# Three-Component Microseismic Data Denoising Based on Re-Constrain Variational Mode Decomposition
## 4 Field Microseismic Data  
The field three-component microseismic event data were acquired in the hydraulic fracturing of a shale gas reservoir. The original waveforms of this microseismic event are shown in the first row of Figure 9. These microseismic data include the P-wave and S-wave of the microseismic event. The range of the P-wave is approximately from the 1000 th sampling point to the 1450 th sampling point, and the range of S-wave is approximately from the 1450 th sampling point to the 2000 th sampling point. Because of the formation mechanism of microseismic events, there is a partial overlap between the P-wave and S-wave. The SNR of the vertical component is better than that of the two horizontal components. The noise reduction results obtained using VMD and re-constrain VMD are shown. Comparing the waveforms between the original data and denoising results, the random noise was effectively suppressed by both methods.  
As shown in Figure 9, these two methods could suppress random noise to a certain extent. According to the original waveform of the horizontal component H1, the SNR of the P-wave is relatively low. Comparing the H1 components of the two groups' denoising results, it can be seen that the P-wave waveforms of microseismic data were not well recovered using VMD. Correspondingly, the waveforms of the P-wave were more obvious after using re-constrain VMD. It can also be seen that the background noise was better suppressed after using re-constrain VMD, but this effect was controlled by the background noise intensity of the vertical component.  
Because the P-wave is commonly used to analyze the polarization direction of microseismic events, we selected from the 1000 th point to the 1070 th point as the time window range, which is indicated by the red lines in Figure 9. The data in the time window were used to analyze the polarization and to judge the denoising effect. After denoising using VMD and re-constrain VMD, the polarization characteristics of the denoising results are shown in Figures 10 and 11.  
Figure 9: Comparison of denoising results between VMD and re-constrain VMD. (**a**) The original waveforms of a three-component microseismic event. (**b**) The denoising result obtained using VMD. (**c**) The denoising result obtained using re-constrain VMD.  
Figure 11: The hodograms of microseismic event data after denoising using re-constrain VMD. The black dotted lines represent the polarization characterization before denoising, and the red solid lines represent the polarization characterization after denoising. (**a**) The horizontal component H1 vs. horizontal component H2 hodogram. (**b**) The horizontal component H1 vs. vertical component V hodogram. (**c**) The horizontal component H2 vs. vertical component V hodogram.  
Figure 10: The hodograms of microseismic event data after denoising using VMD. The black dotted lines represent the polarization characterization before denoising, and the red solid lines represent the polarization characterization after denoising. (**a**) The horizontal component H1 vs. horizontal component H2 hodogram. (**b**) The horizontal component H1 vs. vertical component V hodogram. (**c**) The horizontal component H2 vs. vertical component V hodogram.  
In this study, the enhancement of polarization information by denoising was evaluated by determining whether the polarization characteristic of the curve of the hodogram was clear, compact, and self-consistent. It was largely straightforward to determine whether the polarization characteristic of the curve was clear, except for the H1-H2 hodogram in Figure 10. Regarding compactness, all hodograms in Figure 11 are better than those in Figure 10. Regarding self-consistency, according to the positive or negative correlation among the three components, the hodograms in Figure 11 are better than those in Figure 10.