Generation mechanism of slow earthquakes: Numerical analysis based on a dynamic model with brittle ductile mixed fault heterogeneity

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2010jb008188, 2011 Generation mechanism of slow earthquakes: Numerical analysis based on a dynamic model with brittle ductile mixed fault heterogeneity Ryoko Nakata, 1 Ryosuke Ando, 2 Takane Hori, 1 and Satoshi Ide 3 Received 29 December 2010; revised 5 May 2011; accepted 2 June 2011; published 20 August [1] Various characteristics have been discovered for small, slow earthquakes occurring along subduction zones, which are deep nonvolcanic tremor, low frequency earthquakes (LFEs), and very low frequency earthquakes (VLFs). In this study, we model these slow earthquakes using a dynamic model consisting of a cluster of frictionally unstable patches on a stable background. The controlling parameters in our model are related to the patch distribution and the viscosity of both the patches and the background. By decreasing patch density or increasing viscosity, we observed the transition in rupture propagation mechanism, that is, from fast elastodynamic interactions characterized by an elastic wave propagation to slow diffusion limited by viscous relaxation times of traction on fault patches and/or background. Some sets of these geometrical and frictional parameters collectively explain the moment rate functions, source spectra, and scaled energy of observed slow earthquakes. In addition, we successfully explain both parabolic and constant velocity migrations in the case of the diffusion limited rupture. Therefore, the observed various characteristics of tremor, LFEs, VLFs, and, potentially, slow slip events, may be essentially explained by our simple model with a few parameters describing source structures and frictional properties of brittle ductile transition zones along plate boundaries. Citation: Nakata, R., R. Ando, T. Hori, and S. Ide (2011), Generation mechanism of slow earthquakes: Numerical analysis based on a dynamic model with brittle ductile mixed fault heterogeneity, J. Geophys. Res., 116,, doi: /2010jb Introduction [2] Deep nonvolcanic tremor, which radiate seismic waves that exceed background noise levels in the range of 1 10 Hz, occur in many subduction zones and are often accompanied by slow slip events (SSEs) [Rogers and Dragert, 2003; Obara et al., 2004]. Low frequency earthquakes (LFEs) are similar to tremor but occur in isolation in tremor sequences and have larger amplitudes [Katsumata and Kamaya, 2003]. Shelly et al. [2007] demonstrated that tremor can be explained as swarm like sequences of LFEs. In southwestern Japan, very low frequency earthquakes (VLFs) with seismic waves dominant in the range of 0.02 to 0.05 Hz also occur simultaneously. The moment magnitudes of LFEs, VLFs, and SSEs are approximately 1, 3 3.5, and 6 7, respectively, and the corresponding seismic moments are proportional to event durations. This scaling relation, together with the similarity of focal mechanisms, suggests that these phenomena are 1 Earthquake and Tsunami Research Project for Disaster Prevention, Japan Agency for Marine Earth Science and Technology, Yokohama, Japan. 2 Active Fault and Earthquake Research Center, National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan. 3 Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. Copyright 2011 by the American Geophysical Union /11/2010JB different aspects of a single process summarized as slow earthquakes [Ide et al., 2007b]. [3] Slow earthquakes are considered as shear slips on the plate interface [Ide et al., 2007a] and have different characteristics from those of ordinary earthquakes. For example, the velocity spectra of seismic waves from small slow earthquakes, tremor, LFEs, and VLFs are almost flat in a broad frequency range of 0.01 to 10 Hz [Ide et al., 2008], while those of ordinary earthquakes have a 1/f decay at high frequencies, where f is frequency. While the waveforms of ordinary earthquakes include P and S phases with clear onsets, tremor waveforms seldom have an observable P phase; even the S phase often lacks a clear onset and grows gradually. The seismic energy of slow earthquakes is much smaller than that of ordinary earthquakes of equivalent seismic moment. Quantitatively, the scaled energy, which is seismic energy divided by seismic moment, [Kanamori and Rivera, 2006] is approximately [Ide et al., 2008; Maeda and Obara, 2009] and [Ide and Beroza, 2001] for slow and ordinary earthquakes, respectively. The tremor sources migrate with a velocity of roughly 10 km/d ( 0.1 m/s) in the strike direction of the subducting plate [Obara, 2002], and 100 km/h ( 10 m/s) in the dip direction [Shelly et al., 2007]. The slow along strike migrations of tremor episodes are sometimes accompanied by the rupture propagation of SSEs [Rogers and Dragert, 2003; Obara et al., 2004]. 1of15

