## 4 Discussion
### Uncertainty Analysis and Error Propagation  
The quantification of lidar uncertainties by error propagation is necessary for evaluating the quality of retrieved lidar products such as aerosol backscatter or number concentration. The purpose here is to evaluate the error propagation during the fog-oil event. The uncertainties in the lidar products result from uncertainties in the pre-processing, i.e., DC and BG, and in the geometric and radiometric calibration, i.e., O(r) and K\({}_{0}^{*}\).  
We assess in the following the total uncertainties by error propagation in the different lidar products presented along the manuscript, namely attenuated backscatter U(r), backscatter profiles \(\beta_{\text{aer}}(\text{r})\), and aerosol number concentration \(\text{N}_{\text{p}}(\text{r})\). The noise of the variables is assumed uncorrelated, but the variables have a certain degree of dependency since calibration parameters were measured using the same instrumental setup. The measurements were not performed under the same exact experimental conditions (i.e., statistical and ambient changes could be considered over time); however, the use of the same Lambertian surface to retrieve the calibration parameters was found to be a stable and reliable method.  
The lidar uncertainties are estimated by error propagation for instantaneous RCS profiles by deriving the Equations (15), (21) and (24), as  
\[\delta\text{RCS}(\text{r})=\left[\frac{\delta\text{DC}}{\langle\text{DC} \rangle}+\frac{\delta\text{BG}}{\langle\text{BG}\rangle}\right]\text{RCS}( \text{r}) \tag{25}\]  
where \(\delta\)DC and \(\delta\)BG are the uncertainties associated to the pre-processing quantities DC and BG.  
\[\delta\text{U}(\text{r})=\left[\frac{\delta\text{RCS}}{\langle\text{RCS} \rangle}+\frac{\delta\text{K}_{\text{o}}^{*}}{\langle\text{K}_{\text{o}}^{*} \rangle}+\frac{\delta\text{O}(\text{r})}{\langle\text{O}(\text{r})\rangle} \right]\text{U}(\text{r}) \tag{26}\]  
\[\delta\beta(\text{r})=\left[\left(1+2\text{LR }\beta(\text{r})\text{U}(\text{r})/ \text{U}^{\prime}(\text{r})\right)\frac{\delta\text{U}(\text{r})}{\text{U}( \text{r})}\right]\ \beta(\text{r}) \tag{27}\]  
Figure 7: Single attenuated backscatter profiles measured by the Colibri micro-lidar for fog-oil plumes at multiple time-resolution (1 ms, 10 ms, 20 ms).  
\[\delta\mathrm{N}_{p}(\mathrm{r})=\left[\frac{\delta\beta(\mathrm{r})}{\beta}\right] \ \mathrm{N}_{p}(\mathrm{r}) \tag{28}\]  
where \(\delta\mathrm{K}_{\mathrm{o}}^{*}\) and \(\delta\mathrm{O}(\mathrm{r})\) are the uncertainties associated to the calibration quantities \(\mathrm{K}_{\mathrm{o}}^{*}\) and \(\mathrm{O}(\mathrm{r})\). The fact that Equation (27) has a derivative term in the denominator may drive to some singularities in the \(\delta\beta\) calculation, but in any case can not be associated as error propagation of \(\mathrm{K}_{\mathrm{o}}^{*}\) or \(\mathrm{O}(\mathrm{r})\) over \(\beta(\mathrm{r})\) or \(\mathrm{N}_{p}(\mathrm{r})\).  
Figure 8 summarizes the error propagation results for typical micro-lidar products from Section 3. As a result, the total error propagation over \(\beta(\mathrm{r})\) and \(\mathrm{N}_{p}(\mathrm{r})\), considering both the pre-processing and calibration, leads to consider a mean relative mean error lower than 14%. This error analysis yields a significant insight into the robustness of the proposed calibration method for retrieving aerosol products in the short-range using the Colibri micro-lidar.