## Abstract

Leakages in pipelines can cause severe hazards to industries, the environment and people. For the purpose of an accurate identification of the leakage location, a transient-based leakage detection method using multiple signal classification (MUSIC)-like is applied to this paper. The localization is achieved by a one-dimensional search of leak location along the pipe, which means it involves low computational cost. The performance of the MUSIC-Like method in the cases of a single leak and multiple leaks is discussed by comparison with three spectral-based methods. In the single-leak case, the MUSIC-like algorithm provides precise localization estimation even for a high level of noise. For the multiple-leak case, the MUSIC-like method is superior to the other three methods. It is capable of identifying all leaks where the leak-to-leak distance is less than half the shortest probing wavelength. Therefore, the MUSIC-like method has an excellent performance in leak detection and location.

## HIGHLIGHTS

The pipeline transient model for leak location is obtained via transfer matrix analysis.

The multiple signal classification (MUSIC)-like method is used to solve the leak localization problem.

The method provides accurate localization estimation for the low SNR environment for the single-leak case.

In the two-leak case, the method can still identify all leaks where the leak-to-leak distance is less than half the shortest probing wavelength.

## INTRODUCTION

Pipelines play an important role in providing convenient and economical modes of transportation for a large amount of oil, water and gas in industry. According to the research (Colombo & Karney 2002), pipelines are the safest means of transportation, but this does not mean that they are risk-free. Ensuring the integrity and reliability of the pipeline infrastructure has become a crucial requirement. The main threat considered in the water supply system is the occurrence of leaks. Regardless of their sizes, pipeline leakages are a principal problem because they can have a quite significant impact. These effects exceed the costs including downtime and maintenance expenses, harm to citizens' security and even environmental damage (Lopezlena & Sadovnychiy 2019). Therefore, there is an urgent need for a feasible method to accurately identify leakage points in order to reduce the waste of water resources and minimize the detrimental effects of the leakage.

Research on leak detection and location has been going on for decades, and a variety of available commercial leak detection techniques have been developed, ranging from simple physical inspection to acoustic technology (Colombo *et al.* 2009), such as vibration signal (Sepideh *et al.* 2017), negative pressure wave (Zhang *et al.* 2019) and acoustic waves (Jin *et al.* 2014). Transient-based technology has generated significant attention in leak detection and localization over the past decade. The transient-based leak detection method works as follows: to make use of introduced hydraulic pressure wave(s) and identify the time-frequency signal characteristics of transient pressure in the leakage pipeline measured at a specific location to determine the leak location.

Specifically, any change in the physical structure of a pipe or system (e.g., blockage, leakage, roughness transition, contraction or expansion) produces a wave reflection on an incident transient signal, which in some way alters the flow and pressure response of the system (Liu *et al.* 2015). The pressure response signal in fluid conduits measured at a specific location is changed by its interaction with the physical system as it propagates and reflects throughout the system as a whole. Consequently, it contains useful information about the conduit's properties and state. This principle constitutes the foundation of a series of transient-based defect detection methods (TBDMs) (Wang & Ghidaoui 2018). These TBDMs can be classified into four categories. The first method is the inverse transient-based method (ITM) (Vitkovsky *et al.* 2007; Soares *et al.* 2011; Stephens *et al.* 2013), which attempts to calibrate the object function between measured and simulated data. The second method is the frequency response-based method (FRM) (Lee *et al.* 2005; Duan 2016a; Scola *et al.* 2017), which utilizes the variations of frequency response to detect leaks. Thirdly, the transient damping-based method (TDM) is to exploit pressure decay features to identify leaks (Wang *et al.* 2002; Covas *et al.* 2005). Finally, the transient reflection-based method (TRM) is to investigate some characteristics of pressure trace (Covas *et al.* 2015; Sun *et al.* 2016).

As shown in the previous literature, the current implementation schemes of defect detection methods based on fluid transient are robust for simple pipeline systems with large signal-to-noise ratio (SNR) or single leak, but the real world pipeline environment contains numerous noise sources, such as turbulence, mechanical devices, dynamic flow control, traffic and many other activities. Wang *et al.* (2019) proposed a spectral-based methodology which can identify leaks based on one-dimensional search. However, it was found that two close leaks cannot be separately identified. Therefore, a transient-based method that can detect leaks in a high level of noise and identify close leaks is desirable.

