Background

Seismicity has become high in almost all parts of Japan since the 2011 magnitude (M) 9.0 Tohoku-Oki earthquake (e.g., Ishibe et al. 2011; Toda et al. 2011). The 2016 Kumamoto earthquakes occurred under these circumstances, leading to a memorable event that once again caused devastating damage in Japan. The Kumamoto earthquakes in the Kyusyu district began with an M6.5 quake on April 14, 2016, at 21:26, on the Hinagu fault zone (FZ), which was followed by numerous shocks including an M6.4 quake on April 15, at 00:03. Eventually, on April 16, at 01:25, an M7.3 quake occurred on the Futagawa FZ. The probability of the occurrence of an M7.0 or so earthquake on the Futagawa FZ had been estimated by the ERC to be 0–0.9% within 30 years (Earthquake Research Committee 2016a), which was rather high among the probability at active faults in Japan. In a press conference that was held just after the M6.5 quake, the Japan Meteorological Agency (JMA) warned of the possibility of large aftershocks that should bring further damages. However, no information on an increased probability of M7 or larger quakes was announced by the JMA during the period between the M6.5 quake and the M7.3 quake. This is because, according to the ERC (1998) protocol (see also ERC 2016b), the JMA had not considered the occurrence of larger earthquakes after an M6.5 quake. Moreover, it usually takes more than one day after a mainshock occurrence for the JMA to issue an announcement of the probability of large aftershocks.

The objective of this study was to investigate if any indication of the occurrence of the M7.3 quake had not appeared in the space–time changes in the rate of occurrence and magnitude-number distribution of shocks after the M6.5 quake on April 14. We found that M3 or larger shocks decreased during the period of one day preceding the M7.3 quake and that the b value (the ratio of small to large events) became pronouncedly high, corresponding to a decrease in comparatively large shocks, especially in the area near the hypocenter of the M7.3 quake.

Tremendous efforts have been made to find effective signatures that indicate a large earthquake occurrence in the near future. A temporal decrease in the b value may be a candidate (e.g., Suyehiro et al. 1964; Nanjo et al. 2012) while another may be seismic quiescence (e.g., Wiemer and Wyss 1994; Sobolev and Tyupkin 1997). It is known that aftershock activity becomes significantly quiescent preceding large aftershocks (Mtasu’ura 1986). However, any signatures ever proposed do not seem to be universal, since accelerating seismicity (e.g., Bowman et al. 1998) and an increase in the b value (Smith 1981) were occasionally reported.

We examined the spatial and temporal trends in the rate of occurrence of comparatively large (M ≥ 3) shocks and the b value in the seismic activity following the M6.5 quake on April 14. We found that the b value in the activity in the northern area became significantly high before the M7.3 quake on April 16 and that this feature corresponded to a decrease in large (M ≥ 3) shocks. We consider that the high b value and the decrease in comparatively large shocks were precursors to the M7.3 quake and propose what caused the observed changes in the space–time seismicity.

Methods and data

Activity following the M6.5 quake is well modeled by two statistical laws: the Gutenberg–Richter (GR) frequency–magnitude law (Gutenberg and Richter 1944) and the Omori–Utsu (OU) aftershock decay law (Utsu 1961).

The GR law is given as log10 N = a − bM, where N is the number of earthquakes per unit time with magnitudes greater than or equal to M, a describes the productivity of the regional seismicity, and the b value is the ratio of small to large events. A high b value indicates a larger proportion of small earthquakes, and vice versa. In the laboratory and in the Earth’s crust, the b value is known to be inversely dependent on differential stress (Scholz 1968, 2015). In this context, measurements of spatial temporal changes in the b value could be used to discover highly stressed areas where future ruptures are likely to occur (Schorlemmer and Wiemer 2005; Nanjo et al. 2012). To consistently estimate b values over space and time, we employed the entire-magnitude-range (EMR) technique (Woessner and Wiemer 2005), which also simultaneously calculates the completeness magnitude M c , above which all events are considered to be detected by a seismic network. EMR applies the maximum-likelihood method (Aki 1965) in computing the b value to events with magnitudes above M c . Uncertainties in b and M c were computed by bootstrapping (Schorlemmer et al. 2003, 2004).

