A Small Leak Detection Method Based on VMD Adaptive De-Noising and Ambiguity Correlation Classification Intended for Natural Gas Pipelines

In this study, a small leak detection method based on variational mode decomposition (VMD) and ambiguity correlation classification (ACC) is proposed. The signals acquired from sensors were decomposed using the VMD, and numerous components were obtained. According to the probability density function (PDF), an adaptive de-noising algorithm based on VMD is proposed for noise component processing and de-noised components reconstruction. Furthermore, the ambiguity function image was employed for analysis of the reconstructed signals. Based on the correlation coefficient, ACC is proposed to detect the small leak of pipeline. The analysis of pipeline leakage signals, using 1 mm and 2 mm leaks, has shown that proposed detection method can detect a small leak accurately and effectively. Moreover, the experimental results have shown that the proposed method achieved better performances than support vector machine (SVM) and back propagation neural network (BP) methods.


Introduction
Due to increasing demands for natural gas, gas pipeline construction has developed rapidly. However, for a variety of natural and artificial reasons small leakage accidents happen occasionally in pipelines, which cause tremendous economic losses and casualties [1,2]. In order to provide for the safe operation of pipelines, small leak detection in pipelines should be studied [3,4]. Nowadays, pipeline leak detectors mainly determine the occurrence of leaking accidents based on delivery pressure and flow rate changes, which are influenced by various factors, such as transporting materials and transporting conditions [5,6]. When a leak occurs, there is an air-structure coupling between the high-velocity escaping gas and the tube wall surrounding the leak location, thereby causing stress waves that propagate along the body of the pipe on both sides of the leak hole. This phenomenon is known as stress wave emission and is considered to be a general acoustic emission. The acoustic emission (AE) technique realizes dynamic non-destruction testing according to AE source characteristics (e.g., defect type, size and position), which has advantages of high sensitivity and all-weather real-time detection [7][8][9]. Therefore, in this paper the AE technique was used for small leak detection in natural gas pipelines.
The complicated surrounding environment, big noise disturbances and non-stationary characteristics of collected signals in small pipeline leaks, require a signal de-noising method in through construction and solution of variational problems [17]. The VMD algorithm transfers the signal decomposition process to the variational framework. Hence, the VMD decomposition process is an optimal-solution processing for a constrained variational process. For constrained-variation problems, the augmented Lagrange function is introduced to transform the constrained variation problem into an unconstrained variation problem.
In order to obtain an optimal solution, the alternate direction method of multipliers (ADMM) is used to calculate the saddle point of the augmented Lagrange function, which is the optimal solution of the constrained-variation equation [18]. The saddle-point problem is solved by the alternate renewal of components and Lagrange multiplier, i.e., by solving the optimal-solution problem of the variational problem.
The VMD algorithm continuously updates the modes in the frequency domain, and finally transforms them to the time domain through the Fourier inversion transformation. The specific algorithm is as follows: (1) Initialize û 1 k , ω 1 k , λ 1 , and n = 0. λ is the Lagrange multiplier, {u k } = {u 1 , . . . , u k } represents K components set after decomposition, and {ω k } = {ω 1 , . . . , ω k } represents the center frequencies set of components after decomposition.
where τ is the update parameter of the Lagrangian multiplier.
2 ≤ ε , end the whole loop, output the k modal components u k , otherwise, repeat steps from Equation (2) to Equation (4).

