## 2 Data and Methods
### Evaluation Statistics  
Both the 24 h accumulation and the spatiotemporal distributions of rainfall are evaluated for WRF-only and the fully coupled WRF/WRF-Hydro modeling system. While calculating average rainfall, the relative error (RE) was first evaluated:  
\[\text{RE}=\left|\frac{P_{s}-P_{o}}{P_{o}}\right|\times 100\%, \tag{2}\]  
where \(P_{s}\) and \(P_{o}\) are the simulated and observed 24 h cumulative rainfall values in the two study areas calculated based on rain gauges [48]. For the evaluation indices of the spatial and temporal rainfall distributions, five indices which refer to [28] are used, i.e., the critical success index (CSI), the probability of detection (POD), the false alarm ratio (FAR), the root mean square error (RMSE), and the mean bias error (MBE). The meanings of the above indices and the range values are provided in Table 4. Among these indices, FAR, CSI, and POD are categorical, and RMSE and MBE are continuous indices [49; 50]. The categorical indices were as follows:  
\[\text{POD}=\frac{1}{N}\sum\
olimits_{i=1}^{N}\frac{NA_{i}}{NA_{i}+NC_{i}}, \tag{3}\]  
\[\text{FAR}=\frac{1}{N}\sum\
olimits_{i=1}^{N}\frac{NB_{i}}{NA_{i}+NB_{i}}, \tag{4}\]  
\[\text{CSI}=\frac{1}{N}\sum\
olimits_{i=1}^{N}\frac{NA_{i}}{NA_{i}+NB_{i}+NC_{i }}, \tag{5}\]  
The calculation of categorical indices is based on a rain/no rain contingency table, as shown in Table 5. The NA, NB, NC, and ND represent the total hit numbers of the eligible judgment, whether the simulated and observed values in an observation period or at the observation position were greater than 0.01 mm (judgment of whether rainfall occurred) and \(\text{NA}=\sum\
olimits_{i=1}^{N}NA_{i}\), \(\text{NB}=\sum\
olimits_{i=1}^{N}NB_{i}\), \(NC=\sum\
olimits_{i=1}^{N}NC_{i}\), \(\text{ND}=\sum\
olimits_{i=1}^{N}ND_{i}\). When evaluating on a temporal dimension, the rainfall simulations and observations of different time steps were compared at the same rainfall gauges \(i\), and \(N\) referred to the number of the rain gauges. When evaluating on a spatial dimension, the rainfall simulations and observations of the different rain gauges were compared with the same hourly time step \(i\), and \(N\) (\(N\) = 24) referred to the number of time steps. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Index** & **Range** & **Optimal Value** & **Meaning** \\ \hline CSI & 0–1 & 1 & Proportion of correctly simulated rainfall frequency to all possible rainfall situations \\ POD & 0–1 & 1 & Proportion of observed rainfall being correctly simulated \\ FAR & 0–1 & 0 & Proportion of false positives in simulated rainfall events. \\ RMSE & 0–\(\infty\) & 0 & Mean square error of the simulations \\ MBE & \(-\infty\)–\(\infty\) & 0 & Average error of the simulations \\ \hline \hline CSI: critical success index; POD: probability of detection; FAR: false alarm ratio; RMSE: root mean square error; \\ MBE: mean bias error. & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Descriptions and ranges of the five indices for the rainfall spatial and temporal distribution. There is a general evaluation of simulated performance on the basis of the correctness of rainfall occurrences in categorical indices, and continuous indices are provided for a more quantitative calculation of the simulated error. The continuous indices were calculated as follows:  
\[\text{RMSE}=\sqrt{\frac{1}{M}\sum\
olimits_{j=1}^{M}\left(P_{sj}-P_{oj}\right) ^{2}}, \tag{6}\]  
\[MBE=\frac{1}{M}\sum\
olimits_{j=1}^{M}\left(P_{sj}-P_{oj}\right), \tag{7}\]  
When the indices were calculated in the temporal dimension, \(M\) (\(M\) = 24) was the number of time steps, and \(P_{sj}\) and \(P_{oj}\) were the simulated and observed average areal rainfall of the study catchment at each time step \(j\). When the indices were calculated in the spatial dimension, \(M\) was the total number of each rain gauges, which is 8 for the Fuping catchment and 11 for the Zijingguan catchment and \(P_{sj}\) and \(P_{oj}\) were the simulated and observed values of accumulated rainfall at each rain gauge \(j\).