# Reconstruction of Rainfall Field Using Earth-Space Links Network: A Compressed Sensing Approach
## 4 Results and Discussion  
Firstly, we used the designed ESL networks to obtain the sparse distribution of rain rates in the area. The CS method was then used to reconstruct the sparse rain rates to obtain a continuously distributed rainfall field in the area. To verify the feasibility and performance of reconstructing the rainfall fields using the ESL networks with CS, we used the rainfall products of the Climate Prediction Center Morphing Technique (CMORPH) published by the National Oceanic and Atmospheric Administration (NOAA) to establish the real rainfall fields in the area, and we then reconstructed the real rainfall fields of CMORPH using the ESL networks with CS. Because the experimental area was gridded to 1 km, we used the spatial interpolation algorithm to process the resolution of the rainfall fields of CMORPH from 8 \(\times\) 8 km\({}^{2}\) to 1 \(\times\) 1 km\({}^{2}\), so as to ensure the spatial resolution of the rainfall fields in the area was 1 \(\times\) 1 km\({}^{2}\). To further illustrate the performance of the CS method for reconstructing rainfall fields, the results of the reconstruction of rainfall fields by the CS method are compared with the reconstruction results of the IDW algorithm. Additionally, the root mean square error (RMSE), the mean bias (MB) and the correlation coefficient (CC) are used to quantitatively evaluate the error, unbiasedness and correlation between the reconstructed rainfall fields and the real rainfall fields, respectively [22]. These quantitatively evaluated indicators are given by  
\[\text{RMSE} =\sqrt{\frac{1}{n}\frac{n}{i=1}^{n}\big{(}R_{i}^{\prime}-R_{real,i} \big{)}^{2}} \tag{19}\] \[\text{MB} =\frac{1}{n}\frac{n}{i=1}^{n}\big{(}R_{i}^{\prime}-R_{real,i} \big{)} \tag{20}\]  
Figure 3: Schemes follow the same formatting. Distribution of the ESL network in Jiangning district, Nanjing: (**a**) ESL network 1; (**b**) ESL network 2; (**c**) ESL network 3; (**d**) ESL network 4. The red points with the yellow outline represent the locations of the ground antennas of the ESLs. and  
\[\text{CC}=\frac{\text{cov}\big{(}R_{i}^{\prime}\,,R_{real,i}\big{)}}{\sigma_{R^{ \prime}}\cdot\sigma_{R_{real}}} \tag{21}\]  
where \(\text{cov}(R^{\prime},R_{real})\) is the covariance between the reconstructed rainfall field \(R^{\prime}\) and the real rainfall field \(R_{real}\), and \(\sigma_{R}\) and \(\sigma_{R}\) are the standard deviations of the reconstructed rainfall field and the real rainfall field, respectively. To verify the feasibility of reconstructing a real rainfall field using an ESL network, a rainfall event on 10 August 2019 was reconstructed using the designed ESL network 1, which has the minimum number of links. Rainfall field 1 to rainfall field 4 depict the development of the rainfall event from 01:00 to 04:00 with intervals of one hour in Figure 4. The rainfall fields of the rainfall event were then reconstructed using ESL network 1, and the results are shown in Figures 5 and 6 for the cases of no noise and systematic noise, respectively. By simple comparison of Figures 4 and 5, it can be found that the systematic noise has essentially no significant effect on the reconstruction results of the IDW algorithm, while there is a slight effect on the reconstruction results of the CS method. Additionally, by comparing the real rainfall fields in Figure 4 with the reconstructed rainfall fields in Figures 5 and 6, it can be shown that both the IDW algorithm and CS have the capacity to reconstruct the basic distribution characteristics of the rain rates of the rainfall fields and to accurately monitor the movement of the rain cluster from southeast to northwest in that rainfall process. However, there are significant local distortions in the rainfall fields reconstructed by the IDW algorithm which reflect the local differences between the reconstructed rainfall fields and the real rainfall fields in terms of the structure.