# Denoising Method of Pipeline Leakage Signal Based on VMD and Hilbert Transform

Authors: Zhu Jiang; Junyi Xie; Jing Zhang; Xiang Zhang
Journal: Journal of Sensors
 (2023)

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## Abstract

In order to solve the problem of low localization accuracy of water supply pipeline leakage detection using acoustic emission signals, a noise reduction algorithm based on variational mode decomposition (VMD) and Hilbert transform is proposed in this paper. Firstly, the leakage signal is decomposed into several intrinsic mode functions (IMF) using VMD, and the number of decomposition layers is determined, and the IMF component is transformed by Hilbert to get the marginal spectrum, and the noise are preliminarily filtered according to the marginal spectrum characteristics and the cross-correlation coefficient. Secondly, the signal denoising is realized by using filtering technology and cross-correlation coefficient. Finally, the time delay estimation of the denoised signal is estimated and the location of the leakage point is calculated based on the positioning principle. In order to verify the performance of the algorithm, a water supply pipeline experiment platform for leakage simulation and leak location experiment was conducted. The cross-correlation algorithm, VMD-correlation coefficient algorithm, and the proposed algorithm are analyzed through simulation and experiment. The results show that compared with the other two algorithms, the proposed algorithm can effectively remove the noise in the leakage signal and improve the accuracy of leakage location.


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## Body

## 1. Introduction

With economic development, global demand for water is growing at a rate of 1% a year, and this rate is set to accelerate significantly in the future. Water supply pipelines play a critical role in urban economic development, but corrosion, aging, and external interference factors may cause pipeline leakage [1–3]. The leakage of water supply pipes will lead to the waste of water resources and cause economic losses. In addition, it will cause secondary disasters, resulting in road cave-in and building collapse. Moreover, the pollution sources enter the pipelines through the leak point and will pollute the water quality, endangering the safety of users [4]. Therefore, it is of extraordinary significance to explore accurate detection methods of water supply pipeline leakage and take timely and effective measures to reduce economic losses and accelerate urban construction.In pipeline leakage detection technology, the methods used commonly can be separated into hardware detection and software detection. Manual patrol and gas tracing belong to hardware detection. Among them, the principle of manual patrol is simple and easy to operate. However, the detection speed is slow, and buried pipelines cannot be detected. The detection accuracy of tracer gas is high, but it will pollute the environment and affect the life of users [5]. In addition, the hardware detection technology is unable to detect leakage online, so the detection efficiency is not good. In recent years, more and more attention has been paid to the research and application value of software inspection technology, for example, fiber optic sensing technology and acoustic emission technology [6–8]. The former is safe and reliable, but it needs to lay a lot of optical fibers, and the cost is high [9]. The latter is a technology to detect the vibration signal generated by the interaction between the water at the leakage point of the pipeline and the surrounding medium [10]. This technique has the characteristics of fast detection speed and high positioning accuracy. Meanwhile, the pipeline can be inspected online in real time. Therefore, it has good research and application value in leak detection technology. Considering the above factors, the study of this paper will be based on acoustic emission technology.In the process of detecting and locating water supply pipeline leakage by using the acoustic signal caused by leakage, due to the interference of the surrounding environment (site construction, vehicle driving, etc.), the actual leakage acoustic signals are inevitably mixed with a large amount of noise [11]. These noises will increase the calculation error and further reduce the accuracy of leakage location [12]. Therefore, in the use of acoustic signal method for leakage location, it is essential to choose the suitable method for signal noise reduction processing.The filter can only suppress the noise in the specified frequency band well, but when the acoustic signal and the noise characteristic band are unknown or in the same frequency band, it is difficult to achieve the expected effect. Compared with filter, wavelet decomposition has a better denoising effect [13]. However, noise is a disordered signal, so it is very difficult to select wavelet basis function and threshold [14]. Compared with wavelet decomposition, empirical mode decomposition (EMD) solves the problem of basis function and adaptively decomposes according to the signal. However, EMD suffers from two serious problems: modal aliasing and endpoint effects [15, 16], which reduces the noise suppression effect. The variational mode decomposition (VMD) method was first proposed by Ragomiretskiy et al. in 2014 [17]. Different from the cyclic screening stripping method used in the EMD algorithm [18], in this method, the optimal solution is solved by iterative variational model, and the center frequency and bandwidth of each IMF component are obtained [19]. Therefore, the method can adaptively complete the frequency domain segmentation of signal and the valid separation of each part, without the problem of mode aliasing [20–23]. However, when the noise energy in the signal is high, the traditional VMD-correlation coefficient denoising algorithm often takes the IMF component containing noise as the real signal, and the IMF component containing low-energy real signal as the noise, which cannot accurately screen the IMF component, so the denoising effect is poor.On this basis, in order to better screen out effective feature IMF components, this paper introduces VMD and Hilbert into acoustic signal denoising research and proposes a novel VMD-Hilbert transform algorithm. In this algorithm, the leakage signal is decomposed into multiple IMF components by using VMD, and the spectral characteristics of each IMF component are obtained by using the characteristics of the Hilbert edge spectrum. Filter out some noise according to the cross-correlation coefficient. The Chebyshev filter is then used to filter the remaining noise. Finally, the location of the leakage point is calculated by the time delay of denoising signal estimation with high correlation coefficient. Through simulation and experiment, the algorithm is compared with the cross-correlation algorithm and VMD-correlation coefficient algorithm, and its advantages are analyzed.The structure of this article is as follows: The background of water supply pipeline leakage detection is introduced in Section1; Section 2 introduces the principle of leakage location and the principle and flow of VMD-Hilbert transform algorithm in detail. In Section 3, the VMD-Hilbert transform algorithm is compared with the cross-correlation algorithm and VMD-correlation coefficient algorithm through simulation experiments, and the advantages of the VMD-Hilbert transform algorithm are analyzed. Section 4 introduces the experimental setup. Section 5 is the experimental results, analyzing the effect of VMD-Hilbert transform algorithm. Finally, summarized in Section 6.


## 2. Method

### 2.1. Principle of Leakage Location Based on Acoustic Signal

When the pipeline leaks, the leakage vibration sound signal will propagate upstream and downstream of the pipeline from the leak point. Due to the different spacing between the leakage point and the sensors located upstream and downstream of the pipeline, there is a time gap between the leakage vibration acoustic signal and the upstream and downstream sensors. The location of the leakage point can be obtained by the length of the pipeline, the speed of the vibration sound wave, and the time gap of the propagation [24].Figure1 shows the calculation principle diagram of the leakage point location. Assume that the distance between the two sensors is L, and the distance between leakage point and sensor A is XA. So, the distance between leakage point and sensor B is L−XA. Assume that the speed of the sound wave is V. Therefore, the time it takes for the sound wave to reach sensor A is t1t1=XA/V, and the time it takes for the sound wave to reach sensor B is t2t2=L−XA/V. Let the time difference be Δt=t1−t2, then the distance between the leakage point and sensor A can be expressed:
(1)XA=12L+VΔt,where L can be obtained by measurement and V can take empirical value. Therefore, the key factor affecting the location of leakage is Δt.Figure 1
Leakage location principle.Equation (1) is the basic formula based on delay location. Both the negative pressure wave techniques used in reference [24] and the acoustic emission technique used in this paper are developed based on this principle. In the research process of reference [24], Liu et al. collect the pressure signal in the water supply pipeline through the pressure sensor, and the leakage will trigger the negative pressure wave and thus generate the pressure abrupt inflection point. Liu et al. study the negative pressure wave signal denoising method to achieve the accurate extraction of the inflection point, so as to improve the calculation accuracy of Δt in formula (1), and finally improve the positioning accuracy of the leakage point. In this paper, our research object is acoustic emission vibration signal collected by an acoustic emission sensor. In order to improve positioning accuracy, different noise reduction algorithms will be used to process the signals.Δt is mainly calculated according to delay estimation based on cross-correlation function. The principle is to use the correlation between two signals to determine the delay. Assuming that the original signal released by the acoustic emission source is Sη, η is an integer from 1 to N, and N is the number of sample points included in the sample; then, the expressions of signals xAη and xBη is obtained:
(2)xAη=α1Sη−τ+β1η,xBη=α2Sη+β2η,whereα1 and α2 are the attenuation coefficients of signal propagation in two directions, respectively and β1 and β2 are the noise doped in the two signals, respectively; then, the cross-correlation function of the two groups of signals is
(3)Rxyτ=∑η=1NxAηxBη.The independent variable of the cross-correlation functionRxy is the number of delay sample points τ, which represents the dot product of a column of signals translated τ along the horizontal axis with another signal. The value of τ is an integer in the interval −n+1,n−1. Since noise β1η and β2η are not correlated by themselves, when Rxy reaches the maximum, the correlation between the two columns of signals is the strongest. The corresponding τ1 is the number of delay samples corresponding to the desired delay. In this case, the coincidence degree of xAη and xBη after translation is the largest. Obviously, if the leakage source is located at the midpoint of the two sensors, the value of τ1 is 0. The delay can be obtained from the number of delayed samples by the following equation:
(4)Δt=τ1f,wheref is the signal sampling rate, and its reciprocal is the sampling time interval of sample points. After the delay is obtained, the location XA of the leakage source can be calculated according to formula (1). However, in the actual signal, the existence of noise will inevitably affect the correlation of the two signals, causing a big error in the result of Δt, which significantly impacts the accuracy of positioning. Therefore, the noise reduction method of the acoustic signal will be further studied in this paper.


