**2. What makes low-frequency tremors/earthquakes migrate along strike?**

In this section, we investigate the mechanisms of slow-earthquake migration by comparing observational results and numerical simulation results. One of possible mechanism is chain reaction of numerous small asperities (Fig. 1), which is based on the activity of small repeating earthquakes triggered by afterslip of large interplate earthquakes (Matsuzawa et al., 2004).

Fig. 1. A schematic model of chain-reaction between asperities after Matsuzawa et al. (2004). Yellow stars represent rupture initiation points of slip events.

On the other hand, some simulation results show slow-earthquake migration is reproduced by models with frictional properties almost uniform along strike and constant effective normal stress independent of depth under initial stress uniform along strike (e.g., Liu & Rice, 2005). Their simulation results suggest that an essential condition for the slow-earthquake migration is neither heterogeneous frictional properties nor heterogeneous initial stress distribution, if effective normal stress is constant at low value and independent of depth.

In the case of effective normal stress proportional to depth for a model with uniform frictional properties, slow-earthquake migration does not occur (e.g., Hirose & Hirahara, 2002). Note that above all simulation studies adopt the same friction law. These previous simulation results raised the following question: For an asperity model with low effective normal stress around 30 km depth, are small asperities essential for the slow-earthquake migration?

In this section, we formulate several test models in order to understand the necessary conditions and characteristics for slow-earthquake migration.

#### **2.1 A test model of chain reaction between numerous small asperities**

A test model consists of a planar plate interface dipping at 15 degrees from the free surface in a homogeneous elastic half-space (Fig. 2) with a periodic boundary condition along the strike direction. The plate interface is divided into 1,024 (strike) × 293 (dip) cells.

repeating earthquakes triggered by afterslip of large interplate earthquakes (Matsuzawa et

Fig. 1. A schematic model of chain-reaction between asperities after Matsuzawa et al. (2004).

On the other hand, some simulation results show slow-earthquake migration is reproduced by models with frictional properties almost uniform along strike and constant effective normal stress independent of depth under initial stress uniform along strike (e.g., Liu & Rice, 2005). Their simulation results suggest that an essential condition for the slow-earthquake migration is neither heterogeneous frictional properties nor heterogeneous initial stress distribution, if

In the case of effective normal stress proportional to depth for a model with uniform frictional properties, slow-earthquake migration does not occur (e.g., Hirose & Hirahara, 2002). Note that above all simulation studies adopt the same friction law. These previous simulation results raised the following question: For an asperity model with low effective normal stress around 30 km depth, are small asperities essential for the slow-earthquake

In this section, we formulate several test models in order to understand the necessary

A test model consists of a planar plate interface dipping at 15 degrees from the free surface in a homogeneous elastic half-space (Fig. 2) with a periodic boundary condition along the

Yellow stars represent rupture initiation points of slip events.

conditions and characteristics for slow-earthquake migration.

effective normal stress is constant at low value and independent of depth.

**2.1 A test model of chain reaction between numerous small asperities** 

strike direction. The plate interface is divided into 1,024 (strike) × 293 (dip) cells.

al., 2004).

migration?

Fig. 2. A 3-Dimensional model of a subduction plate boundary with frictional parameter *a-b* (Eq. 3). Cool color represents asperity.

Slip is assumed to occur in the pure dip direction and to obey the quasi-static equilibrium between shear and frictional stresses by introducing a radiation damping term (Rice, 1993):

$$
\mu\_i \sigma\_i = \sum\_{j=1}^{N} K\_{ij} (\mu\_j(t) - V\_{\text{pl}} t) - \frac{G}{2\mathcal{J}} \frac{\text{d}\mu\_i}{\text{d}t} \tag{1}
$$

Here, the subscripts *i* and *j* denote the location indices of a receiver and a source cell, respectively. The left hand side of equation (1) describes frictional stress, where *μ* and *σ* is friction coefficient and effective normal stress, respectively. The right hand side describes the shear stress in the *i*-th cell caused by dislocations, where *Kij* is the Green's function for the shear stress (Okada, 1992) on the *i*-th cell, *N* is the total number of cells, *V*pl is the relative speed of the two plates, *t* denotes time, *G* is rigidity, *β* is the shear wave speed. *Kij* is calculated from the quasi-static solution for uniform pure dip-slip *u* relative to average slip *V*pl*t* (Savage, 1983) over a rectangular dislocation in the *j*-th cell. Parts of the first term of the right-hand side are written as convolutions, by exploiting the along-strike invariance of the Green's function, and efficiently computed by the Fast Fourier Transform (e.g., Rice, 1993; Liu & Rice, 2005).

