Improving GEDI L2B Leaf Area Index Estimation Using a Four-Scale Geometric Optical Model in Temperate Forests

Highlights

What are the main findings?

• The study reveals the effect of Global Ecosystem Dynamics Investigation (GEDI) Level 2B products on measuring Leaf area index (LAI) in temperate forest areas.
• We use a four-scale model method to make it more in line with the actual distribution of forests, significantly improving the accuracy of GEDI’s estimation of leaf area index.

What are the implications of the main findings?

• Our method assumes that the canopy distribution within the GEDI footprint is non-uniform, which is more in line with the actual forest conditions.
• This is of great significance for the development of spaceborne lidar systems.

Abstract

LAI is a critical parameter for forest management and global ecosystem monitoring. GEDI provides global-scale vegetation structure data, yet its L2B LAI product often exhibits systematic biases. This study investigates the Maoer Mountain forest in China, utilizing a total of 60 validated GEDI footprints as the primary dataset. To address the limitations of the standard GEDI L2B algorithm, which assumes a horizontally uniform canopy, we integrated a four-scale geometric optical model to characterize canopy clumping effects. This model was employed to simulate the geometric proportions of sunlit/shaded canopy and ground components within each footprint to derive a footprint-specific clumping index, thereby refining the gap rate estimates. The accuracy of the revised leaf area index was rigorously verified by using the measured data from the sample plots in the Maoer Mountain area. The results indicate that the original GEDI L2B data underestimates LAI, with a mean absolute error (MAE) of 1.79 m²/m², a root mean square error (RMSE) of 1.47 m²/m², and a bias of −1.25 m²/m². After correcting for canopy clumping, accuracy improved significantly, reducing the MAE to 0.65 m²/m² and the RMSE to 0.82 m²/m², while effectively mitigating underestimation. These findings demonstrate that accounting for non-uniform canopy distribution effectively reduces errors, providing a robust methodological basis for high-precision LAI retrieval using spaceborne lidar. Despite these improvements, this method still has certain limitations: the model’s performance is constrained in extremely steep terrain due to waveform aliasing and in fragmented vegetation areas where sub-footprint heterogeneity is high. Future research should incorporate topographic corrections and multi-source data fusion to enhance the model’s robustness in complex landscapes.

1. Introduction

The forest ecosystem is the largest and most critical component of the terrestrial biosphere, playing a vital role in human survival and socioeconomic activities [1,2]. General Secretary Xi Jinping proposed that China’s carbon dioxide emissions will peak by 2030 and achieve carbon neutrality by 2060, and the forest carbon sink is an important part of achieving this goal [1]. Forest parameters are important parameters for carbon sink estimation, so it is of great significance to accurately determine forest parameters over a large range [3].

Leaf area index, defined as one-half of the total green leaf area per unit ground surface area, serves as a fundamental biophysical parameter in terrestrial ecosystems. It characterizes canopy structure and governs critical physiological and physical processes, including photosynthesis, transpiration, respiration, and precipitation interception. By quantifying the exchange of matter and energy at the vegetation–atmosphere interface, LAI plays a pivotal role in the forest carbon cycle. Consequently, the accurate, large-scale quantification of LAI is indispensable for effective forest management and ecosystem modeling [1,2,3,4,5]. Current methodologies for LAI estimation are generally categorized into direct and indirect approaches. Direct methods involve manual data acquisition through destructive sampling or litterfall collection to geometrically measure leaf area [6,7,8,9]. While these methods offer high accuracy and serve as validation baselines, they are labor-intensive, time-consuming, and spatially limited to small plot scales. In contrast, indirect methods infer LAI by measuring canopy light transmission or gap rate. These techniques are non-destructive, cost-effective, and highly efficient, making them significantly more suitable for characterizing LAI across large spatial scales [10].

With the proliferation of remote sensing technology, high-spatial-resolution estimation of leaf area index at broad scales has become a focal point of ecological research. Remote sensing methodologies for LAI retrieval are generally categorized into passive and active techniques. Passive remote sensing typically derives LAI from multi-spectral imagery via two primary approaches: empirical relationships based on vegetation indices (VIs) and physical inversion using radiative transfer models (RTM). Global operational products, such as MODIS-LAI and GLOBCARBON-LAI, are predominantly generated using these methods [11,12,13]. A critical challenge in passive optical inversion is separating the contribution of the forest canopy from the understory background. Early studies on products such as MODIS demonstrated the necessity of separating the influence of the understory background in the inversion algorithm of tree canopy leaf area index [14,15]. To address these structural complexities, the four-scale geometric optical model was developed [8]. By explicitly accounting for within-crown gaps and inter-crown clustering, this model better approximates realistic forest architecture. The model has demonstrated versatility in various applications: As the four-scale model was developed, scholars combined it with MISR data to characterize the seasonal dynamics of forest background reflectance globally [16]; regarding LAI inversion, researchers successfully inverted the leaf area index of artificial forests in Gansu Province by combining the four-scale model and Thematic Mapper (TM) images. The results showed that the inversion results were in good agreement with the measured data [17]; in subsequent research, scholars established scale transformation functions for photochemical reflectance indices (PRI) through the four-scale model [18]. Furthermore, a global LAI estimation method that combines a four-scale model with lookup table (LUT) technology has been proposed. However, this approach simplifies the forest background reflectance to a fixed value based on land cover type, failing to account for pixel-scale variations [19].

Despite the advantages of passive remote sensing in providing continuous spatiotemporal coverage, intrinsic limitations remain [20,21]. A primary constraint is the saturation of VIs at high biomass levels, where sensitivity to LAI variations diminishes asymptotically [9,22,23]. Consequently, the stability of passive methods can be compromised, and model-dependent discrepancies may introduce significant uncertainties into the final retrieval results [24,25].

Active remote sensing, particularly LiDAR, has emerged as a robust tool for LAI retrieval. Depending on the platform and spatial scale, LiDAR systems are categorized into terrestrial (TLS), airborne (ALS), and spaceborne (SLS) platforms. Compared to passive optical sensors, LiDAR offers superior canopy penetration capabilities and significantly mitigates signal saturation issues, thereby achieving higher accuracy in dense forests [26]. Among these, spaceborne LiDAR is particularly advantageous for large-scale ecological monitoring due to its extensive coverage and ability to capture vertical structural profiles.