2 [4] So far, there is no physical model available that encompasses all of the above characteristics. Ando et al. [2010] proposed a model that explains the anisotropy of the tremor source migration speed and the spectral property of small slow earthquakes in which dynamic ruptures occur on frictionally unstable patches triggered by a passing stress pulse of an SSE. Despite the success in explaining some features, their model has some ambiguity because the rupture propagation behavior varies significantly depending on the parameter values in their model. [5] In this study, using a slightly modified version of the dynamic model by Ando et al. [2010], we confirm that their model s results are qualitatively robust over a wide range of parameter space and include the viscosity and patch geometry as key parameters. We also show that dynamic properties such as rupture propagation velocity and scaled energy can change quantitatively and depend on the parameter values that are assumed. 2. Method [6] Following Ando et al. [2010], we constrain our model of the brittle ductile transition zone where SSEs and small slow earthquakes occur (Figure 1, top). The model consists of a number of small frictionally unstable patches in an otherwise stable background region. These unstable patches are assumed to be clustered so as to constitute LFE or VLF sources. Clustered LFE sources are assumed to be the constituent parts of tremor sources. In this study, we investigated the behavior of one clustered source, while three sources were used by Ando et al. [2010]. We also consider these patches as a theoretical constraint to the reproduction of slip pulses, through the pinning effect, that is, the effects of fault heterogeneity [Day et al., 1998]. The patches are on the fault plane located in an infinite, homogeneous, elastic, and isotropic medium. [7] The dynamic rupture process and seismic wave radiation are simultaneously solved by the dynamic boundary integral equation method [Ando and Yamashita, 2007] in a full 3 D space with a triangular mesh [Tada, 2006]. [8] We assume that the unstable patches are tectonically stressed and are triggered by the stress increase ahead of a slow slip front. Thus, the initial shear traction is a superposition of homogeneous stress A 0 on the patches and stress increase ahead of a slow slip front. The slow slip front propagate on sources in the work of Ando et al. [2010], then rupture propagation speed was related to the propagation speed of slow slip front. In this study, however, in order to isolate the spontaneous features of rupture propagation, we assume that the slow slip front does not propagate after the initiation of a rupture during the entire simulation. Therefore, initial shear traction is given by ( i ¼ o þ A w ½1 þ cosf ðx X o Þ=L w gš=2 ðif jx X o j< L w =2Þ ¼ o ðelsewhereþ with o ¼ A o ¼ 0 ðon patchesþ ðbackgroundþ; ð1þ ð2þ where X is the position along the strike; X o is the location of the slow slip front fixed at one edge of the patch cluster (white dashed line in Figure 1, bottom, while X 0 = V prop t in the work of Ando et al. [2010]); A w and L w are the amplitude and width, respectively, of the load due to the slow slip front; and A o is the initial traction level for the patches. We used A w = 0.1, L w = 10, and A o =1 A w = 0.9 in order to easily trigger the rupture of unstable patches by slow slip front. [9] We focus on understanding the effects of fault heterogeneity independent of individual friction laws. It is well known that frictional instability or stability is primarily determined by slip and/or velocity weakening or hardening effects. To introduce these characteristics in a simple form, we assume that the fault strength t s is given by the following fault constitutive law: 8 < : s ¼ p ðif V ¼ 0Þ ¼ r þ V ðif V > 0Þ ; ð3þ where t p and t r are the peak and residual strengths, respectively; h (h 0) is the effective viscosity factor; and V is the slip velocity. The viscous damping term hv, which describes the velocity hardening effect, is a necessary factor that represents the plastic flow behavior of rocks emerging at the brittle ductile transition depth [Scholz, 2002]. The parameter values define the characteristics of the fault heterogeneity. In general, on the patches, we assume the relation t p t r >0 to induce the instability with instantaneous weakening by slipping. In contrast, on the background, t p = t r is assumed to lead to stability. The choice of h defines the additional stabilization effect. For given parameter sets, the same values are used within each domain of patches and background. [10] We defined the values t p = A w + A o such that the rupture is triggered by the slow slip front. To avoid a large disturbance by the triggering procedure, small incremental stress is assumed such that (A w t r )/(A o t r ) = 1/10. [11] We present physical quantities in nondimensional form: space is nondimensionalized by Ds, the side length of a regular triangular mesh to discretize the fault; time, by Ds/V p ; shear traction, by strength drop t p t r ; slip, by Ds(t p t r )/m; and viscosity, by h = 1/(m/V p ), where V p is the P wave speed. [12] We perform a parameter study of the relevant parameters in the following way. First, introducing the velocity hardening term is equivalent to the introduction of a characteristic time scale because of viscous relaxation and diffusion. Therefore, we choose h as a control parameter for the domains of both the patches h patch and the background h back. Second, the spatial patch distribution is the characteristic determining fault heterogeneity and controls the degree of elastic interaction among the patches. Therefore, we also define the patch density and sizes as control parameters, as detailed below. Here, we give less regard to the transient effects of the slipweakening behavior usually described by the slip weakening distance D c and the strength excess t p t i. Although these parameters determine the onset and the transient acceleration of a spontaneous rupture through the energy balance, the stable solution for spontaneous rupture is propagation at an elastic wave speed [Andrews, 1976]. Hence, to realize 2of15

3 Figure 1 3of15