In Eq. (1), the effective normal stress *σ* is given by

$$
\lambda \sigma\_i(z) = \kappa(z) (\rho\_{\text{rock}} - \rho\_{\text{w}}) \text{g.z.} \tag{2}
$$

where *ρ*rock and *ρ*w are the densities of rock and water, respectively, *g* is the acceleration due to gravity, and *z* is the depth. The function *κ*(*z*) is a super-hydrostatic pore pressure factor, as given in Fig. 3. We assume that a high-pore-pressure system locally exists around a depth of 30 km based on the high-*V*p/*V*s zones in southwestern Japan (e.g., Shelly et al., 2006). The increase in pore pressure is probably due to the dehydration derived from the change in facies in the slab (e.g., Hacker et al., 2003). Ariyoshi et al. (2007) estimated that the value of *κ* is 0.1 for the deeper part (>30 km depth) based on the post-seismic slip propagation speed. On the basis of the stress field observation in northeastern and southwestern Honshu, Japan, Wang & Suyehiro (1999) suggested that the apparent frictional coefficient is approximately 0.03, which is consistent with *κ* = 0.1.

The friction coefficient *μ* is assumed to obey an RSF law (Dieterich, 1979; Ruina*,* 1983), as given by

$$
\mu = \mu\_0 + a \log \left( V / V\_0 \right) + b \log \left( V\_0 \theta / d\_c \right), \tag{3}
$$

$$\mathbf{d}\theta/\mathbf{d}\mathbf{t} = \mathbf{1} - V\theta/d\_{\rm cr} \tag{4}$$

where *a* and *b* are friction coefficient parameters, *dc* is the characteristic slip distance associated with *b*, *θ* is a state variable for the plate interface, *V* is the slip velocity, and μ0 is a reference friction coefficient defined at a constant reference slip velocity of *V*0.

We consider a model with close-set numerous small asperities on the deeper outskirt of a great asperity, as proposed by Dragert et al. (2007). In the present study, an asperity denotes a region with *a-b* = γ < 0, following Boatwright & Cocco (1996). The plate interface is demarcated into five parts, as shown in Figs. 2 and 3: (i) one large asperity (LA), (ii) 90 small asperities (SAs), (iii) a shallow stable zone, (iv) a deep stable zone, and (v) a transition zone (γ ~ +0). The values of frictional parameters as described in the caption of Fig. 3 are based on rock laboratory results (e.g., Blanpied et al., 1998), which will be discussed later.

The constant parameters in the present study are *V*pl = 4.0×10–2 m/yr (or 1.3×10–9 m/s), *G* = 30 GPa, β = 3.75 km/s, *ρ*rock = 2.75×103 kg/m3, *ρ*w = 1.0×103 kg/m3, *g* = 9.8 m/s2, *V*0 = 1 μm/s, μ0 = 0.6, and Poisson's ratio ε = 0.25.

#### **2.2 Simulation results of chain reaction effect on low-frequency event migration**

Fig. 4 shows the spatial distribution of slip velocity about 37 years before the origin time of a megathrust earthquake in the large asperity. The recurrence interval and magnitude of the megathrust earthquake in our simulation is 116 years and *M*w 7.9, respectively, where seismic slip is defined as slip faster than 1 cm/sec. The large asperity (LA) is strongly locked while a slow-earthquake occurs in some of small asperities (SA).

