## 4 Seismic Response Characteristics of Layered Sites
### Results of Time Domain Analysis
#### 4.1.2 Dynamic Response Characteristics of Layered Sites  
To study the dynamic amplification effect of the layered sites, the time history curves of waves were extracted from the measuring points in the models, and their dynamic response characteristics were studied by analysing the PGA. For many measurement points in the models, typical measurement points in Model 2 are taken as examples (Figure 3b). Figure 8: Acceleration distribution during wave propagation in Model 1 when the WE wave is input in the \(x\) direction: (**a**) Model 1 (t = 1.80 s; t = 1.95 s; t = 2.05 s), (**b**) Model 2 (t = 1.95 s; t = 2.15 s; t = 2.30 s). (Unit: m/s\({}^{2}\)). Figure 7: The input WE wave (0.08 g): (**a**) Time history, (**b**) Fourier spectrum. Their acceleration-time history and their corresponding Fourier spectra are shown in Figure 9. The PGA in Model 2 increases with elevation, which indicates that elevation has an obvious magnification effect. In addition, the input seismic wave is a WE wave, whose waveform and frequency components are very complex when propagating in layered sites; hence, a new analysis parameter of PGA is proposed. The seismic signals collected were decomposed by the EMD method to obtain a series of IMFs, and the PGA of the IMF with a certain high resolution was selected as the parameter to analyse the dynamic amplification effect of the layered site (Figure 10a). The IMF has local transient characteristics, which can better reflect the characteristics of the original signal. The dynamic response of the layered sites was analysed by comparing the \(\text{MPGA}_{\text{(IMF)}}\) and \(\text{M}_{\text{PGA}}\). The \(\text{M}_{\text{PGA(IMF)}}\) and \(\text{M}_{\text{PGA}}\) refer to the ratio of \(\text{PGA}/\text{PGA}_{\text{(IMF)}}\) of a certain point to point A at the bottom of the models. Taking the input 0.08 g horizontal WE wave as an example, IMFs and their instantaneous frequency curves can be obtained after EMD of the WE wave (Figure 10). The first six IMFs can contain the information of the original signal. IMF2 has the largest amplitude, and its frequency components in the instantaneous frequency curve are rich and easy to identify. Therefore, IMF2 is selected to obtain the \(\text{M}_{\text{PGA(IMF)}}\), earthquake Hilbert energy and marginal spectrum. The PGAs and their magnification coefficients of the models are shown in Table 2 and Figures 11 and 12. The \(\text{M}_{\text{PGA}}\) and \(\text{M}_{\text{PGA(IMF)}}\) of the models increased with elevation, indicating that there was a significant elevation dynamic amplification effect in layered sites under earthquake excitation. Figures 11 and 12 show that the sequences of the \(\text{M}_{\text{PGA}}\) and \(\text{M}_{\text{PGA(IMF)}}\) of the models are as follows: Model 1 < Model 2 < Model 3. This suggests that the seismic magnification effect of the horizontally layered site is greater than that of the homogeneous site and that that of the tilted layered site is the largest; that is, the stratum angle has an influence on the seismic amplification effect of the layered sites. Figure 9: Seismic response of typical points (B, C, D, F, G, and H) at the surface of Model 2 when inputting the WE wave in the \(x\) direction: (**a**) acceleration-time history, (**b**) Fourier spectrum. Figure 11: Change in PGA of the models when input in the \(x\)-direction: (**a**) PGA, (**b**) PGA(IMF). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Elevation/m**} & \multirow{2}{*}{**Measuring Points**} & \multicolumn{3}{c}{**PGA/g**} \\ \cline{3-5}  & & **Model 1** & **Model 2** & **Model 3** \\ \hline
0.5 & A & 0.095 & 0.103 & 0.113 \\
3 & B & 0.101 & 0.121 & 0.131 \\
5.5 & C & 0.106 & 0.132 & 0.145 \\
6.5 & D & 0.115 & 0.139 & 0.154 \\
9 & E & 0.128 & 0.154 & 0.164 \\
11.5 & F & 0.137 & 0.164 & 0.183 \\
12.5 & G & 0.148 & 0.184 & 0.198 \\
14 & H & 0.161 & 0.228 & 0.241 \\
15.5 & I & 0.170 & 0.231 & 0.247 \\ \hline \hline \end{tabular}
\end{table}