## 3 Enhanced Mathematical Model with Atmospheric and Environmental Effects
### Path Loss vs. Transmitter-Receiver (Tx-Rx) Separation Distance  
Figures 8 and 9 model compare different path loss models as a function of the transmitter-receiver (Tx-Rx) separation distance. These models illustrate the path loss for both high and low receiver antenna heights using various path loss models. The two-ray model is employed to account for the ground reflection in addition to the direct LOS path. This model is calculated separately for high and low receiver heights to observe how the reflection affects path loss at different altitudes. Figure 8 illustrates the relationship between path loss and the separation distance between the transmitter (Tx) and receiver (Rx) for a high receiver height. The graph is essential for comprehending the predictions of route loss by different propagation models at different distances, particularly in situations with raised receiver positions. This is particularly critical for 5G-satellite network applications and elevated terrestrial receivers. The FSPL model, is the simplest model and serves as a baseline, showing a continuous and gradual increase in path loss as the Tx-Rx separation increases. The two-ray model, shown by the red curve, considers both the direct path and a reflected path (typically from the ground) between the transmitter and receiver. This model introduces a more complex behaviour, particularly noticeable in the oscillatory pattern of the path loss. The periodic fluctuations are due to constructive and destructive interference between the direct and reflected paths, which result in alternating regions of higher and lower path loss. This model is particularly important in environments where ground reflection plays a significant role. Furthermore, from Figure 8, the log-distance path loss model, depicted by the blue curve, is a more empirical model that accounts for factors like the environment and the terrain. It shows a more gradual increase in path loss compared to the FSPL model and tends to be smoother than the two-ray model. This model is vital for analysing urban or suburban environments where multiple obstacles affect the signal propagation. The high receiver height in this simulation likely reduces the impact of ground reflections to some extent, as indicated by the fact that the two-ray model's oscillations decrease in amplitude as distance increases. However, the high receiver height also means that any interference effects are more pronounced over longer distances, as the direct and reflected paths can differ significantly in length, leading to greater phase differences. The two-ray model's  
Figure 8: Path loss vs. Tx-Rx separation (high Rx). ability to predict the oscillatory behaviour of path loss due to multipath interference is crucial for designing robust communication links, particularly in 5G-satellite integrated networks, where such effects can be pronounced. This model is particularly relevant for ground stations communicating with satellites, where reflections from the ground or water surfaces can lead to significant variations in signal strength. Figure 9 provides a detailed analysis of the path loss experienced at various separation distances between the transmitter (Tx) and receiver (Rx) when the receiver is at a lower elevation. This is crucial for understanding how different propagation models behave under the condition of a low receiver height, which is a common scenario in ground-based IoT devices or terrestrial communication systems. The analysis involves comparing several path loss models, namely, the free-space path loss (FSPL) model, the two-ray model, the log-distance path loss model, and the parabolic equation path loss model. The FSPL model is represented by the green dashed line and assumes an idealized scenario where the signal propagates in a straight line through free space without any obstructions or reflections. The path loss increases steadily with the distance between the Tx and Rx. This model is often used as a baseline because it only considers the distance and frequency but not the environment.