Current methodologies for estimating LAI from spaceborne LiDAR primarily rely on modeling the interaction between laser pulses and forest structure. The most prevalent approach involves deriving gap rates based on the Geometric Optical and Radiative Transfer (GORT) model [27,28] or applying the Beer–Lambert Law to characterize laser transmission through the canopy. The efficacy of the GEDI, a new-generation spaceborne LiDAR system, has been extensively validated across diverse biomes. For instance, scholars have demonstrated GEDI’s effectiveness in measuring LAI in temperate forests of southeastern Australia using ALS as a benchmark [29]. Similarly, scholars have validated GEDI LAI estimates in semi-arid tropical savannas, establishing a foundation for its application in heterogeneous landscapes [30]. Furthermore, the application of GEDI in measuring LAI of different forest types in the continental United States was also comprehensively evaluated, and the factors affecting the retrieval accuracy were analyzed [31].

Although GEDI has great potential for estimating LAI, its current application still faces two key bottlenecks that affect high-precision forest monitoring work. Firstly, most existing studies have always focused on “product accuracy assessment” rather than “improvement of the basic algorithm”. More fundamentally, there are mechanism problems in the underlying physical principles of the GEDI L2B product algorithm. The inversion framework for estimating LAI through Beer’s law assumes that the vegetation distribution is randomly distributed horizontally. This idealized clumping essentially ignores the ubiquitous tree crown clumping effect, and in reality, there are multi-scale spatial clumping phenomena of leaves within and between tree crowns, which are significant characteristics of complex natural forests. Therefore, the clumping of uniform distribution is significantly different from the non-uniform distribution of the actual forest structure, resulting in significant systematic deviations, especially in dense forest areas, where the estimation of leaf area index (LAI) has always been underestimated [32,33]. Solving this theoretical limitation through structural modeling based on physical principles is crucial for fully leveraging the potential of spaceborne laser radar observations.

To address the existing challenges in estimating leaf area index using active and passive remote sensing techniques, this study investigated whether combining a geometric optics model with onboard laser radar data could generate more accurate leaf area index products. By introducing a model that is closer to the actual forest structure and takes into account the clumping effect between tree canopies, the study evaluated whether it is possible to robustly achieve high-precision leaf area index inversion [34,35].

The purposes of this study are:

(i)

To improve the accuracy of estimating LAI by using the method of the four-scale model and correct the clumping of uniform canopy distribution of GEDI L2B products.

(ii)

To reduce the accuracy errors caused by waveform stretching due to vegetation coverage and slope; these factors blur the boundary between the surface and the tree canopy, thereby causing deviations in the estimation of the gap rate.

2. Materials and Methods

2.1. Study Area

The study area is the Maoer Mountain forest area. Maoer Mountain forest is located in Shangzhi City, Heilongjiang Province, China. The landform belongs to a low mountainous and hilly area, which belongs to a temperate humid area. The terrain gradually rises from south to north, with an altitude range of 250~805 m. The vegetation in the study area is inlaid with natural secondary forests such as precious broad-leaved forest, poplar birch forest, and oak forest, and plantations such as Korean pine, larch, and Pinus sylvestris. mongolica. The forest type is temperate coniferous and broad-leaved mixed forest dominated by broad-leaved trees [36]. The specific location of the study area is shown in the Figure below (Figure 1).

The forest of Maoer Mountain has a unique lower vegetation layer, which includes various shrubs and herbaceous plants. This layer is at its most lush during the growing season. Although the seasonal changes in the lower vegetation may cause errors in passive optical remote sensing, the use of GEDI space-based laser radar (NASA Goddard Space Flight Center, Greenbelt, MD, USA) can achieve better penetration effects, thereby enabling a more accurate depiction of the vertical structure.

2.2. Data

2.2.1. GEDI Data

GEDI is the world’s first spaceborne laser radar equipped with multiple linear laser altimeters, used for high-resolution measurement of forest vertical structure. It is mainly employed to accurately measure the tree canopy height, vertical structure, and ground elevation of forests in tropical and temperate regions. GEDI provides full waveform data, with a total of eight beam tracks, including four full-power beams and four coverage beams. The size of each footprint is about 25 m, the center points of the footprints are 60 m apart, and the cross-track spacing is 600 m [28]. Compared with the footprint size of ICESat/GLAS, about 70 m and 170 m track spacing, GEDI provides a significantly higher sampling density. This high density is particularly advantageous in heterogeneous forests where structural parameters vary over short distances [27,37], and the effect is better when combined with other types of data, such as Landsat, TanDEM-X, etc., which is more suitable for the observation of forest structure and understory topography (Figure 2). Since 2019, GEDI has released level 4 products [38]. In order to explore the performance of using optical models to detect leaf area index and to improve the measurement accuracy, this study selected the GEDI L2A and L2B data from July in the Maoer Mountain area in 2022 for the experiment. The Python 3.12 script used to process the spot data is GEDI_Subsetter [39].

Since GEDI data will be affected by many factors during observation, resulting in low-quality and unusable data, high-quality data were selected according to the parameters quality_flag = 1 and sensitivity > 0.9. To ensure that the observed data represented forest areas, landsat_treecover > 0 and modis_treecover > 0 were used as selection criteria [40]. GEDI L2B level data provides the effective leaf area index calculated based on Beer’s law, so the experiments are all about the effective LAI.

2.2.2. Measured LAI Data

The diameter of the GEDI footprint is approximately 25 m. Therefore, a 30 m × 30 m square plot is set with the center position of the GEDI footprint as the center of the plot; due to the randomness of the GEDI footprint positions and considering the complexity and accessibility of the forest, it is necessary to set up sample plots as widely as possible to cover various canopy densities and forest types. A total of 60 plots were set up. In this study, the LAI-2200 canopy analyzer (LI-COR Biosciences, Lincoln, NE, USA) was used to measure LAI. The LAI-2200 measured the transmitted light of the tree canopy from five different angles through a “fisheye” optical sensing sensor with a vertical field of view of 148° and a horizontal field of view of 360°, which were 7°, 23°, 38°, 53°, and 68°, respectively. The parameters, such as the leaf area index and porosity of the canopy, were calculated through the radiative transport model of the canopy. Its measurement efficiency is relatively high. The measured leaf area index is the effective leaf area index, and the mean LAI of the measurement points at different scales is the effective LAI of that scale [41]. As a mature temperate secondary forest, its canopy structure remains relatively stable without significant disturbances. Due to the lag in the release of GEDI data, this measurement was carried out in July 2023 [40]. The statistical information of the sample plots is presented in

Table 1. Overview of sample points.

