3.3 Deviation of the seismicity from Omori’s law
We investigated the seismicity rate of the Kagoshima Bay earthquake sequence after the mainshock. Figure 1d shows the seismicity rates of the MJMA ≥1.0 events in the area surrounding the hypocenter of the mainshock (red frame in Fig. 1b). The seismic rate was obtained by calculating the reciprocal of the time required to generate ten earthquakes that were arranged in chronological order. Based on Fig. 1d, the seismicity rate decreases by the power of the elapsed time immediately after the mainshock, as described by the modified Omori law (Utsu, 1961). The seismicity rate abruptly increases ~44 days after the mainshock, which corresponds to the occurrence of the largest aftershock (ML 4.4), suggesting that the increase is due to secondary aftershocks. A period with a high seismicity rate started approximately 20 to 40 days after the mainshock; the seismic activity was temporarily strong despite the absence of large aftershocks.
Based on maximum likelihood estimation, we obtained the following parameters for the ETAS model: \(K_{0}=\ 34.205\),\(c=1.3163\times 10^{-2}\), \(p=1.0685\), \(\alpha=1.5078\), and\(\mu=2.9603\times 10^{-2}\). Based on Ogata (1992), the range of α-values is 0.35–0.85 for swarm seismicity and 1.2– 3.1 for non-swarm seismicity. The α value estimated for the seismic activity in Kagoshima Bay is within the latter range.
In Fig. 9, the cumulative number of earthquakes simulated using the estimated model parameters is compared with the observations. Overall, the number predicted based on the ETAS model matches the observations. However, the simulated number of earthquakes is lower than the observed number 20–40 days after the mainshock. To quantitatively examine the magnitude of the discrepancy between the model and observations, we performed residual analysis using the transformed time, similar to Ogata (1988). Figure 9c shows that the discrepancy between the model and observations is high at a transformed time between 1.000 and 1.500, corresponding to the period of 20–40 days after the mainshock. This deviation is significant at the 95% significance level based on the assumption of a uniform distribution.