## 3 Results and Discussion
### Probability of Detection (PoD) with Inter-Granular Cracks
Figure 24: Signals in pristine condition: different number of grains for aluminum. Figure 25: Signals in damaged condition with a 2 [mm] crack: 100 grains and 5 different morphologies for aluminum. Figure 27 presents the signals obtained under the damaged condition with varying crack orientations, namely 0\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\). Notably, the backscattered signal from the crack with a 0\({}^{\circ}\) orientation exhibited the highest amplitude, while the amplitude decreased as the crack angle increased. This behavior can be attributed to the fact that the wave is perpendicular to the 0\({}^{\circ}\) crack, thus resulting in maximum signal strength. Furthermore, the duration of the backscattered signal increased with an increasing crack angle. This is because of the decrease in signal amplitude, which requires more time for the same energy to be dissipated. Figure 28 displays the signals obtained for a 2 [mm] crack length, in which different numbers of grains were considered. As depicted in the figure, the signals exhibited excellent agreement in both amplitude and the ToA. Figure 27: Signals in damaged condition with different orientation: 0\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\) crack for aluminum. To eliminate the effect of multiple boundaries reflecting the signal, there were two choices to select with respect to the inspection interval (time window):  
1. Up to the first back-wall reflection signal (time interval in red arrows in Figure 29);
2. Up to the damaged backscattered signal (time interval in green arrows in Figure 29). In this study, the selected time interval of inspection was the first one and, in the case that the first wave packet was ambiguous and could not be clearly identified, the time window for the first wave packet was manually determined based on the signal characteristics. Let \(D_{i}\) denote the damaged signal and let \(P_{i}\) denote the corresponding pristine state signal. The residual signal \(r_{i}\), was obtained by subtracting the pristine signal from its damaged signal: \(r_{i}=D_{i}-P_{i}\). Many existing approaches for damage detection involve extracting damage-sensitive features from UGW signals and establishing an index that indicates the presence of damage (commonly referred to as the Damage Index (DI)). The damage index was then compared to a threshold value, above which the presence of damage was signified. The damage detection strategy involved comparing the current signal with its baseline signal and quantifying the changes. In this case, the current signals were recorded under the same environmental and loading conditions, and any deviation from the baseline signal was attributed to the presence of damage. The magnitude of the deviation was quanti  
Figure 28: Signals in damaged condition with a 2 [mm] crack: different number of grains for aluminum. Figure 29: Time interval of inspection. fied by the DI. The damage index used in this analysis was the reduction in correlation coefficient between the signal and its baseline, noted as \(DI_{SDC}\):  
\[DI_{SDC}=SDC=1-\left|\frac{Cov(P_{i},D_{i})}{\sigma_{P_{i}},\sigma_{D_{i}}} \right|=1-\frac{\sum_{t=0}^{T}\left(D_{i}-\sigma_{D_{i}}\right)\left(P_{i}- \sigma_{P_{i}}\right)}{\sigma_{P_{i}},\sigma_{D_{i}}}, \tag{20}\]  
where \(Cov(P_{i},D_{i})\) is the covariance of the signals and \(\sigma_{I_{i}}\) is the standard deviation of the signals, respectively. This damage index takes into consideration both the difference in amplitude and in the ToA. There are many ways to evaluate the threshold value. In this study, the threshold value was evaluated as follows:  
1. Fixing the morphology;
2. Changing the frequency with a mean of 1 [MHz] and standard deviation of 5% (uncertainty on frequency);
3. Changing the location of the actuation signal (uncertainty on location).