4 Table 1. Parameters of Patch Distribution Model r p L space D L gap Model 1 a Model 1 b Model 1 c Model 2 a Model 2 b Model 2 c Model 3 a Model 3 b Model 3 c Model 4 d persistent, slow, spontaneous rupture with only this slipweakening framework, one needs trivial fine tuning of parameters at an unstable critical point, being, as an illustrative example, the strength drop rate (t p t r )/D c equal to the stiffness for t p t i = 0 [e.g., Kanamori and Rivera, 2006]. We will rather focus on the robust behavior of the system controlled by the above nontrivial factors and eliminate the transient behavior during the acceleration phase. [13] As shown in Figure 1 (middle), a source region consists of circular patches in a square area of length L, which is fixed at 27.7Ds (round source area was used by Ando et al. [2010]). The distribution of the patches is characterized by the mean density r p, size d, and spacing L space, which are related by d = L space r p. Then, the gap distance between patches is given as L gap = L space (1 r p )=L space d. In section 3, we discuss the dependence of rupture process characteristics on patch distribution by using L gap and r p.we added randomness to their locations, shifting each patch by a random distance with a Gaussian distribution of standard deviation 0.3. We systematically investigated the dynamic rupture propagation for two control parameter combinations. We show the results of ten patch distribution models (Figure 1, bottom) in which we can elicit the characteristic effects of patch geometry. The actual values of these parameters are listed in Table 1. The unstable patches are shown by white in each panel, denoting their initially high traction. [14] In order to investigate the effects of the viscosity, we changed both h patch and h back from 0 to 40. It should be noted that in some cases, the viscosity is set to be unrealistically high on patches relative to the background; however, this is only for investigating the system behavior dependence on viscosity. When we investigate the effects of the patch distribution, we assumed h patch = h back =1. [15] Although we could not deal with a wide range of parameter space owing to limited computational capability, we were sufficiently able to capture critical system behavior (as will be shown below), which is the transition from fast ruptures, controlled primarily by elastic wave speed, to slow ruptures, controlled by diffusion. 3. Results 3.1. Dependence on Patch Distribution [16] As a reference, we first investigate the behavior of our ten patch distribution models (Figure 1, bottom), assuming a combination of viscosities such that h patch = h back = 1. The moment rate functions obtained for the ten distribution models show an initial rise at T 0.5 and decay from T =50to 100, except models 2 a, 3 a, and 4 d (Figure 2b). Figure 2c shows the slip amount at T = 120. The large slip area, which is colored red to yellow, is seen on the unstable patches. A background region, colored blue to green, has a small slip because of its frictionally stable properties. [17] In this section, we show two typical examples of the rupture process. One is a pulse like rupture, as in distribution model 2 b, and the other is a crack like rupture, as in distribution model 1 c. [18] In distribution model 2 b, the moment rate function has a trapezoidal shape obtained by superposition of sequential single peak functions (Figure 2b). Figure 3a shows snapshots of the rupture process of this model. The arrival of the prescribed slow slip front is seen as a dark red band along the left edge of the left panel, showing traction. Once a patch ruptures, the traction on the patch decreases. The distribution of slip rates in the middle panels shows that slip occurs within limited bands in the form of pulses. We can see that a pulse like rupture of patch propagates laterally across the source. The propagation of the rupture to all patches ends at T = The moment rate function decays smoothly and stress relaxation continues after this time (Figure 2b). Then we can see that the final slip amplitude is the largest in the center of each patch at T = in model 2 b (Figure 3a). This pulse like rupture is characterized by an almost trapezoidal moment rate function as seen in distribution models 2 b and 3 b. [19] In another typical example, model 1 c, the moment rate function has a triangular shape (Figure 2b). Figure 3b shows snapshots of the rupture process in this distribution model. The distribution of slip rates shows a more expanded area of relatively high velocity behind (the left side) the rupture front, which is parallel to the dip direction. The rupture propagation to all patches ends at T = 45.25, which is shorter than the previous case. Finally, at T = 170.0, large slips is seen on patches located at the center of the source (Figure 2c). These are the characteristics of simplified ordinary earthquakes. A source composed of a single patch is the usual model for ordinary earthquakes, as seen in distribution Figure 1. Schematic diagrams of this study. (top) Schematic diagram of strength heterogeneity on a fault in the brittle ductile transition zone (modified from work by Ando et al. [2010]). The black circles illustrate unstable patches. The rectangle enclosed by a broken line is the region of this simulation. This region indicates the area of the middle panel. (middle) Initial patch distribution in a source before randomness is added. The square enclosed by a dotted line is the source area with side length L. The center of the patches can be distributed with spacing L space within this area. The white circles of diameter d represent unstable patches in the otherwise relatively stable background region colored gray with side lengths X and Y. The background length between patches, L gap, depends on L space and d. (bottom) Patch distribution models with initial conditions of traction of the rupture process at time T = 0. White spots are locations of unstable fault patches. White dashed lines show the position of X 0 (the location of the fixed slow slip front). 4of15

