## 2 Development of a Statistical Green's Function
Therefore, the site characteristics of the other sites can be obtained as a ratio relative to this reference. Because we extracted the site amplification factor from the observed spectra at YMGH01 relative to the bottommost layer, whose S-wave velocity (Vs) is more than 3400 m/s, the obtained site characteristics of the other sites are considered to be absolute values from the seismological bedrock. A more complete picture of the delineated spectral characteristics with additional recent events can be found in [PERSON] et al. [23], where we obtained statistical properties to model the source, path, and site characteristics. In this study, however, we used directly a source spectrum separated from a selected moderate-sized event without using any moment-source spectra regressions in our statistical Green's function method to predict strong ground motions for mega-thrust earthquakes. The time-varying characteristics of seismic motion are characterized by the envelope form and its duration or phase characteristics. In this study, the time-varying characteristics are modeled by the envelope function W(t) of [PERSON] [12] and we identified its parameters. [PERSON]'s model is specified by the duration T\({}_{\rm d}\), the time to the maximum amplitude T\({}_{\rm r}\), \(\varepsilon\) the ratio of T\({}_{\rm r}\) to T\({}_{\rm d}\), and \(\eta\) the ratio of the amplitude at time T\({}_{\rm d}\) relative to the maximum amplitude. The equation of W(t) is given below. \[\rm W(t)=a\times t^{b}\times\exp(-ct)\times H(t) \tag{2}\]  
\[\rm a= \left(\frac{2.7182}{\varepsilon\times T_{d}}\right)^{b} \tag{3}\]  
\[\rm b= \frac{-\varepsilon\times\ln\eta}{\left[1+\varepsilon\times(\ln \varepsilon-1)\right]} \tag{4}\]  
\[\rm c= \frac{b}{\varepsilon\times T_{d}} \tag{5}\]  
H(t) in Equation (2) represents the unit Heaviside step function, and a in Equation (3) is so specified that the maximum value of the function W(t) is 1. First, the observed waveform was selected for 120 s from the onset of S-wave, and a cosine taper of 2 s was applied before and after the extracted part. The maximum value of the observed waveform was normalized to 1, and the section-wise maximum of its absolute amplitude per 1 s was taken and used to fit the envelope to W(t) with the fixed \(\eta\) of 1/20. The fitting target was the entire observation waveform, and the optimal value was searched by varying \(\varepsilon\) in the  
Figure 1: Source locations and mechanisms of (**a**) crustal earthquakes, (**b**) plate-boundary earthquakes, and (**c**) intraplate earthquakes, and (**d**) positions of observed sites and regions of different attenuations referring to the volcanic front [18]. range of 0.05 to 0.95 in every 0.02 increments. Next, the obtained \(\mathrm{T_{d}}\) and \(\mathrm{T_{r}}\) were regressed on the magnitude \(\mathrm{M}\) and the hypocentral distance \(\mathrm{X}\). In order to eliminate the possibility of a mutual trade-off between \(\mathrm{M}\) and \(\mathrm{X}\), a two-step regression analysis method was used. As an example of the modeling of the time-varying characteristics, Figure 2 shows the relationship between \(\mathrm{T_{r}}\) (or \(\mathrm{T_{d}}\)) and magnitude \(\mathrm{M}\) or the distance \(\mathrm{X}\), as well as an example of the statistical envelope identified for a specific observed record. The resultant \(\mathrm{T_{r}}\) and \(\mathrm{T_{d}}\) are shown in Equations (6) and (7). \[\mathrm{logT_{r}} =0.0943\times\mathrm{M}+0.5109\times\mathrm{logX}-1.0396 \tag{6}\] \[\mathrm{logT_{d}} =0.0884\times\mathrm{M}+0.6868\times\mathrm{logX}-0.7946 \tag{7}\]