Expansion of aftershock areas caused by propagating post-seismic sliding

Transcription

1 Geophys. J. Int. (2007) 168, doi: /j X x Expansion of aftershock areas caused by propagating post-seismic sliding Naoyuki Kato Earthquake Research Institute, University of Tokyo, Tokyo , Japan. Accepted 2006 October 3. Received 2006 August 21; in original form 2006 March 24 1 INTRODUCTION It is known that the aftershock areas of large earthquakes expand over time. Mogi (1968) reported that the aftershock areas of great shallow earthquakes at subduction zones expand overtime, and suggested that the aftershock area at 1 day after the main shock provides a good approximation of the coseismic slip area. Tajima & Kanamori (1985) analysed moderate- to large-scale earthquakes observed from 1963 to 1980 and determined the aftershock areas at different time intervals following the main shock using an objective method of defining the aftershock area on the basis of the spatial distribution of the energy released from the aftershocks. The authors found that for some cases 1-yr aftershock areas are several times larger than 1-day aftershock areas. They also found significant variation in the SUMMARY It is known that the aftershock areas of large earthquakes often expand over time. This expansion indicates that the stress-increase associated with a main shock gradually propagates outward from the main shock rupture area. It is likely that this propagating stress-increase is caused by the propagation of post-seismic sliding, which has been detected for many large earthquakes from geodetic observations, mainly global positioning system (GPS). In the present study, I perform a numerical simulation of the expansion of the aftershock area caused by the propagation of post-seismic sliding. The model fault exists within an infinite elastic medium and is loaded at a constant displacement rate. The frictional stress on the fault obeys a laboratory-derived rate- and state-dependent friction law. Non-uniformity of the frictional constitutive parameters is introduced to the model fault plane to represent a large asperity for a large earthquake (main shock) and many possible nucleus sites for aftershocks around the main shock asperity. Negative values of A B are assigned to areas of seismic slip for the main shock and aftershocks, while positive values of A B are assigned to regions where aseismic sliding occurs. Here, A B is defined by dτ ss /d ln V that describes the velocity (V) dependence of steady-state friction stress τ ss. Seismic slip may be nucleated for negative A B, and aseismic slip occurs for positive A B. Slip histories on the model fault plane are simulated through numerical integration of the friction law under the condition of uniform shear loading of a constant velocity. The simulation can reproduce aftershocks triggered by stress-increases related to postseismic sliding and expansion of the aftershock area. The rate of expansion of the aftershock area decreases with increasing distance from the main shock asperity or increasing value of A B in the velocity-strengthening region. This finding suggests that the A B value for a plate interface can be estimated from aftershock data. The simulation indicates that the 7-day and 30-day aftershock areas are and per cent larger, respectively, than the 1-day aftershock area. These numbers are approximately consistent with observations from large earthquakes at subduction zones. Key words: aftershocks, creep, earthquake-source mechanism, fault slip, seismicity. expansion patterns of aftershock areas, and suggested that a large expansion in the aftershock area tends to occur within regions of lower seismic coupling where small asperities are sparsely distributed. By analysing relocated aftershock hypocentres of large shallow earthquakes that occurred from 1977 to 1996, Henry & Das (2001) obtained similar results in terms of the expansion of aftershock areas with time, and noted that expansion is especially significant for thrust earthquakes at subduction zones. Wesson (1987) reported that the aftershock area of a moderate earthquake along the San Andreas Fault, California, expanded with time, and that significant post-seismic sliding was observed by two creepmetres located across the fault. Wesson suggested that expansion of the aftershock area was caused by the fracture of small asperities due to propagating post-seismic sliding, and proposed a mechanical model for GJI Seismology C 2006 The Author 797

2 798 N. Kato post-seismic sliding on the assumption of viscous materials for creeping fault zones. Nadeau & McEvilly (1999) estimated spatio-temporal variations in aseismic slip rate along the Parkfield segment of the San Andreas Fault, California, from small repeating earthquakes that are thought to have been caused by stick-slip on small asperities located within creeping regions, assuming that the earthquake occurrence rate at an asperity is proportional to the aseismic slip rate around the asperity. Igarashi et al. (2003) examined small repeating earthquakes on the subducting plate boundary along the Japan Trench and found that the activity of such earthquakes increased following large interplate earthquakes. Igarashi et al. (2003) suggested that small repeating earthquakes activated by large interplate earthquakes are triggered by accelerating aseismic sliding associated with the large earthquakes. On the same basis, Uchida et al. (2004) used data from small repeating earthquakes to estimate the outward propagation of post-seismic sliding associated with the 1994 Sanrikuoki Earthquake (M w 7.7): an interplate earthquake that occurred along the Japan Trench. Their study indicates that expansion of the aftershock area was caused by propagating post-seismic sliding, suggesting that the expansion pattern of the aftershock area is related to frictional properties that control post-seismic sliding on the plate interface. Significant post-seismic sliding associated with the 1994 Sanriku-oki Earthquake was detected from GPS observations (Heki et al. 1997; Yagi et al. 2003), although the spatial resolution of post-seismic sliding estimated from small repeating earthquakes is much better than that estimated from GPS data. Uchida et al. (2004) further examined the activity of small repeating earthquakes following the M and M Sanriku-oki Earthquakes and suggested that the small repeating earthquakes were triggered by post-seismic sliding, which were actually detected by geodetic observations. Tse & Rice (1986) and Marone et al. (1991), among others, showed that