture process (Figure 10i,j,k). Kato (2004) suggested that by conducting quasi-dynamic simulations, such a concentration of rupture fronts may cause radiation of a seismic signal.
Figure 11 represents the moment and moment rate functions for the cL events where α = 2.5 and β = 3. Note that the event with the largest moment rate is the first event that has an almost uniform preseismic shear stress and state variable. This event is not realistic in a sense that its preseismic condition is largely set by the artificially imposed initial conditions of the simulation. Later events, whose preseismic conditions are spontaneously set by interseismic processes, are dissimilar to the first one, and their source-time functions approach those of an event in a limit cycle, which we refer to as a characteristic behavior. Compared to the case with a larger scale ratio (Figure 9), rupturing of Patch S (Figure 10a,b,c,d) is not clearly reflected in the source-time functions. The rupture propagation guided by the creeping rim of Patch L (Figure 10e,f,g) is similar to long strike-slip ruptures in terms of the high aspect ratio except that the propagation direction rotates. In such a rupture, the width of the seismogenic zone gives the compliance of the system (Kanamori and Anderson 1975; Romanowicz 1992), and rupture propagation at a constant speed would cause a constant moment rate. Indeed, the moment acceleration in this stage (e to g in Figure 11b) is not significant. The moment acceleration takes its maximum value not during two-dimensional expansion of the rupture, but near the end of the event when the rupture fronts concentrate on the perimeter of Patch L.
The characteristics of moment rate and moment acceleration functions for cL events vary with the scale ratio as discussed later.
Delayed cascade-up sequence pS-dcL
An S event is sometimes followed shortly by an L event before the afterslip settles down to the interseismic level. Those S events were classified as precursory S (pS) events.
An arrested small rupture expands quasistatically (Figure 12a,b,c,d), and the following L event may be initiated by acceleration inside it including a part of the boundary of Patch S (Figure 12d,e). It expands by re-advancing the arrested rupture front (Figure 12e,f), and it spans Patch L (Figure 12g,h,i). Such a sequence of earthquakes can be regarded as delayed cascade-up.
Figure 13a,b represents the moment rate and moment acceleration functions during the pS-dcL sequence, respectively. pS events tend to have larger seismic moment and afterslip than nS events (Noda et al. 2013a). The onset of the dcL event is characterized by milder moment acceleration similarly to an LL event. As reported by Noda et al. (2013a), the preseismic acceleration before a dcL event is similar to what is expected for a large nucleation if the time between a pS event and the following dcL event is long enough. As shown in Figure 13d,e, the acceleration before the dcL event takes place in a region that is somewhat smaller than the large nucleation, but definitely larger than the small nucleation. It seems then that Patch S just after an arrested pS event does not work as an unstable patch, and small nucleation does not occur in it.
Large nucleation during an afterslip pS-LL
An afterslip of an S event propagates through being guided by the creeping region inside Patch L (Figure 14a,b,c,d,e,f,g). If there is a large-enough creeping region to accommodate the large nucleation, then the afterslip may accelerate there and host a large nucleation that leads to an L event (Figure 14h,i,j,k,l). Similar to the pS-dcL sequence, the background slip rate before the small nucleation tends to be larger than nS events (Noda et al. 2013a) (compare Figures 3a and 14a). Additionally, pS-events before LL events tend to be larger than nS events (compare Figures 3e and 14e), similarly to pS events before dcL events. Figure 15a,b shows the moment rate and acceleration functions, respectively. Compared with a typical nS event that ruptures the same Patch S, the moment acceleration is quite complex while the rupture tries to cascade-up (unsuccessfully) right after having spanned Patch S.
There is a clear causality we can recognize between pS events and the following LL events if we can see the spatio-temporal distribution of the slip rate. Noda et al. (2013a) suggested that unusually large afterslip and its acceleration would provide hints for detecting this type of earthquake sequence, although a clear threshold in the bulk afterslip (moment rate) amplitude does not exist.
