# A Novel 3D Analytical Scattering Model
## 2 Geometry of the Proposed Analytical Channel Model
\(\mathrm{P_{9}}\) is the centre of this upper face ellipse. A line drawn in the \(x\)-\(z\) plane (for a value of the \(y\) coordinate taken as zero) parallel to the \(x\)-axis representing the rooftop level (or the sea-waves' crest level) of the scattering region intersects the bounding ellipsoid \(E_{\tau}\) at points \(\mathrm{p_{1}}\) and \(\mathrm{p_{3}}\). We can obtain the \(x\)-coordinate of the intersection points \(\mathrm{p_{1}}\) and \(\mathrm{p_{3}}\) by substituting the equation of the line into the equation of the bounding ellipsoid \(E_{\tau}\). The major dimension \(a_{e}\) of the ellipse formed at the upper face of TESR along the \(x\)-axis can thus be obtained from the calculated \(x\)-coordinate of the intersection points. The expression for \(a_{e}\) is given below. \[a_{e}=\frac{\sqrt{-2\;a_{\tau}^{2}\;b_{\tau}^{2}\;\big{(}-a_{\tau}^{2}-b_{\tau }^{2}+2\;\sigma_{c}^{2}-4\;\sigma_{c}\;z_{o}+2\;z_{o}^{2}+(a_{\tau}^{2}-b_{ \tau}^{2})\;\cos(2\theta_{\mathrm{los}})\;\big{)}}}{a_{\tau}^{2}+b_{\tau}^{2}+ (-a_{\tau}^{2}+b_{\tau}^{2})\;\cos(2\theta_{\mathrm{los}})}. \tag{3}\]  
The expression for minor dimension \(b_{e}\) obtained in the same manner is given below,  
\[b_{e}=b_{\tau}\;\sqrt{\frac{a_{\tau}^{2}+b_{\tau}^{2}-2\;\sigma_{c}^{2}+4\; \sigma_{c}\;z_{o}-2\;z_{o}^{2}+(-a_{\tau}^{2}+b_{\tau}^{2})\;\cos(2\theta_{ \mathrm{los}})}{a_{\tau}^{2}+b_{\tau}^{2}+(-a_{\tau}^{2}+b_{\tau}^{2})\;\cos( 2\theta_{\mathrm{los}})}}. \tag{4}\]  
The horizontal shift of GS from the centre point is shown by \(\Delta x\). The expression for \(\Delta x\) can be calculated as below by using the coordinates of the GS's position. \[\Delta x=\frac{(a_{\tau}^{2}+b_{\tau}^{2})\;x_{o}-(a_{\tau}^{2}-b_{\tau}^{2}) \;x_{o}\;\cos(2\theta_{\mathrm{los}})+(a_{\tau}^{2}-b_{\tau}^{2})\;(\sigma_{c }-z_{o})\;\sin(2\theta_{\mathrm{los}})}{a_{\tau}^{2}+b_{\tau}^{2}+(-a_{\tau}^{ 2}+b_{\tau}^{2})\;\cos(2\theta_{\mathrm{los}})}, \tag{5}\]  
The shift of bounding ellipsoid \(E_{\tau}\) in the \(x\)- and \(z\)-axis is represented by the parameters \(x_{o}\) and \(z_{o}\), respectively, with respect to the base of AS, which can be obtained as,  
\[x_{o}=\frac{d_{\mathrm{los}}}{2}\;\cos\theta_{\mathrm{los}}, \tag{6}\]  
\[z_{o}=h_{\mathrm{AS}}-\frac{d_{\mathrm{los}}}{2}\;\sin\theta_{\mathrm{los}}. \tag{7}\]  
The angles \(\beta_{\mathrm{min}}\) and \(\beta_{\mathrm{max}}\) are angular limits of the elevation angle observed at AS, which can be defined as,  
\[\beta_{\mathrm{min}}=\arctan\left(\frac{h_{\mathrm{AS}}-\sigma_{c}}{d-\Delta x +a_{e}}\right) \tag{8}\]  
\[\beta_{\mathrm{max}}=\left\{ \begin{array}{ll}\arctan\left(\frac{h_{\mathrm{ AS}}-\sigma_{c}}{d-\Delta x-a_{e}}\right)&,&d>a_{e}+\Delta x.\\ \frac{\pi}{2}&,&d\leq a_{e}+\Delta x.\end{array} \right. \tag{9}\]The angles \(\phi_{\min}\) and \(\phi_{\max}\) represent the lower and upper angular limit of the azimuthal angle observed at AS, which can be defined as,  
\[\phi_{\max}^{+} \tag{10}\] \[\phi_{\min}^{-}\ \bigg{\}}=\left\{ \begin{array}{ll}\pm\arctan \left(\mp\frac{b_{\rm c}}{\sqrt{-a_{\rm c}^{2}+(d-\Delta x)^{2}}}\right)&,&d>a _{\rm c}+\Delta x.\\ \\ \pm\pi&,&d<a_{\rm c}+\Delta x.\end{array} \right.\]  
A scattering object is assumed (defined) to be a lossless re-radiating element, which redirects (reflects) a part of the incident wave's power directly towards the receiver. The distance of a certain scattering point \(\rm S_{p}\) from AS and GS is \(r_{a}\) and \(r_{b}\), respectively.