## 6 Sensitivity Analysis of Cloud Droplet Effective Radius Retrievals  
To assess the realistic errors in cloud droplet effective radius retrievals that can occur, a rigorous sensitivity analysis is performed focusing on errors that may result due to incorrect atmosphere/aerosol  
Figure 8: Regression plots between LWP derived in different ways (**a**) Closure comparison between averaged measured and derived LWP, and (**b**) Averaged _versus_ Instantaneous LWP. assumptions below the cloud as well as imprecise surface albedo. Other variables such as geometric cloud base and cloud extent are less important and will not be considered here. Also, it is not possible to analyze all mechanisms individually since the inversion of each variable taken separately is not stable. Hence, we adopt the following approach for performing the sensitivity analysis of retrieved Ref:  
1. Fix the \"reference\" state (mean atmosphere, mean surface) along with a reference cloud (COD\({}_{\text{ref}}\), Ref\({}_{\text{ref}}\)) to provide the algorithm with reference values for LWP\({}_{\text{ref}}\) and Tdiff\({}_{\text{ref}}\). 2. Given the reference state, we now perform the retrieval sequentially for each atmosphere state, \(i\). 3. The retrieval looks for the solution set which simultaneously satisfies \[\left\|{LWP_{i}(COD,R_{\text{eff}})-LWP_{\text{ref}}}\right\|<30\,\text{g}m^ {-2}\ (\text{LWP error})\] (12) \[\text{and}\ \left\|{Tdiff_{i}(COD,R_{\text{eff}})-Tdiff_{\text{ref}} \right\|<0.05\,\text{*}\,Tdiff_{\text{ref}}}\right.\] (13)
4. The above inequalities define a set of COD and R\({}_{\text{eff}}\) cells in the inversion domain which are possible solutions. However, all such solutions should not be weighted equally since some of the solution cells fit the measurement better than others. 5. For each cell in the set, COD(k,l) and R\({}_{\text{eff}}\)(k,l), we assign a weight \(w_{k,l}^{i}\) which measures the discrepancy between the reference measurement (LWP\({}_{\text{ref}}\), Tdiff\({}_{\text{ref}}\)) and the model (for the _i_th atmosphere) evaluated at the cell (k,l). In other words, we weight the cell by the reciprocal distance the cell model is from the reference measurement in \"measurement\" space given by \[\left(w_{k,l}^{i}\right)^{-1}=\left\|{LWP_{i}(COD(k),R_{\text{eff}}(l))-LWP_{ \text{ref}}}\right\|^{2}+\left\|{T_{diff}^{i}(COD(k),R_{\text{eff}}(l))-T_{ diff}^{\text{ref}}}\right\|^{2}\] (14)
6. This process is performed sequentially for all atmosphere cases and the cumulative weights (summed over _i_) result in the 2D distribution of the admissible (COD, R\({}_{\text{eff}}\)) retrieval: \[\left(w_{k,l}\right)=\sum_{i}\left(w_{k,l}^{i}\right)\] (15)
7. Exactly, the same procedure is repeated for the surface albedo variability, which results in another 2D distribution function for this process. 8. In order to combine the uncertainty from 2 separate processes each taken independently, the result is just the 2D convolution of the individual 2D probability density functions (PDFs) of each process taken separately and can be implemented using a suitable 2D convolution function. 9. Once the combined PDF is generated, the final error associated with either parameter is the marginal distribution given by, \[P(R_{\text{eff}})=\int\limits_{\stackrel{{\sim}}{{\to}}}^{\infty}P( COD,R_{\text{eff}})d(COD)\] (16)
10. Finally, given the 1D marginal distributions, the mean and standard deviation can be obtained directly from the appropriate numerical integrals over the 1D distributions \[\overline{R_{\text{eff}}}=\int\limits_{\stackrel{{\sim}}{{\to}}} (R_{\text{eff}})P(R_{\text{eff}})d(R_{\text{eff}})\] (17) \[std\left(R_{\text{eff}}\right)=\sqrt{\int\limits_{\stackrel{{ \sim}}{{\to}}}^{\infty}\left(R_{\text{eff}}-\overline{R_{\text{eff}}} \right)^{2}P(R_{\text{eff}})d(R_{\text{eff}})}\] (18)In performing this sensitivity analysis, we fix the \"reference\" state of cloud properties corresponding to those values which are most likely be influenced by aerosols within the aerosol-cloud interaction.