## 2 Field Measurement of Rainfall from OEL  
### Inversion of Rain Intensity by a Single OEL  
For a single OEL, the received signal power \(P_{r}\) is related to the power of satellite transmitter \(P_{e}\), the gain of antennas \(\eta_{G}\), the free-space power loss \(\eta_{F}\), the tropospheric attenuation \(\eta_{T}\) and the power of noise \(P_{n}\), which can be written as:  
\[P_{r}(t)=\eta_{G}(t)\cdot\eta_{F}(t)\cdot\eta_{T}(t)\cdot P_{e}(t)+P_{n}(t) \tag{1}\]  
where the \(\eta_{T}\) refers to the attenuation caused by rainfall \(\eta_{R}\) and other atmospheric factor \(\eta_{O}\) such as oxygen, water vapor, liquid water in cloud and scintillation. Therefore, in terms of the signal to noise ratio (\(SNR\)), its expression in decibels can be given by:  
\[SNR(t)=10\cdot\log\frac{\eta_{G}(t)\eta_{O}(t)P_{e}(t)}{P_{n}(t)}-A_{R}(t)-A_ {F}(t) \tag{2}\]  
where \(A_{R}\) and \(A_{F}\) are the rainfall- and free-space attenuation in decibels, respectively. As for the noise power \(P_{n}\), it can be described as:  
\[P_{n}(t)=k_{B}\cdot B\cdot\left(T_{sys}(t)+T_{sky}(t)\right) \tag{3}\]  
where \(k_{B}\) is the Boltzman's constant, \(B\) is the bandwidth of OEL signal, \(T_{sys}\) and \(T_{sky}\) denote the noise temperature of receiving system and sky. During rainy period, the sky noise collected by antenna mainly derives from rainfall. The corresponding noise power is related to the rain intensity and antenna directivity. Because the sampling period of satellite signal is very short in our experiment (about 6 s), it is reasonable to assume that the change of \(T_{sys}\), \(\eta_{G}\) and \(P_{e}\) is negligible. Therefore, the Equation (2) can be rewritten as:\[SNR(t)=V(t)-F_{n}(t)-A_{R}(t)-A_{F}(t) \tag{4}\]  
where \(F_{n}\) is defined as Equation (5), and if we know the \(T_{sys}\), the \(F_{n}\) will only relate to \(T_{sky}\). Moreover, the change of \(V\) mainly depends on the attenuation caused by no-rain factor and its expression is given by Equation (6). \[F_{n}(t)=10\cdot\log\left(\frac{T_{sys}+T_{sky}(t)}{T_{sys}}\right) \tag{5}\]  
\[V(t)=10\cdot\log\frac{\eta_{G}\eta_{G}(t)P_{e}}{P_{0}} \tag{6}\]  
where \(P_{0}\) is written as:  
\[P_{0}=k_{B}\cdot B\cdot T_{sys} \tag{7}\]  
For a fixed solid angle \(\Omega\), the relation between the sky noise temperature \(T_{sky}(\Omega)\) and rain attenuation in this direction is described as [38]:  
\[T_{sky}(t,\Omega)=t_{m}\left(1-10^{-\frac{A_{B}(t)}{10}}\right) \tag{8}\]  
where \(t_{m}\) is the path-average temperature. Considering the effect of antenna beam, the received sky noise can be expressed by [38]:  
\[T_{sky}(t)=\frac{\int G(t,\Omega)T_{sky}(t,\Omega)d\Omega}{\int G(t,\Omega)d\Omega} \tag{9}\]  
where the \(G(t,\,\Omega)\) is the antenna gain for a particular direction. Therefore, the sky noise \(T_{sky}\) is a function of rain attenuation \(A_{R}\). Moreover, the free-space attenuation \(A_{F}\) relates to the signal frequency and the distance from satellite to antenna. The specific relation among them can refer to literature [39]. For a GEO earth-space link, the value of \(A_{F}\) is assumed as a constant. According to ITU-R P.838 [40], the rain specific attenuation \(\gamma_{rain}\) is a function of rain intensity \(R\) in a power-law manner that is:  
\[\left\{ \begin{array}{l}\gamma_{rain}=\alpha R^{\beta}\\ A_{R}=\gamma_{rain}\cdot\left(h_{0}+0.36\right)/\sin\theta\end{array} \right. \tag{10}\]  
where the coefficients \(\alpha\) and \(\beta\) relate to signal frequency, polarization, temperature, and raindrop size distribution, \(\theta\) is the elevation of antenna and \(h_{0}\) is the height of \(0\,^{\circ}\)C isotherm. Most of existing studies on rainfall inversion by OLE depend on this relation. During the no-rain period, the value of \(SNR\) is given by:  
\[SNR_{no-rain}=V(t)-F_{n}(t)-A_{F}(t) \tag{11}\]  
where we consider this value as a reference level \(SNR\)fer that is used to obtain the rain attenuation. In our previous work, the issue, determining reference level during rain period has been addressed [28].