# First Lidar Campaign in the Industrial Sites of Volta Redonda-RJ and Lorena-SP, Brazil
## 3 Lidar Retrieval Methodology  
The expected elastic return signal from a pure molecular atmosphere can be written as  
\[P_{m}(\lambda,z) \propto \frac{1}{z^{2}}\;\beta_{mol}(\lambda,z)\;\times\exp\left[-2\int_{ 0}^{z}\alpha_{mol}(\lambda,z^{{}^{\prime}})dz^{{}^{\prime}}\right], \tag{1}\]  
where \(\lambda\) is the wavelength of the radiation, \(z\) is the altitude above ground level (a.g.l. ), and \(\alpha_{mol}(\lambda,z)\) and \(\beta_{mol}(\lambda,z)\) are the molecular extinction and backscatter coefficients. The molecular extinction coefficient is computed as  
\[\alpha_{mol}(\lambda,z)=N_{s}\,\sigma_{ray}(\lambda,CO_{2})\;\frac{P(z)/T(z)}{ P^{std}/T^{std}}, \tag{2}\]  
where \(N_{s}\) is the molecular number density for standard atmosphere (\(2.5469\times 10^{25}\) m\({}^{-3}\)), \(\sigma_{ray}(\lambda,CO_{2})\) is the total Rayleigh scattering cross section per molecule, written as a function of the wavelength and the \(CO_{2}\) concentration factor \(CO_{2}\), and \(P(z)\) and \(T(z)\) are the pressure and temperature for a given altitude. For calculation of the molecular backscatter and extinction atmospheric components, we considered the same approach done by [PERSON] et al. [55], taking into account the \(CO_{2}\) and total water vapour concentration (which is considered small and does not significantly affect the above calculations). For \(\lambda=532\) nm and \(CO_{2}=392\) ppm, the Rayleigh cross section is \(\sigma_{ray}=5.1706\times 10^{-31}\) m\({}^{2}\), considering the King correction factor calculated as given in [55; 56; 57] in order to correct for dispersion and scattering of light due to the effects of molecular anisotropy. For these calculations, we used the depolarization values of the gases \(O_{2}\), \(N_{2}\), \(CO_{2}\) and \(Ar\) according to [PERSON] [57]. In addition, the molecular backscatter coefficient is obtained by  
\[\beta_{mol}(\lambda,z)=\frac{\alpha_{mol}(\lambda,z)}{4\pi}\;P_{ray}(\theta, \lambda,CO_{2}), \tag{3}\]with the depolarization function  
\[P_{ray}(\theta,\lambda,CO_{2})=\frac{3}{4(1+2\gamma)}\Big{[}(1+3\gamma)+(1-\gamma) \cos^{2}\theta\Big{]}, \tag{4}\]  
where \(\gamma=\rho_{n}/(2-\rho_{n})\) and \(\rho_{n}\) is the depolarization factor [56]. The proportionality factor of Equation (1) can be obtained by comparison with the background (\(\mathcal{BG}\)) subtracted elastic signal, \(P(\lambda,z)~{}-~{}\mathcal{BG}\). Both the proportionality factor \(\mathcal{A}\) and the background \(\mathcal{BG}\) can be obtained by performing the linear fit,  
\[P(\lambda,z_{0})=\mathcal{A}\,P_{m}(\lambda,z_{0})+\mathcal{BG}. \tag{5}\]  
Notice that the linear relation is only true if no aerosols are present in the region around \(z_{0}\) selected for the regression. While this could indicate that a region of very high altitude would be a better one to perform the fit, the decrease of the signal-to-noise ratio of the lidar measurement as a function of altitude makes difficult the choice of the best region. In this work, the regions chosen to perform the fit were the ones that produced a good quality fit and did not result in non-physical solution for the aerosol backscatter coefficient. By defining the range corrected signal (RCS) as  
\[S(z)=[P(\lambda,z)-\mathcal{BG}]\times z^{2}, \tag{6}\]  
we can compare the scaled molecular simulated signal with the RCS, \(S(z_{0})\), to check if the lidar apparatus was properly aligned. The expected lidar return when aerosols are present in the atmosphere is  
\[P(\lambda,z)~{}~{}\propto~{}~{}\frac{1}{z^{2}}\,\beta(\lambda,z)~{}\times\exp \left[-2\int_{0}^{z}\alpha(\lambda,z^{{}^{\prime}})dz^{{}^{\prime}}\right], \tag{7}\]  
with \(\beta=\beta_{mol}+\beta_{ear}\) and \(\alpha=\alpha_{mol}+\alpha_{ear}\), where \(\beta_{ear}\) and \(\alpha_{ear}\) are the aerosol backscatter and extinction coefficients. The retrieval of aerosol profiles was performed by applying the Klett-Fernald-Sasano (KFS) inversion method ([28]).