post-seismic sliding can be modelled using the laboratory-derived rate- and state-dependent friction laws developed by Dieterich (1979) and Ruina (1983). According to the friction laws, seismic slip can nucleate on a fault with velocity-weakening frictional properties, while aseismic sliding occurs for the case of velocity-strengthening frictional properties. When a nearby earthquake leads to an increase in shear stress within a region of velocitystrengthening frictional properties upon a fault plane, post-seismic sliding is generated within the velocity-strengthening region to relax the increased shear stress. Using a mechanical model with a rate- and state-dependent friction law, Kato (2004) simulated postseismic sliding that triggers a subsequent earthquake. Kato s simulation results suggest that time-dependent expansion of the aftershock area can be modelled in terms of the triggering of aftershocks due to propagating post-seismic sliding. In the present paper, I perform numerical simulations of the expansion of the aftershock area to explore factors that control patterns of aftershock expansion. The frictional stress on the model fault plane is assumed to obey a rate- and state-dependent friction law, and many small patches of velocity-weakening frictional properties, representing possible nuclei for small earthquakes, are distributed on the model fault plane in areas where aseismic sliding takes place. These weak patches enable the simulation to generate aftershocks triggered by aseismic sliding. I then examine the dependence of the expansion velocity of the aftershock area on frictional parameters of the model fault plane. To examine the factors that determine the nature of aftershock expansion patterns, the simulation was run repeatedly with varying frictional parameters upon the model fault plane. It is known that the static stress changes in Coulomb failure stress due to main shock faulting significantly affect the spatial distribution of aftershocks (e.g. King et al. 1994). Most of these aftershocks took place off the main shock fault plane. In the present numerical study, the off-fault aftershocks are not taken into consideration. 2 DESCRIPTION OF THE MODEL The model used in the present simulation is essentially the same as that developed by Kato (2004). A square planar fault is assumed within an infinite elastic medium with a rigidity of 30 GPa, Poisson s ratio of 0.25, and S-wave speed of 3.0 km s 1. The xy-plane is taken to be the fault plane, and the fault is shear-loaded so that two blocks located across the fault move relative to each other in the x-direction at an average relative plate velocity of V pl. The model fault plane is divided into (= ) equal-area square cells of m each with uniform slip in the x-direction. An analytical solution of static shear stress due to uniform slip on a square cell in an infinite uniform elastic medium is used to obtain shear stress due to fault slip. A periodic boundary condition is assumed for efficient computation of shear stress using a 2-D FFT technique. Quasi-static equilibrium is assumed between shear stress, derived from relative plate motion and fault slip, and the frictional stress on the fault, where the radiation damping quasi-dynamic approximation is used to obtain solutions during high-speed seismic slip (Rice 1993). The frictional stress τ on the fault is assumed to follow the composite rate- and state-dependent friction law proposed by Kato & Tullis (2001): τ = τ 0 + A ln(v/v 0 ) + B ln(v 0 θ/l), (1) dθ/dt = exp( V/V c ) (V θ/l) ln(v θ/l), (2) where V is sliding velocity; θ is a state variable; A, B, L and V c are constants; V 0 is an arbitrary reference velocity; τ 0 is a reference stress corresponding to the steady-state frictional stress at V = V 0 ; and V c is a cut-off velocity of healing, under which time-dependent strengthening of friction is significant (Kato & Tullis 2001). In the present study, V c = 10 8 ms 1, following Kato & Tullis (2001), and V 0 = V pl. τ 0 does not affect the simulation results because perturbation of the shear stress from the reference stress is crucial in the present model. In the rate- and state-dependent friction law, frictional characteristics are described by the parameters A, B and L. A represents the direct effect of sliding velocity on frictional stress, B represents time-dependent strengthening of frictional stress and L represents a characteristic slip distance over which frictional stress evolves at a step change in sliding velocity. The steady-state frictional stress τ ss can be defined for dθ/dt = 0 as follows: τ ss = τ 0 + (A B) ln(v/v 0 ), (3) when V V c. A B represents the velocity dependence of the steady-state frictional stress. Seismic slip can nucleate for A B < 0 (velocity-weakening friction) because the steady-state friction decreases with accelerating slip, while aseismic sliding takes place for A B > 0 (velocity-strengthening friction). A spatially nonuniform distribution of A B is introduced to the model fault plane to simulate earthquake sequences triggered by stress associated with aseismic sliding, where A is assumed to be uniform and B is variable. Fig. 1 shows the distribution of circular Patches 1 41, each with velocity-weakening frictional properties ( A B < 0), upon the model fault plane. These velocity-weakening patches represent potential nuclei of seismic slip. The radius of Patch 1 is 15 km, while those of Patches 2 41 are 3 km. The main shock

3 Expansion of aftershock areas 799 the critical size. The critical length l c for a square-shaped fault is given by Kato (2004) as follows: l c = 7 2 3π G L, (4) (B A) Figure 1. Locations of velocity-weakening (A B < 0) patches (asperities) on the model fault plane. Patch 1 is the main shock asperity, while Patches 2 41 are aftershock asperities. In the remaining region of the fault, A B > 0 for aseismic sliding. Letters A H denote observation points for postseismic sliding. In the outermost 5-km perimeter of the model, the A value is increased by 3 MPa to prevent rupture from propagating over the model boundary. asperity is intended to be represented by velocity-weakening Patch 1, and aftershock asperities by Patches The density of aftershock asperities is set to decrease in inverse proportion to distance from the centre of the main shock asperity, while the direction azimuth of each aftershock asperity is randomly selected. The aftershock asperities are numbered according to distance from the main shock asperity. On the remaining parts of the fault, velocity-strengthening friction (A B > 