# Optimal Estimation Retrieval of Aerosol Fine-Mode Fraction from Ground-Based Sky Light Measurements
## 2 Model and Methods  
### Aerosol Model  
In line with many studies [30; 31], a bimodal lognormal function is adopted to describe the volume PSD of spherical aerosol particles in the following form:  
\[\frac{dV}{d\ln r}\ =\ \sum\
olimits_{\mathrm{i}=1}^{2}\frac{V_{0}^{\mathrm{i} }}{\sqrt{2\pi}\ln\sigma_{\mathrm{g}}^{\mathrm{i}}}\exp\left[-\frac{\left(\ln r -\ln r_{\mathrm{v}}^{\mathrm{i}}\right)}{2\ln^{2}\sigma_{\mathrm{g}}^{\mathrm{ i}}}\right], \tag{1}\]  
where \(V\) is the particle volume, \(r\) is the radius, \(dV/d\ln r\) represents the volume contained in each logarithmic particle size interval, \(r_{\mathrm{v}}\) and \(\sigma_{\mathrm{g}}\) represent the volume geometric median radius and the geometric standard deviation, respectively, and the superscript \"i\" denotes the aerosol mode; hereafter,we use \"f\" and \"c\" to represent fine and coarse mode, respectively. Then, we get the total aerosol volume concentration \(V_{0}\) (with unit of \(\mu\)m\({}^{3}\mu\)m\({}^{-2}\)) from the sum of \(V_{0}^{\mathrm{f}}\) and \(V_{0}^{\mathrm{c}}\). Furthermore, the fine-mode volume fraction is calculated as  
\[\mathrm{FMF_{v}}\ =\ V_{0}^{\mathrm{f}}/V_{0}. \tag{2}\]  
Then, the spectral AOD is derived from the columnar volume concentration as [32]  
\[\tau_{\mathrm{a}}(\lambda)\ =\ \tau_{\mathrm{a}}^{\mathrm{f}}(\lambda)+\tau_{ \mathrm{a}}^{\mathrm{c}}(\lambda)\ =\ \frac{3V_{0}^{\mathrm{f}}Q_{\mathrm{ext}}^{\mathrm{f}}(\lambda)}{4\tau_{ \mathrm{eff}}^{\mathrm{f}}}+\frac{3V_{0}^{\mathrm{c}}Q_{\mathrm{ext}}^{ \mathrm{c}}(\lambda)}{4\tau_{\mathrm{eff}}^{\mathrm{c}}}, \tag{3}\]  
where \(Q_{\mathrm{ext}}\) represents the aerosol extinction efficiency factor, which is defined as the ratio between the extinction cross section and the geometric cross section. Based on the hypothesis of aerosol spherical particles, \(Q_{\mathrm{ext}}\) is calculated by the Mie code. \(\tau_{\mathrm{a}}^{\mathrm{f}}\) and \(\tau_{\mathrm{a}}^{\mathrm{c}}\) represent the fine-mode and coarse-mode spectral AOD respectively, and the total AOD (\(\tau_{\mathrm{a}}\)) is the sum of \(\tau_{\mathrm{a}}^{\mathrm{f}}\) and \(\tau_{\mathrm{a}}^{\mathrm{c}}\). The effective radius (\(r_{\mathrm{eff}}\)) and the effective variance (\(v_{\mathrm{eff}}\)) are converted from the geometric parameters by the relationship as  
\[\left\{ \begin{array}{l}r_{\mathrm{eff}}\ =\ r_{\mathrm{v}}\exp\!\left(-\frac{1 }{2}\ln^{2}\sigma_{\mathrm{g}}\right)\\ v_{\mathrm{eff}}\ =\ \exp\!\left(\ln^{2}\sigma_{\mathrm{g}}\right)-1\end{array} \right.. \tag{4}\]  
Correspondingly, the aerosol spectral fine-mode fraction (FMF\({}_{\mathrm{o}}\)) and the Angstrom exponent (AE) are calculated as  
\[\mathrm{FMF_{o}}(\lambda)\ =\ \tau_{\mathrm{a}}^{\mathrm{f}}(\lambda)/\tau_{ \mathrm{a}}(\lambda), \tag{5}\]  
\[\mathrm{AE}\ =\ \ln\frac{\tau_{\mathrm{a}}(\lambda_{1})}{\tau_{\mathrm{a}}( \lambda_{2})}/\ln\frac{\lambda_{1}}{\lambda_{2}}, \tag{6}\]  
where \(\lambda_{1}\) and \(\lambda_{2}\) in Equation (6) represent two specific wavelengths, e.g., 670 and 865 nm. Actually, the definition for the proportion of fine aerosols to total aerosols is not exactly in the same form. Table 1 lists the definitions of FMF used in different studies. Despite the inconsistency between these definitions, all of them can represent well the proportion of fine particles in the atmosphere. The FMF\({}_{\mathrm{v}}\) and FMF\({}_{\mathrm{o}}\) used in this paper are derived from the bimodal aerosol model and defined from the physical and optical perspectives separately.