## 3 Retrieval Method
### The State Vector and A Priori Uncertainty  
In our analysis, the state vector \(\mathbf{x}\) consists of aerosol optical parameters and surface parameters. The real and imaginary parts of the refractive index at the reference wavelength and the a priori of the mode radius and variance for lognormal PSD corresponding to the four mono-modal aerosol types are taken from the Optical Properties of Aerosols and Clouds (OPAC) database [45]. The land surface model parameter c (the fraction of vegetation cover) is also included in the state vector. Generally, the state vector includes six parameters: AOD, the real part and the imaginary part of the refractive index, PSD mode radius and variance and the fraction of vegetation surface albedo. The aerosol loading profile is assumed to take a Gaussian-shaped distribution, which is described by two parameters: the peak height and the half width of the profile. The AOD at the reference wavelength, aerosol profile parameters and surface parameters are consistent in simulations for each aerosol type. Table 2 lists the a priori and corresponding a priori uncertainties (square-root of the variance) for all state vector elements. To describe the fast and dramatic change of aerosol due to human activity, the a priori uncertainty of AOD is assumed 100%. The real part of refractive index of each type of aerosols has a small difference and changes little; thus, an assumption of 0.15 a priori uncertainty is available for each aerosol type. On the contrary, the uncertainty of the imaginary part of the refractive index is more difficult to estimate and is assumed large. The values are based on a previous study [38] and change with aerosol type. Similarly, the a priori uncertainties of the PSD mode radius and variance also follow the assumptions in [38]. Considering the relatively accurate estimation about the surface albedo, the a priori error of vegetation fraction is 0.2.  
The measurement vector \(\mathbf{y_{s}}\) for each viewing geometry is established using simulated (synthetic) CAPI measurements, including radiances \(\mathbf{I}\) for the five CAPI bands and polarization quantities \(\mathbf{Q}\) and \(\mathbf{U}\) from Bands 2 and 5 (Table 1). Thus, the measurement vector for single-angle viewing is:  
\[\mathbf{y_{s}}=[\mathbf{I_{band1}},\mathbf{I_{band2}},\mathbf{Q_{band2}}, \mathbf{U_{band2}},\mathbf{I_{band3}},\mathbf{I_{band4}},\mathbf{I_{band5}}, \mathbf{Q_{band5}},\mathbf{U_{bands5}}]^{T}. \tag{11}\]  
To improve retrieval information using observations at multiple angles, we extend the measurement vector by concatenating individual measurements:  
\[\mathbf{y}=[\mathbf{y_{s1}}^{T},\mathbf{y_{s2}}^{T},\mathbf{y_{s3}}^{T},\cdots ]^{T} \tag{12}\]  
where the subscripts s1, s2, s3,... indicate observations at different viewing angles. Jacobians for the various bands and viewing geometries are concatenated in the same way.  
In this study, we consider an ideal scenario and ignore calibration errors and systematic errors. The measurement noise is assumed to follow the SNR of each channel (Table 1). When considering the multi-angle observation, the instrument noise is accumulated in the same way as the measurement vector.