# Data Assimilation of AOD and Estimation of Surface Particulate Matters over the Arctic
## 2 Experiments
### CMAQ-Derived and Assimilated AODs  
The radiant energy is attenuated in the atmosphere by aerosol scattering and absorption. The attenuation is expressed by the extinction coefficient (\(\sigma_{\text{ext}}=\sigma_{\text{sct}}+\sigma_{\text{abs}}\), unit: Mm\({}^{-1}\)), which depends on the particulate composition with respect to particle size [81]. Figure 2: Scatter plots between hourly \(\tau_{AERONET}\) and \(\tau_{MODIS}\) for (**a**) April, (**b**) May, (**c**) June, (**d**) July, (**e**) August, (**f**) September, and (**g**) all months. Their statistical analysis of R (correlation coefficient), S (slope), N (number of data), MB (mean bias), NMB (normalized mean bias, %), and LOA (index of agreement) was presented. With consideration of all modeled particulate species, the CMAQ-derived AOD was calculated by integrating the aerosol extinction coefficient vertically from the surface to the top-of-atmosphere:  
\[\tau_{CMAQ}=\sum\limits_{i=1}^{N}(\sigma_{sct}+\sigma_{abs})\Delta z_{i}=\sum \limits_{i=1}^{N}(\sigma_{ext})\Delta z_{i} \tag{3}\]  
where, \(\Delta z_{i}\) and N are layer thickness at each layer and 17 layers, respectively. The AOD was calculated using the extinction coefficients estimated from the empirical \"reconstructed mass-extinction method\" as following Equations (4) and (5). We used the formula of [PERSON] et al. [82] instead of the original Interagency Monitoring of PROtected Visual Environments (IMPROVE) extinction equation because of the possibly significant contributions of sea salt to extinction over the oceans [82; 83]. Light scattering by sea salt aerosol was not considered in the original IMPROVE formula. Additionally, our domain covers a wide range of ocean, as shown in Figure 1.  
\[ \begin{array}{ll}\sigma_{ext}(Mm^{-1})&\approx\sum\limits_{s=1}^{N}\pi_{s,dry }f_{s}(RH)[C_{s}]\\ &\approx 2.2\times f_{F}(RH)\times\left[(NH_{4})_{2}SO_{4}\right]_{F}+4.8 \times f_{L}(RH)\times\left[(NH_{4})_{2}SO_{4}\right]_{L}\\ &+2.2\times f_{F}(RH)\times\left[NH_{4}NO3\right]_{F}+4.8\times f_{L}(RH) \times\left[NH_{4}NO3\right]_{L}\\ &+2.8\times\left[OM\right]_{F}+6.1\times\left[OM\right]_{L}\\ &+10.0\times\left[EC\right]_{F}\\ &+1.7\times f_{SS}(RH)\times\left[SS\right]\\ &+0.6\times\left[CORS\right]+0.6\times\left[SOIL\right]\end{array}  \tag{4}\]  
\[ \begin{array}{ll}f_{F}(RH)=f_{L}(RH)=f_{SS}(RH)=a_{1}\times\frac{RH}{1-RH} +a_{2}\text{ for }RH\geq b_{1}\\ f_{F}(RH)=f_{L}(RH)=f_{SS}(RH)=1\text{ for }RH<b_{1}\end{array}  \tag{5}\]  
where,\(f_{s}(RH)\) denotes the hygroscopic growth factor for ammonium sulfate, ammonium nitrate, and sea salt as a function of relative humidity (RH, %). Specifically, \(f_{s}(RH)\) is classified into \(f_{F}(RH),f_{L}(RH)\), and \(f_{SS}(RH)\) for fine, large, and sea salt particles, respectively. The \(f_{s}(RH)\) is also function of coefficients of the \(a_{1}\) and \(a_{2}\) in Equation (5), which are 0.4175 and 1.2312 for fine aerosol (\(F\)), 0.2381 and 1.3276 for large aerosol (\(L\)), and 0.2913 and 2.3308 for sea salt (\(SS\)). The reference relative humidity (\(b_{1}\)) is 37% for fine and coarse aerosol and 46% for sea salt. C\({}_{s}\) represents the concentrations of species (s), which are organic mass (OM), element carbon (EC), sea salt (SS), coarse mass (CORS), and soil-derived mass (SOIL), respectively. \(\pi_{S,dry}\) is the dry mass extinction coefficients (m\({}^{2}\) g\({}^{-1}\)) of the species (s). Here, we used the constant values of \(\alpha_{S,dry}\) for ammonium sulfate, ammonium nitrate, sea salt, and organic mass calculated from the Mie theory at 550 nm under the dry condition [82]. Many investigators have used the data assimilation technique by combing modeling and observations to produce an optimal estimated AOD of the evolving state of the system [38; 42; 84]. We performed the data assimilation every hourly using the OI (optimal interpolation) method expressed as the following Equations (6)-(9). For the data assimilation, the MODIS data were collected every hour within a 30 min in time differences.