Figure 4 compares the spatial distributions of SDRs with the epicenters of LFEs (Arai et al. 2016; Nakamura 2017) and the slip rates of cumulative SSEs from July 2010 to February 2013 (Kano et al. 2018b). Notably, our analysis suggests the existence of a large SDR region at depths of 20–25 km that is complementarily distributed with LFE epicenters on the shallower side and the large slip area of SSE on the deeper side. This may reflect that frictional properties vary with depth along the plate interface, although it is uncertain whether the shallow plate interface where the LFEs occur is coupled or not due to the low resolution (Additional file 1: Fig. S1b). The depth boundary between large SDR and SSE areas was ~ 25 km, which corresponds to the structural boundary between the crust and mantle wedge in the overriding plate, as investigated by a seismic reflection survey (Figure 3 in Arai et al. 2016); the large SDR areas are located below the arc crust, while the large SSE areas are located below the mantle wedge. Although the profile in the seismic survey (indicated by the purple line in Fig. 4) was located ~ 50 km east of Ishigaki Island, the plate geometry does not change significantly in the along-strike direction, and thus, the fault slip style along the plate interface may be related to the structure in the overriding plate. A similar relation between the structure in the overriding plate and fault slip style was indicated by seismic tomography (Yamamoto et al. 2018). Yamamoto et al. (2018) discussed the high Vp/Vs of oceanic crust inferred below the SSE region, indicating the existence of fluid that contributes to SSE activities. In addition, their result may suggest an along-dip variation of Vp/Vs, that is, lower Vp/Vs values near the large SDR region compared to the SSE region. However, considering the spatial resolution of tomography (~ 30 km in the horizontal direction), further investigation will be essential to discuss an along-dip variation of the velocity structure and its relation to fault slip style.
Figures 3a and 4 show small SDRs in the main slip area of the SSEs. As already mentioned, the secular velocities are the long-term average crustal movements that include the effects of both stress release and accumulation in the SSE region. Therefore, small SDRs in the SSE region indicate that the accumulated stress during the inter-SSE period are fully released by the repeated SSEs, as suggested by Kano et al. (2018b).
Our results reliably indicate SDRs of 17–47 mm/year to the south of the Yaeyama Islands. However, if we consider a 1 − σ estimation error, the SDRs on the significant patches range from 0.09 to 71 mm/year. In the following discussion, to roughly investigate the magnitude of possible fault slips, we used the representative SDR value of 33 ± 24 mm/year calculated by averaging the SDRs for the subfaults to the south of the Yaeyama Islands (the black rectangle in Fig. 4). This value corresponds to 26 ± 19% of the relative plate motion (~ 13 cm/year) of the PH relative to the SR. This result indicates that SDRs exist despite the low coupling ratio, suggesting the accumulation of elastic strain for future possible earthquakes.
One possibility to release the slip deficit could be a seismic slip. As there are no records of magnitude 8 class earthquakes around the Yaeyama Islands for the 250 years following 1771 (Ando et al. 2018), it can be stated that if this region accumulates a slip deficit at the same rate in the 2010s analyzed in this study as for the 250-year period, the expected slip could be 8.4 ± 6.1 m. If we assume a rigidity of 30 GPa and the source region indicated by the black rectangle in Fig. 4 with a uniform slip of 8.4 m (we refer to this fault model as model I), this leads to an earthquake with a seismic moment of 8.1 ± 5.9 × 1020 Nm corresponding to an Mw of 7.5–8.0. As mentioned before, the resolution of the subfaults shallower than the large SDRs is low (Additional file 1: Fig. S1b), thus it may be possible that the source region extends to shallower depths. If we extend the source region (~ 120 km wide) near the trench axis (we refer to this fault model as model II as indicated in Fig. 4) to reach the location where Nakamura (2009) estimated the source model of the 1771 Yaeyema earthquake, the Mw becomes larger, up to ~ 8.3, assuming the same uniform slip as in model I.
Usami (2010) reported from historical documents that Ishigaki Island experienced a ground shaking corresponding to the JMA seismic intensity of IV in the 1771 Yaeyama earthquake as summarized in Ando et al. (2018). Supposing that either models I or II caused this earthquake, both models could result in the large ground shaking in the Ishigaki Island as reported by Usami (2010). Tsunami deposits indicated that the 1771 earthquake generated large tsunamis leading to maximum runups of ~ 27 m on the eastern coast of Ishigaki Island (e.g., Ando et al. 2018). As mentioned above, model II includes a part of the source region of Nakamura (2009), which was determined to explain the recorded tsunami heights by numerical simulation. Therefore, model II can generate tsunamis as well although the amplitude of shallow slip needs to be further considered for the quantitative comparison of tsunami run-up heights. On the other hand, model I is unlikely to generate large tsunamis because it is an interplate earthquake at depths of 20–25 km. Thus, a possible scenario to cause such a large tsunami may be that a submarine landslide occurred in addition to the earthquake itself. Imamura et al. (2001) proposed a model in which the submarine landslide triggered by an intraplate earthquake caused the large 1771 tsunami. Although the location of this source model is ~ 40 km away from the eastern edge of the large SDR area (indicated by a yellow square in Fig. 4), if the interplate earthquake in model I resulted in a submarine landslide in the same location, the large tsunami run-up height could be explained. Other models of the 1771 tsunami were proposed (Hsu et al. 2013; Okamura et al. 2018). Based on seismic reflection surveys, Hsu et al. (2013) proposed that the tsunami was related to a thrust fault along the ~ 450 km long mega-splay fault in the trench parallel direction east of 125.5° E close to the Ryukyu Trench. On the other hand, Okamura et al. (2018) discovered a seafloor depression on the seafloor located above the source model of Nakamura (2009). This depression was interpreted to be caused by the collapse of the accretionary prism and the resulting rotational slide. They showed that the collapse can explain the 1771 tsunami regardless of the coupling state of the shallow plate interface. Both models indicated a tsunami source close to the Ryukyu Trench; however, our results cannot judge these possible scenarios, as we only considered the SDRs along the subducting plate interface, and onland GNSS data cannot resolve the coupling of splay faults located close to the trench.