### 2.2. VMD-Hilbert Transform Algorithm

#### 2.2.1. VMD

In the VMD method, the calculation process of the optimal solution of the variational model can be expressed as [25]
(5)minuk,ωk∑k∂tδt+jπt∗ukte−jωkt22s.t.∑kukt=xt,whereuk=u1,⋯,uk is the k IMF components of input signal xt after VMD decomposition, and ωk=ω1,⋯,ωk is the center frequency of the k component.The penalty factorα and the augmented Lagrange function are introduced to obtain the expression of the optimal solution of the variational model:
(6)Luk,ωk,λ=α∑k∂tδt+jπt∗ukte−jωkt22+xt−∑kukt22+λt,xt−∑kukt,whereα refers to the penalty factor and λ refers to the Lagrange multiplier.In the formula,uk and ωk are obtained by iterating the values of uk, ωk, and λ.


#### 2.2.2. Hilbert Transform

The output response of a continuous time signal through a linear system with impulse responseht=1/πt is equivalent to the Hilbert transform. Therefore, the Hilbert transform expression of input signal xt is obtained:
(7)1π∫−∞+∞xtt‐τdτ=1π∫−∞+∞xt−ττdτ=xt∗1πt.After VMD decomposition, the expression of input signalxt is obtained:
(8)xt=∑i=1kukt+rkt,whereukt is the IMF component of xt after VMD decomposition, and rkt is the residual component.We apply Hilbert transform to each IMF component:(9)Hukt=1π∫−∞+∞uktt−τdτ=1π∫−∞+∞ukt−ττdτ=ukt∗1πt.We construct analytic signal:(10)zkt=ukt+jHukt=aktejθkt.Among them,(11)akt=uk2t+H2ukt,θkt=arctanHuktukt.The corresponding instantaneous frequency is(12)fkt=12πdθktdt.The analytical expression ofxt is obtained:
(13)xt=∑i=1kaktejθkt.According to Equations (10) and (12),
(14)ukt=Reaktej2π∫fktdt,whereRe denotes the real part. When akt is represented in the coordinate plane, the Hilbert time spectrum of IMF component ukt can be obtained:
(15)Hkt,f=akt,f=fkt,0,f≠fkt.The overall Hilbert time spectrum ofxt is analyzed, and the Hilbert spectrum is obtained:
(16)Ht,f=∑k=1nHkt,f.The Hilbert spectrum describes how the amplitude of a signal varies with time and frequency over the total frequency range.Further, we define the Hilbert marginal spectrum by the Hilbert spectrum:(17)hf=∫0THkt,f,whereT is the signal sampling time. The Hilbert marginal spectrum reflects the statistical situation of the amplitude of the signal in the whole frequency segment and also describes the actual frequency component of the signal.Considering that Fourier transform is the most commonly used signal processing method, the differences between Fourier transform and Hilbert transform are researched.If the input signalxt by using Fourier transform, it can be expressed as
(18)xt=∑i=1∞akejωkt,whereak and ωk are constants. In formula (13), akt and θkt are variables. Therefore, compared with Fourier transform, Hilbert transform can be applied to nonstationary signal processing. At the same time, the frequency and amplitude can be separated by Hilbert transform to realize the variable frequency and amplitude description of the signal.In order to more intuitively understand the difference between the Fourier spectrum and Hilbert spectrum, we choose to use a set of data collected by sensor A for analysis, and the results are shown in Figure2. Figure 2(a) is the signal collected by sensor A, Figure 2(b) is the Fourier spectrum of IMF1, Figure 2(c) is the Fourier spectrum of IMF2, and Figure 2(d) is the marginal Hilbert spectra.Figure 2
Frequency spectrum analysis.
(a)
Signal collected by sensor A(b)
Fourier spectrum of IMF1(c)
Fourier spectrum of IMF2(d)
Marginal Hilbert spectraIt can be seen from Figures2(b) and 2(c) that when Fourier spectrum is used to analyze signals, all IMF components cannot be drawn on a single diagram and can only be analyzed separately. It can be seen from Figure 2(d) that all IMF components can be plotted on a single graph using edge Hilbert spectrum, which improves efficiency. At the same time, we can also observe the amplitude of all IMF components on a single graph and more effectively select “half of the maximum energy amplitude.”


#### 2.2.3. VMD-Hilbert Transform Algorithm

When there is high-energy and high-frequency band noise in the signal and the low-frequency band noise signal is similar to the leakage signal frequency band, in order to achieve good denoising effect, a novel method named VMD-Hilbert transform algorithm is proposed in this paper.Figure3 gives the noise reduction process of the VMD Hilbert transform algorithm, and the detailed steps are as follows:
(1)
The leakage signalxAη collected by sensor A is decomposed by VMD, and the initial decomposition layer k is set as 3 (after experimental verification, the residual component of the leakage signal decomposed is background noise, and the center frequency is expressed as 0. Therefore, the initial decomposition level k=3 is appropriate), the penalty factor α=2000, and the accumulation tolerance c=1e−6 (Penalty coefficient α determines the bandwidth of the IMF. The smaller the α value, the greater the IMF bandwidth. Excessive bandwidth will cause some signals to mix together; the higher the α value is, the smaller the IMF bandwidth is. If the bandwidth is too small, some signals will be lost. The common value range of α is 1000~3000. c is one of the stop criteria for VMD decomposition. Generally, 1e−6 ~ 5e−6 can be adopted.) After decomposition, IMF1, IMF2, ⋯, IMF k are obtained, let Δ be the absolute value of the difference between adjacent IMF components(2)
Set the number of decomposition layersk=k+1 and repeat step (1), until the absolute value of Δ, the difference between adjacent IMF components of xAη after VMD decomposition, is less than 200 (as demonstrated by a large number of experimental results; when Δ<200, the leakage signal can get the best decomposition effect); then, the number of decomposition layers kA at this time is judged to be the best, and the best decomposition layers kB can be obtained by the same processing of xBη(3)
AfterxAη and xBη are decomposed by VMD, residual components are removed from the IMF components, and Hilbert transform is performed to obtain Hilbert edge spectrum by using formula (9) and formula (14)–(17). The energy amplitude of each IMF component is obtained through the marginal spectrum. The components whose energy amplitude is greater than half of the maximum energy amplitude are selected for reconstruction (half of the maximum energy amplitude can be regarded as the basis for identifying the main components of the signal), and the coefficient h obtained after the normalization of the cross-correlation function of the reconstructed signal is calculated. The high-energy noise signal is discriminated by the value of h(4)
The decomposed IMF components are reconstructed into low-frequency (0-1000 Hz) and high-frequency (1000-2000 Hz) components. When the coefficienth1 obtained by the normalized cross-correlation function of the low-frequency reconstructed signal is larger than the coefficient h2 obtained by the normalized cross-correlation function of the high-frequency reconstructed signal, it indicates that the high-energy noise signal exists in the high-frequency band. Therefore, the low-frequency IMF component with less noise is reconstructed to obtain the reconstructed signals xA′η and xB′η(5)
The reconstructed signalsxA′η and xB′η are low-pass filtered through Chebyshev filter. Firstly, the passband is set to n characteristic frequency bands, and the difference between the left and right boundaries of each passband is 100 Hz. The signals xAnη and xBnη of N characteristic frequency bands are obtained after filtering, and the coefficient Rn is obtained after the cross-correlation function normalization of xAnη and xBnη is calculated(6)
Select the groupRmax with the largest Rn, the corresponding passband is the characteristic frequency band of the leakage signal, output denoising signal xAmaxη and xBmaxη, Rmax is the correlation coefficient of the two denoising signalsFigure 3
Noise reduction process of the VMD Hilbert transform algorithm.After noise reduction by using the VMD Hilbert algorithm, we can use formula (4) to calculate Δt and then use formula (1) to calculate the leak point location XA.


## 2.1. Principle of Leakage Location Based on Acoustic Signal

When the pipeline leaks, the leakage vibration sound signal will propagate upstream and downstream of the pipeline from the leak point. Due to the different spacing between the leakage point and the sensors located upstream and downstream of the pipeline, there is a time gap between the leakage vibration acoustic signal and the upstream and downstream sensors. The location of the leakage point can be obtained by the length of the pipeline, the speed of the vibration sound wave, and the time gap of the propagation [24].Figure1 shows the calculation principle diagram of the leakage point location. Assume that the distance between the two sensors is L, and the distance between leakage point and sensor A is XA. So, the distance between leakage point and sensor B is L−XA. Assume that the speed of the sound wave is V. Therefore, the time it takes for the sound wave to reach sensor A is t1t1=XA/V, and the time it takes for the sound wave to reach sensor B is t2t2=L−XA/V. Let the time difference be Δt=t1−t2, then the distance between the leakage point and sensor A can be expressed:
(1)XA=12L+VΔt,where L can be obtained by measurement and V can take empirical value. Therefore, the key factor affecting the location of leakage is Δt.Figure 1
Leakage location principle.Equation (1) is the basic formula based on delay location. Both the negative pressure wave techniques used in reference [24] and the acoustic emission technique used in this paper are developed based on this principle. In the research process of reference [24], Liu et al. collect the pressure signal in the water supply pipeline through the pressure sensor, and the leakage will trigger the negative pressure wave and thus generate the pressure abrupt inflection point. Liu et al. study the negative pressure wave signal denoising method to achieve the accurate extraction of the inflection point, so as to improve the calculation accuracy of Δt in formula (1), and finally improve the positioning accuracy of the leakage point. In this paper, our research object is acoustic emission vibration signal collected by an acoustic emission sensor. In order to improve positioning accuracy, different noise reduction algorithms will be used to process the signals.Δt is mainly calculated according to delay estimation based on cross-correlation function. The principle is to use the correlation between two signals to determine the delay. Assuming that the original signal released by the acoustic emission source is Sη, η is an integer from 1 to N, and N is the number of sample points included in the sample; then, the expressions of signals xAη and xBη is obtained:
(2)xAη=α1Sη−τ+β1η,xBη=α2Sη+β2η,whereα1 and α2 are the attenuation coefficients of signal propagation in two directions, respectively and β1 and β2 are the noise doped in the two signals, respectively; then, the cross-correlation function of the two groups of signals is
(3)Rxyτ=∑η=1NxAηxBη.The independent variable of the cross-correlation functionRxy is the number of delay sample points τ, which represents the dot product of a column of signals translated τ along the horizontal axis with another signal. The value of τ is an integer in the interval −n+1,n−1. Since noise β1η and β2η are not correlated by themselves, when Rxy reaches the maximum, the correlation between the two columns of signals is the strongest. The corresponding τ1 is the number of delay samples corresponding to the desired delay. In this case, the coincidence degree of xAη and xBη after translation is the largest. Obviously, if the leakage source is located at the midpoint of the two sensors, the value of τ1 is 0. The delay can be obtained from the number of delayed samples by the following equation:
(4)Δt=τ1f,wheref is the signal sampling rate, and its reciprocal is the sampling time interval of sample points. After the delay is obtained, the location XA of the leakage source can be calculated according to formula (1). However, in the actual signal, the existence of noise will inevitably affect the correlation of the two signals, causing a big error in the result of Δt, which significantly impacts the accuracy of positioning. Therefore, the noise reduction method of the acoustic signal will be further studied in this paper.


