## 2 Methodology
### Method for Selecting the Optimal Rain Gauge Network  
The combination of two or more rain gauge stations that maximizes conditional entropy or joint entropy is considered an optimal combination. In this study, the observation data of unselected rain gauge stations were predicted using the selected rain gauge network and OK to evaluate the suitability of the selected combination of rain gauge stations. This is because the precipitation of unselected stations is quantitatively calculated when the selected rain gauge network is operated. The MAE, RMSE, bias, and Corr calculated from the differences between the observed and predicted values of unselected rain gauge stations were considered as the measures of evaluation. \[\text{MAE}=\ \frac{1}{n}\ \sum_{i=1}^{n}\ \Big{(}|z(x_{i})-z(x_{i})|\Big{)}, \tag{9}\]\[\text{RMSE}=\ \frac{1}{n}\ \sum_{i=1}^{n}\ \sqrt{\left[\hat{z}(x_{i})-z(x_{i}) \right]^{2}}, \tag{10}\] \[\text{bias}=\ \frac{1}{n}\ \sum_{i=1}^{n}\ \left(\hat{z}(x_{i})-z(x_{i}) \right),\] (11) \[\text{Corr}=\ \frac{\sum_{i=1}^{n}\left(z(x_{i})-\overline{z(x)} \right)\left(\hat{z}(x_{i})-\overline{\hat{z}(x)}\right)}{\sqrt{\sum_{i=1}^{n} \left(z(x_{i})-\overline{z(x)}\right)^{2}\sum_{i=1}^{n}\left(\hat{z}(x_{i})- \overline{z(x)}\right)^{2}}}, \tag{12}\]  
where n is the number of data, \(\hat{z}(x_{i})\) is the predicted value at an arbitrary position \(x_{i}\), and \(z(x_{i})\) is the observed value. In addition, \(\overline{\hat{z}(x_{i})}\) is the average of the predicted values and \(\overline{z(x_{i})}\) is the average of the observed values. The optimal combination according to the determined number of stations can be obtained using the maximum value of entropy. A previous study presented a method of obtaining the optimal number of stations using the maximum informational content [34]. The study selected the combination with the maximum informational content among the various combinations of rain gauge stations based on joint entropy. The prediction performance of the selected combination, however, could not be reflected. This study proposes a method of determining the optimal number of stations through the RMSE calculated using OK. As the number of rain gauge stations increases, the RMSE decreases due to the improvement in the spatial interpolation performance. In other words, the appropriate number of rain gauge stations can be determined by evaluating the degree of the RMSE reduction due to the increase in the number of stations. First, we considered the difference in the RMSE between the optimal combination of i rain gauge stations and that of i - 1 stations. The minimum value of the RMSE (\(\text{RMSE}_{\text{min}}\)) was obtained when there were n - 1 rain gauge stations, and the maximum value of RMSE (\(\text{RMSE}_{\text{max}}\)) was obtained when there was one rain gauge station. When the number of rain gauge stations increased by one, the average RMSE reduction was obtained from the range of RMSE (\(\text{RMSE}_{\text{max}}-\text{RMSE}_{\text{min}}\)), as follows. \[\overline{\text{RMSE}}=\ \frac{\text{RMSE}_{\text{max}}-\text{RMSE}_{\text{ min}}}{n-1}. \tag{13}\]  
Assuming that \(S_{i}\) is the reduction ratio of the RMSE of i rain gauge stations, \(S_{i}\) is calculated using the following equation. \[S_{i}=\ \frac{\text{RMSE}_{i-1}-\text{RMSE}_{i}}{\text{RMSE}}\,\text{ where }i=1,2,\cdots,n-1. \tag{14}\]  
In other words, when \(S_{i}\) is larger than 1, the number of rain gauge stations can be considered an efficient number because the RMSE reduction for i rain gauge stations is relatively larger than the average RMSE reduction. In addition, we calculated the RMSE reduction ratio \(R_{i}\) for i rain gauge stations. \(R_{i}\) represents the RMSE reduction when i rain gauge stations are operated, compared to the total RMSE reduction.