Changes in the b value in and around the focal areas of the M6.9 and M6.8 earthquakes off the coast of Miyagi prefecture, Japan, in 2021

We investigated changes in the b value of the Gutenberg-Richter's law in and around the focal areas of earthquakes on March 20 and on May 1, 2021, with magnitude (M) 6.9 and 6.8, respectively, which occurred off the Pacific coast of Miyagi prefecture, northeastern Japan. We showed that the b value in these focal areas had been noticeably small, especially within a few years before the occurrence of the M6.9 earthquake in its vicinity, indicating that differential stress had been high in the focal areas. The coseismic slip of the 2011 Tohoku earthquake seems to have stopped just short of the east side of the focus of the M6.9 earthquake. Furthermore, the afterslip of the 2011 Tohoku earthquake was relatively small in the focal areas of the M6.9 and M6.8 earthquakes, compared to the surrounding regions. In addition, the focus of the M6.9 earthquake was situated close to the border point where the interplate slip in the period from 2012 through 2021 has been considerably larger on the northern side than on the southern side. The high-stress state inferred by the b-value analysis is concordant with those characteristics of interplate slip events. We found that the M6.8 earthquake on May 1 occurred near an area where the b value remained small, even after the M6.9 quake. The ruptured areas by the two earthquakes now seem to almost coincide with the small-b-value region that had existed before their occurrence. The b value on the east side of the focal areas of the M6.9 and M6.8 earthquakes which corresponds to the eastern part of the source region of the 1978 off-Miyagi prefecture earthquake was consistently large, while the seismicity enhanced by the two earthquakes also shows a large b value, implying that stress in the region has not been very high.


Introduction
An earthquake of magnitude (M) 6.9 occurred on March 20, 2021, at 18:09, on the Pacific coast of Miyagi prefecture, northeastern Japan, and successively, an M6.8 4 earthquake occurred on May 1, 2021, at 10:27, at about 50 km south of the M6.9 earthquake (Figures 1a,b) (Earthquake Research Committee (ERC), 2021a,b,c). Both earthquakes were located on the periphery of the slip region of the 2011 M9.0 Tohoku earthquake (Figure 1a), almost corresponding to the down-dip end of the interplate coupling zone between the overriding continental and the subducting Pacific plates (Igarashi et al., 2001). As can be seen in Figure 1a, the foci of the two 2021 earthquakes were located in the area where the coseismic slip (orange contours) of the 2011 Tohoku earthquake was relatively large and the afterslip (black contours) during 7 months just after the earthquake occurrence was small compared to the surrounding region . Interestingly, it was reported that the focus of the M6.9 earthquake was situated at the border point where the interplate slip during the period from 2012 through 2021 inferred by the analysis of repeating earthquakes was large on the northern side and notably small on the southern side (Tohoku University, 2021b).
When the M6.9 earthquake occurred on March 20, the ERC (2021a,b) pointed out that the focal area was located in the western part of the source region of the so-called off-Miyagi prefecture earthquake (Figure 1b), an interplate earthquake that has occurred 5 sequentially at intervals of about 38 years, the most recent one being the M7.4 earthquake in 1978 (green contours in Figures 1a,b; Yamanaka and Kikuchi, 2004).
Note that the focal area of the M6.8 earthquake on May 1 was located at the west of the slip region of the M7.2 earthquake that occurred at off-Miyagi prefecture in 2005 (purple contours in Figures 1a,b;Yaginuma et al., 2006). There was an overlap with the southeastern part of the source region of the 1978 off-Miyagi prefecture earthquake ( Figure 1b).
Here, we report the results of our analysis on spatio-temporal changes in the b value of the Gutenberg-Richter's (GR) law (Gutenberg and Richter, 1944) in and around the focal areas of the M6.9 and M6.8 earthquakes and in the source regions of the 1978 and 2005 off-Miyagi prefecture earthquakes, and discuss the implications of these results, noting the stress state on the plate interface in these regions.

