## 1 Introduction  
Forest canopies play a vital role in determining the energy cycle, regulating the climate, and providing habitats for various species; most of these functions are affected by foliage respiration and photosynthesis [1; 2; 3]. Therefore, it is essential to quantitatively describe the foliage in the canopy. The leaf area index (LAI) and vertical foliage profile (VFP) are two key canopy structure parameters that quantify the distribution of foliage in the horizontal and vertical directions, respectively. Accurate estimation of the LAI and VFP is essential for promoting understanding of the role of forest canopies in terrestrial ecosystems [4; 5; 6]. Due to the importance of the LAI and VFP, various efforts have been made to estimate them from a variety of data sources. Passive remote-sensing data are some of the broadest data sources for LAI estimation because of their easy availability and global observation capabilities [7; 8; 9; 10; 11]. However, the LAI from passive remote-sensing data suffers from some limitations. For example, the LAI tends to saturate in dense forests because of the limited penetrability of sunlight [12; 13; 14]. In addition, passive remote-sensing data do not include detailed vertical measurement information, which limits its application areas, such as VFP inversion. In contrast, measurements from light detection and ranging systems (lidar) have good penetration ability and can provide detailed vertical observations from the top of the canopy to the ground. The above reasons promote the application of lidar observations to forest structure parameters [15; 16; 17], especially the LAI and VFP estimations [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. In general, lidar systems can be classified into three types depending on the platform, including terrestrial, airborne, and spaceborne lidar. Due to the limited spatial coverage, terrestrial and airborne lidar are only suitable for small- or middle-scale research. The spaceborne lidar has global observation capabilities, which supports its utility as a data source for global-scale research. Existing spaceborne lidar systems include the Geoscience Laser Altimeter System (GLAS) on board the Ice, Cloud and land Elevation Satellite (ICESat), ICESat-2, and Global Ecosystems Dynamics Investigation (GEDI), of which the ICESat GLAS is the most widely used spaceborne lidar system because it is the first spaceborne lidar for Earth observation. More importantly, the office of ICESat GLAS can provide free and rich datasets corresponding to the lidar observation, which makes it an ideal data source for estimating the LAI and VFP at the global scale [23; 24; 25; 26; 27; 28]. Meanwhile, methods that serve to estimate the LAI or VFP using GLAS data come into rescue. Currently, two types of method have been developed in accordance with GLAS data to estimate the LAI or VFP: empirical methods and physical methods. The empirical methods were developed based on a statistical regression relationship between the GLAS-derived metrics and field-measured LAI [23; 24]. Although these empirical methods are easy to implement and understand, they cannot be free from the need for the associated field data. Thus, this type of method is not adapted for application in large-scale LAI mapping because the empirical relationships usually vary with locations, species, canopy structures, etc. Physical methods, which are constructed around the gap fraction obtained based on radiative transfer theory [25; 26; 27; 28], open the door to large-scale LAI or VFP mapping using lidar data. For example, the LAI and VFP have been successfully retrieved over the entire state of California based on a geometric optical and radiative transfer (GORT) model [25]. By summarizing the current physical methods [25; 26; 27; 28], we find that all methods are constructed depending on the received waveform information.