Unlike the results of Arai (2021), the b-value in the rift-axis of the southern Okinawa Trough was almost the same as the b-value in the Ryukyu arc. One reason for this is that the previous study did not consider the effect of spatial variation in MC. In the YA, the MC was approximately 2.0 (Fig. 2b). However, when the earthquake swarms occurred, the MC increased to 3.0–3.5 (Fig. 2b). If the MC is estimated to be smaller than the actual value, the FMD slope will be calculated according to the magnitude range, including undetectable small earthquakes, resulting in an estimated b-value that is lower than the true one. For these reasons, the low b-values (< 0.6) in the YA and MI, which were observed in the previous study, were apparent values. After an appropriate MC was used, the area of low b-values (below 0.8) almost disappeared (Fig. 3a). In the previous study, since the b-value was calculated with constant MC across the region, without considering the temporal and spatial heterogeneity of MC, a lower b-value appeared.
Another factor is the discontinuity between MD and MV around M4.0, which led to an apparent decrease in the number of earthquakes around MJ 4.0 and a local flattening of the FMD slope (Fig. 1d). For example, consider that the discontinuity between MD and MV occurs around MJ 4.0. In this case, if the MC is sufficiently smaller than MJ 4.0, the discontinuity will have little effect on the b-value. However, when the MC increases and becomes close to MJ 4.0 due to earthquake swarm activity, the estimated b-value would be smaller. The MD and MV discontinuity effects do not affect the estimation of b-values in most areas. However, in the October 2002 earthquake swarm area in YA, the b-values calculated using the MJ were different from those using MV; the very low b-values calculated using MJ would be due to the high MC and splitting of MD and MV.
MC is high (1.6–2.8) in both the southern and central Okinawa Trough (Additional file 2: Figures S5a and b). Contrastingly, in the Ryukyu arc, MC is low (0.9–1.8) in both the southern and central Ryukyu arc (Additional file 2: Figure S5c and d). This indicates that the seismic network is located only in the islands; thus the detection capability is high in the vicinity of the islands, while it is low away from the islands, such as in the Okinawa Trough. The value of MC calculated using earthquakes along the entire southern Okinawa Trough rift-axis is 1.6. However, each grid’s MC is considerably variable, ranging from 1.6 to 2.8. This variation would be caused by the low earthquake detection capability of the Okinawa Trough and the effect of a temporary increase in MC due to earthquake swarm activity. Another notable feature of MC in the Okinawa Trough is that the MC for the entire region corresponds to the lower end of the MC calculated for each grid. This indicates that even if the MC for the whole area is sufficiently small, it would not be reasonable to use that MC for the whole area when estimating the b-values for each grid in the region.
In the case of the October 2002 earthquake swarm in the YA area, the large number of earthquakes with detectable events in the seismic waveforms but not determined hypocenters was another factor that contributed to the large MC. Indeed, there were many earthquakes in the 2002 earthquake swarm whose waveforms were recorded but whose epicenters have not been determined. When the earthquake swarm began at 6:00 Japan Standard Time (JST) on October 24, a few earthquakes below MJ 3.0 were determined, and the MC was approximately 4.0 (Fig. 5a). F-net waveforms at station IGK (epicentral distance from the earthquake swarm area was approximately 100 km) at 7:13–7:45 JST showed several small spindle-shaped waveforms whose hypocenters were not determined by the JMA (Fig. 5b). These waveforms were similar to those of the MJ 3 class earthquakes that immediately preceded and followed. Therefore, the spindle-shaped waveforms should also correspond to earthquakes that occurred in the earthquake swarm area. For example, between 7:16 and 7:20 and between 7:35 and 7:42, the waveform at IGK showed multiple spindle-shaped waveforms (blue lines in Fig. 5b). Assuming that the hypocenters were within the earthquake swarm area, these Mv were estimated to be 2.2–2.7, based on the maximum amplitude of the spindle-shaped part (2000–6000 nm/s). This means that the earthquakes of Mv 2.2 to 2.7 were distinguishable from the noise in IGK, but their hypocenters could not be determined. In the waveform at station YNG (epicentral distance from the earthquake swarm area was approximately 140 km), the spindle-shaped waveforms corresponding to those at IGK were very difficult to detect because they were obscured in the noise. As P and S phases could not be picked up at any stations other than those near Ishigaki Island, there observations were not sufficient to determine the hypocenters.
