## 3 Results
\begin{split}\text{MEE}&=\left(\left(a\times d_{gv} ^{b}\right)^{-2}+\left(c\times d_{gv}^{d}\right)^{-2}\right)^{-\frac{1}{2}}= \left\{\left(\left(\left(\delta_{1}\sigma_{g}+\varepsilon_{1}\right)\times d _{gv}^{(\delta_{2}\sigma_{g}+\varepsilon_{2})}\right)^{-2}+\right.\right.\\ &\left.\left.\left(\left(\delta_{3}\sigma_{g}+\varepsilon_{3} \right)\times d_{gv}^{(\delta_{4}\sigma_{g}+\varepsilon_{4})}\right)^{-2} \right\}^{-\frac{1}{2}}.\end{split}  \tag{10}\]  
Figure 6 shows the determined MEE coefficients (\(a\), \(b\), \(c\), and \(d\)) as a function of geometric standard deviation. For AMS (Figure 6a,b), AMN (Figure 6c,d), and sea salt (Figure 6e,f), the MEE\({}_{L}\) values were approximated using the same equation (MEE\({}_{L}=2.5d_{gv}^{-1}\)), which means that \(c\) and \(d\) were \(2.5\) and \(-1\), respectively, regardless of geometric standard deviation. For EC, the MEE\({}_{S}\) values were approximated to be MEE\({}_{s}=12d_{gv}^{0.2}\) regardless of geometric standard deviation. The remaining coefficients (\(a\) and \(b\) for AMS, AMN, and sea salt, and \(c\) and \(d\) for EC) were linear as a function of geometric standard deviation, as shown in Figure 6. Table 3 summarizes the determined coefficients \(a\), \(b\), \(c\), and \(d\) of each aerosol component as a function of geometric standard deviation. The determined coefficients of strongly absorbing EC were distinct from non-absorbing other aerosol species. Based on Figures 4-6 and Table 3,  
Figure 5: Same as Figure 4, but for sea salt (SS, **left column**) and elemental carbon (EC, **right column**). the MEE can be approximated as analytical equations for each component of polydispersed aerosol particles. For example, the \(\text{MEE}_{\text{S}}\) and \(\text{MEE}_{L}\) values of AMS can be approximated as a simple form of  
\[\text{MEE}_{\text{S}}=a\times d_{gv}^{b}=\left(-30\sigma_{g}+83.33\right) \times d_{gv}^{\left(-\sigma_{g}+3.5\right)}\text{ and }\text{MEE}_{L}=c\times d_{gv}^{d}=2.5\times d_{gv}^{-1}. \tag{11}\]  
Figure 6: The determined mass extinction efficiency coefficients \(a\) and \(b\) as a function of geometric standard deviation in Equation (6) for ammonium sulfate (**a**,**b**), ammonium nitrate (**c**,**d**), and sea salt (**e**,**f**). For elemental carbon coefficients, \(c\) (**g**) and \(d\) (**h**) are determined. The extinction coefficient now can be expressed as a function of the size distribution parameters (i.e., geometric volume-mean diameter and geometric standard deviation):  
\[ \begin{split} b_{ext}=\sum_{i}\text{MEE}_{i}\times M_{i}& =\sum_{i}\left\{\left(\left(\mathfrak{s}_{1,i}\sigma_{g,i}+\varepsilon_{1,i }\right)\times d_{gv,i}^{(\mathfrak{s}_{2,i}\sigma_{g,i}+\varepsilon_{2,i})} \right)^{-2}+\left(\left(\mathfrak{s}_{3,i}\sigma_{g,i}+\varepsilon_{3,i} \right)\times\right.\\ &\left.d_{gv,i}^{(\mathfrak{s}_{4,i}\sigma_{g,i}+\varepsilon_{4,i })}\right)^{-2}\right\}^{-1/2}\times\text{M}_{i}.\end{split}  \tag{12}\]  
To represent absorbing properties, the MAE of EC can also be approximated using the same approach of  
\[b_{abs}=\text{MAE}_{EC}\times\text{M}_{EC}, \tag{13}\]  
where MAE is expressed as follows:  
\[\text{MAE}=\left\{\left(7\times d_{gv,i}^{0.08}\right)^{-2}+\left(\left(0.5 \sigma_{g}+0.25\right)\times d_{gv,i}^{(0.6\sigma_{g}-2.6)}\right)^{-2}\right\} ^{-\frac{1}{2}}. \tag{14}\]  
Figure 7 shows the comparison of the approximated MAE and that using Mie theory for EC with varying geometric standard deviations of 1.5, 2.0, and 2.5 were considered to represent polydispersed aerosol size. The harmonic mean-type analytical expression for MAE proposed in this study (i.e., Equation (14)) also agreed with the results of the Mie theory-based calculations. To test the accuracy of the newly developed MEE approximation, the simple forcing efficiency (SFE) [5,21] of individual aerosol species was calculated.