0) is assigned for aseismic sliding. The values of A, B and L in the main shock asperity (Patch 1), aftershock asperities (Patches 2 41) and the remaining regions, are shown in Table 1. The values of the friction parameters are varied to explore the effect of parameter values on aftershock expansion patterns given fixed locations and asperity radii. When friction follows a rate- and state-dependent friction law, seismic slip can nucleate at A B < 0. The critical size of the slip nucleation zone can theoretically be derived (e.g. Dieterich 1986), where unstable slip occurs when the size of the slip region exceeds where G is rigidity. Rice (1993) showed that the computation cell size must be much smaller than l c to obtain reliable numerical results from simulations. In the present study, the cell length of 500 m is much smaller than the minimum l c value of 3.2 km, which ensures numerical stability. Similarly, the critical fault radius r c of the slip nucleation zone is obtained for a circular fault: r c = 7π G L. (5) 24 (B A) Kato (2004) showed that seismic slip occurs when the radius of a velocity-weakening region is larger than r c, while episodic aseismic slip events (slow earthquakes) occur when the radius of the velocityweakening region is a little smaller than r c. For a velocity-weakening patch with a radius much smaller than r c, continuous stable sliding occurs. The values of r c are <8.2 km for the main shock asperity (Patch 1) and 2.7 km for the aftershock asperities (Patches 2 41), which are smaller than the velocity-weakening patch radii of 15 and 3 km for the main shock asperity and the aftershock asperities, respectively. Accordingly, seismic slip is expected to occur both within the main shock asperity and within the aftershock asperities. Within 5 km of the edge of the model fault plane, the value of A is increased by 3 MPa. This increased A value reduces the effect in which slip at an edge of the model fault plane also appears at the opposite edge due to the periodic boundary condition; a larger A value tends to arrest rupture propagation. For the initial conditions, the frictional stress τ is given by τ 0 + (A B)ln(V /V 0 ), with the sliding velocity V = 0.1V pl over the fault plane. The relative plate velocityv pl is set at 0.1 m yr 1. The histories of slip and shear stress at each cell on the model fault plane are obtained through numerical integration of the friction eqs (1) and (2) coupled with static equilibrium of elastic solid using a Runge-Kutta method. Consult Kato (2004) for details of the simulation method. 3 SIMULATION RESULTS 3.1 Propagating post-seismic sliding and aftershocks We examine the characteristics of simulated propagating postseismic sliding in this subsection, and discuss the effect of frictional properties on aftershock area expansion pattern in the next Table 1. Values of A, A B and L for the model fault plane and the recurrence interval T r, seismic moment M 0, maximum coseismic slip u max and equivalent radius R s of the coseismic slip area of simulated large earthquakes at Patch 1. See text for definitions of T r, M 0, u max and R s. Patch 1 Patch 2 41 Remaining region Simulated large earthquakes Case A A B L A B L A B L T r M 0 u max R s (MPa) (MPa) (m) (MPa) (m) (MPa) (m) (yr) (Nm) (m) (km)

4 800 N. Kato Figure 2. Spatial distribution of coseismic slip for a large earthquake in Patch 1 for Case 3, where coseismic slip is defined as slip with a slip rate equal to or greater than 10 mm s 1. The inner grey circle indicates the boundary between velocity-weakening (A B < 0) and velocity-strengthening (A B > 0) regions of Patch 1. The area within the outer grey circle, with the radius R s, corresponds to the area in which coseismic slip is equal to or greater than 0.1u max, where u max is the maximum value of coseismic slip. subsection. The recurrence of large earthquakes within Patch 1 and small earthquakes within Patches 2 41 are simulated in each case listed in Table 1, where the recurrence interval of large earthquakes within Patch 1 is much longer than those of small earthquakes within Patches 2 41 because of greater coseismic slip within Patch 1. To avoid the effect of artificial initial conditions, I discarded the simulation result for the first earthquake within Patch 1 at the beginning of each simulation run. The characteristics of simulated large earthquakes within Patch 1 are listed in Table 1. The recurrence interval T r is the time interval between the second and third earthquakes at Patch 1, while the seismic moment M 0, the greatest coseismic slip u max, and the radius R s of the coseismic slip area are those of the second earthquake at Patch 1. Coseismic slip is defined as slip that has a slip rate equal to or greater than 10 mm s 1. Fig. 2 shows the spatial distribution of coseismic slip associated with a large earthquake in Patch 1 for Case 3. Significant coseismic slip occurs within the velocity-weakening (A B < 0) region, and it propagates to a small degree into the velocity-strengthening region. R s is defined as the radius of the circular area that is equal to the area of coseismic slip 0.1u max during a large earthquake at Patch 1. The R s value is larger than the radius of the velocity-weakening region of Patch 1. Fig. 3(a) shows a part of the simulated slip histories for 20 velocity-weakening patches for Case 4 (Table 1), where the large simulated earthquake at Patch 1 is the second event from the start of the simulation. Aftershocks at the small Patches 2 41 related to the second earthquake at Patch 1 are used in the present analysis. Fig. 3(a) indicates that simulated earthquakes at Patch 1 and at small patches close to Patch 1 take place almost simultaneously, while in the time interval immediately following the Patch 1 earthquake, earthquakes are not triggered at small patches located far from Patch 1. Fig. 3(b) shows simulated slip histories at an expanded timescale at Patches 2 21 for the time interval immediately following the large earthquake shown in Fig. 3(a). This figure clearly demonstrates that earthquakes at small patches occur following the large earthquake, and that the time interval from the large earthquake to a small Figure 3. (a) Simulated slip histories for Case 4 at the centres of 20 velocity-weakening patches over a time interval that contains a large earthquake at Patch 1 (large asperity) and several earthquakes at each velocity-weakening patch. (b) Simulated slip histories at small velocity-weakening Patches 2 21 immediately following the large earthquake at Patch 1 shown in (a).

