## 3 Analysis of Experimental Data and Results
### Model Validation  
In order to validate the developed AE model, model predictions of crack growth rate were compared against the validation experimental data set. For a given value of AE count rate, a prediction of crack growth rate was estimated based on the developed AE-based model. Available information captured from AI7075-T6 experimental data was used as the input to the Bayesian estimation procedure, as the model was developed based on AI7075-T6 data. The Bayesian estimation approach updated the model prediction with the experimental results. With this Bayesian estimation inference, uncertainties in the experimental values were propagated in the model and resulted in model prediction uncertainty assessment. The prediction results were then compared against the true crack growth rates obtained by validation experiments. In the proposed model validation methodology [31], both model prediction and experimental results are considered to be estimations and representations of the true values, given some error as it is shown in Equations (8) and (9):  
\[\frac{X_{i}}{X_{e,i}}=F_{e,i};\ F_{e}\ \sim LN(b_{e},s_{e}) \tag{8}\]  
and  
\[\frac{X_{i}}{X_{m,i}}=F_{m,i};\ F_{m}\ \sim LN(b_{m},s_{m}) \tag{9}\]  
where \(X_{i}\) is the true value, \(X_{e,i}\) and \(X_{m,i}\) indicate the experimental results and model prediction, respectively. \(F_{e}\) is the multiplicative error of the experiment with respect to true value, and \(F_{m}\) is the multiplicative error of the model prediction with respect to true value. A multiplicative error of the experiment with respect to the model prediction is defined by Equation (10):  
\[\frac{X_{e,i}}{X_{m,i}}=\frac{F_{m,i}}{F_{e,i}}=F_{t,i} \tag{10}\]  
Since both \(F_{m,i}\) and \(F_{e,i}\) distributions are lognormal, the distribution of \(F_{t,i}\) would also be lognormal with mean and standard deviation of \((b_{m}-b_{e})\) and \(\sqrt{s_{m}^{2}+s_{e}^{2}}\), respectively. In this approach, the likelihood used is shown in Equation (11):  
\[L(X_{e,i}/X_{m,i},b_{e},s_{e}|b_{m},s_{m})=\prod_{i=1}^{n}\frac{1}{\sqrt{2\pi} \left(\frac{X_{e,i}}{X_{m,i}}\right)\sqrt{s_{m}^{2}+s_{e}^{2}}}exp\left(- \frac{1}{2}\times\frac{\left[ln\left(\frac{X_{e,i}}{X_{m,i}}\right)-(b_{m}-b _{e})\right]^{2}}{s_{m}^{2}+s_{e}^{2}}\right) \tag{11}\]  
Using the validation sets of data, the validation approach was implemented and the results are discussed in this section. For more information about the validation method, refer to the paper by [PERSON] [31]. For simplicity, the distributions of model predictions were reduced to a mean value and compared one-to-one with the experimental results. The mean and standard deviation of \(F_{e}\), which are \(b_{e}\) and \(s_{e}\), were determined from the unbiased experimental error of \(\pm 1\%\). The values determined were \(-0.00002\) for \(b_{e}\) and \(0.002\) for \(s_{e}\). The summary statistics for the marginal posterior _pdf_ of parameters \(b_{m}\) and \(s_{m}\) as well as the distribution of \(F_{m}\) are presented in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Mean** & **Standard Deviation** & **2.5\%** & **Median** & **97.5\%** \\ \hline \(b_{m}\) & 0.009 & 0.011 & \(-0.013\) & 0.009 & 0.031 \\ \(s_{m}\) & 0.059 & 0.008 & 0.045 & 0.058 & 0.078 \\ \(F_{m}\) & 1.011 & 0.061 & 0.894 & 1.009 & 1.141 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Model validation statistic summary. The model uncertainty bounds for the crack length estimation can be determined from the 2.5 and 97.5 percentile of the multiplicative error of \(F_{m}\). The resulting upper bound on reality was calculated as 14%, while the lower bound is \(-10\%\). These results are presented graphically in Figure 10. It can be noticed that the mean values of the model prediction are standing on the upper bounds of the experimental error, and the value of \(F_{m}\) for the model prediction is greater than 1. The value of \(F_{m}\) around 1.01 suggests a very small bias in the AE model to under predict the true crack growth rate.