2.4 Detection of aseismic processes from seismicity
The Epidemic Type Aftershock Sequence (ETAS) model (Ogata, 1988), which
is based on the superposition of the modified Omori law (Ustu, 1961),
can be used to explain mainshock–aftershock seismicity. The ETAS model
assumes that the seismicity rate is the sum of the background rate of
independent events, \(\lambda_{0}\), and aftershocks triggered by each
event, \(\lambda_{i}(t)\):
\(\lambda\left(t\right)=\lambda_{0}+\sum_{i:t_{i}<t}{\lambda_{i}\left(t\right).}\)(6)
Based on the modified Omori law, each earthquake can trigger its own
aftershock sequence (Utsu et al., 1995):
\(\Lambda_{i}\left(t\right)=\frac{K_{0}}{\left(c+t-t_{i}\right)^{p}}e^{\alpha\left(M_{i}-M_{\min}\right)},\)(7)
where \(t_{i}\) is the occurrence time; \(M_{i}\) is the magnitude of
each event, \(i\), that occurred prior to time \(t\); \(M_{\min}\) is
the magnitude of completeness of the earthquake catalogue; \(K_{0}\),\(c\), and \(p\) are constants; and \(t\) is the time that has elapsed
since the main event.
We applied the ETAS model to the seismicity observed after the mainshock
in Kagoshima Bay and investigated the difference between the simulated
and observed seismicity. The results show that the foreshock activity
cannot be explained by the ETAS model, likely because aseismic processes
mainly controlled the foreshock activity. We used the timings and
magnitudes of the earthquakes listed in the JMA catalogue. The lower
limit of the magnitude, MC, was set to 1.0. Figure S4
shows the magnitude–frequency distribution. The distribution follows
the Gutenberg–Richter law (Gutenberg & Richter, 1944) when
MJMA ≥ 1.0. The SASeis2006 algorithm by Ogata (2006) was
used to estimate the model parameters and calculate the residuals of the
ETAS model.