# Recognition of Rebar in Ground-Penetrating Radar Data for the Second Lining of a Tunnel
## 2 Materials and Methods  
The tunnel consists of different media layers, as shown in Figure 1. Here, the external pressure on the tunnel is nonlinear and asymmetric [22]. As part of the secondary lining's support structure, rebar was used to reinforce the lining and improve the bearing capacity of the lining. The essential principle of the GPR system is for the antenna transmitter (T) to emit high-frequency electromagnetic waves as brief broadband pulses to the target medium. These electromagnetic waves are subsequently reflected by the medium and received by the radar antenna receiver (R). The electromagnetic wave emitted by T is reflected at the electromagnetic wave impedance interface formed by the rebar and concrete. Part of the  
Figure 1: The main structural components of the tunnel media. reflected wave penetrates the tunnel's surface and is received by R, and the other part is reflected again by the tunnel surface back to the rebar. This process is shown in Figure 2a. The electromagnetic wave emitted by GPR passes through the tunnel medium. The propagation velocity and attenuation coefficient of the wave are expressed as  
\[\beta=\frac{\sigma}{2}\sqrt{\frac{\mu}{\epsilon}} \tag{1}\]  
where \(\epsilon\) is the relative permittivity, \(\sigma\) is the electric conductivity and \(\mu\) is the magnetic conductivity. As the electromagnetic wave propagates from concrete with a low dielectric constant to the rebar with a high dielectric constant, the waveform of the reflected wave is changed. The reflection coefficient describes the reflection of an electromagnetic wave when it reaches an interface between media with different electromagnetic properties [18; 21]. Therefore, the reflection coefficient can be used to analyze the characteristics of the medium. The reflection coefficient is expressed as  
\[\gamma=\frac{\eta_{2}\cos\alpha_{1}-\eta_{1}\cos\alpha_{2}}{\eta_{2}\cos\alpha _{1}+\eta_{1}\cos\alpha_{2}} \tag{2}\]  
where, \(\alpha_{1}\) and \(\alpha_{2}\) are the angle of incidence of the electromagnetic waves and the angle of reflection of the electromagnetic waves, respectively, and \(\eta_{1}\) and \(\eta_{2}\) are the wave resistance of the two media, which are calculated as follows:  
\[\eta_{1}=\sqrt{\frac{\mu_{1}}{\epsilon_{1}}},\eta_{2}=\sqrt{\frac{\mu_{2}}{ \epsilon_{2}}} \tag{3}\]  
In the actual process of detection, the reflection coefficient can be approximately expressed as  
\[\gamma=\frac{\sqrt{\epsilon_{1}}-\sqrt{\epsilon_{2}}}{\sqrt{\epsilon_{1}}+ \sqrt{\epsilon_{2}}} \tag{4}\]  
The permittivity of the material has a significant role in determining the particular velocity of the electromagnetic waves. The electromagnetic wave starts from a transmitting antenna and travels through the material at a velocity. The wave spreads outward and propagates downward until it hits an object with different electrical properties from the surrounding medium, and thus it is scattered from the object and detected by the receiving  
Figure 2: (**a**) Principle of GPR detection of rebar in concrete. (**b**) The rebar shows a hyperbolic feature in the B-scan. (**c**) The black line corresponds to the A-scan in Figure 1b, where the reflection points of the rebar in the A-scan correspond to the hyperbolic features. antenna. The interval between the propagation of the wave from the transmitting antenna to the receiving antenna is called the propagation time \(t\):  
\[t=\frac{\sqrt{4z^{2}+d^{2}}}{v} \tag{5}\]  
where \(d\) is the distance from the transmit antenna to the receiving antenna, \(z\) is the depth between the ground surface and the reflective object and \(v\) is the speed of propagation of an electromagnetic wave in a medium, which can be expressed as  
\[v=\frac{c}{\sqrt{e}} \tag{6}\]  
where \(c\) is the velocity of electromagnetic wave propagation in air, taken to be 0.300 m/ns.