## 2 Materials and Methods
### Preprocessing of the Raw Lidar Data
Starting from the lidar equation  
\[S(z)=\frac{h}{2}\frac{C}{z^{2}}\frac{\beta(z)}{4\pi}exp[-2\int_{0}^{z}\sigma_{ ext}(z^{\prime})dz^{\prime}] \tag{1}\]  
we can express the signals on the two detectors, if we neglect the extinction and only consider the crosstalk as originating from the optical elements (polarizer and viewport), as  
\[S_{\parallel}(z)=g_{1}\frac{1}{z^{2}}\beta_{\parallel}(z)(1-CT)\quad\text{ and}\quad S_{\perp}(z)=g_{2}\frac{1}{z^{2}}\Big{(}\beta_{\perp}(z)+CT\beta_{ \parallel}(z)\Big{)} \tag{2}\]  
where \(g_{1}\) and \(g_{2}\) are the gain factors of the two detectors, and CT is the crosstalk from the parallel channel to the perpendicular channel. We neglect the crosstalk from perpendicular to parallel channel. In order to facilitate the interpretation of the signals and the detection of clouds, we divide both signals by the molecular backscatter coefficient and multiply by \(z^{2}\) and we get  
\[r_{\parallel}(z)=g_{1}\frac{\beta_{\parallel}(z)(1-CT)}{\beta_{\text{mol}}(z )}\quad\text{and}\quad r_{\perp}(z)=g_{2}\frac{(\beta_{\perp}(z)+CT\beta_{ \parallel}(z))}{\beta_{\text{mol}}(z)} \tag{3}\]  
The molecular backscatter coefficient was calculated using local temperature and pressure provided by radio soundings and, where these were not available, from the NCEP (National Centers for Environmental Predictions). We can normalize these expressions to 1, where no aerosols are present (typically between 26 and 30 km). The normalized expressions become  
\[r^{\prime}_{\parallel}(z)=\frac{r_{\parallel}(z)}{rn_{\parallel}}\quad\text{ and}\quad r^{\prime}_{\perp}(z)=\frac{r_{\perp}(z)}{rn_{\perp}} \tag{4}\]  
where\[rn_{\parallel}=g_{1}\frac{\beta_{\rm mol\parallel}}{\beta_{\rm mol}}(1-CT)\quad \text{and}\quad rn_{\perp}=g_{2}\bigg{[}\frac{\beta_{\rm mol\perp}}{\beta_{\rm mol }}+CT\frac{\beta_{mol\parallel}}{\beta_{\rm mol}}\bigg{]} \tag{5}\]  
With some algebra, we can now obtain expressions for \(R\) and \(\beta_{\perp}\)  
\[R(z)=\frac{(1-CT)}{1+\delta_{\rm mol}}\bigg{[}r_{\parallel}^{\prime}(z)+r_{ \perp}^{\prime}(z)\frac{\delta_{\rm mol}+CT}{(1-CT)}\bigg{]} \tag{6}\]  
and  
\[\beta_{\perp}(z)=\frac{\delta_{\rm mol}+CT}{1+\delta_{\rm mol}}\bigg{[}r_{ \perp}^{\prime}(z)-r_{\parallel}^{\prime}(z)\frac{CT}{\delta_{\rm mol}+CT} \bigg{]}\beta_{\rm mol}(z) \tag{7}\]  
where \(\delta_{\rm mol}=\beta_{\rm mol\perp}/\beta_{\rm mol\parallel}\). In our case, using an optical bandpass filter centered on the laser wavelength (532 nm) with an FWHM of 2 nm, \(\delta_{\rm mol}\) is 0.007 ([18]). Now, we can see that the crosstalk CT can be written as  
\[CT=\frac{\frac{\xi_{1}}{\beta_{2}}\frac{rn_{\perp}}{rn_{\parallel}}-\delta_{ \rm mol}}{1+\frac{\xi_{1}}{\beta_{2}}\frac{rn_{\perp}}{rn_{\parallel}}} \tag{8}\]  
The two parameters \(rn_{\parallel}\) and \(rn_{\perp}\) can be determined from the calibration process for aerosol-free regions, so we only need the ratio of the two gain constants of the two detection channels, which can be determined, e.g., by switching the detectors or by more sophisticated methods ([17]). After correcting the two channels for extinction and range, two optical parameters can be extracted, taking the crosstalk into account. The backscatter ratio \(R\), defined as the ratio of the total backscatter coefficient and the molecular backscatter coefficient, accounts for the scattering by particles (aerosol) with respect to molecular scattering. This implies that \(R=1\) if no aerosols are present. The second parameter accounts for the depolarization caused by solid particles and can be described by several parameters [19]. The linear volume depolarization is defined as the ratio of the perpendicular backscatter coefficient and the parallel backscatter coefficient, while the aerosol depolarization can be expressed as the ratio of the perpendicular backscatter coefficient and the parallel backscatter coefficient for aerosol only. We use the perpendicular backscatter coefficient \(\beta_{\perp}\) as a measure for depolarization to facilitate the comparison with CALIOP data.