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Large nucleation before large earthquakes is sometimes skipped due to cascade-up—Implications from a rate and state simulation of faults with hierarchical asperities


Corresponding author: H. Noda, Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan. (hnoda@jamstec.go.jp)


[1] Does a large earthquake have a large quasi-static preparation? A hierarchical asperity model in which a large tough patch (Patch L) contains smaller fragile patches (Patch S) enables a large earthquake to start with only small preparation because of cascade-up rupture growth. We realized such a model in a rate-and-state framework by heterogeneous distributions of the state evolution distance, and its consequences are investigated by earthquake sequence simulations. Focus is put on elementary processes, the interaction between two scales: one Patch L containing one Patch S is simulated, with their size ratio as a parameter. If Patch S is larger than the nucleation size of Patch L, the system falls into a limit cycle consisting of only one earthquake that starts with Patch S nucleation and grows dynamically to rupture entire Patch L. If Patch S is considerably smaller than the nucleation size of Patch L, small earthquakes never dynamically cascade-up, and the large earthquakes are initiated by large quasi-static nucleation. In between, large earthquakes start in various ways: by large nucleation, dynamic cascade-up, or delayed cascade-up. In the final stage of quasi-static nucleation, the preseismic moment release rate increases roughly inversely proportional to the time-to-failure tf with its amplitude depending on the nucleation size. For a Patch S rupture to cascade up, strength in the adjacent region must have been reduced, manifested by a higher creep velocity before the Patch S nucleation starts following 1/tf acceleration. Large nucleation sometimes has a precursory small earthquake characterized by larger afterslip than nonprecursory ones.

1 Introduction

[2] Whether a large earthquake has a large preparation process or not has been intensively debated [e.g., Ellsworth and Beroza, 1995] and is of great interest for many seismologists who investigate the source processes and try to find a way to apply geophysical observations to earthquake hazard mitigation. According to Griffith's energy balance theory, a crack of a critical size (nucleation size) is required before a rupture expands spontaneously in a stressed elastic medium [e.g., Scholz, 2002]. In the case of an earthquake rupture on a preexisting fault, a compact region of preslip may act as a critical crack [e.g., Scholz, 2002]. A key question is how large are the nuclei that ultimately grow to become large earthquakes. The critical size of such a nucleus is inversely proportional to the square of the stress drop, and proportional to the fracture energy Gc of the surrounding region, which is about half the product of the breakdown strength drop and the slip-weakening distance Dc [e.g., Rice, 1980].

[3] In laboratory experiments on dynamic rupture, Dc is controlled by irregularities that are statistically quantifiable independently of the size of the slipping region [e.g., Ohnaka and Shen, 1999] because laboratory surfaces are nominally flat (i.e., scope-scale dependence of roughness [e.g., Russ, 1994] due to fractal topography being limited to short scales that are much smaller than the size of the slipping region [e.g., Ohnaka, 2003] over which microscopic processes are averaged to give macroscopic friction). Exhumed fault surfaces [e.g., Candela et al., 2012] have self-affine topography over a much broader range of length scale. Although there are differences in the fault zone structures between experimental and natural faults (i.e., existence of damage zone and gouge layer) and a direct comparison may not be fair, natural faults may have a much larger nucleation size than in laboratory experiments.

[4] Observations of dynamic earthquake ruptures of various sizes (Mw 1–8) [e.g., Abercrombie and Rice, 2005; Tinti et al., 2005; Yamada et al., 2005, 2007] suggest a strong scaling of Gc with the rupture dimension. Recent studies of radiated energy [e.g., Baltay et al., 2011] show that Gc occupies a good fraction of the available energy (i.e., the released strain energy minus dissipation by residual friction). The fracture energy Gc comes into play in determining not only the nucleation size but also the rupture speed. Such dissipative earthquake ruptures are also supported by the general observation that an earthquake's rupture velocity usually remains around 70–80% of the limiting elastic speed, which implies that Gc represents about half of the available energy when interpreted in terms of a dynamic energy balance [e.g., Freund, 1990]. Until recently, a high Gc for large earthquakes has been proposed on the basis of observation of dynamic ruptures; however, Kato [2012a] has also suggested that the Gc, in a static sense, that matters during nucleation is large. This was estimated by way of earthquake cycle simulations of the large subduction earthquakes loaded by an edge dislocation due to stable aseismic slip of the deeper extension. The wide variability of Gc in natural earthquakes, together with its value being not far below the quasi-static energy release rate (proportional to the instantaneous size of the rupture if stress drop is uniform), seems to provide a natural mechanism to stop earthquake rupture that determines the eventual size of individual earthquakes [e.g., Ide and Aochi, 2005].

[5] Some authors take such scale dependence of Gc as evidence that large earthquakes occur on a tough fault (i.e., high-Gc as a material property) and hence argue that the critical nuclei for large earthquakes should be large, on the order of 1/10 of the final rupture dimension [Matsu'ura et al., 1992; Shibazaki and Matsu'ura, 1998; Ohnaka, 2000, 2003]. Note that an underlying assumption is that the potential stress drop is more or less uniformly distributed on the fault so that the distribution of Gc (and not the stress drop) determines the rupture extent. There are mainly two lines of criticism against the notion of a large nucleus for a large earthquake.

[6] The first line is based on the recent recognition of significant weakening of a fault at coseismic slip rates; a seismologically observed large Gc may emerge only during a dynamic rupture due to the remarkable strength drop at high slip rates over long slip [e.g., Rice, 2006]. If it happens during dynamic rupture propagation, the nucleation size can no longer be constrained by the seismologically observed Gc. Abercrombie and Rice [2005] suggested that observed Gc as a function of average slip of earthquakes of different sizes can be taken as the slip-weakening relation of the fault. Rice [2006] showed that such a relation can be explained by thermal pressurization of the pore fluid [e.g., Sibson, 1973] and that the apparent slip-weakening distance Dc is always a good fraction of the total coseismic slip. Both the seismological Gc and the strength drop are not material constants and depend on the history of slip. If the nucleation takes place frequently, it may sometimes succeed in growing up if the fault around it is ready enough. The remarkable weakening at coseismic slip rates allows a rupture to propagate spontaneously at a level of shear stress substantially lower than what is expected by Byerlee's law with lithostatic and hydrostatic ambient stresses on the fault [Rice et al., 2009; Noda et al., 2009; Noda and Lapusta, 2010a], which is consistent with low stress level observed on some major active faults [e.g., Lachenbruch and Sass, 1980; Townend and Zoback, 2004; Hsu et al., 2009; Hasegawa et al., 2011].

[7] The other line is based on the heterogeneous distribution of the fault constitutive properties such as Gc or Dc. Motivated by the dependence of Dc on the geometrical irregularity of a fault that is self-affine [e.g., Scholz, 2002], Ide and Aochi [2005] assumed a hierarchical distribution of Dcin a linear slip-weakening friction law such that a larger patch with larger Gc has smaller patches with smaller Gc in it. They initiated ruptures from the smallest patches and produced large earthquakes by successive rupture growth to larger length scale (cascade-up) by chance (for details, please see the next section). Such ruptures have much smaller nuclei than what is expected from observed (or averaged) Gc. They also reproduced the above-mentioned observations on natural earthquakes, that is, scale-dependent Gc and reasonable rupture speed.

[8] One of the important differences between these two lines is the different mechanism for generating small earthquakes (or more precisely, how small earthquakes are arrested). In the former, a small earthquake is arrested because it enters a region which is not ready enough. Hence, small earthquakes tend to occur near the boundary of creeping and locked regions (streaks and holes) [Lapusta and Rice, 2003; Barbot et al., 2012]. A barrier in terms of the material properties may not be needed at the end of the small earthquakes, and stress heterogeneity plays the central role. On the other hand, in the latter a small earthquake is a rupture of a small patch having small Gc. The idea that there are patches of rather uniform properties (such as Gc) with characteristic earthquakes is essentially an “asperity model” [Lay and Kanamori, 1981]. Existence of small repeating earthquakes [e.g., Vidale et al., 1994; Bürgmann et al., 2000; Igarashi et al., 2003; Chen and Lapusta, 2009; Hori and Miyazaki, 2010; Uchida and Matsuzawa, 2011] probably indicates the importance of patchy distribution of physical properties on the fault. Evaluating which viewpoint better explains the natural seismic activities is an important future task not only in observational seismology but also in studies of friction of natural faults including the scale dependency of friction.

[9] In the present study, we consider the latter scenario for a fault governed by a rate- and state-dependent friction law (RSF) and investigate the consequence. This is a natural extension of the model by Ide and Aochi [2005] enabling us to simulate the whole seismic cycle (interseismic, preseismic, coseismic, and postseismic periods). In this case, growth of the preslip region to the critical size for a large earthquake may be achieved during dynamic rupture of a small patch, in which case the small rupture may “cascade up” to a large earthquake in one spell. As a result, a large earthquake can occur without a quasi-static formation of a large nucleus beforehand [e.g., Fukao and Furumoto, 1985; Ellsworth and Beroza, 1995; Ide and Aochi, 2005]. Or instead, a large nucleation may occur before a large earthquake if the condition allows.

[10] The problem of “own (quasi-static) nucleation” versus “cascade-up” remains a controversial issue since it was raised in relation with the observation of seismic initial phases [e.g., Umeda, 1990; Iio, 1992; Ellsworth and Beroza, 1995; Shibazaki and Matsu'ura, 1998] and clustering of small immediate foreshocks [e.g., Ohnaka, 1993; Shibazaki and Matsu'ura, 1995, 1998; Dodge et al., 1995, 1996]. Quasi-static nucleation on a frictional fault has been modeled in many earlier studies [e.g., Tse and Rice, 1986; Dieterich, 1992; Matsu'ura et al., 1992; Lapusta et al., 2000], but interaction between seismogenic processes at different scales was treated only cursorily [e.g., Shibazaki and Matsu'ura, 1995; Hori and Miyazaki, 2010, 2011]. Some of those studies are based on quasi-static or quasi-dynamic simulations so that the inertial effects are not fully accounted for even though they are undoubtedly important in considering dynamic cascade-up. In contrast, Ide and Aochi [2005] proposed a multiscale heterogeneity of the Gc distribution and modeled the dynamic rupture process (i.e., with the inertial effects) involving interaction between different scales, including cascade-up. However, as elaborated in the next section, quasi-static slip was not involved in their modeling, so own nucleation was precluded in their study. As a straightforward attempt to evaluate relative importance between own nucleation versus cascade-up, we perform numerical experiments of a fault with a multiscale Gc heterogeneity using rate- and state-dependent friction law (RSF), which allows slow slip at stresses below the nominal frictional strength [Nakatani, 2001].

[11] We first review the concept by Ide and Aochi [2005] in section 2. Section 3 is devoted to the description of the model studied and methodology of simulations. In section 4, some key properties of earthquakes are introduced and a classification of earthquakes in the present study is summarized before describing the properties of the earthquake sequences in section 5. The discussion and conclusion follow in sections 6 and 7. Example earthquakes of different types are shown in section A in detail.

2 Background—Small Fragile Spots in a Large Tough Asperity

[12] Figure 1a is a simplified version of the multiscale Gc distribution proposed by Ide and Aochi [2005], where a lower Gc is indicated by a darker shade of gray. The large solid circle (radius RL=9) is the rupture area of a large earthquake, which we call Patch L. As mentioned earlier, we assume that this region has a high fracture energy (say, Gc=9), and the corresponding nucleation size is fairly large, say, inline image(a dotted circle to the left of the gray scale). It is further postulated that some parts of Patch L are locally fragile, so that small earthquakes can occur there. In Figure 1a, we placed three such fragile spots (Patches S1, S2, and S3), with a small radius RS=2 and proportionally low Gc(=2), resulting in a proportionally small nucleation size of inline image. For usual physical quantities, this manner of spatial variation may be unnatural. But for the critical slip distance Dc (proportional to Gc), this is very plausible as Ide and Aochi [2005] has argued using fractal topography of natural faults, where fractality holds up to a large scale beyond the rupture sizes of smaller earthquakes. The assumed general proportionality between the patch radius and Gc is to explain the observed scaling of Gc with earthquake size as mentioned earlier. The assumed relatively small difference between the nucleation size and patch radius (1:3) is consistent with the observed rupture speed and energy partitioning also mentioned earlier.

