## 2 Three-Dimensional Ray-Tracing Model
### Path Correction of the Multipath Transmission Model
\tag{20}\]  
Finally, the decision function is obtained:  
\[f(x)=\text{sgn}(\overrightarrow{\omega^{*}}^{T}\overrightarrow{x}+b^{*})= \text{sgn}\left[\sum\limits_{i=1}^{m}\alpha_{i}y_{i}(\overrightarrow{x_{i}},\overrightarrow{x})+b\right] \tag{21}\]  
For the problem of linearly inseparable support vector machines, the introduction of slack variables \(\zeta\) (\(\zeta_{i}\geq 0,i=1,2,\cdots m\)) and penalty factor \(C\) (\(C>0\)) transforms the objective function and constraints into the dual problem as follows:  
\[\left\{ \begin{array}{l}\max\!L(\overrightarrow{\omega},b, \overrightarrow{a})=\sum\limits_{i=1}^{m}a_{1}-\frac{1}{2}\sum\limits_{i=1}^{m }y_{i}y_{j}\alpha_{i}\alpha_{j}K(\overrightarrow{x_{i}},\overrightarrow{x_{j }})\\ s.t.\sum\limits_{i=1}^{m}a_{i}y_{i}=0\\ C\geq a_{i}\geq 0,i=1,2,\cdots m\end{array} \right. \tag{22}\]  
In the equation, \(K(\overrightarrow{x_{i}},\overrightarrow{x_{j}})\) represents the kernel function. The final classification decision function is obtained:  
\[f(\overrightarrow{x})=sgn(\sum\limits_{i=1}^{m}\alpha_{i}y_{i}K( \overrightarrow{x_{i}},\overrightarrow{x})+b) \tag{23}\]  
The Least Squares Support Vector Machine (LS-SVM) replaces the inequality constraints with equality constraints and transforms the quadratic programming problem into solving a set of linear equations. For individual outliers, the variable \(\xi\), is introduced, and \(\gamma\)is the penalty factor defined as the regularization parameter. The expression of LS-SVM is as follows:  
\[\left\{ \begin{array}{l}\min(\overrightarrow{\omega},\xi)=\frac{1}{2}\left\| \overrightarrow{\omega}\right\|^{2}+\frac{\gamma}{2}\sum\limits_{i=1}^{l}\xi_{ i}^{2}\\ s.t.y_{i}=\overrightarrow{\omega}^{T}\cdot\phi(\overrightarrow{x_{i}})+b+\xi_{i},i=1,2,\cdots i \end{array} \right. \tag{24}\]  
where \(\phi(\overrightarrow{x})\) is the mapping of the sample eigenvector to a high latitude. The Lagrange function is introduced to obtain:  
\[L(\overrightarrow{\omega},b,\xi_{i},\alpha_{i})=\frac{1}{2}\overset{\to}{ \omega}\overset{T}{\omega}\overset{\to}{+}\frac{\gamma}{2}\sum\limits_{i=1}^ {N}\xi_{i}^{2}-\sum\limits_{i=1}^{N}\alpha_{i}(\overrightarrow{\omega}^{T} \phi(\overrightarrow{x_{i}})+b+\xi_{i}-y_{i}) \tag{25}\]  
In Equation (25), \(\alpha_{i}\) is the Lagrange multiplier. Taking the partial derivative of \(\overrightarrow{\omega},b,\xi_{i},\alpha_{i}\) and eliminating \(\overrightarrow{\omega},\xi\), we obtain the matrix:  
\[\left[ \begin{array}{ccc}0&\
u^{T}\\ \
u&K+\gamma^{-1}I\end{array} \right] \begin{bmatrix}b\\ \alpha\end{bmatrix} = \begin{bmatrix}0\\ Y\end{bmatrix}  \tag{26}\]  
In Equation (26), \(\
u= \begin{bmatrix}1&1&\cdots&1\end{bmatrix} ^{T}\), I is the identity matrix, \(\overrightarrow{\alpha}= \begin{bmatrix}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{N} \end{bmatrix} ^{T}\), \(Y= \begin{bmatrix}y_{1}&y_{2}&\cdots&y_{N}\end{bmatrix} ^{T}\). Where \(K\) is an \(N\)-dimensional square matrix. \(K_{il}=\phi(\overrightarrow{x_{i}})^{T}\phi(\overrightarrow{x_{j}})=K( \overrightarrow{x_{i}},\overrightarrow{x_{j}})\), and the predictive function for the LS-SVM model is as follows:  
\[f(\overrightarrow{x})=\sum\limits_{i=1}^{N}a_{i}K(\overrightarrow{x_{i}}, \overrightarrow{x_{j}})+b \tag{27}\]  
Among them, \(K(\overrightarrow{x_{i}},\overrightarrow{x_{j}})\) represents the Gaussian radial kernel function, and its expression is as follows:  
\[K(\overrightarrow{x_{i}},\overrightarrow{x_{j}})=\exp\left[-\left\| \overrightarrow{x_{i}}-\overrightarrow{x_{j}}\right\|^{2}/2\sigma^{2}\right]\]  
where \(\sigma\) is the bandwidth of the Gaussian radial kernel function. Sample set \(D\) was established, which included the influencing factors as signal frequency \(f\), tunnel section shape and size \(a\), path length \(d\) from the transmitting end, and vector representation as \(\overrightarrow{x}=(f,a,d,\cdots)\). The output y is the correction factor \(\Delta L\).