Forest Type	Number of Plots	Mean Height/(m)	Height Range/(m)	Stand Density/(Trees/ha)
Broad-leaved forest	40	13.12	5.95–24.01	100–180
Coniferous forest	20	15.24	6.12–27.07	100–180
Total	60	14.18	5.95–27.07	100–180

.

2.2.3. Auxiliary Data

Using the small plot survey data of the Maoer Mountain Forest Farm of Northeast Forestry University in 2023, the data package contained tree species types, dominant tree species, forest age, slope position, slope orientation, soil type, etc., within each small plot. The tree species type at the small plot where the GEDI footprint was located was extracted by applying Arcgis 10.8, and we used this as the basis for subsequent experiments.

In July 2023, an airborne lidar (DJI, Shenzhen, Guangdong, China) was used to measure the footprint area. The drone was equipped with a DJI M300RTK from DJI Zenmuse L1, with an RTK positioning accuracy of 1 cm ± 1 pm in the horizontal direction and 1.5 cm ± 1 pm in the vertical direction. The point cloud data is denoised, and then the ground points are classified by using the improved progressive encryption triangulation network filtering algorithm to create the irregular triangulation network (TIN) of the ground points. This TIN is used to linearly interpolate the DEM elevation on the 1 m grating grid, and the slope of the DEM data is analyzed to generate the slope data of the study area position [42,43,44], as the data supports the subsequent analysis of the influence of slope on the determination of leaf area index.

2.3. Method

2.3.1. Canopy Gap Rate Measured by GEDI

The GEDI L2B level data was followed by LAI products obtained from the 3D GORT model, which is a geometric radiative transfer model proposed by Ni-Meister in 2001 [27] to characterize the probability that incident beams with different angles reach a given point at a certain canopy height without being scattered. Specifically, Equations (1)–(3) are used to characterize the effect of vegetation and pulse at the land surface ($R_g$), canopy ($R_v\left(0\right)$), and any height ($R_v\left(z\right)$):

$$R_g=J_0{\rho }_{g}P\left(0\right)$$

(1)

$$R_v\left(0\right)=J_0{\rho }_v\left(1-P\left(0\right)\right)$$

(2)

$$R_v\left(z\right)=J_0{\rho }_v\left(1-P\left(z\right)\right)$$

(3)

where $J_0$ is the energy of the laser reaching the top of the canopy, ${\rho }_g$ and ${\rho }_v$ are the reflectivity of the ground and vegetation, and $P\left(0\right)$ is the total gap rate in the GEDI footprint. The clearance rate $P\left(z\right)$ at any height and the total clearance rate $P\left(0\right)$ in the footprint can be concluded (Equations (4) and (5)) [40]:

$$P\left(z\right)=1-{\displaystyle \frac{R_v\left(z\right)}{R_v\left(0\right)+R_g\times \frac{{\rho }_v}{{\rho }_g}}}$$

(4)

$$P\left(0\right)=1-{\displaystyle \frac{1}{1+\frac{R_g}{R_v\left(0\right)}\times \frac{{\rho }_v}{{\rho }_g}}}$$

(5)

2.3.2. LAI Estimation Under Uniform Canopy Distribution

Beer’s law is based on the clumping of horizontal uniformity and vertical stratification of continuous vegetation, and obtains the relationship among crown clearance rate, quenching coefficient, clumping index, leaf inclination, and LAI. Equation (6) is used to establish the relationship between clearance rate and LAI [45]:

$$P\left(z,\theta \right)=e^{-G\left(\theta \right)\times \Omega \left(\theta \right)\times {\displaystyle \frac{LAI\left(z,\theta \right)}{\mathit{cos}\left(\theta \right)}}}$$

(6)

where $P\left(z,\theta \right)$ is the gap rate at the observation zenith angle $\theta$ and height $z$, $G$ is the projection coefficient, and $\Omega$ is the clumping index. GEDI uses Beer’s law to estimate the effective LAI, assuming that the near-zenith observation is cos($\theta$) ≈ 1, the blade projection is a spherical distribution $G$ = 0.5, and the blades inside the footprint are randomly distributed $\Omega$($\theta$) = 1. Therefore, GEDI’s Formula (7) using Beer’s law can be obtained:

$$P\left(z\right)=e^{-0.5\times LAI\left(z\right)}$$

(7)

This process is the process for GEDI to obtain effective LAI. However, the spatial distribution of the forest involves the spatial structure of multiple crowns, including not only the gap rate of overlapping elements in the crown, but also the overlap and gap rate between crowns within the population. Beer’s law assumes that the crown is horizontally uniform and vertically layered, but in fact, the spatial distribution of the canopy is uneven, and the distribution of the crown and the distribution of the leaves inside the crown must not be independent of each other. Therefore, GEDI will have a certain error when it uses Beer’s law to obtain the LAI value at the footprint scale.

2.3.3. LAI Estimation Under Non-Uniform Canopy Distribution

This study has developed a GEDI LAI correction method based on a physical model, with the core being the coupling of the observed variables of the GEDI L2B product with a four-scale geometric optics model. Firstly, the canopy gap rate in the nadir direction observed by GEDI is taken as the primary boundary condition to ensure that the model simulation is consistent with the real observation in terms of macroscopic optical properties. Secondly, the canopy relative height parameters extracted from the GEDI L2A product are used to constrain the key geometric parameters in the four-scale model. On this basis, the four-scale model abandons the traditional random distribution clumping of Beer’s law and introduces the internal clumping effect of trees and the distribution pattern among tree crowns to jointly represent the non-uniform distribution of the canopy layer, decomposing the gap rate observed by GEDI into the inter-tree gap rate determined by the spatial distribution of trees and the intra-tree gap rate determined by the leaf clumping within the tree canopy. Through iterative optimization, the gap rate simulated by the four-scale model approaches the measured value by GEDI, and at this point, the output LAI of the model is the corrected LAI.

