## 2 Materials and Methods  
### Hydrodynamic Model  
Based on the Delft-3D hydrodynamic model, the mathematical model of storm surge was established by comparing the Fujita-Takahashi model and the wind field model obtained by ERA5 reanalysis. The Delft3D hydrodynamic simulation software developed by Deltares Research Institute in the Netherlands was adopted as the mathematical model. Delft3D model can be used to calculate and analyze two- and three-dimensional tidal current, flood, wave, sediment, salinity, temperature, oil spill, and pollutant concentration, and is suitable for rivers, lakes, and oceans. Delft3D adopts incompressible fluid N-S equation based on Boussinesq assumption and static pressure assumption. The orthogonal curvilinear coordinate system was adopted in the horizontal direction (\(\xi\), \(\eta\)), and \(\sigma\) coordinates were introduced in the vertical introduction. The numerical grid transformation was inherently understood through the conversion of grid vertex coordinates from the physical domain to the computational domain, as depicted in Figure 1. This discretization process facilitated the incorporation of the transformed equations into the computational framework, enabling the effective modeling and analysis of the system under consideration ([PERSON] and [PERSON], 1994) [19]. The \(\sigma\) coordinate system is defined as:  
\[\sigma=\frac{z-\delta}{d+\delta}=\frac{z-\delta}{H} \tag{1}\]  
where \(z\) is the vertical coordinate of physical space; \(\delta\) is the water level above the base level (\(z=0\)); \(D\) is the water depth; \(H\) is the total water depth. The average continuity equation along water depth is:  
\[\frac{\partial\delta}{\partial t}+\frac{1}{\sqrt{G_{\eta\eta}}\sqrt{G_{\xi \xi}}}\frac{\partial\big{(}(d+\xi)U\sqrt{G_{\eta\eta}}\big{)}}{\partial\xi}+ \frac{1}{\sqrt{G_{\eta\eta}}\sqrt{G_{\xi\xi}}}\frac{\partial\big{(}(d+\xi)V \sqrt{G_{\eta\eta}}\big{)}}{\partial\eta}=0 \tag{2}\]  
For the two-dimensional model of averaging water depth, the momentum equations in the \(\xi\) and \(\eta\) directions are:  
\[ \begin{array}{c}\frac{\partial u}{\partial t}+\frac{v}{\sqrt{G_{\xi\xi}}} \frac{\partial u}{\partial\xi}+\frac{u}{\sqrt{G_{\eta\eta}}}\frac{\partial u }{\partial\xi}+\frac{w}{\sqrt{G_{\eta\eta}}}\frac{\partial u}{\partial\xi}- \frac{w}{\sqrt{G_{\eta\eta}}\sqrt{G_{\xi\xi}}}\frac{\partial\sqrt{G_{\eta\eta }}}{\partial\xi}+\frac{v^{2}}{\sqrt{G_{\eta\eta}}\sqrt{G_{\xi\xi}}}\frac{ \partial\sqrt{G_{\xi\xi}}}{\partial\eta}+fv\\ =-\frac{1}{\rho_{0}\sqrt{G_{\xi\xi}}}P_{\xi}+F_{\xi}+\frac{1}{(d+\xi)^{2}} \frac{\partial}{\partial\sigma}\left(v_{V}\frac{\partial w}{\partial\sigma} \right)+M_{\xi}\end{array}  \tag{3}\]  
\[ \begin{array}{c}\frac{\partial v}{\partial t}+\frac{u}{\sqrt{G_{\xi\xi}}} \frac{\partial w}{\partial\xi}+\frac{v}{\sqrt{G_{\eta\eta}}}\frac{\partial v}{ \partial\eta}+\frac{w}{d+\xi}\frac{\partial w}{\partial\sigma}-\frac{w}{ \sqrt{G_{\eta\eta}}\sqrt{G_{\xi\xi}}}\frac{\partial\sqrt{G_{\eta\eta}}}{ \partial\xi}+\frac{u^{2}}{\sqrt{G_{\eta\eta}}\sqrt{G_{\xi\xi}}}\frac{\partial \sqrt{G_{\xi\xi}}}{\partial\eta}+fu\\ =-\frac{1}{\rho_{0}\sqrt{G_{\eta\eta}}}P_{\eta}+F_{\eta}+\frac{1}{(d+\xi)^{2} }\frac{\partial}{\partial\sigma}\left(v_{V}\frac{\partial w}{\partial\sigma} \right)+M_{\eta}\end{array}