## 2 Development of a Statistical Green's Function  
To predict strong ground motions over a wide frequency range, [PERSON] and [PERSON] [17] developed a method to create a statistical Green's function that is valid in the frequency range from 0.12 to 20 Hz based on the strong motion database in Japan. Their empirical modeling of the dataset based on the generalized inversion technique of [PERSON] and [PERSON] [18] covered the entirety of Japan and investigated 110 earthquakes (with 25 inland earthquakes, 34 subduction-zone intraplate earthquakes, and 51 subduction-zone interplate earthquakes) between August 1996 and March 2005, where the magnitude M \(\geq\) 5.5, the hypocentral depth \(\leq\) 60 km, the maximum acceleration \(\leq\) 200 cm/s\({}^{2}\), and the maximum distance to a hypocenter \(\leq\) 400 km. Crustal earthquakes are defined by a hypocentral depth less than 25 km and an inland hypocenter location. Plate-boundary earthquakes have a hypocentral depth of more than 25 km, are low-angle reverse fault according to the fault mechanism solution by Harvard CMT [19] (before 1997) or F-net [20] (after 1997), and the slip direction is along the plate motion. The intraplate earthquakes have a hypocentral depth of more than 25 km and the fault mechanism solution is not a low angle reverse fault as in the interplate earthquakes. The K-NET, KiK-net [21], and Japan Meteorological Agency (JMA) network [22] data observed at 1684 sites for 110 earthquakes were used to separate source, path, and site terms in both spectral and envelope shapes first, and then the convolution scheme for a statistical Green's function from a specific source valid up to a long period (<8 s) was derived. Using the estimated source, path, and site characteristics to evaluate strong ground motion over the entirety of Japan allows us to generate a quantitative input seismic motion for design or damage evaluation at any strong motion observation sites that considers the earthquake type and regional dependence in the source and path characteristics. Figure 1 shows how the entirety of Japan is separated into six regions and where measurement points are located. Regions are determined based on the location distribution of crustal, intraplate, and interplate earthquakes, regional attenuation difference across the volcanic front, and the Itoigawa-Shizuoka Tectonic Line, as in Kawase and Matsuo [18]. Details of the Green's function construction and its validation exercise, including modeling of time-varying characteristics, can be found in Ho and Kawase [17], so here we only briefly explain the scheme used. First, the logarithm of the observed Fourier spectra Fij as a function of frequency from an event i at a site j can be expressed as  
\[\text{logF}_{\text{ij}}=\text{logS}_{\text{i}}-\text{n}_{\text{(i)}}\text{logX} _{\text{ij}}+\sum_{\text{k}}\text{b}_{\text{(i)k}}\text{X}_{\text{ijk}}+\text {logG}_{\text{j}} \tag{1}\]  
where Si is the source spectrum, n(i) is the geometrical spreading factor, b(i)ik is the region-specific intrinsic and scattering attenuation, and Gj is the site factor. We use l(i) to describe earthquake types (C: crustal, l: intraplate, and B: plate boundary), while we use \(k\) to describe regions with different attenuations (Figure 1d). Xij is the hypocentral distance, whereas Xijk is a part of the distance within the k-th region. The statistical characteristics of the Fourier spectra are extracted in the range of 0.12 to 20 Hz, in which the root mean square values of NS and EW components are used to reduce the influence of the radiation characteristics from the source. Fourier spectra were smoothed using a 0.05 Hz Parzen window. Because at least one constraint condition is required for the spectral inversion based on Equation (1), the site coded as YMGH01 of KiK-net is used as the reference observation site, and the amplification factor calculated by the 1D multiple-reflection theory [8] from the inverted ground structure at YMGH01 is given as a constraint.