5 Figure 2. (a) Same as the bottom panels in Figure 1. (b) Moment rate functions obtained for 10 patch distribution models with hpatch = hback = 1 in the simulation. (c) Slip amount at T = 120. The same color scale is used in all figures. (d) Source spectra of moment acceleration functions. Note that the values in these figures are nondimensionalized. 5 of 15

6 Figure 3. (a) Snapshots of the rupture process from T = 1.0 to 60.0, and T = for h patch = 1 and h back =1 of model 2 b. The three panels in each snapshot represent (left) shear traction, (middle) slip velocity, and (right) slip. (b) Same as Figure 3a but for model 1 c. The color scale of slip is 1.6 times larger than that of Figure 3a. model 4 d. Simulated results for model 1 c are similar to these characteristics. This crack like rupture is characterized by a triangular shaped moment rate function. [20] These rupture process characteristics depend on patch distribution, which is determined by two control parameters r p and L gap. When r p is large and L gap is small as in distribution model 1 c, the moment rate function shows a triangular shape with small fluctuations during rupture time and large slips occur on patches located at the center of the source. In model 2 b, slip amount on the upper left patches is larger than the lower left corner because the patch density is high on the upper left corner in model 2 b. On the other hand, when r p is small and L gap is large as in model 2 b, the moment rate function shows a trapezoidal function obtained by superposition of sequential single peak functions and the final slip amplitude is the largest in the center of each patch. [21] Additionally, when r p is too small and L gap is too large, such as in distribution models 2 a and 3 a, rupture stops propagating halfway through. In the following analysis, we use seven distribution models and exclude these two models. [22] Figure 2d shows the moment acceleration spectrum, that is, the Fourier spectrum of the differentiation of the moment rate function. This function is proportional to the farfield velocity waveform. When the shape of the moment rate function is trapezoidal (a superposition of sequential singlepeak functions), as is the case for models 2 b and 3 b, the Fourier spectra are almost constant at low frequency. On the 6of15

7 Figure 4. Moment rate functions obtained for hpatch = 1 and hback = 0.3, 1, 5, 20, and 40 for model 2 b. Different colors of the lines represent different values of hback. The black line shows the results for hpatch = 1 and hback = 1, which is the same as that in Figure 2b. Figure 5. Same as Figure 3a but from T = 1.0 to for hback = of 15

8 Figure 6. Same as Figure 4 but for hback = 1 and hpatch = 1, 5, 20, and 40. The different colors represent different values of hpatch. Figure 7. Same as Figure 3a but from T = 1.0 to for hpatch = 20. The color scale of slip velocity is one eighth of that in Figure 3a. 8 of 15

9 Moreover, peak slip velocity is lower in Figure 7 than in Figure 5. [27] By compiling the peak slip velocity as a function of the viscosity of the patches (Figure 8), we can see that the peak velocity is delimited by the viscosity alone, without additional parameters such as cutoff velocities [e.g., Shibazaki and Shimamoto, 2007]. The observed inverse linear relation is predicted on the basis of the form of the viscous damping term t s = t r + hv, indicating that the velocity induced by stressing becomes inversely proportional to the viscosity. The different kinds of dependence on the velocity, such as logarithmic or power law, change the above relation quantitatively but not qualitatively. Figure 8. Maximum slip velocity as a function of patch viscosity, h patch. As an example, the same patch distribution is considered (patch distribution model 2 b) with the variation of h patch. The different color codes represent different h patch (same as Figures 6 and 9a). other hand, when the shape of the moment rate function is triangular (a single peak function), as is the case of patch distribution model 4 d, the Fourier spectra decay as 1/f. Therefore, when r p is relatively small and L gap is large, the moment rate function has a trapezoidal shape and the source spectrum is flat at low frequency Dependence on Background Viscosity [23] We investigated the effect of the background viscosity using the distribution model 2 b using various values of h back. We changed h back from 0.3 to 40, keeping h patch at unity. Figure 4 shows the resultant moment rate functions. As h back increases, the amplitudes of the moment rate function and the rupture duration decrease and increase, respectively. [24] When we use higher values of h back = 20 and 40, the small triangular parts of the moment rate function are not always superimposed, but are instead sometimes isolated. In other words, the rupture of patches occurs intermittently. As shown in Figure 5, rupture takes a long time to propagate in the background owing to slower viscous relaxation. We can regard these cases as examples of a sequence of multiple events, rather than a single event Dependence on Patch Viscosity [25] We further investigated the effect of the patch viscosity, using the same distribution model 2 b with various values of h patch. We changed h patch from 1 to 40, keeping h back at unity. [26] Figure 6 shows the calculated moment rate function. Comparison of