## 2 Proposed Method  
### Inversion Model  
In this paper, the electromagnetic wave tomography inversion model is established for detecting the distribution of abnormal areas within the working face in a coal mine. Due to the different electrical properties (resistivity and dielectric constant) of different media, their absorptivity to electromagnetic waves is also different. Coal with high resistivity has weak absorption of electromagnetic waves, which is beneficial to its penetration. The anomalous structures with lower resistivity, such as faults, collapse columns, and voids, have stronger shielding effects on electromagnetic wave absorption, resulting in significant energy attenuation [22; 23; 24]. Therefore, the distribution of abnormal structures within the working face can be predicted by analyzing the variation of electromagnetic field intensity in the working face. The survey of EMT is carried out between two tunnels of the working face; the electromagnetic wave with a certain frequency is emitted in one roadway and received in another roadway. The fixed points method is always adopted for observation, which is fast, efficient, and easy to coordinate. The transmitter is relatively fixed in a certain period of time, and the receiver observes its field strength point by point in a certain range, which is shown in Figure 1.  
The \(H_{i}\) received by the acquisition station of the \(i\)-th ray is  
\[H_{i}=H_{0}-\beta r-\ln r \tag{1}\]  
where \(H_{0}\) is the original field strength emitted at the signal source, \(\beta\) is the attenuation coefficient, and \(r\) is the length of the \(i\)-th ray. After the detection area is discretized as shown in Figure 1, \(x_{j}\) is the attenuation coefficient of the \(j\)-th grid, and \(d_{ij}\) is the interception of the \(i\)-th ray passing through the \(j\)-th grid. Therefore, in the path of \(i\)-th ray, the expression of \(\beta r\) can also be described as  
\[\beta r=\sum_{j=1}^{n}d_{ij}x_{j}. \tag{2}\]  
If \(y_{i}\) is defined as  
\[y_{i}=H_{0}-H_{i}-\ln r, \tag{3}\]  
then  
\[y_{i}=\sum_{j=1}^{n}d_{ij}x_{j}. \tag{4}\]  
Assuming that the total number of rays is \(A\) and the total number of grids is \(B\), the matrix equation can be obtained from multiple observations with different signal sources:  
\[\left[ \begin{array}{cccc}d_{11}&d_{12}&\ldots&d_{1B}\\ d_{21}&d_{22}&\ldots&d_{2B}\\ \vdots&\vdots&\vdots&\vdots\\ d_{A1}&d_{A2}&\ldots&d_{AB}\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{B}\end{array} \right]=\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{A}\end{array} \right]. \tag{5}\]  
It can also be expressed as  
\[[\text{D}][\text{X}]=[\text{Y}] \tag{6}\]  
where [D] is the (\(A^{*}B\)) matrix operator, which is the intercept of each ray in all meshes; [Y] is the A dimensional constant vector acquired from the EMT survey, which is related to the measured field strength; [X] is the \(B\) dimensional attenuation coefficient vector, which is the goal of tomographic inversion. Limited by field construction conditions, the number of rays is always far less than the number of grids. Therefore, an approximate solution of a sparse, ill-conditioned Equation (6) must be found. In this paper, the value of [X] is required when the objective function (\(F\)) reaches the global minimum  
Figure 1: Fixed point observation sketch. according to the AMPGA. \(F\) is defined as the root mean square error (_RMSE_) of the value of the attenuation coefficient between that computed and that acquired from the survey:  
\[F=\sqrt{\frac{1}{A}\sum_{i=1}^{A}{(y_{i}-\sum_{j=1}^{B}{d_{ij}x_{j}})^{2}}}. \tag{7}\]  
On this basis, this paper attempts to improve the effect of tomographic inversion by adding the prior information to constrain the iteration direction. The prior information is acquired from the tunnels, and the mathematical model that needs to be reconstructed after introducing a prior constraint is described by  
\[\left\{