## 2.2. VMD-Hilbert Transform Algorithm

### 2.2.1. VMD

In the VMD method, the calculation process of the optimal solution of the variational model can be expressed as [25]
(5)minuk,ωk∑k∂tδt+jπt∗ukte−jωkt22s.t.∑kukt=xt,whereuk=u1,⋯,uk is the k IMF components of input signal xt after VMD decomposition, and ωk=ω1,⋯,ωk is the center frequency of the k component.The penalty factorα and the augmented Lagrange function are introduced to obtain the expression of the optimal solution of the variational model:
(6)Luk,ωk,λ=α∑k∂tδt+jπt∗ukte−jωkt22+xt−∑kukt22+λt,xt−∑kukt,whereα refers to the penalty factor and λ refers to the Lagrange multiplier.In the formula,uk and ωk are obtained by iterating the values of uk, ωk, and λ.


### 2.2.2. Hilbert Transform

The output response of a continuous time signal through a linear system with impulse responseht=1/πt is equivalent to the Hilbert transform. Therefore, the Hilbert transform expression of input signal xt is obtained:
(7)1π∫−∞+∞xtt‐τdτ=1π∫−∞+∞xt−ττdτ=xt∗1πt.After VMD decomposition, the expression of input signalxt is obtained:
(8)xt=∑i=1kukt+rkt,whereukt is the IMF component of xt after VMD decomposition, and rkt is the residual component.We apply Hilbert transform to each IMF component:(9)Hukt=1π∫−∞+∞uktt−τdτ=1π∫−∞+∞ukt−ττdτ=ukt∗1πt.We construct analytic signal:(10)zkt=ukt+jHukt=aktejθkt.Among them,(11)akt=uk2t+H2ukt,θkt=arctanHuktukt.The corresponding instantaneous frequency is(12)fkt=12πdθktdt.The analytical expression ofxt is obtained:
(13)xt=∑i=1kaktejθkt.According to Equations (10) and (12),
(14)ukt=Reaktej2π∫fktdt,whereRe denotes the real part. When akt is represented in the coordinate plane, the Hilbert time spectrum of IMF component ukt can be obtained:
(15)Hkt,f=akt,f=fkt,0,f≠fkt.The overall Hilbert time spectrum ofxt is analyzed, and the Hilbert spectrum is obtained:
(16)Ht,f=∑k=1nHkt,f.The Hilbert spectrum describes how the amplitude of a signal varies with time and frequency over the total frequency range.Further, we define the Hilbert marginal spectrum by the Hilbert spectrum:(17)hf=∫0THkt,f,whereT is the signal sampling time. The Hilbert marginal spectrum reflects the statistical situation of the amplitude of the signal in the whole frequency segment and also describes the actual frequency component of the signal.Considering that Fourier transform is the most commonly used signal processing method, the differences between Fourier transform and Hilbert transform are researched.If the input signalxt by using Fourier transform, it can be expressed as
(18)xt=∑i=1∞akejωkt,whereak and ωk are constants. In formula (13), akt and θkt are variables. Therefore, compared with Fourier transform, Hilbert transform can be applied to nonstationary signal processing. At the same time, the frequency and amplitude can be separated by Hilbert transform to realize the variable frequency and amplitude description of the signal.In order to more intuitively understand the difference between the Fourier spectrum and Hilbert spectrum, we choose to use a set of data collected by sensor A for analysis, and the results are shown in Figure2. Figure 2(a) is the signal collected by sensor A, Figure 2(b) is the Fourier spectrum of IMF1, Figure 2(c) is the Fourier spectrum of IMF2, and Figure 2(d) is the marginal Hilbert spectra.Figure 2
Frequency spectrum analysis.
(a)
Signal collected by sensor A(b)
Fourier spectrum of IMF1(c)
Fourier spectrum of IMF2(d)
Marginal Hilbert spectraIt can be seen from Figures2(b) and 2(c) that when Fourier spectrum is used to analyze signals, all IMF components cannot be drawn on a single diagram and can only be analyzed separately. It can be seen from Figure 2(d) that all IMF components can be plotted on a single graph using edge Hilbert spectrum, which improves efficiency. At the same time, we can also observe the amplitude of all IMF components on a single graph and more effectively select “half of the maximum energy amplitude.”


### 2.2.3. VMD-Hilbert Transform Algorithm

When there is high-energy and high-frequency band noise in the signal and the low-frequency band noise signal is similar to the leakage signal frequency band, in order to achieve good denoising effect, a novel method named VMD-Hilbert transform algorithm is proposed in this paper.Figure3 gives the noise reduction process of the VMD Hilbert transform algorithm, and the detailed steps are as follows:
(1)
The leakage signalxAη collected by sensor A is decomposed by VMD, and the initial decomposition layer k is set as 3 (after experimental verification, the residual component of the leakage signal decomposed is background noise, and the center frequency is expressed as 0. Therefore, the initial decomposition level k=3 is appropriate), the penalty factor α=2000, and the accumulation tolerance c=1e−6 (Penalty coefficient α determines the bandwidth of the IMF. The smaller the α value, the greater the IMF bandwidth. Excessive bandwidth will cause some signals to mix together; the higher the α value is, the smaller the IMF bandwidth is. If the bandwidth is too small, some signals will be lost. The common value range of α is 1000~3000. c is one of the stop criteria for VMD decomposition. Generally, 1e−6 ~ 5e−6 can be adopted.) After decomposition, IMF1, IMF2, ⋯, IMF k are obtained, let Δ be the absolute value of the difference between adjacent IMF components(2)
Set the number of decomposition layersk=k+1 and repeat step (1), until the absolute value of Δ, the difference between adjacent IMF components of xAη after VMD decomposition, is less than 200 (as demonstrated by a large number of experimental results; when Δ<200, the leakage signal can get the best decomposition effect); then, the number of decomposition layers kA at this time is judged to be the best, and the best decomposition layers kB can be obtained by the same processing of xBη(3)
AfterxAη and xBη are decomposed by VMD, residual components are removed from the IMF components, and Hilbert transform is performed to obtain Hilbert edge spectrum by using formula (9) and formula (14)–(17). The energy amplitude of each IMF component is obtained through the marginal spectrum. The components whose energy amplitude is greater than half of the maximum energy amplitude are selected for reconstruction (half of the maximum energy amplitude can be regarded as the basis for identifying the main components of the signal), and the coefficient h obtained after the normalization of the cross-correlation function of the reconstructed signal is calculated. The high-energy noise signal is discriminated by the value of h(4)
The decomposed IMF components are reconstructed into low-frequency (0-1000 Hz) and high-frequency (1000-2000 Hz) components. When the coefficienth1 obtained by the normalized cross-correlation function of the low-frequency reconstructed signal is larger than the coefficient h2 obtained by the normalized cross-correlation function of the high-frequency reconstructed signal, it indicates that the high-energy noise signal exists in the high-frequency band. Therefore, the low-frequency IMF component with less noise is reconstructed to obtain the reconstructed signals xA′η and xB′η(5)
The reconstructed signalsxA′η and xB′η are low-pass filtered through Chebyshev filter. Firstly, the passband is set to n characteristic frequency bands, and the difference between the left and right boundaries of each passband is 100 Hz. The signals xAnη and xBnη of N characteristic frequency bands are obtained after filtering, and the coefficient Rn is obtained after the cross-correlation function normalization of xAnη and xBnη is calculated(6)
Select the groupRmax with the largest Rn, the corresponding passband is the characteristic frequency band of the leakage signal, output denoising signal xAmaxη and xBmaxη, Rmax is the correlation coefficient of the two denoising signalsFigure 3
Noise reduction process of the VMD Hilbert transform algorithm.After noise reduction by using the VMD Hilbert algorithm, we can use formula (4) to calculate Δt and then use formula (1) to calculate the leak point location XA.


## 2.2.1. VMD

In the VMD method, the calculation process of the optimal solution of the variational model can be expressed as [25]
(5)minuk,ωk∑k∂tδt+jπt∗ukte−jωkt22s.t.∑kukt=xt,whereuk=u1,⋯,uk is the k IMF components of input signal xt after VMD decomposition, and ωk=ω1,⋯,ωk is the center frequency of the k component.The penalty factorα and the augmented Lagrange function are introduced to obtain the expression of the optimal solution of the variational model:
(6)Luk,ωk,λ=α∑k∂tδt+jπt∗ukte−jωkt22+xt−∑kukt22+λt,xt−∑kukt,whereα refers to the penalty factor and λ refers to the Lagrange multiplier.In the formula,uk and ωk are obtained by iterating the values of uk, ωk, and λ.