The four-scale geometric optical model was proposed in 1997 [46,47] on the basis of Beer’s law. The model was established based on the northern forest. The reflectance of the vegetation surface was obtained from the perspective of the sensor, and the canopy BRDF (Binomial Reflectance Distribution Function) was simulated. The four-scale model comprehensively considered the uneven distribution of the canopy and the presence of the canopy overlap effect, so as to calculate the background visibility probability. The four-scale model simulates the bidirectional reflection characteristics of forest canopy from four scales of leaves, branches, crowns, and communities. Specifically, there are the following four scales (Figure 3):

(1)

1-scale: turbid media.

(2)

2-scale: randomly distributed discrete objects containing turbid media.

(3)

3-scale: non-random discrete objects containing turbid media.

(4)

4-scale: non-random discrete objects with internal structures (such as branches and shoots).

Figure 3. Four-scale schematic diagram. 1-scale: turbid media. 2-scale: randomly distributed discrete objects containing turbid media. 3-scale: non-random discrete objects containing turbid media. 4-scale: non-random discrete objects with internal structures (such as branches and shoots).

Figure 3. Four-scale schematic diagram. 1-scale: turbid media. 2-scale: randomly distributed discrete objects containing turbid media. 3-scale: non-random discrete objects containing turbid media. 4-scale: non-random discrete objects with internal structures (such as branches and shoots).

They are characterized by Formula (8):

$$R=R_T{P}_T+R_G{P}_G+R_{ZT}{P}_{ZT}+R_{ZG}{P}_{ZG}$$

(8)

In the formula, $R_T$ and $P_T$ represent the reflectance of the light canopy component and the area rate of the component, $R_G$ and $P_G$ represent the reflectance of the light background component and the area rate of the component, $R_{ZT}$ and $P_{ZT}$ represent the reflectance of the shadow canopy component and the area rate of the component, $R_{ZG}$ and $P_{ZG}$ represent the reflectance of the shadow background component and the area rate of the component. The sum of the proportion of shadow background and light background in the total surface area is the background visibility probability ($P_G$ + $P_{ZG}$), which is also the gap rate [1].

Considering the gap within the tree crown and the overlap between the tree crowns, the gap rate can be expressed by Equation (9):

$$P={\sum }_{i=1}^k{P}_{tj}\left(V_g\right)P\left(z\right)+P_{t0}$$

(9)

where $i$ is the number of trees within the application scale of the model, $P_{tj}$ ($V_g$) is the probability that there are $j$ trees overlapping in the line-of-sight direction, $P\left(z\right)$ is the gap rate in the canopy obtained by using Beer’s law (it is the result of Formula (7)), and $P_{t0}$ is the gap rate without tree overlapping.

When the four-scale model is applied to LAI estimation of the GEDI footprint scale, more parameters are used, which is more in line with the real forest distribution in the footprint. The input parameters of the model mainly fall into three categories: plot parameters, tree structure parameters, and leaf and background spectral characteristics parameters. In the experiment, through the collected data parameters, a look-up table with an LAI step size of 0.1 is established [48,49,50]; the specific input parameters are shown in

Table 2. Input parameters of four-scale model.

Input Parameter	Coniferous Forest	Broad-Leaved Forest
Sample plot range/m2	900	900
Number of trees	100, 120, 150, 180	100, 120, 150, 180
Neyman clustering	3	2
Crown shape	Combination of cone and cylinder	Ellipsoid
Crown height	5/11 GEDI Canopy height	7/11 GEDI Canopy height
Trunk height	6/11 GEDI Canopy height	4/11 GEDI Canopy height
Crown radius	1	1.5
The clumping index	0.8	0.8
Slope	The actual slope of the sample plot	The actual slope of the sample plot
LAI	0.5–10 (step = 0.1)	0.5–10 (step = 0.1)
Solar zenith angle	38	38
Observation zenith angle	0	0
Relative azimuth	0	0

. To improve the inversion accuracy, a cost function was constructed to search for the optimized LAI:

$$J\left(LAI\right)=\sqrt{{\left(P_{sim}\left(LAI\right)-P_{obs}\right)}^2}$$

(10)

where J is a cost function, $P_{sim}$ is the background visual probability simulated by the four-scale model, which is the P in Equation (9), and $P_{obs}$ is the canopy gap rate measured by GEDI, which is the $P\left(z\right)$ in Equation (7).

The retrieved LAI value is determined as the optimal solution that satisfies $LAI=argmi{n}_{LAI}\left(J\right)$. This rigorous minimization ensures the selected value statistically best represents the observation. Finally, this result was verified through corresponding measured data. The parameters were set according to previous studies, and the specific input parameters are shown in Table 2 [17,40].

2.3.4. Selection of Influencing Factors

In order to explore the factors influencing the estimation of leaf area index, this study focused on terrain, vegetation type, and vegetation coverage. Since GEDI is a full-waveform dataset with a large footprint, based on past experience, excessive slope can lead to waveform aliasing and thus affect the extraction of observation parameters [51]. Therefore, the slope data generated from ALS data is used to verify the influence of slope on GEDI and LAI corrected by the four-scale model. In addition to terrain, due to the different leaf and growth distributions of different tree species types, their clumping degrees and shapes also vary. Therefore, in the experiment, broad-leaved tree species and coniferous tree species were studied, respectively. Another factor is vegetation coverage, which is a key factor for the accuracy of laser radar measurement. The reason for choosing this factor for analysis is that as the vegetation coverage increases, the types and quantities of vegetation elements contained within the footprint gradually increase, leading to an increase in the complexity of the canopy structure and its spatial distribution, and a more dense distribution. Due to the GEDI L2B standard algorithm being based on the idealized clumping of uniform distribution of the canopy in the horizontal direction, it fails to fully consider the canopy clumping effect that is commonly present in natural forests. In areas with higher coverage, the deviation between this “uniform distribution” clumping and the actual structure of the forest will be significantly amplified, thereby causing systematic errors in the inversion of LAI. Therefore, this study compared the estimation accuracy under different vegetation coverage levels to evaluate the improvement effect of the four-scale model in handling structural heterogeneity.