Any completeness estimation method is subject to errors, particularly during productive seismicity times such as investigated in this study. To confirm implications of this study, we compared M c by the EMR method with that by other three methods. Results of the comparison are shown in Appendix 1 and Fig. A1 in Additional file 1.

The OU law is given as λ = k/(c + t)−p, where t is the time since a mainshock occurrence, λ is the number of aftershocks per unit time at t with magnitudes greater than or equal to a cutoff M, and c, k, and p are constants. p = 1 is generally a good approximation (Omori, 1894). From the view point of the rate- and state-dependent friction law (Dieterich 1994), variability in p can be observed: p > 1 and p < 1 for rapidly and slowly decreasing stress, respectively. When stress increases with time, p = 1 at t ≫ 0. If this friction law is valid in the Earth’s crust, then the p values could be used to infer stress history. Similar to the GR case, we used a maximum-likelihood fit to determine the parameters for this law. Uncertainties in the p value were computed by bootstrapping.

Our dataset is the earthquake catalog maintained by JMA. From this catalog, we separated events associated with the 2016 Kumamoto earthquakes after the occurrence of the M6.5 quake on April 14. Owing to a large increase in the number of borehole seismic stations (Obara et al. 2005), event delectability has been greatly improved, which lowers the minimum magnitude for catalog homogeneity (Nanjo et al. 2010). Since 2000, the minimum magnitude is about M = 0–1 in the Kyushu district.

Small events in clusters such as swarms and aftershocks are often missed in the earthquake catalog, as they are “masked” by the coda of large events and overlap with each other on seismograms. According to previous cases (e.g., Helmstetter et al. 2006; Nanjo et al. 2007), M c depends on time t. In creating Fig. 1, we used a moving window approach, whereby the window covered 200 events (red). M c decreased with t from 2.9 and reached a constant value at around 2.2. Relatively large events occurred early in the sequence, and the mean magnitude of these events evolved to small values over time (gray). The time-dependent decrease in M c is consistent with the data. To examine if the occurrence of the M6.4 quake affected completeness, events before the M6.4 quake were excluded from the dataset and M c was assigned as a function of t since the M6.4 quake (blue). Immediately after the M6.4 quake, the estimate of M c was as high as 2.4. M c decreased with t and reached a constant value at around 2.2. We were unable to see any significant deviation of the M c  − t pattern since the M6.4 quake (blue) from since the M6.5 quake (red). This is interpreted as an indication that the effect of the M6.4 quake on M c was weak.

Fig. 1
figure 1

Plot of M c as a function of relative time to the M7.3 quake (vertical solid line) since the M6.5 quake (red) and since the M6.4 quake (blue) in which a 200-event window was used. Uncertainty was according to the bootstrapping (Schorlemmer et al. 2003, 2004). Also included in this figure is a M–time diagram. Vertical dashed lines indicate the relative times, 0.98 and 0.67 days, to the M7.3 quake. Red stars M6-class quakes (M6.5, M6.4), yellow star M7.3 quake

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Results

As a preliminary analysis to modeling the OU law, the cumulative number of M ≥ 3 earthquakes was calculated as a function of relative time to the M7.3 quake over the entire period of activity (Fig. 2a). Taking M c in Fig. 1 into consideration, we truncated the catalog at M = 3 in advance to discard all data that may have been inhomogeneous. Visual inspection of the cumulative curve indicates that there are two kinks at the times 0.98 and 0.67 days before the M7.3 quake, suggesting that transitions to lower occurrence rates of M ≥ 3 shocks happened twice. We divided all shocks into two groups: shocks in the northern area (higher than 32.72° in latitude) and those in the southern area (lower than 32.72° in latitude). We employed the same plotting procedure as for Fig. 2a to show that there were again two kinks at 0.98 and 0.67 days before the M7.3 quake in both groups (Fig. 2d, g). Figure 2d, g shows that the timing of the kinks did not correspond to the occurrence of the M6.4 quake (star at 1.05 days preceding the M7.3 quake), indicating that this quake did not affect the seismic activity as a whole.

