## 3 Error Estimation for the Acoustic Maps
### Error Associated with the Stratified Sampling Method: \(\sigma_{sample}^{2}\)  
In stratified spatial sampling, the sample is split up into strata (sub-samples) in order to decrease variances in sample estimates, to use partly non-random methods applied to sub-groups or clusters, or to study strata individually [42]. The central limit theorem states that, given a sufficiently large sample size \(n\), from a population of size \(N\) with a finite level of variance \(\sigma_{pop}^{2}\), the mean \(\overline{x}\) of all samples taken from the same population will be approximately equal to the mean of the population (expected value). Furthermore, all of the samples will follow an approximate normal distribution pattern, with all variances \(\sigma_{x}^{2}\) being approximately equal to the variance of the population divided by each sample's size \(n\), that is \(\sigma_{x}^{2}=\sigma_{pop}^{2}/n\). The maximum error \(\sigma_{sample}\), which is the largest expected deviation of the sample mean from the population mean with the stated confidence level \(1-\alpha\) (for \(1-\alpha=95\%\), the amplitude of the Gaussian distribution is \(Z_{\alpha}\sigma_{x}\) with \(Z_{\alpha}=1.96\)), is \(\sigma_{sample}=Z_{\alpha}\sigma_{x}\). The minimum number of elements of a sample \(n_{min}\) for a correct estimation of the mean of the population within an accuracy \(\pm\sigma_{sample}\) is  
\[n_{min}= Z_{\alpha}^{2}\ \sigma_{pop}^{2}/\sigma_{sample}^{2}. \tag{10}\]  
In general, \(\sigma_{pop}^{2}\) is unknown, so we used in its place the sample variance \(\sigma_{x}^{2}\), i.e.,  
\[n_{min}= Z_{\alpha}^{2}\ \sigma_{x}^{2}/\sigma_{sample}^{2}. \tag{11}\]  
In our case, the overall sample size consists of 192 measurements to be divided into each cluster. According to Equation (11), the actual expected deviation \(\sigma_{sample}\) for the two clusters with a number of measurements, \(n_{c1}\), \(n_{c2}\), respectively, reads  
\[\sigma_{sample_{i}}= Z_{\alpha}\ \sigma_{i}/\sqrt{n_{ci}}\ (i=1,2) \tag{12}\]  
Figures 7-9 show the expected deviation \(\sigma_{sample}\) calculated for a temporal discretization \(\tau=(5,15,60)\) min.  
Figure 7: Expected deviation \(\sigma_{sample}\) of the sample mean from the population mean calculated for a temporal discretization \(\tau=5\) min.  
To provide a homogeneous uncertainty, we chose different updating times for the day and night hours.  
In Table 2, the mean expected deviation \(\overline{\sigma_{sample}}\) calculated within the three time intervals T\({}_{\text{h}}\), defined as T\({}_{\text{h}}\) = (07:00-21:00), (21:00-01:00), and (01:00-07:00), for the three discretization times, defined as \(\tau\) = (5, 15, 60) min is reported for each cluster. The time interval subdivision allows us to estimate an overall mean \(\overline{\sigma_{sample_{1}}}\) of 0.37 dB and an overall mean sample \(\overline{\sigma_{sample_{2}}}\) of 0.33 dB for Clusters 1 and 2, respectively. As is apparent, the sampling error can be neglected as compared to those reported in Table 2.  
Figure 8: Expected deviation \(\sigma_{sample}\) of the sample mean from the population mean calculated for a temporal discretization \(\tau\) = 15 min.  
Figure 9: Expected deviation \(\sigma_{sample}\) of the sample mean from the population mean calculated for a temporal discretization \(\tau\) = 60 min.  
Table 3. Since the stratified sampling method represents an intrinsic error due to the finite number of measurements, we take the same value (mean value) \(\sigma_{sample}\big{(}\overline{x}_{g}\big{)}=0.35\) dB for each group \(g\).