Corresponding author: S. Ozawa, Geospatial Information Authority of Japan, Kitasato-1, Tsukuba, Ibaraki, Japan, 305-0811. E-mail: (email@example.com)
 The time evolution of afterslip on a plate boundary experiencing interplate earthquakes is expected to show logarithmic decay. The global positioning system network in Japan has been monitoring transient deformation since the occurrence of two large interplate earthquakes with moment magnitudes of 6.8 and 6.7 in the Hyuga-nada area, southwest Japan, in 1996. The spatial and temporal evolution of aseismic interplate slip based on crustal deformation data indicates that afterslip followed the two earthquakes and gradually declined to background rates by around 2004 with total moment magnitude of 7.3. However, quasi-periodic slow slip events suddenly began within the afterslip area in 2005 with approximately one year duration and two-year recurrence interval. The moment magnitudes of the three slow slip events since January 2005 range from 6.7 to 6.8. This differs greatly from the expected behavior of logarithmic decay over time. Both velocity-strengthening and velocity-weakening rate-and-state modes have been implicated as the cause of afterslip, whose location is complementary to the main shock area of velocity-weakening, while a slow slip event occurs in the velocity-weakening area with different frictional properties from those of an afterslip area. In light of the seemingly different frictional properties, the coexistence of afterslip and slow slip events in the same area would provide additional information about precisely how the plate interface is behaving. The monitoring of these slow slip events should give the clues to understanding the coexistence of long-term afterslip and slow slip events and the increasing risk of earthquakes in neighboring areas.
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 In the cycle of most interplate earthquakes, afterslip follows a main shock and decays over time in a logarithmic manner [Marone et al., 1991; Perfettini and Avouac, 2004]. Afterslip is thought to occur under frictional conditions of velocity-strengthening [e.g., Marone et al., 1991; Perfettini and Avouac, 2004] or velocity-weakening with a fault patch size much smaller than the critical fault radius [e.g., Dieterich, 1992; Kato and Hirasawa, 1996]. The afterslip area is usually complementary to the main shock area of velocity-weakening [e.g., Miyazaki et al., 2004; Ozawa et al., 2004]. After the subsidence of afterslip, interplate coupling recovers to the state before the earthquake and strain accumulates until the next earthquake. Because of the development of continuous GPS networks in many places, we can observe how interplate coupling recovers after interplate earthquakes on the basis of crustal deformation data [e.g., Ozawa et al., 2007a, 2012]. Understanding the recovery process of interplate coupling on a plate boundary would yield clues to understanding the above-mentioned earthquake cycle and the friction characteristics of the plate boundary in more detail, which are still elusive. Here, we introduce a remarkable event that occurred in the recovery process after two large earthquakes, which may suggest the possibility of unknown frictional properties.
 The Hyuga-nada area is part of the Pacific Ocean lying offshore of the east of Kyushu Island, southwest Japan, as shown in Figure 1. The Philippine Sea plate is subducting from the Nankai trough beneath the Amurian plate in the northwest direction at a rate of 5 to 7 cm/yr in this region (Figure 1) [Sella et al., 2002]. Because of the subduction of the Philippine Sea plate, there are many interplate earthquakes in this area (Figures 1b and 2). Most shallow earthquakes in the Hyuga-nada region are characterized by low-angle thrust faulting, which is directly related to the thrusting of the Philippine Sea plate. We can observe the high seismicity in the Hyuga-nada area in Figure 1b. Magnitude-7-class low-angle thrust earthquakes in the Hyuga-nada area of southwestern Japan occur repeatedly at a time interval of around 10 to 20 years on the plate interface of the subducting Philippine Sea plate (Figure 2) [Shiono et al., 1980]. As shown in Figure 2, the magnitude of earthquakes in the Hyuga-nada area ranges up to 7.5, which was recorded for the 1968 earthquake, on the scale of the Japan Meteorological Agency. The asperity area of these earthquakes is distributed from north to south offshore of the Pacific coast of Kyushu Island, southwest Japan (Figure 2). The last two large interplate earthquakes occurred in 1996 on 19 October (moment magnitude (Mw) 6.8) and 3 December (Mw 6.7) (see Figures 1 and 2) [e.g., Yagi et al., 1999], which were followed by postseismic deformation. Figure 1b shows the epicenters of the 1996 Hyuga-nada earthquakes determined by the Japan Meteorological Agency and the slip distributions of these earthquakes estimated by Yagi et al. . The estimated afterslip on the plate boundary based on postseismic deformation indicates that the afterslip area was deeper than the coseismic slip area and that the afterslip abated gradually [e.g., Yagi and Kikuchi, 2003; Miyazaki et al., 2003] and had subsided by 2004. However, the GPS network in Japan (GEONET), which is operated by the Geospatial Information Authority of Japan, detected quasi-periodic transients near the Hyuga-nada area, which suddenly started in approximately January 2005. This time evolution of ground displacements is markedly different from the expected logarithmic decay after the main shock. Furthermore, the transients since January 2005 suggest the occurrence of slow slip events in the afterslip area on the plate boundary. The coexistence of afterslip and slow slip events in the same area will provide additional information on the frictional property on the plate interface, because the frictional property of a slow slip event area, which is usually assumed to be velocity-weakening, is thought to be different from that of the afterslip area [e.g., Marone et al., 1991; Perfettini and Avouac, 2004]. In support of this conventional view, there have been no clearly observed cases of the afterslip of a large earthquake being followed by slow slip events in the same area. In this study, we estimate the spatial and temporal evolution of aseismic slip on the plate boundary in the Hyuga-nada area since 1997 by applying a method that uses a time-dependent inversion technique [McGuire and Segall, 2003; Ozawa et al., 2007b].