Fig. 2
figure 2

Plots of the cumulative number (a, d, g), the number (i.e., the rate of occurrence) (b, e, h), and the change in the number (i.e., the change in the rate of occurrence) (c, f, i) of M ≥ 3 earthquakes as a function of relative time (days) to the M7.3 quake (vertical solid line). a–c seismicity in the entire area, d–f seismicity in the northern area (higher than 32.72° in latitude), and g–i seismicity in the southern area (lower than 32.72° in latitude). We used a time window of 0.08 days. See also Fig. A2 in Additional file 1 for other time windows of 0.05 and 0.10 days. Vertical dashed lines indicate relative times 0.98 and 0.67 days to the M7.3 quake

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We also checked the change in the number (i.e., the change in the rate of occurrence) of M ≥ 3 shocks. To do this, we first plotted the number (the rate of occurrence) of M ≥ 3 shocks in the entire area as a function of relative time to the M7.3 quake in Fig. 2b, where we counted M ≥ 3 shocks in each time window of 0.08 days. We then plotted the change in the number (the change in the rate of occurrence) of those shocks in the same area as a function of relative time to the M7.3 quake in Fig. 2c. Also included in this figure are the mean and standard deviation of the change in the number of M ≥ 3 shocks. At 0.98 days relative to the M7.3 quake, the change in the number was negative and smaller than the standard deviation, while the change at 0.67 days before the M7.3 quake was also at negative but within the margins of the standard deviation. This shows that the transition to a lower occurrence rate at 0.98 days relative to the M7.3 quake was significant, but the transition at 0.67 days might not have been. The analysis was performed for the northern area (Fig. 2e, f) and southern area (Fig. 2h, i) separately, obtaining the same feature. We confirmed that the above-described result is robust, by sampling different time windows (0.10 and 0.05 days) and by noting that the general features are the same for all the cases (Fig. A2 in Additional file 1). A preliminary analysis of the change in the number of M ≥ 3 shocks suggests that the decay of seismic activity over time was faster than that predicted by the OU law, typically with p = 1.

Figure 3 shows a good fit of the OU law with p = 1.49 ± 0.29 to the activity with M ≥ 3 for the entire area during the period between the M6.5 and M7.3 quakes (t = 0–1.16 days). Note that this value is significantly larger than a typical value (p = 1). For a short time period (t = 0–0.18 days, equivalent to 1.16–0.98 days preceding the M7.3 quake), p = 0.46 ± 0.14 is significantly below 1. For an intermediate period (t = 0–0.49 days, corresponding to 1.16–0.67 days preceding the M7.3 quake), p = 0.99 ± 0.26 lies between these two p values. The difference in the p value between the short and long periods is statistically significant, showing that decay of seismic activity became faster as period increased, as expected from a preliminary analysis. The influence of period on p is shown in the inset of Fig. 3 (data in black). We conducted the same analysis for shocks that occurred in the northern and southern areas. The results are plotted in the inset of Fig. 3 (data in cyan and pink, respectively). The difference in the p value between these areas is significant for periods longer than 0.5 days, where the p value was not plotted for the southern area for periods longer than one day because the model-fitting analysis did not converge. It can be said that the rapid decrease in the number of M ≥ 3 shocks in the northern area was responsible to the rapid decay in the activity observed over the entire area.

