## 5 Uncertainty Estimation
### Inversion Error  
The LUT approach reduces the computational burden but can lead to errors during the inversion process because of a limited set of entries. Figure 11 shows the inversion error as a function of the reference value. The retrieval error is defined as the difference between the retrieved value and the reference condition and is shown as symbols with error bars in Figure 11. The error bar indicates the 1-\(\sigma\) range of retrieval results together with mean values (symbols) for each aerosol type. Depending on SZA, AOD, SSA and ALH dimensions of simulation, the average and standard deviation were obtained from 108 retrievals for each set of AOD and SSA conditions and 81 retrievals for each ALH condition. The inversion error investigation starts from an a priori state retrieval with the assumption that all measurement conditions and model parameters are exactly estimated. Because the a priori states of AOD and SSA are estimated using the two-channel method in which ALH is assumed, the inversion error test is performed in two ways by changing the ALH assumption from the reference condition to an arbitrary value of 3 km. The results of the two simulations are shown in the upper and lower panels of the Figure 11.  
In general, the fitting errors in AOD and SSA are not significant. When the a priori ALH is assumed as the reference condition (Figure 11a), the average errors in AOD are \(-0.01\), \(-0.01\) and \(-0.01\) for HAF, dust and NA aerosols, respectively and a maximum error of \(-0.02\) is found for the NA type. While the average error for NA aerosols is relatively higher than the error for absorbing aerosols, the standard deviation for dust is generally higher than that for the other types. The inversion errors in  
Figure 11: Inversion error of the OE method for (**a,d**) AOD, (**b,e**) SSA and (**c,f**) ALH retrieval, without errors in model parameters. The a priori ALH is based on actual conditions in the upper panels but an a priori ALH of 3 km is assumed in the lower panels. The retrieval errors are estimated from 108-member ensembles of AOD and SSA retrieval and 81-member ensembles for ALH retrieval for a given set of reference conditions. Symbols indicate the aerosol type (diamonds: HAF, triangles: dust, squares: NA) and error bars represent the standard deviation of the smoothing error for the reference test dimension.  
SSA (Figure 11b) are below 0.001 on average and the deviation in SSA for each aerosol type is almost negligible. Although the magnitude of the error is smaller than that of AOD, it is not negligible due to the narrow range of SSA values, approximately 0.7 to 1.0. In comparison with AOD and SSA, ALH shows a wider range of retrieval errors from 0.19 to 0.27, 0.10 to 0.15 and 0.02 to 0.07 for HAF, dust and NA aerosols, respectively, because the sensitivity of the backscattered radiance to the ALH is relatively lower than the contribution of AOD and SSA.  
Figure 11d-f shows that changing the a priori ALH state increases the inversion error in the order ALH > SSA > AOD. Although changes in the average errors for AOD and SSA are insignificant, an increasing 1-\(\sigma\) range indicates that the error can increase depending on the measurement geometry. The large variation (1-\(\sigma\) range) for NA aerosols indicates that the retrieval accuracy increases with increasing absorption. Moreover, the ALH inversion error for NA aerosols is relatively higher compared with other types because high aerosol absorptivity increases the sensitivity to ALH. Figure 11f also shows that the retrieval error in ALH significantly increases with an increase of the difference between the reference ALH condition and the assumed a priori value. The algorithm uses the grid-averaged \(Z_{\text{aer}}\) value as the a priori state for ALH to take account of the spatial variation of ALH but temporal variations cannot be applied in real time. The uncertainty of the a priori state assumption is likely a major cause of the ALH retrieval uncertainty.