## 2 Materials and Methods
### Improved CEEMDAN with Kullback-Leibler Divergence  
During the signal collection using the \(\Phi\)-OTDR system in practical environments, the lack of prior knowledge and poor SNR make it difficult for the wavelet and Fourier transform to work. CEEMDAN was developed based on EMD decomposition. It decomposes the time-series signal into several IMF components with adaptability [20]. These IMF components have little influence on each other and differ from the original signal, which can simplify the coupling of signal feature information [27]. By adding adaptive white noise, the unique margin is calculated to obtain the IMF component. The decomposition process is then complete [28; 29]. To measure the similarity between each IMF component and the original signal, this paper introduces the Kullback-Leibler divergence (KL). It optimizes the CEEMDAN process. In short, it enhances the signal characteristics and provides support for the \(\Phi\)-OTDR system performance optimization. Firstly, the \(E_{k}(\cdot)\) is difined as the \(k\)th mode component generated by the decomposition of the conventional EMD algorithm. The \(\widetilde{C_{k}}\) is difined as the mode component generated by CEEMDAN. \(N\) is Gaussian white noise. \(\sigma\) is the standard deviation of the noise. Then, the steps of CEEMDAN-KL decomposition are as follows:  
1. [label=(0)]
2. Conduct the experiment of the signal \(x(t)+\sigma_{0}N^{i}(t)\)\(m\) times, where the first modal component is: \[\widetilde{C}_{1}(t)=\frac{1}{m}\sum\
olimits_{i=1}^{m}E_{1}^{i}(t)= \overline{E}_{1}(t).\] (6)
3. Obtain the unique residual signal: \[r_{1}(t)=x(t)-\widetilde{C}_{1}(t).\] (7)
4. Do the experiment \(m\) times with the signal \(r_{1}(t)+\sigma_{1}E_{1}\big{(}N^{i}\big{)}\) and break it down until \(E_{1}(\cdot)\) is obtained. Based on this, the second empirical modal component is calculated: \[\widetilde{C}_{2}(t)=\frac{1}{m}\sum\
olimits_{i=1}^{m}E_{1}\big{(}r_{1}(t)+ \sigma_{1}E_{1}\big{(}N^{i}\big{)}\big{)}.\] (8)
5. For \(k=2,\dots,\ K\), calculate the residual signal and the corresponding empirical mode component, where the calculation steps are consistent with step 3, and the KL divergence values between each component \(\widetilde{C_{k}}(t)\) and the original signal \(x(t)\) are calculated as follows: \[r_{k}(t)=r_{k-1}(t)-\widetilde{C}_{k}(t),\] (9) \[\widetilde{C}_{k-KL}=f_{KL}(\widetilde{C}_{k}(t)|x(t)),\] (10) \[\widetilde{C}_{k+1}(t)=\frac{1}{m}E_{k}\big{(}r_{k}(t)+\sigma_{k}E_{k}\big{(} N^{i}\big{)}\big{)}.\] (11)
6. Execute step 4 until the residual cannot be decomposed. The judgment criteria for termination are: 1. [label=()] 2. The number of extremum points of the residual signal should not exceed two. 2. The component combination with the smallest mean square error of the decomposed component KL value. Here, from the perspective of mathematical statistics, the best reconstruction effect is when the KL standard deviation of the chosen components is less than that of half of all KL components: \[\widetilde{x}(t)=\sum_{k=1}^{K^{\prime}}\widetilde{C}_{k}(t).\] (12)  
To gain a better understanding of this algorithm, we experimented with a standard knocker. The data of a standard knocker was collected with a frequency of 10 Hz using the \(\Phi\)-OTDR system. After standardization, the CEEMDAN-KL feature enhancement was adopted. IMFs meeting the KL selection criteria were selected to complete the reconstruction of the original signal, as shown in Figure 3, and the comparison effect is shown in Figure 4.  
Figure 4: Comparison of the standard signal enhancement based on CEEMDAN-KL: (**a**) Original signal and (**b**) Signal after the enhancement. Figure 3: The relationship diagram between the intrinsic mode functions (IMFs) and KL. In Figure 3, the abscissa is the IMFs obtained after the decomposition. The ordinate indicates the KL values of the IMFs from the original signal. This step eliminates the IMFs components with a large difference, and the remaining components express the characteristics of the original signal more clearly.