Fig. 5 shows time history of velocity field before and after the megathrust earthquake. Figures 2 and 3 suggest slow-earthquake migration at the migration rate of 0.3~3 km/day, which is driven by the chain reaction of small asperities. On the other hand, these figures also show that slow-earthquake migration does not usually occur in the region without small asperities, where slip velocity is largely stable at values comparable to *V*pl.

Green's function, and efficiently computed by the Fast Fourier Transform (e.g., Rice, 1993;

where *ρ*rock and *ρ*w are the densities of rock and water, respectively, *g* is the acceleration due to gravity, and *z* is the depth. The function *κ*(*z*) is a super-hydrostatic pore pressure factor, as given in Fig. 3. We assume that a high-pore-pressure system locally exists around a depth of 30 km based on the high-*V*p/*V*s zones in southwestern Japan (e.g., Shelly et al., 2006). The increase in pore pressure is probably due to the dehydration derived from the change in facies in the slab (e.g., Hacker et al., 2003). Ariyoshi et al. (2007) estimated that the value of *κ* is 0.1 for the deeper part (>30 km depth) based on the post-seismic slip propagation speed. On the basis of the stress field observation in northeastern and southwestern Honshu, Japan, Wang & Suyehiro (1999) suggested that the apparent frictional coefficient is approximately

The friction coefficient *μ* is assumed to obey an RSF law (Dieterich, 1979; Ruina*,* 1983), as

 d*θ/*d*t* = 1 – *Vθ/d*c, (4) where *a* and *b* are friction coefficient parameters, *dc* is the characteristic slip distance associated with *b*, *θ* is a state variable for the plate interface, *V* is the slip velocity, and μ0 is a

We consider a model with close-set numerous small asperities on the deeper outskirt of a great asperity, as proposed by Dragert et al. (2007). In the present study, an asperity denotes a region with *a-b* = γ < 0, following Boatwright & Cocco (1996). The plate interface is demarcated into five parts, as shown in Figs. 2 and 3: (i) one large asperity (LA), (ii) 90 small asperities (SAs), (iii) a shallow stable zone, (iv) a deep stable zone, and (v) a transition zone (γ ~ +0). The values of frictional parameters as described in the caption of Fig. 3 are based on

The constant parameters in the present study are *V*pl = 4.0×10–2 m/yr (or 1.3×10–9 m/s), *G* = 30 GPa, β = 3.75 km/s, *ρ*rock = 2.75×103 kg/m3, *ρ*w = 1.0×103 kg/m3, *g* = 9.8 m/s2,

Fig. 5 shows time history of velocity field before and after the megathrust earthquake. Figures 2 and 3 suggest slow-earthquake migration at the migration rate of 0.3~3 km/day, which is driven by the chain reaction of small asperities. On the other hand, these figures also show that slow-earthquake migration does not usually occur in the region without

**2.2 Simulation results of chain reaction effect on low-frequency event migration**  Fig. 4 shows the spatial distribution of slip velocity about 37 years before the origin time of a megathrust earthquake in the large asperity. The recurrence interval and magnitude of the megathrust earthquake in our simulation is 116 years and *M*w 7.9, respectively, where seismic slip is defined as slip faster than 1 cm/sec. The large asperity (LA) is strongly locked

small asperities, where slip velocity is largely stable at values comparable to *V*pl.

reference friction coefficient defined at a constant reference slip velocity of *V*0.

rock laboratory results (e.g., Blanpied et al., 1998), which will be discussed later.

*V*0 = 1 μm/s, μ0 = 0.6, and Poisson's ratio ε = 0.25.

while a slow-earthquake occurs in some of small asperities (SA).

*σi*(*z*) = *κ*(*z*)(*ρ*rock – *ρ*w)*gz*, (2)

*μ* = *μ*0 *+ a* log (*V/V*0) *+ b* log (*V*0*θ/d*c), (3)

Liu & Rice, 2005).

given by

In Eq. (1), the effective normal stress *σ* is given by

0.03, which is consistent with *κ* = 0.1.

