## 3 Vibration Wave Splitting and Loading
### Separation and Filtering of Vibration Waveforms
\(T_{s}(w_{l},b)\) is calculated exclusively at the central \(w_{l}\) of \([\omega_{l}-\Delta\omega/2,\omega_{l}+\Delta\omega/2]\) within the specified frequency range, as indicated in Formula (8). \[T_{s}(\omega_{l},b)=\frac{1}{\Delta\omega}\sum_{a_{k}:|\omega(a_{k},b)-\omega _{l}|\leq}\frac{\Delta\omega}{2}W_{s}(a_{k},b)a^{-1.5}\Delta a_{k} \tag{8}\]  
The high-frequency components of the signal are filtered using a Butterworth filter to suppress noise while retaining the primary components of the signal. The difference equation for the Butterworth filter is presented in Formula (9) [22]. \[y[n]=\sum_{i=1}^{N}a_{i}y[n-i]+\sum_{j=0}^{M}b_{j}x[n-j] \tag{9}\]  
In the formula, \(x[n]\) represents the input signal sequence; \(y[n]\) denotes the filtered signal; and \(a_{i}\) and \(b_{j}\) are the feedback and feedforward coefficients of the filter, respectively, which are associated with the filter's order. The filtered signal is separated from the P-wave and S-wave through signal reconstruction, as illustrated in Formula (10). Each component of the signal undergoes a discrete synchrosqueezing wavelet transform on the overlapping component \(l\in L_{k}(t)\) and discrete time \(t_{nt}=t_{0}+m\Delta t\), where \(\Delta t\) represents the sampling time interval. \[s_{k}(t_{m})=2C_{\phi}^{-1}\Re\bigg{(}\widetilde{T}(\omega_{l},t_{m})\bigg{)} \tag{10}\]  
In the formula, \(C_{\phi}^{-1}\) represents the phase difference constant associated with the chosen wavelet, whereas the real component \(\Re\) of the discrete synchrosqueezing transform within the designated frequency band is utilized to reconstruct the real component \(t_{m}\). The original signal is recovered from the rearranged frequency domain signal using the inverse synchrosqueezing wavelet transform, as demonstrated in Formula (11). \[x(t)=\frac{1}{C_{\phi}}\int\limits_{-\infty}^{\infty}S_{x}(t,f)\psi_{f}(t)df \tag{11}\]  
In the formula, \(C_{\phi}\) is the normalized constant of the wavelet function, \(\psi_{f}\) represents the wavelet function in the frequency domain, and \(S_{x}(t,f)\) is a time-frequency spectrum that is compressed synchronously. In accordance with the research requirements, the code for data segmentation based on microseismic data has been developed using MATLAB (R2023a(9.14)), facilitating the noise filtering and separation of seismic wave data. The concave-convex wavelet is employed in the process of synchronous compression changes. After the synchronous compression wavelet transformation and noise reduction, a high-resolution time-frequency domain representation of the vibration wave is produced. The results are depicted in Figure 6. Based on the distinct frequency components of P-waves and S-waves, these waves are extracted individually. The frequency range of P-waves, as determined by short-time Fourier transform analysis, is identified to be between 35 Hz and 82 Hz, which is notably higher than the frequency range of S-waves, which spans from 18 Hz to 30 Hz. [PERSON] et al. [23] observed that only low-frequency, high-energy waves can potentially damage rock masses,which is why waveforms above 90 Hz are filtered out. The study by [PERSON] et al. [24] demonstrates that to accurately represent the propagation of vibration waves in the FLAC\({}^{\text{3D}}\) model, the non-physical effect of superimposed oscillations is avoided. The maximum mesh size, denoted as \(\Delta l\), required for modeling should be less than approximately one-tenth to one-eighth of the wavelength corresponding to the highest frequency component of the input wave, as indicated in Formula (12). \[\Delta l\leq(\frac{1}{8}-\frac{1}{10})\frac{C_{\text{s}}}{f_{\text{max}}} \tag{12}\]  
In the formula, \(C_{\text{s}}\) is the velocity of the S-wave, and \(f_{\text{max}}\) is the maximum frequency. The vibration wave has been subjected to filtering in MATLAB. However, after the separation of the P-wave and S-wave for input into the FLAC\({}^{\text{3D}}\) model, residual velocity resulting from baseline drift continues to be present even after the vibration has ceased.