## 3 Numerical and Experimental Verification
### Numerical Verification  
As shown in Figure 4, the numerical example considered specimens with various concrete cover thicknesses (100, 200 mm) and different embedment lengths of the anchor bolt (100, 200, 300 mm). Table 1 lists the designation of the specimens according to the embedment length of the anchor bolt and the cover thickness. In the designation, the numbers after \"E\" and \"C\" indicate, respectively, the embedment length of the anchor bolt and the cover thickness in mm. The measurement of \(x_{i}\) was computed using Equation (2) so that the arrival time \(t_{i}\) became minimal for the position \(d_{i}\) of the receiver. A minimization problem is defined to generate measured values for the numerical example using Equation (2). \[\underset{x_{i}}{\text{Min}}\overline{t_{i}}=\frac{b}{V_{s}}+\frac{x_{i}}{V_{s }}+\frac{\sqrt{\left(d_{i}-x_{i}\right)^{2}+c^{2}}}{V_{c}}\text{ subjected to }x_{i}\leq e \tag{7}\]  
where the embedded depth of the anchor, the cover thickness, and the exposed length of the anchor are as given for each specimen. The adopted velocity of the ultrasonic pulse is 3800 m/s in concrete and 5900 m/s in the anchor [36; 37; 38; 39]. The value of \(t_{i}\) corresponding to \(x_{i}\) calculated by Equation (7) is accepted as the exact measured value. In order to consider the case where perfect measurement is not \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Embedment Depth (mm)**} \\ \cline{3-5}  & & **100** & **200** & **300** \\ \hline Cover Thickness & 100 & E100C100 & E200C100 & E300C100 \\ (mm) & 200 & E100C200 & E200C200 & E300C200 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Designation of the specimens in the numerical verification. Figure 4: Specim for numerical verification. achieved due to the presence of voids or irregularities in concrete, the exact measured values were mixed with a uniformly distributed white noise of \(\pm\)3% of the measurement. Figures 5 and 6 present the results of the simulation for the cases with a cover thickness of 100 and 200 mm, respectively. The convergence value where \(x_{i}\) starts to converge was determined by the engineer's decision. The final value \(x^{*}\) was decided to be the average of the transferred distance of the ultrasonic pulse through the embedded anchor bolt after the convergence. As shown in Figures 5 and 6, the scattering of the test data is very wide before the introduction of regularization, so it is impossible to determine the point at which the value begins to converge. Therefore, it is necessary to find a convergence value by reducing the scattering degree of test data by introducing regularization and using an appropriate regularization factor. In this study, an optimal value of 0.02 for the regularization factor was obtained empirically and used for solving Equation (6). If a regularization factor higher than 0.02 was used, the scattering degree can be reduced, but there was a disadvantage in that the data convergence is delayed. As shown in Figure 6c, when a regularization factor of 0.02 was used, the convergence value could be determined within the range of the x axis (0-1000 mm) in the case of the E300C200 specimen. Therefore, the regularization factor of 0.02 was used for the equal comparison of all specimens. The determination of the regularization factor was discussed at the end of the present section. The resulting estimated embedment depth of the anchor bolt for the specimens E100C100, E200C100, E300C100, E100C200, E200C200, and E300C200 was, respectively, 100.1, 202.2, 298.4, 102.9, 199.9, and 315.9 mm, which showed respective errors of 0.1, 1.1, 0.5, 2.9, 0.0, and 5.3% compared to the exact depths of 100, 200, 300, 100, 200, and 300 mm. Table 2 summarizes the results. Even though a white noise of 3% was introduced, the error remained below 5.3% in all six cases. At the exception of case E300C200, the error remained at an ignorable level. Consequently, the proposed method appeared to provide good results against the white noise disturbance of the measuring device.