Accurate Estimation of Forest Canopy Height Based on GEDI Transmitted Deconvolution Waveforms

Highlights

What are the main findings?

• The Inverse Deconvolution of Transmitted Waveforms (IDTW) algorithm effectively mitigates waveform broadening and overlap in GEDI data caused by the non zero half width of transmitted laser pulses (NHWTLP).
• The developed rh model outperforms the RH model across various conditions (beam types, terrain slopes, and canopy closures) in estimating forest canopy height(FCH).

What is the implication of the main finding?

• IDTW reduces dependence on LiDAR hardware pulse width precision while maintaining high canopy height accuracy, providing a cost effective solution for global forest structure monitoring.
• Combining IDTW with terrain slope constraints enables robust estimation of canopy height in complex terrain and high closure forests, supporting accurate assessment of forest carbon stocks and carbon cycle dynamics.

Abstract

Accurate estimation of the forest canopy height is crucial in monitoring the global carbon cycle and evaluating progress toward carbon neutrality goals. The Global Ecosystem Dynamics Investigation (GEDI) mission provides an important data source for canopy height estimation at a global scale. However, the non-zero half-width of the transmitted laser pulses (NHWTLP) and the influence of terrain slope can cause waveform broadening and overlap between canopy returns and ground returns in GEDI waveforms, thereby reducing the estimation accuracy. To address these limitations, we propose a canopy height retrieval method that combines the deconvolution of GEDI’s transmitted waveforms with terrain slope constraints on the ground response function. The method consists of two main components. The first is performing deconvolution on GEDI’s effective return waveforms using their corresponding transmitted waveforms to obtain the true ground response function within each GEDI footprint, thereby mitigating waveform broadening and overlap induced by NHWTLP. This process includes constructing a convolution convergence function for GEDI waveforms, denoising GEDI waveform data, transforming one-dimensional ground response functions into two dimensions, and applying amplitude difference regularization between the convolved and observed waveforms. The second is incorporating terrain slope parameters derived from a digital terrain model (DTM) as constraints in the canopy height estimation model to alleviate waveform broadening and overlap in ground response functions caused by topographic effects. The proposed approach enhances the precision of forest canopy height estimation from GEDI data, particularly in areas with complex terrain. The results demonstrate that, under various conditions—including GEDI full-power beams and coverage beams, different terrain slopes, varying canopy closures, and multiple study areas—the retrieved height (rh) model constructed from ground response functions derived via the inverse deconvolution of the transmitted waveforms (IDTW) outperforms the RH (the official height from GEDI L2A) model constructed using RH parameters from GEDI L2A data files in forest canopy height estimation. Specifically, without incorporating terrain slope, the rh model for canopy height estimation using full-power beams achieved a coefficient of determination (R²) of 0.58 and a root mean square error (RMSE) of 5.23 m, compared to the RH model, which had an R² of 0.58 and an RMSE of 5.54 m. After incorporating terrain slope, the rh_g model for full-power beams in canopy height estimation yielded an R² of 0.61 and an RMSE of 5.21 m, while the RH_g model attained an R² of 0.60 and an RMSE of 5.45 m. These findings indicate that the proposed method effectively mitigates waveform broadening and overlap in GEDI waveforms, thereby enhancing the precision of forest canopy height estimation, particularly in areas with complex terrain. This approach provides robust technical support for global-scale forest resource assessment and contributes to the accurate monitoring of carbon dynamics.

1. Introduction

The Earth’s forest ecosystems play a crucial role in the monitoring and study of the global carbon cycle [1]. According to investigations, global forests store approximately 662 billion tons of carbon and absorb 2.4 billion tons of carbon dioxide annually, with a net carbon sink of about 1.1 ± 0.8 Pg C [2]. Additionally, a 2024 study revealed that the gross primary production of global plants via photosynthesis was 31% higher than previously expected values [3]. Therefore, the large-scale, multi-temporal, and high-precision monitoring of forest resources is essential in advancing global carbon cycle research [4].

The accurate estimation of the forest canopy height (FCH) is a key indicator in monitoring forest carbon storage and studying the global carbon cycle. Currently, when classified by data type, spaceborne Earth observation satellites used for forest resource monitoring fall into three categories: full-waveform Earth observation satellites, LiDAR point cloud Earth observation satellites, and optical imaging Earth observation satellites [5,6,7]. Given the vertical three-dimensional structure of forests and the dense distribution of forest canopies, utilizing full-waveform spaceborne LiDAR with strong penetration capabilities for forest resource investigations can fully leverage the advantages of this technology in estimating carbon sequestration in forest ecosystems. This approach also provides an important data foundation for monitoring the carbon cycle in global forest ecosystems [8,9].

Following the launch of the first full-waveform spaceborne LiDAR—namely the Ice, Cloud, and Land Elevation Satellite–Geoscience Laser Altimeter System (ICESat-GLAS)—the satellite collected a total of 1,984,210,719 laser footprint points during its operational period from 2003 to 2009. This satellite achieved groundbreaking results in terms of FCH [10], canopy cover [11], forest biomass [12], and the carbon cycle [13]. To enable high-resolution monitoring of the Earth’s three-dimensional structure and fundamentally improve the characterization of carbon cycle processes, NASA launched the Global Ecosystem Dynamics Investigation (GEDI) Earth observation satellite in 2018. GEDI is capable of monitoring global-scale forest structural parameters within the latitude range of 51.6°N to 51.6°S and has been widely applied in estimating parameters such as FCH [14,15], canopy cover [2,16], forest biomass [17,18,19], and forest primary productivity [1,20].

The key to accurately estimating FCH using spaceborne LiDAR full-waveform data is to address waveform broadening and reduce the overlap between canopy and terrain echoes [21]. However, due to the near-Gaussian distribution characteristics of GEDI’s transmitted waveforms, even on level ground, the returned laser pulses exhibit waveform broadening, with a half-width of approximately 15 ns. This broadening further exacerbates the overlap between canopy and terrain echoes [22]. Moreover, in non-flat forested areas, particularly in steep-slope regions, the small vertical gap between the forest canopy and the underlying terrain intensifies the overlap between GEDI’s canopy and terrain echoes, thereby increasing the difficulty in accurately estimating FCH [23].

To address the issues of waveform broadening and waveform overlap between the satellite LiDAR canopy and terrain echoes caused by non-flat terrain, which subsequently reduce the FCH estimation accuracy, previous studies on FCH estimation based on spaceborne LiDAR waveform data often incorporated terrain slope parameters [24,25]. As Lefsky found, the waveform length parameter of spaceborne LiDAR is easily affected by the terrain slope, so he introduced terrain slope parameters into the FCH estimation model [26]. Nie et al. found that correcting the terrain slope parameter through the projection characteristics of the footprints on the ground can reduce the waveform broadening caused by non-flat terrain, thereby improving the accuracy of FCH estimation [27]. However, spaceborne LiDAR echo waveforms still exhibit waveform broadening and waveform overlap caused by the non-zero half-width of the transmitted laser pulses (NHWTLP). Estimating FCH solely based on the spaceborne LiDAR waveform length and terrain slope parameters can partially address waveform broadening and overlap caused by the terrain slope. However, it does not eliminate the effects of waveform broadening and waveform overlap caused by NHWTLP, which hinders the accurate estimation of FCH.

