## 4 Results Analysis and Discussion  
Parameter Estimation and Uncertainty Quantification of Traffic Noise Prediction Model at Scenario A\({}_{1}\)  
Site A is a special site located along Assiniboine Ave to Wascana Pkwy in the City. Before 4 October 2013, this way was an old pavement, and then it was repaved. Consequently, the empirical traffic noise prediction model, expressed as Equation (20) was applied to Site A under both old and new pavement conditions. In this study, Site A with the old pavement conditions is denoted as scenario A\({}_{1}\). Before the data assimilation process, initial ensembles would be sampled from predefined intervals. As presented in Table 1, the ensembles of A and B were uniformly sampled from predefined intervals. Figure 4a presents the generated traffic noise prediction function based on those initial samples. In Figure 4, the green lines are generated based on the initial and final ensemble values of A and B; the red stars indicate the actual measurements; the red line displays the curve of Model (20) with A and B estimated through the maximum likelihood estimation method. As can be seen from Figure 4a, large uncertainty exists in noise prediction. Based on the noise measurement in scenario A\({}_{1}\), the uncertainty in the traffic noise prediction model can be significantly reduced. As presented in Table 1, the mean values of the updated coefficients are closer to their corresponding values obtained through maximum likelihood estimation compared to the initial ones. Further, standard deviations of the updated coefficients are reduced dramatically throughout the data assimilation process, and thus the uncertainty of the coefficients has been greatly mitigated. Figure 5 depicts the variations of coefficients A and B versus time. Significant uncertainty is found to be associated with the coefficients at the early stage of the assimilation process, though it is generally reduced as time proceeds. Eventually, the uncertainty of the traffic noise prediction model with the updated coefficients was significantly reduced, as shown in Figure 4b. Figure 6 presents a comparison between the forecasts of the traffic noise prediction model with updated coefficients and the actual measurements. This indicates that the predicted interval, with a confidence level of 0.1, can cover most measurements at Site A under old pavement conditions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multicolumn{4}{c}{**A**} & \multicolumn{4}{c}{**B**} \\ \cline{2-9}  & **Initial** & **Final** & **Mean** & **MLE a** & **Initial** & **Final** & **Mean** & **MLE** \\ \hline A\({}_{1}\) & [0.1, 20] & [11.5, 12.2] & 11.9 & 12.1 & [\(-\)10, 30] & \(-\)7.1 & \(-\)7.1 & \(-\)9.5 \\ A\({}_{2}\) & [0.1, 20] & [6.7, 7.2] & 7.0 & 5.1 & [\(-\)20, 50] & [10.5, 11.4] & 10.8 & 23.4 \\ B & [0.1, 20] & [4.5, 4.9] & 4.7 & 5.2 & [\(-\)20, 50] & [26.8 In this study, the Pearson correlation coefficient (R\({}^{2}\)) is employed to compare the performance of the proposed nested ensemble filtering method and traditional maximum likelihood estimation method. Since the traffic noise prediction model still contains some uncertainty after the data assimilation process, leading to uncertain forecasts for noise emission levels, the mean values of these uncertain forecasts will be applied to calculate the R\({}^{2}\) values. As can be seen in Table 2, both the NEF and MLE approaches can perform well in estimating the unknown parameter in Model (20) at Site A with old pavement. Figure 4: Initial (**a**) and updated (**b**) traffic noise prediction curves in scenario A\({}_{1}\). Figure 5: Variations of parameters (**a**) and (**b**) in Model (20) during the data assimilation process under scenario A\({}_{1}\).