The MUSIC-like method is a signal processing methodology which is suitable for noisy environments and unknown parameter estimation problems. It has been successfully used in many array signal processing applications to localize signal sources (Zhang & Ng 2010; Reddy *et al.* 2013). The MUSIC-like approach has also been applied to sound source localization in harsh underwater environments (Lim *et al.* 2017). Borijindargoon *et al.* (2019) applied the MUSIC-like algorithm to source localization in electrical impedance tomography. Inspired by the aforementioned views, the present paper applies a transient model-based MUSIC-like to pipeline leak localization, which is able to detect leaks in a noisy environment and identify two close leaks.

The paper is organized as follows. It begins with the hydraulic transient model description and then introduces the MUSIC-like method. To verify the validity of the method, numerical simulations are conducted. The performance of the MUSIC-like method for single leak and multiple leaks is discussed and the results are shown. Finally, conclusions are drawn.

## BASIC THEORY

### Model description

In this section, the model of transient wave propagation in a pipeline is introduced. The configuration of the considered pipe system is illustrated in Figure 1. A single, horizontal pipeline is bounded by two reservoirs, whose coordinates are *x* = *x ^{U}* = 0 and

*x*=

*x*=

^{D}*l*, respectively. A pressure sensor is assumed to be positioned near the downstream node whose coordinate is denoted by

*x*=

*x*. Let

^{M}*x*(

^{L}*x*<

^{L}*x*) be the unknown location of the leak, and and are, respectively, the steady-state discharge and head at the leak. The steady-state discharge of the leak is related to the lumped leak parameter by , where

^{M}*g*is the gravitational acceleration and

*z*denotes the elevation of the pipe at the leak. The lumped leak parameter

^{L}*s*=

^{L}*C*stand for the leak size, in which

^{d}A^{L}*C*is the discharge coefficient of the leak and

^{d}*A*is the flow area of the leak orifice.

^{L}*q*and

*h*, respectively, represent the oscillations of discharge and pressure head due to a rapid change in flow setting (e.g., valve operation). Let

*a*denote the wave speed,

*g*the acceleration due to gravity,

*A*area of pipeline,

*x*ɛ [

*x*,

^{U}*x*) ∪(

^{L}*x*,

^{L}*x*] the distance from the upstream node and

^{D}*t*the time label.

*t*, and

*q*and

*h*are given in the frequency domain.where

*ω*is the angular frequency. Here,

*R*is the steady-state friction resistance term being

*R*= (

*fQ*)/(

_{0}*gDA*

^{2}) for turbulent flow, in which

*f*is the Darcy–Weisbach friction factor,

*Q*is the steady-state discharge, and

_{0}*D*is the pipe diameter. Note that the friction resistance term is obtained by linearization, because the term for turbulence pipe flow is second-order nonlinear, which is difficult to convert directly into the frequency domain (Duan

*et al.*2018). Equations (3) and (4) are solved with boundary condition of the head

*h*(

*x*) and discharge

^{U}*q*(

*x*) at

^{U}*x*and the head and mass conservation condition across the leak. The quantities at

^{U}*x*can be computed via the transfer matrix method after considering linearized forms of the friction resistance term and orifice equations, and the results can be obtained as follows (Lee & Vítkovský 2010; Chaudhry 2014):

^{M}*NL*stands for no leak,is the propagation function,is the characteristic impedance.

*x*but independent of the size

^{L}*s*of the leak.

^{L}*x*

*=*

*x*near the downstream for a given angular frequency

^{M}*ω*(

_{j}*j*= 1,2,…,

*J*) is assumed to follow the theoretical expression from Equation (9) plus a noise term:whereand

*n*follows additive independent Gaussian random noise with 0-mean and covariance

_{j}*σ*

^{2}.

