## 2 Methods
### Extinction Coefficient (\(b_{ext}\)) Calculations for Polydisperse Aerosols  
Generally, the contribution of aerosol optical properties (scattering and absorption) differ owing to the characteristics of each chemical compound. Figure 1 shows a schematic diagram of how to obtain source-based aerosol optical properties of polydisperse aerosol particles from the SMP model and Mie theory [15]. Three steps are required. First, size-resolved aerosol optical properties (aerosol extinction and the scattering and absorption coefficients) can be obtained on the basis of input variables such as the chemical-species-based mass concentration of the PM2.5 measurement, lognormally assumed aerosol size distribution, aerosol densities, and wavelength (550 nm in this study) using Mie theory. Second,the SMP receptor model can derive source-based mass concentrations. Third, from size-resolved aerosol mass extinction coefficients using Mie theory and source-based mass concentrations, source- and size-resolved mass extinction efficiencies can be estimated using the MLR model [10]. As outlined above, the \(b_{ext}\) value at a given size distribution was calculated with Mie theory, and the source-based MEE was calculated from the size-resolved \(b_{ext}\) and source-based mass concentration. To calculate the aerosol extinction coefficient (\(b_{ext}\)), the mass concentrations of inorganic ions, EC, OM (WSOM, HULIS, WISOM), NaCl, and crustal component, as well as aerosol properties (size distributions, densities, refractive index) are necessary as input data. Here, we basically used the same refractive index for OM; only the HULIS refractive index was sensitively investigated. Table 1 lists the refractive indices at a wavelength of 550 nm and the densities of the various aerosol components used in this study [16; 17; 18; 19; 20; 21]. We assumed a log-normal aerosol size distribution. An organic aerosol can be divided into water-soluble (HULIS and WSOC) and water-insoluble components. With these assumptions and initial data, we calculated the total aerosol extinction coefficient based on Mie theory. From the total \(b_{ext}\) and total mass concentration, the MEE, MSE, and MAE were obtained under the assumption that the aerosol was internally mixed. Figure 2 shows the sensitivities of mass efficiencies (MEE, MSE, and MAE) as a function of the \(\mathrm{d_{g}}\) and the GSD of internally mixed aerosols. The \(\mathrm{d_{g}}\) of 0.1-2.5 \(\mathrm{\SIUnitSymbolMicro m}\) and the GSD of 1.5-2.5 were considered. As shown in Figure 2, the calculated scattering, absorption, and extinction efficiencies had different values for the different GSDs and \(\mathrm{d_{g}}\). Generally, the mass efficiencies had a peak value around \(\mathrm{d_{g}}\) = 0.3 \(\mathrm{\SIUnitSymbolMicro m}\) for a GSD of 1.5. However, the \(\mathrm{d_{g}}\) at which mass efficiencies had their maximum value decreased as the GSD increased. (Supplementary Figure S2 shows the sensitivity of the MEE for EC, inorganic ion, OM, NaCl, and crustal aerosol.) \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{m_{r}}\) & \(\mathbf{m_{i}}\) & **Density (g cm\({}^{-3}\))** \\ \hline EC & 1.75 & 0.44 & 1.00 \\ Inorganic ions & 1.43 & 0.00 & 1.7 \\ OM & 1.53 & 0.00 & 1.2 \\ NaCl & 1.5 & \(1.0\times 10^{-8}\) & 2.2 \\ Crust & 1.53 & \(1.1\times 10^{-3}\) & 2.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Refractive indices and densities of externally mixed aerosols [16; 17; 18] at \(\lambda\) = 550 nm. Figure 1: Schematic diagram showing the steps to calculate the source-based aerosol optical properties of polydisperse aerosols using the Solver for Mixture Problem (SMP) model and Mie theory.