Method
We exploited the GR law, log10N = a -bM, where N is the number of events equal to or above M, and a and b are constants (Gutenberg and Richter, 1944). Globally, 6 on average, b ~ 1, but locally, b values show substantial spatial and temporal variation.
In some cases, the proportion of earthquakes with large magnitudes is higher (b < 1), in others, the proportion of earthquakes with small magnitudes exceeds the average expectation (b > 1) (Figures 2-5).
To estimate b values consistently over space and time, we employed the EMR (Entire-Magnitude-Range) technique (Woessner and Wiemer, 2005), which simultaneously calculates the completeness magnitude Mc, above which all events are considered to be detected by the referential seismic network. EMR applies the maximum likelihood method represented by Eq. (6) of Utsu (1999) (e.g., Aki, 1965;Utsu, 1965) when computing the b value to events with magnitudes greater than Mc. We calculated the b values (Figures 2-5), provided that we found a minimum of 20 events with magnitudes greater than Mc for a given sample. We evaluated the uncertainty of the maximum-likelihood estimates of the b value, as described in Shi and Bolt (1982). The difference in b is not considered to be significant if the test proposed by Utsu (1992Utsu ( , 1999 is not passed. If logPb, the logarithm of the probability that the b values are not different, is equal to or smaller than -1.3 (logPb ≤ -1.3), then the difference in b is significant (Schorlemmer et al. 2004;Nanjo and Yoshida, 2017). A fit of the GR law to observations for three circle areas is given in Figure 3c, where the b value was estimated by the maximum likelihood method (Aki, 1965;Utsu, 1965Utsu, , 1999, rather than a coefficient of the GR law. The b value is smaller for the circle 1 (b = 0.4 ± 0.1) than for the circle 2 (b = 1.6 ± 0.1), taking an intermediate value for the circle 3 (b = 0.9 ± 0.2), indicating the significant difference in b among the three circle areas. This significance is further supported by the Utsu test (Utsu, 1992(Utsu, , 1999Schorlemmer et al. 2004;Nanjo and Yoshida, 2017), revealing logPb1,3 (the logarithm of the probability that the b values for the circles 1 and 3 are not different) to be -3.6, and logPb2,3 for the circles 2 and 3 to be -4.5 ( Figure 3).
The b value is known to be sensitive to differential stress and its inverse correlation with differential stress has been evidenced by many laboratory and field studies (Mogi, 1963;Scholz, 1968Scholz, , 2015Lei, 2003;Goebel et al., 2013).
Investigation into space-temporal variation in b values to probe the stress state in the Earth's crust (Smith, 1981;Narteau et al., 2009) has been applied to locate asperities (Hirose et al., 2002;Yabe, 2003;Tormann et al., 2015;Nanjo 8 and Yoshida, 2018), and to estimate frictional properties (e.g., Sobiesiak et al., 2007;Ghosh et al., 2008) on the plate interface along subduction zones. Foreshocks have been known to show small b values (Suyehiro et al., 1964;Gulia and Wiemer, 2019). Patches with small b values on active faults have been observed to coincide with locations of subsequent large earthquakes Nanjo et al., , 2019Nanjo, 2020).

Data
We used the Japan Meteorological Agency ( In the first period, the local b value for the seismicity around the eventual M6.9 hypocenter was b = 0.8 ± 0.3 (light blue), nearly a global average (b ~ 1) (Figure 3d).
However, the b value in the second period became 0.6 ± 0.2 (blue), and it decreased to a low value (0.4 ± 0.1) in the last period. As shown in Figure 2b, the decrease in the b value around the focus of the 2021 M6.9 earthquakes in the third period relative to the 11 first period was statistically significant (Utsu, 1992(Utsu, , 1999Schorlemmer et al., 2004;Nanjo and Yoshida, 2017). The decrease in the b value in region A, a region surrounding focal areas of the 2021 M6.9 and M6.8 earthquakes, is also seen from Figure S4   On the other hand, the b value in region B had been relatively large compared to the b value in region A throughout the whole period before the M6.9 earthquake, although region B showed a rather large range in b extending from 0.6 to 1.2 ( Figure 5). It is also 13 worth noting that the large-b-value state in region B has been continuing not only after the occurrence of the M6.9 earthquake on March 20, but also after the M6.8 earthquake on May 1. We suppose this result suggests that the differential stress in the eastern part of the source region of the 1978 off-Miyagi prefecture earthquake has been relatively low throughout the entire study period.

Discussion
By making maps and cross sections of the b value in the region around the foci of the March 20 and May 1, 2021, off-Miyagi prefecture earthquakes, we found that the b value in their focal areas had been considerably and consistently small and that the value became as low as 0.4-0.6 within a few years before their occurrence (Figures 2   and 3). The distinct small b-value spot corresponded to the eventual M6.9 hypocenter (Figures 2 and 3). This close match between the area of low and decreasing b-values and the eventual M6.9 and M6.8 hypocenters supports the idea that the b value may be a stress meter for the Earth's crust. Our finding also indicates that the stress before the M6.9 and M6.8 earthquakes had been high in the eventual focal areas, and the differential stress there had been heightened as time progressed. Here, it is to be noted that the coseismic slip of the 2011 Tohoku earthquake (orange contours in Figure 1a) stopped just short of the east side of the focus of the M6.9 earthquake and the afterslip of the 2011 Tohoku earthquake in 7 months (black contours in Figure 1a and Iquique (Chile) (Wiemer and Wyss, 2002;Nanjo et al., 2012Tormann et al., 2012Tormann et al., , 2015Schurr et al., 2014;Nanjo, 2020).

