## 4 Discussion on the Initialization of Dynamap and Its Implementation
### Clustering of the 24 Recording Stations: \(\delta_{\mathrm{C1,2}}^{\tau}(t)\)  
In the following, we discuss a second procedure for updating the acoustic maps based on the two-cluster expansion scheme, which should be more accurate as it uses all the 24 stations to determine \(\delta_{g}^{\tau}(t)\) simultaneously. The clustering method cannot be implemented until a sufficient amount of noise data from the 24 stations have been recorded. That is, from each station, we need to record \(\mathrm{L_{eqT,i}}(t)\) and obtain sufficiently accurate mean values for each interval of time \(\tau\). Then, the clustering calculation can be performed.  
Once the compositions of Clusters 1 and 2 have been found (meaning that there are \(N_{1}\) stations in Cluster 1, \(k_{1}\) = (1,..., \(N_{1}\)), and \(N_{2}\) stations in Cluster 2, \(k_{2}\) = (1,..., \(N_{2}\)), such that \(N_{1}+N_{2}\) = 24), we need to rearrange the variations obtained from Equation (14) according to the indices \(C_{1,k1}\) and \(C_{2,k2}\), which we denote as \(\delta_{\mathrm{C1,k1}}^{\tau}(t)\) and \(\delta_{\mathrm{C2,k2}}^{\tau}(t)\) within each cluster, \(C_{1}\) and \(C_{2}\). Then, we calculate the mean variations, \(\delta_{\mathrm{C1}}^{\tau}(t)\) and \(\delta_{\mathrm{C2}}^{\tau}(t)\), for each cluster according to,  
\[\delta_{\mathrm{C1}}^{\tau}(t)= \frac{1}{N_{1}}\Sigma_{k1=1}^{N_{1}}\ \delta_{\mathrm{C1,k1}}^{\tau}(t) \tag{15}\] \[\delta_{\mathrm{C2}}^{\tau}(t)= \frac{1}{N_{2}}\Sigma_{k2=1}^{N_{2}}\ \delta_{\mathrm{C2,k2}}^{\tau}(t)\]  
where \(C_{1,k1}\) and \(C_{2,k2}\) are the \(N_{1}\) and \(N_{2}\) indices of stations belonging to Cluster 1 and Cluster 2, respectively. Here, the allowed ranges of variation of (\(\delta_{\mathrm{C1,k1}}^{\tau}(t)\), \(\delta_{\mathrm{C2,k2}}^{\tau}(t)\)) are (compared to Equation (14)) similarly determined using [\(Q_{1,\mathrm{C1,2}}^{\tau}(t)-1.5\cdot\mathrm{IQR}_{\mathrm{C1,2}}^{\tau}(t)\), \(Q_{3,\mathrm{C1,2}}^{\tau}(t)+1.5\cdot\mathrm{IQR}_{\mathrm{C1,2}}^{\tau}(t)\)], calculated for all stations (\(k_{1},k_{2}\)) within each cluster, \(C_{1}\) and \(C_{2}\), respectively. Then, the mean variation \(\delta_{g}^{\tau}(t)\) associated with each group \(g\) can be calculated using the formula  
\[\delta_{g}^{\tau}(t)= \overline{\beta}_{1}(\overline{x}_{g})\ \delta_{\mathrm{C1}}^{\tau}(t)+ \overline{\beta}_{2}(\overline{x}_{g})\ \delta_{\mathrm{C2}}^{\tau}(t) \tag{16}\]  
where the parameters \(\overline{\beta}_{1}(\overline{x}_{g})\) and \(\overline{\beta}_{2}(\overline{x}_{g})\) for each group \(g\) are reported in Table 1 for the entire city of Milan. Here, the value \(\overline{x}_{g}\) represents the mean non-acoustic parameter associated with group \(g\), and \(\overline{\beta}_{1}(\overline{x}_{g})\), \(\overline{\beta}_{2}(\overline{x}_{g})\) the corresponding probabilities to belong to Clusters 1 and 2, respectively. Again, in the extreme case in which more than 12 of the stations of the entire network have exceeded their A.I.'s (for instance 6 stations per cluster fail to remain within their A.I.'s), we are in the presence of a global network extreme event, so their data cannot be rejected but need to be included in the updating procedure of the maps. The number of excluded stations (here initially taken as 12) will be reconsidered during the development of Dynamap.