2.3.5. Accuracy Assessment

The LAI data measured in the sample plots were used to evaluate the accuracy of GEDI L2B LAI data and GEDI LAI data corrected by the four-scale model, so as to determine the effect of GEDI LAI data and the four-scale model on its correction. To evaluate the stability of the estimated values within the sample space, leave-one-out cross-validation was conducted. The statistical contents include: mean deviation bias, mean absolute error MAE, coefficient of determination R², and mean square error (RMSE) [52,53]. The specific process mechanism is as follows (Figure 4):

$$Bias={\displaystyle \frac{1}{n}}\times {\sum }_{i=1}^n\left(x_i-y_i\right)$$

(11)

$$MAE={\displaystyle \frac{1}{n}}\times {\sum }_{i=1}^n\left|x_i-y_i\right|$$

(12)

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(x_i-y_i\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}$$

(13)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}\times {\sum }_{i=1}^n{\left(x_i-y_i\right)}^2}$$

(14)

To ensure the objectivity of the assessment, all 60 measured data points are regarded as an independent validation dataset. Since the four-scale model used in this study is based on physical principles, the input parameters are independent of the validation data. To evaluate the stability of the estimated values within the sample space, leave-one-out cross-validation was conducted. In each iteration, one sample is retained as the validation point, while the remaining samples provide background information for the error distribution analysis.

3. Results

3.1. The Accuracy Under Uniform and Non-Uniform Distribution of the Canopy

By matching the GEDI footprint with measured LAI data, the results of all samples can be obtained (Figure 5).

The effective LAI of GEDI L2B level data in the Maoer Mountain area was verified using the measured data of the sample plots. It can be known from the results that the R² of the effective LAI measured by GEDI L2B data in the Maoer Mountain forest area is 0.86, and the RMSE is 1.47 m²/m²; it can be seen that its bias is −1.25 m²/m². There is a phenomenon that GEDI underestimates LAI, and the error in calculating LAI is large. If the product is used directly, there will be a large gap from the actual situation.

The gap rate parameter pgap_theta was extracted from the data at the GEDI L2B level, which is the gap rate within the footprint calculated by applying the GORT model. It corresponded to the background visibility probability of the established four-scale model lookup table to obtain the corresponding corrected LAI value of the four-scale model. And compared with the measured data, the results under the clumping of non-uniform distribution of the canopy were obtained (Figure 6).

The LAI of GEDI data corrected by the four-scale model was verified using the measured data. It can be known from the results that after the correction using the four-scale model, the R² of GEDI for measuring LAI is 0.88 and the RMSE is 0.82 m²/m². Compared with the original GEDI L2B-grade product, there has been a significant improvement in accuracy. Furthermore, the results of the measured leaf area index no longer show a situation where the values were severely underestimated as a whole. It can be seen that if the canopy is regarded as a non-uniform distribution and the clumping effect between the leaves and the canopy is considered, estimating LAI will have a very good effect.

In order to verify the optimization effect of the four-scale model on the GEDI product, this study conducted a paired-sample T-test on the original GEDI LAI, the LAI corrected by the four-scale model, and the ground-measured LAI (

Table 3. Significance analysis before and after LAI correction.

Comparison	p	Sig
G vs. M	3.73 × 10−18	***
F vs. M	0.025	*
G vs. F	3.13 × 10−11	***

Note: Sig represents the significance level; * indicates that the result is statistically significant at the 0.05 level; *** indicates that the result is highly significant at the 0.001 level; G represents the LAI value of GEDI L2B data; F represents the corrected LAI value using the four-scale model; M represents the measured LAI value.

).

The results show that there is a highly significant difference (p < 0.001) between the original GEDI L2B product and the measured values. After correction by the four-scale model, there is also a highly significant difference (p < 0.001) between the LAI and the original GEDI values, proving that the numerical gain produced by the model correction has statistical significance. Although there is still a certain difference between the corrected values and the measured values (p = 0.025), compared to the p-value of approaching 0 before correction, the significance level has significantly increased.

3.2. Strong and Weak Beam Influence

The GEDI data is classified into weak beams and strong beams, and the energy of the strong beam is 3.3 times that of the weak beam [45]. In dense sample areas, the waveform of the weak beam experiences severe energy attenuation after penetrating the deep canopy layer, making it difficult to accurately identify the ground echo from the background noise. This leads to deviations when calculating the gap rate. The four-scale model redefines the physical relationship between waveform energy distribution and vegetation structure by introducing the clumping effect parameter. Compared to the traditional algorithm that assumes uniform distribution, it has stronger robustness against waveform distortion caused by signal attenuation and can effectively compensate for the systematic error of the weak beam in detecting deep gaps. According to the beam, the beam values 0000, 0001, 0010, and 0011 are weak beams, and the beam values 0101, 0110, 1000, and 1011 are strong beams. The ability of GEDI to measure LAI with strong and weak beams in the Maoer Mountain study area and the measured LAI results after correction by the four-scale model were evaluated (

Table 4. Effect of measured LAI with strong and weak beams.

Beam Type	R2	Bias/(m2/m2)	MAE/(m2/m2)	RMSE/(m2/m2)
GEDI strong beam	0.89	−1.14	1.16	1.33
GEDI weak beam	0.82	−1.73	1.73	1.92
Four-scale corrected strong beam	0.90	−0.13	0.6	0.74
Four-scale corrected weak beam	0.86	−0.66	0.87	1.09

).

It can be known from the results that the performance of the strong beam in determining LAI is better than that of the weak beam. For the GEDI L2B level data, the R² of the weak beam is 0.82, and the RMSE is 1.92 m²/m². Strong beam R² = 0.89, and RMSE = 1.33 m²/m². After correction using the four-scale model, the effect of the weak beam was improved to R² = 0.86 and RMSE = 1.09 m²/m²; the effect of the strong beam has been enhanced to R² = 0.90 and RMSE = 0.74 m²/m². When using the data, it is advisable to choose the data with a strong beam as much as possible. If the data with a weak beam is used directly, the error will be relatively large. After correction by the four-scale model, there will also be obvious effects. The LAI determination effect of the weak beam has also reached a relatively good level.

3.3. Slope

The slope data generated from ALS data is used to verify the influence of slope on GEDI and LAI corrected by the four-scale model. The slopes are divided into six groups: 0–10°, 10–15°, 15–20°, 20–25°, 25–30°, and greater than 30° (Figure 7).

The GEDI L2B products were statistically analyzed, and the LAI accuracy was determined under different slope conditions after correction using the four-scale model (

Table 5. Effect of slope on GEDI and LAI measured after correction with four-scale model.