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It is notable that the b value for the events after the M6.9 earthquake on March 20, which occurred mainly near the southern end of the slip area had been as small as 0.5-0.6 (blue) (bottom panel of Figure 2a and left panel of Figure 3b). We should have focused our attention more on the observation that the events after the M6.9 earthquake showed a small b value and the rupture of the quake had not covered the whole small-b-value area that had existed before the quake, although it is not certain if we could have foreseen the occurrence of the M6.8 earthquake before May 1.
The finding that the b value that had appeared to be small before the M6.9 earthquake was still low after the M6.8 earthquake is somewhat an enigma ( Figure 5). This might mean that the two earthquakes had not fully unloaded the stress on the pre-existing asperity as interpreted for the Parkfield earthquake in 2004(Tormann et al., 2012. However, we consider that such an interpretation cannot be applied to our case, because the rupture areas of the two earthquakes on March 20 and May 1 seem to cover almost entirely the small-b-value zone that had existed before their occurrence (bottom panel of Figure 2a). One possible explanation may be that some patches with high stress had remained unruptured, and events after the M6.8 earthquake have been occurring there. The results of seismic source analysis of the M6.9 and M6.8 earthquakes (Tohoku University, 2021a), which indicate a rather complex rupture process, seem to support this idea. Moreover, we would like to point out the occurrence of an M5.8 earthquake on April 18 at the far end of the rupture area of the earthquake on May 1 (Figure 5a and Figures S3a,c of Additional file 1). The sequential occurrence of the M6.9, M5.8 and M6.8 earthquakes in the small-b-value zone that had been observed before these earthquakes indicates that there had existed at least three high-stress asperities. This also seems to imply that additional smaller patches might have remained unruptured. ERC (2021a), after a meeting held on March 22, commented that it was necessary to pay attention to the occurrence of another large earthquake that might result in a similar or even stronger seismic intensity during the period of one week, especially in a few days. We suppose that, in the background of this caution, ERC (2021a) concerned about the occurrence of the so-called off-Miyagi prefecture earthquake that has been occurring sequentially at intervals of about 38 years, and whose probability of occurrence within the next 30 years was estimated to be about 60-70% as of January 1, 2021 (ERC, 2021a,b). In addition, Nakata et al. (2016), based on numerical simulation, suggested that the time interval between the M~9 earthquake and the subsequent earthquake off the coast of Miyagi prefecture would become shorter than the average recurrence interval during the later stage of the M~9 earthquake cycle. As was pointed out in the Introduction, the two 2021 earthquakes occurred in the western part of the source region of the 1978 M7.4 off-Miyagi prefecture earthquake, and the focal area of the M6.8 earthquake is located west of the source region of the 2005 earthquake that occurred off the coast of Miyagi prefecture, fracturing the southern part of the source region of the 1978 earthquake ( Figure 1b). Therefore, it might not be unreasonable for ERC (2021a) to have been anxious about the possibility of the occurrence of a large earthquake on the east side of the focal areas of the two 2021 earthquakes.
Concerning this anxiety, we would like to note that the b value in the region had consistently been rather large before the two 2021 earthquakes and that the seismicity induced there by those earthquakes has been showing a large b value as well (Figures 4   and 5). This indicates that stress in the region on the east side of the focal area of the two 2021 earthquakes had not been so high and the low-stress state has been continuing.
Therefore, we conjecture that the probability of occurrence of a large earthquake in the 18 adjacent region in the very near future may not be so large, although it is necessary to continue to watch for any signal that indicates change in local stress in the region.

Conclusions
This study revealed that the b value in and around the focal areas of the M6.9 and M6.8 earthquakes that occurred off the Pacific coast of Miyagi prefecture, northeastern Japan, on March 20 and May 1, 2021, respectively, had been considerably low before their occurrence. The b value in the vicinity of the M6.9 earthquake decreased to around 0.4 in the last few years. On the other hand, the b value on the east side of the focal areas that corresponds to the eastern part of the source region of the 1978 off-Miyagi prefecture earthquake had been relatively large during the whole period that was investigated. This result implies that the stress in the region had not been as high as the stress in the focal areas of the two earthquakes in 2021 and that the low stress state there has been continuing.

Ethics approval and consent to participate
Not applicable.

Competing interests
The authors declare that they have no competing interests.