# Assessment of the Length and Depth of Delamination-Type Defects Using Ultrasonic Guided Waves
## 4 Verification of Proposed Delamination Depth and Length Estimation Technique
### Verification with 2D Model
#### 4.1.1 Estimation of Defect Depth  
In order to estimate the depth of the defect, the procedure described in Section 2 is employed. The reference dependency is calculated according to Equations (9)-(11) for defect lengths \(l_{1}=10\) mm, \(l_{2}=11\) mm, \(\ldots\), \(l_{\rm N}=100\) mm and defect depths \(h_{11}=0.5\) mm, \(h_{12}=1\) mm and \(h_{13}=1.5\) mm for the same set of excitation frequencies (\(f_{1}=50\) kHz, \(f_{2}=60\) kHz, \(\ldots\),\(f_{\rm n}=200\) kHz). Then, \(h_{\rm opt}\) is calculated by using Equation (4) in order to estimate the depth of the damage. The graphic illustration of \(u_{\rm std}(h,l)\) at different lengths and depths is presented in Figure 13. The criteria to estimate the actual depth of the defect are based on the selection of the minimum of \(u_{\rm std}(h,l)\). The estimated \(h_{\rm opt}\) values for sample No. 1 and sample No. 2 at different depths of the defect are summarized in Table 1.  
It can be observed that, for sample No. 1 (Figure 13a), \(h_{\rm opt}\) is close to zero for depth \(h_{12}=1\) mm (the dash-dot line in Figure 13a). Hence, the estimated depth of delamination for sample No. 1 is 1 mm, which corresponds to the actual depth of the damage. Similar observations may be made for sample No. 2. \(h_{\rm opt}\) possesses the lowest values at depth \(h_{13}=1.5\) mm (the dashed line in Figure 12b), which is in agreement with the actual depth of the damage. In the presented case, the actual depth of the  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Sample No. 1** & **Sample No. 2** \\ \cline{2-3}  & \(h_{\rm opt}\) & \(h_{\rm opt}\) \\ \hline \(h_{11}=0.5\) mm, \(h_{21}=3.5\) mm & 0.033 & 0.137 \\ \hline \(h_{12}=1\) mm, \(h_{22}=3\) mm & **0.018** & 0.091 \\ \hline \(h_{13}=1.5\) mm, \(h_{23}=2.5\) mm & 0.035 & **0.036** \\ \hline \hline \end{tabular}
\end{table}
Table 1: \(h_{\rm opt}\) values for sample No. 1 and sample No. 2 which are used as a criterion to estimate the depth of delamination.  
Figure 13: Deviation values of difference between the experimental and reference dependency: the results for sample No. 1 with a defect located 1 mm below the surface (**a**), the results for sample No. 2 with delamination situated at a depth of 1.5 mm (**b**).  
damage was included in the database. In order to evaluate the performance of the method in case of absence of actual depth in database, another set of magnitude patterns has been generated with depth increments of 0.4 mm (\(h_{11}\) = 0.4 mm, \(h_{12}\) = 0.8 mm, \(h_{13}\) = 1.2 mm, \(h_{14}\) = 1.6 mm, \(h_{15}\) = 2 mm). In the case for sample No. 1 the minimum values of the \(h_{\text{opt}}\) were close for both \(h_{12}\) = 0.8 mm and \(h_{13}\) = 1.2 mm, however the absolute minimum was obtained for depth \(h_{12}\) = 0.8 mm. For sample No. 2 \(h_{\text{opt}}\) possesses the lowest values at depth \(h_{14}\) = 1.6 mm. The amount of error of course depends on the depth increments in the database. Such database in general is computationally inexpensive (in this case U\({}_{\text{A0 ref}}\)(\(11\times 5\times 90\)), hence can be calculated small depth increments according to the desired depth estimation accuracy.