## 1 Introduction  
The aerosol extinction coefficient \(b_{ext}\) is the key parameter that determines the aerosol optical depth by integrating over the light path; hence, it is important for the calculation of aerosol radiative forcing and the criterion of air pollution. The visibility is inversely proportional to the aerosol extinction coefficient and hence crucial for the safety and meteorological observations. Theoretically, with an assumption of the spherical shapes of the aerosol particles, the extinction coefficient of one species of distributed aerosol particles is calculated as  
\[b_{ext}=\int Q_{ext}\frac{\pi}{4}d_{p}^{2}n(d_{p})dd_{p}, \tag{1}\]  
where \(b_{ext}\) is the extinction coefficient, \(Q_{ext}\) is the single particle extinction efficiency, \(d_{p}\) is the particle diameter, and \(n(d_{p})\) is the number particle size distribution. The Mie theory-based calculations use Equation (1) for the extinction coefficient computation by integrating the Mie scattering libraries (e.g., \(Q_{ext}\)) over an observed particle size distribution. Equation (1) can be rewritten as  
\[b_{ext}=\int Q_{ext}\frac{\pi}{4}d_{p}^{2}n(d_{p})dd_{p}=\int\frac{\pi\rho_{p}d _{p}^{3}}{6}n(d_{p})Q_{ext}\frac{3}{2\rho_{p}d_{p}}dd_{p}, \tag{2}\]  
where \(\rho_{p}\) is the particle density. Equation (2) implies that the aerosol extinction coefficient is a product of aerosol mass concentration and mass extinction efficiency (MEE, see also Figure 1), which is also known as the reconstruction method (e.g., [1,2]). Physically, aerosol MEE represents the amount of light extinction by aerosol for a given mass of aerosol in a unit volume of air [1,2]. Figure 1 shows the calculated extinction efficiency and mass extinction efficiency as a function of particle size for various aerosols using a Mie code [3] at a wavelength \(\lambda=550\) nm (Table 1). The strong dependence of both extinction efficiency (blue lines) and MEE (red lines) on the particle size and composition is revealed as also reported in previous studies (e.g., [4,5]). Several numerical models (e.g., the Community Multiscale Air Quality (CMAQ) modeling system [7]) provide empirical approaches, such as the reconstruction method (e.g., Equation (2)), that use mass concentrations of aerosol chemical compositions to calculate total aerosol optical depth with precalculated lookup tables. For example, [PERSON]'s reconstruction method [8] of scattering and absorption by aerosol particles used in the Interagency Monitoring of Protected Visual Environments (IMPROVE) protocol is given by following form with an external mixing:  
\[b_{ext}=\sum_{i}\text{MEE}_{i}\times\text{M}_{i}=3f(\text{RH})\times[\text{ AMS}]+3f(\text{RH})\times[\text{AMN}]+4[\text{OC}]+10[\text{EC}]+1[\text{ Soil}]+0.6[\text{CM}], \tag{3}\]  
where \(i\) indicates aerosol species and \(\text{M}_{i}\) is the mass concentration of species \(i\). The scattering enhancement factor \(f(\text{RH})\) is the ratio between dry and wet scattering as a function of relative humidity (RH) and wavelength. The constants of 3, 3, 4, 10, 1, and 0.6 are the MEE of ammonium sulfate (AMS), ammonium nitrate (AMN), organic carbon (OC), elemental carbon (EC), soil, and coarse mass (CM), respectively [6,8]. The square brackets indicate the mass concentration of each component. This \begin{table}
\begin{tabular}{l l l l} \hline  & **Real Part (\(m_{r}\))** & **Imaginary Part (\(m_{i}\))** & **Density (g cm\({}^{-3}\))** \\ \hline Ammonium sulfate & 1.53 & 0.00 & 1.77 \\ Ammonium nitrate & 1.55 & 0.00 & 1.73 \\ Elemental carbon & 1.95 & 0.79 & 1.0 \\ Sea salt & 1.55 & 0.00 & 1.9 \\ \hline \end{tabular}
\end{table} 
Table 1: Aerosol refractive indices (\(m_{r}\) + \(in_{i}\)) at a wavelength of 550 nm [5,6] and densities for external mixing. equation assumes that contributions to total ambient light scattering are from ammonium sulfate, ammonium nitrate, organic carbon, soil, and coarse mass.