## 3 Vibration Wave Splitting and Loading
### Separation and Filtering of Vibration Waveforms  
The vibration waves identified by the microseismic monitoring system generally comprise a combination of P-waves and S-waves, accompanied by a certain degree of noise. The presence of this noise complicates the process of isolating the waveform. Recent advancements in signal processing technology have facilitated the growing application of the synchrosqueezed wavelet transform within the domain of seismic signal processing. This technique has been utilized to enhance time-frequency resolution and to improve the accuracy of signal feature extraction. By employing the technique of instantaneous frequency redistribution and estimation [19; 20], it is feasible to attain a high-resolution representation in the time-frequency domain. This approach facilitates the differentiation and separation of P-waves and S-waves that possess distinct frequency components. Because microseismic monitoring systems collect extraneous noise, Butterworth filters are commonly employed in noise processing due to their advantages in preserving waveform fidelity [21]. Ultimately, the processed P-waves and S-waves are reconstructed from the time-frequency domain signal back to the time domain signal by applying the inverse synchrosqueezing wavelet transform. Drawing upon the foundational principles of the synchrosqueezed wavelet transform in conjunction with the Butterworth filter, the following sections present a comprehensive discussion of the methodologies employed for the extraction and separation of features within vibration waveforms. The acquired vibration wave signal is expressed as illustrated in Formula (5) [19]. \[f(t)=\sum_{k=1}^{K}A_{k}(t)\cos(2\pi\varphi_{k}(t))+e(t) \tag{5}\]  
In the formula, \(A_{k}(t)\) denotes the amplitude of this component; \(\varphi_{k}(t)\) is the instantaneous phase of the above components; and \(e(t)\) is a white noise with zero mean and variance of \(\sigma^{2}\). Initially, the original vibration waveform that has been acquired is subjected to a continuous wavelet transform [20]. The continuous wavelet transform of the vibration wave signal \(f(t)\) is expressed in Equation (6). \[W_{s}(a,b)=\frac{1}{\sqrt{a}}\int f(t)\psi^{*}(\frac{t-b}{a})dt \tag{6}\]  
In the formula, \(\psi^{*}\) is the complex conjugate of the mother wavelet, and \(b\) denotes the time offset applied to the mother wavelet, and will also be scaled by the scale factor \(a\). The continuous wavelet transform involves the convolution of the signal \(f(t)\) with a scaled and translated mother wavelet. The wavelet coefficients of the signal \(f(t)\) are derived by scaling and shifting the mother wavelet. \(W_{s}(a,b)\) represents the coefficients of the signal ft after the continuous wavelet transform and describes the characteristics of the signal at different scales \(a\) and positions \(b\)[20]. When \(W_{s}(a,b)\
eq 0\), use Formula (7) to calculate the instantaneous frequency \(W_{s}(a,b)\):  
\[\omega_{s}(a,b)=\frac{-j}{W_{s}(a,b)}\frac{\partial W_{s}(a,b)}{\partial b} \tag{7}\]  
where \(\frac{\partial W_{s}(a,b)}{\partial b}\) represents the rate of change of wavelet coefficients in the b direction of displacement, which is utilized to describe the local frequency variation of the signal. The final step is to map the speed plane of the signal to the time-frequency plane; specifically, \((b,a)\) is converted to \(w_{s}(a,b)\). The methodology that integrates continuous wavelet transform with instantaneous frequency reassignment is known as the synchrosqueezing transform technique [20]. Given that the variables \(a\) and \(b\) are discretized, a time step of \(\Delta a_{k}=a_{k-1}-a_{k}\) is established for the computation of \(a_{k}\) across all \(W_{s}(a,b)\). Additionally, a frequency band of \(\Delta\omega\) is selected due to its primary concentration of energy within the signal, thereby facilitating a more effective capture of the signal's essential characteristics.