To address the issues of waveform broadening and waveform overlap caused by NHWTLP in spaceborne LiDAR echo waveforms, previous approaches have primarily focused on adjusting the waveform length parameters and waveform estimation models to mitigate the effects of such broadening and overlap [28,29]. For example, Masato Hayashi introduced the leading edge and trailing edge parameters, which helped to reduce the waveform broadening caused by NHWTLP [30]. Xing et al. found that, when the waveform length parameter is used as a model factor, the logarithmic model has stronger applicability than the linear model in estimating FCH, and the application of the logarithmic model can, to some extent, mitigate waveform broadening caused by non-flat terrain and NHWTLP [31]. However, the above studies did not describe the issues of waveform broadening and waveform overlap caused by NHWTLP, nor did they conduct an in-depth investigation or propose effective mitigation strategies.

The objective of this study is to address the waveform broadening and overlap issues in GEDI data caused by the non-zero half-width of the transmitted laser pulse and the presence of non-flat terrain and to develop a forest canopy height estimation model applicable to complex topographic conditions, thereby improving the accuracy of GEDI waveform data for forest resource surveys. The main contributions of this research are summarized as follows. (1) A transmitted–received GEDI waveform deconvolution algorithm is proposed to effectively mitigate the waveform broadening and overlap between canopy and ground returns, both of which are exacerbated by the non-zero half-width of the transmitted laser pulse of the spaceborne LiDAR. This algorithm reduces the influence of the transmitted pulse half-width on GEDI return waveforms and relaxes the hardware requirements on the spaceborne LiDAR transmitter to minimize the transmitted pulse width. (2) A forest canopy height estimation model is developed based on the GEDI ground response function, with terrain slope incorporated as a model parameter. This approach addresses the waveform broadening and overlap of the ground response function induced by the terrain slope, thereby reducing the adverse impact of non-flat terrain on the accurate estimation of the forest canopy height. This study on the estimation of the forest canopy height under different beam types, terrain slopes, and canopy cover conditions confirms the applicability of the proposed algorithm for forest canopy height estimation.

2. Materials

2.1. Study Area

Considering that the latitude can influence the FCH estimation accuracy by altering forest vegetation types [22], we selected eight sites in North America, spanning different latitudes, as the study areas for data analysis. This was conducted to verify the applicability of the proposed IDTW, as well as the effectiveness of combining ground response functions and terrain slope parameters in addressing waveform broadening and overlap issues. Figure 1 shows the distribution map of these sites, each located in a different U.S. state. The names of the sites are PUUM, DSNY, JERC, UKFS, BLAN, WLOU, NOGP, and STEI, with the corresponding states being Hawaii, Florida, Georgia, Kansas, Virginia, Colorado, Texas, and Michigan, respectively.

Table 1. Detailed statistical data for different sites.

Site Name	Longitude and Latitude Range	Acquisition Time	Number of Footprints	Tree Height Range	Terrain Slope Range	Canopy Density Range
PUUM	19.48°N~19.63°N, 155.46°W~155.21°W	1 Jan 2020	2411	0.00~26.91	0.22~40.42	0.00~100
DSNY	28.02°N~28.15°N, 81.50°W~81.36°W	1 Sep 2021	2683	0.00~29.39	0.06~13.04	0.00~100
JERC	31.15°N~31.33°N, 84.58°W~84.35°W	1 Sep 2021	5331	0.69~99.62	0.00~20.13	0.00~100
UKFS	38.94°N~39.10°N, 95.27°W~95.12°W	1 Jul 2020	2078	0.00~27.21	0.02~35.88	0.00~100
BLAN	39.03°N~39.15°N, 78.11°W~77.93°W	1 Aug 2021	6097	0.00~40.98	0.04~44.16	0.00~100
WLOU	39.86°N~39.91°N, 105.95°W~105.88°W	1 Aug 2021	468	0.02~16.96	2.72~40.25	0.00~88
NOGP	45.76°N~45.85°N, 90.14°W~90.02°W	1 Jun 2020	2277	0.00~27.35	0.09~20.65	0.00~100
STEI	45.43°N~45.57°N, 89.63°W~89.36°W	1 Sep 2020	10,754	0.00~28.91	0.04~28.00	0.00~100

shows information such as time, latitude and longitude, and topographic slope for different stations.

2.2. GEDI Data

The GEDI satellite was successfully launched on 5 December 2018. It enables high-resolution monitoring of the vertical structural information of forests, filling the gap in 3D structural observations among NASA’s observational assets. It also facilitates the statistical analysis of the global-scale ecosystem carbon content within the latitude range of 51.6°N to 51.6°S.

GEDI product files are divided into four levels based on the types of data storage [22], and the GEDI L1B and L2A datasets were used in this study. The GEDI L1B product datasets provide the spot number of the center of each laser footprint, as well as the longitude, latitude, time, transmitted waveforms, and echo waveforms. The GEDI L2A product datasets provide information on the spot number, longitude, latitude, time, waveform length parameters, and canopy cover.

Since the release of the GEDI L2A product datasets, studies on FCH estimation based on GEDI echo waveforms have primarily used the waveform length parameter RH_N provided by these datasets as a model factor, overlooking the underlying data processing of GEDI echo waveforms. To address the unavoidable issues of waveform broadening and waveform overlap in waveform data processing, this study proposes an IDTW algorithm from the perspective of processing GEDI echo waveforms. The construction of the IDTW algorithm requires the use of the transmitted waveforms and echo waveforms from the GEDI L1B product datasets.

NASA’s official team defines the RH_N parameter as the vertical distance from the ground peak point to the point where the energy proportion within the effective echo waveform reaches N, with the end point of the effective echo waveform serving as the reference starting point. The effective echo waveform is defined as the segment of the GEDI echo waveform between its start and end points. It should be noted that RH_N represents the waveform length parameter of the GEDI echo signal.

2.3. Airborne Data

In large-scale forest resource monitoring studies based on spaceborne LiDAR waveform data, airborne LiDAR point cloud data are often used as measured data to validate the estimation results regarding forest structure parameters due to their high resolution and convenient acquisition method. To meet the needs of various environmental, climatic, and ecosystem management applications for long-term and multi-scale observation data, the United States has established the National Ecological Observatory Network (NEON). As one of the world’s largest ecosystem observation platforms, NEON aims to reveal the response mechanisms of ecosystems to climate change, land use change, and human activities through long-term monitoring at multiple scales and dimensions. Its data cover 81 ecological sites across the United States, integrating three core data sources—airborne remote sensing, ground observation, and aquatic monitoring—thus forming a complete observation system from molecules to landscapes. As of 2022, this network has accumulated a large number of high-precision observation datasets on the forest structure, carbon stock, and ecosystem dynamic changes. These datasets have been widely applied in climate change research, forest management, and biodiversity conservation [32,33,34]. To validate the applicability of the proposed algorithm in FCH estimation, this study utilized the digital terrain model (DTM) and canopy height model (CHM) data provided by NEON.

