## 3 Methodology
### The Spatiotemporal Fusion Model  
The model diagram of the spatiotemporal fusion of precipitation data using ConvLSTM based on GPM satellite observation data and ground-based spatiotemporal interpolation data calculated by the F-SVD method is shown in Figure 2b.  
At a point \(s\) in space, nine surrounding grid points, including itself, are selected to form the GPM satellite-observed precipitation matrix \(G^{t}\) and the ground spatiotemporal interpolated precipitation matrix \(I^{t}\) at time \(t\). These matrices are defined as follows:  
\[\begin{split} G^{t}=\left[\begin{array}{ccc}g_{s-4,t}&g_{s-3,t}&g _{s-2,t}\\ g_{s-1,t}&g_{s,t}&g_{s+1,t}\\ g_{s+2,t}&g_{s+3,t}&g_{s+4,t}\\ \end{array}\right]\\ I^{t}=\left[\begin{array}{ccc}i_{s-4,t}&i_{s-3,t}&i_{s-2,t}\\ i_{s-1,t}&i_{s+1,t}\\ i_{s+2,t}&i_{s+3,t}&i_{s+4,t}\\ \end{array}\right]\end{split} \tag{8}\]  
where \(g_{s,t}\) and \(i_{s,t}\) are the GPM satellite observation data and ground-based spatiotemporal interpolation data at time \(t\), point \(s\), respectively.  
For the spatiotemporal precipitation fusion method, the GPM satellite observed precipitation matrix and the ground spatiotemporal interpolated precipitation matrix for the past six time periods need to be inputted into the constructed ConvLSTM to obtain the spatiotemporal fused precipitation \(p_{s,t}\) at point \(s\) on space at time \(t\) in the following form:  
\[p_{s,t}=f_{convlstm}\left[\left(G^{t-5},G^{t-4},\ldots,G^{t}\right),\left(I^{t -5},I^{t-4},\ldots,I^{t}\right)\right] \tag{9}\]  
The structure of the ConvLSTM network model constructed for spatiotemporal precipitation fusion is presented in Figure 4. The figure illustrates the precipitation fusion process from input to output at a specific time point, with rectangles representing different neural network levels and rounded rectangles indicating the format of input, output, and intermediate variables. The input data have a tensor of size 6 \(\times\) 3 \(\times\) 3 \(\times\) 2, where the dimensions represent time, row, column, and the number of channels, respectively. This tensor stores the data of GPM and ground-based spatiotemporal interpolation. The output data have a three-dimensional tensor of size 1 \(\times\) 1 \(\times\) 1, with dimensions of time, rows, and columns, respectively, which holds the fused precipitation.  
Figure 3: Structure of ConvLSTM cell.  
With the rain gauge as the central point, the nine surrounding grid points were selected. The input data for the ConvLSTM model comprised the GPM data and ground spatiotemporal interpolation data (F-SVD) of the past six moments, while the output was the precipitation observed at the station. The hourly precipitation data from 2006 to 2014 were used as the training set, and the hourly precipitation data from 2014 to 2018 were used as the testing set to evaluate the model's accuracy.  
The main steps of the multi-source precipitation spatiotemporal fusion algorithm (FS-ConvLSTM) are as follows:  
1. Download GPM satellite observation data with a spatial resolution of 0.1\({}^{\circ}\) and a temporal resolution of 0.5 h. Process the data to obtain precipitation data with a time interval of 1 h.
2. Collect and organize the rain gauge observation data of the study watershed and interpolate them into 0.1\({}^{\circ}\)\(\times\) 0.1\({}^{\circ}\) spatial grid point data using the F-SVD method.
3. Construct input sample sets based on GPM and interpolation results of F-SVD, normalize the data, and use the data from 2006 to 2014 as the training set and the data from 2014 to 2018 as the testing set.
4. Train the model at 15 rain gauges with precipitation observations as the true value and mean squared error as the loss function to minimize the training loss.
5. Apply the trained model to each grid point within the study watershed for precipitation fusion.