# Unmanned Vehicles' Placement Optimisation for Internet of Things and Internet of Unmanned Vehicles
## 5 Towards Algorithm Development: Propagation Concepts, Path Loss and Communication Graph
### Air-to-Ship Channel (A2S)  
Usually, air-to-ship channels are characterised by the LoS path and reflection path [74]. The latter is due to the reflection of radio waves on the water surface. In addition, due to reflections, there is a possibility that deep nulls appear when the signals compose destructively [75]. In the literature, according to our research, there are many empirical models proposed for an air-to-sea channel [63; 76], which is a particular example of a water channel, and especially for ultra high-frequency (UHF) and very high-frequency (VHF) communications where large antennae are used, but very few deterministic models or models adapted to other water environments such as big rivers or lakes or to other communication technologies exist. Moreover, few models can be applied to different communication frequencies. A couple of those models are the two-ray model and the three-ray model. To consider the LoS and reflected paths, the two-ray model given by Equation (7) can be used [63; 77]. \[PL_{2-ray}=20\cdot log_{10}\left(\frac{4\pi fd}{c}\right)-10\cdot log_{10}\left[ 2\sin\left(\frac{2h_{t}h_{r}f}{cd}\right)\right]^{2} \tag{7}\]  
Based on the model, another parameter, the critical distance, can be computed. The critical distance (Equation (8)) reveals the distance threshold after which the path loss increases with \(d^{4}\). \[d_{c}=\frac{4h_{t}h_{r}c}{f} \tag{8}\]  
By also considering the evaporation due to the existence of a duct layer of height, \(h_{d}\), the three-ray model is obtained (Equation (9)). \[PL_{3-ray}=20\cdot log_{10}\left(\frac{4\pi fd}{c}\right)-10\cdot log_{10}[2(1 +\Delta)]^{2} \tag{9}\]Equation (10) represents the relationship used to compute \(\Delta\). \[\Delta=2\sin\left(\frac{2\pi h_{t}h_{r}f}{cd}\right)\sin\left[\frac{2\pi(h_{d}-h_ {t})(h_{d}-h_{r})f}{cd}\right] \tag{10}\]  
Tabel 4 reviews the parameters mentioned in two-ray and three-ray path loss models. Based on Equations (7) and (8), we performed the following simulations for the transceivers in Table 1.  
In Figure 3, the simulated path loss, link budget and critical distance for a UAV-USV link are emphasised. The simulations were performed using MATLAB 2019b for communication between a USV and a UAV. The simulation parameters and their values are given in Table 5.  
The technical solution implies the use of the communication modules in [57]. The height of the USV antenna varied from \(h_{t}=0.5\) m to \(h_{t}=5\) m when higher USVs were implied in the simulations. The height of the UAV varied from 5 m to 120 m (the maximum allowed altitude to comply with the UAV regulations). The results are given in Figures 3-5. It can also be observed from the simulations that, at certain distances, \(d\), the UAV-USV communication path loss has maxima, due to the destructive composition of the radio signals. Mathematically, it can be proven that the argument of the sinusoidal function in the two-ray model, \(\frac{2h_{t}h_{r}f}{cd}\), reaches values close to \(\pi\) for certain combinations \((d,h_{t},h_{r})\). Thus, the value of the sinus function tends towards 0, while the logarithm tends towards \(-\infty\), Consequently, the overall path loss would tend towards \(\infty\). When \(h_{t}=0.5\) m (Figure 3) or \(h_{t}=2\) m (Figure 4), there are several maxima that exceed the link budget, at specific values of \(h_{r}\), but in the case of higher values of \(h_{t}\) (Figure 5) the number of maxima exceeding the link budget increases further. From the simulations, it can be observed that a very important task of the algorithm resides in the estimation of the optimum tuple \((d,h_{t},h_{r})\) which will be translated to tuple \((\phi_{i},\lambda_{i},\phi_{j},\lambda_{j},e_{i},e_{j})\) in order to minimise the path loss and avoid the occurrence of signal loss due to the destructive composition of the signals. Different recommendations may arise from the simulations. For \(h_{t}=5\) m, the results are given in Table 6.