2 Data and Analytical Procedure
 Global positioning system data were analyzed for daily positions with Bernese GPS software (version 5.0). We adopted the F3 solution [Nakagawa et al., 2008], which uses the final orbit and Earth rotation parameters of the International GNSS Service and provides a higher SNR than the previous F2 position time series [Hatanaka et al., 2003]. We transformed the daily F3 coordinates into local displacement time series (east-west, north-south, and up-down components) with respect to the Fukue site (Figure 1b) for GPS sites near the Hyuga-nada area. We used the east-west (EW), north-south (NS), and up-down (UD) components at selected GPS sites near the Hyuga-nada area. We first estimated and removed annual components from the raw data using the following polynomial function and trigonometric functions:
where u(t) is a time series, t is time, and n and i are integers. The first term is a polynomial function, while the second term consists of trigonometric functions.
 We estimated these components from data between 1 January 1997 and 20 July 2010. The degree of the polynomial function and the overtone of the trigonometric functions are estimated on the basis of Akaike's information criterion [Akaike, 1974]. After removing only the annual components without removing the polynomial function, we estimated the linear trend from the data between 1 October 2007 and 1 March 2009, during which no transient displacements occurred. Although 1.5 years is short compared with the entire period of data, we consider that the adopted steady state for this period is satisfactory for emphasizing the results, because an afterslip, a steady state, and a periodic slow slip were clearly detected from this adopted steady state as deviations. We refer to the procedure for removing these components from the raw data as detrending, and we refer to the estimated steady crustal deformation rates having a linear trend and the corresponding slip deficit rate as the steady state. After the detrending, we smoothed the position time series by averaging over three days to reduce errors. The position time series with the linear trend component at the selected GPS sites in Figure 3 are shown in Figure 4, and the detrended position time series without a linear trend component are shown in Figure 5. The time series with the linear trend component show westward displacements for most of the time due to the subduction of the Philippine Sea plate (Figure 4), while the detrended position time series show eastward displacements for most of the time (Figure 5). Because the detrended position time series are deviations from the linear trend estimated for the period between 1 October 2007 and 1 March 2009, a zero slope in Figure 5 indicates that the ground displacement rates are the same as those given by the linear trend.