5 Expansion of aftershock areas 801 Figure 4. Simulated slip histories at the centres of Patches 7, 11, 15, 21, 34 and 41 and Points A H for the time interval immediately following the large earthquake at Patch 1 shown in Fig. 3(a). The locations of observation points are shown in Fig. 1. earthquake tends to increase with increasing distance of the small patch from Patch 1. Fig. 4 shows simulated slip at small Patches 7, 11, 15, 21, 34 and 41, and Points A H within the velocity-strengthening region (Fig. 1) for the time interval immediately following the large earthquake shown in Fig. 3(a), where the slip histories are placed in order of distance from Patch 1. Within the velocity-strengthening region, significant post-seismic sliding propagates outward from Patch 1 as indicated from the slip histories at Points A, B and C in Fig. 4. The amplitude of post-seismic sliding decreases with increasing distance from Patch 1. The small earthquakes at Patches 7, 11, 15 and 21 appear to be triggered by an increase in shear stress associated with propagating post-seismic sliding. The data in Fig. 4 indicate that sliding rates at Points F, G and H, which are located far from Patch 1, are barely affected by the large earthquake at Patch 1. Simulated earthquakes at small Patches 34 and 41 (Fig. 4) are not regarded as earthquakes that were triggered by the main shock (aftershocks), but are considered to be generated by steady loading associated with relative plate motion. The time intervals from the main shock to these small earthquakes ( 5 yr) are nearly one-third of the inherent recurrence intervals (15.3 ± 1.2 yr), where the inherent recurrence interval is obtained from simulated earthquakes at Patches 2 41 for a time interval that does not include large earthquakes at Patch 1. The scatter of the recurrence intervals of simulated earthquakes at small patches far from Patch 1 is smaller than that for small asperities located close to the large asperity. This difference in the regularity in earthquake recurrence arises from differences in the degree of stress perturbations associated with post-seismic sliding following large earthquakes at Patch 1. Figure 5. (a) Simulated slip histories at Point B for Cases 2, 4 and 5 immediately following large earthquakes at Patch 1. The dashed line indicates the average relative plate velocity (= 0.1 m yr 1 ) for reference. (b) Simulated slip histories at Point D for Cases 2, 4 and 5 immediately following large earthquakes at Patch 1. The dashed line indicates the average relative plate velocity (= 0.1myr 1 ) for reference. See Fig. 1 for the locations of Points B and D. Simulated time histories of post-seismic sliding at Points B and D (see Fig. 1) following large earthquakes at Patch 1 are shown in Figs 5(a) and (b), respectively, for Cases 2, 4 and 5, where A B values are different in each of the velocity-strengthening regions. Fig. 5 clearly shows that post-seismic sliding propagates far from the coseismic slip region of a main shock over just a short time interval when A B is small (Case 2), and that the propagation speed and amplitude of post-seismic sliding within the velocity-strengthening region decrease with increasing A B value. These results suggest that the A B value for the velocity-strengthening region affects the characteristics of the expansion of the aftershock area, provided that expansion of the aftershock area is caused by the propagation of stress due to propagating post-seismic sliding. For larger values of A B, shear stress increases with increasing sliding velocity, as understood from eq. (3) for steady-state frictional sliding, which is approximately achieved for post-seismic sliding with slip larger than the characteristic slip distance L, as will be shown below. Thus, A B controls the degree of shear stress increase due to post-seismic sliding, and therefore, also controls resistance to slip. This indicates that post-seismic sliding is diminished for larger values of A B. The sliding rate is similar to the relative plate velocity V pl for a period following the 1 2 yr transient post-seismic phase in Cases 4 and 5, while it is significantly smaller than V pl following rapid

6 802 N. Kato Figure 6. Simulated post-seismic histories of slip (solid lines) and shear stress reduced by the steady-state frictional stress τ τ ss (dotted lines) at (a) Point B in Case 2, (b) Point B in Case 5, (c) Point B in Case 11 and (d) Point D in Case 11. Red dotted lines indicate fitted curves of the theoretical post-seismic slip histories (eq. 6) to simulated histories. The dashed line in (d) indicates the average relative plate velocity (= 0.1myr 1 ) for reference. post-seismic slip phase in Case 2. This partial locking in regions of velocity-strengthening frictional properties is caused by excess slip during a post-seismic period; the degree of partial locking is greater for smaller values of A B or closer to a locked asperity. Simulated post-seismic histories of slip and shear stress reduced by the steady-state frictional stress τ ss are shown in Fig. 6. Assuming a single degree of freedom spring-block model with rate- and statedependent friction, Scholz (1990) and Marone et al. (1991) derived a theoretical time function of post-seismic sliding as follows: u(t) = u 0 ln(t/t 0 + 1) + pt, (6) where t is the time from the earthquake occurrence and u 0, t 0 and p are constants. This theoretical logarithmic equation shows a good fit to observed post-seismic creep on ground surfaces (Marone et al. 1991). Although a sudden shear stress increase at the time of earthquake occurrence is assumed in their model, in the present simulation, shear stress gradually increases as post-seismic sliding approaches an observation point. Despite this difference, the simulated time functions of post-seismic sliding are well approximated by eq. (6), as shown in Fig. 6. In applying eq. (6) to simulated post-seismic sliding, time-shifts