The DTM and CHM were derived from airborne LiDAR point cloud data provided by the National Ecological Observatory Network (NEON). After denoising, point classification, and spatial interpolation, the point clouds were processed into DTM and CHM products with a spatial resolution of 1 m. These datasets accurately characterize terrain variation and canopy height information, providing high-precision baseline data for terrain slope analysis and forest resource monitoring. Figure 2 illustrates the observation tracks of GEDI full-power and coverage beams within the DSNY site, along with the spatial coverage of the NEON data. To minimize potential biases in FCH estimation due to temporal mismatch, NEON datasets acquired within one month of the GEDI waveform collection were selected as reference data in this study.

2.4. Canopy Cover Data

To comprehensively analyze the applicability of the algorithm proposed in this study for accurate FCH estimation under different canopy covers, it is crucial to select a forest canopy closure product with sufficient canopy information to ensure the reliability of the research results.

The Global 30 m Landsat Tree Canopy Version 4 (TCC) dataset is a component of the Landsat land cover data products, providing global land cover and land type information [1,35]. The TCC land type product divides the ground into several components, including tree canopy cover, non-tree cover, urban/built-up areas, wetlands, and others. TCC Tree Canopy Cover provides high-resolution and consistent data on tree canopy cover globally, supporting various ecological research, environmental monitoring, and forest resource management and applications. Moreover, the TCC data, with a resolution of 30 m, can fully cover GEDI laser footprints (each with a diameter of 25 m). Therefore, TCC Tree Canopy Cover was used as measurement data in this study.

3. Methods

The waveform broadening and waveform overlap caused by NHWTLP and non-flat terrain are important factors that reduce the accuracy of FCH estimation. To address the issues of waveform broadening and overlap caused by NHWTLP in spaceborne LiDAR, this study proposes the IDTW algorithm, which combines the principle of convolving the satellite LiDAR laser pulse with the ground response function to retrieve the echo waveform. The IDTW algorithm mainly includes the construction of the initial ground response function, the development of the GEDI echo convolution convergence function, and the noise reduction and noise removal of GEDI waveform data, as well as the construction and optimization of the ground response function. To address the waveform broadening and overlap caused by non-flat terrain in spaceborne LiDAR, this study combines the GEDI ground response function obtained through IDTW processing and terrain slope parameters from DTM data to eliminate terrain slope information in the GEDI ground response function, thereby enabling high-precision FCH estimation. The technical roadmap of the research method is shown in Figure 3.

3.1. Ground Response Function Construction via IDTW

3.1.1. IDTW Algorithm

Based on waveform simulation principles [36], it is found that the GEDI echo waveform can be regarded as the convolution of the transmitted waveform and the ground response function, as shown in Equation (1). Therefore, by performing the deconvolution of the transmitted waveform on the GEDI echo waveform, we can obtain the GEDI ground response function, as shown in Equation (2):

$$W=F\ast A$$

(1)

$$A=W/F$$

(2)

where $W$ is the spaceborne LiDAR echo waveform. $F$ is the spaceborne LiDAR transmitted waveform. $A$ is the ground response function within a spaceborne LiDAR footprint. $\ast$ is the convolution processing. $/$ is the deconvolution processing.

However, the measured transmitted waveforms of the GEDI instrument contain various types of radar noise, primarily including current noise generated by the detectors and receiving circuitry, shot noise arising from the statistical fluctuations of photon detection, and thermal noise induced by the operating temperature of the device. These noise sources cause deviations between the measured transmitted waveforms and their theoretical counterparts, thereby affecting the precise estimation of FCH. Therefore, it is necessary to denoise the measured transmitted waveforms prior to performing deconvolution on the GEDI return waveforms. The noise reduction algorithm is as follows:

$$W_{emit\_\_after}=W_{emit}-W_{emit}(1:10)$$

(3)

where $W_{emit\_\_after}$ is the noise-reduced transmitted waveform of GEDI. $W_{emit}$ is the raw GEDI transmitted waveform. $W_{emit}(1:10)$ is the average amplitude of the first 10 frames of the raw GEDI transmitted waveform. In Figure 4, the brown waveform and the blue waveform are the raw and noise-reduced GEDI transmitted waveforms, respectively.

The GEDI measured echo waveforms contain not only radar noise but also atmospheric noise, causing them to significantly deviate from the GEDI theoretical echo waveforms. Therefore, in this study, noise reduction was also applied to the GEDI measured echo waveforms prior to IDTW. The noise reduction function is as follows:

$$W_{echo\_after}=W_{echo}-W_{echo}(1:100)$$

(4)

where $W_{echo\_after}$ is the noise-reduced GEDI echo waveform, and $W_{echo}$ is the GEDI echo waveform before noise reduction. $W_{echo}(1:100)$ refers to the mean amplitude of the first 100 frames of the GEDI echo waveform before noise reduction. In Figure 5, the brown waveform and the blue waveform represent the GEDI echo waveforms before and after noise reduction, respectively.

To construct the transmitted waveform deconvolution function, it is essential to gain an in-depth understanding of the interaction mechanism between the GEDI transmitted waveform and the ground response function. This process is illustrated in Figure 6 (schematic diagram of GEDI return waveform acquisition) and is described as follows. First, regarding the basic characteristics of the transmitted waveform, as shown in Figure 6a, the laser pulse emitted from the satellite exhibits an approximately cylindrical beam shape. Figure 6b indicates that the transmitted waveform within the beam follows a Gaussian distribution. For ease of visualization, its amplitude (referring to the transmitted waveform) is represented by the brightness of the red curve, where higher amplitudes correspond to higher brightness values. Second, for the propagation of the transmitted waveform and its interaction with the ground, Figure 6c, in which the curve is depicted by a red line, illustrates that the pulse energy decreases with increasing distances from the beam spot center. Figure 6d shows that, when the transmitted laser pulse approaches the ground, the elevation positions of the waveforms within the beam remain essentially consistent. Figure 6e demonstrates that, upon contacting the surface objects within the footprint, the transmitted laser pulse causes the waveforms occluded by these objects to be reflected—carrying object-specific information back to the satellite receiver—and that objects at different elevations introduce time delays in the reflected waveforms. Finally, regarding the formation of the return waveform, Figure 6f–h present the statistical process used to derive the amplitude values of the GEDI return waveform. Given that the return waveform represents the vertical energy distribution resulting from the interaction between the transmitted pulse and surface objects, we first rotate Figure 6d counterclockwise by 90° and then remove the object information from Figure 6f to obtain the reflected waveforms at different vertical vegetation layers. By summing the reflected waveforms in each frame, we derive the amplitude values of the return waveform, as shown in Figure 6h. A more detailed description can be found in the Appendix A. Based on this in-depth analysis of the interaction mechanism between the transmitted waveform and the ground response function, we expand the convolution function for the return waveform (Equation (5)) into the following explicit form:

$$\begin{array}{c}W(m)={\displaystyle \sum _{i=1}^{m+e_m-1}}f(i)\ast A(m-i+e_m)\\ W(n)={\displaystyle \sum _{i=1}^{I28}}f(i)\ast A(n-i+e_m)\\ W(p)={\displaystyle \sum _{i=1}^{N+I27-p+I}}f(128-i+1)\ast A(p-i+e_m)\end{array}$$