*h*(

*x*) and

^{U}*q*(

*x*) are assumed to be known. Here, the upstream is connected to a reservoir, thus

^{U}*h*(

*x*) = 0 is reasonably assumed. The discharge

^{U}*q*(

*x*) can be estimated if a measurement station near the upstream boundary, whose location is denoted by , is available. Assuming there is no leak between

^{U}*x*and and using the pressure head measurement

^{U}*h*() at and the boundary

*h*(

*x*), the discharge at upstream

^{U}*q*(

*x*) can be solved via condition

^{U}*h*(

*x*), the discharge at upstream

^{U}*q*(

*x*) can be solved viathat is,

^{U}### MUSIC-like algorithm for leak detection

**w**is the solution to the optimization problem,

**R**

_{CM}is the estimation of the correlation matrix,

**G**(

*x*) is a function to be tuned to estimate the actual leak location,

^{L}*β*is a control parameter which introduces certain relaxation to the constraint, and

*c*is an arbitrary constant which is not critical to the results of the optimization process. While the objective function minimizes the output power, it can be shown that the quadratic equality constraint regulates the desired signal power in relation to the objective function. Therefore, it was proposed to constrain the desired signal power at the potential leak location with an

*L*

_{2}norm constraint on the weight vector which guarantees a meaningful result.

- 1.
Select

*J*frequencies*ω*_{1}_{,}*ω*_{2}_{,…,}*ω*._{J} - 2.
Estimate the discharge at the upstream

*q*(*x*) according to Equation (15).^{U} - 3.
Compute head differences Δ

**h**as the data._{n} - 4.
Calculate the correlation matrix

**R**_{CM}via Equation (21). - 5.
Determine an appropriate

*β*according to Equation (26). - 6.
Calculate

**Ɓ**= (**G**(*x*)^{L}**G**(^{H}*x*) +^{L}*β***I**)^{−1}**R**_{CM}. - 7.
Derive the eigenvector with respect to the minimum eigenvalue of

**Ɓ**. - 8.
Estimate the leakage location by the spatial power spectrum based on Equation (25).

*λ*of (

_{min}**G**(

*x*)

^{L}**G**

*(*

^{H}*x*) +

^{L}*β*

**I**)

^{−1}

**R**

_{CM}, and the leakage localizations are identified from the peaks of the spatial power spectrum given by

*β*is a function of the correlation matrix eigenvalue (Zhang & Ng 2010), and its expression is:wherein

*L*

*=*

*JM*, the value of

*ξ*is the minimum eigenvalue of correlation matrix

_{L}**R**

_{CM}, and

*ξ*

_{L}_{−1}is the second smallest eigenvalue of

**R**

_{CM}.

## EXPERIMENTAL RESULTS

In this section, we provide extensive simulation results to verify the validity of the proposed algorithm. We also compare the MUSIC-like with other leak localization methods. The cases of a single leak and multiple leaks with white noise are considered, respectively.

### Numerical setup

The setup of the considered pipe system is shown in Figure 1. A valve is situated at the downstream of the single pipeline and there are two pressure sensors located at around the downstream. Assuming that an impulse wave is generated by rapidly closing and opening the valve, the boundary conditions *h*(*x ^{U}*) = 0 and

*q*(

*x*) = 1 are given. In the forward problem, the transient wave propagation simulation in the frequency domain is accomplished by using the transfer matrix method in Equation (9). Another pressure sensor at is used to estimate

^{D}*q*(

*x*) according to Equation (15). The major parameters are shown in Table 1.

^{U}Head of the upstream reservoir | H_{1} = 25 m |

Head of the downstream reservoir | H_{2} = 20 m |

Length of the pipe | l = 2,000 m |

Speed of the wave | a= 1,200 m |

Diameter of the pipe | D= 0.5 m |

Darcy–Weisbach friction factor | f = 0.02 _{DW} |

The steady-state discharge | Q_{0} = 0.0153 m^{3}/s |

Head at the upstream boundary | h(x) = 0 ^{U} |

Discharge at the downstream boundary | q(x) = 1 ^{D} |

Head of the upstream reservoir | H_{1} = 25 m |

Head of the downstream reservoir | H_{2} = 20 m |

Length of the pipe | l = 2,000 m |

Speed of the wave | a= 1,200 m |

Diameter of the pipe | D= 0.5 m |

Darcy–Weisbach friction factor | f = 0.02 _{DW} |

The steady-state discharge | Q_{0} = 0.0153 m^{3}/s |

Head at the upstream boundary | h(x) = 0 ^{U} |

Discharge at the downstream boundary | q(x) = 1 ^{D} |

### Single leak

In this section, estimation of a single leak is considered with the location *x ^{L}* = 800 m and the lumped leak parameter (effective leak size)

