## 4 Seismic Response Characteristics of Layered Sites
### Results of Time-Frequency Domain Analysis  
The propagation characteristics of seismic energy in the time-frequency domain can be studied by analysing the variation of seismic Hilbert energy spectrum characteristics and the amplitude of seismic waves. The seismic response characteristics of layered sites can be further studied from the perspective of energy. The HHT is used to clarify the influence of (1) the interface between the soft rock mass and gravel soil and (2) the interface between the soft and hard rock masses on the change in wave energy in the layered sites. The latter is also used to study the change in wave energy during the wave propagation process in the surface gravel soil. Taking points A-I of the three strata in Model 2 as examples, according to the analyses of their Hilbert energy spectra (Figure 15), the seismic Hilbert energy of the horizontally layered site is mainly distributed within 24-25 s and 7-13 Hz. It should be noted that the PHSA (peak Hilbert energy spectrum amplitude) is within 9-10 Hz, suggesting that the low-frequency band has a great influence on the seismic response of the horizontally layered site. Figure 15a shows that in the hard rock, the Hilbert energy spectrum of points A-C shows single peak characteristics, the spectrum amplitude near the PHSA is small, and the frequency component is not rich. When waves are transferred from Layers 1 (hard rock) to 2 (soft rock), the Hilbert energy spectrum characteristics near the interface between soft and hard rock masses show obvious changes. Figure 15b shows that the influence of the interface between Layers 1 (hard rock) and 2 (soft rock) on the wave energy transmission is mainly manifested as the change in frequency components near the PHSA. As waves pass through the stratigraphic interface, the frequency components around the PHSA become more abundant, from a single peak to multiple peaks, but the PHSA and its occurrence time and frequency are not basically changed. This is because refraction and reflection effects of waves occur near the soft rock/hard rock stratum interface due to lithology changes, leading to frequency components becoming abundant near the PHSA. In addition, to investigate the change in energy when waves propagate from bottom to top in the weak gravelly soil layer, the Hilbert energy spectrum of points G-I is shown in Figure 15c. When the wave enters Layer 2 (soft rock) from Layer 3 (gravelly soil), the shape and peak value of the energy spectrum above the interface between gravelly soil and soft rock change greatly. Compared with Layer 2, the PHSA of the seismic wave is significantly magnified from the bottom to the top of the gravelly soil layer, and the multipeak characteristics of the energy spectrum are more obvious. The frequency components near the PHSA of seismic waves become more abundant, and the energy spectrum amplitude near the PHSA shows a significant amplification effect. The amplitude of the Hilbert energy spectrum in the high-frequency band (28-36 Hz) showed an obvious amplification effect, but the appearance location of the seismic Hilbert energy peak had no basic change. This phenomenon suggests that the interface between gravel soil and soft rock magnifies the peak energy of seismic waves, but  
Figure 14: Change of the PFSA in Model 2 when inputting the WE wave in the \(x\) direction at different natural frequencies: (**a**)\(f_{1}\)-\(f_{5}\) and (**b**)\(f_{6}\)-\(f_{8}\). the interface has no effect on the appearance location of the PHSA of seismic waves. When waves propagate in the gravel soil layer, the energy spectrum amplitude of the peak value and its vicinity is significantly amplified, especially in the high-frequency band (\(f_{6}\)-\(f_{8}\)). This indicates that the horizontally layered site can be analysed from the perspective of high-frequency components.