## 2 Material and Methods
### Framework  
The returned lidar signal for bi-static and multi-axial systems originates from the two-way attenuated backscattering by particles in the atmosphere at an angle determined by the transmitter-receiver geometry [22]. The elastic lidar equation under the single-scattering approximation assumes that the incident wave is scattered only once and multiple-scattering events are excluded. This well-known equation can be directly derived from the radiative transfer equation and can also be seen as the link-budget that provides the backscattered power \(\mathrm{P}(\mathrm{r})\) at a given range \(\mathrm{r}\) as:  
\[\mathrm{P}(\mathrm{r})=\mathrm{K_{o}}\,\mathrm{O}(\mathrm{r})\,\mathrm{U}( \mathrm{r})\,\mathrm{r}^{-2}\qquad[\mathrm{W}] \tag{1}\]  
where:  
\(\mathrm{K_{o}}\) is the radiometric lidar constant which depends on the transmitted laser pulsed energy \(\mathrm{E_{o}}\) or power \(\mathrm{P_{o}}\), the duration of the laser pulse \(\tau\), and the receiver aperture area \(\mathrm{A}\) as:  
\[\mathrm{K_{o}}=\mathrm{P_{o}}\frac{c\tau}{2}\mathrm{A}=\mathrm{E_{o}}\frac{c} {2}\mathrm{A}\qquad[W\,m^{3}\,sr] \tag{2}\]  
O(r) is the range-dependent overlap function which is a correcting function for measurements where the field of view of the receiver does not completely overlap the emission beam. U(r) is the attenuated backscattering function defined as:  
\[\mathrm{U}(\mathrm{r})=\beta(\mathrm{r})\exp\left(-2\int_{0}^{\mathrm{r}} \alpha(\mathrm{r}^{\prime})\mathrm{d}\mathrm{r}^{\prime}\right)\qquad[ \mathrm{m}^{-1}\,\,\mathrm{sr}^{-1}] \tag{3}\]with \(\alpha(r)\) and \(\beta(r)\) are the total extinction and backscattering coefficients defined as:  
\[\beta(r)=\sum_{\gamma=\mathrm{aer,mol,sur}}\beta_{\gamma}(r)\quad[m^{-1}\;sr^{-1}] \qquad\quad\alpha(r)=\sum_{\gamma=\mathrm{aer,mol}}\alpha_{\gamma}(r)\quad[m^{-1}] \tag{4}\]  
where the subscripts \(\mathrm{aer,mol}\), and sur describe the aerosol, molecular, and surface components, respectively. For sake of clarity, the wavelength dependence of these parameters is omitted. Different physical quantities must be derived from light-scattering for retrieving the lidar products of interest. The aerosol extinction and backscattering coefficient profiles, \(\alpha_{\mathrm{aer}}\) and \(\beta_{\mathrm{aer}}\), are derived from the aerosol extinction and differential backscattering cross-sections \(C^{\mathrm{ext}}\) and \(\mathrm{d}C^{\mathrm{bac}}\) as:  
\[\alpha_{\mathrm{aer}}(r)=\int_{R_{\mathrm{min}}}^{R_{\mathrm{max}}}N_{\mathrm{ p}}(r,R)C^{\mathrm{ext}}(r,R)\mathrm{d}R\quad[m^{-1}] \tag{5}\]  
\[\beta_{\mathrm{aer}}(r)=\int_{R_{\mathrm{min}}}^{R_{\mathrm{max}}}N_{\mathrm{ p}}(r,R)\mathrm{d}C^{\mathrm{bac}}(r,R)\mathrm{d}R\quad[m^{-1}\;sr^{-1}] \tag{6}\]  
where \(N_{\mathrm{p}}\) is the aerosol or particle number concentration per unit volume of an isotropic scattering medium formed by an ensemble of randomly oriented particles of radius \(R\) with \(R_{\mathrm{min}}\) and \(R_{\mathrm{max}}\) the minimum and maximal radii. Consequently, these aerosol cross-sections are the key radiative quantities for aerosol lidar applications [23] and are essential for evaluating microphysical properties such as aerosol number concentration from lidar profiles. The particle extinction cross-section \(C^{\mathrm{ext}}\), expressed in unit of surface area, and differential backscattering cross-section \(\mathrm{d}C^{\mathrm{bac}}\), expressed in unit of surface area per unit of solid angle, are defined as:  
\[C^{\mathrm{ext}}(r,R)=Q^{\mathrm{ext}}(r,R)C^{\mathrm{geo}}(r,R)\quad[m^{2}] \tag{7}\]  
\[\mathrm{d}C^{\mathrm{bac}}(r,R)=\left(\frac{d\sigma^{\mathrm{sca}}(r,R)}{d \Omega}\right)_{\theta=\pi}=\ \frac{1}{4\pi}Q^{\mathrm{bac}}(r,R)C^{\mathrm{geo}}(r,R)\quad[m^{-1}\;sr^{-1}] \tag{8}\]  
where \(Q^{\mathrm{ext}}\) is the extinction efficiency, \(Q^{\mathrm{bac}}\) is the backscattering efficiency and \(C^{\mathrm{geo}}\) is the geometrical cross-sections defined as \(\pi R^{2}\) for a spherical particle of radius \(R\).