cases of high viscosities for the background and the patches, for example h back = 20 (Figure 4) and h patch = 20 (Figure 6), reveals that the moment rate functions for the latter case are smoother than those for the former case. However, the rupture times are almost the same. The difference between high h back with h patch = 1 and high h patch with h back = 1 is well differentiated by comparing the snapshots (Figures 5 and 7); when h patch is relatively high (Figure 7), it takes longer to decrease both the slip velocity and the traction on the patches than when h patch is relatively low (Figure 5) Rupture Velocity Determined by Patch Distribution and Viscosities: Emergence of Two Different Propagation Mechanisms [28] In this simulation, we found that the rupture duration depends on the patch geometry and viscosity for a fixed size fault. In other words, these parameters control the rupture velocity, which is a fundamental characteristic in rupture dynamics. [29] We calculated rupture propagation velocity V rup by dividing the source size by rupture duration. The duration T r = T e T s is obtained by observing a moment rate function, where T s is the initial risetime and T e is the time when the moment rate function starts to decay without any fluctuations that are due to the rupture of patches. The rupture propagated to the end of the source at T = T e. Regarding the source size, we used the distance along the strike between the points on patches located at the left ( = X 0 ) and right ends of the source area. [30] Figure 9a shows the relation between rupture velocity and viscosity for seven distribution models with several h back values from 0 to 40. In this figure, we can observe the two characteristic regimes of the rupture velocity. One regime has the rupture velocity converging toward V s, i.e., the S wave speed under the condition of high r p, and small L gap. This regime corresponds to the mechanism of rupture propagation dominated by elastodynamic interactions between patches. Hereafter, we call this rupture process elastodynamic rupture. The other regime is that in which the rupture velocity decay is inversely proportional to the viscosity of the background or patches, as 1/h, above a corner value, which increases as L gap decreases and r p increases. Regarding the decay rate, the effect of the patch viscosity is shown to be the same as that of the background viscosity. These dependences on the viscosity are due to a relaxation process of the traction on the patches and background. Hereafter, we call this rupture process diffusion limited rupture. [31] Figure 9b shows the same rupture velocity plotted against L gap (see Figure 1, middle). It should be noted that L gap shown here relates to the mean values of the random distribution. Although the ratio of the total length of L gap and the source size is comparable to that of patches, the remarkable result obtained here is that the overall rupture velocity decays as the inverse square of L gap, i.e., V rup = 1/L 2 gap, under the condition of diffusion limited rupture as is explained below (see section 4.4 and 4.5.). [32] In summary, in this dynamic model, the rupture tends to be dominated by elastodynamic interaction when the viscosity is relatively small and patch gap is relatively short with 9of15

10 Figure 9. (a) Dependence of rupture propagation speed (V rup ) for seven patch distribution models on viscosity of background (squares, triangles, and inverted triangles) or patch (stars). The gray scaled strength represents the difference of L gap noted at the lower left. Velocities are nondimensionalized by V p. Inclined and horizontal broken lines indicate slope of inverse linear viscosity (1/h) and S wave speed, respectively. (b) V rup versus the background length of two patches, L gap. The different color codes and shapes of symbols represent different h back and r p, respectively. Inclined broken line indicates slope of inverse square of patch gap (1/L 2 gap ). relatively high patch density. This case leads to the moment rate function becoming triangular and characterized by a crack like rupture, similar to ordinary earthquakes. Otherwise, the rupture is delimited by diffusion with the moment rate function becoming trapezoidal, as characterized by a pulse like rupture, similar to small slow earthquakes. However, we consider a purely elastic medium without viscoelasticity and take into account the inertia effect, so that the interaction of all the elements in the model region is always elastodynamic even in the case of diffusion limited rupture. The diffusional effect arises from the viscosity only on the fault. 4. Discussion 4.1. Source Spectrum [33] As discussed by Ando et al. [2010], the simulated source spectrum reproduces the observed flat acceleration spectrum when the moment rate function is trapezoidal by superposition of sequential single peak functions. In this study, we restricted conditions to a flat spectrum in a wide range of parameter space, finding the correlation between the flat spectrum and the slower rupture velocity given by sparse patch distributions. [34] In addition, when the value of h back is large enough to isolate the rupture of some patches, the moment rate functions are shown to be partly trapezoidal owing to pulse like ruptures. If we select the time window during the trapezoidal section, the moment acceleration spectrum will be flat. Furthermore, in actual faults, small patches, which are not modeled in our simulation, may exist around large patches, and waveforms will have multiple peaks Shape of the Moment Rate Function [35] Ide et al. [2008] showed that the seismic