## 6 Error Analysis and Correlation Matrix for Aerosol Retrieval
### Retrieval Error and Its Components  
From Figures 4 and 5, it can be seen that the patterns of posterior errors and the corresponding smoothing errors are similar for all aerosol types. This implies that the a priori uncertainties rather than measurement noise are the dominated component of the posterior error. We also note that among all retrieved parameters, the imaginary part of the refractive index and the PSD mode radius have the largest posterior errors in nadir mode, i.e., 78% and 70% on average. Therefore, the retrieval accuracy of these two parameters cannot be guaranteed. Actually, for aerosols with little absorption, i.e., sea salt and sulfate, the DFS of the imaginary part of the refractive index are close to zero (Figure 3c,d), and the posterior errors are almost 100% (Figure 4l,p and Figure 5l,p); therefore, this parameter cannot be retrieved from measurements. In other words, its information is mainly from a priori, and the measurement noise has little effect on this parameter (Figure 4j,n and Figure 5j,n).  
In addition to a posteriori error, measurement noise and smoothing errors, forward model parameter uncertainties from two aerosol profile parameters are also investigated in Figures 4 and 5. The apparent low values of these errors for sea salt and sulfate indicate that the inaccuracy in depicting the shape of the aerosol profile has little impact on the retrieval uncertainty for these two aerosols. The corresponding model parameter errors for dust and soot are larger, mainly because radiation in the UV band is sensitive to the height of aerosols with absorption.  
Furthermore, we also note that in Figure 4, when the scattering phase function is larger at certain observation angles, lower retrieval errors are derived. However, purely noise-related errors are more random. From Figure 5, it is found that the use of three measurements in retrieval can effectively reduce posterior and smoothing errors, especially for those parameters with large errors. The posterior error of the PSD mode radius is reduced the most (23% on average), while the least improvement is achieved for the real part of the refractive index (less than 1%). Among all of the aerosols, the improvement of the multi-angle mode is most apparent for dust (76%), in relation to the large variability of its scattering phase function over the range of our sampled scattering angles. Similarly, the least improvement is achieved for soot (only 10%) due to both the small scattering property and the comparatively uniform phase function.  
Figure 4: Different error estimates (1\(\sigma\)) of aerosol parameters at different observation geometries for all aerosols. Each row represents one type of aerosol: dust (**a**–**d**), soot (**e**–**h**), sea salt (**i**–**l**) and sulfate (**m**–**p**), respectively. The range of ordinate represents percentage.  
Figure 5: Different error estimates (1\(\sigma\)) of aerosol parameters at different solar zenith angles in nadir and multi-angle mode. Others are the same as Figure 4.  
If another two measurements (forward VZA 30\({}^{\circ}\) and backward VZA 30\({}^{\circ}\)) are added in multi-angle retrieval, the retrieval errors will be reduced to a greater extent. Figure 6 shows that smoothing errors, and posterior errors are all improved again when five measurements are used in the retrieval. Measurement noise is also reduced, except for soot. Thus, adding more measurements in aerosol retrieval is not an efficient method to improve measurement noise for soot. For dust, which had the greatest improvement in multi-angle retrieval, the smoothing errors are less than 0.1% and almost can be ignored; meanwhile, the average posterior errors are as low as 3% when the number of measurement angles reaches five. The posterior errors for soot, sea salt and sulfate decrease by 4%, 6% and 2% on average, respectively. Finally, we find that a larger number of retrieval measurements can result in less smoothing and posterior errors, impacting on aerosols, especially dust.