*s*= 1.0 × 10

^{L}^{−4}m

^{2}is assumed. The transient wave propagation is simulated, and the measurements obtained at = 2,000 m and = 1,800 m constitute the measured heads. The resonant and antiresonant frequencies

*ω*=

*κω*,

_{th}*κ*= 1,2,…,31, are used for leakage detection, where

*ω*=

_{th}*aπ/*2

*l*. Here,

*M*= 2 stands for the number of the pressure sensors used. The sample size of the correlation matrix

**R**

_{CM}is 620, i.e.,

*N*

*=*620. With these parameters, the MUSIC-like algorithm is applied and compared with the other three algorithms, namely, MFP, Capon's beamforming (BF) and MUSIC (Wang

*et al.*2019). In principle, the MUSIC-like method, MFP and Capon belong to beamforming methods, while MUSIC is the representative of subspace-based methods which require knowledge of the number of sources to perform accurate subspace decomposition of the correlation matrix. The three aforementioned beamforming algorithms correspond to different designs of the weighting vector as a function of the candidate leak location

*x*. In this case, the spatial power spectrum function reaches maximum at the actual leak position. When the signal-to-noise ratio is relatively high, SNR is set to be 0 dB. The results are shown in Figure 2. According to the leakage location results presented in Figure 2, different from the MFP method, which has a comparatively wide main lobe and a very high secondary lobe at around 1,600 m, the three methods, Capon's BF, MUSIC and MUSIC-like successfully achieve narrow peaks and remove all the side lobes. The above three methods perform better than MFP in suppressing side lobes and fluctuations. For relatively smaller SNR, the case with SNR = −40 dB is displayed in Figure 3. It is observed that as the noise levels increase, the performance of various algorithms decreases, which is reflected in the fact that the four algorithms all return a result with side lobes, but the leak can still be roughly located. Although all the four methods can accurately estimate the leak, the presence of higher side lobes affects the accuracy of the leak location because they are mistaken for leaks, especially if the number of leaks is unknown. The methods MUSIC and MUSIC-like perform better than the others in suppressing side lobes and fluctuations. It is worth noting that the MUSIC algorithm needs to estimate the number of leaks beforehand, but the MUSIC-like method has no requirement for estimating the number of leaks. The above results obtained by using MUSIC are based on the correct estimation of the leak number. However, it is difficult to accurately estimate the number of leaks in low SNR environment, which results in the performance degradation of the algorithm. As shown in Figure 4, the actual number of leaks is one, and the estimated number of leaks is three. The MUSIC method has more obvious peaks than MUSIC-like

^{L}**.**In summary, the MUSIC-like method is more suitable for the low SNR environment compared with the other three methods.

### Multiple leaks

^{−4}m

^{2}and = 1.2 × 10

^{−4}m

^{2}. Sensor placement is the same as that in the case of a single leak. The SNR of measurement noise at all the frequencies is assumed to be 0 dB, and the sample size

*N*= 10

*JM*= 620. The results are displayed in Figure 5. It is clear that there is a local maxima near each actual leak in each figure, that is, all the four methods can accurately localize the two leaks. However, all algorithms have side lobes, especially the MFP method which has relatively wide main lobes and some side lobes, in particular a very high side lobe at around 1,800 m. The presence of high side lobes can interfere with leak localization because they may be mistakenly identified as leaks, especially if the number of leaks is unknown. The MUSIC-like algorithm and the other two algorithms have almost the same ability to successfully suppress side lobes. The three methods perform better than the MFP method. In order to test and compare the performance of the MUSIC-like algorithm to the other three methods, the root-mean square error (RMSE) of each leak location estimation with various SNR is plotted in Figure 6. The RMSE is calculated aswhere

*K*= 10 for our simulation,

*x*is the actual leak location and represents the estimated value of the

^{L}*i*th trial. The SNR is varied from −40 to 0 dB. The results in Figure 6(a) clearly show that the RMSE of the first leak location estimation decreases as SNR increases and the error of MUSIC-like is smaller than that of other methods. Figure 6(b) has almost the same results. The localization error of MUSIC-like is slightly large for the second leak but still within an acceptable range. There are numerous local maxima for MUSIC-like methods when the SNR is smaller, as displayed in Figure 7(a). The SNR is assumed to be −40 dB. Under such circumstances, a range of spatial power spectra are used as the threshold values to generate the receiver operating characteristic (ROC) curves (Yang & Boccelli 2016) to depict the trade-off between the actual leaks and false alarms. Figure 7(b) shows the ROC curve with the threshold varied. The curve is close to the upper left corner, which shows that the accuracy of the algorithm is high. In a noisy environment, without knowing the number of leaks, it is observed that high TPR would occur as FPR increased, which is a trade-off that utilities must make when selecting the threshold to determine the occurrence of leaks.

