## 2 VLC System Model for Underground Mining
### Channel Gain for the Underground Mine VLC Environment  
Given the characteristics involved in a mining environment, the location of the \(T_{x}\) on the roof of the mining scenarios, we will call the optical receivers \(R_{x}\). These will be located at the top of the mining workers' helmets, which agrees with the scenario of the proposed VLC model I2M. Due to the mobility that workers have in mining environments, the optical link will be different according to the location at some time of the \(R_{x}\). Consequently, several gain components for the optical channel should be considered. In our manuscript, the LoS path is adopted as the first and principal gain component, as it is the largest source of energy obtained from the \(T_{x}\). Short-distance LoS links are often modeled as linear attenuation and delay [11]. Also, these are considered non-frequency selective and the path loss depends on the square of the distance between the \(T_{x}\) and the \(R_{x}\)[11]. Therefore, the impulse response of the LoS component applied to underground mining environments can be written as [11]  
\[h_{0}(t)=\frac{A_{eff}(m+1)}{2\pi d_{0}^{2}}\cos^{m}(\phi_{0})\cos(\theta_{0}) \delta(t-\frac{d_{0}}{c}), \tag{3}\]  
where \(A_{eff}\) is the effective signal collection area, define as \(A_{eff}=A_{p}T_{s}(\theta_{0})\ g(\theta_{0})\). Here, \(A_{p}\) is the physical area of the \(R_{x}\), \(T_{s}(\theta_{0})\) is the optical filter gain, \(g(\theta_{0})\) is the optical concentrator gain, exposed in Section 2.4, \(d_{0}\) is the distance between \(T_{x}\) and \(R_{x}\), \(\phi_{0}\) is the LoS transmission angle, \(\theta_{0}\) is the reception angle, \(c\) is the speed of the light in free space, and \(\delta(t-\frac{d_{0}}{c})\) is the Dirac's function, which represents the signal propagation delay. Here, it is assumed that \(\phi_{0}<90^{\circ}\) and \(\theta_{0}<\Theta\), where \(\Theta\) is the \(R_{x}\) FoV. We can also define the channel DC gain of the LoS path, according to the following expression [11]:  
\[H_{0}=\int_{-\infty}^{\infty}h_{0}(t)dt=\frac{A_{eff}(m+1)}{2\pi d_{0}^{2}}\cos ^{m}(\phi_{0})\cos(\theta_{0})rect(\frac{\theta_{0}}{2\Theta}), \tag{4}\]where \(rect(\cdot)\) knows as the rectangular function, represents the signal reception decision function. Not only does the LoS path contribute energy in mining scenarios, since in these hostile environments multiple factors affect the energy received in the \(R_{x}\), such as non-uniform walls, dust particles or extracted materials, reflectivity of the ceiling, walls or others objects, dimensions of the environment, position and orientation of the \(T_{x}\) and \(R_{x}\), among others; these factors also create NLoS and diffuse links. Therefore, the optical path loss is more difficult to predict for NLoS links. To model the gains from the reflections that may exist in an underground mining environment, non-smooth surfaces can be modeled by dividing their surface area into a number of small surface elements, where each area reflects a fraction of incident light energy. Further, each small surface element can be taken as a differential reflective element, which, through its integration, will give us the total reflective area. In general, the fraction of energy is proportional to the reflection coefficient of the surface material. The light behavior in an underground mining environment can be modeled in two parts. The first is the path of light between a \(T_{x}\) and a \(q^{th}\) reflective area, whose impulse response is given as [9; 11; 16]. \[h_{1,q}(t)=\int\frac{A_{ref}(m+1)}{2\pi d_{T_{x},q}^{2}}\cos^{m}(\phi_{1})T_{ s}(\theta_{1})g(\theta_{1})\cos(\theta_{1})rect(\frac{\theta_{1}}{\Theta}) \delta(t-\frac{d_{T_{x},q}}{c})dA_{ref}, \tag{5}\]  
where \(A_{ref}\) is the reflective surface area, \(d_{T_{x},q}\) is the distance between \(T_{x}\), and the \(q^{th}\) reflective area. In the second, the \(q^{th}\) reflective area behaves like a light source, which is reflected on to the \(R_{x}\).