Slope/(°)	R2	Bias/(m2/m2)	MAE/(m2/m2)	RMSE/(m2/m2)
0–10	0.96/0.97	−0.51/0.21	0.67/0.45	0.76/0.55
10–15	0.99/0.96	−0.84/−0.21	0.84/0.52	0.91/0.68
15–20	0.94/0.98	−0.88/−0.33	0.88/0.36	0.99/0.47
20–25	0.96/0.95	−1.59/−0.29	1.59/0.58	1.65/0.79
25–30	0.95/0.86	−1.77–0.84	1.77/1.00	1.84/1.12
30–	0.65/0.70	−2.45/−0.49	2.45/0.78	2.52/1.00

Note: The data in the Table are the accuracy statistics of LAI determined by GEDI/corrected by the four-scale model.

).

It can be known from the results that, under the clumping of uniform canopy distribution, the GEDI data has a good estimation effect of LAI at relatively gentle slopes (0–20°). However, with the increase in the slope, when the slope is greater than 20°, the measurement error increases significantly, and at steep slopes (greater than 30°), the error is relatively large, reaching 2.52 m²/m². The reason for consideration is that as the slope increases, the large spot of the GEDI data undergoes distortion, and the echo signal is significantly affected. After correction using the four-scale model, the overall accuracy of the data has improved, especially when the slope is large, and the effect is obvious. In areas with a steep slope of more than 30°, the RMSE decreases by 1.52 m²/m², and the effect is extremely obvious.

3.4. Tree Species Type

By differentiating between coniferous and broad-leaved forests, the study examines the effects of GEDI measurements on leaf area index and their corrections under the two forest types (Figure 8).

The GEDI L2B products were statistically analyzed, and the LAI accuracy was determined under two tree species types after correction using the four-scale model (

Table 6. The influence of tree species types on GEDI and the determination of LAI after correction using the four-scale model.

Tree Species	R2	Bias/(m2/m2)	MAE/(m2/m2)	RMSE/(m2/m2)
Broad-leaved forest	0.92/0.90	−0.98/−0.24	1.01/0.59	1.17/0.77
Coniferous forest	0.89/0.88	−1.70/−0.60	1.70/0.85	1.81/0.96

Note: The data in the Table are, respectively, the accuracy statistics of GEDI’s determination of LAI/after correction using the four-scale model.

).

It can be known from the results that the effect of GEDI data on determining the LAI of broad-leaved tree species is better than that of coniferous tree species. Under the clumping of uniform canopy distribution, the RMSE of GEDI for determining the LAI of broad-leaved tree species is 1.17 m²/m², and the effect is significantly better than the RMSE of 1.81 m²/m² of coniferous tree species. After correction using the four-scale model and considering the clumping effect between canopies, the accuracy of both types of data was significantly improved. The RMSE of broad-leaved tree species was 0.77 m²/m², and that of coniferous tree species increased to 0.96 m²/m². It can be seen that after correction using the four-scale model, the data accuracy of both regions has been significantly improved. The accuracy of determining coniferous tree species has been improved to close to that of broad-leaved tree species, with good results.

3.5. Vegetation Coverage

To verify the influence of vegetation coverage on GEDI and LAI corrected by the four-scale model, the vegetation coverage was divided into five groups: 0–20%, 20–40%, 40–60%, 60–80%, and 80–100% (Figure 9).

The GEDI L2B products were statistically analyzed, and the LAI measurement accuracy was determined under different vegetation coverage conditions after correction using the four-scale model (

Table 7. The influence of vegetation coverage on GEDI and the measured LAI after correction using the four-scale model.

Vegetation Coverage/%	R2	Bias/(m2/m2)	MAE/(m2/m2)	RMSE/(m2/m2)
0–20	0.88/0.69	−0.79/−0.39	0.79/0.47	0.97/0.63
20–40	0.71/0.87	−1.13/0.66	1.32/0.66	1.29/0.91
40–60	0.99/0.87	−1.65/−0.38	1.65/0.49	1.61/0.51
60–80	0.77/0.97	−1.69/−0.27	1.69/0.43	1.67/0.53
80–100	0.45/0.73	−1.89/0.55	1.89/0.55	2.12/0.94

Note: The data in the Table are, respectively, the accuracy statistics of GEDI’s determination of LAI/after correction using the four-scale model.

).

It can be known from the results that with the increase in vegetation coverage, the RMSE of LAI measured by GEDI data decreases from 0.97 m²/m² to 2.12 m²/m². The denser the area, the worse the effect of LAI measured by GEDI data will be. When the coverage exceeds 80%, the RMSE exceeds 2 m²/m², and there will be a large error. After correction using the four-scale model, the accuracies of different vegetation coverage groups have all improved. The improvement effect is most obvious in densely populated areas. The RMSE of all groups is less than 1 m²/m². Meanwhile, the effect is the best in the medium vegetation coverage area (40–80%). Using the four-scale model to correct GEDI data can obtain relatively accurate results, even in densely forested areas.

4. Discussion

4.1. LAI Determination Under Different Canopy Distribution Clumping

Since GEDI is a large-footprint lidar dataset with a footprint diameter of 25 m, its coverage is relatively wide. Although the measurement of LAI using on-site methods, such as the leaf collection method, has relatively high accuracy, it is limited by human and material resources, and it is difficult to conduct large-scale measurements. Although using optical leaf area index products for reference can cover a wide area and save a lot of manpower and material resources, problems such as insufficient accuracy and low resolution will cause great limitations [28,31]. Therefore, obtaining more accurate LAI using GEDI data is of great significance for forest management and operation.

The product data of GEDI L2B is calculated based on the GORT model through the change in energy transmission to obtain the gap rate, and then the effective LAI at the footprint scale is derived based on Beer’s Law. Beer’s Law assumes that the forest canopy is uniformly distributed horizontally and that the leaves within the footprint are randomly distributed. Under this clumping, the results obtained using the measured data reveal that the LAI observation accuracy of the product data is not very high, and it is difficult to meet the requirements of high-precision operations. Considering the relatively complex structure of forests, the canopy is not uniformly distributed, and the leaves are mostly clustered. Therefore, using Beer’s Law to estimate LAI will cause significant errors [54,55,56].

