## 2 Methodology
### Direct Model  
Lidar signals from both air- or space-platform can be expressed by the lidar equation [44]. The backscattered lidar signal \(S_{v}\) above the ground, for a nadir measurement taken at a height above ground level (agl) \(h\) in the forest (with a ground altitude \(z_{ground}\)) and a wavelength \(\lambda\), is given by [44]:  
\[\begin{split} S_{v}(\lambda,h)\,\approx&\,K(\lambda) \cdot E(\lambda)\cdot\frac{1}{\left(Z_{p}-\left(h+z_{ground}\right)\right)^{2}} \cdot BER(\lambda,h)\cdot\alpha_{FOT}(\lambda,h)\\ &\cdot\exp\left[-2\left(\frac{FOT(h)}{2}\cdot\eta(\lambda,h)+ \tau(h)\right)\right]\end{split} \tag{1}\]  
where we supposed that the atmospheric backscattered part in the canopy is negligible compared to the larger backscattered part of the canopy. The instrumental constant \(K\left(\lambda\right)\) and the laser energy \(E(\lambda)\) are defined in the \"_instrument model_\" (Section 2.2.1, Equation (3)). \(FOT\) is the forest optical thickness (_cf._ Section 2.3) and \(\tau\) is the total atmospheric optical thickness as defined in the \"_Atmosphere model_\" (Section 2.2.2). The backscatter to extinction ratio \(BER\) (\(\lambda\), \(h\)) and the multiple scattering coefficient \(\eta\) are defined in the \"_Surface characteristics_\" (Section 2.2.3). The \(BER\), which is linked to tree species, can be also interpreted as the probability of photons backscattered after the interaction of the laser beam and the forest materials. \(Z_{p}\) is the altitude agl of the platform, which is defined in Section 2.2.4. The canopy extinction coefficient \(\alpha_{FOT}\left(\lambda\right)\), \(h\)) is defined as the sum of the absorption and scattering coefficients, which can be obtained from airborne lidar measurements through the inverse model for different forest biomes. The canopy extinction coefficient can be considered as the same at both NIR and UV wavelengths because leaves are large scatters compared to the wavelength.  
Figure 2: Block diagram of the end-to-end simulator. Both the standard deviations and the bias of the tree top height (_TTH_), the quadratic mean canopy height (_QMCH_) and the extinction coefficient (\(\alpha\)) are computed using a Monte Carlo approach. _TTH\({}_{0}\)_, _QMCH\({}_{0}\)_ and \(\alpha_{0}\) are initial values derived from the real lidar signal \(S\). \(\widehat{TTH}\), \(\widehat{QMCH}\) and \(\tilde{\alpha}\) are estimated values derived from the simulated lidar profile \(\tilde{S}\) for each statistical draw.  
The integrated range-corrected ground return, \(R_{g}\), was used to represent the ground echo (GE). The GE waveform, considered as following a Gaussian distribution [45], can be calibrated by using the returned laser pulse at nadir over a flat surface. We defined g (\(h\)) as a normalized Gaussian distribution and \(\Delta Z_{GE}\) as the equivalent width of the GE, the integrated ground return can be simulated by introducing the surface reflectance (\(\rho_{g}\)) as follows:  
\[\left\{\begin{array}{c}R_{g}\big{(}\lambda,Z_{p}\big{)}=\int_{\frac{\Delta Z _{GE}}{2}}^{\frac{\Delta Z_{GE}}{2}}S_{g}(z)\cdot(Z_{p}-z_{ground})^{2}\cdot dz \\ R_{g}\big{(}\lambda,Z_{p}\big{)}=K(\lambda)\cdot E(\lambda)\cdot\exp\left[-2( \frac{FOT(0)}{2}\cdot\eta(\lambda,0)+\tau(0))\right]\cdot\frac{\rho_{g}}{\pi \cdot\Delta Z_{GE}}\cdot\int_{\frac{\Delta Z_{GE}}{2}}^{\frac{\Delta Z_{GE}}{ 2}}\text{g}(z)\cdot dz\end{array}\right. \tag{2}\]  
Different noises were also considered in the direct model, and the detailed noise sources are discussed in Appendix A.