# Estimating Flood Inundation in Urban Areas Using a Scenario Generation Method and Inundation Graphs
## 2 Methodology
### Scenario Generation and Application  
The proposed scenario generation method is presented in Figure 1. First, this scenario generation includes the estimation of rainfall based on high return periods using widely used theoretical statistical distributions known as the Gumbel distribution and Generalized Extreme Value (GEV) distribution. The second part involves flood inundation modeling using the introduced two-dimensional flow model. For the rainfall estimation, the method utilizes return periods of up to 500 years based on the regional rainfall data that were collected using local hydrologic data from the Flood Control Offices. Since there is limited data availability for extreme precipitation, the precipitation data for extreme floods were recreated using the two distributions. Considering methods used by former research on flood scenario generation, the amount of runoff that can occur from extreme rainfall for each scenario was calculated taking into account the recent high rainfall. After examining the maximum rainfall data by year of the rainfall station in the target area, these data were used as sample data to estimate the variables of the Gumbel and GEV distribution, which can be used for rainfall calculation frequency [31; 32]. Equation (3) is the CDF for the Gumbel distribution, and the parameters \(\alpha\) and \(\xi\) can be estimated using Equations (4) and (5) [33],  
\[F_{X}(x)=\exp\left[-\exp\left(-\frac{x-\xi}{\alpha}\right)\right] \tag{3}\]  
\[\mu_{X}=\xi+0.5772\alpha \tag{4}\]  
\[\sigma_{X}^{2}=\frac{\pi^{2}}{6}\alpha^{2}\approx 1.645\alpha^{2} \tag{5}\]  
where \(\mu_{X}\) is the sample mean and \(\sigma_{X}^{2}\) is the sample variance. Equation (6) is the CDF for the GEV distribution, and the GEV parameters were computed using the L-Moments method [34]. The parameters \(\alpha\), \(\xi\), and \(\kappa\) can be estimated with Equations (7) and (8).  
\[F_{X}(x)=\exp\left[-\left(1-\frac{\kappa(x-\xi)}{\alpha}\right)^{1/\kappa}\right] \tag{6}\]\[\mu_{X}=\xi+\Big{(}\frac{\alpha}{\kappa}\Big{)}[1-\Gamma(1+\kappa)] \tag{7}\] \[\sigma_{X}^{2}=\Big{(}\frac{\alpha}{\kappa}\Big{)}^{2}\Big{[} \Gamma(1+2\kappa)-\big{\{}\Gamma(1+2\kappa)\big{\}}^{2}\Big{]} \tag{8}\]  
Using the estimated variables, the Gumbel and GEV distribution suitable for the rainfall data from the rainfall observation station was constructed, and the rainfall in the designated return period for each scenario was calculated by calculating the inverse function of the cumulative probability of the Gumbel and GEV distribution. Then, using the rainfall from the return periods, the empirical rational flow formula for discharge was utilized to estimate the discharge in the target area. Then the discharge values were used as the scenario boundary conditions for the introduced two-dimensional model. This calculation of inundation elevation through simulation would be based on various scenario-based model boundary condition changes to conduct multiple versions of simulating scenarios. The database storage of results was undertaken using the results of the two-dimensional flow model, so for the upstream water depth or the target point of interest, additional two-dimensional flow modeling would be less necessary since modeling was pre-conducted.