(5)

$$f(i)={\displaystyle \frac{E}{\sqrt{2\pi }\delta }}exp(-{\displaystyle \frac{{(i-e_m)}^2}{{2\delta }^2}})$$

(6)

where $W\left(m\right)$ is the amplitude value of the m-th echo waveform. $e_m$ is the frame value corresponding to the peak point of the transmitted waveform. $m$, $n$, $p$ are the frame values of the echo waveform. The sum of these three amplitude values gives the total length of the echo waveform. Moreover, $m\le 128-e_m$, $128-e_m<n\le N-128+e_m$, 128 are the total lengths of the transmitted waveform. N denotes the total number of vertical layers of ground information within the footprint, divided at 0.15 m intervals. $\ast$ represents the multiplication operation. $f\left(i\right)$ is the amplitude value of the transmitted waveform in the i-th frame, in mJ. $A\left(i\right)$ is the horizontal projection area of the vegetation in the i-th layer in the vertical direction. $E$ is the energy value of the transmitted waveform. $\delta$ is the standard deviation of the transmitted waveform, in ns.

3.1.2. Construction of the Ground Response Function

When constructing the GEDI ground response function based on the process of IDTW, it is difficult to obtain the final GEDI ground response function through a single inverse deconvolution of the GEDI echo waveform. Instead, multiple iterations of inverse deconvolution processing are required. The results are validated by comparing the maximum amplitude difference between the GEDI convolved echo waveform (the echo waveform obtained by convolving the GEDI ground response function with the transmitted waveform) and the GEDI measured echo waveform. If this difference is smaller than the noise threshold, the IDTW process is considered successful.

Additionally, based on the principle of obtaining the echo waveform by convolving the transmitted waveform with the ground response function [37], the initial ground response function directly affects the number of iterations in the IDTW process. Therefore, in this study, the initial ground response function is constructed by dividing the amplitude of the GEDI echo waveform by the peak value of the emitted waveform, based on the principle that the peak of the emitted waveform has the greatest impact on the amplitude of the echo waveform. The formula is as follows:

$$\mathrm{A}\left(\mathrm{i}\right)=\mathrm{W}\left(\mathrm{i}\right)/\mathrm{f}\left({\mathrm{e}}_{\mathrm{m}}\right)$$

(7)

In addition, the research data in this study consist of multi-site, multi-beam GEDI echo waveforms, which results in a large data volume. To improve the processing efficiency of GEDI waveform data during the inverse deconvolution iterative process, this study attempts to convert the one-dimensional ground response function into two-dimensional matrix data. First, the initial ground response function is diagonalized. The formula is as follows:

$$\left(\begin{array}{cccc}{a}_1& 0& 0& 0\\ 0& a_2& 0& 0\\ 0& 0& \ddots & 0\\ 0& 0& 0& a_N\end{array}\right)=\mathrm{diag}(\mathrm{G})$$

(8)

where $a_i$ is the projected area of the ground object in the i-th layer within the GEDI footprint.

After diagonalizing the initial ground response function, the GEDI convolved echo waveform can be regarded as the sum of the column terms resulting from the multiplication of the two-dimensional ground response function and the transmitted waveform. The convolution formula is as follows:

$$\mathrm{W}=\mathrm{sum}\text{\_}\mathrm{column}\left(\begin{array}{ccccccc}\mathrm{f}(1)*a_1& \dots & \mathrm{f}(128)*a_1& & & & \\ & \mathrm{f}(1)*a_2& \dots & \mathrm{f}(128)*a_1& & & \\ & & \mathrm{f}(1)*a_3& \dots & \mathrm{f}(128)*a_1& & \\ & & & ⋮& \dots & ⋮& \\ & & & & \mathrm{f}(1)*a_N& \dots & \mathrm{f}(128)*a_N\end{array}\right)$$

(9)

where $\mathrm{sum}\_\mathrm{column}$ is the summation of the matrix column terms. $N$ denotes the total number of vertical layers of ground information within the footprint, divided at 0.15 m intervals.

There is a significant deviation in the amplitude values between the GEDI convolved echo waveform and the measured echo waveform, as shown in Figure 7. The GEDI convolved echo waveform is obtained by convolving the initial ground response function with the transmitted waveform. This indicates that the initial ground response function differs greatly from the actual ground response function. To obtain the actual GEDI ground response function, this study attempts to construct a GEDI echo convolution convergence function to reduce the amplitude difference between the convolved echo waveform and the measured echo waveform. However, a significant amount of noise data still remains in the noise-reduced GEDI echo waveform, as shown in Figure 8. If the convolution convergence function is constructed using a GEDI echo waveform that contains noise data, the noise signals in the waveform may be mistakenly identified as valid signals, reducing the acquisition accuracy of the ground response function. Therefore, denoising the GEDI echo waveform is necessary before constructing the GEDI echo convolution convergence function.

When removing noise from the noise-reduced GEDI echo waveform, it is considered that the GEDI echo waveform is formed by the superposition of transmitted waveforms that approximately follow a Gaussian distribution [36]. Therefore, this study uses a Gaussian fitting function to remove noise from the noise-reduced GEDI echo waveform. The Gaussian fitting algorithm is as follows:

$${\mathrm{W}}_{\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{d}}=\sum _{\mathrm{i}=1}^{\mathrm{L}}\mathrm{a}\ast \mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{{(\mathrm{x}-\mathrm{b})}^2}{2{\mathrm{c}}^2}})$$

(10)

where ${\mathrm{W}}_{\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{d}}$ is the denoised GEDI echo waveform. $\mathrm{L}$ is the number of iterations for the Gaussian fitting of the GEDI echo waveform, which is set to 8 in this study. $\mathrm{a}$ is the peak value of the Gaussian-fitted waveform. $\mathrm{b}$ is the frame number corresponding to the peak point of the Gaussian-fitted waveform. $\mathrm{c}$ is the standard deviation of the Gaussian-fitted waveform.

After removing noise from the noise-reduced GEDI echo waveform, to reduce the amplitude deviation between the GEDI convolution echo and the actual GEDI measured echo waveform, this study constructs the GEDI echo convolution convergence function by constraining the ratio of the difference between the convolution echo and the measured echo to the ground response function, leveraging the one-to-one correspondence between the GEDI convolution echo and the ground response function. The formula is as follows:

$${\mathrm{G}}_{d+1}={\mathrm{G}}_d+{\mathrm{G}}_d/W_d\ast W_{-}dif$$

(11)

$$W_{dif}=W_{denoised}-W_d$$

(12)

where ${\mathrm{G}}_{d+1}$ is the GEDI ground response function obtained after the $d+1$ iteration calculation. $W_d$ is the GEDI convolved echo obtained after the d-th iteration calculation. $W_{dif}$ is the difference function between the noise-reduced GEDI echo waveform and $W_d$.

