# Probabilistic Modeling of Dust-Induced FSO Attenuation for 5G/6G Backhaul in Arid Regions
## 3 Results and Discussions
The Lognormal distribution provided reasonable fits compared to the other distributions, particularly for dense and moderate dust conditions, with \(R^{2}\) values of 0.78 and 0.85, respectively. However, its performance declined under light dust, where the model underestimated the peak of the distribution, resulting in a higher RMSE of 0.0025 and a lower \(R^{2}\) of 0.87 compared to the other models. The LogL for the Lognormal distribution yielded the highest LogL value under dense dust conditions, indicating the best statistical likelihood among all fitted models under dense dust. Although commonly used for positively skewed data, the Gamma and Weibull distributions demonstrated a moderate performance across all conditions. The Gamma distribution exhibited a stronger fit under light dust (\(R^{2}=0.93\)), while the Weibull distribution achieved its best performance under light dust (\(R^{2}=0.95\)), albeit with weaker fits in the dense and moderate ranges. They also demonstrated comparable performance in terms of the LogL metric. The Exponentiated Gamma distribution showed improved flexibility over the standard Gamma, with better fitting under light dust (\(R^{2}=0.95\)), although its performance was still below Johnson SB's in all cases. It achieved a moderate-to-high \(R^{2}\) across conditions. In terms of the LogL test, the Exponentiated Gamma distribution showed the weakest performance among all models. It is worth mentioning that all data processing and statistical analyses were conducted using MATLAB software, ver. 2022b (MathWorks Inc., Natick, MA, USA). The flexibility in the Johnson SB distribution enables it to outperform simpler models when fitting attenuation data with sharp peaks and heavy tails. This bounded and highly configurable distribution provides a powerful mechanism to capture the empirical characteristics of attenuation under all observed dust regimes. Table 2 presents the estimated parameters \(\gamma\), \(\delta\), \(\lambda\), and \(\xi\) for each visibility condition. These parameters were selected to optimize the fit based on the histogram shape and statistical performance indicators (e.g., the RMSE, MAE, and \(R^{2}\)). The variation in parameter values reflects the changing statistical nature of the attenuation across different dust levels. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Distribution**} & \multicolumn{4}{c}{**Dense Dust**} & \multicolumn{4}{c}{**Moderate Dust**} & \multicolumn{4}{c}{**Light Dust**} \\ \cline{2-11}  & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) & **LogL** & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) & **LogL** & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) & **LogL** \\ \hline Lognormal & 0.0001 & 0.0000 & 0.78 & \(-7.85\) & 0.0009 & 0.0007 & 0.85 & \(-5.55\) & 0.0025 & 0.0018 & 0.87 & \(-4.50\) \\ Gamma & 0.0002 & 0.0001 & 0.66 & \(-7.96\) & 0.0010 & 0.0008 & 0.80 & \(-5.55\) & 0.0019 & 0.0013 & 0.93 & \(-4.45\) \\ Weibull & 0.0002 & 0.0001 & 0.57 & \(-8.00\) & 0.0012 & 0.0009 & 0.70 & \(-5.58\) & 0.0016 & 0.0010 & 0.95 & \(-4.42\) \\ Exp. Gamma & 0.0002 & 0.0001 & 0.67 & \(-209\) & 0.0008 & 0.0006 & 0.87 & \(-6.15\) & 0.0016 & 0.0009 & 0.95 & \(-366\) \\ Johnson SB & 0.0000 & 0.0000 & 0.99 & \(-9.68\) & 0.0005 & 0.0003 & 0.95 & \(-5.40\) & 0.0016 & 0.0009 & 0.95 & \(-4.42\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of distribution fit metrics across dust visibility conditions. To assess the feasibility of unified probabilistic modeling over the full range of dust-induced attenuation, we fitted the entire dataset without classifying it into discrete visibility regimes. Specifically, we compared the Johnson SB distribution--which demonstrated strong performance in earlier segmented cases--with the negative exponential distribution proposed in [17]. Figure 3 illustrates the resulting histogram and fitted PDFs for both models using a consistent bin count of 100, which provides a good trade-off between histogram resolution and fitting stability.