moment rate function determined in the low frequency range has a remarkably similar shape to the squared high frequency velocity waveform. Figure 10 shows the squared velocity VLF waveforms, which are the summation of three component seismograms that are filtered between 2 and 8 Hz, instead of the moment rate function of VLFs. The moment rate functions with h back = 1 and 5 of model 2 b (Figure 4) seem similar to the observed VLF waveform. There are a number of characteristic variations in waveforms of slow earthquakes, such as the template event of LFEs [Shelly et al., 2007]. Although we do not consider a wide range of parameter space and some frictional parameters (e.g., strength excess, slip weakening distance), there are some observed waveforms that appear to correspond to the moment rate function obtained from our simulation. As we stated in section 2, we would like to focus on more fundamental outcomes than the effect of slip weakening distance from the brittle ductile mixed heterogeneity in this study. The most important outcome is the transition of the two different rupture modes and that is why we less regards the finiteness of the slip weakening distance, and also effects of the other types of frictions. Either cases, note that the flat moment acceleration spectra is basically due to the multiple peaked shape of the moment rate function, which is not sensitive to the shapes of individual peaks themselves corresponding to individual patch ruptures as long as they are crack like (this aspect had been discussed by Ando et al. [2010]). The aim of this study is not to search for any observed waveforms that show good correlation to simulated moment rate functions or to estimate patch geometry, viscosity and other frictional parameters from our dynamic model. However, we should note that it might be possible to explain observed waveforms within reasonable ranges of some parameter space Energy Moment Ratio [36] The seismic energy moment ratio is observed to be very different between ordinary and slow earthquakes. Here, we discuss the dependences of the seismic energy moment ratio on the patch distribution and viscosities. 10 of 15

11 Figure 10. Squared velocity very low frequency waveforms at KIS, which is the summation of three component seismograms, filtered between 2 and 8 Hz. Data is from the work of Ide et al. [2008]. Each event time is the time of local maximum of low frequency waves (JST). [37] We calculated seismic energy dimensionally using the simulated nondimensional moment rate function with the following approximate equation: E 1 10Vs 5 Z _M O ¼ VS; 2dt M O ð4þ where the density r is 2700 kg/m 3, S wave velocity V S is 0.58V p, and V p is 6 km/s. _Mo is the moment rate. m is the rigidity of 30 GPa, and V and S are the slip velocity and slip area, respectively, where V = V V p (Ds/m), and S = S (Ds) 2. V and S denote the nondimensionalized values and Ds is the stress drop of patches. Finally, we obtain the relations Mo / DDs 3 E / D 2 Ds 3 : From equations (6) and (7), we can see that the energymoment ratio E/Mo is proportional to Ds and independent of Ds, while the moment and energy depend on Ds. If we provide the appropriate value of Ds that is estimated by observations of slow earthquakes, then we obtain E/Mo dimensionally from our simulated results. [38] Observations show that the stress drop of VLFs is estimated to be roughly 3 kpa if the source sizes are 3 5 km [Matsuzawa et al., 2009], and stress change due to a teleseismic surface wave to trigger tremor is approximately 10 kpa [e.g., Miyazawa et al., 2008]. In our simulation, if we assume Ds = 10 kpa and Ds = 100 m (that is, L = 2.77 km) for seven patch distribution models with h patch = h back = 1, then we obtain Mo value of (Figure 11, right), which is close to the value observed for smaller VLFs ( [Ide et al., 2007b]). Likewise, E/Mo is calculated to be 10 8 (Figure 11, right), which is smaller than that of ordinary earthquakes (10 5 to 10 4 ) and larger than that of observed slow earthquakes (10 10 to 10 9 ). In addition, ð5þ ð6þ ð7þ Figure 11 shows the dependences of the energy moment ratio and the moment on the viscosities. If we assume Ds = 10 kpa, these results suggest large dependence of E/Mo on the patch viscosity. E/Mo is calculated to be in the range of to 10 8 (Figure 11, left), which includes the range of observed slow earthquakes. On the other hand, dependences of E/Mo and Mo on the background viscosity are moderate and small, respectively. The large dependence of E/Mo on the patch viscosity is easily explainable by the fact that the fastest slip occurs exclusively on the patches. [39] Our simulation results suggest that the small energymoment ratio observed for slow earthquakes is primarily attributable to the small stress drop and/or high patch viscosity. Although it is hard to separate these competing effects simply by the observation of E/Mo, observational characterizations regarding the shape of the moment rate functions will make it possible to single out the effect of the patch viscosity as previously discussed Migration Velocity of LFE/Tremor [40] As shown in Figure 9, rupture velocity significantly varies with the patch distribution and viscosity. For example, rupture velocity with h back = 20 for model 3 b(l gap = 3.324; light triangle) is one order lower than that for distribution model 1 b (L gap = 1.108; dark triangle) (Figure 9a). Furthermore, rupture propagation velocity with h back = 40 for model 2 b (L gap = 2.216; medium gray triangle) is one order lower than that with h back =1. [41] Although no direct observations of rupture velocities of individual LFEs/tremor or VLFs have been performed, observations of the migration velocity of these events have been performed. To consider the relationships between rupture velocities within individual events and migration velocities of multiple events, it is interesting to recall the shape of moment rate functions for the cases of high h back (Figure 4). In these cases, although each event tends to be more isolated, some event episodes involve multiple patches in densely distributed areas. Owing to this close patch distribution, the rupture velocity within the episode is higher than the overall migration velocity in the simulation. In fact, this situation appears to be well correlated with migration 11 of 15