As the magnitude increases, the number of seismic stations available to detect seismic waves increases. It is possible that the difference in the number of used seismic stations leads to a bias in the hypocenter location and affects the b-value. To investigate the bias between the number of seismic stations and hypocenters, I examined whether the hypocenter distribution of the same magnitude range is different with a different number of stations (Additional file 2: Figure S6). In area A, for both M2.0–3.0 and M3.0–4.0 earthquakes, epicenters with many seismic stations were concentrated in the south. In contrast, the north part of area A gathered epicenters with fewer seismic stations, i.e., smaller magnitude earthquakes tended to be distributed in the north part of the earthquake cluster. M2.0–3.0 earthquakes are distributed farther north than M3.0–4.0 earthquakes. Therefore, a high b-value area is generated on the northern edge of the earthquake cluster (Fig. 3a and Additional file 2: Figure S6c). In area B, for earthquakes of magnitude 2.0–3.0, epicenters with many seismic stations were distributed north of epicenters with a smaller number of seismic stations. M3.0–4.0 earthquakes are distributed north of the M2.0–3.0 earthquake cluster. Therefore, larger earthquakes were more likely to be distributed to the north and smaller earthquakes are distributed in the south, resulting in high b-values in the southern part of the cluster (Fig. 3a and Additional file 2: Figure S6c). The bias in the hypocenter location due to the difference in the number of seismic stations could be responsible for the anomalous b-values at the edges of the earthquake cluster.
The b-values in the central and southern Okinawa Trough (0.7 to 1.2) were within the b-values observed at the mid-ocean ridge (MOR). The b-values at Rodriguez Triple Junction in the Indian Ocean ranged from 1.1 to 1.46, with higher b-values reported for the segment axis with higher spreading rates (Katsumata et al. 2001). b-values at 23°N in the Mid-Atlantic Ridge (MAR) ranged from 0.75 to 1.2 (Toomey et al. 1988). The b-values at 26°N in the MAR also ranged between 0.9 and 2.25 (Kong et al. 1992). Since both b-values in the MAR were estimated using seismic moments, the b-values obtained from seismic moments were converted to the magnitude case to discuss here. Seismic activity at the MOR is often interpreted as magmatic or tectonic (Bohnenstiehl et al. 2008). Higher b-values at the MOR are also thought to reflect magmatic activity (Tolstoy et al. 2001) or strong tectonic heterogeneity (Bohnenstiehl et al. 2008). Applying the example at the MOR to the Okinawa Trough, a b-value of less than 2.0 at the Okinawa Trough suggests that earthquake swarms at the Okinawa Trough might be strongly tectonically influenced or might be occurring at locations where the structural heterogeneity is not extremely strong.
However, the improvement in the spatio-temporal resolution of seismic activity may reveal the spatio-temporal heterogeneity of b-values. In the geothermal area, the b-value varies in complex ways. In the Yellowstone volcanic field, b-values are normal to high in most areas, with localized low b-value regions (Farrell et al. 2009). For the 2008–2009 Yellowstone earthquake swarm, the b-value was 1.1 (Glazner and McNutt 2021). This earthquake swarm was interpreted to have been triggered by the movement of magmatic fluid or poroelastic stress pulses (Farrell et al. 2010). In other cases, the 2014 southeastern Long Valley Caldera swarm showed a temporary decrease in b-values when the swarm activity (~ 1 km scale) occurred on faults that reflected the regional stress field, followed by an increase in b-values when the swarm activity occurred on the nearby non-parallel faults (Shelly et al. 2016). The spatial resolution of ~ 10 m and the fact that MC captures earthquakes as small as -0.4, even very small earthquakes, capture fine-scale statio-temporal b-value change. In the Okinawa Trough, if the accuracy of hypocenter determination is improved, it would be possible to distinguish between areas of decreasing b-value due to increased stress and areas of high b-value due to hydrothermal activity in the swarm seismic activity area. For example, the YO has a higher b-value than the other Okinawa Trough rift-axis (Fig. 3a). This area is located on the Daiyon-Yonaguni knoll, where active hydrothermal activity has been reported (Ishibashi et al. 2015). This means that even in the rift-axis of the Okinawa Trough, the b-value is higher where hydrothermal activity is active. Since dyke intrusion and hydrothermal activity are occurring in the rift-axis of the Okinawa Trough, these might have resulted in a narrow distribution of low and high b-value regions. However, this might not be detectable due to the insufficient accuracy of hypocenter determination. Therefore, it is necessary to develop a seismic observation system on the seafloor in the Okinawa Trough.