 We applied square-root information filtering [Ozawa et al., 2007b] to the detrended time series (Figure 5) using the inversion technique of McGuire and Segall  for the period between 1 January 1997 and 20 July 2010. Because we used detrended data, the estimated aseismic interplate slip is the deviation from the steady state. The 60 GPS sites used in the filtering analysis are plotted in Figure 3. We weighted the EW, NS, and UD displacements with a ratio of 1:1:1/5, considering the standard deviations estimated from ordinary Kalman filtering. To reduce the computational burden, we sampled position time series every 10 days. We checked that this sampling retains the characteristic features of the original detrended time series averaged over three days. We incorporated the inequality constraint that the slip is within 45° of the plate convergence direction of the Amurian plate relative to the Philippine Sea plate, which is southeastward [Miyazaki and Heki, 2001], following the method of McGuire and Segall . In this filtering analysis, we incorporated the spatial roughness of the slip distribution [McGuire and Segall, 2003]. Hyperparameters that scale the spatial and temporal smoothness and the inequality were estimated by maximizing the log-likelihood of the system [Kitagawa and Gersch, 1996; McGuire and Segall, 2003]. In this analysis, a fault patch and a slip distribution on a fault patch are represented by the superposition of the following third-degree B-spline functions [Ozawa et al., 2001]:
where X(ξ1,ξ2), Y(ξ1,ξ2), and Z(ξ1,ξ2) represent the EW, NS, and UD coordinates of a fault patch, respectively. ξ1 and ξ2 are parameters that roughly increase in the strike and dip directions, respectively (see Figure 6). Ni(ξ1) and Nj(ξ2) are the third-degree B-spline functions of parameters ξ1 and ξ2, respectively. These spline functions consist of knots that are roughly aligned in the strike and dip directions for Ni(ξ1) and Nj(ξ2), with Ni(ξ1) and Nj(ξ2) having peaks at the i-th knot in the strike direction and the j-th knot in the dip direction, respectively (see Figure 6).
 The adopted fault patch is represented by a parametric spline surface that consists of spline functions Ni(ξ1) and Nj(ξ2) with 15 knots in the strike direction and 15 knots in the dip direction, as shown in Figure 6 [Ozawa et al., 2001]. The spacing of knots in the fault patch is approximately 14 km. The plate boundary model is from Hirose et al. . The slip distribution is represented by the following third-degree B-spline functions, whose superposition constitutes the fault patch [Ozawa et al., 2001]:
where Ux and Uy represent the fault slip in the EW direction and NS direction, respectively. uxij and uyij are coefficients to be estimated and Ni(ξ1) and Nj(ξ2) are the same B-spline functions as those used to represent a fault patch. The slip in the UD direction Uz is calculated from nx × Ux + ny × Uy + nz × Uz = 0, where nx, ny, and nz represent the EW, NS, and UD components of a unit normal vector of a fault patch. The coefficients of the B-spline functions (uxij, uyij) that represent the slip distribution on the fault patch are estimated in this inversion [Ozawa et al., 2001].
 In addition, we estimated the slip deficit rate or the steady state for the period between 1 October 2007 and 1 March 2009 by Yabuki and Matsu'ura's method [Yabuki and Matsu'ura, 1992]. The GPS sites and fault patch used in this analysis are shown in Figures 3 and 7, respectively. The number of GPS sites used was 126. The fault patch is represented by a parametric spline surface, which consists of spline functions with 20 knots in the strike direction and 20 knots in the dip direction (Figure 7), representing the plate interface estimated by Hirose et al. . As shown in this figure, the dip angle of the subducting plate surface becomes steep from around 40 to 60 km depth. The backslip or slip deficit direction is constrained to be within 45° of the plate motion of the subducting Philippine Sea plate in this analysis [Ozawa et al., 2011]. The slip distribution on the fault patch is represented by the superposition of the spline functions that represent the fault patch, as in the case for filtering analysis [Ozawa et al., 2001].
 Our analysis shows that immediately after the 1996 Hyuga-nada earthquakes, postseismic deformation occurred and gradually decayed until around 2004, as shown in Figure 5, which shows detrended position time series. As mentioned above, detrended position time series are deviations from the linear trend for the period between 1 October 2007 and 1 March 2009. Thus, a zero slope in Figure 5 indicates that the ground displacement rates are the same as those given by the linear trend. Figure 8 shows a snapshot of ground displacements with the linear trend. Southwestward displacements ranging from 1 to 2 cm/yr are observed in the Hyuga-nada area. The southeastward displacements in southern Kyushu are ascribed to the backarc spreading of the Okinawa trough [e.g., Takayama and Yoshida, 2007]. For comparison, Figure 9 shows detrended ground displacements. As shown in this figure, we can observe clear eastward postseismic deformation immediately after the 1996 earthquakes and its decay to 2004 (Figures 9a–9h) as observed in Figure 5.
 Although there is oscillatory deformation from 1997 to 2004 in the detrended position time series at several sites (Figure 5), the following inversion result does not show a clear episodic change as has been observed since January 2005, except for the years of 1999 and 2002 (Figure 10). Because of this point and the small number of GPS sites in the early period of GEONET operation from 1996, we consider that the deformation at selected GPS sites from 1997 to 2004 can be mostly explained using the afterslip model, which exhibits logarithmic decay, although we cannot rule out the possibility of the occurrence of small slow slip events for this period, corresponding to small oscillatory fluctuations of the EW, NS, and UD components at several GPS sites (Figures 5 and 9).