The actual internal structure of the forest is intricate and complex. The four-scale model estimates parameters under the assumption that the forest canopy is non-uniformly distributed. Therefore, after obtaining the gap rate through the GORT model, the four-scale model is used to replace Beer’s Law in calculating LAI [37]. This method is more in line with the actual situation inside the forest. The four-scale model regards the forest as a series of discrete geometric objects rather than a uniform turbid medium. By taking into account the overlap of tree canopies and the distribution of leaf clusters in sunny and shady areas, this model successfully decomposes the gap rate into components representing the complexity of the structure between tree canopies and within tree canopies. Therefore, compared to Beer’s law, it better conforms to the distribution conditions of forests in real situations, and the results obtained have been significantly improved. The overall retrieval accuracy improved significantly, with the total RMSE dropping from 1.47 m²/m² to 0.82 m²/m² (Figure 5 and Figure 6). This result is of great significance for LAI estimation of the large footprint data of GEDI, significantly enhancing its application in the field of LAI estimation. It proves the feasibility of using this method to correct GEDI data and can also be promoted in the acquisition of other parameters in the future.

The four-scale model has significantly improved the estimation accuracy of the leaf area index, but it still has some inherent limitations in practical applications. The main issue lies in its high-dimensional parameter space. Although these structural parameters can be physically defined, obtaining the individual values of each parameter for a large number of GEDI footprints is overly complex in practical operation. Therefore, it is usually necessary to adopt the method of setting fixed values for multiple parameters based on forest types that are being calculated. Although this simplification improves the computational efficiency, it inevitably introduces systematic uncertainty due to the neglect of structural heterogeneity. Finding the optimal balance between model complexity and operational scalability remains a key focus of future research.

4.2. The Clumping Index

In fact, the canopy and leaves in the forest are not uniformly distributed. To improve the estimation of canopy structure and solar radiation, scholars have introduced the clumping index ($\Omega$). It is defined to quantitatively characterize the deviation degree between the true spatial distribution of canopy leaves and the random distribution, in order to quantitatively measure the degree of canopy deviation from the random distribution and correct the estimated leaf area index results. The leaves in the forest are mostly clustered, and the clustering index is mostly not 1 [57,58]. When the GEDI sensor estimates the LAI within the footprint using Beer’s Law, the leaves are regarded as randomly distributed. Directly setting the clustering index as 1 will cause certain errors [37,59,60,61]. Therefore, during calibration, the interior of the forest within the footprint is regarded as clustered and distributed. Introduce the parameter of the clumping degree index between blades.

The existence of the clumping index will directly affect the gap rate. In experiments, the clumping index is generally related to the forest type. Based on experience, the clumping index of broad-leaved forests is generally set at 0.8, and that of coniferous forests is set at 0.6 [43,58,62]. In addition to the clumping effect existing within the tree canopy, there is also a clumping effect between the tree canopies. Therefore, when GEDI’s L2B product estimates LAI using Beer’s Law, it neither considers the clumping distribution of leaves within the footprint nor the clumping effect between the tree canopies. As a result, there will be a large error. After correction using the four-scale model, the clumping effect at these two scales was solved, and the result will be more accurate [63].

Although applying the clumping index of tree species types to Beer’s law can improve the results through empirical means, the four-scale model provides a more profound physical explanation for this improvement. By modifying the clumping of uniform distribution of the canopy layer, the four-scale model can more accurately reflect the way laser pulses interact with the hierarchical structure of trees, including not only the clumping effect within a single tree but also the gaps between different canopy layers and the clumping phenomena within them.

Due to the different actual conditions in each study area, to ensure accuracy, it would be more effective to measure the concentration index on-site using instruments such as Trac in the study area to obtain the true concentration index. However, similar methods all consume a lot of manpower and material resources, and many study areas are rather complex and not suitable for entry; and there are certain requirements regarding the weather [64,65]. It is difficult to achieve large-scale measurement. Therefore, the following research can start from a detailed study of the clumping index to find a more effective method for measuring the accurate clumping index of the study area, so as to further improve the correction accuracy.

4.3. Analysis of the Influence of Different Factors on LAI Determination

The GEDI data is the full-waveform data of a large footprint. When determining LAI, it will be affected by multiple factors. This study analyzed the influence of different factors on the GEDI L2B product itself and the corrected results from various perspectives. It can be seen that the specific situations are as follows.

The overall determination of LAI by the GEDI L2B product is relatively low, and the same situation occurred in the past determination of forest canopy height by GEDI [40]. Considering that the reason is the systematic error of the spaceborne lidar itself, the spaceborne lidar GEDI underestimates when observing tree height and LAI. The bias value of LAI measurement was −1.25 m²/m². However, after correction using the four-scale model, underestimation has been significantly reduced, and its bias value was −0.23 m²/m², which also demonstrated the applicability of this correction method. The overall low measurement effect can still be corrected by the statistical model method, but correcting only at the mathematical level makes it difficult to achieve more persuasive results. Although the four-scale model method can improve the problem of the overall low measurement results, if this problem is to be completely solved, it should start from the principle of GEDI footprint observation and correct from the system problems of the laser itself. It can reduce the problem of low results caused by systematic errors to a greater extent.

The research results of strong and weak beams show that the effect of the GEDI L2B product in determining LAI with strong beams is significantly better than that with weak beams. Compared to the weak beam, the strong beam has greater penetrating power. It can be seen from the results that the error of weak beams is larger, and its RMSE is close to 2 m²/m² (Table 4). After correction using the four-scale model, the RMSE of the strong beam decreased to below 1 m²/m², and the RMSE of the weak beam also decreased significantly, with an RMSE of 1.09 m²/m². This result is even better than that of the strong beam before correction. Therefore, when screening data for high-precision operations, the strong beam should be chosen as much as possible. However, the weak beam after correction still yields good results. If a denser and larger amount of footprint data is needed, using the data without distinguishing the beam type after correction can also achieve highly accurate results.

In the experiment, the tree species types within the footprint were distinguished and divided into coniferous tree species and broad-leaved tree species for comparative experiments. It can be seen that GEDI data is more accurate than coniferous tree species in determining the area index of deciduous tree species. The reason is that the LAI of coniferous tree species is relatively low and there may be more outliers. After correction, the RMSE of the two types of data was below 1 m²/m², and the results were closer. Since both types of tree species have their own distribution situations, unifying their clumping degree indices will cause significant errors. The four-scale model takes into account the leaf clumping degrees and the clumping degrees between the crowns of the two types of tree species, respectively. To make the determination results of each type of tree species closer to the actual situation, whether in coniferous forests or broad-leaved forests, the data all have good effects.

