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.