# A Novel 3D Analytical Scattering Model
## 4 Shape Factors
The SF angular constriction \(\gamma_{\varphi_{m}}\) gives a measure of the concentration of multipaths around two directions. The value for \(\gamma_{\varphi_{m}}\) ranges from zero to one, with one denoting the arrival of exactly two multipath components from different directions and zero denoting no clear bias in two arrival directions [28]. The angular constriction for the azimuth and elevation angles observed from AS with respect to the height of AS is plotted in Figure 6a,b, respectively. Figure 5: The behaviour of the angular spread observed at AS: (**a**) in the azimuth plane; (**b**) in the elevation plane (\(h_{\text{GS}}=30\) m, \(d=3\) km and \(\sigma_{c}=\sigma_{t}=20\) m). The angular constriction w.r.t. the azimuth angles increases with the increase in the height of AS; however, the rate of increase decreases with the increase in \(\tau_{\rm max}\). The angular constriction w.r.t. the elevation angles gradually increases with the increase in the height of AS. Moreover, with the increase in \(\tau_{\rm max}\), it decreases. The azimuthal and elevational angular spread observed from GS with respect to the height of AS for different values of \(\tau_{\rm max}\) is shown in Figure 7a,b, respectively. The azimuthal angular spread is observed to be constant (approximately) with respect to the increase in height of AS; however, it increases with the increase in \(\tau_{\rm max}\). An exponential increase is observed in the angular spread in the elevation plane seen at GS with the increase in the height of AS. Furthermore, the rate of the increase in the angular spread decreases with the increase in \(\tau_{\rm max}\). The angular constriction for the azimuth and elevation angles for different values of the height of GS observed from GS with respect to the height of AS is plotted in Figure 8a,b, respectively. The angular constriction remains almost constant with respect to the azimuth angles with the increase in the height of AS as long as the increase in the height of AS is observed. Figure 6: The behaviour of the angular constriction observed at AS: (**a**) in the azimuth plane; (**b**) in the elevation plane (\(h_{\rm GS}=30\) m, \(d=3\) km and \(\sigma_{c}=\sigma_{l}=20\) m). Figure 7: The behaviour of the angular spread observed at GS: (**a**) in the azimuth plane; (**b**) in the elevation plane (\(h_{\rm GS}=70\) m, \(d=3\) km and \(\sigma_{c}=\sigma_{t}=30\) m). as AS lies inside the scattering region; however, it increases with the increase in \(h_{\text{GS}}\). The angular constriction with the elevation angles has a gradual decrease with the increase in the height of AS. However, conversely with the azimuthal side, it decreases with the increase in \(h_{\text{GS}}\). The azimuthal and elevational angular spread observed at the AS side of the link with respect to the variations in the elevational LoS angle \(\theta_{\text{los}}\) and the delay of the longest propagation path \(\tau_{\text{max}}\) is shown in Figures 9a and 10a, respectively. When AS moves away from GS, the elevation LoS angle reduces. Subsequently, the azimuthal angular spread increases, and the elevational angular spread reduces. This is true for all of the values of \(\tau_{\text{max}}\); however, the rate of change is different for different values of \(\tau_{\text{max}}\). The concentration of energy about exactly two physical directions (angular constriction) observed at the AS side independently in both the azimuth and elevation planes is shown in Figures 9b and 10b, respectively. In the azimuth plane, the distribution of energy has a higher bias towards two physical angles for lower values of both the elevation LoS angle and delay span. For elevation LoS angles, \(\theta_{\text{los}}>6^{\circ}\), there is almost no bias in the distribution of energy towards two azimuthal angles. On the other hand, the change in elevation LoS angle and delay span has a very marginal impact on the elevational angular constriction; see Figure 10b.