Fig. 3
figure 3

Number λ (day−1) of M ≥ 3 earthquakes in the entire area as a function of t (days) from the M6.5 quake. Fitting of the OU law to seismicity in the periods t = 0–0.18 (blue), 0–0.49 (green), and 0–1.16 (red) results in (p, c) = (0.46 ± 0.14, 0.00 ± 0.00), (0.99 ± 0.26, 0.02 ± 0.02), and (1.49 ± 0.29, 0.06 ± 0.04), respectively, where c (days) is a constant of the OU law. Inset p value as a function of the length of the analyzed period (days) since the M6.5 quake for seismicity in the entire area (black), in the northern area (cyan), and in the southern area (pink)

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We next examined the spatial distributions in the b value for two periods, 1.16–0.98 and 0.98–0 days prior to the M7.3 quake. The results are shown in Fig. 4a, d, respectively. To create these figures, we calculated b values for events falling within a cylindrical volume with a 5 km radius, centered at each node on a 0.005° × 0.005° grid and plotted a b value at the corresponding node if at least 20 events in the cylinder yielded a good fit to the GR law. We assumed that the characteristic dimension of the node spacing (~500 m) was a larger value than the value of location uncertainty (typically, 275 m) of the JMA catalog (Richards-Dinger et al. 2010). Distributions in the M c value that were simultaneously calculated with a b value at each node are shown in Fig. 4b, e. Overall, the b values increased with time. The difference Δb = b 1.16−0.98 − b 0.98−0 is mostly above 0 (cyan to blue colors in Fig. 4g) and ranges from Δb = −0.05 to Δb = 0.53. A map view shows a zone of large increase in the b value (Δb ≥ 0.3, blue colors), but only in the northern area. Observed changes in the b value are not considered significant if the test proposed by Utsu (1992, 1999) is not passed, as is shown in Fig. 4h. If logP b , the logarithm of the probability that the b values are not different, is equal to or smaller than −1.3 (logP b  ≤ −1.3), then the change in b is significant (Schorlemmer et al. 2004). A map of logP b (Fig. 4h) revealed a zone of the low values (logP b  ≤ −1.3, green colors) in the northern area. This observation is further supported by applying a bootstrapping approach (Schorlemmer et al. 2003, 2004) to the standard deviation of the b values, σ. The average standard deviation \(\overline{\sigma }\) of the early period (1.16–0.98 days prior to the M7.3 quake) is 0.08, and 90% of the σ values are smaller than 0.10 (Fig. 4c). For the later period (0.98–0 days prior to the M7.3 quake), \(\overline{\sigma }\) is 0.12 and 90% of the values are smaller than 0.18. Higher errors of σ ≥ 0.18 (yellow) were obtained only in the northern area. The sums between σ in the early period and σ in the late period are at most 0.3. Comparing Δb with σ, we can say that the increase in the b value in part of the northern area is significant. We conducted the same statistical test as shown in for Fig. 4, but calculated b, M c , and σ values, by sampling the nearest 100 earthquakes at each node (Fig. A3 in Additional file 1). The general feature remained similar to Fig. 4. Our statistical test demonstrates that the observed changes in the b value were physically meaningful and not caused by artifacts.

Fig. 4
figure 4

Results from a test of significance of the difference between the periods 1.16–0.98 days and 0.98–0 days prior to the M7.3 quake. a b values, b M c values, c σ values from 1.16 to 0.98 days. d b values, e M c values, f σ values from 0.98 to 0 days. g Δb, the difference in b values between the periods 1.16–0.98 and 0.98–0 days. h log P b , the logarithm of the probability that the b value for 1.16–0.98 days is different from the b value for 0.98–0 days. Red stars M6-class quakes (M6.5, M6.4), yellow star M7.3 quake, red lines surface traces of active faults, where Hinagu FZ and Futagawa FZ are designated

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If the b value offers an indicator of stress in the Earth’s crust, then it is expected to have increased by the time the M7.3 quake occurred. To examine this possibility, we compared the b values before the M6.5 quake with the b values after the M7.3 quake in a polygon that includes epicenters of the M6.5, M6.4, and M7.3 quakes (Fig. A4 in Additional file 1). The b value was almost stable when b was 0.6–0.8 before the M6.5 quake, consistent with the results obtained by Nanjo et al. (2016). Though the b value immediately after the M7.3 quake fluctuated due to many aftershocks following the quake, the b values after this period were significantly larger than those before the M6.5 quake. This result indicates that measurement of spatial and temporal changes in the b value is useful to infer stress in the Earth’s crust, supporting our interpretation that the significant increase in the b value in Fig. 4 shows a decrease in stress before the M7.3 quake.