Again, Figure 8 plots the power spectra for other cases of leak location where two leaks are located at = 300 m and = 1,200 m with actual sizes = 1.0 × 10^{−4}m^{2} and = 1.2 × 10^{−4}m^{2}. The SNR is set to 0 dB. It can be seen that there is a local maximum at each actual leak location, but there may be higher lobes at other locations, which will affect the determination of leak locations. From the above results, one comes to the conclusion that the location accuracy of multiple leaks is related to the location of the leak. In addition, MUSIC-like, MUSIC and Capon's BF have better interference suppression ability than MFP.

In the following, the case of two close leaks is considered, where = 1,000 m and = 1,020 m with actual sizes = 1.0 × 10^{−4}m^{2} and = 1.2 × 10^{−4}m^{2}. The other simulation conditions for leak localization are the same as in the previous case. The localization results can be seen in Figure 9. In this case, the corresponding shortest probing wavelength is *λ _{min}* = 2

*πa*/(31

*ω*)

_{th}*=*258 m. The distance between the two leaks (20 m) is smaller than half the minimum wavelength (129 m). As shown in Figure 9, there is only one maximum between the two leaks using the three methods of MFP, Capon's BF and MUSIC, which implies that the two leaks cannot be identified separately. This makes it easy to incorrectly determine the situation of two close leaks as that of only one leak without knowing the actual number of leaks. The reason for this is that each leakage in the pipeline creates a reflected signal, and the transient-based leak detection methods rely on the accurate identification and location of reflected signals. The higher bandwidth signals (faster valve closure) produce greater leak detection accuracy and an increased ability to distinguish multiple and close leaks. Low bandwidth signals create overlapping reflections where the reflections from close leaks cannot be distinguished, affecting the accuracy of the leak detection process. However, the MUSIC-like method returns a result that two local maxima can be found to determine the leak location although both lobes are not completely separated, thus avoiding the omission of leakages. The case with SNR = 0 dB is displayed in Figure 10. The case of leak locations = 1,000 m and = 1,040 m with SNR = −20 dB is shown in Figure 11. The MUSIC-like method can still determine that there are two close leaks, while the other three methods have only one local maximum, i.e., only one leak. Meanwhile, the multiple leaks’ location accuracy also relies on data accuracy. In the following, RMSE (from the 100 results) is used to evaluate the localization performance of the MUSIC-like method by SNR testing, as shown in Tables 2 and 3. By comparing Table 2 with 3, it is observed that the localization performance of the algorithm improves as SNR increases. As indicated in each table, it can be found that the estimation error decreases as the sample size

*N*increases. It is concluded that the MUSIC-like method performs better than the other three methods in the too-close case.

x (m)
. ^{L} | RMSE (m) (N = 100)
. | RMSE (m) (N = 200)
. | RMSE (m) (N = 300)
. | RMSE (m) (N = 400)
. | RMSE (m) (N = 500)
. | RMSE (m) (N = 600)
. |
---|---|---|---|---|---|---|

1,000 | 395.0752 | 142.3954 | 27.7968 | 2.2650 | 2.2517 | 2.6981 |

1,020 | 284.6758 | 71.2166 | 1.9596 | 1.1790 | 1.4107 | 1.5297 |

x (m)
. ^{L} | RMSE (m) (N = 100)
. | RMSE (m) (N = 200)
. | RMSE (m) (N = 300)
. | RMSE (m) (N = 400)
. | RMSE (m) (N = 500)
. | RMSE (m) (N = 600)
. |
---|---|---|---|---|---|---|

