## 3 Analysis of the Scattering Phenomenon in Underground Mines and Distribution of Dust Particles
### The 3D Dust Particles Distribution Model  
In this subsection, we present the 3D dust particles distribution model. Figure 2 shows the three-dimensional geometric outline of the UM scenario that relates the LED, the distribution of the dust particles (scatterers) around the receiver, and the PD. Although only one scatter is shown in the Figure 2 for better understanding, there are \(N\) dust particles around the receiver. Consequently, the expressions obtained are applied to the total number of scatterers that make up the distribution. In this context, for a better description of the scenario, the following preliminary and general assumptions about dust particles were considered:  
1. An optical link (transmission path) from the LED to the PD interacts with a single dust particle, being considered an SB scattering model.
2. Suspended dust particles in the UM environment are spherical and uniformly distributed in the mining tunnel, which theoretically tends to infinite.
3. Dust particles behave as reflective isotropic scatterers with similar scatterer coefficients and uniform random phases.  
Based on the features, behavior, and distribution of dust particles in UM tunnels and as a baseline to establish a scattering component included in the UM-VLC channel model, we present a random and independently distributed scatterer distribution within a hemispheric 3D region. This distribution is spatially located around the helmet of the mining worker where the PD is located, which is the central point of the hemispheric area as shown in Figure 2.  
For simplicity and mathematical convenience, the spatial distribution function of the hemispheroid will be derived considering initially, a uniform general spherical probability distribution/density function, which is expressed as follows  
\[f_{x,y,z}^{(sp)}(x_{s},y_{s},z_{s})=\frac{(a+1)r^{a}}{4\pi R^{(a+1)}}, \tag{20}\]  
Figure 2: Proposed hemispherical scatterers spatial distribution around the PD for the downlink Cartesian/spherical coordinates relating the LED, an arbitrary dust particle, and the PD.  
where \(R\) is the sphere radius, \(r\) is the distance from the scatterer \(S\) to the PD, and \(a\geq 0\) is the shape factor. In our case, to obtain a spherical 3D shape, we consider a value of \(a=2\). Therefore expression (20) is reformulated as follows  
\[f^{(sp)}_{x,y,z}(x_{s},y_{s},z_{s})=\frac{3r^{2}}{4\pi R^{3}}. \tag{21}\]  
It is important to mention that outside the spherical region where the scatterers are distributed \(f^{(sp)}_{x,y,z}=0\). The distribution denoted in expression (21) can be easily transformed to derive the corresponding distribution for the hemispheric case considering the following premises:  
* \(f^{(sp)}_{x,y,z}=0\) for all underground azimuth angle (\(\phi\)). Therefore, \(\phi\
otin[0,\pi]\).
* \(f^{(sp)}_{x,y,z}/2\) for any distribution parameterized by \(\phi\). Therefore, \(\phi\in[\pi,2\pi]\).  
In this context, the PDF of the 3D hemispherical distribution of the dust particles is defined as  
\[f^{(le)}_{x,y,z}(x_{s},y_{s},z_{s})=\frac{3r^{2}}{2\pi R^{3}}. \tag{22}\]