# Reconstruction of Rainfall Field Using Earth-Space Links Network: A Compressed Sensing Approach
## 2 Principles of Reconstructing Rainfall Field
### Algorithm for Reconstructing Rainfall Field by ESL Network  
When there are multiple ESLs in the area as shown in Figure 2, the rain rates at the locations of all these ESLs can be obtained. As the rain rates at these different locations represent the distribution of parts of the rainfall field over the whole area, a reconstruction algorithm can be used to reconstruct the complete rainfall field. However, the actual density of the ground antennas of the ESLs is very sparse, which makes the number of ESLs much smaller than the scale required for the refinement of the rainfall field. In this case, the system of equations for reconstructing the rainfall field using an ESL network is underdetermined, and it is difficult to obtain a high-precision rainfall field by ordinary spatial interpolation algorithms. To solve this problem, the CS theory with certain a priori  
Figure 1: Correspondence between the ESL signal and rainfall from 14 June to 15 June 2020. The black line is the SNR of the ESL signal, and the blue line is the rain rate. information is proposed to realize reconstructing the high-precision rainfall field in the area using the ESL network in the case of sparse sampling. The core of reconstructing the rainfall field using the ESL network is how to reconstruct the complete rainfall field in the area from sparsely distributed rainfall information measured by the ESL network, and the process can be represented as  
\[\mathbf{x}=\mathbf{Af}\,,\;\textit{where}\;\mathbf{x}\in\mathbb{R}^{m},\mathbf{A}\in\mathbb{R}^ {m\times n},\mathbf{f}\in\mathbb{R}^{n} \tag{6}\]  
where \(\mathbf{x}\) is the rain attenuation at the locations of the links measured by the ESL network, \(\mathbf{f}\) denotes the rainfall field, and \(\mathbf{A}\) is the measurement matrix that is determined by the structural and density distribution of the ground antennas of the ESL network in the area. In fact, the amount of links \(m\) that form the ESL network is generally much smaller than the scale of the rainfall field \(n\). Therefore, the complete rainfall field is not directly obtained from the sparse rainfall information measured by the ESL network. However, according to the CS theory [29], if \(\mathbf{f}\) has sparse properties under certain conditions and can be represented by a sparse vector, then \(\mathbf{f}\) can be solved by using \(\mathbf{x}\) based on the CS method. The sparse representation of \(\mathbf{f}\) is given by  
\[\mathbf{f}=\sum_{i=1}^{n}\psi_{i}s_{i}=\mathbf{\psi s} \tag{7}\]  
where \(\mathbf{s}\) is the sparse signal of \(\mathbf{f}\), and \(\mathbf{\psi}\) is the sparse basis of \(\mathbf{f}\). Previous studies have shown that various types of signals have good sparsity under a redundant dictionary that is an adaptive sparse basis representation [29; 30]. Therefore, we use a redundant dictionary based on the K-singular value decomposition (K-SVD) method [30] for the sparse representation of the rainfall field \(\mathbf{f}\). As for the solution of the redundancy dictionary, it is a process of establishing the prior relationship between the rainfall field \(\mathbf{f}\) and the sparse signal \(\mathbf{s}\), and the mathematical model is given by  
\[\min_{D,s}\left\|\mathbf{f}-\mathbf{D}\mathbf{s}\right\|_{l}^{2},\;s.t.\;\forall i,\left\| s_{i}\right\|_{0}<\varepsilon_{0} \tag{8}\]  
where \(\mathbf{D}\) is the redundancy dictionary, \(\varepsilon_{0}\) is the constant, and \(\left\|s_{i}\right\|_{0}\) is the \(l_{0}\) norm which refers to the number of non-zero vectors in \(\mathbf{s}\). In order to solve Equation (10), it is converted  
Figure 2: Schematic of joint monitoring of rainfall by multiple ESLs in the area. into an unconstrained mathematical model with the Lagrange multiplier \(\lambda\), and \(\|s_{i}\|_{0}\) is replaced by \(\|s_{i}\|_{1}\), resulting in  
\[\min_{\mathbf{D},\mathbf{s}}\|\mathbf{f}-\mathbf{D}\mathbf{s}\|_{l}^{2}+\lambda\|s_{i}\|_{1} \tag{9}\]  
The redundant dictionary is then updated column by column using the iterative method, in which the \(j\)-th column vector is calculated as  
\[