12 Figure 11. Energy moment ratio and moment of simulated results. (left) Triangles and stars represent the dependence on h back and h patch, respectively, of patch distribution model 2 b (L gap = 2.216). The colors of each symbol represent the different values of h. (right) Inverted triangles, triangles, and squares represent r p = 0.5, 0.6, and 0.7, respectively, for h patch = h back = 1. Different gray scaled colors represent different L gap values. phenomena. From the simulation, we can conclude that the rupture velocities of small slow earthquakes tend to be faster, as determined by the denser local patch distribution, and that the overall migration velocities tend to be slower, reflecting sparser regional average distributions of patches in nature. [42] Moreover our result is useful for quantitative analysis of migration velocity. The observed migration velocities show two ordered anisotropy between along strike and along dip [Shelly et al., 2007]. This anisotropy can be explained by the differences in viscosity and/or patch distribution. However, there may not be anisotropy of viscosity; therefore, our results suggest that the anisotropy of migration velocity is caused by the difference in patch (source) distribution between along strike and along dip as suggested by Ando et al. [2010]. This difference is actually observed in the form of streaky structures subparallel to dip direction [Ide, 2010]. [43] In addition, as shown in section 3.4, the rupture propagation velocity decays as 1/h and 1/(L gap ) 2. We can safely extrapolate these values to a regime four orders slower than V s to obtain the scale of 10 km/d, which is the observed migration velocity in the strike direction, probably reflecting propagation of a SSE [e.g., Obara, 2002]. [44] However, we do not consider the healing effect here. Once a patch rupture, the traction of the patch and background do not increase. Then this model cannot explain that tremor migrates rapidly back along strike such as the Rapid Tremor Reversals found in Cascadia [Houston and Delbridge, 2010]. [45] In this study, we observed the transition in rupture propagation mechanism from fast elastodynamic interactions to slow diffusion limited. Ide et al. [2007b] suggests that there are two modes of slip, either fast or slow. If all Figure 12. (a) Initial condition of traction at T = 0 for model 3 b. The white line shows the cross section plotted in Figure 12b to Figure 12e. (b) Cross section of slip velocity for h patch = 1 and h back = 20 along the strike (X = 0 to 30) at Y = 17 (white line in Figure 12a). The color of each curve represents T = 27.5 to with intervals of T = 12.5 (see the color scale in Figure 12e). (c) Same as Figure 12b but for X = 14 to 22. Because of the spatial discretization, the sampling interval (Ds) is one along strike. Broken lines denote the traction transiently perturbed by waves from broken patches away from this section. (d) Cross section of shear traction along strike (X = 0 to 30) at Y = 17. Yellow backgrounds represent the initial traction on patches. Once slip starts on the left patch, traction around the patch has the value of 0.1, which implies t r = 0.1. (e) (left) Space time plot of rupture front (open circles). The broken parabola curve (red) was obtained by using T / x 2. This suggests that the rupture front propagates on background by diffusion. (right) Schematic figure of Figure 12e, left. The gray shaded area is corresponding to the area of left panel. The superposition of diffusive propagation (red parabolic curve) over L gap explains the constant velocity migration (blue dashed line) over long distances. The rupture propagation of a patch over the distance of yellow shaded area is illustrated by the green line. 12 of 15

13 various frictional properties and patch distribution used in this study exist in nature, and there is no instrumental limitation, we may observe yield not only fast or slow modes but also transitional events. However, we seldom observe such transient events in spatial and temporal scale where no observational problem exists. This fact suggests that all settings in our paper are not necessarily allowed in nature Migration Is Parabolic (Diffusive) or of Constant Velocity? [46] There are contradictory observations of migration patterns: parabolic migration is observed over relatively short distances [Ide, 2010] and constant velocity migration is observed over long distances [Ito et al., 2007]. On the basis of our results regarding diffusion limited rupture, we can Figure of 15