1,000 | 395.0752 | 142.3954 | 27.7968 | 2.2650 | 2.2517 | 2.6981 |

1,020 | 284.6758 | 71.2166 | 1.9596 | 1.1790 | 1.4107 | 1.5297 |

x (m)
. ^{L} | RMSE (m) (N = 100)
. | RMSE (m) (N= 200)
. | RMSE (m) (N = 300)
. | RMSE (m) (N = 400)
. | RMSE (m) (N = 500)
. | RMSE (m) (N = 600)
. |
---|---|---|---|---|---|---|

1,000 | 357.1670 | 330.6980 | 197.3861 | 122.7535 | 38.6375 | 28.4512 |

1,020 | 403.4348 | 291.5390 | 143.9225 | 95.9546 | 11.2601 | 11.0508 |

x (m)
. ^{L} | RMSE (m) (N = 100)
. | RMSE (m) (N= 200)
. | RMSE (m) (N = 300)
. | RMSE (m) (N = 400)
. | RMSE (m) (N = 500)
. | RMSE (m) (N = 600)
. |
---|---|---|---|---|---|---|

1,000 | 357.1670 | 330.6980 | 197.3861 | 122.7535 | 38.6375 | 28.4512 |

1,020 | 403.4348 | 291.5390 | 143.9225 | 95.9546 | 11.2601 | 11.0508 |

### Sensitivity with uncertain wave speed

The application results (Duan 2015, 2016b, 2018) revealed that the transient-based frequency domain methods were much more accurate in the numerical simulations than the realistic experimental cases, which was explained by possible experimental test uncertainties and model inaccuracy in these studies. In practical applications, the wave speed and data measurement are the two dominant factors that significantly affect the applicability and accuracy of the transient-based methods for leakage detection. In the following, sensitivity of the leakage detection method with respect to the uncertainty of wave speed is investigated. The wave speed in the pipeline may generally change by 10–15% for a given pipe system and the assumed wave speed is *a* = 1,000 m/s. The actual wave speed in the simulation for measurement generation is *a* = 850, 875,…,1,150 m/s. Here, the leak location is *x ^{L}* = 400 m, the lumped leak parameter is

*s*= 1.0 × 10

^{L}^{−4}m

^{2}, the SNR is 5 dB and the peak angular frequencies

*ω*=

*ω*

_{th}× (1:3:31). The RMSE of five trials is calculated to evaluate the localization performance. Figure 12 displays the leak detection results with wave speed uncertainties. It is observed that the localization error exhibits a large jump when the wave speed error increases to 7.5%. It can be illustrated with this sentence that a slight change of wave speed alters the resonant frequency and the magnitude of the peaks, which may result in a strong sensitivity of leak localization estimation.

## CONCLUSIONS

In this paper, the problem of pipeline leakage by using fluid transient detection wave is discussed. The transient wave model in a water-filled pipeline is established by the transfer field matrix method deduced by the momentum equation and continuity equation. The MUSIC-like method is applied to detect leaks with a one-leak model and compared with MFP, Capon's BF and MUSIC. The simulation results show that all the four methods are able to accurately locate a single leak. When the number of leaks is correctly given, the MUSIC method has the best ability to suppress side lobes and fluctuations followed by MUSIC-like, MFP and Capon's BF in noisy environments. Otherwise, MUSIC-like is slightly better than MUSIC, and avoids leak number estimation. Meanwhile, the method applied for the case of two close leaks’ situation is investigated. It is able to identify two leaks and roughly determine the leak locations and the errors are acceptable when the sample size of the pressure difference is sufficient; nevertheless, the other three methods determine the situation as that of only one leak. Therefore, this method has high accuracy for leak detection and location. In addition, for the case where the leak-to-leak distance is of the same order or larger than half the shortest probing wavelength, none of the four approaches are robust and are limited by leak locations. The multileak detection method of locating performance that is not limited by leak location will be considered in future studies.

There are many issues, including the influence of the linear approximations of friction resistance term and orifice equations, the effect of input signal bandwidth, system uncertainties and more complicated hydraulic systems that require further investigations to better apply the transient-based leak detection method to practical applications.

## ACKNOWLEDGEMENTS

This paper is supported by the key Science Foundation of the Department of Science and Technology of Jilin Province (Grant No. 20180201081SF, 20190303082SF), science and technology project of The Education Department of Jilin Province (Grant No. JJKH20200983KJ), and the Fund project of The Science and Technology Department of Jilin Province (Grant No. 20200201046JC). Thanks for the permission to publish this paper.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.