## 3 Empirical Scaling Equations vs. Real Records
In the first phase, we fitted the Equation (1) model to a dataset containing only the 406 horizontal acceleration components from the EQINFOS database [28]. We fitted Equation (1) without the coefficient \(c_{5}\) because there were no deep soil locations among the 406 components. As a result, we only calculated coefficients \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), \(c_{6}\), and \(c_{7}\). To maximize the \(R^{2}\) statistics of the \(PSA\) prediction, the \(R_{0}\) values were iteratively changed. Because most of the data were collected at shorter distances, we conducted a secondary analysis utilizing only data collected at epicentral distances of less than 30 km. We calibrated the generated prediction model in the second phase by restricting the coefficient \(c_{5}\) (again through linear regression) based on the extra 30 acceleration components recorded at 10 different deep soil sites in the analyzed region. Table 1 (see Appendix A) gives coefficients for Equation (1) with \(R\) as the epicentral distance based on the data collected at all distances. Table 2 shows the scaling coefficients for Equation (1) with \(R\) as the hypocentral distance. Attenuation of \(PSA\)(\(T\)) with distance is shown in Figures 3 and 4 for four distinct vibration periods, \(T\) = 0.05, 0.10, 0.30, and 0.50 s. Figure 3 presents empirical forecasts \begin{table}
\begin{tabular}{c c c c} \hline
**Shallow Geology** & **Shallow** & **Deep Geology** & **Deep** \\
**Categorical Variables** & **Geology Type** & **Categorical Variables** & **Geology Type** \\ \hline \(S_{L1}=S_{L2}=0\) & \begin{tabular}{c} “Rock” soil: \\ \(s_{L}=0\) \\ \end{tabular} & \(S_{G1}=S_{G2}=0\) & \begin{tabular}{c} Basement (geological) \\ rock: \(s=2\) \\ \end{tabular} \\ \hline \(S_{L1}=1\) & Stiff soil: & \(S_{G1}=1\) & \
\begin{tabular}{c} Intermediate \\ sites: \(s=1\) \\ \end{tabular} \\ \hline \(S_{L1}=0\) & Deep soil: & \(S_{G1}=0\) &
\begin{tabular}{c} Deep geological \\ sediments: \(s=0\) \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 2: The categorical variables employed in this study’s scaling equations for various types of shallow and deep geological site surroundings. for three different deep geology types at deep soil sites (\(s_{L}=2\)). The solid lines represent median empirical estimates based on data recorded at all epicentral distances, whereas the dashed lines represent estimates based on distances under 30 km. Figure 4 presents empirical predictions for three different local soil types at deep geological deposits (\(s=0\)). In Figure 4, we show the median, median \(\pm\) 1 \(\sigma_{log}\), and median \(\pm\) 2 \(\sigma_{log}\) (shaded area) empirical estimates for the deep soil sites. The coefficients corresponding to the categorical variables \(S_{L}\) and \(S_{G}\) can be used to calculate the exact differences between values estimated for different site conditions. For example, if the coefficients from Table A1 are used, the \(PSA\) (0.05 s) estimates (which can be considered as the upper bound for PGA estimates) are \(1/10^{-0.143}=1.39\) times larger at the geological rock (\(s=2\)) than at the deep geological sediments (\(s=0\)), as seen in the top left-hand side plot in Figure 3. This is most likely because short-period waves travel more easily through more compact rocks (e.g., granites and basalts). Conversely, the \(PSA\) (0.50 s) estimates are \(10^{0.137}=1.37\) times larger at the sediments than at the geological rock, as seen in the bottom right-hand side plot in Figure 3.  
Furthermore, by using coefficients from the same Table A1, the \(PSA\) (0.05 s) at deep soil sites is going to be equal to \(10^{-0.058}=0.88\) times the \(PSA\) (0.05 s) for rock soil sites. In other words, the seismic waves will be de-amplified, as shown in the top left-hand side plot in Figure 4. This suggests that energy dissipation in deep soil overcomes local soil amplification, which is consistent with the findings of numerous other research studies of the non-linear behavior of soft sediments under significant ground motion.