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.