resulting from the propagation of post-seismic sliding are taken into consideration. eq. (6) was derived on the assumption that frictional stress is under a steady-state (eq. 3). This condition is achieved when slip is significantly larger than the characteristic slip distance L for a constant sliding velocity. Fig. 6 shows that the simulated post-seismic slip histories are in all cases well approximated by eq. (6) following transient periods following the main shock. During a transient period, shear stress increases due to the approach of post-seismic slip to an observation point. This is followed by a decrease in shear stress related to the onset of significant post-seismic slip at that point. During this period, shear stress τ is significantly larger than the steady-state frictional stress τ ss, and the simulated slip history cannot be explained by eq. (6). Following this transient period, shear stress approaches the steady-state frictional stress τ τ ss via slip > L, and the simulated slip history can be fitted by eq. (6) because the assumption of τ τ ss in the derivation of eq. (6) is met. Fig. 6 shows that greater slip is required to achieve steady-state frictional stress for larger L. In the present simulation, I used the composite law, for which slipweakening behaviour is similar to that of the slip law (Kato & Tullis 2001). For the slowness law, the apparent slip-weakening distance is significantly greater than L; therefore, greater slip is required to achieve the steady state. 3.2 Effects of friction parameters on the expansion pattern of the aftershock area Fig. 7(a) shows the occurrence time of an aftershock within a small velocity-weakening patch, relative to the occurrence time of the main shock at Patch 1, plotted against the closest distance to the centre of a small patch from Patch 1, for Cases 3, 8 and 9. Values of A B in Patch 1 are different in each of the above-mentioned case, while the other parameter values are the same (Table 1). Fig. 7(a) demonstrates that the time intervals from a large earthquake to small earthquakes at small patches tend to increase with increasing distance from Patch 1 for all three cases, although there is

7 Expansion of aftershock areas 803 Figure 7. (a) Occurrence times of simulated earthquakes at small velocity-weakening Patches 2 41, as measured from the occurrence time of a large earthquake at Patch 1, plotted versus the closest distance between Patch 1 and the centre of each small velocity-weakening patch. Data are for Cases 3, 8 and 9. (b) Occurrence times of simulated earthquakes at small velocity-weakening Patches 2 41, as measured from the occurrence time of a large earthquake at Patch 1, plotted versus R/R s, where R is the distance from the centre of Patch 1 to the centre of each small velocity-weakening patch and R s is the radius of the coseismic slip area of the large earthquake at Patch 1. significant difference in vertical position between them. This difference is mainly caused by differences in the coseismic slip amplitude and coseismic slip area of large earthquakes at Patch 1, corresponding to the difference in the A B value within the large asperity. Because the magnitude of the stress drop during seismic slip increases with increasing B A in a velocity-weakening region, the maximum amplitude of seismic slip u max of a large earthquake increases with B A in Patch 1 (Table 1). In addition, seismic slip penetrates into the velocity-strengthening region because of the large stress concentration generated at the rupture front. The distance of seismic slip penetration depends on the magnitude of stress concentration at the rupture front; therefore, it increases with increasing amplitude of coseismic slip. The radius R s of the coseismic slip area increases with B A in Patch 1, as shown in Table 1 (Cases 3, 8 and 9). To correct for the effect of areas of coseismic slip associated with large earthquakes on the pattern of expansion of the aftershock area, R/R s is taken as the abscissa in Fig. 7(b) for plots of the time intervals from the main shock to aftershocks, where R is the distance from the centre of Patch 1 to the centre of each small velocity-weakening patch where an aftershock occurs. There is no significant difference in the expansion patterns of aftershock areas between Cases 3, 8 and 9 in Fig. 7(b). This indicates that the normalization of the distance to a small patch by the radius R s of the main shock coseismic slip area is useful for reducing the effect of the difference in coseismic slip of the main shock. To explore the effect of the A B value in the velocitystrengthening region on the pattern of aftershock-area expansion, Fig. 8 shows the aftershock occurrence time following the main shock plotted against R/R s for Cases 1 5, with variable A B values. The coseismic slip area is larger for smaller values of A B in the velocity-strengthening region (Table 1), because seismic slip readily propagates into the velocity-strengthening region for smaller positive values of A B. Fig. 8 indicates that the pattern of aftershock-area expansion still depends on the A B value in the velocity-strengthening region, even after correcting for the effect of coseismic slip area by taking R/R s as the abscissa. The aftershock occurrence time following the main shock tends to be shorter for Figure 8. Occurrence times of simulated earthquakes at small velocityweakening Patches 2 41, as measured from the occurrence time of a large earthquake at Patch 1, plotted versus R/R s for Cases 1 5 with various A B values within the velocity-strengthening region. smaller values of A B. This is consistent with the observation that the amplitude and propagation speed of post-seismic sliding are smaller for larger A B values (Fig. 5), because the resistance to propagation of post-seismic sliding increases with increasing A B as discussed in the preceding subsection. Earthquakes at small patches with occurrence times greater than 10 8 s( 3 yr) from the main shock are not regarded as aftershocks triggered by propagating post-seismic sliding because small earthquakes occur repeatedly at small patches at recurrence intervals of 15 yr, as described in the preceding subsection. Fig. 8 further indicates that almost all small patches with R/R s < 2 are ruptured due to post-seismic sliding, while those with R/R s > 3 are not ruptured by post-seismic sliding. This suggests that the length of the aftershock area is about two to three times as large as that of the main shock coseismic slip area. Because the time interval from a main shock to an aftershock increases with R/R s, which represents the normalized distance to

