## 5 Results and Discussion
### Leakage Location Analysis  
To verify the reliability of the MLPNN locator, the gas leakage was located by the CCF location method and the MLPNN locator under different conditions, which are maintaining a distance of 2.16 m between the simulated leak orifice and sensor 1 and varying the distance (1.08 m, 2.36 m and 3.54 m) between the simulated leak orifice and sensor 1. In addition, the group speed of the leakage-induced signal was determined by the peak frequency of the leakage-induced signal and the group speed dispersive curve of the fundamental flexural mode. Tables 2 and 3 show the location results obtained by the CCF method and the MLPNN locator. The relative location errors in Table 3 show that the relative location errors increase as the distance between the two sensors increases, which implies that some high frequency components of the leakage-induced signal attenuate as the propagation distance increases. Figure 15a,b show the histograms of the relative location errors in Tables 2 and 3, respectively. The vertical axis represents the number of measurements, and the horizontal axis represents the percentage of relative location error. These two tables show the measurement results of twelve groups by using the proposed method and the cross-correlation function, respectively, with twelve measurements in both Figure 15a,b. Despite the small sample size, all the measurement results are typical and can show the location accuracy of the two location methods. The averages of the relative location errors are 18% and 4%, respectively, which indicates that the average of the relative location errors obtained by the MLPNN locator is reduced by 14% compared to that using the CCF location method. Consequently, the MLPNN locator is more suitable for locating a leak in a gas pipe than the CCF location method is. However, the MLPNN locator suffers the drawback of needing to be trained again if one of the following experimental conditions changes: for example, the distance between the two AE sensors, the geometric parameters of the pipe wall, or the material of the pipe wall. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(L\) (m) & 3.24 & 3.24 & 3.24 & 4.52 & 4.52 & 4.52 & 4.52 & 5.70 & 5.70 & 5.70 & 5.70 \\ \hline \(f\) (kHz) & 124.2 & 123.8 & 123.4 & 125.4 & 105.5 & 103.4 & 107.2 & 106.7 & 104.7 & 104.3 & 104.7 & 103.5 \\ \(v\) (m/s) & 2404 & 2400 & 2397 & 2415 & 2200 & 2173 & 2222 & 2215 & 2190 & 2186 & 2190 & 2175 \\ \(l_{1}\) (m) & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 & 2.16 \\ \(\hat{l}_{1}\) (m) & 1.78 & 1.71 & 1.82 & 1.72 & 1.70 & 1.77 & 1.75 & 1.79 & 1.83 & 1.74 & 1.79 & 1.83 \\ \(\Delta l_{1}\) (m) & 0.38 & 0.45 & 0.34 & 0.44 & 0.46 & 0.39 & 0.41 & 0.37 & 0.33 & 0.42 & 0.37 & 0.33 \\ \(\delta l_{1}\) (\%) & 17.6 & 20.1 & 15.7 & 20.4 & 21.3 & 18.1 & 19.0 & 17.1 & 15.3 & 19.4 & 17.1 & 15.3 \\ \hline \hline \end{tabular} \(L\) is the distance between the two AE sensors, \(f\) is the peak frequency of the leakage-induced signal, \(v\) is the group speed estimation of the leakage-induced signal, \(l_{1}\) is the real distance between the leak and sensor 1, \(l_{1}\) is the distance estimation between the leak and sensor 1, \(\Delta l_{1}\) is the leakage location error, \(\delta l_{1}\) is the relative location error.  
\end{table} 
Table 2: The location results obtained by the CCF location method.