## 5 Numerical Simulation and Verification
### Numerical Investigation of the Proposed Method  
For simplification, only Steps (5) to Step (9) in Section 3 are first checked for noise robustness in this section. Considering that the signal-to noise ratio (SNR) of the actual guided wave signal was poor, artificial noises were added into the numerical signals. A numerical guided wave signal \(\hat{x}(t)\) including two Gabor pulses \(x_{1}(t)\), \(x_{2}(t)\) and a random white noise \(\varsigma(t)\), can be represented as:  
\[\hat{x}(t)=x(t)+\varsigma(t)=x_{1}(t)+x_{2}(t)+\varsigma(t)=\sum_{i=1}^{2}A_{ i}g_{i}(\alpha_{i},u_{i},f_{ci},\theta_{i})(t)+\varsigma(t) \tag{11}\]  
where the SNR of \(\hat{x}(t)\) is defined as:  
\[\text{SNR}=10\log_{10}\frac{\int\abs{x(t_{i})}^{2}dt}{\int\abs{n(t_{i})}^{2}dt }(\text{dB}) \tag{12}\]  
In the first case, the two Gabor pulses in \(\hat{x}(t)\) are not overlapped, which means that the axial length of the reflector is long enough. \(x_{1}(t)\) and \(x_{2}(t)\) represent the reflection signals from the front- and back-edge of a reflector, respectively, while the reflector is set as a defect. The sampling frequency is assumed to be \(f_{s}=3000\) kHz. The Gabor excitation pulse \(S_{i}\) is (\(\sigma\), \(\mu\), \(\theta_{n}\),\(f_{c}\)) = (\(4\times 10^{-6}\) s, \(2\times 10^{-6}\) s, \(\pi\)/2 rad, 64 kHz). The specific parameters of the Gabor pulses \(g_{1}(t)\) and \(g_{2}(t)\) are tabulated in Table 1.  
After calculating in MATLAB, Figure 8 shows the numerical signals with different SNRs as well as comparisons of the results obtained from the proposed method and the correlation coefficient method. The observation time is \(T=7\times 10^{-4}\) s. The dimensions of \(S_{i}(n)\) and \(\hat{x}_{1}(n)\) are \(\Gamma_{i}=181\) and \(\Gamma_{r}=2101\), respectively. The analysis step \(\gamma\) is equal to 1; therefore, the total number \(M=1920\). The horizontal axis represents the number \(M\) of calculations; while the vertical axis represents the signal spatial distances or correlation coefficients. For \(\hat{x}_{1}(t)\) with SNR = 1 dB shown in Figure 8a, the minimum values of the distance curves in Figure 8b are \(D_{1}=\min\{D(S_{i},\hat{x}_{1})\}=95.87\) and \(D_{2}=\min\{D(\overline{\widetilde{S}_{i}},\hat{x}_{1})\}=110.21\) \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Pulses** & \(A_{i}\) & \(\sigma\) & \(u\) **(s)** & \(f_{c}\) **(kHz)** & \(\theta\) **(rad)** \\ \hline \(x_{1}(t)\) & 1 & \(4.0\times 10^{-6}\) & \(3.6\times 10^{-4}\) & 64 & \(\pi\)/2 \\ \(x_{2}(t)\) & 0.5 & \(4.0\times 10^{-6}\) & \(4.0\times 10^{-4}\) & 64 & \(-\pi\)/2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of Gabor pulses \(x_{1}(t)\) and \(x_{2}(t)\). respectively. \(D_{1}<D_{2}\), therefore the front-edge of the reflector is a step down, indicating the reflector as a defect feature. The value of the \(R\)\(ICR\) is equal to 11.56 (\(R\)\(ICR\)\(>\)1.05), thus the judgement is reliable. For \(\hat{x}_{2}(t)\) with SNR = 0.1 dB in Figure 8c, the minimum values in Figure 8d are \(D_{1}=\min\{D(S_{i},\hat{x}_{2})\}=96.7\) and \(D_{2}=\min\{D(\overline{\hat{S}_{i}},\hat{x}_{2})\}=109.12\), respectively. \(D_{1}<D_{2}\), therefore the front-edge of the reflector is a step down, indicating the reflector being a defect feature. The value of the \(R\)\(ICR\) is equal to 10.94 (\(R\)\(ICR\)\(>\)1.05), so the judgement is reliable. By contrast, Figure 8e,f present plots of the correlation curves of the signals based on the [PERSON]'s correlation coefficient method (PCCM), with the Gabor excitation pulse \(S_{i}\) employed as a reference. The results using the PCCM could not represent the differences of \(x_{1}(t)\) (the reflection signal from the front-edge) and \(x_{2}(t)\) (the reflection signal from the back-edge) in amplitudes, and their SNRs are lower. Therefore, the PCCM failed to determine and evaluate the front-edge and the type of the reflector, especially when the pulses are overlapped. Figure 8: _Cont._  
The second case is performed for two overlapped pulses.