Fig. 3. Frictional parameters (*a*, γ (= *a*-*b*), *d*c, *κ* (See Eq. (2)) as functions of distance along the dip direction from the surface on the plate boundary, where (*a*1, *a*2) = (2, 5) [×10–3], (γ1, γ2, γ3, γ4) = (0.5, 0.01, -0.3, 4.9) [×10–3], (*d*c1, *d*c2, *d*c3) = (10, 0.43, 100) [mm], and (*κ*1, *κ*2) = (1.0, 0.1). Half the length of the minor axis (along dip) of the elliptical asperity takes the following values: for LA, (*R*1, *R*2, *R*3) = (35, 36.25, 37.5) [km] and for SA, (*r*1, *r*2, *r*3) = (1.33, 1.5, 1.67) [km], where the aspect ratios for LA and SA are 2.0 and 1.5, respectively. The distance between central points of SA along strike and dip direction is 2 and 2.5 km, respectively. This figure is developed from Ariyoshi et al. (2011a).

Fig. 4. Velocity field on the subduction plate boundary 36.6 years before the megathrust earthquake. Note that 8 of log10(*V*/*V*pl) is about 1 cm/sec. This figure is developed from Ariyoshi et al. (2011a).

Fig. 4. Velocity field on the subduction plate boundary 36.6 years before the megathrust earthquake. Note that 8 of log10(*V*/*V*pl) is about 1 cm/sec. This figure is developed from

Ariyoshi et al. (2011a).

Fig. 5. Spatiotemporal evolution of slip velocities at the "Dip" of 115 km along strike as shown in Fig. 3 before and after the megathrust earthquake in case that numerous small asperities in Fig. 3 are removed for Strike>0. Color scale is the same as Fig. 4. This figure is developed from Ariyoshi et al. (2011a).

After about 4.5 years after the megathrust earthquake in Fig. 5, the largest slow slip event (~ 10 *V*pl at most; much slower than slow-earthquakes in Fig. 5) occurs in the region without small asperities. Since all of slow slip events in the region without small asperities originate from the transition between SA-belt and no-SA (between Strike = 0 and 75 km due to the cyclic boundary condition along strike direction), these slow slip events are triggered by the chain reaction of SA.

#### **2.3 Size effect of small asperities on chain reaction behaviors**

Next, we perform another test model with different size of asperity generating chain reaction as shown in Fig. 6 in order to investigate size effect.

Fig. 6. Spatial distribution of large asperity (LA), middle asperity (MA), small asperity (SA) (*a*-*b* < 0) with identification numbers of the asperities, weak stable (WS; *a*-*b* > 0) and strong stable (SS; *a*-*b* >> 0). Setting fault geometry, elevated pore pressure, and the constant value of other geophysical parameter is the same as section 2.2. This figure is partly derived from Ariyoshi et al. (2009).

Figs. 7a-7k shows simulation results of slip migration driven by chain reaction of asperities. For MA (left panel of Figs. 7a-7e), unilateral chained propagation process is clearly seen for MA, with propagation speed between asperities No. 7 to No. 5 of about 0.2 km/day (Figs. 7b-d). In the time span indicated by the cyan background in Figs. 7l-m, the patterns of stress drop and averaged slip velocity at asperities No. 3 to 5 appear to be relatively similar but quantitatively different. Especially, the amount of stress drop and averaged slip velocity in asperity No. 3 is greater than in the others. In addition, there is not only leftward chained propagation from the asperity No. 10 but also rightward propagation from the asperity No. 2, where the latter is much slower than the former because of the large area locked in the asperity No. 3. In Fig. 7e, both chained propagations cross at the asperity No. 3, which promotes larger stress drop and higher slip velocity. Since similar behavior seems to be seen in the simulation of Liu & Rice (2005), these results suggests that some phenomena generated by introducing along-strike variations of constitutive parameters or non-uniform initial conditions may be represented by interaction between small asperities.

