## 2 Data and Methods
### Estimation Methods for AOD
#### 2.2.2 Other Formulations for Estimating AOD from WRF-CMAQ  
Hygroscopic growth factors are determined by relative humidity; as such, formulations such as Equation (4) are essential for estimating the AOD algorithm. We applied two alternative methods to remove the uncertainty from the IMPROVE method [15]. First, we replaced Equation (4) following the method of [PERSON] [41]:  
\[\text{f(RH)}=\text{ exp}\left(-1-\frac{0.6}{\text{RH}-1.2}-\frac{0.75}{\text{ RH}-1.5}\right) \tag{5}\]  
Equation (5) was obtained by solving the Kohler equation to determine the equilibrium radius of dry sulfate aerosol particles at a given relative humidity and fitting their size distribution with a lognormal size distribution [41]. In most simulations, there is no case where supersaturation exists for RH \(\geq\) 1.2; thus, we did not set an upper limit for RH in Equation (5).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(b_{0}\) & \(b_{1}\) & \(b_{2}\) \\ \hline Spring & \(-0.01097\) & \(0.78095\) & \(0.08015\) \\ Summer & \(-0.18614\) & \(0.99211\) & - \\ Fall & \(-0.24812\) & \(1.01865\) & \(0.01074\) \\ Winter & \(0.34603\) & \(0.81984\) & - \\ Annual & \(0.33713\) & \(0.58601\) & \(0.09164\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Seasonal parameters of \(b_{0}\), \(b_{1}\), and \(b_{2}\) in Equation (4).  
Figure 2: **(a)** Observed aerosol optical depth (AOD) obtained from the geostationary ocean color imager (GOCI) satellite at 01:00 UTC on 24 May 2016. Estimated AOD calculated using the Interagency Monitoring of PROtected Visual Environments (IMPROVE) method [15] with maximum relative humidity (RH) values of **(b)** 93%, **(c)** 96%, and **(d)** 98%.  
Second, we applied the Pitchford method [16], in which AOD is estimated from the empirical reconstructed mass-extinction as:  
\[\text{AOD}=\sum_{i=1}^{N}(b_{ext})_{i}\Delta Z_{i} \tag{6}\]  
where \(N\) is the number of vertical layers in the numerical model, \(\Delta Z_{i}\) is the thickness of each layer, and \(b_{ext}\) is light extinction due to scattering and absorption by particles. In Equation (6), \(b_{ext}\) is calculated using the relation:  
\[b_{ext}\approx 2.2\times\text{f}_{\text{s}}(\text{RH})\times[\text{ Small Sulfate}]+4.8\times\text{f}_{\text{L}}(\text{RH})\times[\text{Large Sulfate}]+2.4\times\text{f}_{\text{s}}(\text{RH}) \times[\text{Small Nitrate}]\] \[+5.1\times\text{f}_{\text{L}}(\text{RH})\times[\text{Large Nitrate}] +2.8\times[\text{Small Organic Mass}]\] \[+6.1\times[\text{Large Organic Mass}]+10\times[\text{Elemental Carbon}]+1\times[\text{Fine Soil}] \tag{7}\]  
\[+1.7\times\text{f}_{\text{ss}}(\text{RH})\times[\text{Sea Salt}]+0.6\times[ \text{Coarse Mass}]+\text{Rayleigh Scattering}\;(\text{Site Specific})\] \[+0.33\times[\text{NO}_{2}(\text{ppb})]\]  
The unit of air pollutants is mg m\({}^{-3}\), and the unit of each coefficient (2.2, 4.8, 2.4, 5.1, 2.8, 6.1, 10, 1, 1.7, 0.6, and 0.33) is expressed as m\({}^{2}\) mg\({}^{-1}\); f\({}_{\text{s}}(\text{RH})\), f\({}_{\text{L}}(\text{RH})\), and f\({}_{\text{ss}}(\text{RH})\) are the small size, large size, and sea salt hygroscopic growth factor functions, respectively. Equations (6) and (7) consider the effects of sea salt on extinction over the ocean and subdivide hygroscopic growth factors according to particle size [28], unlike the IMPROVE algorithm of [PERSON] [15]. They use an organized look-up table (see Tables 1 and 2 in Appendix A) for building f\({}_{\text{s}}\)(RH), f\({}_{\text{L}}(\text{RH})\), and f\({}_{\text{ss}}(\text{RH})\). As indicated in [30], RH values over than 95% are set equal to them at RH = 95%. For simplicity, the IMPROVE, [PERSON] and [PERSON]'s method are referred to as M1, M2 and M3, respectively, later in this paper.