## 2 System Model
### Signal Attenuation Model  
The signal attenuation model used here is the same as those in previous studies [6; 15]. The signal power at a ground receiver is obtained by  
\[P_{r}=\xi_{I}\times\xi_{F}\times\xi_{C}\times P_{t}+P_{n}, \tag{1}\]  
in which the power of the received signal and the power of the transmitted signal at the satellite are denoted by \(P_{r}\) and \(P_{t}\), respectively. The power of noise is denoted by \(P_{n}\), which includes both the system noise and the sky noise. The channel gain in Equation (1) is determined by three major factors: \(\xi_{I}\) is the path-integrated cloud attenuation, \(\xi_{F}\) is the free space power loss (FSPL) and \(\xi_{C}\) represents the attenuation due to all other factors such as the gain of antennas and effects of water vapor. Suppose that there are \(M\) signal samples available at the receiver over the duration the overpass of the satellite. Each sample is acquired from a different and very short (0.05 s in the simulations) time window. For the \(k\)-th sample, the receiver's SNR in decibels is obtained by  
\[\rho(k)=10\cdot\log\frac{\xi_{I}(k)\times\xi_{F}(k)\times\xi_{C}(k)\times P_{ t}(k)}{P_{n}(k)},\;k=1,2,\cdots,\;M. \tag{2}\]  
We define \(A_{I}(k)\) as the path-integrated cloud attenuation in decibels as  
\[A_{I}(k)=-10\times\log(\xi_{I}(k)), \tag{3}\]  
and the free space loss \(A_{F}(k)\) is obtained by  
\[A_{F}(k)=-10\times\log(\xi_{F}(k)), \tag{4}\]  
thus, \(\rho(k)\) can be rewritten as  
\[\rho(k)=10\times\log\frac{\xi_{C}(k)\times P_{t}(k)}{P_{n}(k)}-A_{I}(k)-A_{F}(k). \tag{5}\]  
Let the noise temperature of the receiving system and the sky temperature be denoted by \(T_{sys}(k)\) and \(T_{sky}(k)\), in Kelvin, respectively. In this case, noise power \(P_{n}(k)\) is obtained by [16]  
\[P_{n}(k)=k_{B}\times B\times\Big{(}T_{sys}(k)+T_{sky}(k)\Big{)}, \tag{6}\]  
where \(k_{B}\) is the Boltzmann constant and \(B\) is the bandwidth of the communication system. We define parameter \(C(k)\) as  
\[C(k)=10\times\log\frac{\xi_{C}(k)\times(k)}{k_{B}\times B\times T_{sys}(k)}, \tag{7}\]  
and noise figure \(F_{n}(k)\) as  
\[F_{n}(k)=10\times\log\Bigg{(}\frac{T_{sys}(k)+T_{sky}(k)}{T_{sys}(k)}\Bigg{)}, \tag{8}\]  
according to Equation (5), \(\rho(k)\) is then obtained by  
\[\rho(k)=C(k)-F_{n}(k)-A_{I}(k)-A_{F}(k). \tag{9}\]Parameter \(C(k)\) can be regarded as a baseline value for a receiver, which is mainly determined by the noise temperature of the receiving system, the antenna gain and the power of the transmitted signal. In this model, cloud is the primary source for \(T_{sky}(k)\), which is generally obtained by [17]  
\[T_{sky}(k)=\frac{\int G(k,\Omega)T(\Omega)d\Omega}{\int G(k,\Omega)d\Omega}, \tag{10}\]  
where \(\Omega\) is the solid angle, \(G(k,\Omega)\) is the antenna gain pattern and \(T(\Omega)\) is the noise temperature for a particular direction determined by \(\Omega\), which is obtained by  
\[T=t_{m}\Big{(}1-10^{-\frac{A_{I}}{10}}\Big{)}, \tag{11}\]  
where \(t_{m}\) is the mean path temperature, in Kelvin, and \(A_{I}\) is the total path attenuation for that direction, in dB. We assume that the beam pattern of the antenna is known so the relation between \(T_{sky}\) and \(A_{I}\) can be established using Equations (10) and (11).  
Meanwhile, \(A_{F}(k)\) is a function of the satellite-to-ground path length \(L_{S}\) and the signal frequency \(f\), and is commonly obtained by [18]  
\[A_{F}(k)=20\times\log(L_{S}(k))+20\times\log(f)+92.45, \tag{12}\]  
where the unit of \(L_{S}(k)\) is in km and \(f\) in GHz. Since \(A_{F}(k)\) can be regarded as known, Equation (9) indicates that \(A_{I}(k)\) has the following relation with the SNR estimation:  
\[A_{I}(k)=C(k)-\hat{\rho}(k)-A_{F}(k)-F_{n}(k), \tag{13}\]  
where \(\hat{\rho}(k)\) is the estimated SNR at the receivers.