Determining the iteration count of the GEDI echo convolution convergence function is crucial in evaluating the quality of IDTW. Considering that the noise signal amplitude of the denoised GEDI echo waveform is relatively low, this study extracts the noise threshold from the noise-reduced GEDI echo waveform. When the maximum amplitude difference between the GEDI convolution echo and the denoised waveform is smaller than the noise threshold, the IDTW process is considered complete, and the obtained ground response function is regarded as the real value. The noise threshold extraction formula is as follows:

$$\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}=\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\left({\mathrm{W}}_{\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{o}\_\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r}}\right(1:100)+\mathrm{k}\ast \mathrm{s}\mathrm{t}\mathrm{d}({\mathrm{W}}_{\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{o}\_\mathrm{a}\mathrm{f}\mathrm{t}\mathrm{e}\mathrm{r}}(1:100)\left)\right)$$

(13)

where $\mathrm{threshold}$ is the noise threshold. $\mathrm{mean}$ and $\mathrm{std}$ are the mean and standard deviation of the vector, respectively. $\mathrm{k}$ is the standard deviation multiplier, typically set to 3. Figure 9 shows a comparison between the GEDI convolution echo waveform and the noise-reduced GEDI echo waveform.

3.1.3. Optimization of Ground Response Function

The GEDI ground response function does not carry background noise, making it difficult to establish a background noise threshold based on its amplitude values and subsequently extract waveform feature parameters. Consequently, this study leverages the correspondence between the GEDI ground response function and the surface object information in the GEDI echo waveform [36], attempting to determine the background noise threshold of the GEDI ground response function using the noise-reduced GEDI echo waveform. However, the amplitude of the GEDI ground response function is significantly lower than that of the GEDI echo waveform, as shown in Figure 9 and Figure 10. Therefore, before constructing the background noise threshold for the GEDI ground response function, both the denoised GEDI echo waveform and the GEDI ground response function need to be optimized.

This study attempts to normalize the denoised GEDI echo waveform and the GEDI ground response function to construct a background noise threshold. Considering that the GEDI transmitted waveform has 128 frames [23], this study, based on the principle of GEDI echo convolution, sets the mean value of the first 127 frames of the denoised and normalized GEDI echo waveform to zero to establish the normalization formula. The formula is as follows:

$${\mathrm{W}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}}=(\mathrm{W}-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}(\mathrm{W}(1:127)\left)\right)/\left(\mathrm{m}\mathrm{a}\mathrm{x}\right(\mathrm{W})-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}(\mathrm{W}(1:127))$$

(14)

$${\mathrm{G}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}}=(\mathrm{G}-\mathrm{m}\mathrm{i}\mathrm{n}(\mathrm{G}\left)\right)/\left(\mathrm{m}\mathrm{a}\mathrm{x}\right(\mathrm{G})-\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}(\mathrm{G}(1:127))$$

(15)

where ${\mathrm{W}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}}$ is the normalized waveform data of the denoised GEDI echo waveform, and ${\mathrm{G}}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}}$ is the normalized waveform data of the GEDI ground response function. Figure 11 shows the comparison of the normalized GEDI echo waveform and the GEDI ground response function.

3.2. Construction of FCH Estimation Model Combining the Ground Response Function and Slope Constraints

3.2.1. Extraction of Waveform Length Parameter

To evaluate the effectiveness of the IDTW and the combination of the ground response function with terrain slope parameters in addressing GEDI waveform broadening and waveform overlap, this study conducts relevant analyses. This study estimates the FCH using the waveform feature parameters extracted from the ground response function and verifies the applicability of the proposed method based on the estimation accuracy.

The waveform length parameter extracted based on the GEDI ground response function is consistent with that of the RH_N waveform length parameter extraction method provided by GEDI L2A [14,15].

Considering that normalization applies the same gain to both the valid signal and the noise signal in the GEDI echo waveform, this study constructs the ground response function background noise threshold using the regularized GEDI echo waveform. The calculation formula is as follows:

$${\mathrm{T}\mathrm{h}}_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}=\mathrm{k}\ast \mathrm{s}\mathrm{t}\mathrm{d}\left({\mathrm{W}}_{\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{d}}\right(1:127\left)\right)$$

(16)

where ${\mathrm{T}\mathrm{h}}_{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}$ is the background noise threshold, and $\mathrm{k}$ is a multiple of the standard deviation.

Figure 12 shows a schematic of the ground response function waveform length parameter extraction. In this study, the stop point of the ground response function waveform data is taken as the starting point, and the vertical distance to the stop point for different energy percentage distances in the effective echo waveform is calculated. The energy percentage is $N$ recorded as the waveform length parameter $r{h}_N$, 0 $\le N\le 1$00. It should be noted that $r{h}_N$ represents the waveform length parameter of the ground response function, which is different from $R{H}_N$, the waveform length parameter of the GEDI return waveform.

3.2.2. Construction of FCH Estimation Model

To analyze the applicability of IDTW in addressing waveform broadening and waveform overlap caused by the NHWTLP, this study constructs an FCH estimation model using the parameters $r{h}_N$ and $R{H}_N$ and verifies the effectiveness of IDTW by comparing the estimation accuracy. The FCH estimation model is shown as follows:

$$H=d\ast r{h}_N+e$$

(17)

$$H=d\ast (R{H}_N-R{H}_0)+e$$

(18)

where $H$ is the FCH estimation value. $D$ is the GEDI footprint diameter. $g$ is the terrain slope provided by the DTM product in the NEON data. $d$ and $e$ are the correlation coefficient values.

The GEDI echo waveform contains both vertical structural information about the forest and waveform broadening caused by NHWTLP, differing from the DTM data, which only contain vertical elevation information [14,15]. By applying IDTW to the GEDI echo waveform, the issue of waveform broadening caused by NHWTLP is resolved. Therefore, combining the ground response function with DTM data to construct an FCH estimation model should have stronger applicability in mitigating the waveform broadening and waveform overlap caused by non-flat terrain. In this study, terrain slope parameters were introduced into both the rh and RH models, and the feasibility of combining the ground response function with DTM data for FCH estimation was verified by comparing the estimation accuracy of the two models. The estimation models after introducing the terrain slope parameters are as follows:

$$H=a\ast r{h}_N-b\ast D\ast tand\left(g\right)+c$$

(19)

$$H=a\ast (R{H}_N-R{H}_0)-b\ast D\ast tand\left(g\right)+c$$

(20)

where $R{H}_0$ is the vertical distance from the stop point of the GEDI echo waveform to the peak point of the ground wave, and $tand$ is the tangent function. $a$, b, and $c$ are the correlation coefficient values.

4. Results and Discussion

To address the issues of waveform broadening caused by NHWTLP and non-flat terrain, as well as the waveform overlap between canopy echoes and terrain echoes, this study proposes the following approach. First, this study uses the CHM and DTM data provided by NEON as the measured FCH data and measured terrain slope data, respectively. Then, the waveform length parameter extracted from the ground response function obtained by applying IDTW to the GEDI echo waveform is used as a model factor to estimate the FCH. Finally, the estimated FCH is compared with the FCH derived from the waveform length parameters provided by the GEDI L2A file, in order to verify the applicability of IDTW in solving the issues of waveform broadening and overlap caused by NHWTLP. In addition, the terrain slope parameter g is introduced into the FCH estimation model. By analyzing the accuracy of FCH estimation, this study assesses the feasibility of combining the GEDI ground response function with the terrain slope parameter to address the issues of waveform broadening and overlap caused by the terrain slope.

