## 2 Methodology
### Inverse Model  
In order to retrieve forest vertical structures, it is necessary to detect the intensity peaks of both the canopy and ground echoes in the full-waveform lidar signals. The _TTH_ is calculated as the distance between the first return at the upper surface of the vegetation and the last return from the ground surface. Following the signal processing described in [PERSON] and [PERSON] [42], the ground echo was first detected compared to the noise level, which can be inferred from the signal remaining after the ground echo when only the instrumental noise exists. The canopy echo was then detected, considering the atmospheric signal just above the trees.  
To simplify the calculation, the range-corrected lidar signal is introduced, which is defined as the product of the backscattered lidar signal \(S_{\
u}(h)\) and the square of the distance between the laser emission and the target. The integrated canopy signal \(R_{\
u}(h)\) is defined as the integral of the range-corrected lidar signal from the canopy top _TTH_ to height level \(h\). As \(BER\) is assumed to be constant for all canopy levels (_i.e._, \(BERo\)) and \(\eta\sim\)1 in UV, \(R_{\
u}(h)\) can be expressed, after correction of the atmospheric transmission, as:  
\[\begin{cases}R_{\
u}(h)=\int_{h}^{TTH}S_{\
u}(z)\cdot(Z_{p}-(h+z_{ground}))^{ 2}\cdot dz\\ R_{\
u}(h)=K_{0}\cdot E_{0}\cdot BER_{0}\cdot[1-exp(-FOT(h))]\end{cases} \tag{8}\]  
The \(FOT\) can then be derived as:  
\[FOT(h)=-ln\left(1-\frac{1}{K_{0}\cdot E_{0}\cdot BER_{0}}\cdot R_{\
u}(h)\right) \tag{9}\]  
The \(FOT\), defined as the optical thickness in a forest layer between the considered height \(h\) and the canopy top _TTH_, depends on the canopy extinction coefficient \(\alpha_{FOT}\), which is then derived from Equations (4) and (9) as:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mission Lidar**} & **Our Study** & **CALIPSO 1** & **ICESat 1** & **MERLIN 1** & **ADM-Aeolus 1** & **ISS** \\  & **ULICE** & **CALIOP** & **GLAS** & **IPDA** & **ALADIN** & **-** \\ \hline Orbit altitude z\({}_{p}\) (km) & 0.35 & 705 & 600 & 506 & 400 & 350 \\ Wavelength \(\lambda\) (nm) & 355 & 1064 & 1064 & 1645 & 355 & - \\ Laser Pulse Energy E (mJ) & 0.007 2 & 110 & 73 & 9 & 120 & - \\ QE & 20\% & 33\% & 35\% & 60\% & 85\% & - \\ OE & 26\% & 64\% & 55\% & 65\% & 60\% & - \\ Telescope diameter (cm) & 15 (0.02) & 100 (0.79) & 100 (0.79) & 55 (0.24) & 150 (1.77) & - \\ (surface A (m\({}^{2}\))) & 0.75 & * & 0.15 & * & - & - \\ \hline \hline \end{tabular}  
* the actual \(\Delta\)z cannot be used for forest sampling, and chosen to be 0.75 m; 1 These parameters of spaceborne lidar missions were found out through research papers [53, 57, 58, 59, 60]; 2 Calculated by \(E_{\theta}\)/10\({}^{\text{\tiny{{0D}}}}\) with emitted laser energy \(E_{\theta}\) = 7 mJ and optical densities OD = 3.  
\end{table}
Table 3: Actual parameters for ULICE and referenced spaceborne lidar missions.  
\[\alpha(h)=\frac{\partial\ln\left[1-\frac{1}{K_{0}\cdot E_{0}\cdot BER_{0}}\cdot R_{ p}(h)\right]}{\partial h} \tag{10}\]  
This solution is consistent with the one derived from the statistic consideration made by [PERSON] _et al._[15]. Canopy extinction coefficient is a fundamental parameter used in our direct model to simulate the lidar signal by using the lidar equation under the hypothesis of \"_single scattering_\".  
The _QMCH_ (quadratic mean canopy height) parameter, which can be used to evaluate the aboveground biomass [42], is also a function of the extinction coefficient [15]:  
\[Q\!MCH=\sqrt{\int_{TTH}^{0}\!\alpha(h)\cdot h^{2}\cdot dh} \tag{11}\]