2.3 Estimation of the size of the mainshock source
We estimated the size of the mainshock source based on the circular
crack source model (e.g., Sato & Hirasawa, 1973; Madariaga, 1976). In
this source model, the source radius is related to the S-wave corner
frequency, \(f_{c}\), as follows:
\(r=\frac{\text{kβ}}{f_{c}},\) (2)
where \(r\) is the source radius, \(k\) is a constant, and \(\beta\) is
the S-wave velocity close to the source. Based on a rupture velocity of
0.9\(\beta\), \(k\) is 0.44 in the model of Sato and Hirasawa (1973) and
0.32 in the model of Madariaga (1976) for P-waves (Kaneko & Shearer,
2014). Because the estimated source size depends on the source model, we
computed the fault size using both models. We assumed a \(\beta\) value
of 3.4 km/s.
We used the spectral ratio method (e.g., Imanishi & Ellsworth, 2006) to
estimate the corner frequency of the mainshock. In this method, effects
of the propagation and location on the seismic wave are empirically
removed using the waveforms of an adjacent small earthquake (empirical
Green’s function, EGF, event). Based on the assumption that the source
spectrum, that is, \(S_{j}\left(f\right)\), follows the \(\omega^{2}\)model (Aki, 1967; Brune, 1970), the theoretical ratio between the
velocity spectra of the mainshock, \(v_{i}(f)\), and the EGF event,\(v_{i}^{\text{egf}}(f)\), at station \(i\) can be calculated as
follows:
\(\text{SSR}_{\text{ij}}\left(f\right)=\frac{v_{i}\left(f\right)}{v_{i}^{\text{egf}}\left(f\right)}=\frac{M_{0}}{M_{0}^{\text{egf}}}\frac{R_{\text{θφi}}}{R_{\text{θφi}}^{\text{egf}}}\frac{1+\left(\frac{f}{f_{c}^{\text{egf}}}\right)^{2}}{1+\left(\frac{f}{f_{c}}\right)^{2}},\)(3)
where \(M_{0}\) and \(M_{0}^{\text{egf}}\) are the seismic moments of
the target earthquake and EGF event, respectively; \(R_{\text{θφij}}\)and \(R_{\text{θφi}}^{\text{egf}}\) are their radiation patterns at
station i, respectively; and \(f_{c}^{\text{egf}}\) is the
corner frequency of the EGF event. Based on Eq. (3), \(f_{c}\) can be
estimated from the spectral ratios.
We calculated the spectral ratios by using P-wave velocity waveforms
observed at the 20 stations surrounding the source area (green inverted
triangles in Fig. 1b). The EGF events were earthquakes with M ≥ 2 and a
distance from the mainshock below 1.0 km based on the relocated
hypocenters. The following procedure was performed (Yoshida et al.,
2017):
(1) For the target mainshock and EGF events, the waveforms of the three
components were extracted from a 2.0 s time window starting 0.3 s before
the arrival of the P-wave at each station. The multitaper method
(Thomson, 1982; Prieto et al., 2009) was applied to calculate the
spectra.
(2) For channels with EGF observation spectra with a signal-to-noise
ratio > 4 at all frequencies from 0.5 to 30.0 Hz, the
spectral ratio between the mainshock and EGF event was calculated. We
used waveforms up to 0.3 s before the arrival of the P-waves for the
noise window.
(3) We calculated the geometric mean of the spectral ratios\(\text{GSR}\left(f\right)\) of all channels at each frequency for the
EGF events, which satisfied the above-mentioned criterion at five or
more stations:\(\text{GSR}\left(f\right)=\prod_{i=1}^{N}{\left(\text{SR}_{i}\left(f\right)\right)^{\frac{1}{N}}\ ,}\)(4)
where \(\text{SR}_{i}\left(f\right)\) is the observed spectral ratio
obtained at station i and \(N\) is the number of stations.
(4) By using the grid search and minimizing the evaluation function\(J\), the corner frequencies of the mainshock, \(f_{c}\), and EGF
event, \(f_{c}^{\text{egf}}\), were determined:\(J=\sum_{k=1}^{n_{\text{freq}}}\left|\log{\left(\text{GSR}\left(f_{k}\right)\right)-\operatorname{Alog}\left(\text{NSR}\left(f_{k};f_{c},f_{c}^{\text{egf}}\right)\right)}\right|,\)(5)
where\(\text{NSR}\left(f;f_{c},f_{c}^{\text{egf}}\right)=\frac{1+{(f/f_{c}^{\text{egf}})}^{2}}{1+{(f/f_{c})}^{2}}\),\(n_{\text{freq}}\) is the number of frequencies, and \(f_{k}\) is
frequency (at 0.5 Hz intervals from 0.5 to 30 Hz).\(\ \)The grid search
was performed for \(f_{c}\) and \(f_{c}^{\text{egf}}\) by assuming a
range from 0.1 to 100 Hz at 0.1 Hz steps. The amplitude ratio, \(A\),
was estimated using the least squares method for each grid search step.
We applied the spectral ratio method to 33 EGF candidates. We obtained
spectral ratios for 13 EGF events, which satisfy our S/N ratio and data
criteria. Figure S4 shows the spectral ratios of the 13 EGF events.