On the other hand, the propagation process of SA is different from that of MA in some respects. On the right side of Fig. 7 shows leftward propagation from the asperity No. 2 through No. 31 to No. 22, which is due to the periodic boundary condition. In the time span indicated by a green background in Figs. 7n-o, there are two slow earthquakes for each asperity from No. 28 to 31. Propagation speed between asperities No. 31 to 28, No. 28 to 23 and No. 23 to 22 is about 0.2, 0.15 and 0.03 km/day, respectively, which progressively becomes slower than that of MA. Fig. 7j shows that rightward propagation from the asperity No. 27 and 30 is also seen.

Frictional Characteristics in Deeper Part of Seismogenic Transition Zones on a Subduction Plate Boundary 113

112 Earthquake Research and Analysis – Seismology, Seismotectonic and Earthquake Geology

Next, we perform another test model with different size of asperity generating chain

Fig. 6. Spatial distribution of large asperity (LA), middle asperity (MA), small asperity (SA) (*a*-*b* < 0) with identification numbers of the asperities, weak stable (WS; *a*-*b* > 0) and strong stable (SS; *a*-*b* >> 0). Setting fault geometry, elevated pore pressure, and the constant value of other geophysical parameter is the same as section 2.2. This figure is partly derived from

Figs. 7a-7k shows simulation results of slip migration driven by chain reaction of asperities. For MA (left panel of Figs. 7a-7e), unilateral chained propagation process is clearly seen for MA, with propagation speed between asperities No. 7 to No. 5 of about 0.2 km/day (Figs. 7b-d). In the time span indicated by the cyan background in Figs. 7l-m, the patterns of stress drop and averaged slip velocity at asperities No. 3 to 5 appear to be relatively similar but quantitatively different. Especially, the amount of stress drop and averaged slip velocity in asperity No. 3 is greater than in the others. In addition, there is not only leftward chained propagation from the asperity No. 10 but also rightward propagation from the asperity No. 2, where the latter is much slower than the former because of the large area locked in the asperity No. 3. In Fig. 7e, both chained propagations cross at the asperity No. 3, which promotes larger stress drop and higher slip velocity. Since similar behavior seems to be seen in the simulation of Liu & Rice (2005), these results suggests that some phenomena generated by introducing along-strike variations of constitutive parameters or non-uniform

On the other hand, the propagation process of SA is different from that of MA in some respects. On the right side of Fig. 7 shows leftward propagation from the asperity No. 2 through No. 31 to No. 22, which is due to the periodic boundary condition. In the time span indicated by a green background in Figs. 7n-o, there are two slow earthquakes for each asperity from No. 28 to 31. Propagation speed between asperities No. 31 to 28, No. 28 to 23 and No. 23 to 22 is about 0.2, 0.15 and 0.03 km/day, respectively, which progressively becomes slower than that of MA. Fig. 7j shows that rightward propagation from the asperity

initial conditions may be represented by interaction between small asperities.

**2.3 Size effect of small asperities on chain reaction behaviors** 

reaction as shown in Fig. 6 in order to investigate size effect.

Ariyoshi et al. (2009).

No. 27 and 30 is also seen.

Fig. 7. Examples of chain-reaction between asperities for MA (left panels (a)-(e)) and SA (right panels (f)-(k)) with time history of friction and normalized slip velocity averaged in each asperity (MA; (l)(m), SA; (n)(o)) after Ariyoshi et al. (2009). Italic numbers are identification of asperities. Cyan and green regions correspond to time spans of snapshots for MA and SA, respectively.

These differences from MA are explained as follows. SA has shorter recurrence intervals and smaller moment release because of smaller asperity size with shorter characteristic distance than those of MA. The smaller moment release makes propagation speed slower (0.03 km/day for SA is smaller than 0.2 km/day for LA), which causes that recurrence interval of SA is relatively much shorter than the passage time of aseismic slip across more than twice the SA diameters (10 km). In addition, locking of SA soon after the occurrence of slow earthquakes, due to their short characteristic distance, tends to prevent aseismic slip propagation. Therefore, slow earthquakes occur again soon after the passage of aseismic slip from asperity No. 29 as shown in Figs. 7n-o. This is why propagation process of SA as shown in Figs. 7n-o appears to be bilateral, rather than the unilateral propagation seen in Figs. 7a-7e.