VMD-Based Noise Adaptive Removal Approach
Signals collected by sensors often contain noise interference, which weakens the leakage characteristics. In order to extract the characteristic information accurately, components after VMD have to be de-noised, which increases the signal-to-noise ratio (SNR). Currently, most studies divide components after VMD into noise components and noiseless components. They eliminate noise components and reconstruct noiseless components to realize a de-noising process [24]. Sun et al. presented a multi-wavelet block threshold method to remove the noise components [25]. Kang et al. effectively eliminated high-level noise by exploiting a soft-threshold with adaptively estimated positive and negative noise levels from the time domain AE signal [14,26]. As it is well known, an amplitude is a signal characteristic, so noise information could be gained by amplitude analysis. Probability density function (PDF) means that the probability of some random vibration signal amplitude is within certain range [27]. Therefore, the PDF can be used to distinguish the noise signals. Hence, an adaptive de-noising method was proposed in this paper by combing VMD and PDF, which was used for signals processing and de-noising signals acquisition. Specific steps of the algorithm are: (1) For any data {x i , i = 1, . . . , n}, calculate its PDFf h (x): where k(•) presents the Gaussian kernels function and h is the bandwidth. To prevent overlarge of variance and deviation h is often determined 0.15 times of the prediction confidence interval of variable x. (2) Calculate PDFs of the original signal x(t) and component u k (t) after VMD: where p and Q are PDFs, and K is the number of decomposed components. (3) Calculate distance (L(k)) between two PDFs: (4) Choose noiseless component according to the distance: It can be concluded from Equation (9) that N th − 1 components are noiseless components and the rest of them are noise components.

Algorithm Simulation
In order to verify the algorithm effectiveness, a simulation signal is used. The simulation signal was composed of frequency-modulated and amplitude-modulated signals and random white noises. The corresponding mathematical expressions are: After VMD of the simulation signal, PDFs of decomposed components and the original signal were calculated. VMD results are shown in Figure 1 and the distance between two PDF is shown in Figure 2.
In Figure 2, the PDF of U 3 has the largest distance of the original signal. Therefore, previous two components were chosen as the noiseless components. According to Figure 1, two previous components are noiseless components and the third component is the noise component. The obtained results show that the VMD adaptive de-noising method can eliminate noise interference effectively and obtain noiseless signals.  In Figure 2, the PDF of U3 has the largest distance of the original signal. Therefore, previous two components were chosen as the noiseless components. According to Figure 1, two previous components are noiseless components and the third component is the noise component. The obtained results show that the VMD adaptive de-noising method can eliminate noise interference effectively and obtain noiseless signals.
Mean error square (MES) and SNR are often used to evaluate de-noising effect [28]. EMD and VMD were used for de-nosing process, and obtained results are listed in Table 1. A higher SNR and a smaller MES after de-noised signal indicate a better de-noising effect. As it can be seen in Table 1, EMD has poor de-noising effects due to mode mixing, while VMD achieves good de-noising effects since it can decompose non-stationary signals effectively and acquire the correct components.    In Figure 2, the PDF of U3 has the largest distance of the original signal. Therefore, previous two components were chosen as the noiseless components. According to Figure 1, two previous components are noiseless components and the third component is the noise component. The obtained results show that the VMD adaptive de-noising method can eliminate noise interference effectively and obtain noiseless signals.
Mean error square (MES) and SNR are often used to evaluate de-noising effect [28]. EMD and VMD were used for de-nosing process, and obtained results are listed in Table 1. A higher SNR and a smaller MES after de-noised signal indicate a better de-noising effect. As it can be seen in Table 1, EMD has poor de-noising effects due to mode mixing, while VMD achieves good de-noising effects since it can decompose non-stationary signals effectively and acquire the correct components.  Mean error square (MES) and SNR are often used to evaluate de-noising effect [28]. EMD and VMD were used for de-nosing process, and obtained results are listed in Table 1. A higher SNR and a smaller MES after de-noised signal indicate a better de-noising effect. As it can be seen in Table 1, EMD has poor de-noising effects due to mode mixing, while VMD achieves good de-noising effects since it can decompose non-stationary signals effectively and acquire the correct components.