Figure 1.

Synoptic description of multiscale Gc heterogeneity. (a) Patch L includes multiple smaller patches that are smaller than the nucleation size of Patch L. (b) Patch L includes a smaller patch that is comparable to the nucleation size of Patch L in size.

[13] Using Figure 1a, we synoptically describe the argument of Ide and Aochi [2005] emphasizing “cascade-up”. For simplicity, as they assumed, let us assume for now that the background stress level and frictional properties other than Dc are uniform. First, we consider what happens if Patch S1 ruptures. The earthquake on S1 creates a slipped region of RS=2. This is smaller than the inline imagerequired to propagate into the surrounding tough region of high Gc(=9). So, this rupture ends up with a small earthquake of RS=2. Second, we consider what happens if Patch S2 ruptures. In this case, another fragile inline image Patch S3 is close enough to the slipped region of S2 earthquake and hence S3 ruptures successively. Now we have a contiguous slipped region (S2+S3) with a half-length of about 4, which is greater than the nucleation size inline image for the surrounding tough region. Therefore, the rupture started at Patch S2 cascades up to rupture the entire Patch L, causing a large earthquake of RL=9. A rupture starting from Patch S3 will follow the same story.

[14] There are two important nondimensional parameters characterizing the distribution of Gc. One is the ratio of the sizes of Patch S and Patch L, which we shall call the hierarchical scale gap α:

display math(1)

The other is the ratio between the nucleation size and patch size, which we shall call the brittleness β (see section 3.2 for more explanation):

display math(2)

Of course, β for Patch L and Patch S can be taken differently, but for simplicity, we do not consider such cases in this paper. Although Ide and Aochi [2005] calculated dynamic rupture processes over a much broader scale range, the essence is just as shown in the above exercise. Within their argument, growth into a large earthquake is a matter of chance. A large earthquake is the sequence of cascade-up started from one of the few luckiest small patches whose subsequent ruptures proceeded along a path that always happened to have an appropriate stepping stone to cascade up to the next stage. Smaller earthquakes are regarded as failed attempts that missed an appropriate stepping stone at some earlier stage.

[15] However, we must note that their simulation assumed that the rupture always starts as an instability within one of the smallest patches. This was also the case in their later numerical experiments [Aochi and Ide, 2009], where the buildup of tectonic stress and the stress inhomogeneity due to seismic slip of earlier earthquakes, for example, were incorporated as a bias in selecting which smallest patch to kick. In other words, the possibility of large earthquakes initiating directly from large quasi-static nucleation in the high-Gc part was not considered. Though ruptures at the smallest patches may be of frequent occurrence, if the cascade-up to the largest earthquake requires such an extreme luck, then large earthquakes, which are in any event needed for regions of significant seismic coupling, might be the ones that started directly from a large quasi-static nucleation in the tough part. This point will be discussed further in section 6.3.2 in view of Gutenberg-Richter relation.

[16] Now we contemplate what will happen if we simulate the system of Figure 1a using RSF. A critical difference from the calculation by [Ide and Aochi, 2005] is the possibility of the large quasi-static nucleation. For example, we cannot preclude the possibility that a quasi-static nucleation grows to a critical size inline image in the left-hand part of Figure 1a where Gc is uniformly high. Of course, further smaller fragile spots omitted in Figure 1 would fail dynamically along the way [e.g., Shibazaki and Matsu'ura, 1995], but they are too small to trigger the rupture of Patch L. On the other hand, we cannot say that quasi-static nucleation of a size inline image would occur for sure. For example, if Patch S2 or S3 ruptures frequently, with an interval less than the characteristic time of nucleation's growth to inline imagein the tough part, chances are that Patch L always ruptures by cascade-up from S2 or S3 before the large nucleation is ready. These are just some examples of the issues that we cannot answer without actually simulating the system. Below is a list of specific processes relevant to the cascade-up versus own nucleation on the faults with multiscale Gc heterogeneity.

[17] (P1) Easiness of cascade-up: In addition to the size and density of smaller patches, the evolving stress and strength of Patch L around the small patches should be a factor.

[18] (P2) Temporal frequency of small-patch rupture: In addition to the frictional properties of the small patch, this may be affected also by the loading from aseismic slip in the large asperity, which, unlike the external tectonic loading, is not linear with time.

[19] (P3) Termination of large-patch nucleation by a small earthquake: Nucleation toward a large patch may touch a small patch along the way, triggering a small-patch earthquake, which may cascade up to a large earthquake.

[20] (P4) Bias on large-patch nucleation by small-patch activity: Afterslip of small-patch earthquake may promote large-scale nucleation, attracting its location to the small patch. Then if the small patch heals fast enough, P3 may follow.

[21] (P5) Bias on large-patch nucleation by the existence of a small patch: assuming a short Dc for the small patch generally tends to suppress aseismic slip [Kato, 2003, 2004], suppressing large-scale nucleation around the small patch. The opposite effect may be possible because possible stress concentration at the rim of small patches.

[22] As enumerated above, involvement of aseismic slip can increase the diversity of the ways large and small patches interact, which would depend not only on Dc but also on other RSF parameters. In the present study, we have sought a simplest setting to elucidate major characteristics of interscale interactions through aseismic slip. Through preliminary experiments, we have realized that diverse ways of interaction can be demonstrated using the simplest multiscale system that involves only one small fragile spot in a large tough patch (Figure 1b). Without aseismic slip and under homogeneous stress field as in Ide and Aochi [2005], the fate of a small rupture is completely dictated by the juxtaposition of patches. In case of Figure 1b, only small earthquakes occur if Patch S is smaller than the nucleation size of the large patch inline image (scale gap dominates, α > β), while only the (cascade-up) large earthquakes occur if Patch S is larger than the inline image (brittleness dominates, β > α). However, as we will show, all of the three varieties—i.e., a large earthquake directly occurring from large quasi-static nucleation, a large earthquake cascaded up from a small earthquake, and a small earthquake that does not grow into a large earthquake—occur in the long history simulated under a fixed setting. Very roughly speaking, the variety emerges due to the difference in the readiness of the large patch at the time of the occurrence of the small rupture. Hence, we mainly show the results on the system involving only one small patch (Figure 1b), and a systematic exploration in parameter space is only done on the size of the small patch RS relative to inline image, which exerts a strong control on the ratio among the three varieties. However, by contemplating the specific factors mentioned in the last paragraph, we can easily imagine a few other parameters that should affect the ratio among the varieties. Some aspects that can be explored by further studies along this line will be discussed in section 6.

3 Model Settings

3.1 Rate-State Earthquake Sequence Simulation and Its Time-Length Scaling

[23] We consider a planar fault embedded in a linearly elastic infinite space. A Cartesian coordinate system is chosen such that x1, x2, and x3 are the coordinates along the overall slip direction, normal to the fault, and in the remaining orthogonal direction, respectively. The origin is located at the center of Patch L. The fault is governed by a rate- and state-dependent friction law [Dieterich, 1979; Ruina, 1983]:

display math(3)

where τis the magnitude of the shear stress on the fault; V is the magnitude of the slip rate; σ is the normal stress; f is the friction coefficient; θ is the state variable; and a, b, and L are the parameters representing the magnitude of direct effect, evolution effect, and the characteristic slip distance for the state evolution, respectively. Subscripts 1 and 3 indicate the component of the vector quantity. fref is the steady-state friction coefficient at a reference slip rate Vref. Just note that unlike in 2-D problems, (fref,Vref) matters in 3-D problems in which the perturbation in the shear traction vector is not necessarily parallel to the perturbation in the slip rate vector. The evolution of the state variable θ is governed by the aging law:

display math(4)

[24] The strength of a fault can be defined as [Nakatani, 2001]

display math(5)

The value of Θ depends on the selection of Vref, but its variation is useful in checking how a portion of the fault is aged during locking or weakened due to interseismic creeping and coseismic slip.

[25] At a steady state, the frictional resistance is written as

display math(6)

It is known that a rate-strengthening fault (ab>0) slides only steadily and cannot spontaneously nucleate frictional instability while a rate-weakening fault (ab<0) can nucleate a rupture if it is large enough [e.g., Ruina, 1983; Rubin and Ampuero, 2005].

[26] The only frictional parameter having the dimension of length is L. Since elastodynamics has no length scale, the length scale of the distribution of parameters and system size can be scaled by L without changing the nondimensional solutions. If one chooses length, velocity, and stress as three independent dimensions, increasing the length scale (L and the system size) by a factor of 10 while keeping the other scales causes increases in the moment magnitude by 2 and the time scale by a factor of 10, which is the length scale divided by the velocity scale. For the presentation purpose, we set the length scale such that large earthquakes spanning the Patch L have a moment magnitude of approximately 6. For example, with 10 times longer L and a larger system by the same factor, the large earthquakes become Mw 8 which has 10 times longer recurrence intervals.

[27] A sequence of earthquakes, interaction of elastodynamics and fault friction over a wide range of timescales, can be simulated very efficiently by using a spectral boundary integral equation technique [e.g., Lapusta et al., 2000; Lapusta and Liu, 2009; Noda and Lapusta, 2010b]. The shear traction on the fault is expressed as

display math(7)

inline image is the shear traction without any slip on the fault, μ is the shear modulus, and cs is the S wave speed. φi is a functional term depending on the spatio-temporal distribution of slip, representing the stress transferred by the elastic wave as well as the static field. It is calculated using a fast Fourier transform technique. Every time step, equations (3) and (7) are solved for τi and Vi.

3.2 Parameter Distribution

[28] The simulated model is illustrated in Figure 2, and the parameters are summarized in Table 1. A large patch (Patch L, radius RL=4 km) includes one small patch (Patch S, radius RS). In the present model, periodic boundaries are assumed normal to x1 and x3 directions every 25.6 km. The linearly elastic medium has the S wave speed cs=3 km/s, the shear modulus μ=30 GPa, and the Poisson's ratio ν=1/4. Many of the fault constitutive parameters are assumed to be uniformly distributed on the fault, σ=100 MPa, fref=0.7, Vref=8 cm/yr (≈2.54×10−9 m/s), and a=0.016, where 1 year is defined as 3.1536×107s. The fault is loaded by setting a region of prescribed slip rate Vpl=8 cm/yr near the periodic boundaries:

display math(8)
Table 1. Physical Properties and Model Parameters
Elastic Properties
Shear modulusμ30 GPa
Poisson's ratioν0.25
Shear wave speedcs3 km/s
Frictional Properties
Reference slip velocityVref8 cm / 3.1536×107 s
Steady state friction at Vreffref0.7
Direct effect parametera0.016
Evolution effect parameterbout0.012
Normal stressσe0100 MPa
Brittleness inline imageβ3
Boundary Conditions
Hierarchical gap RL/RSα2-5
Loading rateVpl8 cm / 3.1536×107 s
Spatial periodicity 25.6RL/4
Width of prescribed slip region 0.3RL/4
Selection of Length Scale for Presentation Purpose
Large patch radiusRL4 km
Large nucleation radiusinline imageβ× 4 km
Small patch radiusRSα× 4 km
Small nucleation radiusinline imageαβ× 4 km
Figure 2.

The model setting studied in the present study. A planer fault embedded in a linearly elastic medium has a seismogenic patch that includes another smaller patch with shorter length scale in the friction law L. The elastodynamics is solved by a spectral boundary integral equation method [Lapusta and Liu, 2009; Noda and Lapusta, 2010b].

