# The Sensitivity Analysis of Parameters in the 1D-2D Coupled Model for Urban Flooding
## 2 Method
### Parameter Sensitivity Analysis Method
#### 2.2.2 Parameter Sensitivity Analysis  
1. Local Sensitivity Analysis  
The [PERSON] method [25], also known as the one-at-a-time (OAT) method, is a local sensitivity analysis approach first proposed by [PERSON] in 1991. It evaluates the influence of each input parameter on model outputs by sequentially altering the input parameters and observing the resulting changes in the output. The [PERSON] method determines parameter sensitivity by calculating the impact of each parameter's variation. The sensitivity \(E_{i}\) of a parameter is calculated using the following formula:  
\[E_{i}=\frac{e_{a}-e_{b}}{\Delta x_{i}} \tag{6}\]  
In the formula, \(e_{a}\) represents the accumulated water volume in the model output after the parameter change, \(e_{b}\) represents the accumulated water volume in the model output before the parameter change, and \(\Delta x_{i}\) is the magnitude of the variation in the analyzed parameter \(\Delta x_{i}\). The Morris method employs the mean value and the absolute mean value as criteria for determining sensitivity. The formulas are as follows:  
\[\mu_{i}=\frac{1}{r}\sum_{j=1}^{r}E_{i} \tag{7}\]  
In the formula, \(\mu_{i}\) represents the overall strength of the parameter's influence on the output. A larger mean value indicates that the parameter has a greater average impact on the model output. \(r\) represents the number of sampling iterations. However, since the sign of \(\mu_{i}\) reflects whether the perturbation has a positive or negative impact on the output, the calculated sensitivity \(\mu_{i}\) might underestimate the parameter's influence if its effects alternate between positive and negative (i.e., sometimes increasing the output and other times decreasing it), resulting in a cancellation effect. To eliminate this cancellation effect, \(\mu_{i}^{*}\), the absolute mean value, is introduced to measure parameter sensitivity. Regardless of whether the sensitivity is positive or negative, \(\mu_{i}^{*}\) captures the total influence of the parameter. This adjustment makes the analysis more precise, particularly when a parameter causes significant fluctuations in the model output. \(\mu_{i}^{*}\) better reflects the parameter's impact in such cases. \[\mu_{i}^{*}=\frac{1}{r}\sum_{j=1}^{r}|E_{i}| \tag{8}\]  
In the formula, \(\mu_{i}^{*}\) represents the absolute mean sensitivity of the \(i\)-th input parameter. 2. Global Sensitivity Analysis  
The Sobol method is based on the principle of variance decomposition, which partitions the total variance of the model output into contributions from different input variables \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Parameter** & **Description** & **Unit** & \
\begin{tabular}{c} **Minimum** \\ **Value** \\ \end{tabular} &
\begin{tabular}{c} **Maximum** \\ **Value** \\ \end{tabular} \\ \hline \multirow{4}{*}{Telemac} & Friction & Surface Friction & / & 0.01 & 0.8 \\  & Fc & Stable Infiltration Rate & mm/h & 0 & 100 \\  & Vd & Velocity Diffusion Coefficient & / & \(1.00\times 10^{-6}\) & \(1.00\times 10^{-3}\) \\  & CFL & Courant Number & / & 0.2 & 0.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._and their combinations. For a model with \(k\) input variables \(X_{1},X_{2},\ldots,X_{k}\), where the output is \(Y=f(X_{1},X_{2},\ldots,X_{k})\), the total variance of the model output can be decomposed as follows:  
\[\text{Var}(Y)=\sum_{i=1}^{k}\ V_{i}+\sum_{i<j}\ V_{ij}+\sum_{i<j<l}\ V_{ijl}+ \ldots+V_{12\ldots k} \tag{9}\]  
In the formula, \(V_{i}\) represents the first-order sensitivity of input variable \(X_{i}\), indicating the contribution of \(X_{i}\) alone to the output variance, and \(V_{ij}\) represents the second-order sensitivity, reflecting the contribution of the interaction between \(X_{i}\) and \(X_{j}\) to the output variance. Similarly, \(V_{12\ldots k}\) represents the joint sensitivity of all input variables. \(S_{1}\) is the first-order sensitivity index in the Sobol method.