## 2 Background Study on Pipeline Leak Detection Methods
### Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN)  
CEEMDAN addresses the drawbacks of EEMD [51]. The CEEMDAN process is as follows: the first IMF and the first mode generation are the same as EEMD, and then, from the noise realization, the first residue is obtained. Then, the first EMD mode is computed with the ensemble of residues having different noise realizations. Then, these modes are averaged to obtain the second mode, and the next residue is generated. This process repeats until the stopping criterion. Using Algorithm 3, iterating through steps 4 to 6 is continued until either the EMD cannot decompose the residue while satisfying the IMF criteria or the residue exhibits fewer than three local extrema. The final residue \(r_{k}\) is obtained by the CEEMDAN as in Equation (2):  
\[r_{k}=S(t)-\sum_{k=1}^{K}\widetilde{I}_{k} \tag{2}\]  
The original signal \(S(t)\) is as in Equation (3):  
\[S(t)=\sum_{k=1}^{K}\widetilde{I}_{k}+r_{k} \tag{3}\]```
Step 1: For every \(i=1,\ldots,I\) decompose each \(S^{i}(t)=S(t)+\beta_{0}Wn^{i}\) using EMD until the first mode and compute \(\widetilde{I}_{1}=1\sum_{i=1}^{I}I_{1}^{i}=\widetilde{I}_{1}\) Step 2: For \(k=1\) obtain the first residue \(r_{1}=S(t)-\widetilde{I}_{1}^{\top}\) Step 3: Obtain the first mode of \(r_{1}+\beta_{1}E_{1}(Wn^{i}),i=1,\ldots,I\), define the second CEEMDAN mode using EMD \(\widetilde{I}_{2}=\frac{1}{\tau}\sum_{i=1}^{I}E_{1}(r_{1}+\beta_{1}E_{1}(Wn^{ i}))\), where \(E_{k}(*)\) is the operator produces the \(k^{th}\) mode with EMD. Step 4: For \(k=2,\ldots K\) estimate the \(k^{th}\) residue \(r_{k}=r_{(k-1)}-\widetilde{I}_{k}\). Step 5: Obtain the \(k^{th}\) mode of \(r_{k}+\beta_{k}E_{k}(Wn^{i}),i=1,\ldots,I\) using EMD until defining the \((k+1)^{th}\) CEEMDAN mode \(\widetilde{I_{(k+1)}}=\frac{1}{\tau}\sum_{i=1}^{I}E_{1}(r_{k}+\beta_{k}E_{k}( Wn^{i}))\) Step 6: Go to step 4 for the next \(k\)
```

**Algorithm 3** CEEMDAN algorithm steps

### Mel-Frequency Cepstral Coefficients (MFCC)

The mel-frequency cepstral coefficient (MFCC) method's application is mainly in speech signal processing as a feature extraction technique for emotional state analysis [41]. Its application is extended to pipeline leak detection, as the data collected from a pipeline are a non-stationary signal [10]. The MFCC captures the frequency component from the time domain and represents it in both the time and frequency domains. Figure 1 shows the process of the MFCC. Figure 1 gives detailed information on the process of MFCC extraction. First, the raw input signal is pre-emphasized. After that, the pre-emphasized signal is split into some frames. The frames are passed through the hamming window before performing the discrete Fourier transform (DFT). The DFT converts the time to the frequency domain representation. After the conversion into the frequency domain, the mel filter bank helps capture the frequency component in the way humans perceive the frequencies. After the filter bank application, the discrete cosine transform (DCT) helps convert the data into an aggregate of cosine functions oscillating at distinct frequencies. This DCT conversion helps to obtain MFCCs. Equation (4) gives the formula for the DFT. \[D(k)=\sum_{t=0}^{T-1}I_{d}(t)e^{-j2 nk\frac{t}{t}} \tag{4}\]

where \(t=0,1,2,\ldots T-1\). The spectrum obtained from Equation (6) has a wide range. The leak signal does not apply on a linear scale. The mel filter bank helps to view the frequency level the way humans hear with the help of Equations (5) or (6). \[mel(f)=2595 log_{10}(1+\frac{f}{700}) \tag{5}\]

\[mel(f)=1127 ln(1+\frac{f}{700}) \tag{6}\]

Figure 1: MFCC process. By applying a DCT on the log energy (\(E_{l}\)), 'N' MFCCs are obtained with triangular bandpass filters. Equation (7) gives the formula for the DCT. \[DC_{m}=\sum_{c=1}^{S}(E_{l}cos(m*c-\frac{1}{2})\frac{\pi}{S}) \tag{7}\]

where \(m=1,2,3,\ldots N\), \(S\) is the count of log spectral coefficients, and \(N\) is the obtained MFCC. [PERSON] et al.