 Although detrended postseismic deformation appears to have ended around 2004, quasi-periodic transients with a duration of approximately one year and a recurrence interval of two years began in 2005 (Figures 5 and 9). We can see oscillating signals at many GPS sites after January 2005 in Figure 5, as denoted by black or gray arrows. Gray arrows were used for cases when the signal was less clear than those shown with black arrows. In particular, we can clearly observe quasi-periodic oscillation in the NS component at sites 715, 483, and 088 from January 2005 (Figure 5). Furthermore, the eastward displacements from January 2009 are clear at all the GPS sites in this figure. The reason why we cannot see such clear oscillating signals as the NS components of sites 715, 482, and 099 in Figure 5 at the other GPS sites is that the slow slip events occur in a small area near the Hyuga-nada area, which cause a spatial pattern of crustal deformation that mostly affects sites 715, 482, and 099, as discussed in detail later. As mentioned above, many GPS sites show signals of slow slip events and steady states after January 2005, as shown in Figures 9i–9m in addition to the time series in Figure 5. The spatial patterns of the detrended transients for the periods 2005–2006, 2007–2008, and 2009–2010 in Figures 9i, 9k, and 9m, respectively, show eastward displacement in the inland area and the opening of a gap of approximately 1 cm between the southern and northern parts along the Pacific coastal area. These patterns strongly indicate the occurrence of interplate slow slip events since January 2005. The detrended displacements for the periods 2004–2005, 2006–2007, and 2008–2009 are very small and correspond to a steady state (Figures 9h, 9j, and 9l). These spatial and temporal changes in the detrended crustal deformation in Figures 9i–9m since January 2005 are extremely clear and are not observed before January 2005. Thus, we consider it very likely that relatively large slow slip events started suddenly from around January 2005 with approximately one year duration and two-year recurrence interval, while we cannot observe similar events before January 2005.
 Our filtering analysis shows that immediately after the earthquakes, aseismic interplate slip occurred beneath the Hyuga-nada area and declined until around January 2004 (Figures 10b–10h). The center of the afterslip area seems to be adjacent to the shallower area of high seismicity where the 1996 Hyuga-nada earthquakes occurred (Figure 11a). As shown in Figure 11a, the seismicity near the plate interface is high from the Pacific coast to the east, while it is lower beneath the inland area. As an interesting point, the center of the aseismic slip area seems to become deeper over time (Figures 10b–10d). The aseismic slip magnitude in 1999 is estimated to have increased slightly from that in 1998 in the central Hyuga-nada area (Figures 10b–10d). The aseismic slip from January 2000 to January 2002 is small near the Hyuga-nada area (Figures 10e and 10f). However, a slightly larger slip appears near the Hyuga-nada area for the period between January 2002 and January 2003 during the decay of the afterslip (Figure 10g). We cannot rule out the possibility that this aseismic slip between January 2002 and January 2003 is a small slow slip, as will be mentioned in the discussion section. Between January 2003 and January 2005, the afterslip seems to have ended (Figures 10h and 10i). We refer to the aseismic slip for the period between January 1997 and January 2005 as afterslip in this paper, because the time evolution of aseismic slip and its moment exhibits approximately logarithmic changes for this period (Figures 12 and 13a). The total moment magnitude of afterslip until 2004 amounts to 7.3. This value is much greater than those of the main shocks in 1996. Miyazaki et al.  showed that the moment magnitude of the afterslip of the Hyuga-nada earthquakes in 1996 was 6.9 within one year after the first main shock and exceeded that of the main shocks. The estimated afterslip is consistent with that in previous studies [e.g., Yagi and Kikuchi, 2003; Miyazaki et al., 2003].