8 804 N. Kato Table 2. Radii of aftershock areas normalized to the radius R s of the coseismic slip area, (R/R s ) 1,(R/R s ) 7 and (R/R s ) 30, for 1, 7 and 30 days following the occurrence of a main shock, and ratios of the radii of the 7- and 30-day aftershock areas to that of the 1-day aftershock area, R 7 /R 1 and R 30 /R 1, for Cases 1 5 in the present simulation. A B values are for velocity-strengthening regions, while r denotes the correlation coefficient in the least-squares determination of the relationship between time and R/R s (eq. 7). Case A B (R/R s ) 1 (R/R s ) 7 (R/R s ) 30 R 7 /R 1 R 30 /R 1 r (MPa) an aftershock asperity, the radius of the aftershock area can be expressed as a function of time t from the main shock. It is assumed that R/R s is written as: R/R s = α log t + β, (7) for 10 4 s( 2.8 hr) < t < 10 7 s( 116 days), where α and β are constants. The values of α and β for Cases 1 5 are determined by a least-squares method. Table 2 shows R/R s values (R/R s ) 1, (R/R s ) 7 and (R/R s ) 30 at 1, 7 and 30 days, respectively, following a main shock calculated from eq. (7) using the determined α and β values and correlation coefficients, r, from the least-squares determination. High values of r ( 0.8) for Cases 2 5 indicate that eq. (7) provides a satisfactory account of the time-dependent expansion of the aftershock area for the limited time range of the present simulation. This in turn indicates that the radius of the aftershock area expands logarithmically with time. Since a logarithmic time function (eq. 6) well approximates post-seismic slip amplitude, which generates stress concentration that promotes propagation of postseismic sliding, it may be reasonable to assume that the propagation of post-seismic sliding is approximated by a logarithmic function. A lower value of r (= 0.67) for Case 1 results from the fact that almost all small patches of R/R s > 2 are ruptured within the time interval 1 day < t < 30 days, and the dependence of time on R/R s is not clear (Fig. 8). In Case 1, post-seismic sliding rapidly propagates far from the coseismic slip area because of the small A B value. Accordingly, all the small patches are ruptured within 30 days of the main shock, resulting in little time-dependent propagation of the aftershock area. The data in Table 2 further indicate that the values of (R/R s ) 1,(R/R s ) 7 and (R/R s ) 30 are larger for smaller values of A B. This is because the propagation speed of post-seismic sliding is higher for smaller values of A B, as demonstrated in the preceding subsection. Fig. 9 shows the time intervals from a main shock to aftershocks plotted versus R/R s for Cases 4, 6 and 7, where A = 1.0, 1.5 and 2.0 MPa, respectively, and A B in the velocity-strengthening region is fixed at 0.5 MPa. Fig. 9 indicates that although the aftershock occurrence times are shorter for smaller values of A for R/R s < 1.5, A does not significantly affect the pattern of aftershock-area expansion in the range tested in the present simulation when compared with the effect of A B. Ariyoshi et al. (2005) conducted a simulation of frictional sliding on a subducting plate boundary using a rate- and state-dependent friction law, and reported that the propagation speed of post-seismic sliding increases with a decrease in A. It is possible that A affects post-seismic sliding because A represents the direct effect of velocity on frictional stress (eq. 1), and accordingly, the resistance to accelerating slip during the initial Figure 9. Occurrence times of simulated earthquakes at small velocityweakening Patches 2 41, as measured from the occurrence time of a large earthquake at Patch 1, plotted versus R/R s for Cases 4, 6 and 7 with various A values in the velocity-strengthening region. Figure 10. Occurrence time of simulated earthquakes at small velocityweakening Patches 2 41, as measured from the occurrence time of a large earthquake at Patch 1, plotted versus R/R s for Cases 3, 10 and 11 with various L values in the velocity-strengthening region. stage of slip. For slip > L, the effect of the evolution of state θ cannot be neglected. Therefore, the property of post-seismic sliding may be mainly controlled by A B. The results of Ariyoshi et al. (2005) concerning the effect of A on post-seismic sliding possibly arise from that they observed the onset stages of post-seismic sliding, while the present study considers the later stages of post-seismic sliding. The time intervals from a main shock to aftershocks, plotted against R/R s for Cases 3, 10 and 11, are shown in Fig. 10 to examine the effect of L on the pattern of aftershock-area expansion. For R/R s < 2, the aftershock occurrence times for L = 0.05 m (Case 3) tend to be shorter than those for the other cases with longer L. This may be related to the fact that shorter L leads to rapid stress drop as discussed in the preceding subsection (See Fig. 6). In Case 3, aftershocks occur within 30 days from the main shock, and the remaining small velocity-weakening patches rupture a few years later, mainly due to loading by steady plate motion rather than post-seismic sliding. In contrast, the aftershock sequences for Cases 10 and 11 continue for more than 1 yr.

