## 2 Three-Dimensional Ray-Tracing Model
### Path Correction of the Multipath Transmission Model
According to Equation (11), it can be seen that the total transmission loss in the mine tunnel is composed of two parts: one part is the LOS (Line-of-Sight)-free space propagation loss, which is related to the signal operating frequency, transmission distance \(L_{d}\), antenna transmission gain \(G_{T}\), and \(G_{R}\); the other part is the path loss. However, the multipath channel model established based on the plane waveguide method only considers the losses during signal transmission along the plane and cannot accurately reflect the actual field strength at the receiving point. Therefore, it is necessary to analyze the transmission loss of the zigzag helical ray. By applying [PERSON]'s principle and the first and second tent laws, the total path of the zigzag helical ray can be calculated. In the case of wireless signals from the base station and the receiving end, they are sent directly to the receiving end. However, wireless signals on the top and bottom plates of the tunnel, as well as on both sides, undergo multiple reflections before reaching the receiving end. Considering the scenarios where wireless signals undergo multiple reflections and scattering, the final result can be obtained. \[ \begin{split}\left[A_{g}\right](\text{dB})&=32.44+20 \lg L_{d}+20\lg f-10\lg(G_{T}G_{R})\\ &-20\lg\left|1+\sum\limits_{i=1}^{m}\sum\limits_{l=1}^{n}\prod \limits_{k=1}^{i}R_{ilk}^{(\pm)}\exp(j\Delta\varphi_{il})+\sum\limits_{n=1}^{N }\prod\limits_{k=1}^{K}R_{nk}^{(\pm)}\exp(j\Delta\varphi_{n})\right|\end{split}  \tag{12}\]  
The path loss model is further modified according to Equations (8) and (12), and the following results are obtained:  
\[ \begin{split}\left[A_{g}\right](\text{dB})&=32.44+20 \lg L_{d}+20\lg f+L_{total}-10\lg(G_{T}G_{R})-\\ & 20\lg\left|1+\sum\limits_{i=1}^{m}\sum\limits_{l=1}^{n}\prod \limits_{k=1}^{i}R_{ilk}^{(\pm)}\exp(j\Delta\varphi_{il})+\sum\limits_{n=1}^{N }\prod\limits_{k=1}^{K}R_{nk}^{(\pm)}\exp(j\Delta\varphi_{n})\right|\end{split}  \tag{13}\]  
During wireless signal propagation, various factors contribute to signal attenuation. To enhance the accuracy of measuring propagation distance, additional adjustments can be made to Equation (13) by incorporating the correction factor \(\Delta L\). The formula after correction is as follows:  
\[ \begin{split}\left[A_{g}\right](\text{dB})&=32.44+20 \lg L_{d}+20\lg f+L_{total}+\Delta L-10\lg(G_{T}G_{R})-\\ & 20\lg\left|1+\sum\limits_{i=1}^{m}\sum\limits_{l=1}^{n}\prod \limits_{k=1}^{i}R_{ilk}^{(\pm)}\exp(j\Delta\varphi_{il})+\sum\limits_{n=1}^{N }\prod\limits_{k=1}^{K}R_{nk}^{(\pm)}\exp(j\Delta\varphi_{n})\right|\end{split}  \tag{14}\]  
Through derivation, the \(L_{d}\) expression of wireless signal propagation distance in the well can be obtained:  
\[\left\{ \begin{array}{l}L_{d}=10^{a+b-c-d}\\ a=-1.622-\lg f-\left[A_{g}\right]/2+\lg(G_{T}G_{R})/2\\ b=\lg\left|1+\sum\limits_{i=1}^{m}\sum\limits_{l=1}^{n}\prod\limits_{k=1}^{ i}R_{ilk}^{(\pm)}\exp(j\Delta\varphi_{il})+\sum\limits_{n=1}^{N}\prod\limits_{k=1}^{K}R_{nk}^{( \pm)}\exp(j\Delta\varphi_{n})\right|\\ c=L_{total}\\ d=\Delta L\end{array} \right. \tag{15}\]From the formula, it can be seen that the propagation distance \(L_{d}\) of the signal underground is composed of four parts: the first part is the free propagation loss, which is related to the operating frequency, antenna transmission gain, etc. ; the second part is the transmission loss of the synthesized signal from multiple paths; the third part is related to the specific environment underground, such as tunnel cross-section, roughness, etc. ; and the fourth part is related to the correction factor \(\Delta L\). The correction factor \(\Delta L\) exhibits different values in different underground environments and can be obtained by using the principle of a support vector machine with least squares support.