## 3 Computed Amplification Factors: Main Statistical Characteristics
### General Background of Computed AFs  
This section presents on overview of the computed sets of frequency-dependent _AFs_ and their short- to mid- and long-period average values. These data are essential and constitute the learning set needed to identify the key parameters controlling the site response characteristics. In total, 99,792 _AF_(_T_) curves are computed using (Equation (1)) for the 324 soil profiles under the 14 seismic excitations normalized by 11 _PGA_ levels for the two types of soils (clay and sand). The _AF_ curves for clay are shown in Figure 4, and those for sand are shown in Figure 5. They may be written in the general form of \(AF(P_{k},\theta,S_{l},T_{i},PGA(lo))\), where  
* \(P_{k},\ k=1,\ldots n_{p}\) is introduced to identify the soil profile, \(n_{p}=324\);
* \(\theta=0\) for clay (using the shear modulus degradation curve of clay) and \(\theta=1\) for sand (using the shear modulus degradation curve of sand);
* \(S_{l},\ l=1,\ 14\) is the lth excitation, where the geometrical average of the 14 amplification factors has been computed for each site and for each _PGA_ for the results of Figures 4 and 5;
* \(T_{i},\ (i=1,..100)\) is the ith structural period, and _AF_ values are systematically computed for 100 values, equally spaced between 0.01 and 10 s on a logarithmic period axis;
* \(PGA(lo)\), for the identification of the _PGA_ level, and lo, vary from 1 (for _PGA_ = 0.01 g) to 11 (for _PGA_ = 1.05 g). \begin{table}
\begin{tabular}{c c c c c c c} \hline  & Depth & \(f_{0}\) & Cv & Cv2 & \(V_{sm}\) & \(V_{s30}\) \\ \hline Depth & 1 & 0.5119 & 0.0001 & 0.2584 & 0.0001 & 0.3311 \\ \hline \(f_{0}\) & 1 & 0.2926 & 0.4445 & 0.3744 & 10.7196 \\ \hline Cv & & 1 & 0.8755 & 10.7815 & 1.7815 & 1.7815 \\ \hline Cv2 & & 1 & 0.8755 & 10.7825 & 10.7825 \\ \hline \(V_{sm}\) & & 1 & 0.8755 & 10.7825 & 1.7825 \\ \hline \(V_{s30}\) & & & & 1 & 0.7855 \\ \end{tabular}
\end{table} 
Table 2: Correlations (R\({}^{2}\)) between various site parameters. For instance, \(AF(P_{20},0,S_{8},T_{50},0.1\) g) stands for the \(AF\) obtained at the soil profile P\({}_{20}\) of clay type, subjected to the eighth seismic excitation, \(S_{8}\), for the 50 th period (\(T=\) Log\({}^{-1}\) (\(-2+(50-1)\times(\)Log \(10-\) Log (0.01)) = 0.295 s), and normalized to a \(PGA\) level of 0.1 g (lo = 3). After the \(AF\) is calculated for a particular profile, \(k\), and for 14 seismic excitations, the average site amplification factor \(AF_{m}\) is computed for the soil profile so that  
\[\text{Log}[AF_{m}(P_{k},\theta,T_{i},PGA(lo))]=\left(\frac{1}{14}\right)\sum_{l 0=1}^{14}\text{Log}[AF(P_{k},\theta,S_{l},T_{i},PGA(lo))] \tag{15}\]  
Then, the average amplification factor for each profile, for all intensity levels, is obtained by the following:  
\[\text{Log}(AF_{0}(\theta,T_{i}))=\frac{1}{\text{m}_{\text{P}}}\sum_{k=1}^{ \text{m}_{\text{P}}}[\text{Log}(AF_{m}(P_{k},\theta,T_{i}))] \tag{16}\]  
Figure 4: Average amplification factors for 324 sets of clay soil profiles at 0.1 g and 0.75 g \(PGA\) levels. Figure 5: Average amplification factors for 324 sets of sand soil profiles at 0.1 g and 0.75 g \(PGA\) levels. Consequently, 3564 geometric mean values of the amplification factor are calculated, and stand for each soil profile and type. Simultaneously, for each profile \(P_{k}\), the \(AF\) variability derived from the 11 different average amplification factors is quantified using the corresponding standard deviation:  
\[\sigma_{AF}(P_{k},\theta,T_{i})=\sqrt{\frac{\sum\limits_{l=1}^{11}\left[\log( AF_{m}(P_{k},\theta,T_{i},PGA(lo)))-\log(AF_{m}(P_{k},\theta,T_{i}))\right]^{2}}{11}} \tag{17}\]  
The \(\sigma_{AF}\) values are displayed in Table 3 for all soil profiles. They exhibit a significant period dependence. The maximum variability is observed at around 0.1 s. It hardly decreases at shorter periods as short as \(\sim\)0.03 s, but it decreases significantly at intermediate and long periods. These values are greater, especially at short to intermediate periods with a lower degree.