Ambiguity Correlation Theory
As a non-stationary signal analysis method, the ambiguity function has been widely used due to its advantages in delay-frequency shift plane analysis [23]. Like Cohen's time-frequency distribution, the ambiguity function suffers from a serious cross terms interference in signal analysis and has a heavy computational load. Interference disturbs effective analysis and signal explanation, as well as component parameters extractions, thus causing difficult extraction of the time-frequency signal characteristics. Hence, there must be an internal relationship between eigenvalues of different signal types. Such an internal relationship varies significantly among different types. Correlation coefficients can be used as the internal relationship measure. Different absolute values of correlation coefficients indicate different correlation degrees between signal eigenvalues [29]. According to that, the ambiguity correlation method was proposed in order to avoid interference from cross terms and to reduce the computation load of a time-frequency analysis. Specific steps are as follows: (1) Calculate ambiguity function of signal x(t): where r x (t, τ) is the autocorrelation function of signals, A(τ, θ) is the ambiguity function of signal x(t).
(2) Calculate the correlation function of ambiguity function images of signals x(t) and y(t): In Equation (16), R xy (τ, θ) is the correlation function. (3) Calculate normalized correlation coefficient ρ xy (τ, θ): (4) Select correlation coefficient when τ = 0 or θ = 0: (5) Calculate the ambiguity correlation coefficient ρ: Providing the reliable theoretical basis and method to build the identification system based on the ambiguity correlation coefficient of leakage signals.

Basic Principle of Classifier
The ambiguity correlation classifier (ACC) represents a one-to-one classifier. ACC has advantages of low computational load and no parameter setting. Firstly, two types of signals were processed by the VMD-based adaptive de-noising method in order to get the de-noised signals. Ambiguity functions of de-noised signals A and B, as well as the test signal C, were calculated. The correlation coefficient, I, between the ambiguity function of A and the ambiguity function of C was calculated. Similarly, the correlation coefficient, II, between the ambiguity function of B and the ambiguity function of C was calculated. Afterwards, I and II were compared. In the case that I > II, then C is equal to A; otherwise, C is equal to B. If A is a de-noised signal of a 1 mm leak, B is de-the noised signal of the non-leak situation, and C is an unknown signal. Next they are input into ACC to identify, and C can be determined to be a 1 mm leakage signal or no leakage signal. If C is neither a 1 mm leakage signal nor a non-leak signal, then C continues to be classified as described above. The principle of the ambiguity correlation classifier is shown in Figure 3.

Basic Principle of Classifier
The ambiguity correlation classifier (ACC) represents a one-to-one classifier. ACC has advantages of low computational load and no parameter setting. Firstly, two types of signals were processed by the VMD-based adaptive de-noising method in order to get the de-noised signals. Ambiguity functions of de-noised signals A and B, as well as the test signal C, were calculated. The correlation coefficient, I, between the ambiguity function of A and the ambiguity function of C was calculated. Similarly, the correlation coefficient, II, between the ambiguity function of B and the ambiguity function of C was calculated. Afterwards, I and II were compared. In the case that I > II, then C is equal to A; otherwise, C is equal to B. If A is a de-noised signal of a 1 mm leak, B is de-the noised signal of the non-leak situation, and C is an unknown signal. Next they are input into ACC to identify, and C can be determined to be a 1 mm leakage signal or no leakage signal. If C is neither a 1 mm leakage signal nor a non-leak signal, then C continues to be classified as described above. The principle of the ambiguity correlation classifier is shown in Figure 3.

Small Leak Detection Method
The small leak detection method based on VMD and ACC can eliminate noise interference, extract characteristic components from weak and input reconstructed signals into the classifier in order to realize a small leak detection. The flowchart of this detection method is shown in Figure 4. Specific steps are as follows: (1) Collect experimental data by sensors and get series of U components of collected data from VMD. (2) Select noiseless components according to VMD adaptive de-noising method.
(3) Reconstruct chosen noiseless components and input using the ACC. (4) Collect several groups of data for training and testing, and realize the pipeline small leak detection.

Small Leak Detection Method
The small leak detection method based on VMD and ACC can eliminate noise interference, extract characteristic components from weak and input reconstructed signals into the classifier in order to realize a small leak detection. The flowchart of this detection method is shown in Figure 4. Specific steps are as follows: (1) Collect experimental data by sensors and get series of U components of collected data from VMD.
(3) Reconstruct chosen noiseless components and input using the ACC. (4) Collect several groups of data for training and testing, and realize the pipeline small leak detection.