 After the cessation of the afterslip of the 1996 Hyuga-nada earthquakes around 2004, the coupling in the Hyuga-nada area recovered to that of the steady state (Figure 10i). However, a slow slip event suddenly began in approximately January 2005 and ended in approximately January 2006 (Figure 10j). The center of this slow slip area seems to be in an area adjacent to the shallower high-seismicity area where the 1996 Hyuga-nada earthquakes occurred, as is the case for the afterslip (Figure 11b). The estimated moment magnitude for this period is 6.7, assuming that the rigidity is 30 GPa and Poisson's ratio is 0.25. This value is close to those of the main shocks. The aseismic slip from January 2006 to January 2007 is small (Figure 10k). After this period, a relatively large slow slip occurred in the period between January 2007 and January 2008, as shown in Figure 10l. The aseismic slip is negligible for the period between January 2008 and January 2009 (Figure 10m), corresponding to a steady state. Although we assumed a steady state for the period between October 2007 and March 2009, our filtering analysis shows a steady state from approximately January 2008 to approximately January 2009, as shown in Figures 5 and 10, probably because of temporal smoothing. From January 2009 to July 2010, a slow slip event occurred, as shown in Figure 10n. Thus, these slow slip events estimated since January 2005 seem to have a duration of roughly 1 year and be separated by intervals of about 1 year as mentioned above. The centers of the second and third slow slip areas also seem to be in an area adjacent to the shallower high-seismicity area (Figures 11c and 11d). The moment magnitudes for the transients in Figures 10l and 10n are 6.7 and 6.8, respectively. These values are close to those of the main shocks. The three detected slow slip events since January 2005 occurred in almost the same area, which is within the afterslip area (Figures 10b–10h and 11). The estimated resolution of the slip parameters in the area of slow slip events is greater than 0.9, as shown in Figure 11. The resolution matrix R is expressed as
where H is the coefficient matrix of the observation equations and HT is the transpose of matrix H [Menke, 1989].
 The inverted slip history at selected points on the plate boundary (Figures 10i and 12) shows that postseismic eastward slip occurred immediately after the earthquakes and slowed in 1998. After 1998, the eastward postseismic slip increased in speed, which corresponds to the increase in aseismic slip and the shift of the slip center to a deeper part, as shown in Figures 10b–10d, then showed gradual logarithmic decay over time without any periodicity until 2004, which was followed by fluctuations due to quasi-periodic slow slip events with an approximate duration of one year and recurrence interval of two years (Figure 12). Although the NS slip has shown slight periodicity since 2002, corresponding to the increase in slip for the time period between January 2002 and January 2003 in Figure 10g, the clear quasi-periodic NS slip after 2004 coincides temporally with the EW quasi-periodic slip (Figures 12a–12e). The slip histories at all selected points show similar features to those mentioned above (Figures 12a–12e). The estimated magnitude leveled off around 2004 and suddenly increased in 2005, corresponding to the occurrence of the first slow slip event (Figures 10j and 12f). After the first slow slip event in 2005, the moment magnitude leveled off and then increased twice as a result of the next two slow slip events (Figures 12f and 13a). The estimated model reproduces the observations well, as shown in Figures 5 and 9. With regard to the time interval of the frames in Figure 10, we consider that the one-year interval is suitable in this case, because it clearly shows the start and end of the slow slip events and the steady states (see also Figure 5), although we cannot account for the small fluctuations that oscillate within one year at several sites.
 As mentioned above, the resulting aseismic slip model shows deviation from the steady state for the period between 1 October 2007 and 1 March 2009. The ground displacement and slip deficit rates of the steady state for this period are shown in Figure 14a, while Figure 14b shows the 1σ error distribution. Westward displacements ranging from 1 to 2 cm/yr due to the thrusting of the Philippine Sea plate near the Hyuga-nada area in the northwestward direction are shown in Figure 14a. A slip deficit rate of 2-4 cm/yr is estimated in the Hyuga-nada area with an error of roughly 1 cm/yr (Figure 14a). The large slip deficit rate beneath southwest Shikoku is due to the size of the adopted fault patch and the GPS sites used. This large backslip area moves southeastward if we adopt a large fault patch and use more GPS sites in Shikoku. We consider that the relatively small fault patch and the small number of GPS sites used in the inversion have a small effect on the estimated backslip rate near Hyuga-nada, because southwest Shikoku and the Hyuga-nada area are approximately 100 km apart. The estimated backslip model closely reproduces the observations as shown in Figure 14a. It has been reported that the asperity areas of the past large Hyuga-nada earthquakes are strongly coupled (see Figure 2) on the basis of on a study of repeating earthquakes [Yamashita et al., 2012].
 The time evolution of afterslip is modeled by logarithmic decay,
where U(t) is the slip on a fault; A is related to the stiffness, frictional parameter, and normal stress; t is time; t0 is the time of the main shocks; and Tr is the relaxation time [e.g., Marone et al., 1991]. Because the moment is μSU(t), where μ is the rigidity, S is the area, and U(t) is the slip as discussed above, we can use the above formula to estimate the time evolution of the total moment.
