## 2 Materials and Methods
### Top-Cool Structure Detection  
#### 2.3.1 GPR Reflection Intensity Model  
As shown in Figure 4, assume that all the media in top coal are isotropic and homogeneous and that the top coal contains \(n\) gauge layers of thickness \(L\); the thickness of each coal seam layer is denoted as \(L_{1},L_{3},\cdots,L_{2n+1}\), with a relative dielectric constant of \(\varepsilon_{1},\varepsilon_{3},\cdots,\varepsilon_{2n+1}\) and a conductivity of \(\sigma_{2},\sigma_{4},\cdots,\sigma_{2n}\), while the thickness of each gauge layer is \(L_{2},L_{4},\cdots,L_{2n}\), with a relative dielectric constant of \(\varepsilon_{2},\varepsilon_{4},\cdots,\varepsilon_{2n}\) and a conductivity of \(\sigma_{2},\sigma_{4},\cdots,\sigma_{2n}\). The relative dielectric constant and conductivity of roof rock are \(\varepsilon_{2n+2}\) and \(\sigma_{2n+2}\). Suppose that the GPR wave, \(T_{0}\), is emitted from a GPR antenna at a distance of \(L_{0}\) from the coal seam and undergoes reflection and transmission at the interface between the air layer (first layer); assuming an incidence angle of \(\theta_{0}\) and a refraction angle of \(\beta_{0}\), the reflected wave, \(R_{0}\), is received through the GPR-receiving antenna. The transmitted wave, \(T_{1}\), propagates in the first layer of the medium and undergoes reflection and refraction at the interface between the first and second layers of the medium. Its incidence angle is \(\theta_{1}\), and the refraction angle is \(\beta_{1}\). The reflected wave, \(R_{1}\), will be received by the GPR-receiving antenna. In the same manner, we can obtain the reflected wave, \(R_{2n+1}\), of the \(2n+1\) layer medium and the \(2n+2\) layer medium; then, the total thickness of the coal top is computed based on the two-way travel time of the wave as [32]  
\[L=\sum_{i=1}^{2n+1}\frac{v_{i}t_{i}\cos\beta_{i}}{2} \tag{9}\]  
where \(v_{i}\) is the propagation speed of GPR waves in the \(i\)th layer medium, and \(t_{i}\) is the two-way travel time of electromagnetic waves in the \(i\)th layer medium. The electromagnetic wave intensity of the multi-frequency GPR waves between the air layer and the first layer of the medium can be calculated as  
\[P_{0}=\frac{P_{t}G_{t}}{4\pi(L_{0}/\cos\theta_{0})^{2}} \tag{10}\]  
where \(P_{t}\) is the initial electromagnetic wave intensity emitted by the GPR, and \(G_{t}\) denotes the transmitting antenna gain. The reflected wave at the interface between the air layer and the first layer of the medium is obtained by the radar-receiving antenna, and the electromagnetic wave reflection intensity is expressed as  
\[R_{0}=\frac{P_{t}G_{t}G_{r}\lambda^{2}r_{0|1}}{16\pi^{2}(L_{0}/\cos\theta_{0}) ^{2}} \tag{11}\]  
where \(G_{r}\) represents the receiving antenna's gain; \(\lambda\) denotes the wavelength of the electromagnetic wave in the medium; and \(r_{0|1}\) represents the reflection coefficient in the first layer interface. The electromagnetic wave energy of the GPR traveling between the first and second layers of the top coal can be computed as  
\[P_{1}=\frac{P_{t}G_{t}}{4\pi(L_{0}/\cos\theta_{0}+L_{1}/\cos\beta_{1})^{2}} \cdot t_{0|1}e^{-\frac{\varepsilon_{1}L_{1}}{\cos\beta_{1}}} \tag{12}\]where \(t_{0|1}\) represents the transmission coefficient of the GPR signal transmitting from the air layer into the first layer; \(a_{1}\) denotes the attenuation coefficient of the first layer.