## 3 Models and Methods
### Methods
#### 3.2.2 Transmitted Energy Profile (TEP) Calculation  
The TEP was derived based on material reflectance characteristics. For lidar observation, the observed material reflectance characteristics usually depend on the sensor configuration and the relationship between the detected object and the laser. The sensor configuration can be expressed by the lidar equation [47], which is shown as follows:  
\[R_{return}=I_{trans}\tau_{opt}\frac{A_{scope}}{D^{2}}\cdot\frac{\rho}{\Omega} \tau_{atm} \tag{6}\]  
where \(R_{return}\) is the received return energy, \(I_{trans}\) is the emitted energy, \(\tau_{opt}\) is the optics transmission, \(A_{scope}\) is the telescope area, \(D\) is the range between lidar sensor and ground, \(\rho\) is the reflectance of the target, \(\tau_{atm}\) is the roundtrip atmosphere transmission, and \(\Omega\) is the scattering solid angle. The relationship between the observed object and laser can be given by the physical model of lidar echoes for forest (i.e., Equation (1)). By combining the lidar equation [i.e., Equation (6)] and the physical model of lidar echoes for a forest [Equation (1)], we can obtain an equation to express the observed material reflectance characteristics, which is shown as follows:  
\[R_{n}=I_{n}[1-\exp(-k\cdot LAI_{n})]\cdot\tau_{opt}\frac{A_{scope}}{D^{2}} \frac{\rho}{\pi}\tau_{atm} \tag{7}\]Here, we assumed that the observed surfaces were Lambertian surfaces, so the \(\Omega\) was equal to \(\pi\). According to the relationship of the transmitted energy of two successive recorded layers (i.e., Equation (2)), we rewrote Equation (7); the result is shown as follows:  
\[I_{n}=\frac{R_{n}}{\rho}\cdot\frac{D^{2}\pi}{\tau_{opt}A_{scope}\tau_{atm}}+I_{n+1} \tag{8}\]  
In our application scene, two types of materials were involved: ground and vegetation. Based on Equation (8), the corresponding mathematical expression including these two materials can be given as follows:  
\[I_{0}=\left(\frac{\sum\limits_{i=1}^{n-1}R_{n-1}}{\rho_{ vegetation}}+\frac{R_{n}}{\rho_{ground}}\right)\cdot\frac{D^{2}\pi}{\tau_{opt}A_{scope}\tau_{atm}} \tag{9}\]  
where \(I_{0}\) is the total energy emitted by the lidar, \(\sum\limits_{i=1}^{n-1}R_{n-1}\) is the lidar received energy reflected from the vegetation, \(\rho_{vegetation}\) is the vegetation reflectance, \(R_{n}\) is the lidar received energy reflected from the ground surface, and \(\rho_{ground}\) is the ground reflectance. The values of \(\tau_{opt}\), \(A_{scope}\), \(\tau_{atm}\), \(D\), \(I_{0}\), and \(R_{i}\) (\(I\) = 1,2,... \(n\)) are known, as shown in Table 2. Therefore, if we know the ground position in the recorded received waveform and the reflectance of the observed materials (i.e., \(\rho_{vegetation}\) and \(\rho_{ground}\)), the TEP can be derived from Equation (8). The ground position can be identified from the GLAS received waveform through the Gaussian decomposition method (Figure 2); the process includes filtering, Gaussian decomposition, and determining the ground position. First, the raw GLAS echo waveform was denoised by Gaussian filtering. Then, the denoised GLAS echo waveform was decomposed into multiple Gaussian waveforms using the Gaussian decomposition method. Finally, the peak position of the last decomposed Gaussian waveform was assumed to correspond to the ground surface. Equation (9) shows that if the ground reflectance is known, the canopy reflectance can be derived. Here, we used the average reflectance results (i.e., GLAS reflectance = d_reflectUC * d_refLCor_atm) from 12 non forest-covered GLAS footprints in Saihanba National Forest Park as \(\rho_{ground}\). The location of non forest-covered GLAS footprints is shown in Figure 2, the reflectance corresponding to these GLAS footprints is shown in Figure 4, which has an average result of 0.21 (i.e., \(\rho_{ground}\) = 0.21). This strategy can be close to the actual reflectance of the GLAS laser for the ground.