[29] The fault is assumed to be rate-strengthening outside Patch L. The rate-strengthening region has rate-state parameters bout=0.012 and Lout. Patch L is set at the center of the computational domain and has bin=0.02. Note that about=0.004 and abin=−0.004. The critical nucleation size (radius) of Patch L inline imagecan be written as [Rubin and Ampuero, 2005; Chen and Lapusta, 2009; Lapusta and Liu, 2009]

display math(9)

where LL is the characteristic slip distance for the state evolution inside Patch L, excepting Patch S. The ratio of the patch size to the nucleation size β (equation (2)) is a nondimensional parameter which controls the mode of initiation of earthquakes as demonstrated by, for example, Kato [2003], Kaneko and Lapusta [2008], and Chen and Lapusta [2009]. If β is not significantly larger than 1, seismic slip requires coalescence of creeping front(s). On the other hand, large β enables nucleation inside a creeping region in the seismogenic patch: the compact nucleation region accelerates to the coseismic slip rate [e.g., Lapusta and Liu, 2009; Mitsui and Hirahara, 2011]. Larger β requires a shorter penetration length of creeping fronts inside a seismogenic patch, causing a shorter recurrence interval [Kato, 2012b; Mitsui and Hirahara, 2011] and higher seismic coupling. In this sense, we call β the brittleness of the patch. For convenience, let us define a characteristic slip scale L0 corresponding to nucleation size as large as Patch L:

display math(10)

L0 is about 13.6 mm for RL=4 km. We adopt this to be the characteristic slip distance for the state evolution outside the seismogenic patch, Lout=L0. LL can be written as

display math(11)

[30] The hierarchical structure is realized by heterogeneous distribution of L. Patch S having a characteristic slip LS(<LL) is set inside Patch L in order to model earthquakes smaller by one hierarchical order (Figure 2), as discussed in section 2. Patch S is smaller than Patch L by a factor of α (equation (1)). We assume the same brittleness β for Patch L and S so that the nucleation size (radius) of the small patch:

display math(12)

is given by

display math(13)

Thus, LS is expressed as

display math(14)

Since we use the same a and b for the both seismogenic patches, the fracture energy of the patches is proportional to their radius if the state variable θ or Θ is uniform before an earthquake. Therefore, the present model can be regarded as a rate-state version of the hierarchical slip-weakening model by Ide and Aochi [2005]. However, Vi, τi, and Θ outside a nucleus are never uniform at the onset of seismic slip because of previous earthquakes, aseismic transients, and the nucleation process itself. This is the source of the rich behavior shown later.

[31] The distribution is made smooth using a smoothed boxcar function [e.g., Noda and Lapusta, 2010b] in the radial direction. Patch L and Patch S have smooth rims the width of which is wL=RL/4 and wS=RS/4, respectively.

[32] In this paper, we shall present cases in which Patch S is set near the rim of Patch L (Figure 2), although we have tested other places (e.g., center of Patch L). It has turned out that the location of Patch S matters. For example, if a tiny Patch S is at the center of Patch L, which is mostly locked throughout the interseismic period, it never initiates an earthquake since nucleation requires penetration of creeping motion. By putting Patch S near the rim of Patch L, we maximize the loading to Patch S and its effect on the sequence of earthquakes. The center of Patch S is located in the direction of inline image from the center of Patch L and at a distance such that the smooth rims of the both patches touch each other (Figure 2). Fully understanding the effect of the location of Patch S deserves further study.

[33] The shear stress distribution corresponding to uniform slip and uniformly negligible slip rate on the fault (equation (7)), τ0, is assumed to be consistent with the steady state sliding at the plate rate Vpl:

display math(15)

The fault is assumed to be sliding uniformly at Vpl at the steady state (i.e., θ=L/Vpl) during t<0. We start the simulation by imposing a perturbation of slip in the first direction by 1 cm within a circular region of 1.5 km radius centered at (x1,x3) = (1 km, 1 km) at t=0.

[34] The numerical simulations in the present study are expensive in terms of memory size and CPU time. Because of limited numerical resources, it is difficult to conduct numerical simulations involving too broad a range of scale, such as multiple hierarchical orders or α and β much larger than unity. In addition, it is often the case that the simulation result does not fall into a limit cycle even after tens of earthquakes. Because of these difficulties, we would like to put our focus in the present paper on the effect of α while keeping β=3 for both small and large patches. Further exploration of the parameter space and, more generally, the dependency of the properties of earthquake sequences on the distribution (or the statistics) of L is a future task.

3.3 Numerical Procedure

[35] The numerical procedure is similar to what is used by Noda and Lapusta [2010b]. The only difference is the way of solving equations (3) and (7) for τi and Vi. In the present study, we use the logarithmic law, equation (3), rather than an arcsinh-regularized law [Nakatani, 2001; Rice et al., 2001]. The difference between them is negligible since the friction coefficient f is always much larger than a by more than 1 order of magnitude. Equations (3) and (7) yield

display math(16)

where τφ is the magnitude of a vector τ0i+φi(i=1,3) and Ω() is the Lambert W function. Instead of solving equations (3) and (7) with the Newton-Raphson method as was done in previous studies [e.g., Lapusta et al., 2000; Lapusta and Liu, 2009; Noda and Lapusta, 2010b], we evaluate equation (16) using the method used by Corless et al.[1996].

[36] The fault is discretized by 2048 × 2048 grid points for α>3 and 1536 × 1536 otherwise. The time window for which inertial effects are accounted for is set as the duration for an S wave to propagate for the diameter of Patch L (2RL). By setting the elastic time step to the time for an S wave to propagate for one third of the interval of the nodes, the number of time steps within the time window is 1920 for α>3 and 1440 otherwise. The size of the largest arrays (history storage and kernel) are, in total, 720 GB for α>3 and 304 GB otherwise.

[37] The simulations in the present study are executed on the Earth Simulator 2 with 16 nodes for α>3 and 8 nodes otherwise. The nodes communicate with each other using MPI, and Open MP parallelization is used for the 8 vector processors in each node. In the cases with α>3, the vectorization ratio is 98.237 %, and the parallelization ratio is 99.6289 %.

[38] We have calculated the sequence of earthquakes until the twentieth large earthquake. The simulations take from about a half day for α=2 to 4 days for α=5.

4 Diversity in Simulated Earthquakes

[39] The simulated earthquakes are initiated in various ways. For ease of discussion, we classify the earthquakes according to some earthquake properties most relevant to the “cascade-up” versus “own nucleation.” In this section, we first introduce the properties of earthquakes to which we pay special attention and then summarize the classification. Note that in the present study, the coseismic periods are defined by the spatially maximum slip rate being larger than a threshold of 0.1 m/s =3.942×107Vpl.

4.1 Properties of the Earthquakes

4.1.1 Ruptured Area and Moment Magnitude

[40] From our problem setting (section 3.2), it is straightforward to expect that there are two characteristic sizes of earthquakes, each corresponding to Patch L and Patch S. Patch L (RL=4 km) ruptures as an earthquake of approximately Mw 6, which we call an “L-event.” The characteristic moment magnitude of the earthquakes which rupture only Patch S is then Mw6–2log10α which is 5.40, 5.05, 4.80, and 4.60 for α=2, 3, 4, and 5, respectively. Those smaller earthquakes are called “S-events.” In addition, much smaller earthquakes occur sometimes as a local increase in the slip rate to the defined coseismic slip rate (0.1 m/s) by the coalescence of the creeping front.

4.1.2 Distribution of V at Initiation of Earthquakes

[41] We visually check how the earthquakes are initiated and use that information in classifying the earthquakes. Depending on the L of the nucleation site, two different characteristic nucleation sizes, inline image and inline image, are theoretically expected (equations (9) and (12)). In the following sections, “small” and “large” nucleations mean that the acceleration just before an earthquake takes place in a compact region of the radius comparable to inline image and inline image, respectively. It is sometimes the case that the nucleation involves the boundary of Patch S. In such cases, we classify earthquakes differently. Most of the earthquakes are initiated in one of these three ways.

[42] Kaneko and Lapusta [2008] and Chen and Lapusta [2009] demonstrated that the initiation of earthquake does not necessarily require the nucleation (i.e., acceleration of a compact region comparable to the nucleation size); the slip rate increases locally, in a much smaller region than the nucleation size, by the coalescence of creeping front and then propagate further. However, such a way of earthquake initiation is mostly limited to faults with β close to unity and is of rare occurrence with β = 3.

4.1.3 1/tf Acceleration

[43] From the observational point of view, the spatial imaging of nucleation would be tough due to the limited spatial resolution and the nonuniqueness of the solution of geodetic inverse problems. Therefore, it is probably useful to classify the earthquakes on the basis of bulk properties such as the moment rate or the averaged slip rate over Patch L, Vpatch. Dieterich [1992] simplified the nucleation as an acceleration of a patch of a fixed size and derived the moment rate as a function of time-to-failure tf. We use a similar function as an ideal acceleration in the final nucleation phase and see if the simulated moment rate history is consistent with it.

[44] In a circular crack model with radius inline image, the uniform stress change inside the crack inline imagecan be written as

display math(17)

where inline imageis the average slip inside the crack. Note that the overline denotes the averaged quantity over the nucleation patch, not Patch L. The stiffness of the nucleation patch is

display math(18)

Assuming a spring-slider model with 1 degree of freedom and by neglecting tectonic loading

display math(19)

the rate- and state-dependent friction for the averaged (overlined) quantities

display math(20)

and the constant-weakening limit for the state-evolution equation

display math(21)

then the averaged slip rate inline imagediverges at a finite time as

display math(22)

where tf is the time-to-failure which decreases with time lapse and takes the value of zero at the divergence of inline image. Note that Dieterich [1992] assumes no-healing limit in place of equation (21):

display math(23)

but for the parameter used in the present study (a/bin=0.8), the constant-weakening limit is more appropriate [Rubin and Ampuero, 2005]. The averaged slip rate over Patch L, Vpatch, as a function of tf for the large nucleation is then

display math(24)

This is 1/tf acceleration toward the initiation of the earthquake. For the small nucleation, the crack radius is inline image and its 1/tf acceleration is reflected to Vpatch:

display math(25)

In reality, the average slip rate over Patch L, Vpatch, is bounded. As an earthquake approaches and the slip rate becomes higher, the assumptions taken here become invalid at some point, for example, by inertial effects. These 1/tf accelerations (equations (24) and (25)) do match the calculated history of the fault slip prior to the earthquakes very well as shown later. As mentioned earlier, we define the initiation of earthquakes (tf=0) as the time when the maximum slip rate becomes larger than 0.1 m/s.

[45] The good match by the 1/tf approximation may be understandable dimensionally. First, let us assume the process is quasi-static and the wave speeds are not important. Second, the nucleation is a spontaneous growth of local perturbation so the plate rate Vpl would not matter. Third, the slip direction is uniform and the reference slip rate Vref is not important. These assumptions are adopted implicitly in the derivation above. In consequence, there is no parameter having the dimension of speed which matters. The goal here is to write Vpatch as a function of tf, and the only way to achieve it is [length scale]/tf. If the friction law has other parameters such as characteristic time or slip rate, then the resulting preseismic acceleration may not follow 1/tf.

4.1.4 Existence of Precursory S-events to L-event

[46] When an earthquake nucleated in Patch S fails to cascade up, it is sometimes the case that the following L-event takes place before waiting for the averaged slip rate Vpatch settling down to an interseismic level around 0.1Vpl. In such cases, we regard the S-event as a precursory earthquake to the L-event. The following L-event may be initiated in various ways, and the interval between them depends, roughly speaking, on the way of initiation of the L-event. As we will show, the short interval between the precursory S-events and the following L-events is usually not coincidence: the precursory S-event plays an active role in initiating the following L-event.

4.2 Classification of the Earthquakes

[47] Before looking into the simulated sequences of earthquakes and discussing the dependency of the earthquake statistics on α, we would like to summarize here the types of earthquakes obtained in the simulations and define labels used in tables later. More detailed information on each classification is described in section A with examples.

[48] S-events are classified into two types, precursory S-events (pS) and nonprecursory S-events (nS) as discussed in section 4.1.4 (also see section A1). L-events are initiated mostly by large nucleation (LL) (section A2), by dynamic cascade-up (cL) (section A3), or by delayed cascade-up (dcL; section A4). A cascade-up L-event starts from small nucleation and grows to span Patch L in one coseismic period in which the spatially maximum slip rate remains greater than 0.1 m/s. Sometimes the growth to large rupture is delayed, and the cascade-up process is separated into a precursory S-event and a delayed cascade-up L-event by a short interseismic period in which the maximum slip rate is somewhat less than 0.1 m/s.

