## 2 Methodology for Determination of Beam Pointing Dynamics
### [PERSON]'s Ray-Based Model  
Power transmitted in any transmit sub-beam is scattered by the scatterers present in the corresponding sub volume and is received at the receive antenna in the corresponding received sub-beam. The expression of the power received, for any sub-beam set, is given by [PERSON] et al. [19], using the bistatic radar range equation as follows:  
\[P_{r(i,j)}\ =\ \frac{P_{t}\ G_{t}\ G_{r}\ \sigma_{V(i,j)}\ dV_{C(i,j)}\ \lambda^{2}\ \rho}{\left(4\pi\right)^{3}{R_{t(i,j)}}^{2}{R_{r(i,j)}}^{2}}\ \ \ \text{watts} \tag{1}\]  
Subscript \((i,j)\) denotes \(j\)th transmit and receives the sub-beam set for the \(i\)th elevation angle of antenna beams or the \(j\)th sub volume of the common volume formed by the antenna beams with \(i\)th elevation angle. For example, in Figure 2, \(j\) varies from 1 to M while \(i\) remains 1 since the lowest set of transmit and receive antenna beams is shown in the figure. However, \(i\) can take any value from 1 to \(N\), as shown in Figure 1. The total power received for the \(i\)th set of transmit and receive beam elevation angles is obtained as follows:  
\[P_{r(i)}=\sum_{j=1}^{M}P_{r(i,j)}\ \ \text{watts}\ i\in(1,N) \tag{2}\]  
\(P_{t}\) in Equation (1) is the power transmitted in any transmit sub-beam in watts while \(G_{t}\) and \(G_{r}\) are the gains of transmit and receive antennas, respectively. \(P_{t}\), \(G_{t}\), and \(G_{r}\) are considered equal for all sub-beams. \(\lambda\) is the wavelength in meters, corresponding to the  
Figure 2: Splitting of antenna beams into sub-beams.  
transmit carrier frequency. \(\rho\) is the factor that accounts for the polarization mismatch between transmit and receive sub-beams with a value lying between 0 and 1. The value of \(\rho\) has been taken as 1 in this paper for all sub-beams, assuming no loss due to polarization mismatch. \(R_{t(i,j)}\) and \(R_{r(i,j)}\) are the distances of the sub volume center from transmit and receive antennas, respectively.  
\(\sigma_{V(i,j)}\) is the differential scattering cross section of the sub volume per unit volume of turbulence in the sub volume. \(dV_{c(i,j)}\) is the size of the sub volume. The final expression for \(P_{r(i,j)}\) has been given in [19] as follows:  
\[P_{r(i,j)}=\frac{P_{t}\ G_{t}\ G_{r}\ \pi^{2}\Big{(}\psi_{(i,j)}^{2}\Big{)}\ \big{(}\frac{6\pi}{ \delta z}\big{)}_{(i,j)}^{2}}{21.64\ \lambda^{2}\ \sqrt{R_{t(i,j)}+R_{r(i,j)}^{2}}\ sin \Big{(}\psi_{(i,j)}\Big{)}}\cdot\frac{L_{0}(z)^{\frac{4}{3}}}{\frac{\psi_{(i,j )}}{2}}\ \text{watts} \tag{3}\]  
where \(d\omega\) is the vertical beamwidth of any transmit or receive sub-beam in radians. \(\psi_{(i,j)}\), known as the scatter angle (Appendix A), is the angle in the radians made by the direction of the incident EM ray with a scattered ray. It has been taken as the angle between the lowest rays of the \((i,j)\)th transmit and receive sub-beams in this paper. \(\big{(}\frac{\delta\pi}{\delta z}\big{)}_{(i,j)}\) is the vertical gradient (Appendix B) of the refractive index \(n\) with a height \(z\) above the Earth's surface in the \((i,j)\)th sub volume. \(L_{0}(z)\) is the outer scale of turbulence, in meters, and \(k\) is the wavenumber given as \(2\pi/\lambda\).