# Detecting Reinforced Concrete Rebars Using Ground Penetrating Radars
## 4 Signal Processing
### Rebar Depth and Diameter  
The shape of a hyperbola depends on two parameters [51]: (1) scan spacing, where a smaller scan's spacing (more scans per meter) produces wide hyperbolas, and (2) wave velocity, where higher velocity (lower dielectric) produces wider hyperbolas and vice versa. Targets of larger diameter produce bright reflections. The shape of a hyperbola does not change significantly with target size for any diameter under 50 mm; all such targets are point-like for the radar as their size is a fraction of the wavelength.  
The following assumptions are made to establish the theoretical equation of the hyperbola:  
* Positive peaks of the reflected waves from rebar correspond to the negative peaks of the transmitting wave (phase reverse).
* The transmitting wave reflects at the surface of the rebar in the shortest two-wave travel path.  
Figure 4: The negative peaks times of the direct signals.  
Figure 3: Definition of the negative peak time of the direct wave.  
Figure 5a shows the schematic of reflected wave from a rebar. The shortest two-wave travel path is represented by \(L_{1}\) and \(L_{2}\) as in the following equations:  
\[L_{1}=\sqrt{\left(z+r-r\text{cos}\,\alpha\right)^{2}+\left(X-\frac{S}{2}-r\text {sin}\,\alpha\right)^{2}} \tag{4a}\] \[L_{2}=\sqrt{\left(z+r-r\text{cos}\,\alpha\right)^{2}+\left(X+\frac{S}{2}-r\text {sin}\,\alpha\right)^{2}} \tag{4b}\]  
where: \(\tan\alpha=X/z\); \(X=x_{p}-x\) is the distance between the rebar and the center line of the transmitter and receiver (TR) in the horizontal direction; \(z\) is the depth of the rebar center; \(r\) is the rebar radius; \(S\) is the spacing of the transmitter and receiver; \(x_{0}\) is the horizontal coordinate of the rebar; \(x\) is horizontal coordinate of the antenna.  
Consider the peak point of the hyperbola (point A in Figure 5b); the two-wave travel time is expressed by Equation (5) where \(t_{0}\) is time zero; \(t_{p}\) is the time of the peak of the hyperbola. Consider a point on the hyperbola; the two-wave travel time can be obtained, as can be depicted in Equation (6). Substituting Equations (4) and (5) into Equation (6) results in producing Equation (7).  
\[t_{p}-t_{0}=\frac{\sqrt{4z^{2}+S^{2}}}{V_{s}} \tag{5}\]  
Figure 5: Reflected wave from a rebar. (**a**) Wave travel path. (**b**) Hyperbolic reflection.  
\[t_{i}-t_{0}=\frac{L_{1}+L_{2}}{V_{\rm s}} \tag{6}\]  
\[t_{i}=\left(t_{p}-t_{0}\right)\frac{\sqrt{\left(z+r-r\cos\alpha\right)^{2}+ \left(X-\frac{\rm S}{2}-r\sin\alpha\right)^{2}}+\sqrt{\left(z+r-r\cos\alpha \right)^{2}+\left(X+\frac{\rm S}{2}-r\sin\alpha\right)^{2}}}{\sqrt{4z^{2}+S^{ 2}}}+t_{0} \tag{7}\]  
Equation (7) depicts the hyperbolic curve resulting from a rebar embedded in RC structures and is depicted for various rebar sizes in Figure 6. It is evident that the disparities between these curves are negligible. The suggested steps for solving for the rebar depth (\(z\)) and radius (\(r\)) in Equation (7) are as follows:  
* Estimate the electromagnetic wave velocity.
* Calculate the rebar depth using Equation (5).
* Identify the coordinates at the peak of the hyperbola.
* Calculate the rebar radius using Equation (7).