## 4 Sensitivity Analysis of Optical Measurement Signals to Characteristic Parameters in ASDs  
Because the inverse accuracy of ASDs is relevant to the information obtained from the optical measurement signals, i.e., spectral transmittance signals or angular light-scattering signals, it is necessary to optimize the measurement signals.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Function Types**} & **Parameters** & **True Values** \\ \hline \multirow{2}{*}{Monomodal} & L-N & \((\overline{D},\,\sigma)\) & \((1.0,\,2.99)\) \\ \cline{2-4}  & Gamma & \((\alpha,\,\beta)\) & \((6.0,\,3.0)\) \\ \hline \multirow{2}{*}{Bimodal} & L-N & \((\overline{D}_{1},\,\sigma_{1},\,\overline{D}_{2},\,\sigma_{2},\,n)\) & \((2.5,\,1.\,1.\,6.0,1.2,0.2)\) \\ \cline{2-4}  & Gamma & \((\alpha_{1},\,\beta_{1},\,\alpha_{2},\,\beta_{2},\,n)\) & \((6.0,\,2.\,0.8,2.0,0.15)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The true values of characteristic parameters of ASDs.  
Figure 1: Spectral optical constants of four different kind of aerosols [25; 26].  
Generally, to improve retrieval accuracy, the optical measurement signals are usually selected irrelevant to each other and sensitive to changes of inverse parameters. With this idea, the sensitivity analysis of optical measurement signals to unsolved characteristic parameters in ASDs at a different measurement wavelength or angle is employed to find the optimal measurement wavelength selection region for SEM and optimal measurement angle selection region for ALSM. The sensitivity coefficient is one of the most important characteristic parameters in the sensitivity analysis, which is the first derivative of the optical measurement signals, i.e., \(\tau(\lambda)\) or \(I(\theta)\), to a certain inverse characteristic parameter. The sensitivity coefficient at every wavelength or angle \(\chi_{a}(\eta)\) is defined as [27]:  
\[\chi_{a}(\eta)=\frac{\partial\phi(a,\eta)}{\partial a}=\frac{\phi(a+\Delta a, \eta)-\phi(a-\Delta a,\eta)}{2\Delta a}\,,\;\phi=\tau\,\text{or}\,I,\;\eta= \lambda\,\text{or}\,\theta \tag{11}\]  
where a denotes the inverse characteristic parameters in ASDs; \(\Delta\) represents a tiny change and is set at 0.5% in this study, \(\Delta\lambda\), \(\Delta\theta>0\); \(I(\theta)\) is the angular light-scattering measurement signals; \(\tau(\lambda)\) is the spectral transmittance obtained in the SEM. It is obvious that to improve the retrieval accuracy, the optical measurement signals with larger absolute values of sensitivity coefficient should better be selected.  
Figures 2 and 3 depict the sensitivity coefficient for different monomodal ASDs with respect to spectral transmittance signals and angular light-scattering signals, respectively. From Figure 2, it can be found that increasing the measurement wavelength will reduce the sensitivity coefficient in general, so to ensure the retrieval accuracy, the measurement wavelengths in this study are selected from \([0.2,\,1.4]\,\upmu\)m. From Figure 3, it is obvious that the value of the sensitivity coefficient will decline with an increase in the measurement angles. According to References [28; 29], for a small angle, the scattered intensities are weakly dependent upon the particle's composition but strongly dependent upon the particle's roughness and size, so the measurement angles in this study are selected from \([0,30]\) deg to obtain more accurate retrieval results. Similar conclusions can be obtained in studying bimodal ASDs, and the measurement wavelengths and angles are also selected from \([0.2,\,1.4]\,\upmu\)m and \([0,30]\) deg, respectively. The details in studying the bimodal ASDs are not shown here and interested readers can contact us.  
Figure 2: _Cont._