Coseismic visibility of a small fragile patch involved in the rupture of a large patch - implications from fully dynamic rate-state earthquake sequence simulations producing variable manners of earthquake initiation
© Noda et al.; licensee Springer. 2014
Received: 8 January 2014
Accepted: 15 April 2014
Published: 12 May 2014
Frictional properties on a fault cannot be uniform, and one idealization would be a hierarchical asperity concept in which a large, tough patch (Patch L, radius R L and nucleation size R L c ) includes small, fragile patches (Patches S, radius R S and nucleation size R S c ), which is consistent with the scale-independent properties of earthquakes. At the onset of large earthquakes, a minor but quick signal from an immediately preceding rupture is sometimes observed during the so-called slow nucleation phase before the moment acceleration starts increasing linearly with time. Understanding what causes such seismic characteristics is important in assessing heterogeneity on a fault. In earthquake sequence simulations with hierarchical distribution of the state-evolution distance, large earthquakes spanning Patch L may be initiated by cascade-up rupture growth from Patch S, by their own large nucleation, and by delayed cascade-up, with their occurrence ratio depending on parameters characterizing the distribution (e.g., ‘scale ratio’ α = R L /R S and ‘brittleness’ β = R L /R L c = R S /R S c ). In the present paper, we compared the coseismic moment rate and acceleration functions between different types of ruptures and between different values of α. The events that started from small nucleation showed quick onset in these functions compared with those from large nucleation. In a cascade-up large earthquake, a small wave from a small rupture spanning Patch S preceded the main wave from the main rupture if α was larger than or comparable to β. This condition is similar to that for the appearance of small events in the simulated history that are nucleated in Patch S and fail to cascade-up. If α > > β, we no longer have cascade-up large events. That is, Patch S behaves as a unit of rupture for α > > β while it merely serves as internal inhomogeneity of Patch L rupture for α < < β. The transition occurs gradually with α over the intermediate range α ~ β.
KeywordsEarthquake Rate-state friction Hierarchical asperity
A nucleation phase in the seismogram is defined as a signal before the recorded ground velocity starts increasing linearly with time. Such a linear acceleration is a characteristic of self-similar rupture expansion in a three-dimensional problem. As summarized by Yoshida and Kato (2005), the nucleation phase may contain clues for understanding the fault behavior in the preseismic preparation stage. Systematic dependence on the earthquake size has been sought in the initial part of seismograms, but the results are mixed. We address them below in the context of multi-scale heterogeneity on which the present simulations are based.
A large earthquake may start from a large nucleation that is consistent with the large fracture energy observed (Venkataraman and Kanamori 2004; Kato 2012). If nucleation processes of earthquakes of different sizes can be scaled in the same manner as the coseismic processes, then a larger earthquake should have a longer nucleation phase. Some seismological studies have yielded consistent results (e.g., Iio 1992; Ellsworth and Beroza 1995; Beroza and Ellsworth 1996). Other studies (e.g., Ishihara et al. 1992; Nakatani et al. 2000) have reported that larger earthquakes show larger slopes in the nucleation phase, which indicates that the space-time scaling in the nucleation phase is different from that of the main rupture. Yet other studies have reported that the shape and amplitude of the nucleation phase do not depend on the size of the earthquakes (e.g., Mori and Kanamori 1996). Those apparently contradicting results for different datasets may represent variety in the earthquake generation processes, which might depend on geological settings, macroscopic loading processes, or some other factors.
It is occasionally reported that a minor burst in the velocity seismogram is observed during the nucleation phase (e.g., Ellsworth and Beroza 1995; Beroza and Ellsworth 1996). What kind of fault heterogeneity causes such seismic characteristics? Under what circumstances does such a minor signal appear before the main one? These are important questions aimed at understanding the potential scale-dependency of the fault friction and assessing the possibility of earthquake prediction by detecting precursory deformation.
