## 2 Materials and Methods
### Flood Susceptibility Mapping
#### 2.4.3 Analytical Hierarchy Process (AHP)
\begin{array}{l}A^{*}=\left\{\left(\left.\begin{array}{c}max_{ij}\\ \end{array}\right|j\in J\right),\left(\left.\begin{array}{c}min_{ij}\\ \end{array}\right|j\in J^{\prime}\right)\right\}\\ A^{-}=\left\{\left(\left.\begin{array}{c}min_{ij}\\ \end{array}\right|j\in J\right),\left(\left.\begin{array}{c}max_{ij}\\ \end{array}\right|j\in J^{\prime}\right)\right\}\end{array}  \tag{12}\]  
As FR and WoE increase, the flood susceptibility increases since the values of flood conditioning factors are hybridized by FR and WoE, and thus, the flood conditioning factors were processed as benefit-based criteria. Step 5: Calculating the distance of each alternative to the positive ideal (\(S_{i}\)*) and negative ideal (\(S_{i}\)*) solutions  
Step 6: Calculation of relative proximity values (closeness coefficient) to the ideal solution  
\[C_{i}=\frac{S_{i}^{-}}{S_{i}^{*}+S_{i}^{-}} \tag{13}\]Step 7: Classifying the study area using classification methods, namely equal, quantile, natural break, and geometric intervals, regarding flood susceptibility in descending order as flood susceptibility decreases from very high susceptibility to low susceptibility. An ideal solution is defined as a collection of ideal levels in all attributes considered. In TOPSIS, the main goal is the best alternative being as close as possible to the ideal solution and as far as possible from the anti-ideal solution [80]. Therefore, the best result depends on the distance computed. So, the effect of various distance metrics should be investigated. In this study, five different distance metrics were tested, where the distances of each alternative to the positive ideal (\(S_{i}^{*}\)) and negative ideal (\(S_{i}^{-}\)) solutions are computed accordingly. In this study, the distance metrics used for the computation of closeness coefficients are given below. * Euclidean Distance  
Euclidean distance is a metric used to determine the shortest (path) distance between two sets of variables by calculating the square root of the sum of squares of the differences. The Euclidean distances between alternative \(i\) and PIS or NIS are as follows, respectively:  
\[S_{i}^{*}=\sqrt{\sum_{j=1}^{n}\left(v_{ij}-v_{j}^{*}\right)^{2}} \tag{14a}\]  
\[S_{i}^{-}=\sqrt{\sum_{j=1}^{n}\left(v_{ij}-v_{j}^{-}\right)^{2}} \tag{14b}\]  
* Manhattan Distance  
Manhattan distance, also known as snake distance or city block distance, is the sum of the real distances between two points where each distance is always a straight line. The amounts of PIS or NIS are calculated, respectively, as  
\[S_{i}^{*}=\sum_{j=1}^{n}\left|v_{ij}-v_{j}^{*}\right| \tag{15a}\]  
\[S_{i}^{-}=\sum_{j=1}^{n}\left|v_{ij}-v_{j}^{-}\right| \tag{15b}\]  
* Chebyshev Distance  
Chebyshev distance (or Tchebychev distance), also known as chessboard distance, is the furthest distance between two points along any coordinate dimension. The difference between the Euclidean distance and the Chebyshev distance is that when a right triangle is considered, the Euclidean distance is equal to the hypotenuse length of the triangle, while the Chebychev distance is the total length of the sides of the triangle. Chebyshev distance between alternative \(i\) and the PIS or NIS is given as  
\[S_{i}^{*}=\underset{i}{max}\left|v_{ij}-v_{j}^{*}\right| \tag{16a}\]  
\[S_{i}^{-}=\underset{i}{max}\left|v_{ij}-v_{j}^{-}\right| \tag{16b}\]  
* Jaccard Distance  
Also known as the Tanimoto distance, it is a different variation of the cosine distance and is a measure of how dissimilar two sets are. The main disadvantage of the Jaccard Index is that it is strongly influenced by the size of the data.