## 4 Nowcasting of Rainfall Field  
### Nowcasting of Rainfall Field Based on Deep Learning  
Based on the reconstructed rainfall fields by an OELs network, it is possible to predict the local regional rainfall over a relatively short period of time (e.g., several hours). In this article, we propose a deeplearning-based method to achieve the high spatiotemporal resolution nowcasting of rainfall field. The structure of our designed learning network is shown in Figure 5 and we can observe that the network comprises seven different layers where \(N\) denotes the number of sequential rainfall fields used in the training set. For the sequence input layer, it receives the rainfall fields of training set and then transfers them into next layer in order. The 2D convolutional layer uses a \(3\times 3\) convolution kernel to extract the distribution features of rain intensity and the element in each feature matrix \(C_{t}\) can be described as:  
\[ \begin{split} C_{t}(i,j)=\sum\limits_{p=1}^{S}\sum\limits_{q=1}^{ S}A(p,q)R_{t}(i-p+1,j-q+1)\\ \left\{\begin{array}{l}1\leq i\leq W_{R}-S+1\\ 1\leq j\leq H_{R}-S+1\end{array}\right.\end{split}  \tag{23}\]  
where \(t\) is the \(t\)th rainfall field sample in training set, \(S\) is equal to 3 and denotes the size of convolution kernel \(A\), \(R_{t}\) is the rain intensity of rainfall field, \(W_{R}\) and \(H_{R}\) are the number of columns and rows of rainfall field, respectively. Because the excessive number of elements in the feature matrix \(C_{t}\) may lead to extra training time and poor learning results, a 2D pooling layer is employed to reduce the number and meanwhile retains the main spatial information, the element of new feature matrix \(D_{t}\) is:  
\[ \begin{split} D_{t}(i,j)=\max(B*C_{t}(1-S+Si:Si,1-S+Sj:Sj))\\ \left\{\begin{array}{l}1\leq i\leq W_{C}/S\\ 1\leq j\leq H_{C}/S\end{array}\right.\end{split}  \tag{24}\]  
Figure 5: The architecture of deep learning network for the rainfall field nowcasting: the sequence input layer receives oblique earth-space link (OEL) measurements, the convolutional- and pooling layer extract the spatial feature of rainfall field, the Long Short-Term Memory (LSTM) layer learns the time-dependence of spatial feature vectors and fully connected layer as well as regression layer predicts the regional rainfall for the next time. where \(S\) is equal to 3 and denotes the size of sampling kernel \(B\), \(C\) is the feature matrix, \(W_{C}\) and \(H_{C}\) denote the number of columns and rows of feature matrix, respectively. Then each feature matrix \(D_{t}\) is reshaped as one-dimension vector \(\mathbf{x}_{t}\) and then transferred into Long Short-Term Memory (LSTM) block in time sequence. As can be seen that an output vector \(\mathbf{h}_{t}\) (also known as the hidden state) is generated at each time step \(t\) which reflects the spatiotemporal property of rainfall field, and the calculation of \(\mathbf{h}_{t}\) is:  
\[\left\{ \begin{array}{l}i_{t}=\sigma_{g}(W_{xi}\mathbf{x}_{t}+W_{hh}\mathbf{h} _{t-1}+b_{i})\\ f_{t}=\sigma_{g}\Big{(}W_{xf}\mathbf{x}_{t}+W_{hf}\mathbf{h}_{t-1}+b_{f}\Big{)} \\ g_{t}=\sigma_{c}\Big{(}W_{xg}\mathbf{x}_{t}+W_{hg}\mathbf{h}_{t-1}+b_{g}\Big{)} \\ o_{t}=\sigma_{g}\big{(}W_{x0}\mathbf{x}_{t}+W_{ho}\mathbf{h}_{t-1}+b_{o}\big{)} \\ \mathbf{c}_{t}=f_{t}\otimes\epsilon_{t-1}+i_{t}\otimes g_{t}\\ \mathbf{h}_{t}=o_{t}\otimes\sigma_{c}(\mathbf{c}_{t})\end{array} \right. \tag{25}\]  
where \(i,f,g\) and \(o\) denote the input-, forget-, update- and output gate, respectively, and the functions of them can refer to literature [51]; \(\mathbf{c}_{t}\) represents the cell state; \(\sigma_{g}\) and \(\sigma_{c}\) are the hyperbolic tangent- and sigmoid activation function.