# Statistical Modeling of Wall Roughness and Its Influence on NLOS VLC Channels in Underground Mining
## 3 Simulation Results
### Second VLC Simulation Position
#### 3.2.3 Third Case  
In this analysis, the uniform, exponential, and gamma distributions are used to generate the angles of incidence and irradiance, while the reflection coefficients follow a normal distribution. The results show distinct behaviors in the power received for each distribution. In UNID, the initial maximum power is \(1.6\times 10^{-7}\) W/m\({}^{2}\) (Figure 8), with a relatively homogeneous spatial distribution. Surprisingly, the power increases to \(4\times 10^{-7}\) W/m\({}^{2}\) (Figure 15) in the second position, likely due to uniformly distributed angles creating more effective reflection contributions. EXPD starts with a maximum power of \(1\times 10^{-7}\) W/m\({}^{2}\) (Figure 8), showing a concentrated pattern due to the probabilistic nature of exponential angles. This power rises to \(2.5\times 10^{-7}\) W/m\({}^{2}\) (Figure 15, indicating that the normal reflection coefficients amplify specific regions. For GAMD, the initial maximum power is \(1\times 10^{-7}\) W/m\({}^{2}\) (Figure 8), the lowest among the three, with reflections concentrated in smaller areas. In the second position, the power increases to \(2.6\times 10^{-7}\) W/m\({}^{2}\) (Figure 15), reflecting the controlled dispersion and focused reflections characteristic of gamma angles. Interestingly, all distributions show an increase in maximum power in the second position, contrary to previous analyses. This can be attributed to the normal distribution of the reflection coefficients, which reduces extreme variability and focuses energy in specific regions. The uniform distribution exhibits the largest absolute increase, driven by regular reflection patterns aligning effectively with the receiver. The exponential angles generate a similar 150% increase, as their probabilistic nature concentrates reflections in advantageous directions. The gamma angles show the highest relative increase (160%), balancing dispersion and concentration to amplify specific areas. These variations highlight the interaction between angular distributions and reflection coefficients, where normal coefficients reduce  
Figure 14: Probability density functions for different reflection coefficients. (**a**) PDF using UNID. (**b**) PDF using GAMD. (**c**) PDF using EXPD. dispersion and enhance efficient reflections. The anomalies and increments observed are a direct result of this interaction, with uniform angles producing the most regular distribution and gamma angles offering the highest amplification in specific regions. This behavior underscores the importance of considering both the angular and the reflection distributions when analyzing the propagation of NLOS power. In the first position, the maximum amplitude varies significantly between distributions, with UNID (Figure 9) reaching \(9\times 10^{-9}\), EXPD (Figure 9) achieving \(8\times 10^{-9}\), and (Figure 9) showing a lower peak of approximately \(4.5\times 10^{-9}\). In the second position, the maximum amplitude shifts, with UNID (Figure 16) increasing to \(2.5\times 10^{-8}\), EXPD (Figure 16) reaching \(1.5\times 10^{-8}\), and GAMD (Figure 16) dropping to \(1.5\times 10^{-9}\). These variations occur due to how each angle distribution affects the concentration of the reflected power. The UNID distribution generates more homogeneous angles, leading to higher reflection efficiency and greater channel gain. EXPD, with a higher probability of lower angles, reduces channel gain while maintaining a structured echo response. GAMD, which favors greater angular variability, disperses energy more, reducing the reflected power and the CIR amplitude. The delay of the first significant echo also varies between distributions. In UNID, primary peaks appear around 30-35 ns in the first set of images but shift earlier to 20-25 ns in the second, indicating changes in the dominant reflection paths. In EXPD, the arrival times remain relatively consistent, producing a more stable echo structure. In GAMD, the echoes arrive less uniformly, displaying greater dispersion.