## 2.2.2. Hilbert Transform

The output response of a continuous time signal through a linear system with impulse responseht=1/πt is equivalent to the Hilbert transform. Therefore, the Hilbert transform expression of input signal xt is obtained:
(7)1π∫−∞+∞xtt‐τdτ=1π∫−∞+∞xt−ττdτ=xt∗1πt.After VMD decomposition, the expression of input signalxt is obtained:
(8)xt=∑i=1kukt+rkt,whereukt is the IMF component of xt after VMD decomposition, and rkt is the residual component.We apply Hilbert transform to each IMF component:(9)Hukt=1π∫−∞+∞uktt−τdτ=1π∫−∞+∞ukt−ττdτ=ukt∗1πt.We construct analytic signal:(10)zkt=ukt+jHukt=aktejθkt.Among them,(11)akt=uk2t+H2ukt,θkt=arctanHuktukt.The corresponding instantaneous frequency is(12)fkt=12πdθktdt.The analytical expression ofxt is obtained:
(13)xt=∑i=1kaktejθkt.According to Equations (10) and (12),
(14)ukt=Reaktej2π∫fktdt,whereRe denotes the real part. When akt is represented in the coordinate plane, the Hilbert time spectrum of IMF component ukt can be obtained:
(15)Hkt,f=akt,f=fkt,0,f≠fkt.The overall Hilbert time spectrum ofxt is analyzed, and the Hilbert spectrum is obtained:
(16)Ht,f=∑k=1nHkt,f.The Hilbert spectrum describes how the amplitude of a signal varies with time and frequency over the total frequency range.Further, we define the Hilbert marginal spectrum by the Hilbert spectrum:(17)hf=∫0THkt,f,whereT is the signal sampling time. The Hilbert marginal spectrum reflects the statistical situation of the amplitude of the signal in the whole frequency segment and also describes the actual frequency component of the signal.Considering that Fourier transform is the most commonly used signal processing method, the differences between Fourier transform and Hilbert transform are researched.If the input signalxt by using Fourier transform, it can be expressed as
(18)xt=∑i=1∞akejωkt,whereak and ωk are constants. In formula (13), akt and θkt are variables. Therefore, compared with Fourier transform, Hilbert transform can be applied to nonstationary signal processing. At the same time, the frequency and amplitude can be separated by Hilbert transform to realize the variable frequency and amplitude description of the signal.In order to more intuitively understand the difference between the Fourier spectrum and Hilbert spectrum, we choose to use a set of data collected by sensor A for analysis, and the results are shown in Figure2. Figure 2(a) is the signal collected by sensor A, Figure 2(b) is the Fourier spectrum of IMF1, Figure 2(c) is the Fourier spectrum of IMF2, and Figure 2(d) is the marginal Hilbert spectra.Figure 2
Frequency spectrum analysis.
(a)
Signal collected by sensor A(b)
Fourier spectrum of IMF1(c)
Fourier spectrum of IMF2(d)
Marginal Hilbert spectraIt can be seen from Figures2(b) and 2(c) that when Fourier spectrum is used to analyze signals, all IMF components cannot be drawn on a single diagram and can only be analyzed separately. It can be seen from Figure 2(d) that all IMF components can be plotted on a single graph using edge Hilbert spectrum, which improves efficiency. At the same time, we can also observe the amplitude of all IMF components on a single graph and more effectively select “half of the maximum energy amplitude.”


## 2.2.3. VMD-Hilbert Transform Algorithm

When there is high-energy and high-frequency band noise in the signal and the low-frequency band noise signal is similar to the leakage signal frequency band, in order to achieve good denoising effect, a novel method named VMD-Hilbert transform algorithm is proposed in this paper.Figure3 gives the noise reduction process of the VMD Hilbert transform algorithm, and the detailed steps are as follows:
(1)
The leakage signalxAη collected by sensor A is decomposed by VMD, and the initial decomposition layer k is set as 3 (after experimental verification, the residual component of the leakage signal decomposed is background noise, and the center frequency is expressed as 0. Therefore, the initial decomposition level k=3 is appropriate), the penalty factor α=2000, and the accumulation tolerance c=1e−6 (Penalty coefficient α determines the bandwidth of the IMF. The smaller the α value, the greater the IMF bandwidth. Excessive bandwidth will cause some signals to mix together; the higher the α value is, the smaller the IMF bandwidth is. If the bandwidth is too small, some signals will be lost. The common value range of α is 1000~3000. c is one of the stop criteria for VMD decomposition. Generally, 1e−6 ~ 5e−6 can be adopted.) After decomposition, IMF1, IMF2, ⋯, IMF k are obtained, let Δ be the absolute value of the difference between adjacent IMF components(2)
Set the number of decomposition layersk=k+1 and repeat step (1), until the absolute value of Δ, the difference between adjacent IMF components of xAη after VMD decomposition, is less than 200 (as demonstrated by a large number of experimental results; when Δ<200, the leakage signal can get the best decomposition effect); then, the number of decomposition layers kA at this time is judged to be the best, and the best decomposition layers kB can be obtained by the same processing of xBη(3)
AfterxAη and xBη are decomposed by VMD, residual components are removed from the IMF components, and Hilbert transform is performed to obtain Hilbert edge spectrum by using formula (9) and formula (14)–(17). The energy amplitude of each IMF component is obtained through the marginal spectrum. The components whose energy amplitude is greater than half of the maximum energy amplitude are selected for reconstruction (half of the maximum energy amplitude can be regarded as the basis for identifying the main components of the signal), and the coefficient h obtained after the normalization of the cross-correlation function of the reconstructed signal is calculated. The high-energy noise signal is discriminated by the value of h(4)
The decomposed IMF components are reconstructed into low-frequency (0-1000 Hz) and high-frequency (1000-2000 Hz) components. When the coefficienth1 obtained by the normalized cross-correlation function of the low-frequency reconstructed signal is larger than the coefficient h2 obtained by the normalized cross-correlation function of the high-frequency reconstructed signal, it indicates that the high-energy noise signal exists in the high-frequency band. Therefore, the low-frequency IMF component with less noise is reconstructed to obtain the reconstructed signals xA′η and xB′η(5)
The reconstructed signalsxA′η and xB′η are low-pass filtered through Chebyshev filter. Firstly, the passband is set to n characteristic frequency bands, and the difference between the left and right boundaries of each passband is 100 Hz. The signals xAnη and xBnη of N characteristic frequency bands are obtained after filtering, and the coefficient Rn is obtained after the cross-correlation function normalization of xAnη and xBnη is calculated(6)
Select the groupRmax with the largest Rn, the corresponding passband is the characteristic frequency band of the leakage signal, output denoising signal xAmaxη and xBmaxη, Rmax is the correlation coefficient of the two denoising signalsFigure 3
Noise reduction process of the VMD Hilbert transform algorithm.After noise reduction by using the VMD Hilbert algorithm, we can use formula (4) to calculate Δt and then use formula (1) to calculate the leak point location XA.


## 3. Simulation Experiment

In order to verify the performance of VMD-Hilbert transform algorithm, Gaussian signalsAt with points N=4000 is generated by Matlab function, and the sampling frequency FS=4000 is set. A passband filter of 500~600 Hz is applied to sAt to simulate the sound signal of water supply pipeline leakage. sBt is obtained by delaying 100 sampling points (D=100) of sAt. White noise with SNR of -5 dB is added to sAt and sBt in the range of 700~900 Hz to simulate the noise of low-energy low-frequency band. At the same time, white noise with SNR of -5db is added to sAt and sBt in the range of 1100~1800 Hz to simulate the noise of high-energy and high-frequency band, and the final input signals xAt and xBt are obtained. Then, the time delay of the source signal is D/FS=0.025s.


### 3.1. Cross-Correlation Algorithm

In order to analyze the interference of noise on the simulated leakage signal, a group of leakage data whose leakage point is closer to sensor B is simulated, and the cross-correlation algorithm is used for the simulated leakage data, as shown in Figure4. Figure 4(a) is the analog signal of sensor A, Figure 4(b) is the analog signal of sensor B, and Figure 4(c) is the correlation coefficient of two analog signals.Figure 4
Flow of cross-correlation algorithm.
(a)
Analog signal of sensor A(b)
Analog signal of sensor B(c)
Correlation coefficient of two analog signalsThe correlation coefficient table is shown in Table1.Table 1
Correlation coefficient table.
Correlation coefficientDegree of correlation0 ~ 0.30Weak correlation0.30~0.50Low correlation0.50~0.80Significant correlation0.80~1.00Strong correlationAccording to Figure4(c), the correlation coefficient of the simulated leakage signal is about 0.10, showing a weak correlation, the time difference Δt′=0.1365, the set time difference Δt=0.025s, and the relative positioning error δ=0.1365−0.025/0.025=446%. Since the correlation coefficient is less than 0.3, it is weak correlated, so there is a lot of noise in the leakage signal, which seriously affects the positioning accuracy.


