# Recognition of Rebar in Ground-Penetrating Radar Data for the Second Lining of a Tunnel
## 2 Materials and Methods
In tunnel lining structures, the calibration of the dielectric constant of concrete can be shown in Figure 3. In Figure 3, the dielectric constant of the concrete can be calculated by measuring the propagation time of the electromagnetic wave throughout the tunnel's lining, the time for the electromagnetic wave to reach the bottom of the lining structure via the groove reserved for the lining sidewall and the depth of the groove reserved for the lining side wall. The dielectric constant can be expressed as  
Figure 1:0  
where \(t_{1}\) is the propagation time of the electromagnetic wave in the lining, \(t_{2}\) is the propagation time of the electromagnetic wave from the lining's reserved side wall groove to the lining bottom and \(h\) is the depth of the lining's reserved side wall groove. By collecting data from the tunnel lining, the dielectric constant of the concrete was calculated to be 6.4. Subsequently, the speed of propagation of the electromagnetic wave in the concrete was obtained to be approximately 0.120 m/ns according to Equation (6). The two-way travel time and electromagnetic wave distance from the receiving antenna to the rebar will alter as the GPR proceeds along the survey line direction. Each A-scan contains the reflected signal from the rebar. The reflected signal from the rebar then takes on a curved shape when each A-scan is ordered and concatenated in a temporal sequence to generate a B-scan. The hyperbolic shape of the rebar is one of the important features in identifying rebar. The process of hyperbolic formation of rebar is shown in Figure 1:1  
Figure 1:2 Measurement of the dielectric constant of concrete within a tunnel’s lining structure. GPR emits electromagnetic waves from a horizontal position. The electromagnetic waves are reflected as they propagate from the concrete to the rebar. The receiving antenna receives the return signal, while the receiving signal device feeds back the two-way travel time. This process is continuously repeated and recorded by the recording instrument, which eventually forms the hyperbolic of the rebar in the B-scan as shown in Figure 2b. In the GPR image, the hyperbola of the rebar can be expressed as  
Figure 1:3  
where \(x\) is the initial position when the GPR transmits electromagnetic waves, \(t\) is the start of propagation time, \(x_{0}\) is the position when the transmitting antenna is perpendicular to the rebar, \(t_{0}\) is the two-way propagation time to the rebar, \(v\) is the propagation speed of electromagnetic waves in the medium and \(r\) is the radius of the rebar. The short-term mean value of the A-scan data in the time window should be stable so that the reflected wave can correctly reflect the underground target. Therefore, we performed a DC offset removal process for the A-scan. The DC offset removal can be expressed as  
Figure 1:4  
where \(A(n)\) is the sample point data before processing, \(A_{p}(n)\) is the sample point data after processed, \(n\) is the number of samples in the A-scan and \(N\) is the total number of samples in each A-scan. Equations (1) and (4) lead to the following conclusions: When the wave is transmitted from a low dielectric constant medium to a high dielectric constant medium, the phase of the reflected wave is opposite to the incident. The electromagnetic wave will propagate in a single medium with exponential attenuation. The denseness of the medium is inversely proportional to the attenuation of the radar wave. Therefore, the reflected waveform of the concrete will be attenuated according to a certain ratio. Then, the electromagnetic wave propagates from the concrete to the peak point where the rebar appears. The phase of the peak point is opposite to the incident wave phase, and the total reflection of the rebar prevents part of the electromagnetic wave from propagating, causing the peak point to exhibit a smaller amplitude in the A-scan than the adjacent peak point amplitude of the same phase. The peak point is shown in Figure 2c.