## 2 Principles of Rainfall Field Retrieval by ESLs and HMLs
### Rainfall Field Retrieval by Combined ESLs and HMLs
#### 2.3.2 Rainfall Field Reconstruction by RBF Neural Network  
RBF neural network is a forward propagation network with good performance, which is composed of input layer, hidden layer and output layer. Among them, the transformation from the input layer to the hidden layer is nonlinear, while the transformation from the hidden layer to the output layer is linear [46]. Each unit in the hidden layer of the RBF network can be regarded as a RBF, which is a function that depends only on the distance to the origin. The hidden nodes of the RBF neural network use the distance between the input vector and the center vector as the independent variable of the function and use the RBF as the activation function. Additionally, the farther the input of the neuron is from the RBF center, the lower the activation degree of the neuron is. Thus, the approximation of the objective function only depends on the nearest RBFs, while the RBFs with long distances basically do not work. This local approximation feature enables RBF neural network to have good convergence speed.  
For the reconstruction of rainfall field in the area, the rainfall, at a point in space, has the highest correlation with the nearest rainfall to it. Additionally, this distribution, characteristic of the rainfall field, coincides with the local approximation feature of the RBF neural network. Therefore, we can design the RBF neural network to reconstruct the rainfall field as shown in Figure 4. The geometric meaning is that a continuous two-dimensional rainfall field \(R(x,y)\) is recovered from the sparsely distributed rain rates \(R(x_{i},y_{i})\) retrieved from the ESLs and HMLs. The specific reconstruction function can be expressed as  
\[R(x,y)=\sum_{n=1}^{m}\sum_{i=1}^{N}W_{n}f_{n}(R(x_{i},y_{i})), \tag{23}\]  
where \(m\) is the number of hidden units, \(W_{n}\) denotes the weights from the hidden units to the output layer, and \(f_{n}\) is the hidden layer RBF.  
In this paper, the detailed setup of the RBF neural network used to reconstruct the rainfall field is as follows:  
(1) The Gaussian function is used as the hidden layer RBF and is given by  
\[f_{n}(R(x_{i},y_{i})) = exp(-\frac{\|R(x_{i},y_{i})-C_{n}\|}{2\sigma_{n}{}^{2}}), \tag{24}\]  
Figure 4: Architectural diagram of RBF neural network for retrieving rainfall fields.  
where \(C_{n}\) denotes the center of the RBF in the hidden units, and \(\sigma_{n}\) is the width of the hidden units, which is primarily used to adjust the resolution of the neural network.
2. The K-means clustering is used to determine the network cluster center \(C_{n}\) and the width of the Gaussian function \(\sigma_{n}\).
3. The learning rate of the RBF neural network is set to 0.01, and the minimum error requirement of the objective function and the maximum number of iterations are set to 0.01 mm/h and 5000 times, respectively. The training will be stopped automatically if the minimum error requirement is reached during the training process.
4. The RBF neural network can automatically add the number of hidden units until the training error requirement is reached.