Brittleness as a threshold in the scale ratio
Cascade-up rupture growth is usually considered as a series of discrete step-by-step processes in which we observe ‘a small wave from a small rupture followed by a large wave from a large rupture’ (e.g., Ellsworth and Beroza 1995; Ide and Aochi 2005). Figure 16 shows the moment rate and acceleration functions of cL events for different α from 2 to 4 together with a typical LL event (the 11th event with α = 3 shown in Figure 6) in gray dashed lines for reference. Our parameter study about the scale ratio α has demonstrated that the chance for a large earthquake to show such stereotypical behavior is high under a condition where the scale ratio α is comparable to the brittleness β. Too small α causes almost all the L events to be initiated by cascade-up, but the signal from Patch S is difficult for us to recognize as a discrete small preceding wave. For example, in the case with α = 2 in Figure 16, the height of the positive peak in the moment acceleration is larger for the rupture of Patch S than for the rupture of the rest. A continuous distribution of L that increases proportionally with the distance from the center of Patch L can be regarded as a limiting case of α → 1 from above. Such is not ‘hierarchical’ any more, but it still is consistent with the self-similar rupture growth with a constant rupture speed. As α increases, the coseismic behaviors of the cL events become apparently more consistent with the stereotypical behavior and a peak in the moment rate for the rupture of Patch S becomes more distinguishable. Too large of an α will cause most of the large earthquakes to be initiated by their own nucleation; we no longer obtain a cL event with α = 5 at least within our simulated history worth of 20 cycles of L events (Noda et al. 2013a) (Figure 2).
With increasing α, Patch S starts having its characteristic earthquakes from around α ~ β (Figure 2b). This condition is similar to that for the small wave preceding cL events to become distinguishable from the large main one. The scale ratio α represents a factor by which the fracture energy increases when a rupture cascades-up. It affects whether a rupture spanning Patch S successfully cascades-up or not, and if successful, how much the rupture speed and thus moment release decelerate through the equation of motion of the rupture front (Freund 1990). The nucleation size in the simulated cases can be estimated from the condition for the energy release rate of the expansion of the nucleus being equal to the fracture energy. Let us consider an ideal case where α is slightly smaller than β and the heterogeneous preseismic conditions are neglected. In such a case, a rupture nucleated in Patch S spans it, decelerates to near-zero rupture speed at the rim of Patch S, and again spontaneously grows outwards. If Patch S has so small a scale ratio that it does not have its characteristic earthquake and a rupture spanning, it always cascades-up, then it is not working as a unit of rupture and would be better considered as an internal structure of Patch L rather than a discrete patch. The brittleness β may give a threshold in the scale ratio α in order for a fragile inclusion to work as a discrete patch in a larger patch.
The location of the Patch S is another important factor. Even if α > β, the Patches S which are far enough from the rim of Patch L cannot interact with the invading creep front, and hence, a small nucleation is not produced in them. We tried a preliminary simulation (data not shown) with Patch S at the center of Patch L. During the rupture of Patch L, local acceleration of the rupture front occurred at Patch S, but like in the LL event plotted in Figure 6, this behavior was not clearly recognized in the source-time function, though seismic analyses with spatial resolution, such as back projection or waveform inversion, may be able to see it.
It should be noted that the range of the scale ratio we have simulated may not be wide enough to extrapolate the abovementioned observation of the simulated results to natural faults. What if there are many tiny fragile patches spaced densely enough so that a large nucleation would necessarily include or interact with them? Such a simulation is numerically very expensive and has not been done yet, but we speculate that an ongoing large nucleation may host many tiny earthquakes that would look like clustered foreshocks (Ohnaka 1993; Shibazaki and Matsu'ura 1995; McLaskey and Kilgore 2013). Those fragile patches inside a large nucleation may also act as sources of the small preceding waves within the nucleation phase of the main rupture. Although the resulting seismogram may have a similarity to the cL events with large α demonstrated in the present study, a large nucleation (or preslip) would be present there and produce the small ruptures as a side-effect. Such behavior is not regarded as the cascade-up (Dodge et al. 1996).