## 5 Uncertainty Estimation
### Estimation of Model Parameters  
While the inversion error is related to uncertainty caused by the non-linearity of radiation between LUT entries, the model parameter error is related to the accuracy of input parameters, which affect the simulated spectrum. In the UV-visible algorithm, the model parameters include surface reflectance and aerosol properties, which are defined by the size distribution and refractive index. Because results are obtained after solving the Levenberg Marquardt equation, the inversion error is also included in the estimated uncertainty but the sensitivity to the a priori ALH assumption is removed by assuming the reference ALH condition. The errors were estimated using the developed algorithm with assumptions of surface reflectance or aerosol type that differed from the conditions used to calculate the synthetic radiation. Sensitivity to surface reflectance errors was estimated for four error conditions with surface reflectance deviations of \(-40\%\), \(-20\%\), \(20\%\) and \(40\%\) from 0.06, which was assumed for the radiance simulation. The sensitivity to aerosol type was tested by selecting the wrong aerosol type during the retrieval process. For each set of error conditions, 108 retrievals for each set of AOD and SSA conditions and 81 retrievals for each ALH were applied, as described above. The sensitivity to surface reflectance error is first analyzed by applying systematic errors within \(\pm 40\%\) and the results are shown in Figure 12 and listed in Table 5. The error range is defined based on the error range of the OMLER product reported by [PERSON] et al. (2018). Their study revealed that an overall 2% error in the TOA reflectance causes 40% errors in LER at 320 nm and 5% errors at 500 nm. Assumptions related to surface pressure, total ozone concentration and the bidirectional reflectance distribution function effect can also increase the LER error. Figure 12 shows the distribution of retrieval errors in retrieval products as a box-and-whisker plot and the median values are summarized for each reference condition in Table 5. While the inversion error in AOD is less than \(-0.01\) for dust (see previous section), the median value of retrieval errors increases to 0.25 when a \(-40\%\) error is applied (Figure 12a). In general, underestimation of surface reflectance results in an overestimation of aerosol extinction along the optical path and thus leads to an overestimation of AOD. This effect is most significant when the AOD is 0.3 and decreases with increasing AOD (Table 5). The error is most significant for NA aerosols (light blue colored box in Figure 12a), because of different sensitivities of the normalized radiance to changes in AOD for the various aerosol types. In the aerosol model used in this study, the radiance change related to the increase of NA AOD is lower than that of the absorbing AODs. Therefore, a higher AOD is required to offset the underestimation of surface contributions to radiance under NA aerosol conditions. Similarly, an underestimation of surface reflectance is expected to lead to an overestimation of SSA. However, because the OE finds the best solution to fit the measured spectra by updating state vectors simultaneously, the results of a reference test cannot be simply interpreted. Figure 12b shows that the retrieval errors in SSA due to surface reflectance errors are not much higher than the effects of inversion errors for absorbing particles. The SSA errors under NA conditions are significantly higher than those for other types and have a wider range. Because the contribution of aerosol particles to upward radiance depends on AOD rather than SSA, the effects of surface reflectance uncertainty are mostly offset by AOD. The overestimation of surface reflectance results in a larger error in SSA than the underestimation in general, because the sensitivity to aerosol extinction decreases as the surface reflectance approaches the critical reflectance ([PERSON] et al., 2018).