# Finding Optimal Stations Using Euclidean Distance and Adjustable Surrounding Sphere
## 4 Proposed Methods
### Fixed Surrounding Sphere (FSS)  
The fixed surrounding sphere (FSS) method was inspired by the original sphere of influence (SOI) [12]. We applied such an idea for determining the area surrounding each station without using pollution concentration data. The constant value is identified to a diameter length of FSS with reference to city air quality monitoring network designing in Seoul city [2]. In Seoul's existing air monitoring network, the 25 stations are located approximately 5 km away from each other. They are not located close to the road, high emission concentration sources, or high-density population areas. On the other hand, these stations are distributed throughout the city. Consequently, we predefined a fixed diameter (\(dm_{s}\)) of FSS to divide the covered and non-covered areas under stations. As shown in Figure 4, the triangle symbols represent stations. The surrounding covered area of the stations is illustrated as a circle shape in Figure 4a and a pie shape in Figure 4b by determining a diameter length. The areas outside represent the non-covered areas. The shapes depend on the position of stations on the map. However, such areas are defined as covered areas, although the shapes of the areas are different.  
The procedure to classify the covered and non-covered areas under a station are described in this subsection. Let station \(S_{j}\) be a location of station and the diameter of \(S_{j}\) be equal to \(dm_{s}\). The location (\(L_{i}\)) is covered (monitored) by a station \(S_{j}\) when the distance from \(L_{i}\) to \(S_{j}\) is less than \(\frac{dm_{s}}{2}\) and also such \(L_{i}\) will be members of all covered areas for \(A_{sj}\). Equation (3) represents the probability \(\mathrm{p}(L_{i},S_{j})\) that a location is covered by station \(S_{j}\).  
\[\mathrm{p}\big{(}L_{i},\ S_{j}\big{)}=\left\{\begin{array}{l}1:\ \text{``covered area''},\ if\ distance(L_{i},S_{j})\ \leq\ \frac{dm_{s}}{2}\\ 0:\ \text{``non $-$ covered area''},\ otherwise\end{array}\right.\in\ A_{Sj} \tag{3}\]  
\(i\) is the index of location from 0 to \(N-1\), where \(N\) represents the number of locations in area \(A\), and \(j\) provides an index of stations in \(S\).  
The Algorithm 1 Fixed Surrounding Sphere (FSS) can be described as follows:  
Figure 4: The (**a**) circle and (**b**) pie shapes indicate the covered areas by stations at the different positions on the map by determining a diameter length.  
```
1:Input: Map; a specified number of stations; index of center of map;
2:Identify the first station at center of map and insert into station list \(S\) = {\(S\)1}.
3:while a specified number of stations is not satisfied do
4:while all \(L\) in the non-covered area have not been tried do
5: Select candidate station (\(L\)) and we temporarily append \(L\) to the station list \(S\).
6: Classify covered and non-covered areas of \(S\) with Equation (3).
7: Calculate the total of the non-covered area and pass value to the #poor variable.
8: Remove the temporarily added \(L\) from \(S\).
9:endwhile
10: #poor values are compared.
11: The \(L\) with the smallest value of #poor will be chosen as the next location of station.
12: \(L\) is permanently appended to the station list \(S\).
13:endwhile
14:return List of stations \(S\). # The output is a list of stations with the first station at \(S\)1. For examples:
15: # \(S\) = {\(S\)1, \(L\)\({}_{a}\), \(L\)\({}_{b}\),..., \(L\)\({}_{n}\)}
```  
**Algorithm 1** Fixed Surrounding Sphere (FSS)