 Here M(t) is the total moment at time t. We applied the above formula to the temporal evolution of the total moment and we estimated μSA and Tr by the linearized least-squares method for the period between 1997 and 2004. The reason why we used the data before January 2005 is that there are large bumps in the estimated moment after January 2005, corresponding to the slow slip events. The result is shown in Figure 13a. The estimated relaxation time, Tr, is 1.5 years. Figure 13b shows the deviation from the logarithmic decay model of equation (9). We can see five steep positive slopes. The steep positive slopes after January 2005 correspond to the slow slip events. The positive slope from 1999 to 2000 corresponds to the increase in slip for this period as shown in Figure 10d. We cannot rule out the possibility that this increase in slip magnitude is due to a slow slip event. However, its spatial pattern seems to be different from that of the slow slip events after January 2005 in that the southern part of the Hyuga-nada area shows relatively large aseisimic slip, which is part of an afterslip. We also cannot rule out the possibility that the positive slope for the period between 2002 and 2003 may be a slow slip, though its spatial pattern seems to be different from the three slow slip events after January 2005 in that the slippage in the central Hyuga-nada area is small and the southern part of the Hyuga-nada area shows slippage with magnitude similar to that of the slippage of the central Hyuga-nada area, which might also be part of an afterslip (Figure 10). Although the total moment shows fluctuation over time before 2005, the slip histories at points near Hyuga-nada show relatively smooth changes over time for the period between 1997 and 2004 (Figures 12 and 13). Thus, we consider that the aseismic slip near Hyuga-nada before January 2005 is mostly explained by logarithmic time evolution except for in the years of 1999 and 2002. The estimated moment of afterslip of medium-size to large earthquakes is usually less than or equal to that of the main shocks [e.g., Chlieh et al., 2007; Melbourn et al., 2002; Ozawa et al., 2012]. Thus, our case is peculiar in that the estimated moment of afterslip is much larger than those of the main shocks. The afterslips of the medium-size earthquakes before the 2011 Tohoku earthquake released much more energy than the main shocks [Suito et al., 2011; Ozawa et al., 2012]. Similarly to the large afterslip in the Tohoku region before the 2011 earthquake, the Hyuga-nada afterslip had decreased the coupling rate to a large extent for a very long time together with slow slip events. In the Tohoku case, there was a possibility that the change in coupling caused by the afterslips of the Mw7-class earthquakes off the Tohoku region, northeast Japan, before the 2011 Tohoku earthquake was a long-term precursor signal of the 2011 earthquake [Suito et al., 2011; Ozawa et al., 2012]. We cannot rule out the possibility that the afterslip of the 1996 Hyuga-nada earthquakes is a long-term precursor signal for the anticipated Hyuga-nada earthquake as was pointed out in the 2011 Tohoku case. Because it seems that the frictional characteristics at the interface of the Pacific plate offshore of the Tohoku region, Japan (Figure 1a), and the Philippine Sea plate offshore of Kyushu, Japan (Figure 1a), are very different from each other, this Hyuga-nada case may provide another constraint on the mechanisms of afterslip processes.
 We provide two interpretations for the three transients since 2005. First, these transients are slow slip events and release part of the interplate energy quasi-periodically. The second interpretation is that these transients are quasi-periodic fluctuations of the slip deficit rate and are continuously accumulating energy from the loading of the Philippine Sea plate, although our analysis shows slow slippage that is slightly larger than the estimated annual slip deficit rate in the central area of the slow slip events (Figures 10, 11, and 14). In the following discussion, we refer to the estimated aseismic slip from 2005 as slow slip, although we cannot rule out the possibility that this aseismic slip is different from an ordinary slow slip that releases subduction-related energy. We consider that the difference between the fault behaviors of slowly releasing strain and only lowering the slip deficit rate on the plate boundary is small.
 Our results give a detailed description of the coexistence of long-term afterslip and subsequent quasi-periodic slow slip events in the same area with markedly different time evolutions. That is, no segmentation appears between long-term afterslip and quasi-periodic slow slip events in this case. However, the frictional properties seem to be very different between the afterslip area and slow slip event area as will be described theoretically later. In addition, there have been no clear observations of afterslip being followed by quasi-periodic slow slip events with a moment equivalent to that of the main shocks in the same area. Thus, our results seem to be inconsistent with the conventional view and many other observations, as mentioned in the introduction. Hereafter, we describe the three proposed mechanisms for slow slip events and one simulation study on the coexistence of afterslip and slow slip events in the Hyuga-nada area.
