## 5 Auto Rebar Picking  
The shape of the hyperbola is a critical factor in GPR data processing, influenced by two parameters: (1) scan spacing, where shorter scan spacings (in/scan) result in wider parabolas, and (2) wave velocity, with lower dielectric constants (higher electromagnetic wave velocity) producing wider parabolas and vice versa. Additionally, larger targets generate brighter reflections; however, the change in the hyperbola shape for different target sizes is nearly negligible for any diameter under 2 in. due to target sizes being a  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Scan Parameters** & **Value** \\ \hline Samples per scan & 2048 \\ Scans per second & 47 \\ Scan per in. & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the GPR device.  
Figure 4: Prestressed concrete box beam survey grid; (**a**) top face transverse (red) and longitudinal (blue) profile lines; (**b**) prestressed concrete box beam top face transverse profile lines (red).  
fraction of the electromagnetic wavelength. The theoretical equations of the hyperbola are derived based on the following assumptions [35]: (1) the signal reflected from a rebar is phase reversal indicating that the positive peaks of the reflected waves correspond to the negative peaks of the transmitted wave; (2) the reflected wave at the surface of the rebar is the shortest two-way travel path.  
The length of traveling path of the signal reflected from a rebar can be obtained in the following equations, as shown in Figure 5:  
\[L_{1}=\sqrt{(z)^{2}+\left(X-\frac{S}{2}\right)^{2}} \tag{1}\]  
\[L_{2}=\sqrt{(z)^{2}+\left(X+\frac{S}{2}\right)^{2}} \tag{2}\]  
where \(X=x_{p}-x\) is the distance between the rebar and the center line of the transmitter and receiver in the horizontal direction; \(z\) represents the rebar depth; \(S\) is T-R offset of the antenna; \(x_{p}\) is the co-ordinate of the rebar in the horizontal direction; \(x\) is the co-ordinate of the antenna in the horizontal direction.  
The relationship between two-wave travel time and path at the peak of the hyperbola is expressed as (Figure 5b):  
\[t_{p}-t_{0}=\frac{\sqrt{4z^{2}+S^{2}}}{V_{s}} \tag{3}\]  
where \(t_{0}\) represents the time zero, indicating when the signal begins emitting into the medium, and \(t_{p}\) represents the time corresponding to the vertical co-ordinate of the peak of the hyperbola. The relationship between two-wave travel time and path can be obtained by employing Equation (3) as follows:  
\[t_{i}-t_{0}=\frac{L_{1}+L_{2}}{V_{s}} \tag{4}\]  
The traveling time of the signal is obtained by introducing Equations (1)-(3) into Equation (4) as in the following equation:  
\[t_{i}=\left(t_{p}-t_{0}\right)\frac{\sqrt{\left(z\right)^{2}+\left(X-\frac{S}{ 2}\right)^{2}}+\sqrt{\left(z\right)^{2}+\left(X+\frac{S}{2}\right)^{2}}}{\sqrt {4z^{2}+S^{2}}}+t_{0} \tag{5}\]  
Equation (5) represents the theoretical hyperbolic curve produced by a rebar embedded in reinforced concrete (RC) structures. However, the variations in hyperbolic curves corresponding to different rebar sizes are negligible.  
Figure 5: (a) EM wave travel path and (b) hyperbolic reflection from a rebar [35].  
The algorithm developed for this study processes GPR data to identify hyperbolic shapes in the radargram and extracts amplitude data for each hyperbolic peak. By comparing the amplitudes of these hyperbolic shapes with theoretical hyperbolas derived from Equation (5) and calculating the sum of squared differences (R\({}^{2}\)), the program can determine whether the hyperbolic shapes represent rebar. Users can adjust the acceptable sum of squared differences to enhance the accuracy of rebar detection by setting a predefined threshold (Q). This study limited the difference between positive amplitude hyperbolic shapes from the GPR data and theoretical curves to 80%. Once the rebar is located by the algorithm, the rebar depth and location can be determined. The algorithm's flowchart is illustrated in Figure 6.