**2. Observed SAF from GIT**

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source to the site, and amplification near the site are represented by the numerical modeling for the whole process from the source to the site. In this method we need a physical model of the medium to represent the wave propagation in the whole path. In other words, we need to calculate the theoretical Green's function for a point source on the fault surface. The other is an empirical method in which we use observed ground motions of a small earthquake as a substitute for the Green's function and sum up all the contributions from the elemental sources on the fault surface. It is called the empirical Green's function method (EGF). If there are no appropriate small earthquake records to be used as the empirical Green's function, we first generate synthetic waveforms based on many records of small earthquakes. It is called the statistical Green's function method (SGF). Because the frequency range for the theoretical approach with coherent nature is limited to the lower end, usually below 1 Hz or lower, while the effective frequency range of EGF or SGF with inherent nature of stochasticity should be higher than that, a hybrid scheme with TGF and EGF or TGF and SGF are used naturally, as has been used in the current

national project for strong motion predictions with specific sources [2]. After the deployment of the dense national strong motion observation networks, namely K-NET, KiK-net, and JMA Shindokei (Instrumental Seismic Intensity) network in Japan, a significant number of data has been accumulated. We can use these data to construct a model of SGF in a broadband frequency range. As long as we can generate the SGF for an arbitrary size of a small earthquake at an arbitrary location of a site in the frequency range of engineering interest, namely from 0.1 Hz to 20 Hz, we need not use a hybrid scheme. Thus we have been analyzing these strong-motion data in Japan by using the generalized spectral inversion technique (GIT) initially developed in 1980's [3, 4] to delineate statistical properties of the three major terms, namely, the source term, path term, and site term [5–8]. The novelty of our approach is that the hypothesized (i.e., extracted) seismological bedrock spectra at a reference site, YMGH01, are used as a reference to calculate site amplification factors at all the other observed sites. Such a separation of observed spectra into three major terms is sufficient to generate SGF at these observed sites. However, strong-motion simulations for the whole region near the seismogenic fault would be still difficult by SGF because we cannot estimate the site term at an arbitrary location other than the observed sites used in GIT. Thus, we need to develop a method to evaluate the site term at an arbitrary location as

When we look at the site term as a function of frequency evaluated at K-NET, KiK-net, and JMA Shindokei network, we found that they show strong spectral fluctuations from 1 to 10 as a normal range of fluctuations and from 1 to 50 at tens of extraordinary sites with various peak frequencies. Several attempts have been made to correlate the primary characteristics of the observed site amplification factor (SAF) with a site proxy or proxies such as the S-wave velocity (Vs) averaged over top xx m, Vs\_xx (e.g., Vs30) or the depth to the layer with the S-wave velocity higher than y.y km/s, Z\_y.y (e.g., Z1.0) [9–13], trying to reproduce primary characteristics of SAF such as the fundamental peak frequency f0 and its peak amplitude A0. Unfortunately, these extracted characteristics are not sufficient to reproduce synthetic seismograms

In what follows, we first introduce the fundamental characteristics of the observed SAF in the horizontal component (hSAF) derived from GIT [7, 8]. Then, we show comparisons of these hSAFs with the 1D theoretical ones calculated from the recently established unified velocity model (UVM) of the National Research Institute for Earth Science and Disaster Resilience (NIED) in the Kanto and Tokai regions. Next, we obtain the modification ratios to reduce the gap between them at the observation points and propose a scheme to evaluate hSAF at an arbitrary point

needed in the SGF summation. We should find a different strategy.

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precisely as possible.

In this section, we briefly introduce the observed horizontal SAF (hSAF) and vertical SAF (vSAF) derived from GIT [7, 8]. Here we introduce only their basic aspects because we are using their results as a starting point.

They restricted events and sites with the Japan Meteorological Agency (JMA)'s magnitude MJMA ≥ 4.5; source depth ≤ 60 km; hypocentral distance ≤200 km; peak ground acceleration ≤ 2 m/s2 ; and a number of observation sites triggered simultaneously for one event ≥3. These selection criteria resulted in 150,468 event-station pairs at 2,593 sites for 1,734 events. Only a relatively short duration of acceleration record from the onset of the S-wave was analyzed (5 s if 4.5 < MJMA ≤ 6; 10 s if 6 < MJMA ≤ 7; 15 s if 7 < MJMA ≤ 8). A Parzen window of 0.1 Hz was used for a minimum level of smoothing. Note that the mainshock of the 2011 Off the Pacific Coast of Tohoku earthquake was excluded because the durations of those records are extraordinarily long.

As mentioned above, the most important feature of their GIT is that they determined the S-wave velocity structure at the reference site (YMGH01) using the transfer function (the spectral ratio and the phase difference) between the surface and the borehole 200 m below and that the observed Fourier spectra on the surface were deconvolved (divided by the amplification factor) to obtain the hypothesized outcrop spectra on the seismological bedrock with Vs of 3,450 m/s. The resultant S-wave velocity profile determined by the matching of the theoretical transfer function to the observed transfer function is quite similar to the original P-S logging data published by NIED [1], only with higher bedrock velocity of 3,450 m/s from 3,100 m/s. After the determination of the velocity profile, Nakano et al. [7] corrected (divided) all the observed spectra at YMGH01 by the calculated 1D S-wave site amplification factor on the surface with respect to the outcrop motion on the bedrock (=twice of the input) and used as the reference spectra in the subsequent GIT analyses, as if they were observed at YMGH01. Thus, their separated site terms, hSAF and vSAF, are considered to be the site amplification factors with respect to the outcrop seismological bedrock, on which there is virtually no site effect. Nakano et al. [7] successfully separated the source spectra and path terms as evidenced by their correspondence to the ω−2 source spectra shapes and Q values similar to the previous studies in Japan.

**Figure 1** shows examples of the observed hSAF at four sites in the Tokai region. We can see significant differences from site to site. Although we did not show vSAF here, the amplitude and its fluctuation of vSAF is much smaller than hSAF, especially below 3 to 4 Hz, that is, below the fundamental peak frequency of vSAF. That is why the earthquake horizontal-to-vertical spectral ratio, eHVSR, which is equal to hSAF/vSAF, tends to be similar to hSAF until the fundamental peak frequency of vSAF. However, to get hSAF from eHVSR, we need to correct vSAF, as recently proposed by Ito et al. [14]. Please note that precisely speaking, vSAF in this paper should be referred to as vSAF\* as in [14] because we use the same reference condition for both hSAF and vSAF as the seismological bedrock spectra in the horizontal component so that we need to have correction due to the vertical-to-horizontal spectral ratio on the seismological bedrock on top of the vertical-to-vertical (i.e., P-wave) site amplification.

**Figure 1.**

*Observed horizontal site amplification factor, hSAF, extracted from strong motions at K-NET, KiK-net, and JMA Shindokei network by GIT after [7].*
