## 2 Methodology for Determination of Beam Pointing Dynamics
### Modified Model  
The expression for the received power given in [PERSON]'s ray-based model requires certain modifications for a more realistic and accurate representation of the tropospheric conditions. The expression for \(\sigma_{V(i,j)}\) used by [PERSON] is the acoustic cross section per unit volume of a random medium given in [21] and is applicable for the propagation of sound waves. However, tropospheric communication involves the propagation of EM waves in a turbulent tropospheric medium. Hence, this paper uses a scattering cross section per unit volume of a random medium applicable for EM waves, given in [21] as follows:  
\[\sigma_{V(i,j)}=2\pi k^{4}\ sin^{2}(X)\ \varphi(i,j)\quad\text{m}^{-1} \tag{4}\]  
where \(\varphi(i,j)\) is the spectral density of refractive index fluctuations in the sub volume. \(k\) is the wavenumber given as \(\frac{2\pi}{\lambda}\), and \(\lambda\) is the transmission wavelength. \(X\) is the angle between the direction of the incident electric field in the sub volume and the direction from the sub volume to the receiver. Since power is received only due to forward scattering toward the receiver, \(X\) is always equal to \(\frac{\pi}{2}\) and, hence, \(sin^{2}(X)\) is equal to 1 for any sub volume. Expression for \(\varphi(i,j)\), based on the Kolmogorov spectrum is given from [21] as follows:  
\[\varphi(i,j)=0.033\ C_{n(i,j)}^{2}\Big{(}\kappa_{(i,j)}\Big{)}^{\frac{-11}{3} }\quad\text{m}^{3} \tag{5}\]  
where \(\kappa_{(i,j)}\) is the turbulence wave number in the sub volume. It is given as \(\kappa_{(i,j)}=2\pi/l\), where \(l\) represents the size of turbulent eddies or scatterers in meters. \(C_{n(i,j)}^{2}\) is the refractive index constant structure representing the strength of atmospheric turbulence in the sub volume, given from [22] as follows:  
\[C_{n(i,j)}^{2}=2.8\ \bigg{(}\frac{\delta n}{\delta z}\bigg{)}_{(i,j)}^{2}L_{0}( z)^{\frac{4}{3}} \tag{6}\]  
The Kolmogorov spectrum is most suitable for modeling \(\varphi(i,j)\), as it is a realistic spectrum, accurately representing the physical characteristics of tropospheric turbulence. As per the Kolmogorov theory, atmospheric turbulence has an outer scale \(L_{0}\) and an inner scale \(l_{0}\). Based on atmospheric studies [23], the value of \(L_{0}\) lies between 10 and 100 m and \(l_{0}\) between 1 and 10 mm. Turbulence characteristics can be divided into three regionsaccording to the size of scatterers relative to these scales viz. : input range, inertial range, and dissipation range [23]. Large-sized eddies (scatterers) are formed in the input range due to the thermal and kinetic energy of the atmosphere. They break down successively into smaller-sized eddies in an inertial range with very little loss of energy. Finally, in the dissipation range, the kinetic energy of the eddies is dissipated through viscous friction when the eddy size becomes very small. The size of the scatterers in inertial range lies between \(l_{0}\) and \(L_{0}\), which is ideal for microwave scattering. Correspondingly, \(\kappa_{(i,j)}\) lies between \(\frac{2\pi}{l_{0}}\) and \(\frac{2\pi}{l_{0}}\). The ray-based model given by [PERSON] assumes the troposphere to be a stationary random medium with the size of scatterers in a given sub volume remaining constant over time. Hence, \(\varphi(i,j)\) has been calculated in [PERSON]'s model for a fixed value of \(\kappa_{(i,j)}\) in a given sub volume. \(\kappa_{(i,j)}\) in the \((i,j)\)th sub volume is given as \(2k\ sin\ sin\ \left(\frac{\Psi_{(ij)}}{2}\right)\), where \(k\) is constant for a given tropospheric system and \(\psi_{(i,j)}\) is constant for a given sub volume. However, turbulence in the troposphere is not stationary, as large-sized eddies successively break down into smaller-sized eddies at all times. Hence, the size of scatterers is not constant but varies with time, lying between the inner and outer scales of turbulence.