4.1. FCH Estimation Results for the GEDI Full-Power and Coverage Beams

The difference in the transmitted laser pulse energy between the GEDI full-power beams and the coverage beams is the key factor causing the significant disparity in echo waveform data quality [38]. To explore the applicability of IDTW and the combination of the ground response function and the terrain slope in addressing waveform broadening and waveform overlap, this study conducted FCH estimation based on two types of echo waveforms separately.

Table 2. Estimation results of FCH from GEDI full-power and coverage beams.

Beam Name		Full-Power Beams	Coverage Beams
Mean terrain slope/°		6.09	6.26
Mean canopy cover/%		57.00	63.59
Mean SNR		22.83	18.56
Number of footprints		18,288	13,811
$\mathrm{r}\mathrm{h}$ model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	96	92
	R2	0.58	0.57
	RMSE	5.23	4.58
$\mathrm{r}\mathrm{h}$_g model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	96	92
	R2	0.61	0.58
	RMSE	5.21	4.55
$\mathrm{R}\mathrm{H}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	98	95
	R2	0.58	0.56
	RMSE	5.54	4.72
$\mathrm{R}\mathrm{H}\_\mathrm{g}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	98	95
	R2	0.60	0.56
	RMSE	5.45	4.71

presents the FCH estimation results for GEDI full-power and coverage beams under different model conditions. From the perspective of either the GEDI full-power beams or the coverage beams, it was found that the FCH estimation results obtained using the rh model were superior to those obtained using the RH model. This suggests that applying IDTW to GEDI echo waveforms can accurately obtain the ground response function, addressing the waveform broadening caused by NHWTLP and thereby reducing the waveform overlap between GEDI canopy echoes and terrain echoes.

Moreover, when the GEDI laser pulse has a half-width of 15 ns, the rh model exhibits higher estimation accuracy compared with the RH model. This indicates that, even with a relatively large half-width of the spaceborne LiDAR (GEDI) laser pulse, the waveform broadening and waveform overlap caused by NHWTLP can still be addressed through IDTW processing.

The rh model shows an improvement in the FCH estimation accuracy for both GEDI full-power and coverage beams after terrain slope is incorporated into it. The analysis demonstrates that terrain slope information is still present in the ground response function. Furthermore, the rh model, after terrain slope parameters are introduced into it, can effectively address the waveform broadening and waveform overlap caused by non-flat terrain.

After the terrain slope parameter is introduced into the RH model, the model shows an improvement in the FCH estimation accuracy for both GEDI full-power beams and coverage beams. However, this improvement is relatively small compared with that of the rh model. The results reveal that the RH parameter also contains terrain slope information; introducing the terrain slope parameter can reduce the waveform broadening and waveform overlap caused by non-flat terrain in GEDI echoes. Nevertheless, the RH parameter is derived from measured GEDI echo waveforms, which already include waveform broadening and overlap caused by NHWTLP. Therefore, introducing terrain slope data into the RH model cannot resolve the issue of waveform broadening and overlap caused by NHWTLP. As a result, although the accuracy of the RH model improves after the terrain slope parameter is incorporated into it, the improvement is relatively small compared with that of the rh model.

In addition, a comparison of the FCH estimation results for GEDI full-power beams and coverage beams reveals that the optimal parameter values for the rh model are lower than those for the RH model. The research results demonstrate the applicability of IDTW in addressing the waveform broadening of GEDI echoes from the perspective of the waveform energy proportion. This study analyzed the composition mechanism of GEDI echo waveforms by combining the GEDI convolution function (Equation (9)). It was found that the amplitude of a GEDI echo waveform is the result of summing the products—for each term—of the projection areas of multi-layered objects and the transmitted waveforms. In other words, there is an overlap of object information between adjacent GEDI echo waveforms. Therefore, compared with the amplitude of the GEDI ground response function, the amplitude of GEDI canopy echoes is higher. The mean FCH mainly corresponds to the elevation located in the left half of the GEDI canopy echo. The elevated amplitude of GEDI canopy echoes reduces the proportion of waveform energy between the waveform start point and the mean FCH point. As a result, when estimating the mean FCH, the optimal RH parameter value in the rh model tends to be lower than that in the RH model.

4.2. FCH Estimation Under Different Terrain Slopes

Terrain slope is an important factor causing waveform broadening in spaceborne LiDAR echo waveforms, increasing the waveform overlap between canopy and sub-canopy terrain echoes, and ultimately reducing the FCH estimation accuracy [38,39]. To explore the applicability of IDTW and the combination of ground response functions with terrain slope parameters in addressing waveform broadening and waveform overlap, this study investigates FCH estimation under different terrain slopes at 10° intervals.

Table 3. FCH estimation results under different terrain slopes.

Terrain Slope Range/°		0~10		11~20		21~30		>30
Beam Name		Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams
Mean terrain slope/°		4.28	4.33	13.37	13.33	23.53	23.19	33.68	32.86
Mean canopy cover/%		53.36	59.99	73.90	79.15	79.50	78.66	77.51	66.41
Mean SNR		22.83	18.82	23.01	17.53	22.81	16.30	20.95	15.23
Number of footprints		15,162	11,192	2711	2323	366	274	49	22
$\mathrm{r}\mathrm{h}$ model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	97	92	95	93	89	91	46	52
	R2	0.58	0.57	0.67	0.58	0.60	0.53	0.78	0.77
	RMSE	5.66	4.59	4.24	4.56	4.78	5.31	3.96	3.51
$\mathrm{r}\mathrm{h}$_g model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	97	92	95	93	89	91	60	52
	R2	0.59	0.58	0.67	0.58	0.61	0.55	0.81	0.77
	RMSE	5.61	4.55	4.24	4.56	4.72	5.17	3.71	3.50
$\mathrm{R}\mathrm{H}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	98	95	96	95	90	58	45	40
	R2	0.58	0.56	0.66	0.53	0.57	0.45	0.69	0.50
	RMSE	5.67	4.64	4.30	4.81	4.94	5.70	4.73	5.19
$\mathrm{R}\mathrm{H}\_\mathrm{g}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	98	95	96	95	90	58	85	40
	R2	0.59	0.57	0.66	0.54	0.59	0.48	0.75	0.50
	RMSE	5.62	4.59	4.30	4.81	4.82	5.54	4.26	5.19

presents the FCH estimation results for GEDI full-power beams and coverage beams under different terrain slopes. Figure 13 and Figure 14 are bar charts showing the FCH estimation accuracy for GEDI full-power beams and coverage beams under varying terrain slopes, respectively. Based on Table 3 and Figure 13 and Figure 14, it is observed that, under different terrain slope conditions, the FCH estimation accuracy is as follows: the rh model outperforms the RH model, the GEDI full-power beams perform better than the coverage beams, and the rh_g model outperforms the rh model, while the RH_g model outperforms the RH model. Additionally, the optimal rh parameter value is lower than that of the RH parameter. The research results are consistent with the FCH estimation results under GEDI full-power beams and coverage beams. The study verifies the applicability of IDTW, as well as the combination of ground response functions and terrain slope parameters, in addressing waveform broadening and waveform overlap from the perspective of different terrain slopes. Moreover, this approach reduces the hardware requirements of spaceborne LiDAR transmitters to narrow the laser pulse half-width.

