# Comparison of Land Cover Categorical Data Stored in OSM and Authoritative Topographic Data
## 3 Materials and Methods
### Method Applied
#### 3.2.3 Compound Correspondence Index Calculation  
TOPSIS is perceived as the most widely used approach among multi-criteria decision analysis (MCDA) [42,43]. This classical MCDA method, formerly developed by [PERSON] and [PERSON] in 1981 [44], addresses complex decision problems involving conflicting goals, uncertainty and different data formats. TOPSIS has gained popularity in a plethora applications because it evaluates real-world problems based on a variety of criteria, both quantitative and qualitative. Simultaneously, it takes into account the mutual distances to positive and negative ideal solutions, and finally orders the rank of preferences based on their relative proximity (a combination of these two distance measures). This study uses the classical TOPSIS method to portray differences in the thematic scope (understood as the area covered by buildings, forests and water reservoirs and the lengths of roads, rivers and railways) of two topographic datasets (BDOT10k and OSM) in the form of a composite index to compare and rank the obtained alternatives.  
The seven standardized steps of the TOPSIS method described in detail by many papers, e.g., [PERSON] and [PERSON] [45], [PERSON] et al. [46], are in general presented below.  
1. Problem description  
The decision relies on the selection of BDOT10k or OSM sets (two alternatives; \(m=2\)) based on the minimal value of the \(k\) criteria (k = 6), as shown in Equation (1):  
\[\begin{array}{l}x_{1}=|BT_{B}-OSM_{B}|,x_{2}=|BT_{F}-OSM_{F}|,x_{3}=|BT_{W}-OSM _{W}|,\\ x_{4}=|BT_{Ro}-OSM_{Ro}|,x_{5}=|BT_{S}-OSM_{S}|,x_{6}=|BT_{Ra}-OSM_{Ra}|,\end{array} \tag{1}\]  
where \(BT\) denotes the BDOT10k dataset; OSM--OpenStreetMap data; subscripts \({}_{B,F},W\)--total area covered by buildings, forest and water bodies in the grid cell, and \({}_{Ro,S,Ra}\)--total length of paved roads, streams, and railways, respectively.  
2. Calculation of the normalized decision matrix (\(n_{ij}\)) using the quotient method (Equation (2)): \[n_{ij}=\frac{x_{ij}}{\sqrt{\sum_{d=1}^{m}x_{ij}^{2}}},i=1\ldots m;j=1\ldots n.\] (2)  
Figure 2: The workflow schema.  
where \(m\) is the number of alternatives, and \(n\) is the number of hexagonal grid cells.
3. Calculation of the weighted normalized decision matrix--Equation (3): \[r_{ij}=w_{j}\times n_{ij}{=1,\;i=1\ldots m;j=1\ldots n}.\] (3)  
The criterion for weighting objects was their recognizability on satellite and aerial imagery, which is the main source for their acquisition [34; 47; 48]. Thus, buildings and forests were given a weight of \(w_{j}=0.25\), paved roads and railways \(w_{j}=0.15\), while water bodies and streams just \(w_{j}=0.10\). Futhermore, these weighting rules are also used in accessibility and passability analyses and, as noted by [PERSON] [49], are extremely important in crisis management.
4. Determine the positive (_PIS_) and negative (_NIS_) ideal solutions, as shown by Equations (4) and (5). \[PIS =\Big{\{}r_{1}^{-},\ldots r_{j}^{-}\ldots r_{n}^{-}\Big{\}},\textit{ where }v_{j}^{+}=\Big{\{}\max\Big{(}r_{j}^{i}\Big{\}}\;\textit{if }j\in B;\min r_{j}^{i}\;\textit{if }j\in J^{\prime}\] (4) \[NIS =\Big{\{}r_{1}^{-},\ldots r_{j}^{-}\ldots r_{n}^{-}\Big{\}},\textit{ where }v_{j}^{+}=\Big{\{}\min\Big{(}r_{j}^{i}\Big{\}}\;\textit{if }j\in B;\max r_{j}^{i}\;\textit{if }j\in J^{\prime}\] (5) As all criteria used are destimulants, the positive ideal solution was computed as \(r_{j}^{i}\) min, while the negative ideal was \(r_{j}^{i}\) max.
5. Calculate the separation measure \(S_{i}\) of each alternative (relative closeness to the positive ideal solution) as (Equations (6) and (7)): \[S_{i}^{+}=\sqrt{\sum_{i}^{\frac{n}{n}\Big{(}r_{ij}-r_{j}^{+}\Big{)}^{2}}};S_{ i}^{-}=\sqrt{\sum_{i}^{\frac{n}{n}\Big{(}r_{ij}-r_{j}^{-}\Big{)}^{2}}}i=1, \ldots,m\] (6)
6. Calculate the closeness coefficient of the alternatives (\(CC_{i}\)) as: \[CC_{i}=\frac{S_{i}^{-}}{S_{i}^{+}+S_{i}^{-}}\] (7)
7. Sort alternatives in descending order, whereby the highest \(CC_{i}\) indicates the best performance in relation to the evaluation criteria.