In the hierarchical asperity concept (Ide and Aochi 2005), a larger and tougher patch (Patch L) has smaller and more fragile patches (Patches S) in it, and such a structure exists in many length scales. An assumption that ‘brittleness’ of the patch (i.e., size of the patches R relative to their nucleation size R c , which is proportional to the fracture energy G c for constant stress drop) is independent of the patch size leads to self-similar characteristics of earthquakes including, for example, typical rupture speeds and radiation efficiencies independent of the earthquake size (e.g., Ide and Beroza 2001; Venkataraman and Kanamori 2004). The minor burst in the velocity seismogram in the nucleation phase may correspond to the small rupture before cascade-up growth, and this has been reproduced by numerical simulations of dynamic rupture growth assuming conformance with the slip-weakening model (Ide and Aochi 2005) and the slip-strengthening-weakening model (Shibazaki and Matsu'ura 1995) with specific distributions of the parameters.
The following interpretation, which may be obvious in a single dynamic rupture propagation with a more-or-less uniform initial condition, has also turned out to be reasonable in earthquake sequences where complicated distributions of initial stress and fracture energy for earthquakes are set spontaneously by preceding earthquake cycles. A patch with larger brittleness β (i.e., smaller fracture energy) can rupture more easily by cascade-up from a smaller scale. Alternatively, if the fracture energy is different by a larger factor α between adjacent scales, then it is more difficult for a small rupture to cascade-up. The competition of α and β may play an important role in determining long-term system behavior including statistics of earthquake size and manner of initiation.
The previous paper by Noda et al. (2013a) mainly reported on the abovementioned overall system characteristics and preseismic behaviors that are different between different types of earthquakes. Note that inertial effects are fully accounted for in their earthquake sequence simulations (Lapusta and Liu 2009; Noda and Lapusta 2010), and the calculated coseismic behavior is realistic in a sense that it is consistent with elastodyanamics. In the present paper, we report on the coseismic behavior of those simulations. Especially, the focus of this paper is on improving our understanding of under what circumstances do small ruptures cause minor seismic signals preceding the main one (i.e., during the nucleation phase).
where τ is the magnitude of the shear traction, f is the friction coefficient, σ is the normal stress, f 0 is the steady-state friction coefficient at a reference slip rate V 0, V is the slip rate, a and b are non-dimensional parameters representing the amounts of the direct and evolution effect, respectively, θ is the state variable, and L is the state-evolution distance. The parameters used are described in Figure 1. In the simulations discussed here, σ = 100 MPa, f 0 = 0.7, V 0 = 8 cm/3.1536 × 107 s, a = 0.016, and b is 0.012 outside the seismogenic patches and 0.02 in them. Note that a/b = 0.8 in the seismogenic patches. A planar fault was embedded in a linearly elastic infinite medium with periodic boundaries (Figure 1). The elastodynamics (the shear modulus μ = 30 GPa, the Poisson's ratio ν = 1/4, and the shear wave speed c s = 3 km/s) was treated by a spectral boundary integral method (e.g., Lapusta and Liu 2009; Noda and Lapusta 2010).
The nucleation size is proportional to L. Since the brittleness is assumed to be the same for Patches L and S, the values of L inside those patches are proportional to the patch radii. The nucleation size which is three times smaller than the patch size in length enables initiation of earthquakes by compact acceleration, the size of which is approximately R c (nucleation), rather than by coalescence of creeping fronts (Chen and Lapusta 2009; Kaneko and Ampuero 2011).
The fault is loaded by prescribing a long-term slip rate V pl = V 0 near the periodic boundary. The simulations were conducted until 20 large earthquakes occurred, which were approximately Mw 6 for the Patch L radius R L = 4 km.
As already mentioned by Noda et al. (2013a), the scaling of the problem is relatively straightforward. With keeping the non-dimensional solution and the scales of speed and stress, an increase in the length scale (radii of the patches and L) by some factor causes an increase in the time scale by the same factor. That is, if the simulated large earthquakes are regarded as approximately Mw 8 by saying R L = 40 km instead of 4 km, then the time quantities (e.g., time between earthquakes and time from initiation of a dynamic rupture) must be multiplied by a factor of 10. We selected the length scale R L = 4 km for presentation purposes.