However, in this study, the FCH estimation accuracy did not decrease with increasing terrain slope, which differs from previous reports [15,40]. Statistical analysis of GEDI data under different terrain slopes revealed that the number ratio of full-power beam footprints in the 0–10°, 10–20°, 20–30°, and >30° intervals was approximately 309:55:7:1, while, for coverage beams, the ratio was about 509:106:12:1. The random distribution of trees increased their vertical structural variability, leading to discrepancies between the estimated and measured FCH values. Additionally, the increase in the number of footprints amplified the variability in deviations between the estimated and observed values. Therefore, even in areas with lower slopes, the FCH estimation accuracy may still decrease due to the larger number of footprints. However, both the number of footprints and the estimation accuracy in the 20–30° slope interval were lower than in the 10–20° interval. A comparison of the GEDI footprint statistics revealed that the mean signal-to-noise ratio (SNR) of waveforms in the 20–30° interval was lower than in the 10–20° interval. Moreover, steep terrain remains a key factor reducing the FCH estimation accuracy. Therefore, under the combined influence of multiple factors, the canopy height estimation accuracy at slopes of 20–30° may be lower than at slopes of 10–20°. In addition, statistical analyses were conducted to compare the footprint-level errors between the 10–20° and 20–30° slope intervals. The results indicated that the errors in the 20–30° interval were significantly greater than those in the 10–20° interval (Welch’s t-test and Mann–Whitney U test, both p < 0.001), with an effect size in the small-to-medium range. These findings provide statistical evidence that an increasing slope reduces the canopy height estimation accuracy. However, given the modest effect size, together with the lower signal-to-noise ratio and the smaller number of footprints in the steeper slope interval, the slope should be considered an important but not exclusive factor, with the footprint count and waveform quality also contributing to the observed decline in accuracy.

Figure 15 is a bar chart showing the improvement in the estimation accuracy of the rh model compared to the RH model for GEDI full-power beams and coverage beams under different terrain slopes. By analyzing Table 3 and Figure 15, it is observed that the improvement increases with the terrain slope. The data illustrate that waveform broadening and waveform overlap, caused by NHWTLP, also increase as the terrain slopes increase. By analyzing Table 3 and Figure 14, it is observed that the improvement magnitude increases with increasing terrain slopes. These results imply that waveform broadening and waveform overlap, caused by the non-zero value of the emitted laser pulse half-width, also increase as the terrain slope increases. Moreover, the study results validate the effectiveness of the proposed IDTW algorithm in addressing waveform broadening and waveform overlap caused by NHWTLP. Additionally, the effectiveness of the algorithm becomes more pronounced as the terrain slopes increase.

In addition, compared to the RH model, the rh model exhibits a greater improvement in estimation accuracy for coverage beams than for full-power beams. It can be observed that the IDTW algorithm proposed in this study can effectively filter out noise data while preserving GEDI signal data. However, GEDI full-power beams have a higher waveform SNR compared to coverage beams. After IDTW processing, although the SNR of the waveform data improves, the increase is relatively small. Therefore, in terms of the FCH estimation accuracy, the improvement for GEDI coverage beams is greater than that for GEDI full-power beams.

4.3. FCH Estimation Under Different Canopy Covers

The canopy cover can affect the extraction accuracy of the waveform length parameter by altering the amplitude of the spaceborne LiDAR echo waveform, thereby reducing the accuracy in estimating the FCH from GEDI waveform data [11,16]. To explore the applicability of IDTW and the combination of ground response functions with terrain slope parameters in addressing waveform broadening and waveform overlap, this study investigates FCH estimation under different canopy coverages at 20% intervals.

Figure 16a,b show bar charts of the FCH estimation accuracy obtained using the rh model and RH model for GEDI full-power beams and coverage beams, respectively. Figure 16c,d present bar charts of the FCH estimation accuracy obtained using the rh_g model and RH_g model for GEDI full-power beams and coverage beams, respectively. Based on

Table 4. FCH estimation results under different canopy covers.

Canopy Cover Range/°		0~20		21~40		41~60		61~80		81~100
Beam Name		Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams	Full-Power Beams	Coverage Beams
Mean terrain slope/°		4.67	4.58	5.46	5.22	4.98	5.11	5.98	6.35	7.35	7.27
Mean canopy cover/%		1.33	1.44	29.15	29.04	49.47	49.21	70.37	70.72	95.52	95.93
Mean SNR		23.27	19.00	23.17	17.93	22.87	17.60	22.84	17.64	22.48	17.55
Number of footprint points		5476	3542	1003	557	1873	1070	1428	982	8508	7660
$\mathrm{r}\mathrm{h}$ model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	90	90	96	83	98	82	97	90	96	92
	R2	0.53	0.53	0.68	0.59	0.61	0.58	0.54	0.52	0.54	0.47
	RMSE	2.95	2.99	5.59	3.50	6.68	3.74	5.89	3.76	4.16	4.27
$\mathrm{r}\mathrm{h}$_g model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	90	90	96	83	98	82	97	90	96	94
	R2	0.53	0.53	0.70	0.59	0.65	0.58	0.57	0.52	0.54	0.47
	RMSE	2.95	2.99	5.37	3.49	6.32	3.74	5.74	3.76	4.14	4.26
$\mathrm{R}\mathrm{H}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	98	93	98	87	98	85	99	93	97	95
	R2	0.36	0.52	0.63	0.57	0.59	0.54	0.54	0.50	0.52	0.41
	RMSE	5.99	3.01	6.03	3.59	6.84	3.92	5.93	3.85	4.25	4.49
$\mathrm{R}\mathrm{H}\_\mathrm{g}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	98	93	98	86	98	85	99	93	97	95
	R2	0.39	0.52	0.65	0.58	0.64	0.54	0.56	0.50	0.52	0.41
	RMSE	5.84	3.01	5.80	3.57	6.47	3.91	5.76	3.84	4.24	4.49

and Figure 16, it is observed that, under different forest canopy coverages, the FCH estimation performance follows a pattern: the rh model outperforms the RH model, and GEDI full-power beams perform better than coverage beams. Additionally, the rh_g model outperforms the rh model, and the RH_g model outperforms the RH model. Moreover, the optimal rh parameter is lower than that of the RH parameter. The research results are consistent with the FCH estimation results for both types of GEDI beams and for different terrain slopes. From the perspective of different forest canopy coverage, the findings validate the applicability of IDTW, as well as the combination of ground response functions and terrain slope parameters, in addressing waveform broadening and waveform overlap. Moreover, this approach reduces the hardware requirements of spaceborne LiDAR transmitters to narrow the laser pulse’s half-width.

