# Application Research of Negative Pressure Wave Signal Denoising Method Based on VMD
## 3 Results and Discussion
### The Laboratory Experiments
#### 3.1.3 Leak Location Results  
The positioning effect of leakage point 1 in Table 1 is analyzed as an example. The denoising results are shown in Figure 6. The denoised signal displays an obvious inflection point of the NPW.  
Figure 5: Comparison of leak signal denoised by four methods.  
The calculation method of time difference \(\Delta t\) is shown in Equation (9):  
\[\Delta t=\frac{n_{1}-n_{2}}{f} \tag{9}\]  
where \(n_{1}\) and \(n_{2}\) are the inflexion points of NPW collected by sensor A and sensor B respectively. The sampling frequency is \(f\) (\(f\) = 2 kHz).  
In order to verify the advantages of our method, four methods are used to locate the leak point 1. In the rectangular box in Figures 7-10, the red and green curves represent the denoised signal and the original leakage signal of sensor A, respectively. The result of sensor B is similar. Therefore, it is not shown in the figure. The result of the low-pass filter method is shown in Figure 7, where \(n_{1}\) = 4106 and \(n_{2}\) = 3984. Substituting \(n_{1}\) and \(n_{2}\) into Equation (9), we obtain \(\Delta t=0.061\) s. According to Equation (1), if \(L\) = 23.34 m, then \(X_{\text{A}}\) = 25.48 m. The real distance is 17.34 m. The positioning error is 34.88%. In Figure 8, the result of wavelet denoising is similar to that produced by the low-pass filter method. \(X_{\text{A}}\) is obtained as 20.72 m. The positioning error is 14.48%. The result of the EMD-based method is shown in Figure 9. There is no obvious inflection point in the figure, and it is difficult to locate the leakage. The result of the proposed method is shown in Figure 10. \(X_{\text{A}}\) is 17.55 m. The positioning error is 0.9%.  
To further verify the leakage location effect of the proposed method, Table 3 shows the location results of all tests using the above methods, where \(X_{1}\) is the positioning result of the low-pass filter method; \(X_{2}\) is the result of the wavelet method; and \(X_{3}\) is the result of the proposed method. \(E_{1}\), \(E_{2}\), and \(E_{3}\), respectively, represent the positioning relative errors of the three methods. \"\(\times\)\" indicates that the location fails. The EMD method failed to locate, so its results are not listed in Table 3.  
Figure 6: Leak signal and its denoised signal.  
Figure 7: The result of the low-pass filter method.  
The pressure signal in the pipeline has non-stationary and non-linear characteristics. However, the low-pass filter method and wavelet denoising rely on experience to select parameters, and their denoising effects on such signals are limited. The method based on EMD distorts the reconstructed signal, due to modal aliasing and end effects. After screening out the effective IMF components and optimizing decomposition layers k, our method achieves effective signal noise reduction.  
Figure 8: The result of wavelet denoising.  
Figure 10: The result of the proposed method.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**\#** & \(\mathbf{L_{\mathrm{A}}}\) & \(\mathbf{X_{1}}\) & \(\mathbf{E_{1}}\) & \(\mathbf{X_{2}}\) & \(\mathbf{E_{2}}\) & \(\mathbf{X_{3}}\) & \(\mathbf{E_{3}}\) (\%) \\  & **(m)** & **(m)** & **(\%)** & **(m)** & **(\%)** & **(m)** & **(m)** \\ \hline
1 & 17.34 & 25.48 & 34.88 & 20.72 & 14.48 & 17.55 & 0.9 \\
2 & 14.47 & \(\times\) & \(\times\) & 15.29 & 3.51 & 15.18 & 3.04 \\
3 & 2.47 & \(\times\) & \(\times\) & 7.94 & 23.44 & 1.82 & 2.78 \\ \hline \hline \end{tabular}  
* is leak point number.  
\end{table}
Table 3: Leak location results.  
Figure 10: The result of the proposed method.  
Figure 9: The result of the EMD method.  
In Table 3, the minimum relative positioning errors of the low-pass filter method and wavelet denoising are 34.88% and 3.51%, respectively, and the positioning cannot be performed when the error is largest. The proposed method achieves a minimum positioning relative error of 0.9%. Therefore, compared with the other three methods, our method has a smaller leakage location error.