Experimental Apparatus and Instrumentations
The experimental setup was composed of sensors, data acquisition cards, bearing pipelines and computer. The pipe was a seamless steel pipe with an outer diameter of 108 mm and wall thickness KHz to 10 KHz. The output was converted to a voltage signal by the charge amplifier. Experimental data were collected by a NI USB-6259 data acquisition card at a rate of 5 Ks/s. The experimental setup is shown in Figure 5. Using the experimental platform, 1 mm leak signals, 2 mm leak signals and the signals under normal conditions (non-leak) were collected, and they are presented in Figure 6.

Experimental Apparatus and Instrumentations
The experimental setup was composed of sensors, data acquisition cards, bearing pipelines and computer. The pipe was a seamless steel pipe with an outer diameter of 108 mm and wall thickness of 4.5 mm; and the leak testing was implemented based on 1 mm and 2 mm holes. Under the test pressure of 1.0-2.0 MPa, pipeline deflation was conducted at a flow rate of 3.1 m/s through internal decompression. The sensitivities of the YD-82 piezoelectric acceleration transducers used were 167.2 pC/ms 2 165.8 pC/ms 2 , 160.4 pC/ms 2 , 189 pC/ms 2 and the frequency response is from 0 KHz to 10 KHz. The output was converted to a voltage signal by the charge amplifier. Experimental data were collected by a NI USB-6259 data acquisition card at a rate of 5 Ks/s. The experimental setup is shown in Figure 5.

Experimental Apparatus and Instrumentations
The experimental setup was composed of sensors, data acquisition cards, bearing pipelines and computer. The pipe was a seamless steel pipe with an outer diameter of 108 mm and wall thickness KHz to 10 KHz. The output was converted to a voltage signal by the charge amplifier. Experimental data were collected by a NI USB-6259 data acquisition card at a rate of 5 Ks/s. The experimental setup is shown in Figure 5. Using the experimental platform, 1 mm leak signals, 2 mm leak signals and the signals under normal conditions (non-leak) were collected, and they are presented in Figure 6. Using the experimental platform, 1 mm leak signals, 2 mm leak signals and the signals under normal conditions (non-leak) were collected, and they are presented in Figure 6.

Acoustic Emission Signal De-Noising
The signal collected by one sensor under 1 mm leak was taken as an example. The obtained VMD results of that signal are shown in Figure 7. Moreover, the signal components after VMD were processed by VMD adaptive de-noising method. The distances between the probability density function of the original signal and the probability density function of components were calculated (Figure 8).

Acoustic Emission Signal De-Noising
The signal collected by one sensor under 1 mm leak was taken as an example. The obtained VMD results of that signal are shown in Figure 7.

Acoustic Emission Signal De-Noising
The signal collected by one sensor under 1 mm leak was taken as an example. The obtained VMD results of that signal are shown in Figure 7. Moreover, the signal components after VMD were processed by VMD adaptive de-noising method. The distances between the probability density function of the original signal and the probability density function of components were calculated (Figure 8). Moreover, the signal components after VMD were processed by VMD adaptive de-noising method. The distances between the probability density function of the original signal and the probability density function of components were calculated (Figure 8). As it can be seen in Figure 8, the probability density function of U5 has the largest distance of the original signal. Therefore, the first four components were chosen as the noiseless components and the remaining three were chosen as the noise components. The first four components were reconstructed, forming a reconstruction signal. Ambiguity function of the reconstructed signal was calculated and the result is presented in Figure 9. In Figure 9, the ambiguity functions of three signals have similar delays and frequency shifts, so they could not be distinguished effectively. As a result, the ACC was proposed for extraction of these three signals' characteristics and realization of small leak detection in pipelines.   As it can be seen in Figure 8, the probability density function of U 5 has the largest distance of the original signal. Therefore, the first four components were chosen as the noiseless components and the remaining three were chosen as the noise components. The first four components were reconstructed, forming a reconstruction signal. Ambiguity function of the reconstructed signal was calculated and the result is presented in Figure 9. As it can be seen in Figure 8, the probability density function of U5 has the largest distance of the original signal. Therefore, the first four components were chosen as the noiseless components and the remaining three were chosen as the noise components. The first four components were reconstructed, forming a reconstruction signal. Ambiguity function of the reconstructed signal was calculated and the result is presented in Figure 9. In Figure 9, the ambiguity functions of three signals have similar delays and frequency shifts, so they could not be distinguished effectively. As a result, the ACC was proposed for extraction of these three signals' characteristics and realization of small leak detection in pipelines.   In Figure 9, the ambiguity functions of three signals have similar delays and frequency shifts, so they could not be distinguished effectively. As a result, the ACC was proposed for extraction of these three signals' characteristics and realization of small leak detection in pipelines.