[49] It is sometimes the case that a large nucleation has a precursory S-event. In this case, the afterslip of the S-event migrates in the creeping region in Patch L for a while and causes large nucleation away from Patch S. On the other hand, a delayed cascade-up event is initiated by re-acceleration including the rim of Patch S where the precursory S-event was arrested.

[50] Although it is of rare occurrence with our selection β = 3, a locked patch may shrink out of existence before being swept by a rupture front. At this point, the coalescence of a creeping front may cause a local increase in the slip rate to even higher than 0.1 m/s defining the coseismic period [Kaneko and Lapusta, 2008; Chen and Lapusta, 2009] (section A5). Such a locally high slip rate can generate an earthquake much smaller than the S-events (Co) or, if the surrounding region is ready enough, prompts large quasi-static nucleation for an L-event (CoL).

[51] In addition, we obtain tiny events which cannot be classified into any categories listed above. Some of them are due to oscillation in the slip rate potentially caused by a resolution problem. Those events are labeled as “Misc” in the tables in the next section, and we do not put our focus on them.

5 Description of Earthquake Sequences

5.1 Without a Small Patch

[52] In this subsection, the sequence of earthquakes without a Patch S is presented for reference. The simulated earthquakes and their classification are tabulated in Table 2. All the earthquakes in this case are L-events which have moment magnitude about Mw 6. The recurrence interval is 17.5±3.2 years where the error represents 2 times standard deviation for 19 intervals. The system did not fall into a limit cycle as long as simulated.

Table 2. List of the Earthquakes Without Patch S, β = 3
1001y 277d 13:29:285.999LL
2019y 137d 01:55:096.019LL
3037y 267d 08:43:176.037LL
4055y 323d 14:42:046.030CoL
5074y 046d 17:05:556.042LL
6092y 039d 00:38:006.027LL
7110y 108d 19:45:216.030LL
8128y 099d 02:15:346.017LL
9142y 330d 02:56:285.977LL
10161y 116d 09:59:126.028LL
11176y 000d 06:57:295.975LL
12194y 222d 15:33:476.050LL
13212y 063d 23:53:116.017LL
14230y 173d 15:25:376.024LL
15251y 085d 02:54:106.037LL
16269y 117d 02:11:596.027LL
17284y 050d 13:03:055.975LL
18299y 028d 16:08:445.988LL
19317y 253d 07:40:036.050LL
20335y 061d 06:50:556.016LL

[53] It is known that the sequence of earthquakes are qualitatively different depending on the value of β as shown by, for example, Kato [2003], Lapusta and Rice [2003], Kaneko and Lapusta [2008], and Chen and Lapusta [2009]; very large β (≫1) causes ruptures to be arrested in the middle of a seismogenic patch, and β comparable to unity or smaller causes the initiation of earthquakes, if possible, by a coalescence of creeping fronts. The brittleness β = 3 typically causes nucleation, not coalescence, in the initiation of earthquakes (Figure 3). The dashed white circles in Figure 3 indicates Patch L. Figures 3a and 3h show that the nucleation size is approximately one third of Patch L in length scale, as expected theoretically. Note that the whole Patch L ruptures in every earthquake.

Figure 3.

Snapshots of the slip rate, the shear stress, and the state variable in the fifth and sixth earthquakes and the interseismic period between them without Patch S. (a) Nucleation of the fifth earthquake. (b) End of the fifth earthquake. (c) Afterslip of the fifth earthquake. (d) Locked patch generated by the fifth earthquake. (e) Shrinking of the locked patch and aseismic transients. (f) An aseismic transient that leads to nucleation. (g) Beginning of acceleration in a region the size of which is comparable to inline image. (h) Nucleation of the sixth earthquake. (i) End of the sixth earthquake.

[54] The earthquakes span Patch L (Figures 3b and 3i), and the duration of the earthquake as defined by Vmax>0.1 m/s is roughly from 3RL/cs to 4RL/cs. The following afterslip expands with stress concentration in front of the outward creeping front, while the coseismically ruptured region is locked (i.e., a slip rate much smaller than Vpl) and supports a shear stress less than σfref, the steady state shear stress at Vpl (Figure 3c).

[55] Interseismically, the locked region shrinks due to inward propagation of a creeping front (Figure 3d). When the creeping region inside the seismogenic patch becomes large enough to host a critical length scale (such that there is an impossibility of coherent steady-state slip [Rice et al., 2001]), the inward propagation of the creeping front becomes nonsteady, and aseismic transients take place (Figure 3e). After the creeping region inside Patch L becomes larger than the nucleation size [e.g., Lapusta and Liu, 2009], one of the aseismic transients grows (Figure 3f), leading to the nucleation of an L-event (Figures 3g and 3h). The final nucleation occurs within a compact region approximately of radius inline image, which is significantly smaller than the area of aseismic transients that germinated the nucleation.

[56] Among the 20 earthquakes simulated, the fourth earthquake is initiated by coalescence of creeping fronts. Figure 4 shows the acceleration of the fault slip prior to the earthquakes, Vpatch as a function of tf. At the early stage of the acceleration (VpatchVpl), the power n where inline image is greater than −1. When nucleation takes place, the power n decreases and Vpatch nears the ideal 1/tf acceleration equation (24) (Figure 4a). When an earthquake is initiated by the coalescence of a creeping front, however, the power n does not decrease to −1 (Figure 4b), although the difference from a large nucleation is not remarkable in this example.

Figure 4.

Acceleration of the fault slip prior to the initiation of earthquakes in the case without Patch S. (a) Nineteen cases that undergoes large nucleation. 1/tf acceleration predicted by the nucleation size inline image is recognized. (b) The fourth earthquake which is initiated by the coalescence of creeping fronts. The acceleration prior to the initiation of the earthquake is milder.

5.2 β > α

[57] This subsection describes simulated sequences of earthquakes in cases with β > α. Because Patch S is larger than the nucleation size of Patch L, inline image, when Patch S ruptures, the energy release rate is already larger than the fracture energy of the surrounding region, provided it is ready for own nucleation. Tables 3 and 4 list earthquakes in cases with α=2 and 2.5, respectively, while β = 3. The recurrence interval of the L-events are 13.0±0.6 years for α=2 and 12.8±3.3 years for α=2.5, significantly shorter than the case without Patch S. The standard deviation for the case with α=2 is very small since the system falls into a limit cycle after only a few L-events. Note that these two sequences fall into limit cycles having only one earthquake that is a cascade-up L-event. The large earthquakes are almost always, except for the third earthquake with α=2.5, initiated by small nucleation in Patch S (Figure 5). In this parameter regime, a fragile patch controls the recurrence interval because a rupture nucleated there almost always cascades up. In this sense, Patch S in these cases may be better regarded as an internal structure of Patch L rather than a small asperity that hosts its characteristic earthquake (S-event). The significant shortening of the L-event interval, contrasting to the cases with α=3 or greater shown later, also supports this view. It may be said that these large Patch S reduces the average Gc over the entire Patch L, which effectively behaves in a more brittle way than expected from the nominal β = 3.

Table 3. List of the Earthquakes With α=2
1000y 353d 16:35:306.021cL
2013y 025d 04:33:595.959cL
3026y 295d 10:14:416.015cL
4039y 160d 02:50:375.987cL
5052y 214d 17:13:256.001cL
6065y 171d 20:23:205.995cL
7078y 178d 01:39:355.998cL
8091y 164d 21:08:015.996cL
9104y 161d 02:27:575.997cL
10117y 153d 11:47:085.997cL
11130y 147d 16:31:065.997cL
12143y 141d 06:39:295.997cL
13156y 135d 06:09:315.997cL
14169y 129d 03:54:205.997cL
15182y 123d 04:22:375.997cL
16195y 117d 05:23:235.997cL
17208y 111d 07:51:215.997cL
18221y 105d 11:20:315.997cL
19234y 099d 15:49:495.997cL
20247y 093d 21:03:255.997cL
Table 4. List of the Earthquakes With α=2.5
1000y 355d 13:12:296.022cL
2011y 154d 09:53:045.358nS
3018y 303d 23:24:496.026LL
4034y 165d 22:02:456.033cL
5049y 149d 23:41:456.011cL
6060y 243d 16:01:335.948cL
7073y 266d 02:30:196.007cL
8085y 082d 16:48:145.951cL
9098y 062d 02:29:366.003cL
10109y 288d 16:46:285.960cL
11122y 153d 07:59:335.990cL
12134y 108d 17:52:225.971cL
13146y 219d 16:12:105.983cL
14158y 215d 22:30:285.976cL
15170y 315d 06:17:355.981cL
16183y 011d 15:35:445.980cL
17195y 077d 04:08:175.981cL
18207y 160d 15:51:515.981cL
19219y 245d 23:02:525.982cL
20231y 312d 18:49:515.981cL
21244y 029d 23:02:295.981cL
Figure 5.

Snapshots of the slip rate, the shear stress, and the state variable in the 20th and 21st earthquakes and the interseismic period between them. α=2.5. (a) Nucleation of the 20th earthquake, which takes place in the small patch. (b) The small patch is first ruptured and the rupture propagates out of it, guided by the creeping region in the large patch. (c) Rupture fronts meet at a point on the perimeter of the large patch. (d) Afterslip region expands. (e) Shrinking of the locked patch and aseismic transients generated in the small patch. (f) Beginning acceleration of a small nucleation, the size of which is comparable to inline image. (g–i) The 21st earthquake, which is identical to the 20th one.

[58] Figure 5 shows the limit cycle in the case with α=2.5, using the 20th and 21st earthquakes as examples. The L-event in the limit cycle is nucleated in Patch S (Figure 5a), ruptures it, and then propagates further outward (Figure 5b). The rupture out of Patch S is guided by the creeping region in Patch L instead of expanding isotropically. Near the end of the earthquake, the rupture front concentrates at a point on the perimeter of Patch L, as reported by Kato [2004] (Figure 5c). Those complex rupture processes make the duration of earthquakes longer than the case without Patch S, 4.22RL/cs with α=2 and 6.88RL/cs with α=2.5 for the earthquakes in the limit cycles, even though the size of Patch L is identical and the moment magnitude differs only modestly.

[59] After an L-event, the afterslip expands outward (Figure 5d) and the locked region shrinks similarly to the case without Patch S (Figure 3). In this case with a Patch S, aseismic transients can start in Patch S at an earlier stage when the creeping region in patch L (but outside Patch S) is still too narrow for transients to start there. Transients that started in Patch S propagate somewhat into the adjacent creeping region in Patch L (Figure 5e), but the creeping region in Patch L is too narrow for it to turn into a large nucleation. Instead, one of the recurring transients in Patch S germinates a small nucleation (Figure 5f). The nucleation results in a cascade-up L-event (Figure 5g) which is identical to the previous one (Figure 5a-c, g-i).

[60] Figures 6a and 6b indicate the 1/tf accelerations in the L-events in the cases with α=2 and 2.5, respectively. With α=2, the L-events are always nucleated in the small patch so that Vpatch always follows the 1/tf acceleration for the small nucleation. The system falls into a limit cycle rapidly, so that most of the curves in Figure 6a overlap each other. With α=2.5, only the third earthquake is an L-event by large nucleation outside Patch S. The acceleration of the fault slip prior to this earthquake is explained by the 1/tf acceleration for the large nucleation. All the other earthquakes are cascade-up L-events and follow the 1/tf acceleration for the small nucleation.

Figure 6.

Acceleration of the fault slip prior to the initiation of the large earthquakes in the case with a small patch (a) α=2 and (b) α=2.5.