Discussion

We found that the rate of occurrence of comparatively large shocks with M3 or larger in the seismic activity after the M6.5 quake on April 14, 2016, decreased significantly preceding the M7.3 quake on April 16. We further demonstrated that the p and b values increased significantly before the M7.3 quake, a feature that is consistent with the decrease in comparatively large shocks. Our analysis of the spatial and temporal changes in the p and b values indicates that the increase in the p and b values was most conspicuous in the northern area of the seismic activity after the M6.5 quake. This finding suggests that stress around the Futagawa FZ had begun to decrease preceding the M7.3 quake, if we considered that a high p value declines an increase in stress (Dieterich 1994) and that a high b value indicates low stress (Scholz 1968, 2015).

The reduction in the normal stress at the Futagawa FZ, i.e., a decrease in compressive stress on the fault, is suggested by an independent observation from the NIED, the National Research Institute for Earth Science and Disaster Resilience (2016). NIED examined the temporal change in the stress field by performing a stress inversion analysis using focal mechanisms of earthquakes in and around the Kumamoto earthquakes. The minimum compressive stress (σ 3) axis lay roughly in a north–south axis before the M6.5 quake. A comparison between the σ3 axis after the M6.5 quake and the σ 3 axis before this quake shows that the axis rotated counterclockwise and became nearly perpendicular to the strike of the Futagawa FZ, which indicates the normal force operating to the fault plane reduced.

In the seismic activity after the M6.5 quake, we found that earthquakes had occurred near the hypocenter of the M7.3 quake in addition to the majority of the seismicity around the Hinagu FZ (Fig. 5a). Figure 5b shows a 5-km-wide cross section nearly perpendicular to the strike of the Futagawa FZ, on which the rupture of the M7.3 quake occurred (ERC 2016a). Note that several events are recognized close to the hypocenter of the M7.3 quake. In the magnitude–time diagram (Fig. 5c) where these events are designated by bold squares, they occurred 0.98 days before the M7.3 event. One possible explanation may be that they were produced by a nucleation process, as suggested by Onaka (1993), of the M7.3 quake. Our finding of the occurrence of earthquakes close to the hypocenter of the M7.3 quake (Fig. 5), combined with the reduction of normal pressure on the Futagawa FZ (NIED 2016), indicates growth of a quasi-static pre-slips before the M7.3 quake. The pre-slips might have relaxed stress around the hypocenter of the M7.3 quake. This stress relaxation provided feedback to reduce the occurrence rate of larger (M ≥ 3) shocks (Fig. 2), causing a significantly larger p value than p = 1 (Fig. 3) and a significant increase in the b value (Fig. 4).

Fig. 5
figure 5

a The epicentral map of earthquakes that occurred during the period from the M6.5 quake to the M7.3 quake. Events marked by dark gray squares were used to create the cross-sectional view in b and the magnitude–time diagram in c. Red stars M6-class quakes (M6.5, M6.4), yellow star M7.3 quake, red lines surface traces of active faults, where Hinagu FZ and Futagawa FZ are designated. b Cross-sectional view of the seismicity. Apart from the majority of the events, outliers (bold squares) can also be observed around the hypocenter of the M7.3 quake. c M as a function of the time (days) until the M7.3 quake (vertical solid line). Outliers seen in b are designated by bold squares. Vertical dashed lines 0.98 days and 0.67 days before the M7.3 quake

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