# Optimal Estimation Retrieval of Aerosol Fine-Mode Fraction from Ground-Based Sky Light Measurements
## 4 Results  
### Retrieval from Synthetic Data  
For the proposed OE-based algorithm, we paid attention to three issues: The dependence on a priori assumptions, the measurement error influence, and the reliability of the aerosol model parameters [51]. Among these three aspects, if reasonable a priori estimates and a priori knowledge are available, observation error is the main source that affects the inversion accuracy. Thus, with the support of relatively accurate a priori knowledge from AERONET products, we mainly made a test on the inversion algorithm by introducing various observation errors. The purpose was to verify the stability of the inversion algorithm and the confidence in the retrieved results. Firstly, simulations of sky scattering radiation measurements were performed by UNL-VRTM. The viewing geometries, including solar zenith angle (SZA), view zenith angle (VZA), and relative azimuth angle (RAA), are listed in Table 4. Different aerosol AOD from 0.1-3.0, with \(\mathrm{FMF_{0}}\) varying from 0.1-0.95, were considered in the simulation. Secondly, to simulate the measurement error, a 5% relative noise following a Gaussian distribution was introduced to the simulation results. After that, the optimal estimation retrieval was performed on the synthetic data superimposed with noise. Figure 4 shows the scatterplots between the inversion results and the true value at 550 nm. The left panel represents the scenario without noise, and the right panel represents the scenario with 5% random noise. The correlation coefficients in the left panel were both greater than 0.99, which indicated a good self-consistency of the OE algorithm. As seen in Figure 4b, the uncertainty of the retrieved AOD increased significantly with the aerosol loading. The mean inversion error was about 4.5% when AOD was less than 2.0, while the mean inversion error was up to 7.76% when AOD was greater than 2.0. As seen in Figure 4d, the mean inversion error of \(\mathrm{FMF_{0}}\) was 4.36%. Evidently, the aerosol retrieval from a single-viewing observation was susceptible to measurement error. Even so, the uncertainties of the inversion results were of the same order of magnitude as the observation uncertainties. In summary, the developed algorithm showed good performance, regardless of the high aerosol loading situation. The simulated radiances, cost function, and gradient vector for each iteration during the retrieval progress are shown in Figure 5. Four cases at different aerosol loadings (AOD = 0.2 and AOD = 1.0 at 550 nm) and fine-mode fraction (\(\mathrm{FMF_{0}}\) = 0.2 and \(\mathrm{FMF_{0}}\) = 0.8) were selected, in which \(\mathrm{FMF_{0}}\) = 0.2 and \(\mathrm{FMF_{0}}\) = 0.8 represented coarse-dominated and fine-dominated aerosol, respectively. As seen in Figure 5a, the model results gradually approached the true value with the iteration. A more than 99% reduction in cost function was achieved, as seen in Figure 5b,c. The Gradient of \(V_{0}\) and \(\mathrm{FMF_{v}}\) approached zero after convergence, as shown in Figure 5d. The details of the inversion process indicated that the OE algorithm had good performance in obtaining the optimal results. Furthermore, the single-viewing multi-spectral observations provided sufficient information to contain the retrieval of \(\mathrm{FMF_{v}}\) theoretically. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Setting** \\ \hline Solar zenith angle (SZA) & 60\({}^{\circ}\) \\ View zenith angle (VZA) & 0\({}^{\circ}\) (vertical upward observation) \\ Relative azimuth angle (RAA) & 0\({}^{\circ}\) (solar principal plane) \\ Aerosol optical depth (AOD) at 550 nm & from 0.1–3.0 \\ Fine-mode fraction (\(\mathrm{FMF_{0}}\)) at 550 nm & from 0.1–0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The parameter settings for the simulation of analog data. Figure 4: The scatter plot between retrieved results and true values at 550 nm, for eighteen different aerosol loadings from AOD = 0.1 to 3.0 and ten different mixed states from \(\mathrm{FMF_{o}}\) = 0.1 to 0.95.