5.3 β α

5.3.1 α=3, 3.5, and 4

[61] With α=3, 3.5, and 4, the system does not fall into a limit cycle as long as calculated, and L-events are initiated in a variety of ways. Tables 5, 6, and 7 list the earthquakes obtained in the simulations. One of the main differences from the cases with α<β is the larger number of S-events. The recurrence intervals of the L-events are 17.0±2.8 years, 17.0±1.2 years, and 16.6±1.7 years for α= 3, 3.5, and 4, respectively. Those values are smaller than the case without Patch S, but the difference is marginal. Although Patch S generates ruptures frequently, it appears that they can cascade up to L-events only when Patch L is so ready that a large own nucleation would have occurred soon, even if it were not stimulated by the S-event.

Table 5. List of the Earthquakes With α=3
1000y 255d 23:11:246.020cL
2009y 307d 09:10:554.971nS
3017y 258d 13:33:555.149pS
4017y 258d 13:34:166.006dcL
5027y 081d 01:07:364.972nS
6035y 279d 17:33:183.578Co
7035y 279d 17:55:276.039LL
8048y 310d 15:21:285.044nS
9053y 108d 10:26:536.018LL
10063y 003d 14:16:374.942nS
11072y 066d 23:33:186.050LL
12089y 225d 19:34:076.016cL
13098y 313d 14:01:394.947nS
14107y 024d 13:00:196.019LL
15121y 056d 03:11:305.314pS
16121y 057d 07:23:535.926LL
17136y 055d 05:39:206.035cL
18145y 225d 06:21:004.911nS
19153y 316d 11:51:296.048cL
20162y 359d 13:38:234.872nS
21171y 094d 04:01:296.038cL
22180y 168d 13:15:004.985nS
23188y 212d 13:27:546.029LL
24201y 054d 21:50:525.256nS
25205y 148d 15:37:176.008LL
26221y 211d 13:40:336.037cL
27234y 308d 08:34:225.199nS
28239y 086d 18:49:426.020LL
29256y 297d 20:35:066.039cL
30266y 021d 06:31:464.958nS
31274y 206d 06:14:506.031LL
32287y 320d 18:51:565.979cL
33298y 273d 19:15:335.096nS
34305y 357d 15:21:346.034cL
35315y 060d 04:44:384.946nS
36323y 159d 14:59:476.037cL
Table 6. List of the Earthquakes With α=3.5
1000y 253d 10:32:565.988LL26186y 316d 14:09:065.059pS
2013y 327d 23:26:574.896nS27186y 316d 14:26:046.002dcL
3018y 112d 16:02:476.037LL28195y 320d 04:45:364.893nS
4030y 125d 22:52:484.588nS29204y 170d 20:53:426.041cL
5034y 173d 00:46:086.007LL30213y 031d 17:34:144.895nS
6044y 182d 16:11:114.742nS31221y 106d 16:39:394.935pS
7052y 129d 02:13:365.054pS32221y 107d 01:43:153.533Misc.
8052y 129d 02:13:536.029dcL33221y 107d 01:43:246.000dcL
9060y 360d 18:38:114.884nS34230y 105d 20:56:064.877nS
10069y 159d 07:21:004.971pS35238y 171d 00:36:134.925pS
11069y 159d 07:35:146.028dcL36238y 171d 02:23:546.021dcL
12077y 302d 19:03:104.719nS37246y 351d 03:49:164.860nS
13085y 102d 16:26:496.015cL38255y 133d 13:06:294.940pS
14094y 168d 17:52:074.894nS39255y 133d 13:07:184.710Misc.
15102y 230d 11:33:416.038cL40255y 133d 13:10:506.011dcL
16110y 359d 06:11:064.797nS41263y 299d 15:55:364.873nS
17119y 098d 15:26:016.018cL42272y 096d 12:09:584.940pS
18128y 233d 02:19:514.915nS43272y 096d 12:37:376.005dcL
19136y 115d 10:27:394.900pS44281y 000d 15:20:294.893nS
20136y 115d 10:57:466.040dcL45290y 125d 05:21:196.053cL
21144y 313d 15:21:184.710nS46298y 244d 06:05:564.777nS
22152y 265d 01:17:516.025cL47306y 237d 03:49:436.021cL
23161y 265d 22:25:124.862nS48315y 312d 11:25:394.892nS
24170y 039d 16:00:596.035cL49323y 228d 08:13:404.893pS
25178y 225d 16:20:414.888nS50323y 228d 18:13:406.037dcL
Table 7. List of the Earthquakes With α=4
1000y 015d 17:43:324.788pS30168y 234d 23:11:564.863pS
2000y 015d 21:17:565.999dcL31168y 235d 01:52:336.013cL
3008y 023d 17:56:004.781nS32176y 310d 13:14:114.787nS
4015y 283d 15:06:374.790pS33184y 140d 03:50:524.765pS
5016y 044d 18:15:495.996LL34184y 235d 06:04:116.012LL
6024y 166d 07:34:204.732nS35192y 344d 01:10:214.766nS
7033y 024d 19:58:516.019LL36200y 235d 09:15:054.784pS
8043y 071d 12:23:414.459nS37201y 083d 13:42:315.998LL
9047y 154d 17:08:264.767nS38209y 322d 21:11:024.777nS
10052y 282d 16:40:266.048LL39217y 175d 06:06:064.790pS
11060y 261d 16:41:454.548nS40218y 054d 00:55:276.009LL
12068y 281d 13:14:174.855pS41229y 101d 00:04:054.712nS
13068y 281d 14:07:363.849Misc.42233y 184d 22:40:206.002LL
14068y 281d 14:07:446.008dcL43242y 033d 11:14:554.754nS
15077y 070d 13:47:324.773nS44250y 155d 07:03:546.024LL
16084y 274d 14:19:364.780pS45258y 171d 00:36:314.742nS
17084y 362d 08:44:266.017LL46267y 052d 08:47:465.999cL
18093y 048d 05:03:584.761nS47275y 118d 02:38:424.768nS
19101y 285d 03:41:586.016cL48283y 090d 02:38:264.802pS
20110y 119d 20:22:014.790nS49283y 140d 00:51:296.011LL
21117y 336d 17:16:024.769pS50291y 095d 20:19:134.670nS
22117y 354d 07:08:533.364Misc.51299y 147d 14:33:184.839pS
23117y 354d 07:08:576.011dcL52299y 147d 14:33:244.461Misc.
24126y 169d 14:18:074.789nS53299y 147d 14:49:334.572pS
25134y 274d 10:13:016.019LL54299y 147d 14:52:204.056Co
26142y 355d 15:54:064.703nS55299y 147d 14:54:515.976cL
27147y 123d 13:23:412.844Co56307y 263d 18:14:114.704nS
28151y 241d 23:15:546.007LL57315y 229d 23:54:186.034cL
29159y 364d 00:22:374.648nS   

[62] The number of L-events initiated by large nucleation is 9, 3, and 12 for α= 3, 3.5, and 4, respectively, not showing a simple monotonic dependency on α. This point will be further discussed later. On the other hand, the number of dynamic cascade-up L-events decreases monotonically with α (10, 8, and 5 for α= 3, 3.5, and 4, respectively). A smaller Patch S (larger α) has smaller energy release rate when it is ruptured, therefore having less ability to rupture Patch L.

[63] An S-event sometimes takes place as a precursor to the following L-event. Figure 7 represents Vpatch before the nonprecursory S-events (left column) and precursory S-events (right column) by black lines with the other earthquakes by light gray lines. They mostly follow 1/tf acceleration for small nucleation. There, however, is a difference between nonprecursory and precursory S-events. Before the 1/tf acceleration becomes dominant (say, at tf=106RL/cs), the value of Vpatch is typically of the order of one tenth of Vpl for nonprecursory S-events. On the other hand, before the precursory S-events, Vpatch tends to be larger, sometimes exceeding Vpl. However, there may not be a clear threshold value of Vpatch at a certain 1/tf as most clearly suggested by the case with α=4 (Figures 7e and 7f).

Figure 7.

Acceleration of the fault slip Vpatch prior to S-events. (a) Nonprecursory S-events (nS) with α=3. (b) Precursory S-events (pS) with α=3. (c) nS with α=3.5. (d) pS with α=3.5. (e) nS with α=4. (f) pS with α=4. Vpatch before the other earthquakes are indicated by light gray lines.

[64] With α=4, one of the nonprecursory S-events occurs during an aseismic transient far from Patch S (Figure 7e). In this case, the S-event does not cascade up or trigger a L-event even though Vpatch is quite high. Vpatch is simply the spatial average of V1 over Patch L, while what really matters in determining whether an earthquake that has ruptured Patch S cascades up (dynamically or with some delay) or not is the readiness of the fault in the vicinity of Patch S. Vpatch is contaminated by moment release information that is irrelevant in determining the condition for cascade-up, and more local information around Patch S would give more directly relevant information.

[65] Before delayed cascade-up L-events, there are precursory S-events by definition. The acceleration after these S-events follows 1/tf acceleration of large nucleation if there is long enough time between the S- and L-events (Figure 8). The time interval between a precursory S-event and a delayed cascade-up L-event distributes in logarithmic spacing, and no characteristic time can be recognized, partly due to small number of the examples simulated.

Figure 8.

Acceleration of the fault slip prior to delayed cascade-up L-events with (a) α=3, (b) α=3.5, and (c) α=4.

[66] Vpatch before the L-events initiated by large nucleation is well explained by 1/tf acceleration for the large nucleation (Figure 9), although it is sometimes perturbed significantly by precursory S-events and rather complex aseismic transients. A few L-events are initiated when the large nucleation touches Patch S (Figure A5). In these cases, the initiation of the L-event is advanced in time, and hence, Vpatch as a function of log10(1/tf) becomes flat at a relatively large tf compared with the other L-events by large nucleation (section A2).

Figure 9.

Acceleration of the fault slip prior to the initiation of the L-events by large nucleation with (a) α=3, (b) α=3.5, and (c) α=4.

[67] Before the cascade-up L-events, the 1/tf acceleration for the small nucleation explains the simulated fault behavior well for many cases (Figure 10) especially for α=3. As αincreases and as Patch S becomes smaller and smaller compared with the nucleation size of Patch L, however, less and less of the cascade-up L-events follow the 1/tf acceleration for the small nucleation. Dynamic cascade-up from a smaller Patch S seems to require a more significant reduction in the fracture energy in the surrounding region beforehand, which is provided by either more significant aseismic transients, large nucleation in its early stage, or the afterslip of an S-event.

Figure 10.

Acceleration of the fault slip prior to cascade-up L-events with (a) α=3, (b) α=3.5, and (c) α=4.

5.3.2 α=5

[68] In the case with α=5, 17 earthquakes are initiated by large nucleation, none is by dynamic cascade-up, two are by delayed cascade-up, and one is by coalescence among 20 L-events (Table 8). The recurrence interval of the L-events is 17.3±2.8 years, which is not significantly shorter than the case without Patch S. It can be said that the existence of this small Patch S hardly affects the timing of L-events.

