## 4 Results
### Delamination Depth Estimation Results  
In this subsection, the results of estimation of the delamination depth obtained using the proposed approach are presented. The results from both the numerical simulations and experiments are summarized in Table 3. Figure 8a shows the history of the objective function value over iterations for Step 1 (P- and S-wave velocity updates) processed with the numerical simulation data. The figure indicates that the optimal P- and S-wave velocities were obtained at 13 iterations, providing the updated P- and S-wave velocities of 3739.2 m/s and 2270.9 m/s, respectively. The updated wave velocities are close to the actual P- and S-wave velocities of 3727 m/s and 2282 m/s, respectively. The objective function over the search space (possible delamination depth) for Step 2 (delamination depth estimation) processed with the numerical simulation data is presented in Figure 8b. In the figure, a global minimum is observed at a depth of 0.06 m, and the estimated delamination depth at the global minimum exactly matches the actual delamination depth.  
The dispersion curves extracted from the numerical simulation data (Figure 7b) and those theoretically computed with the estimated delamination depth are compared in Figure 9. The dispersion curves for the two cases show good consistency with each other.  
Figure 8: Objective history for numerical simulation results: (a) objective history for Step 1 in the data processing approach and (b) objective function for Step 2 over the search space. The actual delamination depth is 0.06 m.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**Numerical Simulations**} & \multicolumn{3}{c}{**Experiments**} \\ \hline \hline  & \multicolumn{3}{c}{Update of the P- and S-wave velocities} \\ \hline \hline  & \(c_{P}\) & \(c_{S}\) & \(c_{P}\) & \(c_{S}\) \\ \hline Initial guess (m/s) & 4000 & 2000 & 4000 & 2000 \\ Updated velocity (m/s) & 3739.2 & 2270.9 & 3580.6 & 2157.1 \\ Actual velocity (m/s) & 3727 & 2282 & & \\ \hline \hline  & \multicolumn{3}{c}{Delamination depth estimation} \\ \hline Estimated depth (m) & 0.06 & 0.0506 \\ Actual depth (m) & 0.06 & 0.06 & 0.06 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of results for the update of bulk wave velocities (\(c_{P}\) and \(c_{S}\) ) and estimation of the depth of concrete delamination.  
Subsequently, the history of the objective function value over iterations for Step 1 processed with the experimental data is presented in Figure 10a. Similar to the numerical simulation results shown in Figure 8a, the optimal P- and S-wave velocities were obtained at 13 iterations. The updated P- and S-wave velocities are 3580.6 m/s and 2157.1 m/s, respectively. Figure 10b shows the objective function over possible delamination depth values for Step 2 processed with the experimental data. A global minimum is observed at a depth of 0.0506 m, providing a good estimation of the delamination depth close to the actual delamination depth (0.06 m). The absolute error in the estimation of the delamination depth was 0.0094 m (15.7% error). The presented numerical simulation and experimental results demonstrate the feasibility and good performance of the proposed ultrasonic data processing approach in the quantitative estimation of concrete delamination.