Small Leakage Detection with ACC
For comparison purposes, mean values and standard deviations (SD) of the ambiguity correlation coefficients of de-noised signals based on VMD and EMD were calculated, and the obtained values are presented in Tables 2 and 3, respectively. According to results presented in Table 2, EMD could not acquire accurate components and eliminate noise interference due to the mode-mixing effect, resulting in a poor differentiation of mean values and SD of the ambiguity correlation coefficient. Thus, it is impossible to recognize different pipeline conditions. On the other hand, the VMD adaptive de-noising method can extract natural modes of vibration and separate noise components accurately. Therefore, the mean values and SD of the ambiguity correlation coefficients could be distinguished clearly. Further comparison was obtained by normal distribution curves of the correlation coefficients' mean values and SD, presented in Tables 2 and 3, and the obtained results are shown in Figures 10 and 11 For comparison purposes, mean values and standard deviations (SD) of the ambiguity correlation coefficients of de-noised signals based on VMD and EMD were calculated, and the obtained values are presented in Tables 2 and 3, respectively.  According to results presented in Table 2, EMD could not acquire accurate components and eliminate noise interference due to the mode-mixing effect, resulting in a poor differentiation of mean values and SD of the ambiguity correlation coefficient. Thus, it is impossible to recognize different pipeline conditions. On the other hand, the VMD adaptive de-noising method can extract natural modes of vibration and separate noise components accurately. Therefore, the mean values and SD of the ambiguity correlation coefficients could be distinguished clearly. Further comparison was obtained by normal distribution curves of the correlation coefficients' mean values and SD, presented in Tables 2 and 3, and the obtained results are shown in Figures 10 and 11, respectively. Comparing the results presented in Figures 10 and 11, it can be concluded that the normal distribution curves can intuitively present the correlation degree and recognition degree of the three Comparing the results presented in Figures 10 and 11, it can be concluded that the normal distribution curves can intuitively present the correlation degree and recognition degree of the three Comparing the results presented in Figures 10 and 11, it can be concluded that the normal distribution curves can intuitively present the correlation degree and recognition degree of the three signals in the ambiguity domain. Using the EMD-based de-noising method, the normal distribution curves could not be distinguished clearly because of the EMD mode mixing phenomenon, while the VMD can eliminate the mode mixing effectively in the decomposition process, thus, the VMD-based de-nosing method can distinguish the three states of the signals and the normal distribution curves.
Thirty groups of data were chosen for training and testing, of which twenty groups were used for training, and ten groups were used for testing. Three classifiers, SVM, BP and ambiguity correlation classifier, were used for data training and testing. Test results are shown in Figure 12. signals in the ambiguity domain. Using the EMD-based de-noising method, the normal distribution curves could not be distinguished clearly because of the EMD mode mixing phenomenon, while the VMD can eliminate the mode mixing effectively in the decomposition process, thus, the VMD-based de-nosing method can distinguish the three states of the signals and the normal distribution curves.
Thirty groups of data were chosen for training and testing, of which twenty groups were used for training, and ten groups were used for testing. Three classifiers, SVM, BP and ambiguity correlation classifier, were used for data training and testing. Test results are shown in Figure 12. In Figure 12, the training times of SVM, BP and ambiguity correlation classifier are 15 s, 25 s and 6 s, respectively. Compared to SVM and BP, the ambiguity correlation classifier achieves better classification effect and requires a shorter training time, which is caused by the absence of parameter setting and the small computational load. Test accuracies of these three classifiers for 60 groups of data, wherein 40 were used for training and 20 were used for testing, are presented in Figure 13.
As it can be seen in Figure 13, all three classifiers can distinguish the different conditions of pipelines, but compared to BP and SVM, the proposed classifier achieves the highest mean recognition rate and realizes the best small leak detection in pipelines, because of its small computational load and no parameter setting.