9 Expansion of aftershock areas 805 shown from observations because aftershock observations usually include both on-fault and off-fault aftershocks. Figure 11. Cumulative number of simulated aftershocks at small velocityweakening patches versus time from the occurrence of a large earthquake in Patch 1 for Case 4. In Fig. 11, the cumulative number of aftershocks at small patches following a main shock at Patch 1 is plotted against the time from the main shock for Case 4. The cumulative number of aftershocks increases logarithmically with time over the time interval from a few days to about 1 yr from the main shock. Omori s law for aftershock occurrence indicates that the aftershock occurrence rate n per unit time interval is inversely proportional to time t from the main shock occurrence (e.g. Utsu et al. 1995): n(t) = K (t + c) 1, (8) where K and c are constants. The cumulative number N of aftershocks at t is expressed by N(t) = K ln(t/c + 1). (9) Fig. 11 indicates that the rate of aftershock occurrence in the present simulation is approximately consistent with Omori s law for the limited time interval. Although just an example case is shown in Fig. 11, logarithmic increases in the cumulative number of aftershocks over limited time intervals are commonly observed in the simulation results. The decay of aftershock occurrence rate with time in the present model is mainly caused by the fact that the amplitude and propagation speed of post-seismic sliding decrease with time. The radius of the aftershock area increases in an approximate relationship with the logarithm of the time interval from the main shock (eq. 7). The density of velocity-weakening patches as a function of distance from the main shock patch also affects the aftershock decay rate because small velocity-weakening patches swept by postseismic sliding are potential sites for aftershocks. The density of velocity-weakening patches is assumed to decrease in inverse proportion to the distance from the centre of the main shock patch. The c value in Omori s law is usually estimated to be much less than 1 day for aftershocks following naturally occurring earthquakes (Utsu et al. 1995), while in the present case the value of c is estimated to be a few days (Fig. 11). This discrepancy suggests that the density of small velocity-weakening patches close to the main shock patch for real plate boundaries is higher than that assumed in the present model. The logarithmic equation (eq. 9) does not explain the simulated aftershocks for times in excess of 10 8 s( 3 yr), as shown in Fig. 11. These earthquakes are not regarded as aftershocks triggered by propagating post-seismic sliding, but can be regarded as quasi-periodic earthquakes resulting from loading by steady plate motion. The present simulation result suggests that the time decay of on-fault aftershocks obeys Omori s law, though it has not been 4 DISCUSSION AND CONCLUSIONS In the present study, the expansion of aftershock areas with time can be explained by the assumption that small asperities upon the fault plane of the main shock are ruptured by increased stress associated with propagating post-seismic sliding. This model is plausible because significant post-seismic sliding has been observed following large interplate earthquakes. Significant expansion of aftershock areas occurs for interplate earthquakes where seismic coupling is low. For a plate interface with higher seismic coupling, the interface is filled with large asperities, as illustrated in Fig. 12(a), and a main shock rupture is arrested when the rupture encounters another stronger or less stressed asperity. Post-seismic sliding cannot propagate along the strike of the fault plane, but propagates updip and downdip. For this reason, the sliding intersects fewer small asperities on the plate interface, and only minor expansion of the aftershock area occurs. On a plate interface with lower seismic coupling, numerous small asperities exist in addition to some large asperities (Fig. 12b). In this case, main shock rupture is arrested when rupture propagates to a velocity-strengthening region where seismic slip cannot occur. Increased shear stress in the velocity-strengthening region associated with the main shock rupture is relaxed by postseismic sliding, which propagates outward from the rupture zone. The post-seismic sliding intersects many small asperities, resulting in the time-dependent expansion of aftershock areas. It is known that more aftershocks tend to occur on a fault plane where relatively small amounts of coseismic slip occur during a main shock (e.g. Mendoza & Hartzell 1988; Yamanaka & Kikuchi 2004). Slip deficit in regions with smaller amounts of coseismic slip must be compensated over time to maintain a uniform longterm slip rate across the fault plane. It is possible that post-seismic sliding performs this role, developing in regions with smaller values of coseismic slip and generating relatively large numbers of aftershocks in such regions. The aftershock area of the 2003 Tokachi-oki Earthquake (M w = 8.0), which occurred on the plate interface along the Kuril Trench, northern Japan, for the period 1 day from the main shock is significantly larger than main shock asperities with large amounts of coseismic slip (e.g. Yamanaka & Kikuchi 2003), and the aftershock area for 30 days from the main shock is almost the same as the 1-day aftershock area (Takahashi & Kasahara 2004). This observation of little time-dependent expansion of the aftershock area is consistent with the spatio-temporal pattern of post-seismic sliding estimated from GPS data. Miyazaki et al. (2004) showed that the area of postseismic sliding for the first few days from the main shock was much larger than significant coseismic slip areas (asperities), and the area of post-seismic sliding did not significantly propagate with time, although the area of most intense post-seismic slip tended to move outwards with time. Matsubara et al. (2005) examined small repeating earthquakes around the source area of the 2003 Tokachi-oki Earthquake and found that the repeating earthquakes were activated throughout an area much larger than the area of coseismic slip. The activated area roughly corresponded in size to the area of significant post-seismic slip. The present simulation indicates that the expansion pattern of the aftershock area is dependant on A B within the velocitystrengthening region of a fault (Fig. 8). For A B = 0.1 MPa (Case 1), the aftershock area propagates rapidly to more than twice the

