## 3 Numerical Experiments
### Sensitivity Studies with Respect to Emission Levels  
The second objective of our particular research is to investigate the sensitivity of the model output (in terms of mean monthly concentrations of several important pollutants) with respect to variations in the input emissions of anthropogenic pollutants. The anthropogenic emission input consists of four different components, denoted as \(\mathbf{E}=(\mathbf{E^{A}},\mathbf{E^{N}},\mathbf{E^{S}},\mathbf{E^{C}})\) as follows:  
\[\mathbf{E^{A}}\mathbf{--}\text{ammonia }(NH_{3}); \mathbf{E^{S}}\mathbf{--}\text{sulphur dioxide }(SO_{2});\] \[\mathbf{E^{N}}\mathbf{--}\text{nitrogen oxides }(NO\ +\ NO_{2}); \mathbf{E^{C}}\mathbf{--}\text{ anthropogenic hydrocarbons}.\]  
The output of the model is mean monthly concentration of the following 3 pollutants:  
\[s_{1} \quad\text{--}\quad\text{ozone }(O_{3});\] \[s_{2} \quad\text{--}\quad\text{ammonia }(NH_{3});\] \[s_{3} \quad\text{--}\quad\text{ammonium sulphate and ammonium nitrate }(NH_{4}SO_{4}\ +\ NH_{4}NO_{3}).\]  
In our particular case, we are interested in sensitivity studies of the mean monthly concentrations of ammonia in Milan. The domain under analysis is the 4-dimensional hypercubic domain \([0.5,1]^{4}\). Second-degree polynomials were employed as an approximation tool [44]. The input data were generated using a specialized version of the UNI-DEM, named SA-DEM, which was specifically developed for sensitivity analysis studies. The results for relative errors and computational times in the approximate evaluation of the quantities \(f_{0}\), total variance, and first-order and total sensitivity indices, using various stochastic approaches for numerical integration, are presented in Table 5, Table 6, and Table 7, respectively. First, it should be noted that the quantity \(f_{0}\) is represented by a 4-dimensional integral, while the other quantities under consideration are expressed as double-dimensional integrals, following the principles of the _correlated sampling_ technique to compute sensitivity measures reliably [38; 45]. The SS-MCA proves to be the most efficient in terms of relative error, with a difference of about 1-3 orders of magnitude (see Tables 1, 2, 5 and 6). A key conclusion drawn from the numerical results in Tables 5-7 is \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Estimated Quantity**} & \multirow{2}{*}{**Reference Value**} & \multirow{2}{*}{**Sobol\({}^{\prime}\) QMCA**} & **MCA-MSS** & **MCA-MSS-S** & **SS-MCA \(N=\)** \\  & & & \(\mathbf{x}\approx 2\ \times\ 10^{-4}\) & \(\mathbf{x}\approx 2\ \times\ 10^{-4}\) & \(\mathbf{x}\approx 2\ \times\ 10^{-4}\) & \(\mathbf{2}\times\mathbf{5}^{\mathbf{5}}=2\times 15,625\)** \\ \cline{3-6}  & & **Rel. Error** & **Rel. Error** & **Rel. Error** & **Rel. Error** \\ \hline \(S_{12}\) & \(6\times 10^{-3}\) & \(1\times 10^{-3}\) & \(3\times 10^{-3}\) & \(2\times 10^{-4}\) & \(5\times 10^{-1}\) \\ \(S_{14}\) & \(5\times 10^{-3}\) & \(2\times 10^{-3}\) & \(9\times 10^{-4}\) & \(2\times 10^{-3}\) & \(8\times 10^{-1}\) \\ \(S_{24}\) & \(3\times 10^{-3}\) & \(5\times 10^{-3}\) & \(4\times 10^{-3}\) & \(1\times 10^{-2}\) & \(5\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relative errors for estimation of sensitivity indices of the second order of chemical rate constants (\(n=2^{14}=16,384\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Estimated Quantity**} & \multirow{2}{*}{**Reference Value**} & \multirow{2}{*}{**Sobol\({}^{\prime}\) QMCA**} & **MCA-MSS** & **MCA-MSS-S** & **SS-MCA \(N=\)** \\  & & & \(\mathbf{x}\approx 2\ \times\ 10^{-3}\) & \(\mathbf{x}\approx 2\ \times\ 10^{-4}\) & \(\mathbf{x}\approx 2\ \times\ 10^{-4}\) & \(\mathbf{2}\times\mathbf{5}^{\mathbf{5}}=2\times 15,625\)** \\ \hline \(S_{2}\) & \(3\times 10^{-1}\) & \(4\times 10^{-3}\) & \(9\times 10^{that all stochastic approaches provide reliable relative errors for sufficiently large sample sizes.