Table 8. List of the Earthquakes With α=5
1000y 007d 16:24:074.607pS36187y 037d 07:29:375.981LL
2000y 011d 09:56:206.003LL37195y 157d 10:50:594.620nS
3006y 109d 18:50:594.529nS38202y 089d 19:41:214.601pS
4013y 348d 13:01:424.605nS39202y 316d 19:53:156.011LL
5017y 333d 22:46:526.020LL40209y 214d 04:33:414.577nS
6028y 204d 11:39:054.345nS41216y 026d 01:13:244.526nS
7034y 045d 12:33:384.593nS42216y 026d 01:13:301.821Misc.
8036y 139d 21:40:356.036LL43216y 026d 01:13:502.914Co
9042y 274d 00:41:3154.555nS44220y 338d 22:23:174.637pS
10050y 226d 19:24:024.606nS45221y 041d 13:04:296.035LL
11055y 041d 19:52:476.035LL46227y 094d 08:53:444.551nS
12063y 024d 11:47:064.544nS47234y 338d 22:18:054.584nS
13070y 168d 09:55:394.578pS48239y 075d 05:04:286.025LL
14071y 234d 22:54:226.018LL49247y 211d 03:10:424.511nS
15079y 243d 23:07:134.300nS50256y 022d 18:04:464.630pS
16086y 301d 17:40:414.628pS51256y 041d 21:16:096.019LL
17088y 028d 20:25:256.017LL52262y 314d 10:13:304.368nS
18101y 004d 19:17:264.496nS53270y 070d 02:31:161.657Misc.
19105y 293d 19:59:366.035LL54270y 070d 02:31:204.611nS
20115y 213d 10:45:364.395nS55274y 211d 06:52:034.625pS
21115y 213d 10:45:451.355Misc.56274y 211d 08:14:462.567Misc.
22122y 061d 05:54:314.614pS57274y 211d 08:14:576.043dcL
23124y 344d 07:55:216.037CoL58280y 249d 18:57:234.490nS
24135y 090d 13:06:004.549nS59288y 060d 12:50:144.606nS
25142y 120d 11:22:474.609pS60292y 201d 19:28:166.024LL
26142y 196d 03:25:013.583Misc.61306y 078d 02:17:522.045Misc.
27142y 196d 03:25:196.011LL62306y 078d 02:17:554.596nS
28149y 079d 09:06:474.555nS63311y 101d 17:11:564.588pS
29157y 037d 16:04:564.616pS64311y 109d 02:22:433.488Co
30157y 039d 00:43:245.992dcL65311y 109d 02:53:143.504Misc.
31165y 129d 02:14:214.629nS66311y 109d 03:01:376.036LL
32172y 104d 00:56:134.590pS67317y 201d 22:55:404.507nS
33172y 320d 12:09:176.006LL68317y 201d 22:55:462.193Misc.
34179y 117d 17:23:254.569nS69325y 033d 08:14:104.598nS
35187y 025d 05:27:354.607pS70329y 108d 21:13:206.021LL

[69] The acceleration prior to the S- and L-events are well explained by 1/tf acceleration for small and large nucleation, respectively (Figure 11). Vpatch(tf) for the nonprecursory and precursory S-events are not much different (Figures 11a and 11b). Many of the L-events have precursory S-events, but Vpatch typically shows 1/tf acceleration for the large nucleation at least for 103RL/cs (Figures 11c and 11d). This means that even if an S-event triggers an aseismic transient that ultimately ends up with an L-event, a large quasi-static nucleation process cannot be skipped.

Figure 11.

Acceleration of the fault slip prior to the initiation of the large earthquakes in the case with α=5. Note that there is no earthquake initiated by cascade-up. (a) The nonprecursory S-events “nS”. (b) The precursory S-events “pS”. (c) The L-events by large nucleation “LL”. (d) The L-events by delayed cascade-up “dcL”.

6 Discussions

6.1 Should a Large Earthquake Have a Large Preparation?

[70] The question we would like to address in the present study is, as mentioned earlier, whether large earthquakes have large preparation processes comparable to the own nucleation or not if we postulate the concept of hierarchical asperity by Ide and Aochi [2005]. The short answer is, “it depends” not only on the distribution of the parameters but also on the situation just before large earthquakes.

6.1.1 Acceleration Before an Earthquake

[71] By the series of simulations with different hierarchical gap α=RL/RS, we have shown that a large enough Patch S almost always causes cascade-up when it ruptures, while a small enough Patch S does not. The transition takes place around α=3, between 2.5 and 5, when the brittleness inline imageis assumed to be 3 (Figure 12). In the transitional regime, L-events are initiated in a variety of ways.

Figure 12.

Statistics of the L-events as a function of α=RL/RS.

[72] L-events initiated by a large nucleation have a preseismic moment release that is scaled by the event length scale to the third and thus have large preparation. Roughly speaking, the fraction of such earthquakes among all L-events increases as the hierarchical gap dominates (Figure 12).

[73] If L-events are initiated by cascade-up or delayed cascade-up, they have much smaller preseismic moment release than those by large nucleation. If brittleness dominates, the acceleration due to a small nucleation preceding cascade-up becomes significant directly from the interseismic level. In this case, we cannot distinguish the size of the earthquake by looking at the preseismic motion even if it is detected.

[74] In the transitional regime, preparation for large earthquakes can be seen even when own large nucleation is skipped. It is true that L-events by cascade-up or delayed cascade-up have much smaller preseismic moment release than what is expected for large nucleation. However, they tend to have higher preseismic moment release rate than simple nonprecursory S-events (Figures 7, 8, and 10) although a clear threshold in the moment release rate (or Vpatch) at a certain time-to-failure tf may not exist.

6.1.2 Nonprecursory Versus Precursory S-event

[75] In addition to the acceleration just before L-events, precursory S-events can be regarded as a part of the preparation for L-events which is obviously larger than the preparation for S-events.

[76] In the transitional regime where the brittleness and the hierarchical gap are comparable, precursory S-events tend to have a larger preseismic moment release rate than nonprecursory ones (Figure 7). In addition, for precursory S-events that fail to cascade-up with some delay and are followed by large nucleation, the background moment release rate is somewhat smaller than precursory S-events before delayed cascade-up (e.g., Figures A1c and A1g).

[77] Nonprecursory S-events and precursory S-events are different not only in the preseismic moment release but also in the postseismic moment release or the afterslip. Figure 13 represents the evolution of Vpatch as a function of time after the S-events ta, until their next earthquakes for the cases with α=3, 3.5, and 4. From Figure 13, we see that precursory S-events are followed by a much larger afterslip than usual (nonprecursory) S-events. It still decays as 1/ta, same as after usual events, but the moment release rate at a given ta is several times higher. Such an abnormally large afterslip may be useful as a telltale sign of high readiness, while it plays a role in prompting large nucleation. If we look at the afterslip following precursory S-events in Figure 13 closely, there is a tendency that the time interval to the next L-events by large nucleation is shorter for stronger afterslip, so the afterslip of small earthquakes can be even a quantitative indicator of the readiness.

Figure 13.

Evolution of Vpatch after S-events. Black and gray lines represent afterslips of precursory and non-precursory S-events, respectively. (a) α=3. (b) α=3.5. (c) α=4.

[78] When large nucleation starts in the area of reduced strength by the strong afterslip, Vpatch turns to acceleration following large nucleation 1/tf curve. This is of course an immediate precursor of a large earthquake. However, we must note that even after this strong afterslip, acceleration does not necessarily proceed to an earthquake straightforwardly as seen from Figure 13, where it sometimes decelerates again before it eventually accelerates further again to proceed to a large earthquake. So, if we cannot tell if the acceleration is the large nucleation or not by its amplitude, false immediate alarms can occur. Miyazaki et al. [2003] reported a transient acceleration and deceleration in the afterslip rate of 1996 Hyuganada earthquakes (two Mw 6.7 events in succession) which were not followed by a great earthquake. At least, however, our simulations suggest that the chance of the acceleration being the true immediate precursor is higher when it happens following an unusually large afterslip.

[79] Three days before the 2011 Mw 9.0 Tohoku-Oki earthquake, Japan, a Mw7.3 foreshock took place near the hypocenter of the main shock. The postseismic behavior of the foreshock has been studied, and several publications have pointed out that the aftershock region expanded and reached the hypocenter of the main shock [e.g., Kato et al., 2012]. Because of the lack of sensitivity in the geodetic inversion there [e.g., Loveless and Meade, 2010, 2011], it is probably difficult to determine the postseismic moment release rate following the foreshock from existing data directly. In addition, we also note that an abnormally large afterslip had been observed over a broad region in the Mw 9 rupture area for several years before the Tohoku-Oki earthquake [Suito et al., 2011]. Also, slow slip events have been observed since November 2008 near the epicenter [Ito et al., 2012]. These are not necessarily short-term precursors on a human time scale. Yet, several years is 1 % of the Mw 9 cycle there [Sawai et al., 2012], and we regard those observations as an indication of the high degree of the readiness in the Mw 9 patch. According to our simulations, such “readiness” involves reduction of fault strength (meaning physical damage as extended in section 6.2), not only the tectonic stress buildup.

[80] We would like to emphasize that we are not arguing that the outcomes here (sections 6.1.1 and 6.1.2) can be of practical use, though they could be. In order to capture the large nucleation or the preseismic acceleration after a precursory S-event (if it exists in nature), we need to resolve a timescale comparable to or orders of magnitude smaller than that for the large nucleation, i.e., tf at which the 1/tf acceleration for the large nucleation meets Vpatch=Vpl (e.g., Figure 8). A dense off-shore network could help, but a quantitative evaluation of the sensitivity and spatio-temporal resolution must be done to see if it can really help or not. In measuring time-integrated values such as displacements by GPS stations, the required resolution of the short time scale is probably tough to achieve. Even if we could monitor the moment release rate perfectly, the recognition of the 1/tf acceleration in advance would be quite difficult though not impossible in principle. Oscillations in the moment release rate or Vpatch (e.g., Figure 13) would cause false alarm, and it is possible for the earthquake initiation to be advanced in time by the interaction with fragile patches as demonstrated in section A (See Figures A1d and A5). Also, temporal evolution of the preseismic moment release rate may be sensitive to specific frictional properties of natural faults. On top of these difficulties, a natural fault of our interest may have such a small hierarchical gap that the earthquakes there may be generated by cascade-up without any detectable preparation, or the assumption we have postulated here that the small earthquakes are characteristic earthquakes of small patches (the hierarchical asperity model) might not be the case [e.g., Lapusta and Rice, 2003; Kaneko and Lapusta, 2008]. In order to address these issues, more constraints from observational studies are required.

6.2 Readiness and Time-Variable Critical Crack Size Around Patch S

[81] Even if the same Patch S ruptures, its fate depends on the condition around it as clearly shown in the simulations presented so far. The size of Patch S relative to the nucleation size of Patch L explains the likelihood of cascade-up L-events, but what determines the fate of individual cases? Or, what does “readiness” really mean?

[82] Cascade-up L-events occur when the high-Gc region around Patch S cannot stop the expansion of a rupture in Patch S. This situation is directly related to the notion of a critical crack size.

[83] The nucleation size equations (9) and (12) can be derived from the estimation of critical crack size under certain assumptions [Rubin and Ampuero, 2005]. Suppose Patch L has a certain state variable θbg and is slipping at a certain slip rate Vbg, and there is a circular potential nucleus in Patch L the radius of which is r, having θnuc and Vnuc. Note that the background values θbg and Vbg correspond to typical interseismic condition and the nucleation values θnuc and Vnuc, to subseismic values just before the beginning of dynamic propagation of a rupture. The stress drop can be written as

display math(26)

Selecting the shear stress inside the nucleus as the reference state of the linear elasticity from which stresses are measured, the energy release due to the existence of the potential nuclei is [e.g., Kanamori and Anderson, 1975]:

display math(27)

Note that the Poisson's ratio is assumed to be 1/4. For a hypothetical isotropic expansion of the potential nucleus, the energy release rate G (the derivative of W with respect to the area of the nucleus) becomes

display math(28)

On the other hand, the fracture energy Gc is given by

display math(29)

For a critical crack r=rc, G=Gc is satisfied, so that rc is

display math(30)

With the selection a/bin=0.8 adopted in the present study, it is known that the constant-weakening limit is a good approximation for spontaneous nucleation [Rubin and Ampuero, 2005]:

display math(31)

This namely means

display math(32)

This is satisfied, for example, for acceleration of slip at nearly steady-state frictional resistance. Then equation (30) leads to

display math(33)

This is only modestly different from the estimate reported by Chen and Lapusta [2009]. If we consider nucleation as the result of a growth of instability in the middle of creeping region, the assumption equation (32) is probably satisfied approximately. Note that on rupturing of Patch S, however, the situation is quite different; Patch S often ruptures when a creeping front invading Patch L interacts with Patch S, and a locked portion ahead of the creeping front is far from the steady-state sliding (e.g., Figure 5).

