## 4 Comparison between Two Models
### Frequency Response of the Models  
#### 4.2.1 Transfer Function  
In order to study the dynamic characteristics of different sites, the transfer function (TF) was chosen as an evaluation indicator [35]. An attempt was made to compare the dynamic characteristics of different regions of the models in the frequency domain. The transfer function is defined as follows: the ratio of the Laplace transform of the output signal to the Laplace transform of the input signal:  
\[TF=\frac{Y(s)}{X(s)}=\frac{L\{y(t)\}}{L\{x(t)\}} \tag{3}\]  
where \(X(s)=L\{x(t)\}\) and \(Y(s)=L\{y(t)\}\), representing the Laplace transforms of the input and output signals, respectively. Figure 7: The acceleration amplification factor (AAF) along the rock height: (**a**) transverse excitation; (**b**) longitudinal excitation. To assess the spectral characteristics of different sites, the transfer functions under white noise excitations (WN1) at AS1, AS2, and A3 were compared. Considering that the ground surface AAF under sinusoidal excitation theoretically approximates the magnitude of the transfer functions under white noise excitation, the AAF at AS1 and AS2 was also compared, as shown in Figure 8. It can be observed that for the HRM model, the transverse fundamental frequency in all three areas was 22.1 Hz, and the longitudinal fundamental frequency was 20.3 Hz; for the JRM model, the transverse fundamental frequency in all three areas was 17.3 Hz, and the longitudinal main frequency was 20.2 Hz. The difference in the fundamental frequency responses of the two models is consistent with the conclusions about the elastic modulus of the models presented in Section 2.4. The following five points can be summarized. 1. The AAF at AS1 and AS2 fell within the range of the ground surface transfer functions, indicating the reliability of the similitude design and model fabrication, ensuring the comparability between the two models. 2. The spectral characteristics of the structures were controlled by the sites. This explains the observations in Figure 7a, where the excitation frequency of 23 Hz of the input case was close to the transverse fundamental frequency of 22.1 Hz for the HRM model; hence, the amplification effect was not significantly reduced. 3. It is worth noting that the presence of the station structure resulted in a slightly smaller amplitude of the ground surface transfer function (AS1) than the free field (AS2), which may be due to the station structures absorbing more input energy, leading to a smaller response at the top of the models. This coincides with the conclusion of the power spectral density analysis in Section 4.4.2 and is also verified in reference [35] regarding the effect of underground structures on the surface. 4. Compared to the HRM model, the JRM model had a significantly lower transverse fundamental frequency, while the longitudinal fundamental frequency remained essentially unchanged. This suggests that the transverse stiffness of the JRM model was reduced due to the influence of joints and discontinuous planes, while the longitudinal stiffness remained unchanged. The discontinuous plane at the top face of the JRM model in Figure 3b might be the primary reason for this. Meanwhile, the transverse fundamental frequency magnitude of the JRM model was 1.51 times that of the HRM model, possibly due to reflections and refractions of seismic waves caused by joints and discontinuous planes, enhancing the dynamic response at the ground surface. Figure 8: Comparison of transfer functions for the two models: (**a**) transverse excitation; (**b**) longitudinal excitation. Comparing the ground surface AS1 and AS2 within the same model, their transverse transfer function curves essentially coincided, with the fundamental frequency and magnitude being nearly identical, further proving that the AS2 location represents a \"free field\". Similarly, the longitudinal transfer functions of AS1 and AS2 followed the same pattern.