## 5 Numerical Simulation
### Retrieval of the Aerosol Size Distribution  
With the help of hybrid SPSO-DE algorithm, the ASDs mentioned above are reconstructed under different random measurement errors by using the SEM and ALSM, respectively, which are showed in Table 4 and Figure 5. Satisfactory retrieval accuracy can be obtained by both SEM and ASLM without measurement errors. The retrieval accuracy deteriorates with more random measurement errors, but are still reasonable even with 10% errors. Comparing the convergence accuracy of retrieval results by using the SEM with that by using the ALSM, it can be found that the robustness of retrieval results by using the ASLM is better than those by using the SEM. Moreover, in studying the bimodal ASDs, there is more satisfactory agreement between the estimated results and the true values of the major peak than those of the other peak.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Different Functions**} & \multirow{2}{*}{**Random Error**} & \multicolumn{2}{c}{**SEM**} & \multicolumn{1}{c}{**ASLM**} \\ \cline{3-5}  & & & **Retrieval Error, \(\delta\)** & **Retrieval Error, \(\delta\)** \\ \hline \multirow{5}{*}{Monomodal} & \multirow{5}{*}{L-N} & 0\% & 0.000125 & 0.000312 \\  & & 5\% & 0.130401 & 0.064109 \\  & & 10\% & 0.215168 & 0.162801 \\  & & 20\% & 0.387621 & 0.265412 \\ \cline{2-5}  & \multirow{5}{*}{Gamma} & 0\% & 0.000322 & 0.000545 \\  & & 5\% & 0.162721 & 0.144108 \\  & & 10\% & 0.187542 & 0.169336 \\  & & 20\% & 0.365729 & 0.287532 \\ \hline \multirow{5}{*}{Bimodal} & \multirow{5}{*}{L-N} & 0\% & 0.000156 & 0.003148 \\  & & 2\% & 0.096542 & 0.085159 \\ \cline{1-1}  & & 5\% & 0.244494 & 0.232532 \\ \cline{1-1}  & & 10\% & 0.356279 & 0.328765 \\ \cline{1-1} \cline{2-5}  & \multirow{5}{*}{Gamma} & 0\% & 0.025242 & 0.090019 \\  & & 2\% & 0.145623 & 0.110330 \\ \cline{1-1}  & & 5\% & 0.181529 & 0.144053 \\ \cline{1-1}  & & 10\% & 0.319732 & 0.298621 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Retrieval results of ASDs under different optical measurement methods.  
Figure 4: Comparison of objective function values of SPSO and hybrid SPSO-DE algorithms in retrieving monomodal Gamma distribution.  
Figures 6 and 7 depict the distributions of the objective function values for monomodal ASDs by using two different type of optical measurement methods. Figure 8 shows the corresponding distributions of estimated results. From Figures 6 and 7, it is obvious that the minimum value region of the objective function is a point, which means the optimal retrieval result is unique, especially for the monomodal Gamma distribution. Moreover, comparing the distributions of objective function by using the SEM and that by using the ALSM, it is obvious that the objective function value by using the SEM is smaller than that by using the ALSM. So, to obtain more accurate retrieval results by using the SEM, the lower objective function value should be obtained during inverse calculation, which will increase the convergence difficulty and time. This phenomenon can explain the reason for the better retrieval accuracies by using the ALSM than those by using the SEM. From Figure 8, it can be found that the distributions of the results retrieved by using the SEM show more dispersion than those retrieved by using ALSM, which also confirms the results obtained above.  
Figure 5: Reproducibility of different ASDs by using different optical measurement methods: (**a**) monomodal L-N distribution; (**b**) monomodal Gamma distribution; (**c**) bimodal L-N distribution; (**d**) bimodal Gamma distribution.  
Figure 8: _Cont._  
Figure 6: Distribution of the objective function values for different ASDs by using the SEM: (**a**) L-N distribution; (**b**) Gamma distribution.  
Figure 7: Distribution of the objective function values for different ASDs by using the ALSM: (**a**) L-N distribution; (**b**) Gamma distribution.