# A Novel 3D Analytical Scattering Model
## 2 Geometry of the Proposed Analytical Channel Model  
The proposed model for A2G/G2A communications is shown in Figure 1, where the AS and the GS are assumed at heights \(h_{\text{AS}}\) and \(h_{\text{GS}}\), respectively. It is assumed that AS and GS have isotropic antennas. The AS is assumed at horizontal distance \(d\) from the GS. The direct line-of-sight (LoS) distance from AS to GS is shown by \(d_{\text{los}}=\sqrt{d^{2}+(h_{\text{AS}}-h_{\text{GS}})^{2}}\), and the elevation angle of the LoS component with the ground plane is \(\theta_{\text{los}}=\arctan\left(h_{\text{AS}}-h_{\text{GS}}/d\right).\) The propagation path delay associated with the longest and shortest paths is denoted by \(\tau_{\text{max}}\) and \(\tau_{o}\), respectively. The sum of the distances from any point on the surface of an ellipsoid to each of its foci points is always the same. Using this geometric property of ellipsoids, the outer and inner boundaries of the effective scattering region can be drawn by the ellipsoids corresponding from \(\tau_{\text{max}}\) and \(\tau_{o}\), respectively. The major and minor axes of the ellipsoid (\(E_{\tau}\)) can thus be obtained from the delay of the longest path (\(\tau_{\text{max}}\)), as below:  
\[a_{\tau}=\frac{c\;\tau_{\text{max}}}{2}, \tag{1}\]  
\[c_{\tau}=b_{\tau}=(1/2)\sqrt{4\;a_{\tau}^{2}-d_{\text{los}}^{2}}. \tag{2}\]  
The vicinity of the AS can be modelled as a scattering-free region due to the fact that the average height of an AS is usually considerably larger than the average rooftop level. The scattering region around GS is thus governed by ellipsoid (\(E_{\tau}\)) truncated by the planes of height equal to the average rooftop level of the surroundings or the average height of the crest and trough of sea-waves, for the cases of GS at the ground surface or the sea surface, respectively. Scattering objects are thus assumed only around the GS confined uniformly distributed within the truncated ellipsoidal-shaped scattering region (TESR) (shaded in grey in Figure 1b). A scattering point is assumed to scatter the incident signal (from any direction) in all directions with equal scattering coefficients and uniform random phases. The average vertical deviations of the crest and trough of the sea-wave from the mean level are shown with \(\sigma_{c}\) and \(\sigma_{t}\), respectively. For the case of communication over the ground surface, the parameter \(\sigma_{c}\) will take the value based on the average rooftop height. The propagation behaviour of electromagnetic (EM) waves through the dense water medium is different from propagation in the air medium, because of water's high permittivity and electrical conductivity. The primary limitation of EM wave propagation in dense water is the high attenuation caused by the conductivity of water [38]. The proposed model also takes the water penetration of EM waves into account for modelling the scattering region; for the low frequency band in the over-water communication scenario. The depth of penetration in meters is represented by \(\psi_{w}\). This parameter may be substituted appropriately according to the frequency range and water conditions and may be substituted as zero to represent no penetration (e.g., for high frequencies). The deviation of the trough and penetration into the water together represent the depth of the scattering region, which is shown by \(\sigma_{d}\) (i.e., \(\sigma_{d}=\sigma_{t}+\psi_{w}\)). For the case of A2G communication over the sea surface, the sea conditions (i.e., height of waves and roughness) are assumed to be time invariant, and scattering from the sea surface is assumed as independent of sea conditions. The angles formed in the azimuth and elevation planes with the direction of the signal corresponding to a certain scattering point \(\mathrm{S_{P}}\) at the GS are symbolized by \(\phi_{b}\) and \(\beta_{b}\) and at the AS are symbolized by \(\phi_{a}\) and \(\beta_{a}\), respectively. The upper face of the TESR is an ellipse whose major and minor axis is \(a_{e}\) and \(b_{e}\).