### 3.2. VMD-Correlation Coefficient Algorithm

The VMD-correlation coefficient algorithm is used to denoise the simulated leakage signal. Starting fromk=3, the Δ after VMD decomposition is calculated until Δ<200, then the center frequency decomposition is appropriate. The center frequencies of each IMF component are shown in Tables 2 and 3, and kA=7 and kB=7 are obtained from Tables 2 and 3. Components with correlation coefficients greater than half of the maximum correlation coefficients are selected for reconstruction. IMF3, IMF4, IMF5, IMF6, and IMF7 are selected for signal reconstruction of sensor A. IMF3, IMF4, IMF5, IMF6, and IMF7 are selected for signal reconstruction of sensor B, and correlation coefficients are shown in Tables 4 and 5. The correlation coefficient of VMD-correlation coefficient algorithm is shown in Figure 5.Table 2
Center frequency of each IMF component ofxAη.
kCenter frequency (Hz)IMF1IMF2IMF3IMF4IMF5IMF6IMF73558.11394.90————4567.91322.01640.70———5548.31203.61441.11690.30——6551.7832.71173.41395.31664.90—7518.9592.2841.41200.11417.51650.30Table 3
Center frequency of each IMF component ofxBη.
kCenter frequency (Hz)IMF1IMF2IMF3IMF4IMF5IMF6IMF73556.71441.20————4549.01234.51636.30———5558.01193.41397.01665.50——6552.5846.11180.31400.01662.80—7521.7588.3851.01191.21473.11708.20Table 4
Correlation coefficients between each IMF components ofxAη and xAη.
KCorrelation coefficient7IMF1IMF2IMF3IMF4IMF5IMF6IMF70.21210.20690.54740.54200.59590.50850.5481Table 5
Correlation coefficients between each IMF components ofxBη and xBη.
KCorrelation coefficient7IMF1IMF2IMF3IMF4IMF5IMF6IMF70.20370.20340.50450.56820.58930.50670.5913Figure 5
Correlation coefficient of the VMD-correlation coefficient algorithm.According to Figure5, the correlation coefficient of simulated leakage signal is about 0.09, showing a weak correlation, time difference Δt′=0.1788s, set time difference Δt=0.025s, and relative positioning error δ=0.1788−0.025/0.025=615.2%. Since the correlation coefficient is less than 0.3, it is weak correlated. Therefore, VMD-correlation coefficient algorithm has poor noise reduction ability, and the leakage signal is still mixed with a lot of noise, which seriously interferes with the location of the leakage point.


### 3.3. VMD-Hilbert Transform Algorithm

We denoise the analog signal by using the VMD Hilbert transform algorithm in Section2.2.3, the VMD Hilbert transform algorithm process is shown in Figure 6. Figure 6(a) is the Hilbert marginal spectrum of sensor A, Figure 6(b) is the Hilbert marginal spectrum of sensor B, Figure 6(c) is the correlation coefficient h of reconstructed signal, Figure 6(d) is the correlation coefficient h1 of low-frequency reconstructed signal, Figure 6(e) is the correlation coefficient h2 of high-frequency reconstructed signal, Figure 6(f) is the denoised signal of sensor A, Figure 6(g) is the denoised signal of sensor B, and Figure 6(h) is the correlation coefficient of signal after denoising.Figure 6
Transformation process of the VMD Hilbert algorithm.
(a)
Hilbert marginal spectrum of sensor A(b)
Hilbert marginal spectrum of sensor B(c)
Correlation coefficienth of reconstructed signal(d)
Correlation coefficienth1 of low-frequency reconstructed signal(e)
Correlation coefficienth2 of high-frequency reconstructed signal(f)
Denoised signal of sensor A(g)
Denoised signal of sensor B(h)
Correlation coefficient of signal after denoisingThe VMD-Hilbert transform algorithm is used to denoise the simulated leakage signal, which is the same as the VMD decomposition method in Section3.2, and kA=7 and kB=7 are obtained. After Hilbert marginal spectrum analysis, components with amplitudes greater than half of the maximum amplitudes are selected for reconstruction. IMF4, IMF5, and IMF6 are selected for the signal reconstruction of sensor A. IMF3, IMF4, IMF5, and IMF6 are selected for the reconstruction of sensor B, and the correlation coefficient of reconstructed signals h=0.09<0.30, so there is high-energy noise in the signal. The correlation coefficient of reconstructed signals is h1=0.96 for the IMF component in the low-frequency band (0-1000 Hz); high-frequency band (1000 Hz-2000 Hz) IMF component reconstruction signal correlation coefficient ish2=0.09;h1>h2, so high-energy noise exists in the high-frequency band, the low-frequency band (0-1000 Hz) IMF component reconstruction is used. The reconstructed signal is low-pass filtered through Chebyshev filter, and 7 kinds of passbands are set, each with a bandwidth of 100 Hz, to obtain 7 different denoised signals. The correlation coefficient is calculated, and the group with the largest correlation coefficient is selected.According to Figure6(h), after denoising the simulated leakage signal by the VMD-Hilbert transform algorithm, the correlation coefficient of analog leakage signal is 0.98, showing significant correlation, time difference Δt′=0.025s, set time difference Δt=0.025s, relative positioning error δ=0.025−0.025/0.025=0%. Compared with the VMD-correlation coefficient algorithm, the correlation coefficient is improved from 0.08 to 0.99. Therefore, the VMD-Hilbert transform algorithm has strong noise reduction ability and can accurately locate the location of the leakage point.When the noise signal-to-noise ratio varies in the range of -20~10 dB, we study the noise reduction effect of these three algorithms. We find that VMD-Hilbert transform algorithm can always achieve the best denoising effect.In order to more intuitively understand the actual frequency range of analog signal before and after denoising, we obtain Figure7 by using fast Fourier transform (FFT). Figure 7(a) is the analog signal spectrum, and Figure 7(b) is the noise reduction signal spectrum. According to Figure 7(a), when the noise signal-to-noise ratio is -5 dB, it is most consistent with the real noise state. Therefore, we conduct a specific analysis of the noise with a signal-to-noise ratio of -5 dB. According to Figure 7(b), after VMD-Hilbert transform algorithm, high-frequency and high-energy noise in the leakage signal is removed, and low-energy and low-frequency noise is also removed. Therefore, the VMD-Hilbert transform algorithm has strong noise reduction ability.Figure 7
Analog signal spectrum diagram.
(a)
Analog signal spectrum(b)
Noise reduction signal spectrum


## 3.1. Cross-Correlation Algorithm

In order to analyze the interference of noise on the simulated leakage signal, a group of leakage data whose leakage point is closer to sensor B is simulated, and the cross-correlation algorithm is used for the simulated leakage data, as shown in Figure4. Figure 4(a) is the analog signal of sensor A, Figure 4(b) is the analog signal of sensor B, and Figure 4(c) is the correlation coefficient of two analog signals.Figure 4
Flow of cross-correlation algorithm.
(a)
Analog signal of sensor A(b)
Analog signal of sensor B(c)
Correlation coefficient of two analog signalsThe correlation coefficient table is shown in Table1.Table 1
Correlation coefficient table.
Correlation coefficientDegree of correlation0 ~ 0.30Weak correlation0.30~0.50Low correlation0.50~0.80Significant correlation0.80~1.00Strong correlationAccording to Figure4(c), the correlation coefficient of the simulated leakage signal is about 0.10, showing a weak correlation, the time difference Δt′=0.1365, the set time difference Δt=0.025s, and the relative positioning error δ=0.1365−0.025/0.025=446%. Since the correlation coefficient is less than 0.3, it is weak correlated, so there is a lot of noise in the leakage signal, which seriously affects the positioning accuracy.


## 3.2. VMD-Correlation Coefficient Algorithm

The VMD-correlation coefficient algorithm is used to denoise the simulated leakage signal. Starting fromk=3, the Δ after VMD decomposition is calculated until Δ<200, then the center frequency decomposition is appropriate. The center frequencies of each IMF component are shown in Tables 2 and 3, and kA=7 and kB=7 are obtained from Tables 2 and 3. Components with correlation coefficients greater than half of the maximum correlation coefficients are selected for reconstruction. IMF3, IMF4, IMF5, IMF6, and IMF7 are selected for signal reconstruction of sensor A. IMF3, IMF4, IMF5, IMF6, and IMF7 are selected for signal reconstruction of sensor B, and correlation coefficients are shown in Tables 4 and 5. The correlation coefficient of VMD-correlation coefficient algorithm is shown in Figure 5.Table 2
Center frequency of each IMF component ofxAη.
kCenter frequency (Hz)IMF1IMF2IMF3IMF4IMF5IMF6IMF73558.11394.90————4567.91322.01640.70———5548.31203.61441.11690.30——6551.7832.71173.41395.31664.90—7518.9592.2841.41200.11417.51650.30Table 3
Center frequency of each IMF component ofxBη.
kCenter frequency (Hz)IMF1IMF2IMF3IMF4IMF5IMF6IMF73556.71441.20————4549.01234.51636.30———5558.01193.41397.01665.50——6552.5846.11180.31400.01662.80—7521.7588.3851.01191.21473.11708.20Table 4
Correlation coefficients between each IMF components ofxAη and xAη.
KCorrelation coefficient7IMF1IMF2IMF3IMF4IMF5IMF6IMF70.21210.20690.54740.54200.59590.50850.5481Table 5
Correlation coefficients between each IMF components ofxBη and xBη.
KCorrelation coefficient7IMF1IMF2IMF3IMF4IMF5IMF6IMF70.20370.20340.50450.56820.58930.50670.5913Figure 5
Correlation coefficient of the VMD-correlation coefficient algorithm.According to Figure5, the correlation coefficient of simulated leakage signal is about 0.09, showing a weak correlation, time difference Δt′=0.1788s, set time difference Δt=0.025s, and relative positioning error δ=0.1788−0.025/0.025=615.2%. Since the correlation coefficient is less than 0.3, it is weak correlated. Therefore, VMD-correlation coefficient algorithm has poor noise reduction ability, and the leakage signal is still mixed with a lot of noise, which seriously interferes with the location of the leakage point.


