## 3 Algorithm Description
### Optimal Estimation  
Based on the theoretical relationship between measured values (\(\mathbf{y}\)), retrieval variables (\(\mathbf{x}\)) and the uncertainties of the observations and the estimation of a priori values (\(\mathbf{x_{a}}\)), the optimal solution, \(\mathbf{x}\), can be described as follows:  
\[\mathbf{x}=\mathbf{x_{a}}+\mathbf{G}(\mathbf{y}-\mathbf{K}\mathbf{x_{a}}), \tag{3}\]  
where \(\mathbf{K}\) is the Jacobian and represents the sensitivity of measured values to the state vector as \(\text{dy}/\text{dx}\). \(\mathbf{G}\) is the gain matrix, which is defined as  
\[\mathbf{G}=\mathbf{S_{a}K^{T}}\Big{(}\mathbf{K}\mathbf{S_{a}K^{T}}+\mathbf{S_ {\varepsilon}}\Big{)}^{-1} \tag{4}\]  
Figure 6: Example of LUT used to estimate a priori AOD and SSA for three different aerosol types: (**a**) HAF, (**b**) Dust and (**c**) NA. The \(x\)-axis represents the normalized radiance at 388 nm and the \(y\)-axis represents the ratio between the radiances at 354 nm and 388 nm.  
and indicates the sensitivity of the retrieval variable to observations taking into account the errors in a priori state (\(\mathbf{S_{a}}\)) and observations (\(\mathbf{S_{\varepsilon}}\)) [22].  
To find the optimal solution, [PERSON] [22] proposed OE theory, which provides a solution that optimizes the balance between the measured spectrum and one that is derived from a priori states by minimizing the following cost function (\(\mathbf{c}\)):  
\[\mathbf{c}=(\mathbf{y}-\mathbf{F(x)})^{T}\mathbf{S_{\varepsilon}}^{-1}( \mathbf{y}-\mathbf{F(x)})+(\mathbf{x}_{a}-\mathbf{x})^{T}\mathbf{S_{a}}^{-1}( \mathbf{x}_{a}-\mathbf{x}). \tag{5}\]  
On each iteration \(\mathbf{i}\), \(\mathbf{c}\) is recalculated by updating \(\mathbf{x_{i}}\) using the Levenberg Marquardt equation,  
\[\Big{(}(\mathbf{1}+\mathbf{\gamma})\mathbf{S_{a}}^{-1}+\mathbf{K_{i}}^{-1} \mathbf{S_{\varepsilon}}^{-1}\mathbf{K_{i}}\Big{)}\mathbf{d}\mathbf{x_{i+1}}= \Big{[}\mathbf{K_{i}}^{T}\mathbf{S_{\varepsilon}}^{-1}(\mathbf{y}-\mathbf{F( x_{i})})+\mathbf{S_{a}}^{-1}(\mathbf{x}_{a}-\mathbf{x_{i}})\Big{]}, \tag{6}\]  
where \(\mathbf{F(x_{i})}\) is the radiance simulated from a priori values and \(\mathbf{d}\mathbf{x_{i+1}}\) represents the change of the state vectors between iteration steps (i.e., \(\mathbf{x_{i+1}}-\mathbf{x_{i}}\)). To control the width of \(\mathbf{d}\mathbf{x_{i+1}}\), the Levenberg parameter (\(\mathbf{\gamma}\)) is applied as a damping factor, which is adjusted at each iteration. In this study, \(\mathbf{\gamma}\) is adjusted from 1.0. \(\mathbf{K}\) and \(\mathbf{F(x_{i})}\) are calculated using VLIDORT.  
After solving for \(\mathbf{d}\mathbf{x_{i+1}}\), a conversion test is conducted to assess the reduction of the cost function by calculating the following quantity:  
\[\mathbf{R}=(\mathbf{c_{i}}-\mathbf{c_{i+1}})/(\mathbf{c_{i}}-\mathbf{c_{i+1, FC}}), \tag{7}\]  
where \(\mathbf{c_{i}}\) and \(\mathbf{c_{i+1}}\) are the cost functions calculated from Equation (5) for each iteration \(\mathbf{i}\) and \(\mathbf{i+1}\) and \(\mathbf{c_{i+1,FC}}\) is calculated assuming that \(\Figure 7: Example of retrieved (a) AOD, (b) SSA and (c) ALLH. The upper panels show the products of the proposed algorithm and the lower panels show the (d) AOD and (e) SSA of the OMI near-UV algorithm and an (f) RGB image from MODIS on 8 April 2006.