## 2 Embedded Length of Anchor Bolt inside Concrete
### Stabilization Using Equation Error Estimation and Regularization  
Equation (2) is reformulated as follows as a second-order equation:  
\[\left(\frac{V_{c}^{2}}{V_{s}^{2}}-1\right)\dot{x}_{i}^{2}-\left[2V_{c}^{2} \frac{1}{V_{s}}\left(t_{i}-\frac{b}{V_{s}}\right)+2d_{i}\right]x_{i}+V_{c}^{2} \left(t_{i}-\frac{b}{V_{s}}\right)^{2}-c^{2}-d_{i}^{2}=0 \tag{3}\]  
From Equation (3), there can be one, two or no solutions for a pair of \(x_{i}\) and \(t_{i}\). In the absence of error in all the variables defined in Equations (2) and (3) like the ultrasonic velocity, the cover thickness, the exposed length of the anchor bolt, and the measurement time and position, there will be one or two roots since it is not possible to have no solution. Figure 3: Theoretical value for the transferred distance of the ultrasonic pulse through the embedded anchor bolt according to the distance of the receiver from top surface. If there is error in the input variables or in the measured values in the equation accommodating only one double root, the equation will have two solutions or no solution. The two-solution case will be discussed later together with the equation accommodating two roots. Nevertheless, for the no-solution case, the extremum of the second-order equation will provide the closest values to the original solution because there exists a solution for the extremum of the second-order equation with a double root. Accordingly, the closest value to the original solution can be obtained by defining Equation (2) as a minimization problem defined by EEE as follows when there is no solution due to measurement errors:  
\[\underset{x_{i}}{\text{Min}}\Pi_{E}=\left(\overline{t_{i}}-t_{i}(x_{i}) \right)^{2}=\left[\overline{t_{i}}-\left(\frac{b}{V_{s}}+\frac{x_{i}}{V_{s}}+ \frac{\sqrt{\left(d_{i}-x_{i}\right)^{2}+c^{2}}}{V_{c}}\right)\right]^{2}\text { for each }i \tag{4}\]  
where \(\overline{t_{i}}\) (s) is the time actually measured by the ultrasonic device. When Equations (2) and (3) have two roots, Equation (4) has also two solutions and one solution is chosen arbitrarily depending on the selection of the numerical technique to solve the minimization problem. This problem is due to the fact that Equation (4) has actually two solutions rather than being a local minimum issue that arises in an optimization problem. Moreover, the inverse problem defined by an EEE like Equation (4) is known to be very sensitive to noises. Therefore, the regularization term is adopted to solve such instability as follows [35]:  
\[\Pi_{r}(x_{i})=(x_{i}-x_{i-1})^{2} \tag{5}\]  
Equation (5) means that the solution \(x_{i}\) of the \(i\)th measurement shall exist near the \(x_{i-1}\) of the precedent \(i-1\)th measurement. As explained for Equation (1), since the receiver is displaced sequentially downward from the top and the measurement done at each location of the receiver, the distance traveled by the P-wave along the anchor bolt also increases gradually. This proves that the value of \(x_{i}\) is close to that of \(x_{i-1}\). Adding Equation (5) to Equation (4) gives:  
\[\underset{x_{i}}{\text{Min}}(\Pi_{E}+\Pi_{r})=\left(\overline{t_{i}}-t_{i}(x_{ i})\right)^{2}+\lambda(x_{i}-x_{i-1})^{2}\text{ for each }i \tag{6}\]  
where \(\lambda\) (s\({}^{2}\)/m\({}^{2}\)) is the regularization factor. The regularization factor is used to control the extent of the regularization effect [35]. The factor does not have any physical meaning but has a unit to match the dimension of Equation (6). A too small regularization factor will make the regularization term failing in playing its role and result in the instability of the solution. In addition, a too large regularization factor will prevent obtaining information about the embedment length of the anchor bolt because the data included in the measurement \(\overline{t_{i}}\) would not be sufficiently reflected in the solution \(x_{i}\). There is still no common technique that has been developed to find the optimal regularization factor.