## 2 Three-Dimensional Ray-Tracing Model
### Path Correction of the Multipath Transmission Model  
The ray optics method is applicable in optical waveguides when the geometric dimensions of the object are much larger than the wavelength. This principle is similar to the propagation of wireless signals in mines. Therefore, using the ray optics analysis method to analyze the propagation of wireless signals in tunnels is an effective approach. Based on the fundamental laws of similarity between super-high-frequency electromagnetic waves and light, as well as light transmission, it is possible to derive the attenuation formula for electromagnetic wave transmission in the optical model [26]. \[A(\mathrm{dB})=10\mathrm{lg}P_{i}/P_{r}=10\mathrm{lg}\left[(\lambda/4\pi)^{2 }\Bigg{|}\frac{G_{d}}{L_{d}}+\sum\frac{G_{i}R_{i}\exp\left(j\frac{2\pi}{ \lambda}(L_{i}-L_{d})\right)}{L_{r}}\Bigg{|}^{2}\right] \tag{9}\]  
where \(A\) is the attenuation of electromagnetic waves. \(G_{d}\) represents the total gain of the transmitting antenna for the Line-of-Sight (LOS) path, and \(G_{i}\) represents the total gain of the receiving antenna for the \(i\)-th path. \(R_{i}\) represents the product of reflection coefficients experienced by the \(i\)-th path. \(L_{d}\) represents the propagation distance for the direct path, and \(L_{i}\) represents the propagation distance for the \(i\)-th path. According to the formula, it can be concluded that the field strength distribution in the tunnel is related to the number of effective rays reaching the receiving point, the lengths of different ray paths, the number of reflections, and the reflection coefficients. According to the ray theory, if the wavelength of the wireless signal is significantly smaller than the cross-sectional size of the mine tunnel, the geometric optical ray method can be employed to determine each path between the transmitter and receiver in the multipath channel. The four reflective effects of the walls in the alley are equivalent to two planar waveguides: One planar waveguide is formed by the left and right walls, creating a vertical waveguide, and the other is created by the upper and lower walls, forming a horizontal waveguide. Without considering reflections and scattering from other obstacles within the alley, the total number of signal paths at the receiving point (\(n\)) includes one refracted path and \(4m\) ray paths reflected by the four alley walls, represented as \(n=4m+1\). By employing Equation (9) and considering the length of each reflection path, it is possible to determine the transmission loss for each ray. From this, the total power at the receiving point and the total transmission loss can be obtained. The total power at the receiving point is given by:  
\[\left\{ \begin{array}{l}P=P_{T}\Big{|}\frac{\lambda}{4\pi L_{d}}\Big{|}^{2}G _{T}G_{R}\Big{|}1+\sum_{i=1}^{m}\sum_{l=1}^{n}\Pi_{k=1}^{i}R_{ik}^{(\pm)}\exp \left(j\Delta\varphi_{il}\right)\Big{|}^{2}\\ \Delta\varphi_{il}=2\pi\Delta L/\lambda\end{array} \right. \tag{10}\]  
In the formula, \(P\) represents the total received power at the receiving point, \(P_{T}\) represents the transmit antenna power, \(L_{d}\) is the direct path length in kilometers, \(f\) is the wireless signal frequency in MHz, and \(G_{T}\) and \(G_{R}\) are the gains of the transmitting and receiving antennas, respectively. \(n\) and \(m\) represent the odd and even numbers of reflection times respectively, \(R_{ik}^{(\pm)}\) represents the reflection coefficient for each reflection, \(\Delta\varphi_{il}\) represents the phase difference between the \(l\)-th reflection path in the \(i\)-th reflection and the direct path \(L_{d}\), \(\Delta\varphi_{il}=2\pi\Delta L/\lambda\). The total transmission loss is calculated as follows:  
\[ \begin{split}\left[A_{g}\right](\text{dB})&=10\lg \frac{P_{T}}{P_{R}}=32.44+20\lg L_{d}+20\lg f-10\lg(G_{T}G_{R})-\\ & 20\lg\left|1+\sum\limits_{i=1}^{m}\sum\limits_{l=1}^{n}\Pi_{k=1}^{ i}R_{ik}^{(\pm)}\exp(j\Delta\varphi_{il})\right|\end{split}  \tag{11}\]  
where \(P_{R}\) is the received power.