## 2 Materials and Methods
### Flood Susceptibility Mapping
#### 2.4.3 Analytical Hierarchy Process (AHP)  
AHP is one of the most frequently used methods of multi-criteria decision-making, enabling the assignment of different weight values to the thematic layers of each conditioning factor to display their relative influences [77]. In this study, the criteria weights of each conditioning factor were implemented by considering W\({}_{\text{j}}\) values obtained from the index of entropy (IoE) and compound factor. W\({}_{\text{j}}\) values were sorted in ascending order by grading from 1 to 9. These compound values were used to generate a pair-wise comparison matrix of flood conditioning factors according to [PERSON]'s [78] scale, from which the criteria weight vectors of flood conditioning factors were obtained using AHP. The consistency of flood conditioning factors was examined by using the inconsistency index (\(I_{i}\)) given as  
\[I_{i}=\frac{\lambda_{max}-n}{n-1} \tag{7}\]  
\[I_{r}=\frac{I_{i}}{R_{i}} \tag{8}\]  
where \(\lambda_{max}\) is the largest eigenvalue of the pair-wise comparison matrix, \(n\) is the number of flood conditioning factors, \(I_{r}\) is the inconsistency ratio, and \(R_{i}\) is the random index. The \(R_{i}\) value can be approximated as 1.45. For \(n\) = 9, it should be noted that the consistency of the factors is trustworthy when \(I_{r}\) is less than 0.10 [78]. In the present study, the consistency of factors is fulfilled since the \(I_{r}\) value was calculated as 0.035. 2.4.4. Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS)  
[PERSON] and [PERSON] developed TOPSIS in 1981 [79] as a simple ranking method based on choosing alternatives that simultaneously have the shortest distance from the positive ideal solution (PIS) and the longest distance from the negative ideal solution (NIS). Thus, the benefit criteria are maximized while the cost criteria are minimized for the PIS and vice versa for the NIS. In this study, the flood hazard susceptibility map of the study area was processed by the TOPSIS method using the criteria weights obtained by using AHP. A total of 25,000 random points correspond to approximately 0.2% of the study area, which is sufficient to catch spatial heterogeneities of the flood conditioning factors in the study area. The FR and WoE values of flood conditioning factors at the randomly generated 25,000 points were used to create a decision matrix, and then it was normalized. Factor weights obtained from AHP were used to calculate the weighted normalized decision matrices. According to TOPSIS, the closeness coefficients are computed for the alternatives based on distance from PIS and NIS, considering the criteria cost or benefit. In many flood susceptibility mapping studies, the Euclidean distance metric is used to compute that distance. However, there are many other distance metrics in the literature that can be used for the computation of distance depending on the scope of the work. After a detailed review of the studies on flood susceptibility mapping, we could not find any research on distance metrics. So, in the present study, the steps given below are followed:  
Step 1: Creation of the decision-making matrix (\(A\))  
\[A_{ij}=\overbrace{ \begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ a_{31}&a_{32}&\cdots&a_{3n}\\ \vdots&\vdots&\vdots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix} }^{Factors} \tag{9}\]  
Step 2: Creating the normalized decision matrix (R) (vector normalization)  
\[r_{ij}=\frac{a_{ij}}{\sqrt{\sum_{k=1}^{m}a_{kj}^{2}}}(k=1,2,\ldots,m\;ve\;j=1, 2,\ldots,n) \tag{10}\]  
Step 3: Creating the weighted normalized decision matrix (\(V\))  
\[\sum_{j=1}^{n}w_{j}=1\;\;\;\;\;v_{ij}=w_{j}.r_{ij} \tag{11}\]  
Step 4: Finding positive ideal (\(A^{*}\)) and negative ideal (\(A^{-}\)) solution vectors  
\[