# 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
\begin{split}\|\mathbf{f}-\mathbf{D}\mathbf{s}\|_{l}^{2}&=\| \mathbf{f}-\sum\limits_{i=1}^{j}d_{i}s_{i}^{T}\|_{l}^{2}\\ &=\|\left(\mathbf{f}-\sum\limits_{i\
eq j}d_{i}s_{i}^{T}\right)-d_{j} s_{i}^{T}\|_{l}^{2}\\ &=\|E_{f}-d_{j}s_{i}^{T}\|_{l}^{2}\end{split}  \tag{10}\]  
where \(d_{j}\) is the \(j\)-th column vector of the redundant dictionary \(\mathbf{D}\), \(s_{j}\) is the \(j\)-th row vector of the sparse signal \(\mathbf{s}\), and \(E_{f}\) is the residual. Furthermore, the singular value decomposition (SVD) is used to express \(E_{f}\) in Equation (10) as  
\[E_{f}=U\!\!\sum\limits V^{\mathrm{T}} \tag{11}\]  
where \(U\) is the left singular matrix, \(V\) is the right singular matrix, and \(\sum\) is the diagonal matrix of singular values. Next, the redundant dictionary and the sparse signal are obtained by  
\[ \begin{cases}d_{j}=U(:,1)\\ s_{j}=\sum(1,1)V^{\mathrm{T}}(1,\,:)\end{cases}  \tag{12}\]  
Therefore, after representing the sparse basis using the redundant dictionary based on the K-SVD method, the rainfall field reconstruction based on the CS method can be given by Equations (6) and (7) as  
\[\mathbf{x}=A\mathbf{\psi}\mathbf{s}=\mathbf{\theta}\mathbf{s} \tag{13}\]  
where \(\mathbf{\theta}\) refers to the sensing matrix. Additionally, to ensure the convergence of the reconstructed rainfall field results, the sensing matrix is required to conform to the restricted isometry property (RIP) [31], as given by  
\[1-\delta\leq\frac{\langle\mathbf{\theta},\mathbf{s}\rangle}{\langle\mathbf{s},\mathbf{s}\rangle }\leq 1+\delta,\delta\in(0,1) \tag{14}\]  
where \(\delta\) is the smallest constant that makes the RIP hold. In general, the RIP is equivalently represented by the coherence of the sensing matrix, which is defined as the maximum absolute value of the inner product of any two column vectors of the sensing matrix, and is denoted by  
\[C(\mathbf{\theta})=\max_{1\leq i\leq j\leq n}\left|\langle\theta_{i},\theta_{j} \rangle\right| \tag{15}\]  
where \(\mathrm{C}(\mathbf{\theta})\) is the coherence of the sensing matrix, and \(\theta_{i}\) and \(\theta_{j}\) are any two column vectors of the sensing matrix. The coherence of the sensing matrix is mainly used to determine the suitability of using the CS method to reconstruct the rainfall field. Moreover, the smaller the coherence of the sensing matrix, the better the performance of the CS method for reconstructing the rainfall field. Additionally, according to the CS theory, the premise of reconstructing the complete rainfall field using the CS method is that the coherence of the sensing matrix has to conform to  
\[\mathrm{C}(\mathbf{\theta})\geq\sqrt{\frac{n-m}{m(n-1)}} \tag{16}\]The lower bound in the above equation is the celebrated Welch boundary [32]. When the actual rainfall field is sparse and the sensing matrix conforms to the RIP, the optimal model of CS can be used to implement the solution for the sparse signal \(\mathbf{s}\)[29]. This computational model is given by  
\[\min\lVert\mathbf{s}\rVert_{0},\ s.t.\ \mathbf{x}=\mathbf{\theta}\mathbf{s} \tag{17}\]  
Here we use basis pursuit (BP) algorithm to solve for \(\mathbf{s}\) in Equation (13), and then the rainfall field can be reconstructed using a redundant dictionary of real rainfall fields.