# Application Research of Negative Pressure Wave Signal Denoising Method Based on VMD
## 2 Methodology
### Signal Noise Reduction Based on VMD
#### 2.2.3 IMF Components Selection and Signal Reconstruction  
When leakage occurs, the pressure signals collected by the sensors are random. When using VMD for signal processing, preserving the true information of the signal is as important as denoising. The conventional method is used to select the effective component through empirical parameters. A disadvantage of this method is that the prior information of signal and noise needs to be mastered. To overcome the above deficiencies in this paper, the effective IMFs are screened using the correlation coefficient. According to [24], in the absence of fixed interference, we can assume that the noise signal collected by the two sensors is irrelevant, while the NPW signals are correlated. One of the two signals acts as a detection signal and the other as a reference signal. Firstly, the detection signal is decomposed into multiple IMF components using VMD. Secondly, we calculate the correlation coefficient between the reference signal and each IMF component. Finally, the effective components containing the pressure signal will be used to reconstruct the signal.  
Let \(x_{1}(t)\) and \(x_{2}(t)\) represent two signals, so:  
\[\left\{\begin{array}{c}x_{1}(t)=s(t)+n_{1}(t)\\ x_{2}(t)=\beta s(t-\Delta t)+n_{2}(t)\end{array}\right. \tag{6}\]  
where \(s(t)\) is the pressure signal, \(\beta\) respects the attenuation factor, and \(n(t)\) is the noise.  
In the process of applying the correlation analysis signals, the IMF component containing the pressure signal shows a strong correlation with the reference signal. These components will be retained. Let \(x_{1}(t)\) be decomposed by VMD into k components: \(u_{1}(t),u_{2}(t),\ldots,u_{k}(t)\), then the correlation coefficient between the \(k\)-th component \(u_{k}(t)\) and \(x_{2}(t)\) can be expressed as:  
\[R_{ux}=\frac{\sum\limits_{i=1}^{n}{(u_{k,i}-\overline{u}_{k})(x_{2,i}-\overline {x}_{2})}}{\left\{\begin{array}{c}\sum\limits_{i=1}^{n}{(u_{k,i}-\overline{ u}_{k})^{2}\sum\limits_{i=1}^{n}{(x_{2,i}-\overline{x}_{2})^{2}}}\end{array}\right\}^{1/2}} \tag{7}\]  
where \(\overline{u}_{k}=\sum\limits_{i=1}^{n}{(u_{k,i})}/n\), \(\overline{x}_{2}=\sum\limits_{i=1}^{n}{(x_{2,i})}/n\), and \(n\) is the length of the signal. According to the definition of correlation coefficient [25], we consider those IMF components with relatively higher correlation coefficients (\(R_{ux}\geq 0.3\)) to be effective components and labeled as \(u^{\prime}_{i}(t)\). The \(u^{\prime}_{i}(t)\) will be used for signal reconstruction. The reconstructed signal is called \(x^{\prime}(t)\):  
\[x^{\prime}(t)=\sum\limits_{i}{u^{\prime}_{i}(t)} \tag{8}\]According to the above method, the reconstructed signal under different \(k\) values can be obtained. Then, the optimal denoising signal can be obtained. In this paper, according to the research results in [24], the range of \(k\) is set to [2,16], and the penalty factor [25]\(\alpha\) is 2000.  
To sum up, the proposed VMD-based leakage signal adaptive noise reduction algorithm is executed as follows:  
1. \(x_{1}(t)\) and \(x_{2}(t)\), respectively, represent two NPW signals propagating along the two pipelines. \(k\), step size, and \(\alpha\) are initialized.
2. The signal is decomposed into \(k\) IMF components. The method in Section 2.2.3 is used to select effective IMF components and reconstruct them.
3. The information entropy of the reconstructed signals is calculated.  
To sum up, the novelty of our proposed method lies in:  
1. The correlation coefficient is used to decompose the IMF components, so that denoising can be achieved.
2. In order to detect the optimal denoising signal, the minimum value of information entropy is obtained by training, and the number of decomposition layers of VMD is determined by this value.  
Next, we will verify the effectiveness of the proposed algorithm through laboratory tests and field tests.