 Since the discovery of slow slip events, many studies have been carried out to model them. In the first model, researchers assumed a change in frictional behavior from velocity-weakening at low slip speeds to velocity-strengthening at higher rates [e.g., Shibazaki and Shimamoto, 2007]. However, the notably limited data for mafic rocks does not support this hypothesis [Segall et al., 2010]. In the second model, slow slip events are modeled under the rate- and state-dependent friction law, which exhibits oscillatory behavior near neutral stability [Liu and Rice, 2007]. Segall et al.  proposed the dilatancy effect as the additional stabilization mechanism required to expand the permissible range for slow slip events to occur in velocity-weakening regions under the rate- and state-dependent friction law. Thus, the model of Segall et al.  is included in the second model. In the third model, it is proposed that slow slip events occur in a velocity-strengthening region induced by external stress perturbations [Perfettini and Ampuero, 2008]. These authors suggested that pore pressure transients could provide the requisite external forcing. It remains unresolved which model reflects the true mechanism of slow slip events. The coexistence of afterslip and slow slip events in the same area and their time sequence observed in our study were not predicted in these simulation studies. Shibazaki and Shimamoto  simulated slow slip events and did not consider afterslip. The models which exploit oscillation under the rate- and state-dependent friction law require very different frictional properties for the afterslip area and for slow slip events. The model of induced slow slip events in a velocity-strengthening area by Perfettini and Ampuero  requires external forcing, the existence of which remains unclear for the Hyuga-nada slow slip events, which mainly started after the cessation of afterslip. Thus, our discovery will provide some constraints on the mechanisms of slow slip events and afterslip in simulation studies. Hereafter, we describe the model by Nakata et al.  for the Hyuga-nada slow slip events and afterslips, in which modeling is based on the rate- and state-dependent friction law.
Nakata et al.  reported that the recurrence of slow slip events and afterslip can be reproduced using the composite law [Kato and Tullis, 2001], which is a type of rate- and state-dependent friction law with a higher cutoff velocity. In their study, they modeled afterslip and slow slip events in Hyuga-nada using the Philippine Sea plate interface estimated by Baba et al.  in this region. They assumed a small characteristic slip distance L (8.4 cm) for the seismogenic fault in the shallow area and a larger L (26.4 cm) for the slow slip area in the deep part near the seismogenic fault. A large L (12 m) was assigned in the background afterslip area that surrounds the seismogenic fault and the slow slip area. The friction parameter A-B in the rate- and state-dependent friction law was set to a negative value for the entire region, which implies velocity-weakening [e.g., Dieterich, 1979]. The critical fault radius, which regulates ordinary earthquakes, slow slip events, and afterslip, is proportional to L [e.g., Kato, 2003; Kato, 2004] as shown below.
 Here Rc is the critical fault radius, μ is the rigidity, A and B are friction parameters, and L is the characteristic slip distance.
 When the fault patch size is much larger than the critical fault radius, ordinary earthquakes occur. In contrast, episodic aseismic slip occurs when the fault patch size is near the critical fault radius. In the case that the fault patch size is much smaller than the critical fault radius, stable sliding occurs [e.g., Nakata et al., 2012]. By using a patch for a slow slip, a patch for an ordinary earthquake, and a surrounding background afterslip area with a suitable critical fault size in three areas, Nakata et al.  reproduced the time evolution of afterslip and slow slip events in the slow slip area. However, the slip history in their results seems to be quantitatively different from the time evolution of aseismic slip in our study, although their results well reproduce our models qualitatively (Figure 12) [Nakata et al., 2012]. For example, their results show that the recurrence time of slow slip events is approximately 30 years, with a recurrence time of approximately 80 years for main shocks. These recurrence periods of slow slip events and main shocks are much longer than our estimate of two years in this study for slow slip events and 10–20 years for main shocks based on historical records.
