## 3 Strong Ground Motion Prediction of the Nankai-Trough Earthquake
### Waveform Summation Scheme  
The strong ground motion waveform of the hypothesized Tonankai earthquake was synthesized using the estimated subduction-zone earthquake elementary source and propagation characteristics, site characteristics of each target site, and the empirical Green's function summation method by [PERSON] [13]. The empirical [PERSON]'s function method is a waveform synthesis method that considers the difference in the seismic moment and \begin{table}
\begin{tabular}{c c} \hline Magnitude M & 8.1 \\ \hline Total area (km\({}^{2}\)) & 14,040 \\ \hline Seismic moment (N m) & \(2.119\times 10^{21}\) \\ \hline Average static stress drop (MPa) & 3.0 \\ \hline Rupture speed (km/s) & 2.7 \\ \hline Strike, dip, and rake (degree) & 250.7/14.0/122.7 \\ \hline Slip in asperities (left/center/right, cm) & 601/850/601 \\ \hline Slip in background (cm) & 299 \\ \hline \end{tabular}
\end{table} 
Table 1: Major fault parameters of the hypothesized Tonankai earthquake. Figure 3: Location of asperities, with higher slip elements (\(\blacksquare\)), the center of the background region (*), and the rupture starting point (\(\bigstar\)) of the seismogenic fault of the hypothesized Tonankai earthquake. stress drop between large and small earthquakes according to the \(\omega^{-2}\) scaling law, which is shown in Equation (8). \[ \begin{split}\mathrm{U(t)}=\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N }\left(\mathrm{r}/\mathrm{r}_{ij}\right)\times&\mathrm{F(t)}^{*} \left(\mathrm{C}\ \mathrm{u}_{ij}(t)\right)\\ \mathrm{F(t)}&=\delta\left(\mathrm{t}-\mathrm{t}_{ ij}\right)+\left(1/n^{\prime}\right)\sum\limits_{k=1}^{N-1}\left[\delta\left\{ \mathrm{t}-\mathrm{t}_{ij}-\left(\mathrm{k}-1\right)\mathrm{T}/\left(\mathrm{ N}-1\right)\mathrm{n}^{\prime}\right\}\right]\\ \mathrm{t}_{ij}=\left(\mathrm{r}_{ij}-\mathrm{r}_{o}\right)/ \mathrm{V}_{s}+\xi_{ij}/\mathrm{V}_{r}\end{split}  \tag{8}\]  
Here, \(\mathrm{U(t)}\) is the mainshock waveform, \(\mathrm{u(t)}\) is the elementary source waveform with a small moment, \(\mathrm{N}\) is the ratio of the fault plane dimension (i.e., the length and width) between large and small earthquakes, \(\mathrm{C}\) is the stress drop ratio, and \(\mathrm{*}\) indicates convolution. \(\mathrm{F(t)}\) is a correction function expressing the difference in slip time function between large and small earthquakes, \(\delta\)(t) is Dirac's delta function, and \(\mathrm{T}\) is the rise time of a large earthquake. Further, \(\mathrm{V}_{s}\) is the S-wave velocity, \(\mathrm{V}_{r}\) is the rupture velocity, \(\mathrm{r}\) is the assumed hypocentral distance between the representative elementary source and the target site, \(\mathrm{r}_{0}\) is the distance between the rupture initiation point and the site, \(\mathrm{r}_{ij}\) is the distance between each elementary source at (i,j) on the fault plane of a large earthquake and the target site, and \(\xi_{ij}\) is the distance between the rupture initiation point and the elementary source, \(\mathrm{n}^{\prime}\) is an arbitrary integer to avoid effects of artifact frequency components from constant superposition. Regarding the correction function \(\mathrm{F(t)}\), [PERSON] et al. [27] used a revised equation as Equation (9) to avoid the dip in the amplitude spectrum of the synthesized waveform, which is adopted here:  
\[ \begin{split}\mathrm{F(t)}&=\delta\left(\mathrm{t}- \mathrm{t}_{ij}\right)+\left\{1/n^{\prime}(1-\exp(-1))\right\}\\ &\times\sum\limits_{k=1}^{\left(\mathrm{N}-1\right)n^{\prime}} \left[\exp\left\{-\left(\mathrm{k}-1\right)/\left(\mathrm{N}-1\right)n^{ \prime}\right\}\times&\delta\left\{\mathrm{t}-\mathrm{t}_{ij}- \left(\mathrm{k}-1\right)\mathrm{T}/\left(\mathrm{N}-1\right)n^{\prime}\right\} \right]\end{split}  \tag{9}\]  
The stress drop is estimated from the relation between the seismic moment \(\mathrm{M}_{0}\) and corner frequency \(\mathrm{f}_{0}\) as Equation (10), where \(\mathrm{\beta}\) is the S-wave velocity of the seismogenic zone.