## 2 Methodology for Determination of Beam Pointing Dynamics
### Modified Model
Hence, in this paper, the turbulence wave number \(\kappa_{(i,j)}\) has been considered a uniformly distributed random variable lying between limits \(\frac{2\pi}{L_{0}}\) and \(\frac{2\pi}{l_{0}}\), in contrast to [PERSON]'s model, which uses a fixed value of \(\kappa_{(i,j)}\) for a given sub volume  
(India), with the common volume of the link formed over Ahmedabad, Gujarat (India). The transmitter location for the second link is near Bhukuradia, Assam (India), and the receiver location is near Bherakuchi, Assam (India), with the common volume of the link formed over Guwahati, Assam (India). The links belong to different climate types. Ahmedabad experiences a hot and semi-arid climate, while Guwahati has a warm and temperate climate. Thus, the results and conclusions obtained in this paper are based on a broad study of tropospheric links belonging to different climate types. The parameters obtained from the Ahmedabad and Guwahati links are given in Table 1. In the case of the Ahmedabad link, the number of sub-beam sets (M) has been taken as 10 for this paper, as it gives a vertical height range of each sub volumes close to 150 m. In the case of the Guwahati link, the number of sub-beam sets (M) has been taken as five due to the narrow beam width of transmit and receive antennas, which gives a vertical height range of each sub volumes close to 180 m. The selection of M for both links has been chosen keeping in mind the fact that the atmospheric conditions within a vertical distance of 200 m remain almost the same, as observed from the meteorological data obtained from the Indian Meteorological Department (IMD). The calculation of the refractive index gradient requires meteorological parameters of atmospheric pressure, temperature, and dew point temperature. These parameters have been obtained from the Ahmedabad and Guwahati observatories of IMD. Meteorological observatories of IMD make radio sonde measurements of these parameters for heights ranging from the Earth's surface up to 20 Kms, two times in a day, at 05:30 a.m. and  
* Location of tropospheric links is not actual, as the links have been software synthesized. ** Antenna gain shown in table factors 3 dB gain reductions due to antenna efficiency of 50%. Actual antenna gain for Ahmedabad link is 42 dB, and for Guwahati, the link is 46.4 dB. *** Values of N and M, for both the links, have been assumed in this paper. \end{table}
Table 1: Parameters for tropospheric links with common volumes over Ahmedabad (Gujarat, India) and Guwahati (Assam, India). 05:30 p.m. However, observatories are not able to record data on all days of the year due to maintenance and other activities. The meteorological data for both observatories were obtained from 2005 to 2021. It was found that the Ahmedabad observatory had the maximum number of recordings for the year 2006, while the Guwahati observatory had the maximum number of recordings in the year 2019. Hence, the meteorological data from the year 2006 were used in the case of the Ahmedabad link, and the year 2019 was used in the case of the Guwahati link for the determination of beam pointing dynamics for these links. The calculation of the received power using the modified model, given in Equation (9), requires the determination of various system and link geometry parameters along with a refractive index gradient. Certain system and link geometry parameters, like transmit power (\(P_{l}\)), transmit and receive antenna gains (\(G_{t}\), \(G_{r}\)), antenna beam widths (\(\omega\), \(\phi\)), and transmission wavelength (\(\lambda\)), are outlined in Table 1. However, other parameters, like the scatter angle (\(\psi_{(i,j)}\)), the distances of sub volume center from transmit and receive antennas (\(R_{t(i,j)}\), \(R_{r(i,j)}\)), the vertical gradient of refractive index (\(\delta n/\delta z\)), the outer scale of turbulence (\(L_{0}(z)\)), and the turbulence wave number (\(\kappa_{(i,j)}\)) need to be calculated. The calculations of these parameters are shown in Appendices A-C.