## 2 Methodology
### PoD Concept for Passive Signals  
The PoD, a function of the damage size, assesses the smallest damage size and combines its quantitative and qualitative parameters [46]. The PoD concept used for the passive NDT methods addresses the construction of a curve based on the related parameters of the damage. Two approaches can be employed to generate PoD curves for determining the detection probability. The detectability of damage with known size, location, or quantity is determined by assessing the accuracy of the simulation results. In this regard, the identification of defects is established by evaluating the ratio of hits to misses or the total number of hits, as given by [PERSON] et al. [47]. The hit/miss approach does not concern itself  
Figure 1: Waveform and spectrum characteristics of an AE signal.  
with signal values but instead predicts the PoD curve based on binary outcomes. In other words, pit defines the ratio of identified defects, as expressed in Equation (1) [47; 48].  
\[PoD_{HM}(n)=\frac{M_{A}(n)}{H(n)\bigcup\,T},\text{ for }A\geq A_{T} \tag{1}\]  
where \(n\) is the sequence of the simulated defect event; \(A\) is the amplitude, which should be equal to or higher than the threshold \(A_{T}\); \(M\) is the missed detections; \(H\) is the number of the detected hits, and \(T\) is the total amount of the missed and detected hits.  
Since this bonded and discrete binary hit/miss model has only two possible values, 0 (miss) or 1 (hit), it cannot bridge the detectability of the in-between flaw sizes [49].  
On the other hand, the second approach involves linking the probability of the signal response with a function by constructing a PoD curve based on a parameter distribution associated with the size of the actual damage data. In this case, when the recorded signal response of the damage is correlated with the characteristic damage size, the initial hypothesis is that the detectability of damage progression, e.g., crack growth, increases with the rise in signal amplitude and energy. Amplitude and energy are related to the release of energy during the source event. Therefore, since the size of the damage is unknown, amplitude and/or energy are defined as independent variables to formulate the PoD model instead of the defect size as the measured variable. In the signal response approach, the PoD function is defined as shown in Equation (2) [45; 50]:  
\[PoD_{SR}(p)=\Phi\bigg{(}\frac{\ln(p)-\mu}{\sigma}\bigg{)} \tag{2}\]  
where \(p\) is the signal parameter, \(\Phi\) is the standard normal cumulative distribution function, \(\mu\) is the mean, and \(\sigma\) is the standard deviation of the \(p\) distribution.  
Another hypothesis is that the cumulative AE energy is linearly proportional to the total crack length [38]. The classical [PERSON] framework for signal response defines a linear regression between a (cumulative AE energy) and \(a\) (crack length) by using the Maximum Likelihood Estimation (MLE). PoD function used in this study, as defined by the [PERSON] framework, is given in Equation (3):  
\[\ln(\hat{\text{a}})=\beta_{0}+\beta_{1}\ln(a)+\delta \tag{3}\]  
where \(\beta_{0}\) and \(\beta_{1}\) are parameters of linear response on cumulative AE energy and \(\delta\) is the random error. The PoD of this approach is defined by Equation (4):  
\[PoD_{SR}(a)=\Phi\Bigg{(}\frac{\ln(a)-\frac{\ln a_{thr}-\beta_{0}}{\beta_{1}} \mu}{\frac{\sigma_{\delta}}{\beta_{1}}}\Bigg{)} \tag{4}\]  
where \(a_{thr}\) is the minimum response threshold and \(\sigma_{\delta}\) is the standard deviation of \(\delta\).