## 3 Numerical Application
### Application of the Optimization  
The procedure is now applied with the simulated signals. The hyper-parameters of PSO are set so \(\alpha=\beta=\gamma=0.5\), with a swarm composed of 40 particles and a maximum of 3 warm restarts. The initial maximum number of iterations is 30, which can go up to 60 if the 3 warm restarts are activated (+10 iterations by restart). The bounds of the exploration space are the same as for the sensitivity analysis in Figure 3. The supervised convergence thresholds of the model parameters are indicated in Table 2. The threshold on the Amax criterion is \(10^{-3}\), as the Amax values are of a unitary order of magnitude. They are set to correspond to a small estimation error on the material properties and on the Amax value. The procedure is statistically evaluated with 100 optimization trials from which we derive several metrics: convergence rate; number of iterations before convergence; error on the echo localization for defect (1); error on the material properties estimation. The convergence rate is defined by the number of optimized images for which Amax \(\geq\) Aref\(-2\) dB, where Aref is the maximum amplitude in the image computed with the actual material properties. The values of the different metrics are presented in Table 3.  
In terms of imaging, the procedure shows very good performances with a success rate of 98%. The estimation of the position of defect (1) is also very good with errors inferior to 1 mm along each direction in the image. The time needed to reach convergence is satisfactory with approximately 21 iterations, which corresponds to the computation of 840 images for one optimization trial. The computations were performed with an 8-core Intel(r) Xeon(r) E3-1240 v6 at 3.70 GHz, with which a TFM image of \(51\times 101\) pixels is computed in 0.83 s. Thus, the 840 images are computed in less than 12 min. However, the estimation of the unknown properties of the weld fails to provide accurate values. This is due to a strong diversity of solutions that is directly related to the use of a single scalar (Amax) to determine the quality of a TFM image. The parameter estimation is then only reliable around the inspected defect, and with 9 unknown correlated parameters with a more or less major contribution, this leads to a great variety of results. As was mentioned in the sensitivity analysis, the position of the reflector at the center of the weld also penalizes the estimation of the less contributing parameters like \(D_{1}\) which is representative of the weld width. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Elasticity** & **Threshold** & **Ogilvy’s Model** & **Threshold** \\ \hline C\({}_{11}\) & 5 GPa & \(T_{1}=T_{2}\) & 0.05 A.U. \\ C\({}_{13}\) & 5 GPa & \(D_{1}=D_{2}\) & 0.1 mm \\ C\({}_{33}\) & 5 GPa & \(\alpha_{1}=\alpha_{2}\) & 0.2\({}^{\circ}\) \\ C\({}_{55}\) & 5 GPa & \(\eta_{1}=\eta_{2}\) & 0.05 A.U. \\ \(\theta_{y}\) & 0.3 \({}^{\circ}\) & & \\ \hline \hline \endDespite the procedure not being able to accurately estimate the weld properties, it remains reliable for imaging as exemplified in Figure 4 where three images are displayed: computed with an isotropic weld model (a); with the actual material properties (b); and an optimized one (c). The optimized image was randomly chosen from the 100 solutions of the optimization trials and is a good representation of the other optimized images. For each image, side-plots contain the horizontal and vertical echodynamic curves (i.e., evolution of the maximum amplitude along each axis). In the optimized image, the degradations due to a wrong reconstruction model are corrected and the defects are now clearly identifiable. Furthermore, the optimized image and the one computed with the actual material properties look very similar. Compared to the degraded image computed with the isotropic reconstruction model (a), the Amax value is increased by 17 dB in the optimized image (c).