## 3.3. VMD-Hilbert Transform Algorithm

We denoise the analog signal by using the VMD Hilbert transform algorithm in Section2.2.3, the VMD Hilbert transform algorithm process is shown in Figure 6. Figure 6(a) is the Hilbert marginal spectrum of sensor A, Figure 6(b) is the Hilbert marginal spectrum of sensor B, Figure 6(c) is the correlation coefficient h of reconstructed signal, Figure 6(d) is the correlation coefficient h1 of low-frequency reconstructed signal, Figure 6(e) is the correlation coefficient h2 of high-frequency reconstructed signal, Figure 6(f) is the denoised signal of sensor A, Figure 6(g) is the denoised signal of sensor B, and Figure 6(h) is the correlation coefficient of signal after denoising.Figure 6
Transformation process of the VMD Hilbert algorithm.
(a)
Hilbert marginal spectrum of sensor A(b)
Hilbert marginal spectrum of sensor B(c)
Correlation coefficienth of reconstructed signal(d)
Correlation coefficienth1 of low-frequency reconstructed signal(e)
Correlation coefficienth2 of high-frequency reconstructed signal(f)
Denoised signal of sensor A(g)
Denoised signal of sensor B(h)
Correlation coefficient of signal after denoisingThe VMD-Hilbert transform algorithm is used to denoise the simulated leakage signal, which is the same as the VMD decomposition method in Section3.2, and kA=7 and kB=7 are obtained. After Hilbert marginal spectrum analysis, components with amplitudes greater than half of the maximum amplitudes are selected for reconstruction. IMF4, IMF5, and IMF6 are selected for the signal reconstruction of sensor A. IMF3, IMF4, IMF5, and IMF6 are selected for the reconstruction of sensor B, and the correlation coefficient of reconstructed signals h=0.09<0.30, so there is high-energy noise in the signal. The correlation coefficient of reconstructed signals is h1=0.96 for the IMF component in the low-frequency band (0-1000 Hz); high-frequency band (1000 Hz-2000 Hz) IMF component reconstruction signal correlation coefficient ish2=0.09;h1>h2, so high-energy noise exists in the high-frequency band, the low-frequency band (0-1000 Hz) IMF component reconstruction is used. The reconstructed signal is low-pass filtered through Chebyshev filter, and 7 kinds of passbands are set, each with a bandwidth of 100 Hz, to obtain 7 different denoised signals. The correlation coefficient is calculated, and the group with the largest correlation coefficient is selected.According to Figure6(h), after denoising the simulated leakage signal by the VMD-Hilbert transform algorithm, the correlation coefficient of analog leakage signal is 0.98, showing significant correlation, time difference Δt′=0.025s, set time difference Δt=0.025s, relative positioning error δ=0.025−0.025/0.025=0%. Compared with the VMD-correlation coefficient algorithm, the correlation coefficient is improved from 0.08 to 0.99. Therefore, the VMD-Hilbert transform algorithm has strong noise reduction ability and can accurately locate the location of the leakage point.When the noise signal-to-noise ratio varies in the range of -20~10 dB, we study the noise reduction effect of these three algorithms. We find that VMD-Hilbert transform algorithm can always achieve the best denoising effect.In order to more intuitively understand the actual frequency range of analog signal before and after denoising, we obtain Figure7 by using fast Fourier transform (FFT). Figure 7(a) is the analog signal spectrum, and Figure 7(b) is the noise reduction signal spectrum. According to Figure 7(a), when the noise signal-to-noise ratio is -5 dB, it is most consistent with the real noise state. Therefore, we conduct a specific analysis of the noise with a signal-to-noise ratio of -5 dB. According to Figure 7(b), after VMD-Hilbert transform algorithm, high-frequency and high-energy noise in the leakage signal is removed, and low-energy and low-frequency noise is also removed. Therefore, the VMD-Hilbert transform algorithm has strong noise reduction ability.Figure 7
Analog signal spectrum diagram.
(a)
Analog signal spectrum(b)
Noise reduction signal spectrum


## 4. Experimental Environment and Leakage Point Arrangement

### 4.1. Experimental Environment

A water supply pipeline leak laboratory is created to test the noise reduction ability of the proposed the VMD-Hilbert leak location algorithm in this paper. The experimental pipeline is connected with the fire hydrant through the fire agricultural high-pressure water hose. The pipe is Q235B carbon steel, with a diameter of 80 mm and a wall thickness of 3 mm. The pressure in the tube is 0.26 MPa~0.28 MPa. According to the pipe size parameters and the experience table that the propagation speedV of the leakage vibration signal is 1278 m/s. Leakage acoustic signals are mainly concentrated in 0 ~ 2000 Hz, and the sampling frequency of the data acquisition card is set to 4000 Hz. The piezoelectric acceleration sensor is used in this experiment, and the sensor parameters are shown in Table 6.Table 6
Sensor parameters.
Model numberYK-YD01Range1 gOutput±5 VPower supply18~28VDCLateral sensitivity<5%Frequency response1 ~ 2.5 kHzVoltage sensitivity5.08 V/gDuring the experiment, the valve on the pipeline is opened to simulate the leakage, and the vibration signal generated by the leakage is picked up by the sensor. The voltage signal passing through the amplifier is collected by a data acquisition card. The computer supplies power to the data acquisition card and analyzes the voltage signal. Laboratory site arrangement is shown in Figure8.Figure 8
Laboratory platform.
(a)
Experimental pipeline(b)
Sensor A(c)
Leakage point 2(d)
Data acquisition equipment


### 4.2. Leakage Point Arrangement

Let the pipe length between sensors A and B beL, and the actual distance between the leak point and sensor A be LA. In this paper, leakage point 1 and leakage point 2 are used for leakage experiment. The specific arrangements are shown in Table 7.Table 7
Leakage point arrangement.
L (m)Leakage pointLA (m)16.95111.1628.28


## 4.1. Experimental Environment

A water supply pipeline leak laboratory is created to test the noise reduction ability of the proposed the VMD-Hilbert leak location algorithm in this paper. The experimental pipeline is connected with the fire hydrant through the fire agricultural high-pressure water hose. The pipe is Q235B carbon steel, with a diameter of 80 mm and a wall thickness of 3 mm. The pressure in the tube is 0.26 MPa~0.28 MPa. According to the pipe size parameters and the experience table that the propagation speedV of the leakage vibration signal is 1278 m/s. Leakage acoustic signals are mainly concentrated in 0 ~ 2000 Hz, and the sampling frequency of the data acquisition card is set to 4000 Hz. The piezoelectric acceleration sensor is used in this experiment, and the sensor parameters are shown in Table 6.Table 6
Sensor parameters.
Model numberYK-YD01Range1 gOutput±5 VPower supply18~28VDCLateral sensitivity<5%Frequency response1 ~ 2.5 kHzVoltage sensitivity5.08 V/gDuring the experiment, the valve on the pipeline is opened to simulate the leakage, and the vibration signal generated by the leakage is picked up by the sensor. The voltage signal passing through the amplifier is collected by a data acquisition card. The computer supplies power to the data acquisition card and analyzes the voltage signal. Laboratory site arrangement is shown in Figure8.Figure 8
Laboratory platform.
(a)
Experimental pipeline(b)
Sensor A(c)
Leakage point 2(d)
Data acquisition equipment


## 4.2. Leakage Point Arrangement

Let the pipe length between sensors A and B beL, and the actual distance between the leak point and sensor A be LA. In this paper, leakage point 1 and leakage point 2 are used for leakage experiment. The specific arrangements are shown in Table 7.Table 7
Leakage point arrangement.
L (m)Leakage pointLA (m)16.95111.1628.28


## 5. Experimental Result

### 5.1. Cross-Correlation Algorithm

In order to analyze the interference of noise on the real leakage signal, a group of data collected at leakage point 1 is analyzed, as shown in Figure9. Figure 9(a) is the signal collected by sensor A, Figure 9(b) is the signal collected by sensor B, and Figure 9(c) is the correlation coefficient of two signals.Figure 9
Cross-correlation algorithm flow at leakage point 1.
(a)
Signal collected by sensor A(b)
Signal collected by sensor B(c)
Correlation coefficient of two signalsAccording to Figure9(c), the real leakage signal correlation coefficient of the two leakage points is about 0.16, showing a weak correlation. The time difference Δt=0.017s and wave speed V=1278m/s can be substituted into formula (1) to obtain XA=19.34m and the relative positioning error δ=19.34−11.16/16.95=48.26%. Since the correlation coefficient is less than 0.3, it is weak correlated, so there is a lot of noise mixed in the leakage signal, which seriously interferes with the location of the leakage point.


