## 5 Results
### Variation in Amplification Factors for Specific Period Ranges Using RBF

Table 5: Standard deviation of model residuals, coefficient of determination \(\rm R^{2}\), and numbers of key neurons in hidden layer determined after training phase for various RBF models at short, mid, and long periods (\(\rm F_{a}\), \(\rm F_{v}\), and \(\rm F_{I}\)) for clay and sand soil profiles for combination of three soil parameters (\(PGA,f_{0}\), and \(V_{S30}\)). Figure 11: Variation in amplification factor at long period \(\rm F_{I}\) with \(f_{0}\) and \(PGA\) for four values of \(V_{s30}\) (150, 270, 450, and 640 m/s) and \(PGA\) (**a**) for clay soil profiles and (**b**) for sand soil profiles. where _we_1, _we_2, and _we_3 can be expressed by  
\[we1=\left(\log(PGA)-\log(IW_{t,1})\right) \tag{28}\]  
\[we2=\left(\log(f_{0})-\log(IW_{t,2})\right) \tag{29}\]  
\[we3=\left(\log(V_{s30})-\log(IW_{t,3})\right) \tag{30}\]  
_we_1, _we_2, and _we_3 are vectors of t lines, while \(b\)1 and \(b\)2 are the biases determined after the training phase. To investigate the performance of the RBF model, we consider only 50% of the data for training and the other for testing. The main results obtained are summarized in Table 6 in terms of the standard deviation \(\epsilon_{\rm RBF}(\theta)\) and coefficient of determination (R\({}^{2}\)). They demonstrate the good performance of the model. The following regression equation between \(V_{s30}\) and \(f_{0}\) is derived from the obtained predictions of the RBF model:  
\[\left(\log(V_{s30})\right)=\left(2.37+0.385\;\mathrm{Log}(f_{0})\right)\pm 0.18 \tag{31}\]  
From the results of Figures 8-11 (and similar results for F\({}_{\rm v}\) and F\({}_{\rm l}\)), we note the following findings:  
* Generally, the amplification factors F\({}_{\rm a}\), F\({}_{\rm v}\), and F\({}_{\rm l}\), are higher for clay-type soil than for sand-type soil. This is particularly true for the range of low frequencies \(f_{0}\) up to 1 Hz. However, for \(f_{0}\) values higher than 3 Hz and for relatively stiff soils, with a \(V_{s30}\) value exceeding 350 m/s, F\({}_{\rm a}\) and F\({}_{\rm v}\) values are slightly higher for sand than for clay. * The amplification factors are higher for low _PGA_ values. This holds true for the factor F\({}_{\rm l}\) and soft soils with low \(V_{s30}\) values. Furthermore, from the results of Figure 9, we can note that the correlations between \(f_{0}\) and \(V_{s30}\) given by Equation (31) are not always respected as the frequency \(f_{0}\) does not increase systematically with the \(V_{s30}\) value at the peak value of the amplification factors. This is explained by the following two factors:  
* Some combinations are not possible in real cases, such as having a soil profile with a fundamental frequency \(f_{0}\) greater than 10 Hz while its \(V_{s30}\) value is lower than 500 m/s. In such situations, the predictions of Equation (26) are somehow extrapolations that are very likely erroneous and meaningless. * The predictions of Equation (26) shall be considered for explaining the global tendency regarding the interactions between different parameters such as the change in the \begin{table}
\begin{tabular}{c c c c} \hline
**Main Result for Amplification at Specific Period** & **F\({}_{\rm a}\)** & **F\({}_{\rm v}\)** & **F\({}_{\rm l}\)** \\ \hline Standard deviation for database & & & \\ (50\% test database) & 0.0479 & 0.004 & 0.0369 \\ \(\epsilon_{\rm RBF,m}(\theta=0)\) (soil type: clay) & & & \\ R\({}^{2}\)(50\% test database) (\(\theta=0\)) & 0.9914 & 0.9667 & 0.9142 \\ Standard deviation for database & & & \\ (50\% test database) & 0.0514 & 0.0436 & 0.0365 \\ \(\epsilon_{\rm RBF,m}(\theta=1)\) (soil type: sand) & & & \\ R\({}^{2}\)(50\% test database) (\(\theta=1\)) & 0.9933 & 0.9794 & 0.9227 \\ \hline \end{tabular}
\end{table} 
Table 6: Performance of RBF model on interpolation, results on terms. Standard deviation of model residuals and coefficient of determination R\({}^{2}\) for database test at F\({}_{\rm a}\), F\({}_{\rm v}\). and F\({}_{\rm l}\) for clay and sand. amplification factors with the _PGA_ level, \(V_{s30}\) (about 600 m/s).