## 2 Methodology
### Direct Model
#### 2.2.3 Surface Characteristics  
**Surface reflectance (\(\rho_{g}\))** was calculated by Equation (2) using the ground echoes of many airborne lidar sampling profiles. The values of \(\rho_{g}\) at 355 nm are found to be similar for all these measured profiles: \(\rho_{g}\)\(=0.022\pm 0.002\) sr\({}^{-1}\). [PERSON] _et al._[48] found out the surface reflectance at 1064 nm: \(\rho_{g}=0.14\pm 0.03\) sr\({}^{-1}\). We will use these values for all our simulations. \begin{table}
\begin{tabular}{c c c c} \hline
**Wavelength** & \(\lambda\)**(nm)** & **355** & **1064** \\ \hline \multirow{8}{*}{Instrument model} & _QE_ & 85\% & 35\% \\  & \(G\) (A\(\cdot\)W\({}^{-1}\)) & \(\sim\)10\({}^{6}\) & \(\sim\)10\({}^{3}\) \\  & _OE_ & 60\% & 65\% \\  & Telescope diameter \(\omega\) (receptor surface A) & 100 cm (\(\sim\)0.8 m\({}^{2}\)) \\  & \(\Delta z\) (m) & 0.75 \\  & \(R_{c}\) (\(\Omega\)) & 50 \\  & \(\text{AOT}_{0}\) & 0.15 & 0.15 \\  & Ideal & \(\text{MOT}_{0}\) & 0.58 & 0.0065 \\ Atmosphere model & & \(\text{COT}_{0}\) & 0 & 0 \\  & Realistic & AOT & \multicolumn{2}{c}{Varied 1  } \\  & \(\rho_{\text{g}}\) (sr\({}^{-1}\)) & 0.022 \\  & \(BER_{\text{simu}}\) (sr\({}^{-1}\)) & 0.0065 & 0.0455 \\  & \(BER_{\text{adjusted}}\) (sr\({}^{-1}\)) & 0.005 & 0.035 \\  & \(\eta\) & 1 & 0.96 \\ \hline \end{tabular}
\end{table} 
Table 1: Instrumental, atmospheric and surface parameters chosen for the simulation. **Backscatter to extinction ratio (_BER_)** was calculated from the scattering phase function (_P_) inside the canopy:  
\[BER=\frac{1}{4\pi}\cdot\omega_{0}\cdot P_{\pi} \tag{6}\]  
where \(\omega_{0}\) is the single scattering albedo of scatters and \(P_{\pi}\) is the backscatter scattering phase function. There is little information about \(P\), especially at the scale at a lidar footprint size of a few meters. As in [PERSON] _et al_. [49], the Bidirectional Reflectance Distribution Functions (BRDF) from passive spaceborne measurements (multi-angular satellite POLDER data) was used to retrieve \(P\) using semi-empirical models, even though they are retrieved with pixels of several kilometers. More detailed BRDF modeling are given in Appendix B. [PERSON] _et al_. [50] show the variation of absorption and scattering with wavelength: there is strong absorption and little scattering in UV, whereas there is \(\sim\)7 times more scattering in NIR. However, regarding the reflectance spectral dependency as documented in the same study, the behavior of the 490 nm (resp. 865 nm) channel seems very close to the one for the UV (resp. NIR) wavelength of 355 nm (resp. 1064 nm). The corresponding \(P_{\pi}\) of the 490 nm (resp. 865 nm) channel can then be used for simulations at 355 nm (resp. 1064 nm). Both mean value and standard deviation of the retrieved \(P_{\pi}\) are given in Table 2 for each considered forest type. They lead to a _BER_ close to \(0.007\pm 0.002\) sr\({}^{-1}\) (resp. \(0.046\pm 0.002\) sr\({}^{-1}\)) in UV (resp. NIR). These _BER_ values were used in the following simulations and supposed constant in the canopy. **Multiple scattering coefficient (\(\eta\))**, at different depths in the scattering layer, is deduced from the ratio between the total lidar signal (including single scattering \(S_{single}\) and multiple scattering \(S_{multiple}\)) and the number of single-backscattered photon as [29]:  
\[\eta(h)=1-\frac{\ln\left(\frac{S_{single}(h)+S_{multiple}(h)}{S_{single}(h)} \right)}{FOT(h)-FOT(h_{t})} \tag{7}\]  
where \(h_{t}\) is the top of the scattering layer. The Multiple scattering (MS) effects in UV are negligible, and the typical value is \(\eta\sim\)1. However the MS effects may become significant compared to the single scattering when switching from UV (355 nm) to NIR (1064 nm). Thus, for the Monte Carlo simulation, the MS coefficient (\(\eta\)) in NIR is derived as \(0.96\pm 0.03\) for different conditions (e.g., different space missions).