# Source-Based Size-Resolved Optical Properties of Carbonaceous Aerosols
## 2 Materials and Methods
### Source-Based Aerosol Optical Properties of Polydisperse Carbonaceous Aerosols  
The source-based receptor model provides information about the aerosol sources. The receptor model can be used to estimate the contribution of each source to the composition-based mass concentration.  
In this study, source-based optical properties were obtained from the source-based mass concentration, size distribution, and refractive index.  
Figure 2 shows the schematic diagram for calculating the source-based optical properties of polydisperse carbonaceous aerosols.  
For the mass concentration of a given species, the extinction coefficient (\(b_{ext,i}\)) and mass extinction efficiency of carbonaceous aerosols can be simplified as follows [16; 22; 23]:  
\[MEE_{i}=\frac{b_{ext,i}}{C_{i}}, \tag{1}\]  
\[b_{ext}=\sum_{i}b_{ext,i}=\sum_{i}[C_{i}\times MEE_{i}], \tag{2}\]  
where \(MEE_{i}\) is the mass extinction efficiency in m\({}^{2}\)/g and C\({}_{i}\) is the mass concentration of the \(i\)th carbonaceous composition (ug/m\({}^{3}\)).  
The \(b_{ext}\) value can be calculated using the Mie theory based on the refractive index and size distribution of the carbonaceous aerosol. A lognormal size distribution was assumed in this study. The lognormal size distribution for mass concentration (\(n_{M}(Ind_{p})\)) can be expressed as a function of the \(d_{g0}\) and geometric standard deviation (\(\sigma_{g}\)) as follows.  
\[n_{M}(Ind_{p})=\sum_{i}\frac{C_{i}}{\sqrt{2\pi ln\sigma_{g}}}exp\left[-\frac{ ln^{2}\left(d_{p}/d_{g0}\right)}{2 ln^{2}\sigma_{g}}\right], \tag{3}\]  
where, \(\sigma_{g}\) is the geometric standard deviation and \(d_{p}\) is particle diameter.  
From \(b_{ext}\) and the mass concentration, the mass extinction efficiency (MEE) can be obtained using Equation (1). Table 1 shows the MEE, mass absorption efficiency (MAE), and refractive index of the carbonaceous species based on different size distributions and refractive indices of HULISs.  
Based on the \(b_{ext}\) value and source-based mass concentration from the receptor model (\(C_{S}\)), the source-resolved optical properties can be obtained. The total mass concentration of the carbonaceous aerosol comprises each carbonaceous aerosol species. In addition, the receptor model can be used to express the mass concentration as a combination of each source ([PERSON] et al., 2018). In this study, we used source-based mass concentrations (\(C_{s}\)) from the PMF receptor model. Subsequently, the source-based MEEs and MAEs were obtained from the source-based mass concentration from the PMF receptor model and \(b_{ext}\) values calculated using the Mie theory [16; 22; 23; 24]:  
\[b_{ext}=\sum_{s=source}\left[C_{s}\times MEE_{s}\right]\),\(b_{abs}=\sum_{s=source}\left[C_{s}\times MAE_{s}\right] \tag{4}\]  
where \(MEE_{s}\) and \(MAE_{s}\) are the mass extinction and absorption efficiencies in m\({}^{2}\)/g, respectively, and \(C_{s}\) is the mass concentration of the sth source-based on the PMF model (ug/m\({}^{3}\)). In this study, the MEEs of each source were obtained using multilinear regression, where the mass concentration of each source from the PMF model is the dependent variable and the aerosol extinction coefficient is the independent variable.  
Subsequently, source-based mass extinction coefficients for source s were calculated as follows:  
\[b_{ext,\;s}=C_{s}\times MEE_{s},\;b_{abs,\;s}=C_{s}\times MAE_{s} \tag{5}\]  
The obtained extinction coefficient is statistically significant at the 99% confidence level. A more detailed methodological description for obtaining the source-based size-dependent optical properties can be found in a previous publication [16].  
Figure 2: Schematic diagram showing the calculation of the source-based optical properties of polydisperse carbonaceous aerosols. HULIS: humic-like substances; WSOC: water-soluble organic carbon; WISOC: water-insoluble organic carbon; EC: elemental carbon; OC: organic carbon; PMF: Positive Matrix Factorization.