## 4 Results and Discussion  
### Logistic Regression Modeling  
For the analysis, the city of Seoul was divided using a grid into 648,174 30 m \(\times\) 30 m squares excluding rivers, and 47,065 of these were found to have inundation traces. Through this process, the grid where flooding had occurred and those where it had not were classified as 1 and 0, respectively, and these values were entered into Y of Equation (5). In addition, the hydrological-topographic factors for each grid were required in the logistic regression equation to estimate the flood vulnerability. This study intends to provide information on the changes in vulnerability according to rainfall, rather than calculating unchanged flood vulnerability for each grid by considering only the topographic factors. Therefore, two logistic regression models were developed and their performance was compared: an equation that used only topographic factors (T) as independent variables, and one that also included hydrological factors (TR). As a result, the logistic regres \begin{table}
\begin{tabular}{c c c} \hline \hline
**AUC** & **Evaluation** \\ \hline
0.9–1.0 & Excellent \\
0.8–0.9 & Very good \\
0.7–0.8 & Good \\
0.6–0.7 & Sufficient \\
0.5–0.6 & Bad \\ \textless{}0.5 & Test not useful \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation of vulnerability according to area under the curve (AUC). \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Inundation Trace**} \\ \cline{2-3}  & **Inundated Area** & **Non-Inundated Area** \\ \hline \multirow{2}{*}{**Flood vulnerability**} & **Vulnerable area** & True positives (TP) & False positives (FP) \\  & **Non-vulnerable area** & False negatives (FN) & True negatives (TN) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overview of 2 \(\times\) 2 confusion matrix. sion coefficients and constant terms of the two equations were determined, as shown in Equations (14) and (15), respectively. \[z=\beta_{0}+\beta_{1}x_{1}+\cdots+\beta_{n}x_{n} \tag{13}\]  
\[ \begin{array}{l}z_{T}=-4.394-1.391\times Elevation-0.120\times Slope\\ \quad+0.049\times Profile\ Curvature+0.070\times Plan\ Curvature\\ \quad+0.335\times TWI-0.147\times SPI+0.240\times Distance\ from\ River\\ \quad-5.746\times Distance\ from\ Manhole\\ \quad-0.093\times Pump\ Capacity\ per\ Area\end{array}  \tag{14}\]  
\[ \begin{array}{l}z_{TR}=-4.486-1.323\times Elevation-0.206\times Slope\\ \quad+0.074\times Profile\ Curvature+0.101\times Plan\ Curvature\\ \quad+0.374\times TWI-0.163\times SPI+0.253\times Distance\ from\ River\\ \quad-5.610\times Distance\ from\ Manhole\\ \quad-0.081\times Pump\ Capacity\ per\ Area+0.503\times Rainfall\end{array}  \tag{15}\]  
With the data for 2001, the flood vulnerability was calculated using the hydrological-topographic factors and the determined regression coefficients, for all grid in the study area (Figure 6). In the figure, a darker color indicates that the area is more vulnerable, while a lighter color indicates that the area is less vulnerable. The flood vulnerability was represented by classifying the probability in the range [0, 1] into five classes using the natural breaks method. The idea of the natural breaks method is to minimize variance among objects within the chosen subsets, and maximize variance between the subsets [48]. The five classes included very high (1.00-0.50), high (0.50-0.34), medium (0.34-0.22), low (0.22-0.13), and very low (0.13-0.02). In addition, as areas with a probability of less than 2% were not evaluated to be vulnerable, a vulnerability level was not assigned to these. occurred in the past coincided. To plot the ROC curve with 10 points, the flood vulnerability of target areas was divided into 10 equal parts using quartiles. If many floods occurred in areas with high vulnerability in the ROC curve (lower side of the \(x\)-axis), the sensitivity would increase, and in particular would increase sharply at the beginning of the curve. Consequentially, as the AUC increases, it can be evaluated that the technique of this study detects flooded areas well.