## 2 Methodology
### Harmonic Mean-Type Approximation  
Although the revised algorithm of [PERSON] et al. [6] reduced biases by using the split-component MEE method for small and large particles, this approach still cannot fully describe the characteristics of polydispersed distributions of aerosol particles. Thus, additional modifications of the reconstruction method for polydispersed aerosol particles are required. A new approach is proposed in this study that MEE can be approximated as a power-law form of geometric volume-mean diameter that is a function of geometric standard deviation. In turn, the MEE is parameterized for various chemical compositions of aerosols with a multimodal lognormal size distribution as a function of geometric volume-mean diameter and geometric standard. In this study, the extinction coefficient of polydispersed aerosol particles are calculated as  
\[\begin{split}& b_{ext}\big{(}d_{ge},\sigma_{g}\big{)}=\text{ MEE}_{AMS}\big{(}d_{ge},\sigma_{g}\big{)}\times f(\text{RH})\times[\text{AMS}]+\text{MEE}_{AMN} \big{(}d_{ge},\sigma_{g}\big{)}\times f(\text{RH})\times\\ &[\text{AMN}]+\text{MEE}_{EC}\big{(}d_{ge},\sigma_{g}\big{)} \times[\text{EC}]+\text{MEE}_{SS}\big{(}d_{ge},\sigma_{g}\big{)}\times f( \text{RH})\times[\text{SS}]+\text{MEE}_{CM}\times[\text{CM}],\end{split} \tag{4}\]  
where the \(SS\) is sea salt (NaCl).  
In this study, the harmonic mean-type approximation was adopted for calculating MEE over an entire particle size range for universal application. The harmonic mean-type approximation (half the harmonic mean) for MEE is described as follows  
\[\text{MEE}=\sqrt{\frac{1}{\frac{1}{\text{MEE}_{s}^{2}}+\frac{1}{\text{MEE}_{L} ^{2}}}}. \tag{5}\]  
The MEE for the small (MEE\({}_{S}\)) and large (MEE\({}_{L}\)) size ranges are represented in a power-law form of the geometric volume-mean diameter:  
\[\text{MEE}_{S}=a\times d_{ge}^{b}\text{ and }\text{MEE}_{L}=c\times d_{ge}^{d} \tag{6}\]  
where \(a\), \(b\), \(c\), and \(d\) are coefficients to be determined. In Equation (5), MEE converges to MEE\({}_{S}\) and to MEE\({}_{L}\) for small and large particle size ranges, respectively. In the intermediate region between small and large particle ranges, MEE is calculated as a harmonic mean-type approximation as described in Equation (5). The harmonic mean-type approximation has been used to estimate the coagulation and condensation kernel in the low Knudsen particle size region [15; 16; 17; 18; 19]. The main advantage of the harmonic mean-type approximation is that it enables the evaluation of an analytical expression for intermediate particle size ranges [18; 19; 20].  
MEE is represented as a function of geometric volume-mean diameter as shown in Equations (5) and (6). A geometric volume-mean diameter (\(d_{gv}\)) is related with a geometric mean diameter (\(d_{g}\)) and a geometric standard deviation (\(\sigma_{g}\)) as follows  
\[\ln d_{gv}=\ln d_{g}+3\ln^{2}\sigma_{g}. \tag{7}\]In this study, polydispersed size distributions of aerosol particles are represented by a lognormal distribution function  
\[n\Big{(}\ln d_{p}\Big{)}=\frac{N}{\sqrt{2\pi}\ln\sigma_{g}}\exp\Bigg{[}\frac{- \ln^{2}\!\left(d_{p}/d_{g}\right)}{2\ln^{2}\sigma_{g}}\Bigg{]}, \tag{8}\]  
where \(N\) is the total number concentration of aerosol particles.  
The coefficients \(a\), \(b\), \(c\), and \(d\) in Equation (6) are represented as a linear expression of geometric standard deviation. For example, the coefficients \(a\), \(b\), \(c\), and \(d\) are a function of geometric standard deviation and represented as  
\[\begin{split} a&=\vartheta_{1}\sigma_{g}+\varepsilon _{1},\\ b&=\vartheta_{2}\sigma_{g}+\varepsilon_{2},\\ c&=\vartheta_{3}\sigma_{g}+\varepsilon_{3},\text{ and}\\ d&=\vartheta_{4}\sigma_{g}+\varepsilon_{4}, \end{split} \tag{9}\]  
where \(\vartheta\) and \(\varepsilon\) are constants to be determined. The same procedure that is used for the calculation of MEE is applicable for those of MAE. In this study, EC was the only absorbing aerosol (see Table 1), and thus, MAE was calculated only for the EC.