## 3 Verification Using Ray Analysis  
To validate the result in Figure 5, we compare the received power at the optimum position \(p_{2}\) along the rocket altitude with the gain of the transmitting antenna. Figure 7 illustrates the relation between the elevation angle and the rocket's altitude in the LoS direction at \(p_{2}\). The blue line expresses the radiation pattern of the transmitting antenna, and the elevation angle at the LoS direction is denoted by \(\theta_{e}\), which is related to the rocket's altitude of h. For example, as \(\theta_{e}\) increases according to the rocket altitude \(h\), the gain along the LoS direction decreases due to the weak pattern of the antenna in the rocket's tail direction. Figure 8 presents the elevation angle and the gain according to the rocket's altitude. The solid line presents \(\theta_{e}\) along \(h\), and the dashed line indicates the LoS direction's gain along the \(h\). As can be seen in Figure 8, \(\theta_{e}\) increases from \(-3.6\) to \(78.7^{\circ}\) when the rocket's altitude \(h\) increases from 0 to 1000 m. At the same time, the gain in the LoS direction decreases from 2.15 to \(-12.0\) dBi. Thus, the \(\theta_{e}\) is proportional to the rocket's altitude, and the gain decreases with the rocket's altitude. The result demonstrates that gain in the LoS direction is related to radiation pattern of the transmitting antenna and rocket's altitude, because the relation of the \(\theta_{e}\) and the gain in the LoS direction is determined by transmitting antenna pattern. Therefore, it can be expected that the increments in path loss are mainly influenced by the radiation pattern of the transmitting antenna, as the rocket's altitude increases. Figure 8: Elevation angle and the gain according to the rocket’s altitude. Figure 7: Relation between the elevation angle and the rocket’s altitude in the LoS direction. The Figure 9 shows the received power and the gain in the LoS direction at \(p_{2}\). The solid line shows the received power when the input power to the transmitting antenna in the rocket is 0 dBm. The received power is \(-86\) dBm at the altitude of 200 m where the path loss is 86 dB. Thus, the received power in Figure 9 is related to the path loss in Figure 5. The dashed line is the antenna gain of the transmitting antenna in the rocket along the LoS direction according to the altitude. Therefore, we found that the envelope of the received power curve is very similar to the gain curve in the LoS direction according to the altitude \(h\). The envelope of the received power gradually decreases due to the weak radiation pattern of the transmitting antenna in the tail direction of the rocket when the rocket's altitude increases. Figure 10 describes the received power intensity and phase differences between direct and reflected rays, which can be written as Equations (2) and (3), respectively [22]:  
\[E_{d}^{k}=\left|E_{1}^{k}\right|e^{-j\beta\phi_{d}} \tag{2}\]  
where \(E_{d}\) is the electric field of a direct ray, which is generally dominant, and the superscript \(k\) is the index of the observed point. \(E_{m}\) represents the total multiple reflected rays, which can be expressed as below:  
\[ \begin{array}{l}E_{m}^{k}=\left|E_{2}^{k}\right|\left\{\prod\limits_{n=1}^{N _{2}}\left(\left|\Gamma_{(2,n)}\right|e^{-j\phi_{(2,n)}}\right)\right\}e^{-j \beta\phi_{2}}\\ +\left|E_{3}^{k}\right|\left\{N_{n}^{3}\prod\limits_{n=1}^{N_{1}}\left(\left| \Gamma_{(3,n)}\right|e^{-j\phi_{(3,n)}}\right)\right\}e^{-j\beta\phi_{3}}\\ +\left|E_{4}^{k}\right|\left\{N_{n}^{4}\prod\limits_{n=1}^{N_{4}}\left(\left| \Gamma_{(4,n)}\right|e^{-j\phi_{(4,n)}}\right)\right\}e^{-j\beta\phi_{4}}\\ =\left|E_{m}^{k}\right|e^{-j\beta\phi_{m}}\end{array}  \tag{3}\]  
where \(N_{p}\) represents the number of \(p\)th rays so that \(\left|\Gamma_{(p,n)}\right|\) and \(\varphi_{(p,n)}\) are the amplitude and phase of nth reflection coefficients for the \(p\)th ray.