## 3 Results and Discussion
### Analysis of Hilbert Spectrum  
In this subsection, we set up a set of homogeneous slope tests without a weak layer as a control group in order to clearly characterize the internal energy response of the slope under seismic wave action and to deeply reveal the influence of the weak layer on the internal energy changes of the slope under earthquake action. The aforementioned examination of the SBRS model with weak layer served as the basis for this. The two sets of dynamic response data are then compared and evaluated using the HHT to further illustrate the effect of the weak layer on the slope during earthquake activity. The HHT, a vibration signal processing method that has a high resolution of vibration signals in the time-frequency domain, was introduced by [PERSON] et al. [37] in 1996. It displays the transitory properties of seismic signals. The two primary parts of the HHT are the Hilbert transform and the empirical modal decomposition (EMD). In essence, the EMD decomposes a non-smooth seismic signal into a residual and several intrinsic mod-al functions (IMFs), each of which needs to satisfy the following specifications: (1) The number of extreme points and zero-crossing points must be equal or less than one during the signal duration. (2) The lower envelope based on the local minimum and the higher envelope based on the local maximum both have an average value of zero. Figure 8 illustrates the precise IMF determination procedure. For instance, Figure 9 displays the IMF components and residuals following EMD processing, as well as the acceleration time-range curve of the smooth-layer slope monitoring M1 under the influence of a 0.1 g seismic wave. Figure 7: Fourier spectrum of different monitoring points on slope surface under action of 0.1 g seismic wave: (**a**) Fourier spectrum of monitoring point M1. (**b**) Fourier spectrum of monitoring point M7. (**c**) Fourier spectrum of monitoring point M12. (**d**) Fourier spectrum of monitoring point M16. (**e**) Fourier spectrum of monitoring point M19. The Hilbert transform is applied to each IMF component by Equation (9), and the result is analyzed by Equations (10)-(13) to obtain the Hilbert spectrum, which better reflects the energy distribution and response characteristics of seismic signals in the time-frequency domain. \[Y(t)=\frac{1}{\pi}P\int_{-\infty}^{\infty}\frac{x(t^{\prime})}{t-t^{\prime}} \mathrm{d}t^{\prime}, \tag{9}\]  
\[a(t)=\left[X^{2}(t)+Y^{2}(t)\right]^{1/2}, \tag{10}\]  
Figure 8: Flowchart of EMD method. Figure 9: IMF component time-course curves and residuals for monitoring point M1. \[\theta(t)=\arctan\biggl{[}\frac{X(t)}{Y(t)}\biggr{]}, \tag{11}\] \[\omega(t)=\frac{d\theta(t)}{dt},\] (12) \[H(t,\omega)=R\sum_{j=1}^{n}a_{j}(t)\mathrm{e}^{i\int\omega_{j}(t) \mathrm{d}t}, \tag{13}\]  
where \(Y(t)\) is the result of HHT, \(P\) is the value of the Cauchy principal component, \(x(t)\) is the original signal, \(H(t,\omega)\) is the Hilbert spectrum, \(R\) is the real part, \(a_{j}(t)\) denotes the magnitude of the \(j\)th-order IMF corresponding to the instantaneous frequency \(\omega_{j}\) at the moment t and is a constant, \(a(t)\) is the instantaneous amplitude, \(\theta(t)\) is the instantaneous phase, and \(\omega(t)\) is the instantaneous frequency. The acceleration signals obtained at various elevation monitoring stations in the SBRS and the homogeneous slope while being impacted by a 0.1 g seismic wave were subjected to EMD and Hilbert transform in this instance. The resultant Hilbert spectrum is displayed in Figures 10 and 11. (1) As can be observed, the distribution range of the Hilbert energy of the monitoring point M5 in the SBRS and the homogeneous slope is primarily concentrated in 2\(\sim\)4 s and 8\(\sim\)10 s on the time axis, respectively, whereas the frequency axis distribution range is primarily concentrated in 10-20 Hz depending on the frequency of the loading seismic wave (Figures 10 and 11).