[84] Although the cascade-up rupture propagation out of Patch S is not isotropic, suggesting that the hypothetical crack expansion cannot give a quantitatively correct criterion for dynamic cascade-up, let us consider how the penetration of a creeping front into Patch L affects the likelihood of cascade-up qualitatively on the basis of equation (30). It is clear from our simulations that when a locked Patch L starts creeping near its rim due to the invasion of a creeping front, both slip rate and shear stress increase while the state variable decreases (e.g., Figures 3 and 5). If the characteristic slip rate and state variable inside the hypothetical nucleus (Vnuc and θnuc) is fixed, then the creeping region has greater potential stress drop Δτ and smaller fracture energy Gc than a locked region. Obviously from equation (30), the critical crack length of the creeping region is smaller than that in the locked region.

[85] In the context of cascade-up, readiness of the fault means how small the critical crack length is. Even if the magnitude of the perturbation to the fault added by a rupture of Patch S is fixed, it may cause an S-event or a cascade-up L-event depending on the critical crack length there at the moment. That is, even an asperity-model sort of system, where the patchy distribution of physical parameters such as Gc plays a central role in determining the size of the earthquakes, may show such variability as demonstrated here because of variable initial conditions for individual dynamic ruptures.

[86] It should be noted that aseismic transients take place during interseismic periods, which indicates that the readiness around Patch S does not monotonically increase with time. Also, the patches have characteristic recurrence intervals [Kato, 2003]. If the natural recurrence periods of Patchs S and L resonate, the dependency of the ratio among the event types on parameters could be complex. The numbers of large earthquakes by large nucleation and delayed cascade-up do not change monotonically with α(Figure 12), which might be an example of this effect.

[87] If a small rupture in Patch S fails to cascade-up dynamically or with some delay, its afterslip may trigger large nucleation. As mentioned earlier, large nucleation occurs in one of the aseismic transients in Patch L. The afterslip of the S-events is an example of aseismic transients, and it may well trigger large nucleation as demonstrated in our simulations. It should be noted that at the beginning of such afterslip, large nucleation has not necessarily started yet. The readiness here is different from what we have discussed so far in this section. The question here is, “Under what circumstances does the next aseismic transient lead to large nucleation?” One condition would be that a large enough region inside Patch L has been swept by an inward creeping front so that the large nucleation can be accommodated. However, because of the oscillatory nature of the aseismic transients whose propagation necessarily causes heterogeneity, it is difficult to quantify the readiness in this context only by averaged properties. Further study is needed to understand the condition for the triggered large nucleation.

6.3 More Complex Parameter Distribution

[88] The parameter distributions studied consists of just two circles and while being very simple, produce surprisingly rich behaviors. It is quite possible that the natural faults have more complex distribution of the fault constitutive parameters. Investigation of the effect of heterogeneous distribution of fault constitutive parameters deserves further study, as well as studies on how the distribution on the natural fault should look. In this section, we shall present some speculations about the effect of more complex distributions.

6.3.1 Location of Patch S

[89] The location of the Patch S may significantly affect the behavior of the system. In the simulations presented, Patch S is always assumed close to the rim of the Patch L in order to maximize its activity. Although systematic parameter studies have not been performed, we have tried a case where Patch S (α=3) is assumed at the center of Patch L, to obtain only cascade-up L-events. By the time when the creeping front reaches Patch S and generates small nucleation there, the region around Patch S becomes ready for cascade-up partly because of the geometrical symmetry. If Patch S is smaller or Patch L is more brittle, the L-events would be initiated by large nucleation even in a similar configuration of Patch S at the center because large nucleation would occur before a creeping front reaches Patch S.

[90] Under what circumstances can small patches have their characteristic earthquakes? In order for them to be loaded efficiently, they should be located near the rim of the large patch. If the hierarchical gap α is sufficiently larger than the brittleness β and if there are many small patches at locations including the rim of a large patch, then small earthquakes may occur frequently outlining a large locked patch. Such a situation is exactly what “streaks and holes” [e.g., Barbot et al., 2012] stands for. On the other hand, if small patches are concentrated near the center of the large patch, then they will not work as asperities having characteristic earthquakes.

6.3.2 Multiple Small Patches

[91] In the study by Ide and Aochi [2005], cascade-up is basically achieved by the help of small fragile patches adjacent to a small rupture. We have not explored such a system intensively partly because of the limited numerical resources available. A preliminary numerical experiment (the geometry of which is depicted in Figure 14) has yielded only one cascade-up L-event via interaction of Patches S2 and S3 before the twentieth L-event, which is the 212th earthquake in the simulation. In addition to α and β, it is obvious that the distance between small patches is an important parameter here, and the parameter study is quite expensive numerically because of the large number of S-events.

Figure 14.

Distribution of multiple small patches simulated. α=5 and β = 3. Patches S2 and S3 rarely cascades up. Also see Figure 2.

[92] In order to have large nucleation which does not interact with small patches, the area inside Patch L (but outside of the small patches) must be large enough to accommodate a circle of radius inline image. Therefore, if Patch L includes enough number of small patches, a large nucleation necessarily interacts with them, and thus, a typical large nucleation and corresponding 1/tf acceleration would be difficult to observe. On the other hand, however, the energy release rate of small ruptures is scaled by RS3, so interaction with very small patches may be negligible in determining the fate of large nucleation.

[93] In the study by Ide and Aochi [2005], the earthquake size distribution similar to the Gutenberg-Richter relation (GR) is recognized for many earthquakes simulated by starting from each of many smallest patches. They also noted that their result is not directly comparable to GR because they were not considering the evolution of the system; a smaller patch may rupture more frequently. In the present study, we do obtain smaller earthquakes (S-events) more frequently than larger ones (L-events) for a value of αgreater than 3.5, even with only one Patch S for one Patch L. However, if the hierarchical gap is not so large, S-events are rare compared with L-events. In an extreme case (α=2), an S-event never occurs. Even with these observations, however, GR cannot be necessarily translated into the statistics of patch distribution of the fault properties. For example, a significant portion of the smaller earthquakes in the GR statistics can occur off the fault plane of the dominant earthquake.

7 Conclusions

[94] In the hierarchical asperity model, which can explain many observational facts such as scaling of fracture energy with the size of earthquake and occurrence of small earthquakes inside the rupture area of a large earthquake, small fragile patches are embedded in a large tough patch, and such a structure exists for a wide range of length scale. Under such conditions, a large earthquake may be generated by cascade-up from a small rupture in one of the fragile patches, or directly by large nucleation in the tough patch. Whether a large earthquake has a large preparation process or not is an important problem, and we have investigated it by means of earthquake sequence simulations. The distribution of the fracture energy is realized in a rate- and state-dependent framework by the heterogeneous distribution of the characteristic slip distance for the state evolution.

[95] We have put our focus on an elementary process that is the interaction of two scales by setting a circular fragile patch (Patch S) within a larger circular tough patch (Patch L). The brittleness of the patches, β (the ratio of the patch size to the nucleation size), is set at 3 so that earthquakes are generated by nucleation and span the patch if there is no strong heterogeneity in it. The simulations are conducted for various ratios αof the size of Patch L to that of Patch S, and its effect on the system behavior has been elucidated. α may be regarded as the parameter quantifying the hierarchical scale gap.

[96] If α is smaller than β or Patch S is larger than the nucleation size of Patch L, then the system falls into a limit cycle having only one earthquake which is a cascade-up large earthquake. On the other hand, if α is much larger than β (α=5 with β = 3), then we obtain no cascade-up large earthquake as long as calculated. In between, the system shows a rich behavior in which large earthquakes (L-events) are generated by large nucleation, by dynamic cascade-up, or by delayed cascade-up. In addition, some of the small earthquakes (S-events) that rupture only Patch S could be regarded as precursory earthquakes to L-events which are triggered by afterslip of precursory S-events. With increasing α, less of the L-events are initiated by dynamic cascade-up.

[97] The preparation process (the acceleration toward initiation of earthquakes) shows different characteristics for different types of earthquake initiation. Nucleation is characterized by a 1/tf acceleration of the moment release rate with its amplitude scaled by the nucleation size to the third power where tf is the time to failure. A large nucleation is characterized by 1/tf acceleration comparable to the nucleation size of Patch L, while S-events are comparable to the nucleation size of Patch S. If a large earthquake follows a small nucleation, the background seismic moment release rate before 1/tf acceleration becomes significant tends to be larger than for nonprecursory small events, indicating that a high degree of preparation (raised stress and/or reduced fracture energy) is required.

[98] By extrapolation of the present results obtained for a limited range of parameter space, we infer the following. If the fault constitutive properties are distributed on the fault such that the hierarchical gap is smaller than the brittleness of patches, then all large earthquakes would be generated through dynamic cascade-up. If that is the case, we would have few small earthquakes and there would be no detectable precursory signal for large earthquakes. On the other hand, if the hierarchical gap dominates, then small ruptures rarely cascade-up and large earthquakes may be initiated by their own nucleation. In between, the system behaves in a complex manner such that large earthquakes are initiated in variety of ways.

[99] Thus, as for the compact quasi-static nucleation immediately preceding the dynamic rupture, the chance to have a large one expected of the large tough patch of the large earthquake depends on the size of the inner fragile patches present. An interesting suggestion for the case with the brittleness comparable with the scale gap is that a repeating characteristic earthquake of the same large patch is sometimes preceded by large immediate nucleation, and sometimes not. On the other hand, within our simulations, it has been noticed that a fairly high degree of readiness is required for the rupture of a small patch to cascade up dynamically to a large earthquake. Within the studied range, it seems that such a “ready” condition is one for which a spontaneous nucleation of the large patch is about to start. Physically, “readiness” seems to correspond to the reduced strength (fracture energy) over a good part of the large tough patch, as indicated by abnormally large afterslip of the small earthquakes that happened to occur in this period. Even if we are not lucky enough to have a small earthquake occurrence as a “test probe,” the region of reduced strength manifests as a region of raised aseismic slip rate. In practice, the limited resolution of geodetic methods and the oscillatory nature of aseismic transients (which, even worse, occurs episodically beginning from a much earlier stage of the cycle) make the geodetic judgment of readiness difficult, resulting in a high ratio of false alarms. However, we emphasize that this “ready” state, which is fairly well restricted in time, is a process that necessarily appears and deserves more attention, though it will never achieve an immediate alert with a low ratio of false alarms expected of large nucleation when it occurs. At last, we note that these implications for earthquake prediction are of course under the assumption that the hierarchical patch model is an appropriate description of natural large faults.

Appendix A:: Detailed Description of Earthquake Types by Examples

[100] In this Appendix, we present examples of each earthquake type defined in the present study. The earthquakes shown in the following subsections are generated in a single sequence simulation with α=β = 3 (Table 5). For this case, the spatio-temporal distributions of slip rate, shear stress, and state variable, and the spatial average of them are shown in the supporting information. Examples of Vpatch as a function of tf for earthquakes presented in this Appendix are summarized in Figure A1.

Figure A1.

Fault acceleration prior to typical earthquakes presented in section A. (a) A nonprecursory S-event. (b) An L-event by large nucleation. (c) A precursory S-event and the following L-event by large nucleation. (d) An L-event by large nucleation which touched Patch S. (e) A cascade-up L-event. (f) A cascade-up L-event after interaction of Patch S and large nucleation in its early stage. (g) A precursory S-event and the following delayed cascade-up L-event. (h) A tiny earthquake by coalescence of a creeping front and the following L-event by large nucleation.

A1 S-events

[101] Figure A2 shows an example of an S-event, the second earthquake in the sequence. This is a nonprecursory S-event. S-events are typically nucleated when the creeping front invading from the rim of Patch L meets and penetrates into Patch S. The moment magnitude of the S-events depends on the size of Patch S or α. The afterslip of S-events usually propagates outside Patch S and inside Patch L, guided by the already creeping region near the rim of Patch L where the strength Θ has been reduced. Vpatch as a function of tf for this earthquake is plotted in Figure A1a, indicating that the fault acceleration prior to the earthquake is well explained by the 1/tf acceleration for the small nucleation (equation (25)).

Figure A2.

An example (the second earthquake with α=3) of an S-event which ruptures only Patch S and the following afterslip. The dashed white circles indicate Patch L and S. (a) An early stage of the small nucleation about 8 days before the earthquake. (b) The beginning of the coseismic period when the maximum slip rate exceeds 0.1 m/s =3.942×107Vpl. (c) The end of the coseismic period. (d–e) The afterslip guided by the creeping region in Patch L and locking of Patch S.

