## 4 Test Results and Discussion
### Distribution of Acceleration Magnification Factors
#### 4.2.1 Distribution of AMFs of the Free-Field System Model Test  
Results of the free-field system tests, shown in Figure 6, indicate that the AMFs of the soil exhibit a \"C\"-shaped distribution along the vertical direction. This distribution pattern, which is characterized by a peak acceleration decrease in the middle soil layer, followed by an increase near the surface, is consistent with the test and numerical simulation results of [PERSON] et al. [20; 27; 28]. To better understand this distribution pattern, frequency response functions \(H_{fre}(i\omega)\) of different embedment measurement points A-2-1 to A-15-1, as shown in Figure 4a, in the free-field system test are presented in Figure 7. \begin{table}
\begin{tabular}{c c c c} \hline
**Loading Case** & **Characteristic Frequency (Hz)** & **Characteristic Damping Ratio (\%)** & **Displacement Amplitude (mm)** \\ \hline WN1 & 6.33 & 2.04 & 2.03 \\ Chichi 0.1 g & 7.30 & 7.69 & 0.93 \\ Chichi 0.25 g & 5.95 & 10.00 & 1.99 \\ Chichi 0.5 g & 5.32 & 10.00 & 4.45 \\ Chichi 0.75 g & 2.86 & 20.00 & 8.28 \\ Chichi 1.0 g & 2.75 & 20.00 & 16.86 \\ Chichi 1.5 g & 1.68 & 33.33 & 22.69 \\ WN2 & 6.45 & 2.0 & 2.25 \\ Kobe 0.1 g & 6.86 & 7.69 & 0.19 \\ Kobe 0.25 g & 6.85 & 7.69 & 0.43 \\ Kobe 0.5 g & 6.66 & 8.33 & 0.68 \\ Kobe 0.75 g & 5.74 & 11.11 & 1.36 \\ Kobe 1.0 g & 5.17 & 18.75 & 4.33 \\ WN3 & 6.43 & 2.0 & 1.96 \\ \hline \end{tabular}
\end{table} 
Table 3: Characteristic parameters of the motion of the caisson under different loading cases. The results in Figure 7 illustrate how seismic waves propagate in the free-field of sandy soil. As seismic waves travel from the bottom to the surface, high-frequency components are attenuated, while low-frequency components are amplified. This amplification is particularly noticeable closer to the surface. This propagation law can provide insights into the distribution of AMFs along the vertical direction. In the middle soil layer, the filtered high-frequency components carry more energy than amplified low-frequency components, resulting in a reduction in the peak acceleration. Near the surface, low-frequency components are further amplified due to reflection and scattering at the soil-air interface, significantly amplifying the peak acceleration of the surface soil layer. Moreover, in Figure 7b, it is evident that the amplitude of \(H_{\mathit{free}}(i\omega)\) at the surface measurement point is higher around 20 Hz under the Kobe wave. This is attributed to the close proximity between the dominant frequency of the Kobe wave, approximately 23.1 Hz, and the second-order frequency of the soil layer, approximately 18.54 Hz. Consequently, the acceleration response near the second-order frequency of the soil layer is amplified. As the table input acceleration increases, trends of AMFs at the surface measurement points vary with different seismic waves. Characteristic parameters of the dynamic response at the surface measurement point A-15-1, as shown in Figure 4a, of the free-field system can be found in Table 4. For the Chichi wave, AMFs at the ground surface decrease as the table input acceleration increases, which is attributed to the lower dominant frequency  
Figure 6: Distribution of AMFs along the vertical direction in different systems under different loading cases: (**a**) Chichi; (**b**) Kobe. Figure 7: Frequency response functions at different depths under different loading cases: (**a**) Chichi; (**b**) Kobe. of the table input acceleration of the Chichi wave, about 12.5 Hz, resulting in greater soil displacement, as shown in Table 4. Consequently, the shear strain of the soil increases, and the soil experiences evident plastic behavior, which can be seen from the fact that the characteristic frequency of the surface response is clearly smaller than the fundamental frequency in the white noise case, leading to a decrease in the dynamic shear modulus and an increase in the damping ratio of the soil, resulting in greater soil energy dissipation and a corresponding decrease in the AMFs at the surface.