10 806 N. Kato Figure 12. (a) Sketch of asperities with velocity-weakening frictional properties located upon a plate interface with a large value of seismic coupling coefficient. The seismogenic zone of the interface is filled with large asperities. (b) Sketch of asperities with velocity-weakening frictional properties located upon a plate interface with a low value of seismic coupling coefficient. Asperities of various sizes are sparsely distributed upon the plate interface. Post-seismic sliding occurs following large earthquakes at large asperities and generates aftershocks as it sweeps over small asperities. length of the area of coseismic slip, while a more distinct expansion pattern of the aftershock area is obtained for A B 0.2 MPa (Cases 2 5). This result suggests that the value of A B on the plate interface around the main shock asperities of the 2003 Tokachioki Earthquake is approximately equal to or smaller than 0.1 MPa. Miyazaki et al. (2004) developed a method for estimating the value of A B upon a fault plane from the relationship between shear stress and the sliding velocity of post-seismic sliding, as determined from GPS observations. Fukuda (2005) used the same method as that employed by Miyazaki et al. (2004) to analyse the 2003 Tokachioki Earthquake, using sliding histories obtained via a sophisticated statistical method that delivers higher slip resolution, especially for the initial onset of slip. Fukuda obtained A B = 0.07 MPa for a point within the post-seismic slip area of the earthquake, which is consistent with the results of the present simulation. Clear time-dependent expansion of the aftershock area has been observed for large interplate earthquakes in the Sanriku region along the Japan Trench (e.g. Uchida et al. 2004). These results are in contrast with results from the Tokachi-oki region, despite the fact that the two regions are adjacent to each other. The present simulation results suggest that the A B values on the plate boundary in the Sanriku region are larger than 0.1 MPa. Henry & Das (2001) obtained the lengths of aftershock areas at 1, 7 and 30 days from main shocks at subduction zones, and reported that the average ratios of the 7- and 30-day lengths of aftershock areas to the 1-day length are 1.31 and 1.43, respectively. The ratios of the radii of the normalized aftershock area for 7-days and 30-days, (R/R s ) 7 and (R/R s ) 30, respectively, to the normalized radius of the 1-day aftershock area, (R/R s ) 1, are shown as R 7 /R 1 and R 30 /R 1 in Table 2. These ratios (R 7 /R 1 and R 30 /R 1 ) are comparable to the

11 ratios reported by Henry & Das (2001). In the present simulation, R 7 /R 1 ranges from 1.14 to 1.22, and R 30 /R 1 ranges from 1.24 to 1.39; these values are slightly less than those obtained by Henry & Das (2001). This discrepancy between model predictions and observations possibly reflects the simplifications involved in constructing the present model. For example, frictional parameters may in fact show spatial variation, and small asperities of various sizes, associated with aftershocks of various sizes, may exist on actual plate boundaries. The obtained values of R 7 /R 1 and R 30 /R 1 are not significantly dependent on A B in the velocity-strengthening region (Table 2), but the ratios of the lengths of aftershock areas to those of coseismic slip areas of main shocks (main shock asperities) clearly increase with a decrease in A B. These ratios may, therefore, be more useful for estimating A B values at plate boundaries. Aftershock sequences simulated using the present model are approximately consistent with Omori s law for the decay of aftershock occurrence rate. Schaff et al. (1998) examined small repeating earthquakes following the Loma Prieta Earthquake on the San Andreas Fault, California, and found that the decay rate of aftershocks was consistent with Omori s law for each of six multiplets, where each multiplet consists of earthquakes upon the same patch of the fault. The authors suggested that the decay rate of aftershocks is related to the logarithmic time function of post-seismic sliding (eq. 6), considering that the stress rate at each patch should be proportional to the sliding velocity of post-seismic sliding. In the present simulation, the aftershock patches are not small enough to enable repeated aftershocks; therefore, it is not possible to discuss the decay rate of aftershocks at each aftershock patch. Many aftershocks occur in regions other than main shock fault planes (e.g. Das & Scholz 1981; King et al. 1994; Hino et al. 2000). These off-fault aftershocks around the main shock fault plane cannot be explained by the present model, in which aftershocks are generated at small asperities that are swept by propagating postseismic sliding upon main shock fault planes. Off-fault aftershocks appear to be better explained by the model proposed by Dieterich (1994), where the rate- and state-dependent response of potential aftershock nuclei to sudden changes in stress associated with a main shock is taken into consideration, and Omori s law of the decay of aftershock occurrence rate is successfully explained. Moreover, some aftershocks may be related to the diffusion of pore fluid following main shocks (e.g. Bosl & Nur 2002; Yamashita 2003) or time-dependent stress changes due to viscoelasticity (e.g. Mikumo & Miyatake 1979). It is likely that there exists more than one mechanism of the generation of aftershocks. The mechanism related to propagating post-seismic sliding, as discussed in the present paper, may be dominant at plate interfaces with a small seismic coupling. ACKNOWLEDGMENTS I am grateful to H.-J. Kümpel and anonymous reviewers for valuable comments. This research was funded by the Japan Society for the Promotion of Science (Project No ). REFERENCES Ariyoshi, K., Matsuzawa, T. & Hasegawa, A., Spatial variation in propagation speed of post-seismic slip on the subducting plate boundary, part 2, Japan Earth Planet. Sci. Joint Meeting, S Bosl, W.J. & Nur, A., Aftershocks and pore fluid diffusion following the 1992 Landers earthquake, J. geophys. Res., 107, 2366, doi: /2001jb Expansion of aftershock areas 807 Das, S. & Scholz, C.H., Off-fault aftershock clusters caused by a shear-stress increase?, Bull. seism. Soc. Am., 71, Dieterich, J.H., Modeling of rock friction 1. 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