# Statistical Modeling of Wall Roughness and Its Influence on NLOS VLC Channels in Underground Mining
## 2 UM-VLC System Model  
UMs exhibit tunnel configurations with structural characteristics that significantly impact VLC systems [17]. These encompass restricted dimensions (namely, width, height, and length), which affect the propagation of light in a manner distinct from that observed in open or conventional indoor environments. Furthermore, personnel and machinery movement can intermittently obstruct the light path, and airborne dust particles further degrade the signal by introducing scattering and absorption effects, which are more pronounced than in typical indoor VLC environments [18]. The composition of the tunnel walls is also a key factor, as different materials can absorb or diffusely reflect the transmitted light, altering the effective received power and the coverage area. While some obstacles and reflective surfaces exist in other VLC environments, the combination of constrained tunnel geometry, frequent obstructions, and particulate interference in UMs introduces a unique propagation environment distinct from conventional indoor scenarios [5]. In addition, the position and orientation of LEDs and PDs, depending on where they are placed, also affect the communication link. For the purposes of this research, a SISO VLC system is assumed, considering an LED pointing down perpendicularly and a PD pointing up in the same way. The focus of the analysis will be on the roughness of the wall and understanding how this affects the NLOS component of the channel in signal reception. For the UM-VLC system model, three components must be taken into account: light sources (transmitters), an irregular sidewall, and light detectors (receivers). The scenario is composed as follows: LED is attached at a specific point \((X_{p}^{T},Y_{p}^{T},Z_{p}^{T})\) [m], but the LOS component is assumed to be blocked by the UM machinery. A PD is placed at \((X_{l}^{R},Y_{l}^{R},0)\) [m]. Then, the last one will receive light from the LED because of reflections that come from the irregular wall represented by the X-Z plane. A grid is placed in X-Z describing the reflection coefficients at \((X_{q}^{S},0,Z_{q}^{S})\) [m]. The important components used in the scenario are represented in Figure 1.  
For the purposes of this article, the main interest is in studying the NLOS component generated by reflections against the tunnel walls and reaching the PD. In this scenario, the standard mathematical model employed to characterize this element is the Lambertian reflection. Here, incoming radiation is evenly scattered in all directions across a hemisphere centered on the normal surface. Consequently, the gain of the NLOS channel between the LED and the PD is represented as follows [5]:  
\[H_{NLOS}=\frac{A_{p}(m+1)}{2\pi}\sum_{q=1}^{Q}\frac{\Delta A_{q}\rho_{q}}{2\pi d _{w}^{2}}cos^{m}(\phi_{llq}^{tilt})cos(\phi_{iq}^{tilt})cos(\phi_{qp}^{tilt}) cos(\phi_{qp}^{tilt})g(\phi_{qp}^{tilt})rect(\frac{\phi_{qp}^{tilt}}{ \Theta}), \tag{1}\]  
Figure 1: Representative UM-VLC system diagram and components. where \(A_{p}\) is the effective area of the PD, \(\Delta A_{q}\) is the area of the reflective element \(q\), \(Q\) is the total number of reflective elements, \(\rho_{q}\) is the reflection coefficient of the wall and the element of study used in the theoretical model presented in the following sections, \(d_{q}\) is the sum of the Euclidean distance between the LED, the wall, and the PD (\(d_{q}=d_{qp}\) + \(d_{lq}\)), and \(rect(\cdot)\) is known as the rectangular function, representing the signal reception decision function.