## 2 Proposed Radio Link Model for Underground Mines
### Path Loss Model  
It is always essential to consider the actual shape of the tunnel cross-sections when designing a wireless sensor network for underground mines. These cross-sections may not always have perfectly rectangular or circular shapes and can often have irregular or complex geometries. A Cartesian coordinate system is utilized to simplify the analysis and design process, with its origin located at the center of the tunnel cross-section. This approach allows the representation of the tunnel geometry in a two-dimensional space, making it easier to position sensor nodes and antennas. The proposed model considers lossy materials and irregular cross-sections in the mine, as shown in Figure 1. The Fresnel zones are important in determining the level of signal attenuation or loss as a signal travels from a transmitter (Tx) to a receiver (Rx). The first Fresnel zone is assumed to be relatively small and is represented by an ellipsoid-shaped region around the direct line of sight between the Tx and Rx, ensuring that most of the transmitted signal energy propagates through the mine tunnel. Equation (1) calculates the free space path loss in the underground mine environment [2]. \[L_{FS}=20\log_{10}(|H_{r}-H_{t}|)+20\log(f_{o})-184.85 \tag{1}\]where \(H_{r}\) and \(H_{t}\) are the Tx and Rx antenna heights from the ground. At lower frequencies, the size of the Fresnel zones is typically smaller, and the angles of incidence between the wave and the walls of the tunnel are also smaller, resulting in relatively low attenuation of the reflected waves. The reflected waves from the tunnel walls contribute significantly to the received power where the size of the Fresnel zones is small because they travel a shorter distance than the direct wave, resulting in less attenuation. This region near the Tx, where the reflected waves are significant, is considered the multi-mode propagation zone. \(L_{M}\) represents the mining loss, which includes different losses experienced in mines, such as tilt angle loss caused by any bend in the mine pathway, roughness loss due to the irregular surface or roughness of mine walls, and obstacle loss resulting from obstacles, such as vehicles or carts, and human movement in the mine that obstruct the propagation of waves. The \(L_{M}\) can be defined as follows:  
\[L_{M}=L_{tilt}+L_{roughness}+L_{obstacle} \tag{2}\]  
The tilt angle loss is calculated as follows:  
\[L_{M}=\frac{4.343\pi^{2}\theta^{2}d}{\lambda} \tag{3}\]  
where \(\theta\) is the tilt angle, which is an attenuation factor assumed to be greater than 90\({}^{\circ}\) for separation distances above 5 m. Similarly, roughness loss is calculated as follows:  
\[L_{roughness}=4.343\pi^{2}h_{r}^{2}\lambda\bigg{(}\frac{1}{w^{4}}+\frac{1}{ \mu^{4}}\bigg{)}d \tag{4}\]  
where \(h_{r}\) is the roughness of the mine walls and is normally considered to be 400 mm\({}^{2}\)[3]. Similarly, \(L_{obstacle}\) is calculated as follows:  
\[L_{obstacle}=\sum_{i=1}^{n}x(i)\beta(i)^{i-1} \tag{5}\]  
In the above equation, the accumulated power reduction due to any obstacles along the signal path can be modeled using an attenuation parameter, \(x(i)\), for the ith obstacle encountered by the signal. This loss can be influenced by various factors, such as the size, shape, and material properties of the obstacle. In addition, the penetration rate of the material of ith obstacle, \(\beta(i)\), can also influence the amount of signal loss, which has a range from 0 to 1. The first obstacle encountered by the signal produces the most significant  
Figure 1: A theoretical model of the mine tunnel cross-section. (**a**) Side view and (**b**) front view. power loss, whereas subsequent obstacles produce less attenuation [2; 3; 4].