[102] S-events are classified into two categories, one is nonprecursory (nS) and the other is precursory earthquakes to L-events (pS). Examples of precursory S-events are shown in section A2 together with the following L-events.

A2 L-events by Large Nucleation

[103] Even if Patch S exists, we sometimes see L-events initiated by large nucleation (LL). An example, the eleventh earthquake, is shown in Figure A3. A nucleation takes place away from Patch S and generates a rupture spanning Patch L. After the L-events, afterslip expands outside Patch L with stress concentration at the outward front while Patch L itself gets locked (i.e., the slip rate becomes orders of magnitude smaller than Vpl). Note that Patch S decelerates faster than Patch L because of faster state evolution. Those L-events show 1/tf acceleration prior to it corresponding to the large nucleation (equation (24), Figure A1b).

Figure A3.

An example (the eleventh earthquake with α=3) of an L-event initiate by large nucleation. The dashed white circles indicate Patches L and S. The color scale is same as in Figure A2. (a) An early stage of the large nucleation about 7 days before the earthquake. (b) Acceleration in the nucleus about 30 min before the earthquake. (c) The beginning of the coseismic period. (d) The end of the coseismic period. (e–f) Afterslip outside Patch L and locking of it.

[104] An L-event by large nucleation is sometimes preceded by a precursory S-event by an interval much shorter than the recurrence of L-events. We define an S-event as precursory if the following L-event occurs before the after slip of the S-event decelerates to the interseismic level. However, examination of each case reveals that the thus-defined precursory S-events indeed play a role in initiating the following L-event. An example is shown in Figure A4. In this example, the L-event is initiated 28 h after the S-event. The quasi-static nucleation to the following L-event occurs in the basically same manner as that for L-events by large nucleation without a precursory S-event. However, it is evident that the afterslip of the precursory S-events has quickly raised the readiness (reduced the strength) of the area for an eventual large nucleation and prompted its timing. Figure A1c indicates that the acceleration of the fault slip prior to the precursory S-event (fifteenth) is explained by the 1/tf acceleration for small nucleation (equation (25)), while the acceleration before the L-event (sixteenth) follows that for large nucleation (equation (24)). Note that those two lines are apparently merged at larger tf than 106RL/cs because the interval between those two earthquakes about 105RL/cs becomes significantly smaller than tf. In the simulations presented in this study, the interval from a precursory S-event and a following L-event by large nucleation is from 2.29×10−4 to 8.42×10−2 of the recurrence of L-events.

Figure A4.

The fifteenth and sixteenth earthquakes with a small patch with α=3. An L-event (sixteenth) is initiated 28 h after a S-event (fifteenth). The color scale is same as in Figure A2. (a) The beginning of the coseismic period of the fifteenth earthquake. (b) The end of the coseismic period of the fifteenth earthquake. (c) Afterslip propagating along the creeping region in Patch L. (d) The beginning of the coseismic period of the fifteenth earthquake initiated by large nucleation. (e) The end of the coseismic period of the sixteenth earthquake.

[105] A large nucleation may interact with Patch S and cause further variation in the manner of initiation of earthquakes. An example is shown in Figure A5. In this case, a large nucleation touched Patch S after the slip rate reached around 1 mm/s. The final acceleration to a seismic slip rate occurred in a region much smaller than inline image (Figure A5c). This earthquake is classified as an L-event by large nucleation for the following reasons. First, in the cascade-up earthquake, a rupture propagates from a smaller length scale to a larger length scale, while on the initiation of the earthquake shown in Figure A5, the rupture propagates mainly into the large nucleus instead of rupturing Patch S first. Second, Vpatch as a function of tf followed the 1/tf acceleration for the large nucleation, though it became flat due to the pre-mature initiation of the earthquake (Figure A1d). It should be noted that this flat Vpatch as a function of log10(tf) does not mean slow acceleration (i.e., small dVpatch/dt). Because the earthquake initiation was advanced by the earlier accident, tf is smaller than what it would be if Patch S did not exist and the large nucleus led to a normal L-event. While the tf is much larger than this advance, the 1/tf acceleration in the figures are little affected. But as tf decreases and becomes comparable to the advance, its effect becomes significant; 1/tf becomes significantly larger than what it would be without the advance. Therefore, Vpatch deviates downward from the 1/tf acceleration for the large nucleation.

Figure A5.

An example (the ninth earthquake with α=3) of the L-events initiated by small nucleation. The dashed white circles indicate Patch L and Patch S. The color scale is same as in Figure A2. (a) A large nucleation takes place outside Patch S. (b) The beginning of the ninth earthquake by the large nucleation at a later stage touching Patch S. (c) A magnified view of (b) showing local acceleration. The dashed black circle indicates the nucleation size of Patch S. (d) The ninth earthquake propagates into the large nucleation instead of rupturing Patch S first as is the case for cascade-up earthquakes. (e) The ninth earthquake spans Patch L.

A3 L-events by Cascade-Up

[106] When small nucleation takes place in Patch S, it may dynamically cascade up, ending up with an L-event (cL) (Figure A6). In the cascade-up earthquakes, Patch S is first ruptured, then the dynamic rupture propagates out of Patch S guided by the creeping region in Patch L, and finally spans Patch L.

Figure A6.

An example (the twelfth earthquake with α=3) of the L-events initiated by small nucleation. The dashed white circles indicate Patch L and Patch S. The color scale is same as in Figure A2. (a) The beginning of the coseismic period of the twelfth earthquake, which is initiated by small nucleation. The earthquake (b) first ruptures Patch S, (c) propagates out of it, (d) and then spans Patch L.

[107] In spite of the large total seismic moment released, the cascade-up earthquakes show fault acceleration prior to them which is smaller than what is expected for large nucleation and basically comparable to the small nucleation for S-events (Figure A1e). Ide and Aochi [2005] suggested that the cascade-up takes place across many hierarchical gaps, resulting in a undetectably small initial phase for even the largest earthquake.

[108] If Patch S is smaller than the nucleation size of Patch L, the adjacent region to Patch S must be prepared enough for dynamic cascade-up to take place (e.g., creeping and consequent reduction in fracture energy). Although the nucleation has a similar length scale comparable to inline image as recognized in the snapshot of the slip rate, the preparation before the small 1/tf acceleration tends to differ somewhat between the cases of nonprecursory S-events and precursory S-events or cascade-up L-events. This issue is discussed in section 6.2. An extreme example, the 34th earthquake, is shown in Figures A7 and A1f. In this case, Patch L was so ready that a large nucleation started (Figure A7a), and then the large nucleation touches Patch S in its early stage, triggering a small nucleation. The fault acceleration is shifted from that for large nucleation to that of small nucleation. Even though a patchy acceleration comparable to the size of large nucleation occurred, we classify this case as a cascade-up L-event because the eventual Patch L rupture was a dynamic cascade-up from a Patch S rupture.

Figure A7.

An example (the 34th earthquake with α=3) of the L-events initiated by small nucleation. The dashed white circles indicate Patch L and Patch S. The color scale is same as in Figure A2. (a) A large nucleation outside Patch S about 17 h before the earthquake. (b) A large nucleation touches Patch S about 7 h before the earthquake, and a creeping front invades Patch S. (c–e) A cascade-up L-event occurs.

[109] Figure A8 presents typical moment release rates as a function of time during coseismic periods and just before them for example earthquakes introduced so far in section A. Note that the moment release rate is proportional to Vpatch. For L-events by a large nucleation, the preseismic moment release rate is much higher than that for S-events. In the ninth earthquake which is initiated by large nucleation touching Patch S, the final acceleration to the coseismic period is much quicker than that for an ordinary large nucleation. Cascade-up L-events have preseismic moment release rate only comparable to S-events, much smaller than L-events by large nucleation. During cascade-up L-events, partial seismic moment comparable to S-events is first released, and then the rest is released shortly after.

Figure A8.

Moment release rate as a function of time after initiation of earthquakes until the end of the earthquakes in characteristic cases. At the initiation of earthquakes, the maximum slip rate is 0.1 m/s.

A4 L-events by Delayed Cascade-Up

[110] Even though an earthquake nucleated in Patch S fails to cascade up dynamically, it sometimes causes an L-event to initiate close to Patch S shortly after. One such case, a sequence of third and fourth earthquakes, is shown in Figure A9. The third earthquake ruptures a region which includes Patch S (Figures A9a and A9b). This is classified as a precursory S-event. The following afterslip does not simply expand while decelerating like in Figure A2, but accelerates within a region adjacent to Patch S including its boundary, and the maximum slip rate again exceeds the threshold 0.1 m/s defining the coseismic period (Figure A9c). The fourth earthquake propagates into Patch S and Patch L at the same time (Figure A9d) and spans Patch L (Figure A9e). This way of initiating a large earthquake can be considered as a quasi-static cascade-up following a failed dynamic cascade-up. The time interval between the precursory S-event and the following L-event is from 3.02×10−8 to 3.97×10−3 of the recurrence of L-events. An important difference from the sequence of a precursory S-event and an L-event by large nucleation intervened by afterslip of the S-event is that large quasi-static nucleation outside Patch S is not required. We shall call it as delayed cascade-up (dcL). The acceleration prior to the fourth earthquake after the third earthquake cannot be explained by the 1/tf acceleration (Figure A1g).

Figure A9.

The third and fourth earthquakes with α=3. An L-event (fourth) is initiated 17 s after a S-event (third). The color scale is same as in Figure A2. (a) The beginning of the coseismic period of the third earthquake. (b) The end of the coseismic period of the third earthquake. (c) The beginning of the coseismic period of the fourth earthquake which is 17 s after Figure A9b. The slip rate increases in the right region of Patch S including its and reaches the threshold slip rate 0.1 m/s defining the coseismic period. (d) The fourth earthquake ruptures the small patch and the large patch at the same time. (e) The fourth earthquake spans Patch L.

A5 Boosted Large Nucleation by Coalescence of Creeping Front

[111] It is sometimes the case that a locked patch shrinks out of existence. The coalescence of a creeping front may cause a local increase in the slip rate even higher than 0.1 m/s [Kaneko and Lapusta, 2008; Chen and Lapusta, 2009]. Such a local high slip rate can generate an earthquake much smaller than the S-events (Co) or prompts large quasi-static nucleation for an L-event (CoL). With our selection of β = 3, the coalescence of creeping front is rare.

[112] Figure A10 represents the sixth and seventh earthquakes. The sixth earthquake is generated by coalescence of a locked patch on route to a large nucleation. With RL=4 km and α=3, moment magnitude of the L-events and S-events are approximately 6 and 5, while the moment magnitude of the sixth earthquake is only 3.578. The sixth earthquake fails to propagate dynamically but accelerates the large nucleation, which leads to an L-event (seventh).

Figure A10.

The sixth and seventh earthquakes with α=3. The color scale is same as in Figure A2. (a) A large nucleation in its early stage eats a locked patch. (b) When the locked patch disappears, the slip rate locally increased above 0.1 m/s (sixth earthquake). (c) This earthquake failed to propagate dynamically. (d–e) The large nucleation is accelerated and leads to a L-event (seventh).

[113] The fault acceleration prior to the sixth and seventh earthquakes is plotted in Figure A1h. When the sixth earthquake is initiated by the coalescence of a creeping front, Vpatch as a function of log(1/tf), where tf is the time to sixth earthquake, follows a line with smaller slope than predicted by 1/tf acceleration. Interestingly, Vpatch as a function of log(1/tf) defined for the seventh earthquake follows roughly the 1/tf acceleration for the large nucleation although perturbed by the sixth earthquake. Therefore, the sixth earthquake can be regarded as a minor perturbation added to the ongoing large nucleation.


[114] We gratefully appreciate discussion with PhD R.C. Viesca on the nucleation process who corrected our careless mistake there. He also gave us a native English check in revision. The comments by anonymous reviewers help improving the manuscript. The Earth Simulator was used for all simulations. This study is 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.