### 5.2. VMD-Correlation Coefficient Algorithm

The leakage signal is denoised by the VMD-correlation coefficient algorithm, which is the same as the VMD decomposition method in Section3.2, and kA=4 and kB=5 are obtained. Components with correlation coefficient is greater than half of the maximum correlation coefficient are selected for reconstruction. IMF1, IMF2, IMF3, and IMF4 are selected for signal reconstruction of sensor A. IMF1, IMF2, IMF3, and IMF4 are selected for signal reconstruction of sensor B. Correlation coefficients are shown in Table 8 and Table 9. The VMD-correlation coefficient algorithm process is shown in Figure 10. Figure 10(a) is the denoised signal of sensor A, Figure 10(b) is the denoised signal of sensor B, and Figure 10(c) is the correlation coefficient of two denoised signals.Table 8
Correlation coefficients between each IMF components ofxAη and xAη.
KCorrelation coefficient4IMF1IMF2IMF3IMF40.54720.51050.43810.3049Table 9
Correlation coefficients between each IMF components ofxBη and xBη.
KCorrelation coefficient5IMF1IMF2IMF3IMF4IMF50.44510.44470.41470.51900.2518Figure 10
VMD-correlation coefficient algorithm flow.
(a)
Denoised signal of sensor A(b)
Denoised signal of sensor B(c)
Correlation coefficient of two denoised signalsCompare Figures10(a) and 10(b) to the original signal shown in Figures 9(a) and 9(b), the signal amplitude of sensor A decreases from -0.5 V ~0.5 V to -0.4 V ~0.4 V, and the signal amplitude of sensor B decreases from -1 V ~1 V to -0.5 V ~0.5 V, which means that the VMD-correlation coefficient algorithm has poor denoising effect.According to Figure10(c), the correlation coefficient of the real leakage signal at leakage point 1 is about 0.14, showing a slight correlation. The time difference Δt=0.016s and wave speed V=1278m/s are substituted into formula (1) to obtain XA=18.86m and the relative positioning error δ=18.86−11.16/16.95=45.42%. Since the correlation coefficient is less than 0.3, it is weak correlated. Therefore, the VMD-correlation coefficient algorithm has poor noise reduction ability, and the leakage signal is still mixed with a lot of noise, which seriously interferes with the location of the leakage point.


### 5.3. VMD-Hilbert Transform Algorithm

The leakage signal is denoised by the VMD-Hilbert transform algorithm. After calculatingkA=4, kB=5. After Hilbert marginal spectrum analysis, same as IMF selection criteria in Section 3.3, paragraph 2, components with amplitudes greater than half of the maximum amplitudes are selected for reconstruction, the signal of sensor A is reconstructed by IMF1 and IMF2, and the signal of sensor B is reconstructed by IMF1, IMF2, and IMF4. The correlation number of reconstructed signals is 0.17<0.30, so there is high-energy noise in the signal. The correlation number of reconstructed signals is h1=0.20 in the low-frequency band (0-1000 Hz) IMF component. High-frequency band (1000 Hz-2000 Hz) IMF component reconstruction signal correlation number is h2=0.0192, h1>h2, so high-frequency band exist in high-energy noise, the low-frequency band (0-1000 Hz) IMF component reconstruction. The reconstructed signal is low-pass filtered through Chebyshev filter, and 7 kinds of passbands are set, each with a bandwidth of 100 Hz, to obtain 7 different denoised signals. The correlation coefficient is calculated, and the group with the largest correlation coefficient is selected, as shown in Figure 11. Figure 11(a) is the denoised signal of sensor A, Figure 11(b) is the denoised signal of sensor B, and Figure 11(c) is the correlation coefficient of two denoised signals.Figure 11
VMD-Hilbert algorithm flow.
(a)
Denoised signal of sensor A(b)
Denoised signal of sensor B(c)
Correlation coefficient of two denoised signalsCompare Figures11(a) and 11(b) to the original signal shown in Figures 9(a) and 9(b), the signal amplitude of sensor A decreases from -0.5 V ~0.5 V to -0.02 V ~0.02 V, and the signal amplitude of sensor B decreases from -1 V ~1 V to -0.1 V ~0.1 V, which means that the method proposed in this paper has a good denoising effect. According to Figure 11(c), the correlation coefficient of the real leakage signal is about 0.5882 after noise reduction using the VMD-Hilbert algorithm, showing significant correlation. The time difference Δt=0.005s and wave speed V=1278m/s are substituted into formula (1) to obtain XA=11.67m and the relative positioning error δ=3.01%.Compared with Figures9(c) and 10(c), the correlation coefficient increases from 0.1577 and 0.1368 to 0.5882. Thus, compared with the cross-correlation algorithm and VMD-correlation coefficient algorithm, the VMD-Hilbert method has stronger noise reduction ability and better localization effect.In order to more intuitively understand the actual frequency range of leakage signal before and after denoising, we obtain Figure12 by using fast Fourier transform (FFT). Figure 12(a) is the leakage signal spectrum, and Figure 12(b) is the noise reduction signal spectrum.Figure 12
Leakage signal spectrum diagram.
(a)
Leakage signal spectrum(b)
Noise reduction signal spectrumAccording to Figure12(a), the collected leakage signal frequency is about 0 ~ 2000 Hz, and the amplitude is about 0 ~ 0.04 V. According to Figure 12(b), after the VMD-Hilbert transform algorithm, the leakage signal frequency is around 500 Hz, the amplitude is around 0 ~ 0.002 V, and most of the high-energy noise of different frequency bands in the signal is removed. Therefore, the VMD-Hilbert transform algorithm has a strong noise reduction ability.


### 5.4. Localization Results of Three Algorithms

After using three different algorithms, the positioning effect is shown in Table10.Table 10
Localization results of different algorithms.
AlgorithmLA (m)XA (m)δ (%)Cross-correlation algorithm11.1619.3648.26VMD-correlation coefficient algorithm18.8645.45VMD-Hilbert algorithm11.673.01As can be seen from Table10, direct positioning of the original signal has a poor effect. Using the VMD-correlation coefficient algorithm, due to the noise interference in the leakage signal, the positioning effect is poor. The VMD-Hilbert transform algorithm is used to achieve a good localization effect.


### 5.5. Experimental Results

In order to test the noise reduction ability of VMD-Hilbert transform algorithm, a large number of experiments have been carried out, and the results are listed in Table11. There are 10 sets of data for leak point 1 and leak point 2. LA represents the real distance from the leak point to sensor A. XA1 represents the location of leakage by the cross-correlation algorithm, and δ1 represents the relative error of location. XA2 represents the leak location after using the VMD-correlation coefficient algorithm, and δ2 represents the relative error of location. XA3 represents the leak location after using the VMD-Hilbert transform algorithm, and δ3 represents the relative error of location.Table 11
Localization results of different algorithms.
L (m)Leakage pointLA (m)XA1 (m)δ1 (%)XA2 (m)δ2 (%)XA3 (m)δ3 (%)16.95111.164.1641.2930.52114.2211.833.9529.40107.6215.6626.579.599.2418.5443.5418.5443.5410.076.4215.3424.698.7913.9611.190.184.9636.584.9636.589.599.244.8037.52-3.5186.539.599.240.9760.140.0165.7910.076.4219.3448.2513.7515.2611.673.0115.8227.5116.1429.4011.673.013.8443.175.6032.8111.190.1828.2810.2311.5210.2311.529.114.925.7614.87-6.3886.507.047.33-1.5958.23-1.9160.116.729.22-2.5563.88-4.1573.318.632.09-2.2362.00-2.5563.889.114.926.4011.105.7614.877.047.335.9213.935.6015.818.160.7311.6720.000.9743.158.160.735.6015.812.8831.847.206.395.7614.877.186.397.683.56Cross-correlation analysis is easy to be disturbed by noise in localization and has poor adaptability. The minimum relative error of the cross-correlation analysis method is 11.10%, shown in Table11. When the noise is very large, this method even locates outside the pipeline, and the positioning result is very unstable. The VMD-correlation coefficient algorithm is also greatly affected by noise. In Table 11, when the noise is small, the minimum relative error is 6.39%. When the noise is large, the relative error is 114.22%, which is also located outside the pipeline, the positioning result is unstable, and the anti-interference ability is poor. The VMD-Hilbert transform algorithm proposed in this paper has strong noise reduction ability and anti-interference ability. In Table 11, the best result of the VMD-Hilbert transform algorithm for leak location is 0.18%, and the worst result is 9.24%. According to the location results of the three algorithms, it is obvious that the VMD-Hilbert transform algorithm has stronger noise reduction performance, smaller leakage location error, and more stable location results.


## 5.1. Cross-Correlation Algorithm

In order to analyze the interference of noise on the real leakage signal, a group of data collected at leakage point 1 is analyzed, as shown in Figure9. Figure 9(a) is the signal collected by sensor A, Figure 9(b) is the signal collected by sensor B, and Figure 9(c) is the correlation coefficient of two signals.Figure 9
Cross-correlation algorithm flow at leakage point 1.
(a)
Signal collected by sensor A(b)
Signal collected by sensor B(c)
Correlation coefficient of two signalsAccording to Figure9(c), the real leakage signal correlation coefficient of the two leakage points is about 0.16, showing a weak correlation. The time difference Δt=0.017s and wave speed V=1278m/s can be substituted into formula (1) to obtain XA=19.34m and the relative positioning error δ=19.34−11.16/16.95=48.26%. Since the correlation coefficient is less than 0.3, it is weak correlated, so there is a lot of noise mixed in the leakage signal, which seriously interferes with the location of the leakage point.


