## 3 Results and Discussion
### Probability of Detection (PoD) with Inter-Granular Cracks
The DI in the pristine state was evaluated with the following:  
\[DI_{pristine}=1-\left|\frac{Cov(P_{i},P_{j})}{\sigma_{P_{i}},\sigma_{P_{j}}} \right|=1-\frac{\sum_{t=0}^{T}\left(P_{j}-\sigma_{P_{j}}\right)\left(P_{i}- \sigma_{P_{i}}\right)}{\sigma_{P_{i}},\sigma_{P_{j}}} \tag{21}\]  
where \(P_{i}\) is the signal in a pristine state with a fixed actuation signal location and 1 [MHz] frequency, while \(P_{j}\) is the signal in a pristine state with uncertainties in both the actuation location and the frequency. As this procedure was followed for each morphology, five different DIs in pristine condition were obtained. Figure 30 shows the DI in a pristine condition for the 100-grain case with a fixed morphology (i.e., #1). The DI exhibited a normal distribution, as illustrated by the black line on the right side of the plot. The dashed red lines refer to one standard deviation or 1\(\sigma\). The DI values ranged from 0.24 to 0.34, with a mean value of 0.29. Figure 31 presents the DI in a damaged condition for the 100-grain case (morphology #1), which was plotted against the crack length. As expected, the results clearly demonstrated that the DI increased with an increase in crack length. The dashed red line corresponds to the threshold value, while the dashed black line represents the mean value of the DI for each crack length. Figure 30: Thresholdfor 100 grains and morphology #1 in the pristine case. There were two main approaches commonly employed to generate PoD curves, namely, hit/miss data and signal response analysis (\"\(\hat{a}\)\" vs. \"\(\hat{a}\)\") [60; 61]. The hit/miss approach provided binary results, indicating whether a flaw of a known size was detected or not. In this analysis, the number of detected cracks, \"n\", was divided by the total number of simulated cracks, \"N\", thereby yielding a reasonable assessment of the inspection system's capability, \(PoD=\frac{n}{N}\). This approach provided a single value representing the entire range of crack sizes. However, since larger cracks are typically easier to detect than smaller ones, cracks are often grouped based on their sizes, and \(\frac{n}{N}\) was calculated for each size range. Figure 32 illustrates the PoD obtained using the hit/miss approach, which was plotted against the crack length. As depicted in the figure, the PoD increased as the crack length increased, reaching its maximum at a crack length of 5 [mm]. The second approach for determining the PoD was of much interest for several practical applications since it gave more information through the recorded defect response \"\(\hat{a}\)\", which was correlated with the defect characteristic dimension \"a\" [60; 61; 63; 64]. This method utilizes regression analysis with Maximum Likelihood Estimation (MLE) to calculate the regression coefficients and their covariance matrix. From the variance of the expected regression response, confidence bounds at a 95% level can be constructed for \"a\". To compute the PoD parameters, the threshold value was determined based on the consideration stated before. The variance (\(var(PoD)\)) was utilized to estimate confidence intervals and the covariance matrix for PoD. The PoD could be generated for each crack by calculating the probability of the signal response when it exceeds the decision threshold value. In this analysis, the target variable was the crack length. Figure 33 depicts the  
Figure 31: Damage index for 100 grains and morphology #1. Figure 24:0 Hit/miss analysis for the 100-grain case. regression analysis for the 100-grain case, where the crack length is represented on the \"\(x\)\" axis and the DI is displayed on the \"\(y\)\" axis. Figure 34 illustrates the PoD curve for the 100-grain case as a function of crack length. The important features of the model were recorded on the curve, including the 95% confidence interval (CI) area, as well as both the lower and upper bound of the PoD curve and the mean value. Furthermore, the \(a_{90}\) value, which is the 95% confidence bound on the \(a_{90}\) estimate, is displayed in the plot.