## 3 Enhanced Mathematical Model with Atmospheric and Environmental Effects
### Analysis of Rain Attenuation in 5G-Satellite Integrated Networks  
The plot in Figure 3 illustrates the rain attenuation versus distance across different frequencies (24 GHz, 28 GHz, and 32 GHz) and rain rates (5 mm/h, 10 mm/h, 20 mm/h, and 50 mm/h). This analysis provides insights into the impact of rain on signal attenuation in 5G-satellite networks, particularly for high-frequency bands commonly utilized in satellite communications. As shown, rain attenuation increases significantly with both frequency and rainfall intensity, highlighting the challenges posed by adverse weather conditions in maintaining reliable communication links over long distances.  
The linear relationship between \(A_{\text{rain}}\) and \(d\) implies that for a given rain rate and frequency, attenuation increases proportionally with distance. This is evident across all plots in Figure 3, where the attenuation curves exhibit a linear trend as distance extends from 0 to 3000 km.  
Higher frequencies result in greater rain attenuation, as demonstrated by the separation of curves for 24 GHz, 28 GHz, and 32 GHz. This effect is due to the proportional relationship between the frequency and the attenuation coefficient \(\gamma_{r}\), which generally increases with frequency. At a given rain rate, the 32 GHz signal experiences the highest attenuation, followed by 28 GHz and 24 GHz. For instance, at a rain rate of 5 mm/h and a distance of 3000 km, the rain attenuation at 32 GHz reaches nearly 4000 dB, while at 24 GHz, it remains below 2000 dB. This demonstrates that for long-distance communication, higher frequencies suffer considerably more loss due to rain attenuation, making them less suitable for adverse weather conditions.  
The influence of rain rate on attenuation is also apparent in Figure 3. As the rain rate increases from 5 mm/h to 50 mm/h, the attenuation grows substantially for all frequencies. The relationship between rain rate \(R\) and specific attenuation \(\gamma_{r}\) follows a power-law function \(\gamma_{r}=kR^{\alpha}\), meaning that even small increases in \(R\) can lead to significant rises in \(\gamma_{r}\), particularly at higher frequencies where \(k\) and \(\alpha\) are larger. This phenomenon is evident at a rain rate of 50 mm/h, where attenuation at 32 GHz approaches \(8\times 10^{4}\) dB at a distance of 3000 km, compared to about \(1.5\times 10^{4}\) dB at 24 GHz under the same conditions.  
The observed pattern indicates that the rain rate plays a critical role in determining signal loss. For example, in a heavy-rain scenario (50 mm/h), even a lower-frequency signal like 24 GHz experiences severe attenuation, which can reach values around \(1.5\times 10^{4}\) dB over long distances. Thus, rain intensity directly impacts link reliability, especially in high-frequency bands where attenuation is already exacerbated by frequency-dependent factors.  
The analysis of rain attenuation reveals essential considerations for 5G-satellite networks, particularly in regions prone to heavy rainfall. Higher frequencies, such as 32 GHz, though useful for bandwidth efficiency, may not be viable for long distances or high-rainfall areas due to the high attenuation values observed. Instead, lower frequencies like 24 GHz, while still impacted by rain, offer relatively lower attenuation, suggesting a trade-off between data rate and weather resilience.  
For IoT applications in smart cities, this trade-off is significant. In dense urban areas, where high-frequency bands are preferred for their data handling capabilities, the presence of rain can substantially degrade the link quality, impacting real-time data transmission essential for critical services. To mitigate this, adaptive frequency management, such as dynamically shifting to lower frequencies during heavy rain, could be implemented to sustain network reliability.  
Figure 3: Rain attenuation vs. distance for different frequencies and rain rates.