## 5.2. VMD-Correlation Coefficient Algorithm

The leakage signal is denoised by the VMD-correlation coefficient algorithm, which is the same as the VMD decomposition method in Section3.2, and kA=4 and kB=5 are obtained. Components with correlation coefficient is greater than half of the maximum correlation coefficient are selected for reconstruction. IMF1, IMF2, IMF3, and IMF4 are selected for signal reconstruction of sensor A. IMF1, IMF2, IMF3, and IMF4 are selected for signal reconstruction of sensor B. Correlation coefficients are shown in Table 8 and Table 9. The VMD-correlation coefficient algorithm process is shown in Figure 10. Figure 10(a) is the denoised signal of sensor A, Figure 10(b) is the denoised signal of sensor B, and Figure 10(c) is the correlation coefficient of two denoised signals.Table 8
Correlation coefficients between each IMF components ofxAη and xAη.
KCorrelation coefficient4IMF1IMF2IMF3IMF40.54720.51050.43810.3049Table 9
Correlation coefficients between each IMF components ofxBη and xBη.
KCorrelation coefficient5IMF1IMF2IMF3IMF4IMF50.44510.44470.41470.51900.2518Figure 10
VMD-correlation coefficient algorithm flow.
(a)
Denoised signal of sensor A(b)
Denoised signal of sensor B(c)
Correlation coefficient of two denoised signalsCompare Figures10(a) and 10(b) to the original signal shown in Figures 9(a) and 9(b), the signal amplitude of sensor A decreases from -0.5 V ~0.5 V to -0.4 V ~0.4 V, and the signal amplitude of sensor B decreases from -1 V ~1 V to -0.5 V ~0.5 V, which means that the VMD-correlation coefficient algorithm has poor denoising effect.According to Figure10(c), the correlation coefficient of the real leakage signal at leakage point 1 is about 0.14, showing a slight correlation. The time difference Δt=0.016s and wave speed V=1278m/s are substituted into formula (1) to obtain XA=18.86m and the relative positioning error δ=18.86−11.16/16.95=45.42%. Since the correlation coefficient is less than 0.3, it is weak correlated. Therefore, the VMD-correlation coefficient algorithm has poor noise reduction ability, and the leakage signal is still mixed with a lot of noise, which seriously interferes with the location of the leakage point.


## 5.3. VMD-Hilbert Transform Algorithm

The leakage signal is denoised by the VMD-Hilbert transform algorithm. After calculatingkA=4, kB=5. After Hilbert marginal spectrum analysis, same as IMF selection criteria in Section 3.3, paragraph 2, components with amplitudes greater than half of the maximum amplitudes are selected for reconstruction, the signal of sensor A is reconstructed by IMF1 and IMF2, and the signal of sensor B is reconstructed by IMF1, IMF2, and IMF4. The correlation number of reconstructed signals is 0.17<0.30, so there is high-energy noise in the signal. The correlation number of reconstructed signals is h1=0.20 in the low-frequency band (0-1000 Hz) IMF component. High-frequency band (1000 Hz-2000 Hz) IMF component reconstruction signal correlation number is h2=0.0192, h1>h2, so high-frequency band exist in high-energy noise, the low-frequency band (0-1000 Hz) IMF component reconstruction. The reconstructed signal is low-pass filtered through Chebyshev filter, and 7 kinds of passbands are set, each with a bandwidth of 100 Hz, to obtain 7 different denoised signals. The correlation coefficient is calculated, and the group with the largest correlation coefficient is selected, as shown in Figure 11. Figure 11(a) is the denoised signal of sensor A, Figure 11(b) is the denoised signal of sensor B, and Figure 11(c) is the correlation coefficient of two denoised signals.Figure 11
VMD-Hilbert algorithm flow.
(a)
Denoised signal of sensor A(b)
Denoised signal of sensor B(c)
Correlation coefficient of two denoised signalsCompare Figures11(a) and 11(b) to the original signal shown in Figures 9(a) and 9(b), the signal amplitude of sensor A decreases from -0.5 V ~0.5 V to -0.02 V ~0.02 V, and the signal amplitude of sensor B decreases from -1 V ~1 V to -0.1 V ~0.1 V, which means that the method proposed in this paper has a good denoising effect. According to Figure 11(c), the correlation coefficient of the real leakage signal is about 0.5882 after noise reduction using the VMD-Hilbert algorithm, showing significant correlation. The time difference Δt=0.005s and wave speed V=1278m/s are substituted into formula (1) to obtain XA=11.67m and the relative positioning error δ=3.01%.Compared with Figures9(c) and 10(c), the correlation coefficient increases from 0.1577 and 0.1368 to 0.5882. Thus, compared with the cross-correlation algorithm and VMD-correlation coefficient algorithm, the VMD-Hilbert method has stronger noise reduction ability and better localization effect.In order to more intuitively understand the actual frequency range of leakage signal before and after denoising, we obtain Figure12 by using fast Fourier transform (FFT). Figure 12(a) is the leakage signal spectrum, and Figure 12(b) is the noise reduction signal spectrum.Figure 12
Leakage signal spectrum diagram.
(a)
Leakage signal spectrum(b)
Noise reduction signal spectrumAccording to Figure12(a), the collected leakage signal frequency is about 0 ~ 2000 Hz, and the amplitude is about 0 ~ 0.04 V. According to Figure 12(b), after the VMD-Hilbert transform algorithm, the leakage signal frequency is around 500 Hz, the amplitude is around 0 ~ 0.002 V, and most of the high-energy noise of different frequency bands in the signal is removed. Therefore, the VMD-Hilbert transform algorithm has a strong noise reduction ability.


## 5.4. Localization Results of Three Algorithms

After using three different algorithms, the positioning effect is shown in Table10.Table 10
Localization results of different algorithms.
AlgorithmLA (m)XA (m)δ (%)Cross-correlation algorithm11.1619.3648.26VMD-correlation coefficient algorithm18.8645.45VMD-Hilbert algorithm11.673.01As can be seen from Table10, direct positioning of the original signal has a poor effect. Using the VMD-correlation coefficient algorithm, due to the noise interference in the leakage signal, the positioning effect is poor. The VMD-Hilbert transform algorithm is used to achieve a good localization effect.


## 5.5. Experimental Results

In order to test the noise reduction ability of VMD-Hilbert transform algorithm, a large number of experiments have been carried out, and the results are listed in Table11. There are 10 sets of data for leak point 1 and leak point 2. LA represents the real distance from the leak point to sensor A. XA1 represents the location of leakage by the cross-correlation algorithm, and δ1 represents the relative error of location. XA2 represents the leak location after using the VMD-correlation coefficient algorithm, and δ2 represents the relative error of location. XA3 represents the leak location after using the VMD-Hilbert transform algorithm, and δ3 represents the relative error of location.Table 11
Localization results of different algorithms.
L (m)Leakage pointLA (m)XA1 (m)δ1 (%)XA2 (m)δ2 (%)XA3 (m)δ3 (%)16.95111.164.1641.2930.52114.2211.833.9529.40107.6215.6626.579.599.2418.5443.5418.5443.5410.076.4215.3424.698.7913.9611.190.184.9636.584.9636.589.599.244.8037.52-3.5186.539.599.240.9760.140.0165.7910.076.4219.3448.2513.7515.2611.673.0115.8227.5116.1429.4011.673.013.8443.175.6032.8111.190.1828.2810.2311.5210.2311.529.114.925.7614.87-6.3886.507.047.33-1.5958.23-1.9160.116.729.22-2.5563.88-4.1573.318.632.09-2.2362.00-2.5563.889.114.926.4011.105.7614.877.047.335.9213.935.6015.818.160.7311.6720.000.9743.158.160.735.6015.812.8831.847.206.395.7614.877.186.397.683.56Cross-correlation analysis is easy to be disturbed by noise in localization and has poor adaptability. The minimum relative error of the cross-correlation analysis method is 11.10%, shown in Table11. When the noise is very large, this method even locates outside the pipeline, and the positioning result is very unstable. The VMD-correlation coefficient algorithm is also greatly affected by noise. In Table 11, when the noise is small, the minimum relative error is 6.39%. When the noise is large, the relative error is 114.22%, which is also located outside the pipeline, the positioning result is unstable, and the anti-interference ability is poor. The VMD-Hilbert transform algorithm proposed in this paper has strong noise reduction ability and anti-interference ability. In Table 11, the best result of the VMD-Hilbert transform algorithm for leak location is 0.18%, and the worst result is 9.24%. According to the location results of the three algorithms, it is obvious that the VMD-Hilbert transform algorithm has stronger noise reduction performance, smaller leakage location error, and more stable location results.


## 6. Conclusion

The application effect of the signal denoising algorithm based on VMD-Hilbert transform in the water supply pipeline leakage location is studied. The main results and findings are summarized below:(1)
The leakage signal can be decomposed by VMD, and the energy distribution of the signal can be accurately obtained by Hilbert transform. The frequency band of high-energy noise can be accurately judged by combining the cross-correlation number. Therefore, the signal denoising algorithm of VMD-Hilbert transform can effectively eliminate the high-energy noise. At the same time, the filter is added to the algorithm, which also has a certain effect on the removal of the rest of the low-energy noise, making the overall denoising effect better(2)
The VMD-correlation coefficient algorithm is vulnerable to external interference. When the noise energy is large, the noise is often mistaken for the real signal. The algorithm has poor anti-interference ability and cannot effectively remove the noise. To solve this problem, a novel method named the VMD-Hilbert transform algorithm is proposed in this paper. Simulation and experimental results show that the VMD-Hilbert transform algorithm has stronger anti-interference ability to noise. In addition, this method has a strong ability to reduce noise of pipeline vibration signals, which can effectively denoise and reduce the overall amplitude of the signal, so as to get a better effect in the cross-correlation analysis, amplify the characteristics of the signal, get a better delay estimation, and reduce the positioning error(3)
Two leakage points are determined by the method proposed in this paper. And the experimental results show that the proposed method has strong anti-interference ability and lower relative error. The best positioning result is only 0.03 m away from the leakage point, while the worst positioning result is 2.02 m away from the leakage point. So this method has great application prospect in acoustic signal-based leakage location technology(4)
The algorithm proposed in this paper is applicable to the conditions of high-energy and high-frequency band noise. In the actual process, there may be a special case that the noise in the high-energy and low-frequency band is mixed with the leakage signal, which will seriously affect the denoising effect and positioning accuracy. Therefore, we can study the denoising algorithm in this special case in the future. At the same time, due to the attenuation characteristics of acoustic signal propagation, it cannot be transmitted too far, which can be further studied in the future



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*Source: 1939606-2023-02-28.xml*