## 3 Application of Uncertainty Quantification Methods and Results
### Results and Discussion of Uncertainty Quantification
#### 3.3.1 Analysis of Quantified Uncertainty
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{6}{c}{**(b) Uncertainty Delta method (UDM).**} \\ \hline \hline \multicolumn{2}{c}{**Procedure**} & \multicolumn{1}{c}{**UMD**} & \multicolumn{1}{c}{**Maximum of**} & \multicolumn{1}{c}{**Final UMD**} & \multicolumn{1}{c}{**Ratio (\%)**} \\ \cline{2-6}  & **to the Final Uncertainty** & **to the Final Uncertainty** & **to the Final Uncertainty** \\ \hline \multicolumn{2}{c}{Initial uncertainty} & 5.33 & 5.33 & - & 113 \\ \hline \multirow{2}{*}{QC} & ORPG & 5.33 & \multirow{2}{*}{5.33} & \multirow{2}{*}{113} \\ \cline{2-6}  & Fuzzy & 4.80 & & \\ \hline \multirow{2}{*}{RPE} & WPMM & 0.18 & \multirow{2}{*}{0.21} & \multirow{2}{*}{5.54} & \multirow{2}{*}{117} \\ \cline{2-6}  & M-P & 0.21 & & & (+4) \\ \hline \multirow{2}{*}{PBC} & LGC & \(-\)0.79 & \multirow{2}{*}{\(-\)0.81} & \multirow{2}{*}{4.73} & \multirow{2}{*}{100} \\ \cline{2-6}  & G/R ratio & \(-\)0.81 & & & (\(-\)17) \\ \hline \multicolumn{6}{c}{Final uncertainty} & 4.73 & 4.73 & 4.73 & 100 \\ \hline \multicolumn{6}{c}{(e) Modified-fractional uncertainty method (M-FUM).} \\ \hline \multirow{2}{*}{**Procedure**} & \multicolumn{1}{c}{**M-FUM**} & \multicolumn{1}{c}{**Maximum of**} & \multicolumn{1}{c}{**Incremental**} & \multicolumn{1}{c}{**Ratio (\%)**} \\ \cline{2-6}  & **to the Final Uncertainty** & **to the Final Uncertainty** & **to the Final Uncertainty** & **to the Final Uncertainty** \\ \hline \multirow{2}{*}{QC} & ORPG & 0.45 & \multirow{2}{*}{0.45} & \multirow{2}{*}{-} & \multirow{2}{*}{126} \\ \cline{2-6}  & Fuzzy & 0.39 & & & \\ \hline \multirow{2}{*}{RPE} & WPMM & 0.44 & \multirow{2}{*}{0.50} & \multirow{2}{*}{0.05} & \multirow{2}{*}{140} \\ \cline{2-6}  & M–P & 0.50 & & & (+14) \\ \hline \multirow{2}{*}{PBC} & LGC & 0.33 & \multirow{2}{*}{0.36} & \multirow{2}{*}{\(-\)0.14} & \multirow{2}{*}{100} \\ \cline{2-6}  & G/R ratio & 0.36 & & & (\(-\)40) \\ \hline \multicolumn{6}{c}{Final uncertainty} & 0.36 & 0.36 & - & 100 \\ \hline \multicolumn{6}{c}{* QC, RPE, and PBC represent quality control, radar precipitation estimation, and precipitation bias correction, respectively. **MEV represents the ME value. MEM: maximum entropy method; WPMM: window probability matching method; M–P. Marshall–Palmer; LGC: local gauge correction.**} \\ \hline \hline \end{tabular}  
* QC, RPE, and PBC represent quality control, radar precipitation estimation, and precipitation bias correction, respectively. **MEV represents the ME value. MEM: maximum entropy method; WPMM: window probability matching method; M–P. Marshall–Palmer; LGC: local gauge correction.  
\end{table} 
Table 3: _Cont._  
Figure 5: Uncertainty quantification and propagation in the radar-based precipitation estimation process using (**a**) the MEM, (**b**) the UDM, and (**c**) the M-FUM. In applying the UDM, the variances of each stage were calculated. Those of the ORPG and fuzzy algorithms in quality control were 206.43 and 121.51 mm\({}^{2}\)/h\({}^{2}\), respectively, those of the WPMM and M-P relations were 145.47 and 254.68 mm\({}^{2}\)/h\({}^{2}\), respectively, and those of the LGC and G/R ratio methods were calculated as 82.45 and 115.58 mm\({}^{2}\)/h\({}^{2}\), respectively. Table 3b and Figure 5b show that the UDM values of the ORPG and fuzzy algorithms in quality control were 5.33 and 4.80, respectively, the UDM values of the WPMM and M-P relations in precipitation estimation were 0.18 and 0.21, respectively and the UDM values of the LGC method and G/R ratio method in post-processing were \(-\)0.79 and \(-\)0.81, respectively. In considering uncertainty in the UDM, the initial uncertainty was 5.33 (same as the UDM value of quality control) and the final uncertainty was estimated to be 4.75, which was the same as that of the post-processing stage. Assuming that the UDM value of the final uncertainty was 100%, the UDM value of the quality control stage was 113% and that of the radar-based precipitation estimation stage was 117%.