# Highway Tunnel Defect Detection Based on Mobile GPR Scanning
## 3 Methodology  
The proposed bi-frequency backward projection (BBP) imaging method comprises two phases: Calculating the response amplitudes (RA) in signals and calculating the correlation between two different frequency GPR data [15]. As shown in Figure 6, the antenna moves from A-scan1 to A-scan_n_ to form a 2D radar image beneath the tunnel surface. The detection zone is divided into \(n\)\(\times\)\(m\) pixels. Consequently, the position of each pixel is given by(\(x=i\)\(\times\)\(dx\), \(z=j\)\(\times\)\(dz\))(\(i\) = 1,2,3,..., \(n\); mboxemphi = 1,2,3,..., _m_). For the pixel in the lining, such as pixel A, the two-way travel-time (TWT) of pixel A at A-scan_k_ can be expressed as the follows:  
\[t_{A,k}=\frac{2\sqrt{\left(\left(i_{A}\times d_{x}\right)-x_{k}\right)^{2}+ \left(j_{A}\times d_{A}\right)^{2}}}{c/\sqrt{\varepsilon_{1}}} \tag{3}\]  
The \(k\) = 1, 2, 3,..., \(n\) is the number of A-scans; The \(c\) is the speed of electromagnetic waves propagating in a vacuum. For the pixel beneath the lining, such as pixel \(C\), the TWT can be expressed as:  
\[t_{C,k}=\frac{2\sqrt{\left(x_{r}-x_{k}\right)^{2}+h^{2}}}{c/\sqrt{\varepsilon _{1}}}+\frac{2\sqrt{\left(\left(i_{C}\times d_{x}\right)-x_{r}\right)^{2}+ \left(j_{A}\times d_{z}\right)^{2}}}{c/\sqrt{\varepsilon_{2}}} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **Test Item** & **Result** \\ \hline
1 & Time stability & 0.1 ns \\
2 & Amplitude stability & 1.6\% \\
3 & Time stability of long time & 0.1 ns \\
4 & Amplitude stability of a long time & 0.1 ns \\
5 & Time-window linearity & 0.9\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The test project results of the GPR antenna stability test. Figure 5: When only the 400 MHz antenna is moved, only changes occur in the **(a)** 400 MHz radar signal, and the **(b)** 900 MHz radar signal remains stable. where \((x_{r},d)\) is the location of the refraction point. The calculation of \(x_{r}\) can be realized by an approximation algorithm, as shown in Figure 7. Consequently, the response amplitude of the pixel could be obtained in the record A-scank as:  
\[RA_{k}=\text{A-scank}(t) \tag{5}\]  
The response amplitude of the pixel at each A-scan can be described as two vectors as follows:  
\[ \begin{split} R_{k}(400\text{ MHZ})&=\left[R\,A_{k,1 }^{400M},RA_{k,2}^{400M}\dots RA_{k,n-1}^{400M},RA_{k,n}^{400M}\right]\\ R_{k}(900\text{ MHZ})&=\left[R\,A_{k,1}^{900M},RA_{ k,2}^{900M}\dots RA_{k,n-1}^{900M},RA_{k,n}^{900M}\right]\\ k=1,2,\dots i\times j\end{split}  \tag{6}\]  
Figure 6: GPR detection process: equipment moves from A-scan1 to A-scank to acquire the information beneath the concrete segment (h is the thickness of the concrete segment, and \(\varepsilon\) is the dielectric constant). Figure 7: A successive-approximation algorithm calculates the position of the refraction point. \(\mathrm{x}\mathrm{r}\) is the horizontal position of the refraction point. In addition, the \(\mathrm{\partial}\) is a minuscule value, which defines the method’s accuracy. The correlation between GPR profiles concerning two different frequencies will be emphasized while calculating the imaging value of pixels. The correlation between RA from B-scans concerning 400 MHz and 900 MHz dominant frequencies will be determined in this process. Specifically, the RAs from two B-scans will be multiplied as Equation (6). \[IV_{k}=sum(R_{k}(400\text{ MHz})\cdot R_{k}(900\text{ MHz}))=\sum\limits_{i=1}^{n}RA_{k,i}^{400 \text{M}}\cdot RA_{k,i}^{900\text{M}} \tag{7}\]  
Since the interference is inevitable, four kinds of radar signals are recorded in the radargram (see in Figure 8): the reflection from the target is represented by \(T\); the noise from the background and GPR antenna is represented by \(N\); and the diffraction and multiple-reflections are denoted by \(D\) and \(M\), respectively. Therefore, the subsurface pixels could be classified into four types: pixels with the target (A), the pixel with diffraction (B), the pixel with multiple reflections (C), and the background.