## 2 Materials and Methods
### Proposed UIM-VLC Channel Model
#### 2.4.3 Proposed Scattering and Absorption Model  
To characterize the effects of dust scattering and absorption in the LOS and NLOS links, transmittance functions \(\zeta_{0}(\lambda)\) and \(\zeta_{w}(\lambda)\) must be defined. Note that in the reference channel, the Rayleigh and Mie scattering regimes contribute exclusively to the scattering-induced NLOS component \(H_{\text{scatter}}\), while the absorption effects are not considered in any of the channel components. In contrast, our proposed channel model considers Mie (and not Rayleigh) scattering and absorption effects in both LOS and NLOS links. The exclusion of the Rayleigh regime can be justified by looking at the ratio of the size of the spherical particles to the wavelength, \(\alpha=\pi D/\lambda\)[30], where \(D\) is the diameter of the particle. If \(\alpha\ll 1\), then the Rayleigh scattering model is valid. When wavelength and particle size are comparable, Mie scattering theory is more accurate, and when \(\alpha\gg 1\), then geometric scattering theory is used [30]. The particle size distribution of airborne particles in underground mines varies depending on the extractive activity, mineral composition, and geology of the mine, but a generalization can be made by looking at several studies conducted in underground coal mines in the United States [31; 32], China [30; 33], and the former Soviet Union [34] and underground gold mines in Alaska, Nevada, and South Africa [35], all of which show that the particle sizes of coal and silica dust particle sizes are in the order of 1 \(\upmu\)m to 100 \(\upmu\)m. Taking into account the wavelengths in the visible spectrum, this range of particle sizes implies that \(\alpha\) takes values in the order of 1 to \(10^{3}\): that means that Mie theory is the preferred regime for describing the scattering and absorption phenomena in underground mines. The attenuation of optical channels in non-ideal environments is usually modeled by the Beer-Lambert law, e.g., underwater optical channels [14; 36; 37], outdoor optical channels degraded by rain and snow [38] and sandstorms [39], and UM-VLC channels attenuated by coal dust [5; 30]. The transmittance function \(T(\lambda)\), according to the Beer-Lambert law, is given by [30]:  
\[T(\lambda)=\exp\biggl{(}-\int_{0}^{L}B_{\text{ext}}(\lambda,\ell)\,d\ell\biggr{)}\,, \tag{82}\]  
where \(L\) is the total length of the optical link, \(\ell\) parameterizes the optical link from \(0\) to \(L\), and \(B_{\text{ext}}\) (\(1/\text{m}\)) is known as the extinction coefficient. This extinction coefficient takes into account the loss of optical power due to scattering and absorption by dust particles. In Mie regime, \(B_{\text{ext}}\) can be expressed as [30; 40]:  
\[B_{\text{ext}}(\lambda,\ell)=\frac{\pi}{4}N_{p}(\ell)\int_{x_{p,\text{min}}}^{ x_{p,\text{max}}}x_{p}^{2}F(x_{p},\ell)Q_{\text{ext}}(x_{p},\lambda,m(\lambda))\,dx_{p}\,, \tag{83}\]  
where \(N_{p}(\ell)\) is the number of particles per unit volume (\(1/\text{m}^{3}\)) at the length \(\ell\) of the optical link, \(x_{p}\) is the particle diameter (\(\mu\)m), \(F(x_{p},\ell)\) is the cumulative percentage undersize mass fraction distribution function at the length \(\ell\) of the optical link. \(x_{p,\text{min}}\) and \(x_{p,\text{max}}\) are the minimum and maximum particle diameters, respectively, and \(Q_{\text{ext}}\) is the extinction efficiency factor defined by Mie theory, which depends on the particle diameter \(x_{p}\), wavelength \(\lambda\), and complex refractive index \(m(\lambda)\).