## 3 Results and Discussions
### Rainfall Simulations by WRF-Only and the Fully Coupled WRF/WRF-Hydro
#### 3.1.2 Indices for the Temporal Rainfall Distribution  
Figure 3 is the temporal results of the five indices for WRF-only and the fully coupled WRF/WRF-Hydro model. For both the fully coupled WRF/WRF-Hydro and WRF-only, it could be determined that the temporal indices results were Type 1>Type 2>Type 3, from the best to the worst. For events in Type 3, the RMSE values in the fully coupled WRF-Hydro were all higher than WRF-only, which might mean a more accumulative error without canceling out the positive and negative errors in time inhomogeneous events.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Type of Storms**} & \multicolumn{2}{c}{**Obs**} & \multicolumn{2}{c}{**Sim(a)**} & \multicolumn{2}{c}{**Sim(b)**} & \multirow{2}{*}{_RE (a)_} & \multirow{2}{*}{_RE (b)_} & \multirow{2}{*}{_ARE (a)_} & \multirow{2}{*}{_ARE (b)_} \\  & **(mm)** & & **(mm)** & & **(mm)** & & & \\ \hline Type 1 & Event 1 & 63.38 & 72.18 & 65.82 & 0.139 & 0.038 & 0.139 & 0.038 \\ Type 2 & Event 2 & 50.48 & 28.51 & 29.33 & 0.435 & 0.419 & 0.435 & 0.419 \\  & Event 3 & 30.82 & 14.41 & 17.75 & 0.532 & 0.424 & & \\ Type 3 & Event 4 & 49.76 & 23.13 & 28.11 & 0.535 & 0.435 & & \\  & Event 5 & 172.17 & 66.13 & 59.03 & 0.616 & 0.657 & 0.515 & 0.477 \\  & Event 6 & 52.06 & 32.46 & 31.6 & 0.377 & 0.393 & & \\ \hline \hline \end{tabular}
\end{table}
Table 6: RE values of the 24 h rainfall accumulations simulation: (**a**) Weather Research and Forecasting model (WRF-only); (**b**) the fully coupled WRF/WRF-Hydro.  
Figure 2: The observed and simulated 24 h rainfall accumulations for the six storm events.  
We further calculated the difference between each index value and its optimal value as the indices errors. That is, for CSI and POD, the indices errors were the positive difference between the corresponding indices and 1; for FAR, RMSE and MBE, and the indices errors were the positive difference with 0.The temporal heat map is shown in Figure 4. Lower indices errors compared with WRF-only indicated that the fully coupled WRF-Hydro improved time distribution of Type1 and Type 2 events. However, in Type 3, Event 3 and Event 6 have the largest (2.393) and second largest (1.887) temporal \(Cv\) values, which might mean a more uneven time distribution in Type 3. In both the fully coupled WRF-Hydro and WRF-only, the CSI, FAR and POD errors for Event 3 were the largest, which meant a worse rainfall occurrences simulation in time. Consistent with this, the RMSE error for Event 3 was the highest, but the decreasing MBE error represented the real random errors canceled out of the total rainfall amount. The indices of Event 6 showed a similar trend with a second worst CSI, FAR and RMSE, but a second best MBE.  
The fully coupled WRF-Hydro got a lower MBE error (0.532) in Event 3 compare with WRF-only (0.419) even if the CSI, POD, FAR and RMSE errors were larger than WRF-only. However, in Event 6, the FAR, RMSE and MBE errors of the fully coupled system were higher than WRF-only and all indices of Event 5(temporal \(Cv\) 1.887) showed that for a 10 h duration flash storm, the fully coupled system had neither a proper temporal rainfall occurrence nor an accurate temporal quantitative calculation  
Figure 4: Temporal heat map of the indices error: (a) WRF-only; (b) the fully coupled WRF/WRF-Hydro.  
Figure 3: Evaluation values of the indices for the temporal rainfall distribution of the six storm events: (a) WRF-only; (b) the fully coupled WRF/WRF-Hydro.  
of rainfall compared with WRF-only. These indicated that the fully coupled modeling system might involve the risk of a larger false alarm in inhomogeneous time than WRF-only.