Slope has a significant impact on large footprint data. GEDI emits laser pulses from the satellite to the ground (Figure 10). After reaching the ground, a circular footprint with a diameter of 25 m is formed. If the ground is not flat and has a certain slope, it will cause the footprint to deform, resulting in confusion between ground echoes and vegetation echoes, thereby causing a large measurement error [53].

Therefore, the study grouped the slopes into gentle slopes, low slopes, medium slopes, and high slopes to investigate the effect of GEDI in determining LAI. The results were similar to those of the GEDI L2A height product, both showing that the product accuracy decreased with the increase in the slope. After the GEDI L2B product data was corrected by the four-scale model, when the slope was less than 20°, the accuracy of measuring LAI performed well. When the slope was higher than 20°, the effect of measuring LAI decreased significantly, with RMSE changing from less than 1 m²/m² to more than 1.65 m²/m². On the steep slope with a slope greater than 30°, its RMSE is even higher than 2.50 m²/m². Compared with the results in a gentle terrain environment, the error of this result is too large, and the GEDI L2B product cannot be used directly on steep slopes. Under the correction of GEDI data, the model takes into account the terrain factor and corrects the errors caused by the slope. Specifically, the gap rate calculated during the correction of the four-scale model is the gap rate of the GEDI footprint after deformation on the slope, rather than the gap rate of the circular footprint on the flat ground. Therefore, the errors caused by the slope problem can be solved. From the results, it can be seen that the footprint on the steep slope still has a relatively high accuracy. The correction method has better solved the accuracy error caused by the slope problem.

It can be known from the research on the height product of GEDI L2A trees that the vegetation coverage will have a certain impact on the footprint data. Therefore, in this experiment, footprints were divided into five groups (low, medium-low, medium, medium-high, and high) for the experiment (Figure 11). The final result is that with the increase in vegetation coverage, the effect of the GEDI L2B product on determining LAI gradually decreases. In areas with high vegetation coverage, its RMSE is greater than 2 m²/m². This result has a large error. If the data is used directly, it is extremely inaccurate. It is considered that as the vegetation coverage increases, the types and quantities of objects in large footprints gradually increase, and the distribution of the crown becomes more complex. The distribution between canopies will be denser. The product itself assumes that the canopy is uniformly distributed horizontally. The clumping effect between the canopies was not considered, and thus the calculated LAI would also have greater errors.

After the correction of the four-scale model, the estimation accuracy under all vegetation coverage conditions has been improved. Among them, the improvement is most significant in areas with dense vegetation. By explicitly considering the clumping effects at the leaf level and the canopy level, this model can achieve good results regardless of the density of the sample plots. Compared to the original product, it has higher accuracy. Moreover, this structural parameterization has particular advantages in multi-layer forests because the understory vegetation usually accounts for a certain proportion of the total LAI. Due to the four-scale model’s explicit consideration of the geometric shadows and gaps between discrete canopies [46], it effectively captures the vertical complexity of the forest stand, which is the main reason for the overall improvement in accuracy compared to the standard GEDI L2B product.

Although the introduction of the four-component geometric optical model significantly improved the estimation accuracy of the GEDI LAI product, there are still certain limitations in its applicability in complex scenarios. Firstly, in areas with extremely steep slopes, the laser pulses emitted by GEDI will cause significant waveform broadening and superposition between the ground and canopy echoes. This phenomenon will lead to errors in the estimation of gap rates. Future research should introduce terrain correction models or utilize the high-density photon data from ICESat-2 for auxiliary verification to reduce the interference of terrain effects on the accuracy of large-spot waveforms. Secondly, in forests with extremely high LAI and a canopy closure approaching 1.0, although the four-scale model takes into account the clumping effect, in dense forest stands where the multiple scattering effect is significant, a single geometric optical clumping may have limitations. There should be subsequent research attempts to couple multiple physical mechanisms using a radiative transfer model to enhance the robustness of the signal saturation area. Thirdly, in fragmented vegetation areas, the GEDI footprint with a diameter of approximately 25 m often contains multiple types of land features or is located at the edge of the forest. In such discontinuous canopies, traditional clumping index characterization may not be able to fully capture the heterogeneity within the pixels. To address this issue, future work will integrate high-spatial-resolution Sentinel-2 or multi-spectral unmanned aerial vehicle data to refine the correction of the clumping index through sub-pixel-scale structural information.

5. Conclusions

The GEDI L2B level data product provides LAI data at the footprint scale and is of great significance in the large-scale estimation of LAI. In this study, the measured data from the Maoer Mountain area were used to verify the performance of GEDI L2B data products in determining LAI under the clumping of uniform horizontal distribution of the forest canopy. Moreover, a four-scale optical model was utilized to explore the correction of GEDI data under the condition of non-uniform distribution of the forest canopy to improve its effect in determining LAI. The influencing factors for the determination of LAI were studied. The results show that the effect of GEDI L2B level data in determining LAI is R² = 0.86 and RMSE = 1.47 m²/m², and the measured results are generally low. After correction, the effect of GEDI L2B data in determining LAI has been significantly improved, with R² = 0.88 and RMSE = 0.82 m²/m². The corrected results have high accuracy. Judging from the influencing factors, this study concludes that the measurement effect of the strong beam of the GEDI L2B product is much better than that of the weak beam. The measurement effect of leaf area index of broad-leaved species is better than that of coniferous species. An increase in slope will lead to footprint deformation, causing greater errors. As the slope increases, the effect of measuring LAI will decrease. An increase in vegetation coverage will lead to a decrease in the accuracy of the measurement. After correction, the measurement accuracy has significantly increased, and the influences caused by beam intensity, tree species type, slope, and vegetation coverage have all significantly decreased, fully demonstrating the rationality and excellent effect of this correction method. The results of this study provide a reference for the determination of LAI effects by GEDI data, offer an excellent data correction method, provide methodometric support for high-precision and large-scale LAI research, and are of great significance for the application of GEDI data and the operation and management of forest resources. However, the conclusions of this study are currently mainly applicable to temperate mixed coniferous and broad-leaved forests, and there are certain limitations in complex environments. Future research will utilize multi-source remote sensing data to enhance the scalability and robustness of the model, in order to be applicable to regional to global applications.