## 3 Experimental Verification
### Localization and Imaging  
The parameters of the group velocities and the arrival times as detailed above were used to realize localization and imaging of the small crack. By combining individual source images obtained from all sensing paths in the network, the final identification image was produced as shown in Figure 8. Based on the ellipse method, the positioning results of the small crack based on linear and nonlinear features are shown in Figure 8a,c, respectively. It can be seen that the positioning results based on two features are both not ideal. The elliptical loci based on linear features do not fall on the position of the small crack at all. Conversely, although certain elliptical loci based on nonlinear features appear on the position of the small crack, the location of the small crack cannot be effectively distinguished from other intersection points.  
When the damage is situated in the propagation path or close to the sensor position, the corresponding elliptical loci cannot be constructed [2]. Thus, the signals received from such propagation paths, namely, the sensing paths AC and BD, should be excluded for accurate damage localization. The re-positioning results based on linear and nonlinear features are shown in Figure 8b,d, respectively. It can be seen that the elliptical loci based on nonlinear features fall accurately on the location of the small crack, while the elliptical loci based on linear features still do not.  
By seeking the intersections of most of these elliptical loci, the small crack position can be localized and highlighted. The center of the small crack is at coordinates (100, 50), and there are seven intersections of most of the elliptical loci based on linear features, all far away from the coordinates (100, 50). However, there are only two intersections of most of elliptical loci based on nonlinear features at coordinates (95.5, 49.5) and (102, 49.5), which are close to the coordinates (100, 50). It is concluded here that the small crack cannot be identified based on linear feature (Figure 8b), and such a failure can be attributed to the minute dimension of the small crack, which causes not much wave scattering (the wavelength of the incident wave is around 20 mm at the selected excitation frequency, greater than the length of the small crack). While the small crack can be identified and localized based on nonlinear feature (Figure 8d). These elliptical loci are thickened for clarity. The actual location of the small crack is highlighted by a red line, while the blue plus signs indicate the intersections.  
Figure 7: Spectrogram of the signals received from the sensing path AB in the current state, and amplitude profiles at the fundamental and double frequency for both the intact and the current state.  
involves the small crack at coordinates (100, 50) and includes but is not limited to the intersections at coordinates (95.5, 49.5) and (102, 49.5). The distance between the two horizontal lines within the crossover area can be used to approximately estimate the length of the small crack (about 1 mm) in the \(y\)-axis direction. Thus, the modified ellipse method utilizing the imperfect group-velocity matching of nonlinear S0 modes can achieve the small crack localization, and the result for damage localization becomes more accurate and more convincing compared with that of the ellipse method. It can be further seen from the identification results that the thus-determined location area can also reveal the size and shape, and the damage level of the small crack to a certain extent.