# Three-Component Microseismic Data Denoising Based on Re-Constrain Variational Mode Decomposition
## 3 Synthetic Data  
We used an attenuated simple harmonic to verify the effectiveness of this method. Figure 2 shows pure attenuated harmonic three-component data with frequencies of 30 Hz and 70 Hz. Figure 3 shows waves contaminated with random noise. The SNR of the horizontal component is set to \(-10\) dB, and the SNR of the vertical component is set to 5 dB. In this study, we used the SNR, which is defined as:  
\[\text{SNR}=10\cdot\lg\left(\frac{\sum\limits_{t=1}^{\text{N}}\text{x}^{2}( \text{t})}{\sum\limits_{t=1}^{\text{N}}\text{n}^{2}(\text{t})}\right) \tag{7}\]  
where x(t) is the pure synthetic signal, n(t) is random noise, and N is the number of sampling points. Figure 4 shows the VMD results of three-component data. Since the parameter k of VMD only impacts the frequency representation performance, it has no noticeable impact on the reconstruction performance [24]. In addition, the proposed method only focuses on the frequency representation of the vertical component, which is viewed as a re-constraint condition. Therefore, the parameter k for each component was set to the same value in this study. Figure 5 shows the decomposition results of the three-component synthetic data obtained by the re-constrain VMD. As shown in Figure 5, the redundant amplitude of the nonlinear relationship information was removed. The remaining modes were also suppressed because they are all random signals and have no direct correlation. To prevent the re-constrained modes of two horizontal components (of which the SNR is low) from being an illusion of noise fitting (in other words, noise is regarded as an effective component of the signal to participate in the reconstruction processing), we  
Figure 1: Flow chart of processing the three-component microseismic data by re-constrain VMD. removed the pure synthetic signal and performed the re-constrain VMD for these two components. Figure 6 shows the corresponding results. For the re-constrained modes in the two horizontal components, there was a weak correlation with the vertical component after re-constrained decomposition, indicating that the re-constrained modes of the two horizontal components were also affected by random noise, and the degree of influence was quite limited. Figure 3: The synthetic data contaminated with random noise. Figure 2: The pure synthetic three-component signal. H1 and H2 denote two horizontal components, V denotes the vertical component. The first and second modes were reconstructed as a result of simple noise reduction. In Figure 7, it can be seen that compared with the VMD results, the results of re-constrain VMD could obtain better polarization characteristics and remove more background noise. Figure 4: The IMFs of VMD for the synthetic three-component data. (**a**) The VMD result of the H1 component. (**b**) The VMD result of the H2 component. (**c**) The VMD result of the vertical component. To better evaluate its denoising effect, we analyzed the polarization characteristics of the 30 Hz signal in the three-component data. The length of the selected time window was an entire period, and the 230 th sampling point was manually selected as the starting point of this window, as shown in Figure 8.  
Figure 5: The IMFs of re-constrain VMD for the synthetic three-component data. (**a**) The re-constrain VMD result of the H1 component. (**b**) The re-constrain VMD result of the H2 component. (**c**) The re-constrain VMD result of the vertical component. Figure 6: The IMFs of re-constrain VMD for the three-component data and the horizontal components without pure synthetic data. (**a**) The re-constrain VMD result of the H1 component without valid data. (**b**) The re-constrain VMD result of the H2 component without valid data. (**c**) The re-constrain VMD result of the vertical component with valid data. Figure 7: _Cont._  
Figure 8: _Cont._  
Figure 1:0 Denoising results of the synthetic microseismic data. (**a**) The synthetic data without noise. (**b**) The synthetic data with noise. (**c**) The denoising result of VMD.