## 3 Models and Methods
### Methods  
#### 3.2.1 Vertical Foliage Profile (VFP) and LAI Calculation  
The basic principle for calculating the VFP and LAI from GLAS lidar data is based on the Beer-Lambert law. According to the Beer-Lambert law, we can establish a relationship between the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Unit** & **Values** \\ \hline Pulse energy & mJ & 73 \\ Half pulse duration & dimensionless & 6 \\ Lidar acquisition rate & ns & 1 \\ Wavelength & nm & 1064 \\ Bandwidth & ns & 0.08 \\ Area of lidar sensor & m\({}^{2}\) & 0.785 \\ Effective telescope diameter & m & 0.95 \\ Sensor altitude & km & 600 \\ Footprint radius & m & 35 \\ Field of view (FOV) radius & m & 48 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Values for the discrete anisotropic radiative transfer (DART) model used in this study are consistent with the GLAS device configuration, as mentioned in the text.  
Figure 3: Overview of the realistic structure scenes constructed in the DART model. (**a**) is a homogeneous turbid scene that is composed of a cube plant canopy and a horizontal ground; (**b**) is a discrete individual tree scene that comprises a 3-D tree model; (**c**) is the 3-D tree model used in the discrete scene.  
leaf area density (LAD) and the degree of penetration of light within unit space, which is the space between two successive recorded layers as follows:  
\[T(\alpha)=e^{-G(\alpha)LAD\Delta(\alpha)} \tag{3}\]  
where \(T(\alpha)\) is the gap fraction of the unit space when the observation zenith angle is \(\alpha\); \(\Delta l(\alpha)\) is the distance of the light traveling through the unit space along the zenith angle \(\alpha\); and \(G(\alpha)\) is the leaf projection coefficient. For GLAS lidar, the vertical resolution is equal to 0.15 m; this means the thickness of the unit space is 0.15 m (i.e., \(\Delta l(\alpha)=0.15\)). Therefore, we only needed the data on the leaf projection coefficient and gap fraction to obtain the LAD.  
The leaf projection coefficient \(G(\alpha)\) is defined as the mean projection of a unit foliage area in the direction of the solar beam. For a canopy with a spherical leaf distribution, that is, the leaf inclination angle is 57.3\({}^{\circ}\), G is equal to 0.5 and independent of the solar zenith angle [46]. Due to the difficulty of estimating real leaf inclination angles, a spherical leaf inclination angle distribution is commonly assumed for a vegetative canopy. This assumption has been followed in studies on the inversion of LAI based on the GLAS data and performs well [25; 27]. Therefore, the projection coefficient \(G(\alpha)\) in this study was set to 0.5. The gap fraction \(T(\alpha)\) of the unit space can be calculated by the TEP (i.e., \(I_{0}\), \(I_{1}\),..., \(I_{n}\)), and the calculation equation is shown as follows:  
\[T_{n}(\alpha)=\frac{I_{n+1}}{I_{n}} \tag{4}\]  
where \(I_{n}\) is the incident energy of the unit space of layer n and \(I_{n+1}\) is the energy passing through this space. The calculation of the TEP was introduced in Section 3.2.2.  
When the LAD is obtained, the corresponding LAI can be calculated by the relationship between the LAD and LAI as follows:  
\[LAI(h)=\int_{m}^{n}LAD(h)dh \tag{5}\]  
where \(m\) and \(n\) are the layers of the LAD. The total LAI requires the integration of the LAD from the canopy top to ground. In addition, the cumulative LAI profiles can be obtained based on Equation (5) by calculating the increment in LAD at each height interval. In this study, we took the LAD profile and cumulative LAI profile as the VFP.