Another possibility through which SDRs are released is by aseismic slip, that is, through SSEs. Although SSEs were recurrently identified with an Mw greater than 6.2, particularly on the deeper side of the large SDR areas (Heki and Kataoka 2008; Nishimura 2014), SSEs including the large SDR region as the source area have not been detected at least since the establishment of GEONET. It may be possible that such a transient slip would occur in the edge of the large SDR region as inferred in the Nankai subduction zone (Kano et al. 2019). Considering the high resolution in the large SDR region, the GNSS observations would be able to detect similar magnitudes of SSEs as those of the deeper ones if an SSE were to occur in the large SDR region. There are three possibilities. One possibility is that quite a large SSE would occur in this region, which has, however, not occurred at least in recent two decades. Supposing that such a large SSE would occur with a recurrence interval of 30 years and release all the slip deficits of 33 ± 24 mm/year, the SSE will have a seismic moment of 9.5 ± 6.9 × 1019 Nm, corresponding to an Mw of 6.9–7.4, which would be certainly detected by the GNSS. In this case, the SDRs can be explained by aseismic slip only. However, large SSEs with such a long recurrence interval have not been observed because of a lack of long-term GNSS observations. Therefore, this possibility is less plausible, although we cannot entirely rule it out. The second possibility is that a long-lived SSE with durations longer than a few decades is occurring; however, this would be difficult to identify from recent GNSS observations. The third possibility is that SSEs would have occurred within the large SDR region, but were too small to be detected by the GNSS. However, it would be difficult to explain the large SDR based on such a series of relatively small SSEs. In this case, the large slip deficit would not be released by an aseismic slip alone but seismically. Although the observations, which are limited in both time and space, cannot resolve which of the three possibilities discussed above is true, continuous GNSS observations as well as seafloor geodetic observations will reveal more about the potential of megathrust earthquakes in the southern Ryukyu region.
Our inversion results indicated significant SDR regions of 17–47 mm/year that would be possibly released by seismic slip. These SDR values correspond to the low coupling ratio of 13–36% calculated from the relative plate motion (~ 13 cm/year) of the PH relative to the SR. Although these values include estimation errors, they suggest that these regions may mostly show a steady sliding, but occasionally result in a dynamic slip. Numerical simulations on earthquake cycles suggested that constant sliding and dynamic slip will occur in the same fault during complex cycles (e.g., Hori and Miyazaki 2011; Nakata et al. 2012; Noda and Lapusta 2013). For example, Hori and Miyazaki (2011) reported that aseismic stable sliding with a speed of ~ 34–62% of the plate subduction rate occurred after a nearby magnitude 7 class earthquake, while the same area indicated a dynamic rupture during a magnitude 9 class earthquake. This complex slip behavior may not be the result of a single source fault but that of a complex interaction between nearby multiple source regions. Although this model is not directly applicable to our study area, coexistence of both fast slip and steady sliding within a low coupling area may imply the existence of another potential seismic source located, for example, in the shallow plate interface or at similar depths on the western or eastern side of the large SDR region revealed in this study. The latter region may be too far from the onshore GNSS stations to investigate plate coupling. Seafloor geodetic observations will be important from this point of view as well.
A possible modification of the proposed SDR model would involve the viscoelastic effect in the asthenosphere, while our modeling is based on the elastic model alone. Ignoring the viscoelastic effect was shown to lead to the overestimation of the lower limit of the locking depth and the amount of SDR (e.g., Li et al. 2015, 2018; Itoh et al. 2021). This is because the viscous relaxation lends itself to a long-wavelength deformation, resulting in movement at more distant stations compared to the elastic model. These results were based on inversion results using secular velocities, while we used distance changes in addition to vertical velocities. Therefore, further analysis is necessary to examine the effect of viscoelasticity on our results. Introducing viscoelasticity is also important while interpreting the temporal variation of vertical velocities. When recurrence intervals of earthquakes are considerably longer than the asthenospheric viscous relaxation time, as in the case of the southern Ryukyu area, decadal-scale temporal variation of interseismic deformation becomes more significant in the later stage of the interseismic period (Sagiya 2015; Hashima and Sato 2017). Hashima and Sato (2017) numerically calculated the vertical deformation associated with the 2011 Tohoku-oki earthquake using a simple two-layered lithosphere–asthenosphere model. Their results indicated that the viscoelastic effect due to coseismic slip is more significant than the effect of interplate locking in the early stage of the interseismic period, while the latter is predominant in the later interseismic period. As a result, temporal variation of vertical velocities at onland stations > 50 km from the interplate coupling area became more significant. In addition, the spatial pattern of the vertical deformation was different from the assumed slip models whether the coseismic slip reaches the asthenosphere (Hashima and Sato 2017). Therefore, how viscoelasticity affects SDR estimation depends on the assumed model. In any case, continuous monitoring of vertical velocities and their decadal-scale temporal change will lead to a more realistic modeling of interplate coupling with a viscoelastic model. In addition, although we only consider the motion due to plate subduction, viscoelastic effects due to back-arc spreading to the north of the Yaeyama Islands will affect the secular velocities. Therefore, this should be taken into account when considering viscoelasticity, which will be subject to future study.