 Whether or not the model of Nakata et al.  for the frictional properties at a plate interface is correct may be evaluated by investigation of the structure of the plate interface in this region by seismic exploration or by any other measures. This kind of exploration of frictional properties will lead to the discrimination between the three main proposed mechanisms for slow slip events, which include the model of Nakata et al. . Analysis of the slip rate and shear stress of the afterslip and slow slip events in the Hyuga-nada area will provide more information on the frictional properties in this region, as was first conducted for the 2003 Tokachi-Oki earthquake of Mw 8.0 [Miyazaki et al., 2004], although this is beyond the scope of this study. We can obtain information about the critical fault radius from the estimated size of slow slip events, which have a radius of roughly 20 km, because a slow slip event occurs when its size is near the critical fault radius under the rate- and state-dependent friction law. Subsequent observations of crustal deformation are also important. If the size and area of slow slip events change over time, our understanding of the nature of the critical fault radius and the slow slip patch in this region will require modification. Furthermore, if the slow slip events vanish in the future, it will suggest that they are induced by an unknown external perturbation. This will require substantial changes to the model proposed by Nakata et al. , in which spontaneous slow slip events occur quasi-periodically in the interseismic period. On the contrary, if slow slip events continue until the next Hyuga-nada earthquake, it will seem that the slow slip events after the cessation of afterslip in the Hyuga-nada area are spontaneous, since there appears to be no external perturbation that induces slow slip events every two years for such a long time in this region except for the loading of the subducting Philippine Sea plate. Thus, the time evolution of aseismic slip will provide us with information on whether the current slow slip events are spontaneous or induced. Furthermore, if we can accurately reproduce observations in simulation studies of the Hyuga-nada afterslip and slow slip events, we may be able to distinguish between the above-mentioned mechanisms proposed for slow slip events and afterslip in the Hyuga-nada area.
 The steady slip deficit rate between October 2007 and March 2009 corresponds to around 30–50% of the plate convergence rate (see Figure 14a) [Sella et al., 2002]. The slow slippage or deviation from a steady slip deficit rate amounts to 4 to 6 cm/yr (Figures 10 and 11) in the Hyuga-nada area. Because slow slippage reaches 4–6 cm in one year and is negligible in another year, the deviation from the 2–4 cm/yr steady slip deficit (Figure 14a) over two years is 4–6 cm, during which the steady slip deficit reaches 4–8 cm. Thus, the total aseismic slip is negligible and the energy balance roughly holds near the central area of the slow slip events from 2005, although the strain will obviously accumulate in each earthquake cycle until the next Hyuga-nada earthquake and its afterslip in these areas.
 It is not clear why slow slip events started after the coupling rate recovered to the steady state. One explanation is that the stress approaches the strength of the plate boundary so that a slow slip event begins to reduce the stress.
 Together with the Bungo long-term slow slip events [e.g., Yagi and Kikuchi, 2003; Ozawa et al., 2007b; Hirose et al., 1999], the Hyuga-nada slow slip events are changing the stress state to one favorable for the occurrence of an anticipated earthquake at the asperity of the 1968 Mw 7.5 earthquake and the 1996 Hyuga-nada earthquakes (Figures 1b and 2). Furthermore, some simulation studies suggest that the recurrence interval of slow slip events gradually becomes shorter until the occurrence of a large earthquake [e.g., Matsuzawa et al., 2010]. In light of these points, ongoing monitoring of the Hyuga-nada slow slip events is important as a natural experiment for deciphering the coupling recovery process, friction properties, and the risk of a nearby large earthquake.
 Afterslip occurred smoothly over time in the Hyuga-nada region before 2004 with a total moment magnitude of approximately 7.3, as shown in Figures 10b–10i, 12f, and 13a, although we cannot rule out the possibility that nonperiodic small slow slip events occurred during this period.
 After 2004, three relatively large slow slip events with moment magnitude ranging from 6.7 to 6.8 occurred in almost the same area, which is within the afterslip area between 1997 and 2004. The total moment magnitude reached 7.4 for the entire period between 1997 and 2010, which is far beyond those of the main shocks. This time evolution of the coupling rate is markedly different from the logarithmic decay expected from modeling studies [e.g., Marone et al., 1991; Perfettini and Avouac, 2004] and observations in many other cases [e.g., Hsu et al., 2006; Ozawa et al., 2012].
 It remains unclear whether the present state, consisting of the steady state for one year and a slow slip event for one year, is a typical interseismic steady state. This will be resolved by observing the time evolution of the future coupling rate. If the slow slip events vanish in the future, it will suggest that they are induced by an unknown external perturbation.
 We are grateful to our colleagues in charge of the GEONET operation and also to our colleagues in the Crustal Deformation Research Division and Shibazaki of the Building Research Institute for helpful discussions. We are also grateful to T. Hori and R. Nakata of the Japan Agency for Marine-Earthquake Science and Technology. All seismicity data were obtained from the Japan Meteorological Agency.