## 2 Methods
### Performance Simulation Using the Return Power Budget Method
#### 2.1.1 Return Power  
The computation of the return power takes into account molecular and aerosol backscattering for the case of the elastic lines [32] (355 nm, 532 nm) and the two N\({}_{2}\) Raman lines (387 nm and 607 nm). Average atmospheric density profiles [9] for the ORM were assumed, as well as average aerosol extinction profiles for the clear night or typical cases of nights affected by calima [21]. Angstrom exponents (\(k=-\ln\left(a_{\lambda_{1}}^{\text{aer}}/\alpha_{\lambda_{2}}^{\text{aer}} \right)/\ln(\lambda_{1}/\lambda_{2})\).) of \(k=1.45\) have been assumed for the clear night and \(k=0.32\) for the case of strong calima [13]. At full overlap, the return power budget \(P_{\lambda_{0}}(R)\) for the elastic lines as a function of range \(R\) and wavelength \(\lambda_{0}\), is then computed using the standard LIDAR equation [33]:  
\[P_{\lambda_{0}}(R)=\frac{E(\lambda_{0})\,c\,A}{2R^{2}}\Big{(}\beta_{\lambda_{ 0}}^{\text{aer}}(R)+\beta_{\lambda_{0}}^{\text{aer}}(R)\Big{)}\cdot\exp\! \left(-2\int_{0}^{R}\alpha_{\lambda_{0}}^{\text{aer}}(r)+\alpha_{\lambda_{0}} ^{\text{aer}}(r)dr\right)\,, \tag{1}\]  
with \(E(\lambda_{0})\) the pulse energy emitted by the laser at wavelength \(\lambda_{0}\), \(c\) the speed of light, and \(A=2.5\) m\({}^{2}\) the effective geometric telescope area. The molecular and aerosol volume backscatter coefficients are \(\beta_{\lambda_{0}}^{\text{aer}}(R)\) and \(\beta_{\lambda_{0}}^{\text{aer}}(R)\), whereas \(\alpha_{\lambda_{0}}^{\text{aer}}(r)=8\pi/3\cdot\beta_{\lambda_{0}}^{\text{arol }}(r)\) is the molecular volume extinction coefficient and \(a_{\lambda_{0}}^{\text{aer}}(r)=LR(r)\cdot\beta_{\lambda_{0}}^{\text{aer}}(r)\) the aerosol volume extinction coefficient, related through the LIDAR ratio \(LR(r)\). We have not applied any overlap correction in Equation (1), because our LIDAR has been optimized for a low region of full overlap, plus near-range optics in the final design [16]. The analysis of the return power signals will start only after full overlap is reached. The return power \(P_{\lambda_{R}}(R)\) for the anti-Stokes Raman wavelength \(\lambda_{R}\) is given by the RL equation [33]:  
\[P_{\lambda_{R}}(R)=\frac{E(\lambda_{0})\,c\,A}{2R^{2}}n_{\text{N}_{2}}(R) \cdot\frac{d\sigma_{\lambda_{R}}(\pi)}{d\Omega}\exp\!\left(-\int_{0}^{R}\! \left(a_{\lambda_{0}}^{\text{arol}}(r)+a_{\lambda_{0}}^{\text{aer}}(r)+a_{ \lambda_{R}}^{\text{aer}}(r)\right)dr\right)\,, \tag{2}\]  
where \(n_{\text{N}_{2}}(R)\) is the number density of the nitrogen molecule and \(d\sigma_{\lambda_{R}}(\pi)/d\Omega\) the range-independent differential Raman scattering cross-section in the backward direction, and the rest of the symbols follow the definition in Equation (1). The background power received \(P_{bg}(\lambda)\) around wavelength \(\lambda\) is calculated as follows:  
\[P_{bg}(\lambda)=L_{bg}\,A\,\,\Omega\,\Delta\lambda\,, \tag{3}\]  
where \(L_{bg}\) is the night sky irradiance (\(2.7\cdot 10^{-13}\) W cm\({}^{-2}\) nm\({}^{-1}\) sr\({}^{-1}\) for the clear moon-less night [31; 34], \(3.0\cdot 10^{-11}\) W cm\({}^{-2}\) nm\({}^{-1}\) sr\({}^{-1}\) for the case of observations under moderate moonlight [35; 36]), \(\Omega=2\pi\cdot(1-\cos(\theta_{1}/2))\) is the solid angle from the telescope FoV \(\theta_{t}=4.4\) mrad--defined as the ratio between the LLG diameter of 8 mm and the telescope focal length of 1.8--and \(\Delta\lambda=10\) nm is the interference filter bandwidth of every channel [16].