# Reconstruction of Rainfall Field Using Earth-Space Links Network: A Compressed Sensing Approach
## 3 Design of the ESL Network  
To verify the feasibility and performance of reconstructing high-precision rainfall fields using an ESL network with CS, we have carried out the design of an ESL network by simulation experiment in Jiangning district, Nanjing. The range of the experimental area is 118.4808\({}^{\circ}\)E-119.1178\({}^{\circ}\)E, 31.6235\({}^{\circ}\)N-32.1165\({}^{\circ}\)N. The whole experimental area of approximately 1581 km\({}^{2}\) is divided into 1581 grids, and the rainfall distribution within each grid is considered to be uniform, so that the spatial resolution of the rainfall fields in the area is 1 \(\times\) 1 km\({}^{2}\). In addition, the average height of the atmospheric 0 \({}^{\circ}\)C layer in the area is approximately 4.67 km [28].  
The ESL network designed is capable of detecting the sparsely distributed rainfall information in the area, and the design of the ESL network directly affects the accuracy of the reconstructed rainfall fields. Additionally, the actual distribution of the ground antennas of the ESL in the area can generally be regarded as a randomly distribution. Therefore, in order to make the rainfall information detected by the designed ESL network as representative of the entire regional rainfall field as possible, we used the K-means clustering algorithm to make the links that form the ESL network distributed uniformly and randomly in the area [33]. The detailed process of the algorithm for designing the ESL network is as follows:  
1. Randomly select \(m\) points from the \(n\) grid points in the experimental area as the initial locations of the antennas of the ESL network.
2. Calculate the distance from all grid points of the area to each antenna position of the ESL network, and divide the area into m classes based on the rule of the shortest distance.
3. The mean value of the area for each class is calculated and used as the center of the new clusters.
4. Repeat steps (2) and (3) until the clustering results no longer change. Following that, \(m\) points in the area are used as locations for the antennas of the ESL network, from which the designed ESL network can be obtained.  
It should be noted that the K-means clustering algorithm is used here to create ESL networks with different densities and structures uniformly distributed, and in practice the networking needs to consider the effect of topography. Additionally, in order to make the sensing matrix determined by the ESL network conform to RIP under CS theory, the number of links \(m\) that form the ESL network needs to conform to the qualification that is given by [29]  
\[\left\{\begin{aligned} m\geq C\cdot k\cdot\ln\left(\frac{n}{k} \right),\\ where\ C=&\frac{1}{2}\log\left(\sqrt{24}+1\right) \approx 0.385\end{aligned}\right. \tag{18}\]  
where \(k\) is the sparsity of the sparse signal, and \(C\) is the constant. With the sparsity \(k\) being set to 10, it is calculated that \(m\geq 19.493\). Therefore, we designed four types of ESL network, with the number of links being 20, 40, 60 and 80, respectively, in the experimental area based on the K-means clustering algorithm, and we named them ESL network 1, ESL network 2, ESL network 3 and ESL network 4 in order, as shown in Figure 3. Additionally, the antennas of the ESL networks receive the horizontally polarized signal of ChinaSat10 at 12.63 GHz, and the Gaussian noise is used to simulate the systematic noise of each link of the ESL networks.