Data sets
Category labels The actual classification The prediction classification In Figure 12, the training times of SVM, BP and ambiguity correlation classifier are 15 s, 25 s and 6 s, respectively. Compared to SVM and BP, the ambiguity correlation classifier achieves better classification effect and requires a shorter training time, which is caused by the absence of parameter setting and the small computational load. Test accuracies of these three classifiers for 60 groups of data, wherein 40 were used for training and 20 were used for testing, are presented in Figure 13.

Discussion
The experimental results indicate that the proposed method can effectively detect small leaks in noisy environments. This improvement is due to the proposed VMD adaptive de-noising and ACC approach. Compared to the recursive "screening" mode of the EMD, VMD can non-recursively decompose a multi-component signals into a number of components and control the decomposition convergence conditions reasonably. Hence, the VMD adaptive de-noised method can effectively eliminate the mode-mixing phenomenon with good noise immunity. Due to its no parameter setting and small computational load, the ACC method achieves better classification effects and shorter training times than the BP or SVM methods. Despite the advantages offered by the proposed scheme for effectively detecting small leaks in natural gas pipelines, it has however certain limitations. The penalty parameters and number of components in the VMD decomposition process need to be prioritized, hence, our future research will focus on investigating how to adaptively set the parameters of the VMD method.

Conclusions
Due to the complicated surrounding environment, large noise disturbances, and weak signals during the small leakage of pipelines, pipeline-leakage signals have unobvious characteristics. A small leak detection method based on VMD and ACC is proposed in this paper. For the non-stationary signals with strong noise, the VMD adaptive de-noising method is proposed according to the signal probability density function in order to process the signal and to acquire the As it can be seen in Figure 13, all three classifiers can distinguish the different conditions of pipelines, but compared to BP and SVM, the proposed classifier achieves the highest mean recognition rate and realizes the best small leak detection in pipelines, because of its small computational load and no parameter setting.

Discussion
The experimental results indicate that the proposed method can effectively detect small leaks in noisy environments. This improvement is due to the proposed VMD adaptive de-noising and ACC approach. Compared to the recursive "screening" mode of the EMD, VMD can non-recursively decompose a multi-component signals into a number of components and control the decomposition convergence conditions reasonably. Hence, the VMD adaptive de-noised method can effectively eliminate the mode-mixing phenomenon with good noise immunity. Due to its no parameter setting and small computational load, the ACC method achieves better classification effects and shorter training times than the BP or SVM methods. Despite the advantages offered by the proposed scheme for effectively detecting small leaks in natural gas pipelines, it has however certain limitations. The penalty parameters and number of components in the VMD decomposition process need to be prioritized, hence, our future research will focus on investigating how to adaptively set the parameters of the VMD method.

Conclusions
Due to the complicated surrounding environment, large noise disturbances, and weak signals during the small leakage of pipelines, pipeline-leakage signals have unobvious characteristics. A small leak detection method based on VMD and ACC is proposed in this paper. For the non-stationary signals with strong noise, the VMD adaptive de-noising method is proposed according to the signal probability density function in order to process the signal and to acquire the noiseless components. Noiseless components were then reconstructed successfully, forming the de-noised signals. Furthermore, in order to overcome the problems of traditional classifiers, such as heavy computational load and abundant parameter settings, the ambiguity correlation classifier is proposed for training and testing of the reconstructed signals. According to obtained experimental results, the proposed method can detect small pipeline leakages and achieve a higher mean detection rate than the classical SVM and BP methods.