# Finding Optimal Stations Using Euclidean Distance and Adjustable Surrounding Sphere
## 4 Proposed Methods
### Euclidean Distance + Fixed Surrounding Sphere (ED + FSS)  
This method combines the concepts Euclidean Distance (ED) and Fixed Surrounding Sphere (FSS). However, the difference between this method and the previous one is the location of the first station. For ED and FSS, the first stations are set at the center of the map, while for ED + FSS, all map locations are tried as a first station. The output of this method is a multi-list of stations in which the first stations are different. Consequently, further evaluation criteria have been introduced to evaluate the best network with a maximum of spatial coverage. We divide ED + FSS into two processes: (i) finding the next stations and (ii) evaluating a maximum coverage percentage while still preserving an appropriate overlapped area for the network. The first process, finding the next stations, is described with Algorithm 2 Euclidean Distance + Fixed Surrounding Sphere (ED + FSS) as follows:```
1:Input: Map; a specified number of stations;
2:For each \(L\) in map do
3: select \(L\) and insert into station list, \(S\) = {\(L\)}. 4:while a specified number of stations is not satisfied do
5:while all \(L\) in the non-covered area have not been tried do
6: Select candidate station (\(L\)) and we temporarily append \(L\) to the station list \(S\). 7: Classify covered and non-covered areas of \(S\) with Equation (3). 8: Calculate \(\mathit{TED}_{\mathit{L}}\) with Equation (2) and stores the current \(\mathit{TED}_{\mathit{L}}\) value in a list. 9: Remove the temporarily added \(L\) from \(S\). 10:endwhile
11: Compare \(\mathit{TED}_{\mathit{L}}\) value in a list
12: The \(L\) with the smallest value of \(\mathit{TED}_{\mathit{L}}\) will be chosen as the next location of station. 13:\(L\) is permanently appended to the station list \(S\). 14:endwhile
15: Store station list \(S\) into multi-list
16:endfor
17:return Multi-list of stations. 18:\(\sharp\) The output is multi-list of stations. For examples:
19:\(\sharp\) {\(S\):= {\(L\), \(L\), \(L\), \(L\), \(\ldots\), \(L\)}, \(S\)\({}_{\mathit{i+1}}\) = {\(L\)+1, \(L\), \(L\), \(L\), \(\ldots\), \(L\)}, \(\ldots\), \(S\)\({}_{\mathit{N}}\) = {\(L\), \(L\), \(L\), \(L\), \(\ldots\), \(L\)}}
```

**Algorithm 2** Euclidean Distance + Fixed Surrounding Sphere (ED + FSS)

After we obtain a multi-list of stations, the next process is to evaluate the maximum coverage percentages of all \(S_{x}\). The ED + FSS will use two criteria, Coverage Percentage (COV) and Weighted Coverage Degree (WCD), for selecting the best network. The descriptions of COV and WCD criteria are outlined in Section 5. ### Euclidean Distance + Adjustable Surrounding Sphere (ED + ASS)

This method is an upgrade from ED + FSS. We change from using a fixed diameter to an adjustable diameter, which depends on a station's covered area proportion. Our proposed method considers economic benefits based on deployment costs and station location with the highest spatial coverage for a specified number of economically feasible stations. The covered area proportion can be adjusted as needed. For example, if the environmental authority has a budget limit for establishing monitoring stations, an arbitrary ratio can be predefined as 50% or 70%. On the other hand, if they require high spatial resolution monitoring or an unlimited budget, the ratio can be predefined as 10% or 30%. As a warmup to ED + ASS, we select a station at the center of the map (\(\mathsf{S}_{\text{center}}\)) for calculating the length of the diameter. Next, we calculate \(\mathit{ED}_{i}\) from all locations (\(L\),\(i\)) on map to the \(\mathsf{S}_{\text{center}}\) with Equation (1). We pass all the \(\mathit{ED}_{i}\) values into a list. Subsequently, we sort the obtained list in ascending order, which means that locations closer to the station with lower \(\mathit{ED}_{i}\) values will be at the beginning of the list. The cut off position corresponds to an arbitrary ratio, as mentioned above. We calculate the cut off position value with Equation (4).