## 2 Principles of Rainfall Field Retrieval by ESLs and HMLs
### Principle of Rainfall Retrieval by Esl  
The basic principle of using ESL to retrieve rainfall is briefly described as follows. When rainfall occurs in the ESL between the satellite and the antenna, the microwave signal of the ESL will be weakened due to the effect of scattering and absorption. Then, the changing process of microwave signal transmission can be monitored by the receiver of the ground satellite antenna. At this time, the microwave rain attenuation can be obtained by using the corresponding extraction algorithm. Finally, the average rain rate on the path of ESL can be obtained by the calculation model between microwave rain attenuation and rainfall [38]. Next, a detailed analysis of the radiation transfer process of ESL is presented. After passing through the atmosphere, the microwave signal sent by the satellite reaches the ground satellite antenna and is transmitted to the receiver for processing. We use \(P_{t}\) and \(P_{r}\) to represent the transmitting power of the satellite and the receiving power of the ground receiver, respectively, and have the following equation [32]:  
\[P_{r}(t)=P_{t}(t)\cdot G\cdot\eta_{A}\cdot\eta_{F}\cdot\eta_{R}+P_{n}(t) \tag{1}\]  
where \(G\) and \(P_{n}\) represent the antenna gain and the noise power of the signal, respectively. \(\eta_{A}\) represents the attenuation coefficient of non-rainfall factors in the atmosphere, mainly related to clouds, gas composition and atmospheric turbulence. Additionally, \(\eta_{F}\) and \(\eta_{R}\) represent the attenuation coefficients of ESL in free space and rainfall, respectively. In  
Figure 1: The process for retrieving rainfall fields in the area by combining ESLs and HMLs. addition, the signal-to-noise ratio (SNR) of the ESL signal received by the ground receiver can be expressed as  
\[\text{SNR}(t)=10\text{log}\frac{P_{t}(t)\cdot G\cdot\eta_{A}\cdot\eta_{F}\cdot \eta_{R}}{P_{n}(t)}. \tag{2}\]  
The free space loss \(A_{F}\) and the path-integrated rain attenuation \(A_{R}\) of ESL are defined as  
\[\left\{ \begin{array}{l}A_{F}(t)=-10\text{log}(\eta_{F})\\ A_{R}(t)=-10\text{log}(\eta_{R})\end{array} \right., \tag{3}\]  
then, Equation (2) can be written as  
\[\text{SNR}(t)=10\text{log}\frac{P_{t}(t)\cdot G\cdot\eta_{A}}{P_{n}(t)}-A_{R} (t)-A_{F}(t). \tag{4}\]  
For noise power \(P_{n}\), it mainly includes sky noise \(T_{sky}\) and system noise \(T_{sys}\), which can be given by  
\[P_{n}(t)=k_{B}\cdot B\cdot\left(T_{sys}(t)+T_{sky}(t)\right), \tag{5}\]  
where \(k_{B}\) is the Boltzman constant, and \(B\) represents the bandwidth of the ESL signal. \(T_{sys}\) can be regarded as constant when the receiver of ESL works stably. In addition, when there is rainy weather, the \(T_{sky}\) received by the receiver mainly comes from the influence of rainfall. Additionally, it can be obtained by  
\[T_{sky}(t)=\frac{\int G(t,\Omega)\cdot t_{m}(1-10)^{-\frac{A_{B}(t)}{\Omega}}d \Omega}{\int G(t,\Omega)d\Omega}, \tag{6}\]  
where \(G\) (\(t\), \(\Omega\)) is the antenna gain pattern at the fixed solid angle \(\Omega\), and \(t_{m}\) is the average temperature of the path [39]. We transform and rewrite Equation (5) as  
\[P_{n}(t)=k_{B}\cdot B\cdot T_{sys}(t).\frac{\left(T_{sys}(t)+T_{sky}(t)\right) }{T_{sys}(t)}, \tag{7}\]  
and define  
\[\left\{ \begin{array}{l}F_{n}(t)=k_{B}\cdot B\cdot\left(\frac{T_{sys}(t)+T_{ sky}(t)}{T_{sys}(t)}\right)\\ P_{0}(t)=k_{B}\cdot B\cdot T_{sys}(t)\\ V(t)=10\text{log}\frac{P_{t}(t)\cdot G\cdot\eta_{A}}{P_{0}(t)}\end{array} \right., \tag{8}\]  
then, Equation (4) can be rewritten as  
\[\text{SNR}(t)=V(t)-F_{n}(t)-A_{R}(t)-A_{F}(t). \tag{9}\]  
When there is no rain, the \(\text{SNR}_{no-rain}\) received by the ESL receiver can be regarded as the SNR caused by factors other than rainfall. Additionally, \(\text{SNR}_{no-rain}\) can be written as  
\[\text{SNR}_{no-rain}=V(t)-F_{n}(t)-A_{F}(t). \tag{10}\]  
In order to extract the rain attenuation caused by rainfall, we use \(\text{SNR}_{no-rain}\) as the attenuation baseline.