We would like to emphasize that modeling of successive earthquake cycles in sequence is essentially important in studying variations in the coseismic behavior. Distribution of shear stress τ and state θ at the initiation of a dynamic rupture could be regarded as the initial conditions for a dynamic rupture. If these are chosen (or tuned) arbitrarily, one may be able to ‘reproduce’ a variety of earthquakes, but it is difficult to judge what kind of scenarios are more likely to be the case than others.
Results and discussion
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
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).
Large earthquakes initiated by large nucleation LL
Large earthquakes initiated by cascade-up rupture growth cL
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.
Large nucleation during an afterslip pS-LL
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
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).
Noda et al. (2013a) conducted earthquake sequence simulations with a rate- and state-dependent friction law and modeled a large, tough patch (Patch L) with a small, fragile patch (Patch S) in it in order to realize the hierarchical asperity concept and to investigate the elementary process involving the interaction between the scales. They reported on the characteristics of the overall system behavior as well as the preseismic processes for different scale ratios (ratio between the radii of the patches). In the present paper, the coseismic behavior from the same simulations was reported. Note that the inertial effects were fully accounted for in the simulations (Lapusta and Liu 2009; Noda and Lapusta 2010).
Earthquakes initiated by small nucleation in Patch S showed a quicker increase in the moment rate and acceleration at the onset of seismograms than those by a large nucleation in Patch L. If the nucleation size of Patch L is comparable to the size of Patch S, then large earthquakes spanning the same Patch L may be generated by the small nucleation and grow via cascade-up (cL), by delayed cascade-up (dcL), or by the large nucleation (LL) in a long simulated history. dcL events, which shortly follow precursory small events (pS), showed milder onset than cL events.
The cascade-up rupture growth is sometimes considered as a series of discrete step-by-step process in which we can 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). Such idealization may be valid only when the scale ratio α is comparable to the brittleness β (ratio between the patch size and the nucleation size). If α < β, eventually all the earthquakes are cL events, but the rupture of Patch S is difficult to recognize in moment rate or moment acceleration functions as a discrete step preceding the main part of the rupture. If α > > β, then all the large earthquakes are initiated by their own large nucleation.
The condition α ~ β is a threshold for the appearance of small events (S events), which are nucleated in Patch S and arrested after spanning it. In the hierarchical asperity concept (Ide and Aochi 2005) in which the brittleness β is independent of the patch size and thus consistent with the self-similar characteristics of the earthquakes (e.g., Ide and Beroza 2001;Venkataraman and Kanamori 2004), the scale ratio α represents the factor of the fracture energy increment that must be overcome for successful cascade-up rupture growth. Even if a rupture cascades-up, the increase in the fracture energy causes a decrease in the rupture speed and thus deceleration of the moment release. Patch S tends to operate as a discrete patch for large α, and its critical value seems to be given by the brittleness β.
Interaction of the large nucleation and tiny patches, which may rupture as a side-effect of the acceleration of the large nucleation, would cause clustered foreshocks as observed for some earthquakes (Ohnaka 1993;Shibazaki and Matsu'ura 1995). This is another possibility for generating the small preceding wave, but the large rupture in this class may not be able to be called a cascade-up rupture. Such a system has not been investigated partly because of the burdensome requirements for computational resources, but it deserves future study.
large event initiated by dynamic cascade-up rupture growth
large event initiated by delayed cascade-up
large event initiated by large nucleation
non-precursory small event
precursory small event
We appreciate the recommendation by T. Matsuzawa, one of the session conveners of the 2013 JpGU annual meeting, for the submission of the present study to PEPS. The Earth Simulator was used for the simulations. This study was partially supported by the Observation and Research Program for the Prediction of Earthquakes and Volcanic Eruptions of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.
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