## 2 The Principle of GPR  
GPR works by actively transmitting an electromagnetic pulse to an underground target, which is reflected when the electromagnetic wave encounters a change in the electromagnetic characteristics of the underground. The GPR system then receives the reflected wave. These waves will be converted into digital signals carrying information about underground targets that can be processed by the host computer. The signals are shown as one-dimensional (A-scan), two-dimensional (B-scan), or three-dimensional (C-scan) data. The three scanning modes are shown in Figure 1.  
A-scan is the echo information of GPR system obtained by single sampling at a certain space point. It can provide accurate electromagnetic characteristics of underground objects at the current location.  
The B-scan is composed of multiple equally spaced A-scan data along the direction of the scan path. It can obtain information on the path. If there is a target underground in a certain area on the path, it will cause a sudden change in the permittivity, resulting in a hyperbolic feature image [6]. Parabolic features are often used to detect subsurface objects.  
The C-scan image is created by combining multiple parallel B-scan images. Three-dimensional information of underground objects can be obtained from C-scan.  
By processing and analyzing these data, the underground objects can be detected and classified.  
When detecting underground targets, taking B-scan data as an example, consider an ideal point target \(A(x_{0},y_{0})\) underground. This is shown in Figures 2 and 3. Suppose that when the GPR system moves along the positive direction of the \(x\)-axis, its distance from target \(A\) is \(d\), then:  
\[d=\sqrt{y_{0}^{2}+\left|x-x_{0}\right|^{2}} \tag{1}\]  
Figure 1: A-scan, B-scan, and C-scan data.  
Figure 2: Schematic of a point-like target.  
For this case, the transmitting and receiving delay t of the electromagnetic wave is as follows:  
\[t=\frac{2d}{v}=\frac{2\sqrt{y_{0}^{2}+\left|x-x_{0}\right|^{2}}}{v} \tag{2}\]  
Rearranging this formula yields:  
\[\frac{t^{2}v^{2}}{4}-\left|x-x_{0}\right|^{2}=y_{0}^{2} \tag{3}\]  
Substituting \(y_{0}=vt_{0}/2\) into the formula yields:  
\[\frac{t^{2}}{t_{0}^{2}}-\frac{\left|x-x_{0}\right|^{2}}{\frac{t_{0}^{2}v^{2}}{ 4}}=1 \tag{4}\]  
The dielectric constant of the subsurface medium is assumed to be \(\varepsilon_{r}\), and the magnetic properties of the underground medium are ignored. Then, the wave velocity of the electromagnetic wave in the underground medium is \(v=c/\sqrt{\varepsilon_{r}}\). Plugging it into Equation (4) yields:  
\[\frac{t^{2}}{t_{0}^{2}}-\frac{\left|x-x_{0}\right|^{2}}{\frac{t_{0}^{2}}{4} \left(\frac{c}{\sqrt{\varepsilon_{r}}}\right)^{2}}=1 \tag{5}\]  
Observing the expression, it can be known that the point target has hyperbolic characteristics in the B-scan image. This is shown in Figure 4.  
Figure 4: Echo image of point target.  
Figure 3: Point target detection.  
When the target is extended to a spherical target with radius \(R\), it is easy to obtain its curve equation as follows:  
\[\frac{\left(t+\frac{2R}{v}\right)^{2}}{\left(t_{0}+\frac{2R}{v}\right)^{2}}-\frac {\left(x-x_{0}\right)^{2}}{\left(\frac{t0v}{2}+R\right)^{2}}=1 \tag{6}\]  
When the GPR system moves at a constant speed along the survey line, the target will show a hyperbolic feature in the data form of B-scan. When performing object recognition and detection, the part of the image with hyperbolic features is also detected. Similarly, the targets under C-scan exhibit hyperboloid features.