## 3 Reconstruction of Rainfall Field by OELs Network
### Simulation Experiment  
To prove the feasibility of the above methods to reconstruct the rainfall field, a simulation experiment has been carried out in Jiangning district. We choose five broadcast satellites to build the OELs network, and the architecture is shown in Figure 2a. To avoid the effect of different power-law relations, the chosen satellites have similar frequencies. The corresponding quantities and parameters are tabulated in Table 1. Considering the heterogeneity of rain intensity in the monitored region, we divide the rainfall field into \(35\times 35\) grid cells with \(1\times 1\) km\({}^{2}\) resolution (as shown in Figure 2b). According to existing studies, we suppose the rain intensity in each cell is uniform, and the change of rainfall in vertical direction is not considered [36]. Therefore, the path-integrated rain attenuation of \(k\)th OEL is described as:  
\[A_{R}(k)=\frac{h_{0}}{L\sin\theta}\sum\limits_{i=1}^{N}\gamma_{i}d_{i} \tag{18}\]  
where the \(N\) is the number of the grid cells traveled through by \(k\)th OEL, \(L\) denotes the horizontal projection of link on the ground and \(d_{i}\) is the projection of OEL on \(i\)th grid cell. Here, it is assumed that the height of 0 \({}^{\circ}\)C isotherm is well known as statistical value of Nanjing equal to 4.67 km. Because the inversed rain intensity is a path-averaged value, we convert it into data point at the middle position of link which is denoted as the blue \"\(\mathbf{\times}\)\" in Figure 2b. In this work, the synthetic rainfall field are derived from the data given by the Climate Prediction Center morphing method (CMORPH) of NOAA [43]. This technique can provide half-hourly global precipitation estimates, which combines the high-quality values derived from LEO satellite passive microwave (PMW) observations with propagation vector matrices obtained from successive GEO satellite infrared radiation (IR) images. Moreover, a typical pencil beam antenna pattern is used to calculate the sky noise temperature [44]. We only consider nine sample directions comprising a main direction of antenna beam center and eight directions that are symmetrically around the center. Four of them are 0.6\({}^{\circ}\) away from the main direction and the remaining four are taken away 1.2\({}^{\circ}\). The values of antenna gain in the eight directions are \(-\)10 dB and \(-\)20 dB, respectively. Based on Equation (9), the sky noise temperature is obtained by:  
\[T_{sky}=\frac{T_{0}+\sum\limits_{i=1}^{4}0.1\cdot T_{i}+\sum\limits_{i=5}^{8} 0.01\cdot T_{i}}{1+4\times 0.1+4\times 0.01} \tag{19}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Satellites** & **Elevation (\({}^{\circ}\))** & **Azimuth (\({}^{\circ}\))** & **Frequency (GHz)** & \(\mathbf{\alpha}\) (\({}^{\circ}\)) & \(\mathbf{\beta}\) (\({}^{\circ}\)) & **Quantities (\({}^{\circ}\))** \\ \hline ApStar7 & 31.5 & 240 & 12.55 & 0.0287 & 1.1119 & 17 \\ AsiaSat5 & 47.87 & 212 & 12.32 & 0.0267 & 1.1276 & 18 \\ ChinaSat10 & 51.75 & 195 & 12.59 & 0.0288 & 1.1216 & 16 \\ AsiaSat9 & 52.67 & 174 & 12.52 & 0.0282 & 1.1242 & 18 \\ ApStar9 & 45.17 & 141 & 11.56 & 0.0212 & 1.1523 & 24 \\ \hline \end{tabular}
\end{table} 
Table 1: The parameters of broadcast satellites chosen to build OELs network. Figure 2: The architecture of OELs network in Jiangning district: (**a**) original distribution of links, and (**b**) rainfall field divided by grid cells where the black line is the OEL and the blue mark denotes the position of inversed rain intensity. where \(T_{i}\) is the sky noise temperature in each sample direction which is calculated by Equation (8). In addition, it is assumed system noise temperature \(T_{sys}\) and path-average temperature \(t_{m}\) are equal to 150 K and 275 K, respectively [38; 45]. Then the noise figure \(F_{n}\) can be given by Equation (5). According to the literature [39], the free-space attenuation \(A_{F}\) of the OELs in Jiangning district is between 205.77 and 206.16 dB.