## 4 Experimental Applications
### Application of the Optimization  
We have little information on the material properties, especially concerning its elasticity. It is therefore a blind-test application of the procedure. The base material is 316L stainless steel where the celerity of longitudinal waves is 5720 m\(\cdot\)s\({}^{-1}\) and we set the density to \(\rho=7.80\) g\(\cdot\)cm\({}^{-3}\). As we can see on the macrograph and photographs, the dendritic structure of the weld is asymmetrical and shows a tilt of about 20\({}^{\circ}\) of the main axis of the grains. Thus, to account for this dissymmetry, the eight Ogilvys' parameters should all be used (\(D_{1}\
eq-D_{2}\), \(\alpha_{1}\
eq-\alpha_{2}\), \(-T_{1}\
eq T_{2}\) and \(\eta_{1}\
eq\eta_{2}\)). We also discard the \(\theta_{y}\) parameter and we introduce a new parameter \(\zeta\), which is the disorientation angle of the main axis of the grains with respect to the \(Z\) axis. The optimization is now conducted on 13 parameters instead of 9. Again, the optimization is evaluated over 100 trials for each case with the same hyper-parameter values as in the numerical application. The imaging area is restricted around defect (1) during the optimization and has the same size and definition as before. The performance metrics are summarized in Table 4 for the two configurations of inspection. As we do not precisely know the material properties, the estimation error of the weld properties is replaced by the dispersion of the results, with respect to the mean values of the parameters within the upper and lower bounds defined for the exploration space. These experimental results are in good agreement with the numerical case. For both weld samples, the convergence rate is once again satisfactory with weak errors on the positioning of the defect echo. We can however remark that the numbers of iterations before satisfying the convergence criteria are different and that the procedure converges faster with an OL60\({}^{\circ}\) configuration than with an OL0\({}^{\circ}\) configuration. Here again, the dispersion of the weld models after the optimization is non-negligible in both cases, although they are found to be better than in the previous simulated applica \begin{table}
\begin{tabular}{c c c} \hline \hline
**Metrics** & **OL0\({}^{\circ}\)** & **OL60\({}^{\circ}\)** \\ \hline Convergence rate (\%) & 84 & 86 \\ Number or iterations & \(30.3\pm 18.6\) & \(12.1\pm 8.4\) \\ Weld model dispersion (\%) & 17.5 & 18.2 \\ \hline Error X (mm) & \(0.7\pm 0.4\) & \(1.0\pm 0.6\) \\ Error Z (mm) & \(0.3\pm 0.2\) & \(0.3\pm 0.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performances of the optimization procedure with experimental data obtained with OL0\({}^{\circ}\) and OL60\({}^{\circ}\) configurations of inspection on two samples of the same weld: values obtained with 100 optimization trials. Figure 7: Macrograph of the weld lent by EDF (**a**). Photographs of the two samples (**b**) et (**c**). The two samples have the same height. tion. The characterization of the material is consequently not accurately achieved because of the non-uniqueness of the solution due to many varying unknown parameters. To exemplify the improvement of the images after the optimization, two final images are randomly picked from the 100 trials of the two inspection cases and are displayed in Figure 8. They are accompanied by the corresponding degraded images, due to the use of the isotropic reconstruction model. The weld model parameters used to compute the optimized images are indicated in Figure 9 along with the associated slowness curves. Figure 8: Images in a 316L stainless steel V-shape weld with experimental signals: isotropic reconstruction model (**a,c**); PSO optimization (**b,d**). The vertical and horizontal echodynamic curves are presented in the side-plots. The optimizations are conducted around SDH (1), and each image is normalized by the maximum amplitude in a 10 \(\times\) 10 mm\({}^{2}\) region around this defect (the amplitude of this defect echo is 0 dB for all images).