14 accurately explain both parabolic migration and constant velocity migration without creating contradictions. We examined the details of the diffusion limited rupture process between two adjoining patches over a short time interval to resolve the short distance parabolic migration. [47] We use the results of model 3 b with h back =20as an example. Figure 12a shows the location of the cross section used in Figure 12b 12e and the spatial initial condition of traction. We examined the rupture process between the two large patches located at X 12 and X 20, respectively. We show the slip velocity along the strike to discuss rupture propagation between two patches in a source (Figure 12b and 12c). [48] In Figure 12c, the rupture front (where the slip velocity is nonzero) gradually approaches the rightward patch with slow decay of the slip velocity because of the viscosity on the fault. At the same time, the traction on the right patch increases to t p = 1 with the arrival of the rupture front (Figure 12d), while the background traction is gradually relaxed. Here, the broken lines denote the traction transiently perturbed by waves from broken patches far from this section. These perturbations do not trigger the rupture here, but will be effective if the rupture front is sufficiently close to the unruptured patch. [49] We found that the space time plot of the rupture front, which propagates on a background shows a parabolic curve (Figure 12e, left). This suggests that rupture propagation on a background in a source is dominated by diffusion, which is the result of the viscous relaxation of the traction on the fault. [50] On the basis of the space time plots of tremor sources, Ide [2010] also suggests that the migration of tremor sources is diffusive. On the other hand, the constant migration speed of tremor over a great distance associated with SSEs [Ito et al., 2007] may be the result of the superposition of diffusive propagation of tremor, as shown schematically in the right panel of Figure 12e. As shown in Figure 4, when the background viscosity (h back ) is high, it takes longer time to propagate on a background than a patch, though the length of patch and background are comparable. As the result rupture duration time on a patch is negligible when we estimate the overall rupture speed. Then, the overall rupture speed is proportional to 1/(duration), that is, 1/(L gap ) 2. Following these considerations, there is no contradiction between parabolic and constant velocity migration. 5. Conclusions [51] We calculated the dynamic rupture propagation for source models of small slow earthquakes with various patch distributions and viscosities based on the dynamic model of Ando et al. [2010], which can reproduce the observed source spectra and the anisotropy of migration speed. In the simulations, diffusion limited rupture is dominant with larger values of viscosity and sparse patch distributions in a source, otherwise rupture by the elastodynamic interaction is dominant. Our results suggest that the dynamic model of Ando et al. [2010] is a robust source model for small slow earthquakes. The energy moment ratio varies significantly with patch viscosity and stress drop. The rupture propagation velocity varies significantly and depends on the patch distribution and/or viscosity. As a result, some parameter sets correctly explain observed characteristics such as spectral properties, parabolic migration, and the energy moment ratio. Therefore, the characteristics derived from observations may provide information about the source structure and frictional properties of our dynamic model of tremor, LFEs, and VLFs. This model will be a powerful tool to understand the generation mechanism and various aspects of slow earthquakes in a simple framework engaging source structures and frictional properties of brittle ductile transition zones along plate boundaries. [52] Acknowledgments. We thank two anonymous reviewers and an associated editor for their helpful comments improving the manuscript. We used seismic data from the National Research Institute for Earth Science and Disaster Prevention. We use Generic Mapping Tools [Wessel and Smith, 1998] for drawing some figures. This work was partly supported by the project Evaluation and disaster prevention research for the coming Tokai, Tonankai and Nankai earthquakes of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. This work was partially supported by MEXT KAKENHI ( ). References Ando, R., and T. 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15 Japan subduction zone, Geophys. Res. Lett., 31, L23602, doi: / 2004GL Rogers, G., and H. Dragert (2003), Episodic tremor and slip on the Cascadia subduction zone: The chatter of silent slip, Science, 300, , doi: /science Scholz, C. H. (2002), The Mechanics of Earthquakes and Faulting, 2nd ed., 471 pp., Cambridge Univ. Press, Cambridge, U. K. Shelly, D. R., G. C. Beroza, and S. Ide (2007), Complex evolution of transient slip derived from precise tremor locations in western Shikoku, Japan, Geochem. Geophys. Geosyst., 8, Q10014, doi: / 2007GC Shibazaki, B., and T. Shimamoto (2007), Modeling of short interval silent slip events in deeper subduction interfaces considering the frictional properties at the unstable stable transition regime, Geophys. J. Int., 171, , doi: /j x x. Tada, T. (2006), Stress Green s functions for a constant slip rate on a triangular fault, Geophys. J. Int., 164, , doi: /j x x. Wessel, P., and W. H. F. Smith (1998), New, improved version of generic mapping tools released, Eos Trans. AGU, 79, 579, doi: / 98EO R. Ando, Active Fault and Earthquake Research Center, National Institute of Advanced Industrial Science and Technology, Central 7, Higashi, Tsukuba , Japan. T. Hori and R. Nakata, Earthquake and Tsunami Research Project for Disaster Prevention, Japan Agency for Marine Earth Science and Technology, , Showa machi, Kanazawa ku, Yokohama , Japan. (nakatar@jamstec.go.jp) S. Ide, Department of Earth and Planetary Science, University of Tokyo, Tokyo , Japan. 15 of 15