## 4 Results
### Flood Vulnerability Indicators  
The flood vulnerability for the 231 administrative districts was ranked in descending order of vulnerability values for each indicator \(FVI_{1}\), \(FVI_{2}\), \(FVI_{3}\), and \(FVI_{4}\) in order to compare the four flood vulnerability indicator outcomes across different aggregation frameworks in Equations (1)-(4). Figure 4 indicates that all four flood vulnerability indicators provided a common spatial pattern for relatively higher vulnerability outcomes in the northwestern part around the Seoul metropolitan area with large population and major facility densities and in the southern coast region more affected by typhoons. Figure 4 and Table 3 denote great similarities for the two pairs of flood vulnerability outcomes, between \(FVI_{1}\) and \(FVI_{3}\) both more highly correlated with sensitivity \(S\), and between \(FVI_{2}\) and \(FVI_{4}\) both more highly correlated with exposure \(E\), respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(FVI_{1}\) & \(FVI_{2}\) & \(FVI_{3}\) & \(FVI_{4}\) & \(E\) & \(S\) & \(A\) \\ \hline \(FVI_{1}\) & - & 0.876 & **0.979** & 0.856 & 0.751 & **0.847** & \(-\)0.441 \\ \(FVI_{2}\) & - & - & 0.790 & **0.937** & **0.829** & 0.586 & \(-\)0.135 \\ \(FVI_{3}\) & - & - & - & 0.788 & 0.666 & **0.899** & \(-\)0.480 \\ \(FVI_{4}\) & - & - & & - & **0.902** & 0.619 & \(-\)0.201 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The comparison of relationships between each two different flood vulnerability indicators, and between flood vulnerability indicators and primary assessment components, measured by the coefficient of correlation.  
For the interpretation of similarities between the two sets of each two flood vulnerability indicators, the framework of each flood vulnerability indicator was taken apart as the shared common element and the other different element, and then correlations were measured between heterogeneous portions of the two indicators producing similar flood vulnerability outcomes. The similarities of \(FVI_{1}\) and \(FVI_{3}\) could be interpreted by a high correlation between different elements of each other, such as \(E\,\times\,A\) and \(E+A\) respectively in \(FVI_{1}\) and \(FVI_{3}\) with the correlation coefficient of 0.910 as shown in Table 4. For the explication of similarities between \(FVI_{2}\) and \(FVI_{4}\), the framework of \(FVI_{2}\) could be expanded as \((S+E)\,\times\,A=S\,\times\,A\,+E\,\times\,A\), and the framework of \(FVI_{4}\) could be expanded as \((S+A)\,\times\,E=S\,\times\,E\,+E\,\times\,A\). There was substantial identity with negligible differences between the flood vulnerability outcomes from each original framework and each expansion form. Hence, the similarities of \(FVI_{2}\) and \(FVI_{4}\) could be explained by a high correlation between different elements of each other, such as \(S\,\times\,A\) and \(S\,\times\,E\) respectively in \(FVI_{2}\) and \(FVI_{4}\) with the correlation coefficient of 0.833 as shown in Table 4. Such high similarities between the each two different indicators corresponded to a high correlation between even heterogeneous portions of the each two indicators. Meanwhile, the flood vulnerability indicators were more affected by the nested portions such that \(FVI_{1}\) and \(FVI_{3}\)  
Figure 4: Spatial distribution of the four flood vulnerability outcomes for the study site: (**a**) Raking orders from \(FVI_{1}\); (**b**) raking orders from \(FVI_{2}\); (**c**) raking orders from \(FVI_{3}\); and (**d**) raking orders from \(FVI_{4}\).  
were more correlated with sensitivity \(S\), and \(FVI_{2}\) and \(FVI_{4}\) were more correlated with exposure \(E\) (or \(E\,\times\,A\)) as denoted in Table 3.