## 1 Introduction
### Research Methodology  
Based on the research work carried out in this paper, the received power from elevated beams can also deliver the highest possible received power if the refractive index gradient is considerably large at higher altitudes, which is contrary to common belief. The refractive index gradient depends on meteorological conditions and is different at different places and at different times of the year. The study of the beam pointing dynamics of a particular link involves obtaining knowledge of received power values for different heights depending on the refractive index gradient at these heights. It helps identify most of the favorable propagation heights for the link at different times of the year. Hence, the research work for this paper involved the identification or development of a tropospheric channel model that is able to determine the received power values for signal propagation at different heights using real-world values of a refractive index gradient rather than statistical or average values. Prominent tropospheric channel models were examined to identify a model that can be directly used or modified for the determination of beam pointing dynamics. [PERSON] and [PERSON] [13] proposed a model based on the scattering theory and the model given by [PERSON] et al. [14] based on the reflection theory, which are the basic models that offer theoretical expressions for a received power over a tropo link. Both of the models assume the common volume to be one homogenous whole with a uniform density of scatterers or reflecting layers, ignoring variations in the propagation characteristics at different heights for simplicity of derivation. Thus, these models cannot be used for the study of beam pointing dynamics. The scattering theory has gained more prominence compared to the reflection theory and has become the basis for the development of various models in the future. [PERSON] used the scattering theory to develop a more comprehensive model [15], dividing common volumes into vertically stacked smaller volumes, to determine the power delay profile (PDP) of tropospheric channels. This method of dividing common volumes is suitable for determining the power received from different heights. However, the received power values obtained for different heights are not realistic as the model assumes the tropospheric refractive index to simply decrease with height in the absence of exact knowledge, making it unsuitable. The models given in the ITU-R recommendations were also considered, as they are widely used for link budgeting and the planning of tropospheric communication links. These models, based on the scattering theory, are able to predict the transmission loss not exceeded for various percentages of time in a year. The Technical Note no. 101 [16], issued by the National Bureau of Standards (NBS) in 1965, gave the first prediction model, which became the basis for various recommendations of the radiocommunication sector of the International Telecommunication Union (ITU-R), such as ITU-R P.452-17 [17], ITU-R P.617-5 [9], and ITU-R P.2001-4 [18]. The model given in [18] is the most wide-ranging model, which calculates the transmission loss as per the climatic zone to which the link belongs. The model considers the average value of the atmospheric refractivity and its gradient over space and time for each zone. However, the refractive index gradient at a particular point and time differs from the zone average. Moreover, the approach used in prediction models is suitable for fixed-beam tropo systems and cannot be applied directly to determine the received power from different heights. The ray-based channel model developed by [PERSON] et al. [19] has an approach similar to the Bello model, as it divides the vertical beam widths of transmit and receive antennas into smaller, vertically stacked contiguous sub-beams, forming vertically stacked sub volumes. The model calculates the individual received powers for each sub-beam set using the refractive index gradient at the height of its corresponding sub volume.