## 3 Numerical and Experimental Verification
### Numerical Verification
Figure 5: Results of the numerical verification for white noise and regularization in specimens with a cover thickness of 100 mm: (**a**) E100C100; (**b**) E200C100; (**c**) E300C100. Figure 6: Results of the numerical verification for white noise and regularization in specimens with a cover thickness of 200 mm: (**a**) E100C200; (**b**) E200C200; (**c**) E300C200. When the problem was solved using Equation (5), which does not apply regularization, numerous outliers occurred as shown in Figures 5 and 6. These outliers can be credited to the fact that a solution that is not the exact solution had to be selected through the optimization of Equation (5) among the two roots of the second order equation in Equation (3). If the result including the outlier is used, the final value \(x^{*}\) cannot be obtained properly. However, the results of Equation (6), which applies regularization, are free of outliers, which eases the assessment of the convergence value. Moreover, the solutions obtained by Equation (5) without the regularization exhibit the wide dispersion, whereas the solutions using regularization show relatively lesser dispersion and smooth continuity that ease the assessment of the convergence value. The comparison of the exact solution with the results of Equation (6) reveals that the regularization tended to delay the convergence. This delay is common when using the precedent solution in the regularization as in Equation (6) but this delay does not affect the final value itself [40]. The exact value of the P-wave transfer velocity, which is one major variable in Equation (2), is often difficult to determine. The most accurate value can be obtained if the velocity is measured directly on site but the measured value is likely to be polluted by various errors like measurement error. Figure 7 plots the results of Equation (6) in the cases where the velocities of the P-wave in the anchor bolt and in the concrete have errors of 1, 2, 3% for specimen E100C100 case. When errors of \(-\)3, \(-\)2, \(-\)1, 1, 2, and 3% were introduced in the P-wave velocity \(V_{s}\) in the anchor bolt, the estimated embedded depth was, respectively, 110.5, 106.6, 102.8, 95.9, 92.7, and 89.7 mm corresponding to estimation errors of 10.5, 6.6, 2.8, 4.1, 7.3, and 10.3%. When errors of \(-\)3, \(-\)2, \(-\)1, 1, 2, and 3% were introduced in the P-wave velocity \(V_{c}\) in concrete, the estimation provided embedment depths of 156.5, 137.8, 118.8, 79.6, 59.4, and 38.9 mm corresponding to estimation errors of 56.5, 37.8, 18.8, 20.4, 40.6, and 61.1%, respectively. The estimated embedment depth of the anchor bolt appeared thus to be very sensitive to the error in the P-wave velocity. Consequently, the velocity should be measured as accurately as possible to apply the proposed method. The estimated results were also seen to be more sensitive to the P-wave velocity in concrete than to that in the anchor bolt because the path of the pulse was longer in concrete than in the anchor bolt. Therefore, the sensitivity can be modified by changing the ratio of the concrete cover thickness to the embedded depth of the anchor bolt. For example, if the thickness of the concrete cover is thin, the length of the P-wave transmitted through concrete is short, so the P-wave velocity of concrete will not have a significant effect on the estimated results. Conversely, if the thickness of the concrete cover is thick, the length of the P-wave transmitted through concrete becomes longer, so the P-wave velocity of concrete will have a great influence on the estimated results. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Specimen ID**} & \multicolumn{2}{c}{**Numerical Verification**} \\  & \multicolumn{2}{c}{**(with 3\% White Noise)**} \\ \cline{2-3}  & **Estimated Value** & **Error** \\  & **(mm)** & **(\%)** \\ \hline E100C100 & 100.1 & 0.1 \\ E200C100 & 202.2 & 1.1 \\ E300C100 & 298.4 & 0.5 \\ E100C200 & 102.9 & 2.9 \\ E200C200 & 199.9 & 0.0 \\ E300C200 & 315.9 & 5.3 \\ \hline \hline \end{tabular}
\end{table}