In addition, the study found that the FCH estimation accuracy obtained using both GEDI full-power beams and coverage beams initially increases and then decreases as the canopy density increases. This study suggests that lower canopy cover can reduce the amplitude of the GEDI canopy echo. When the GEDI waveform length parameter is extracted, the extraction process is prone to delaying the waveform start point and advancing the waveform end point, resulting in a smaller extracted waveform length, which ultimately reduces the accuracy of FCH estimation. However, once the canopy cover exceeds a certain threshold, the laser pulse transmitted by GEDI has difficulty penetrating the forest canopy to acquire understory terrain information. As a result, the waveform end point is mistakenly advanced during the extraction of the GEDI waveform length parameter, leading to a smaller extracted waveform length, which in turn reduces the accuracy of FCH estimation.

4.4. FCH Estimation Under Different Sites

There are significant geographical differences between different sites, leading to variations in climate conditions, vegetation types, terrain slopes, and forest canopy cover [41,42]. To investigate the applicability of IDTW and the integration of the ground response function with terrain slope parameters in addressing waveform broadening and waveform overlap, this study conducted FCH estimation for two types of echo waveforms.

Table 5. FCH estimation results for GEDI full-power beams at different sites.

Site Name		PUUM	DSNY	JERC	UKFS	BLAN	WLOU	NOGP	STEI
Mean terrain slope/°		8.83	2.14	2.96	7.82	7.10	22.27	4.28	7.21
Mean canopy cover/%		64.33	45.63	31.71	35.30	43.97	49.55	86.39	83.88
Mean SNR		21.23	23.69	22.34	21.04	22.56	23.77	21.33	22.72
Number of footprint points		1891	1748	3338	1314	3675	181	1215	4926
$\mathrm{r}\mathrm{h}$ model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	86	82	98	97	93	87	52	95
	R2	0.81	0.84	0.65	0.61	0.75	0.42	0.66	0.62
	RMSE	2.43	2.52	6.09	4.01	4.50	2.42	3.54	3.66
$\mathrm{r}\mathrm{h}$_g model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	86	82	98	97	93	87	52	95
	R2	0.81	0.85	0.65	0.62	0.76	0.51	0.66	0.62
	RMSE	2.39	2.49	6.08	3.98	4.44	2.24	3.53	3.66
$\mathrm{R}\mathrm{H}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	93	84	100	100	96	94	59	98
	R2	0.77	0.86	0.61	0.58	0.75	0.29	0.64	0.58
	RMSE	2.65	2.42	100	4.19	4.57	2.69	3.62	3.85
$\mathrm{R}\mathrm{H}\_\mathrm{g}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	93	84	100	100	96	94	59	98
	R2	0.77	0.86	0.61	0.59	0.75	0.40	0.64	0.58
	RMSE	2.63	2.39	6.40	4.11	4.54	2.24	3.62	3.85

and

Table 6. FCH estimation results for GEDI coverage beams at different sites.

Site Name		PUUM	DSNY	JERC	UKFS	BLAN	WLOU	NOGP	STEI
Mean terrain slope/°		9.13	2.14	2.73	7.31	7.40	17.91	3.91	7.12
Mean canopy cover/%		79.62	37.48	27.93	30.36	51.89	61.55	93.27	82.46
Mean SNR		16.50	17.91	16.41	18.83	17.79	16.06	19.66	17.93
Number of footprint points		520	935	1993	764	2422	287	1062	5828
$\mathrm{r}\mathrm{h}$ model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	94	77	91	93	91	77	41	95
	R2	0.79	0.74	0.62	0.64	0.73	0.64	0.38	0.52
	RMSE	2.34	2.55	5.79	3.74	4.78	2.21	4.25	4.12
$\mathrm{r}\mathrm{h}$_g model	$\mathrm{r}\mathrm{h}\_\mathrm{N}$ optimum parameter	93	77	91	93	91	77	41	95
	R2	0.79	0.74	0.62	0.65	0.73	0.67	0.42	0.52
	RMSE	2.33	2.54	5.78	3.72	4.74	2.12	4.16	4.12
$\mathrm{R}\mathrm{H}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	100	85	100	100	91	92	40	98
	R2	0.74	0.73	0.53	0.58	0.73	0.54	0.34	0.45
	RMSE	2.61	2.58	6.45	4.07	4.79	2.50	4.41	4.44
$\mathrm{R}\mathrm{H}\_\mathrm{g}$ model	$\mathrm{R}\mathrm{H}\_\mathrm{N}$ optimum parameter	100	85	100	100	91	85	40	98
	R2	0.74	0.73	0.53	0.58	0.73	0.59	0.36	0.45
	RMSE	2.61	2.56	6.45	4.05	4.76	2.35	4.34	4.43

present the FCH estimation results for GEDI full-power beams and coverage beams at different sites, respectively. Figure 17 and Figure 18 show bar charts of the FCH estimation accuracy for the rh model and rh_g model, respectively—with the charts generated using GEDI full-power beams and coverage beams.

Based on Table 5 and Table 6 and Figure 17 and Figure 18, this study finds that the FCH estimation accuracy across different sites shows two key trends: first, the rh model generally outperforms the RH model, and GEDI full-power beams perform better than coverage beams; second, the rh_g model outperforms the rh model, and the RH_g model outperforms the RH model. Moreover, the optimal rh parameter value is lower than that of the RH parameter. The research results are consistent with the FCH estimation findings across both GEDI full-power beams and coverage beams, different terrain slopes, and varying canopy cover. From the perspective of different geographical locations, the study validates the applicability of IDTW, as well as the combination of ground response functions and terrain slope parameters, in addressing waveform broadening and waveform overlap. Additionally, this approach reduces the hardware requirements for spaceborne LiDAR transmitters tasked with narrowing the laser pulse half-width.

However, based on Figure 17, the study found that, at the WLOU and UKFS sites, the estimation accuracy of GEDI full-power beams was lower than that of GEDI coverage beams when the rh model was used. For the WLOU site, the study conducted a statistical analysis of the data and found that the average terrain slope for GEDI full-power beams was significantly higher than that for GEDI coverage beams. Since the terrain slope was a key factor affecting the FCH estimation accuracy, this explains why the estimation accuracy of full-power beams at the WLOU site was lower than that of coverage beams.

A statistical analysis of the UKFS site data further revealed two factors: first, the average terrain slope for full-power beams was higher than that for coverage beams; second, the number of footprints of GEDI full-power beams at the UKFS site was approximately twice that of coverage beams. The increased number of footprints introduced greater data randomness, which, to some extent, reduced the accuracy of FCH estimation. These two factors together led to the lower estimation accuracy of full-power beams compared to coverage beams at the UKFS site.

Figure 18 shows that, at the DSNY site, the FCH estimation accuracy—when using the rh model with full-power beams—was lower than that when using the RH model with the same beams. An analysis of the box plots of the terrain slopes for different sites with full-power beams (see Figure 19) revealed that the mean terrain slope at the DSNY site was lower than that of other sites, and its slope distribution was more concentrated. This scenario reduced waveform broadening and waveform overlap caused by NHWTLP. Therefore, the constant term parameter in the RH model can effectively mitigate waveform broadening and waveform overlap at sites with concentrated terrain slope distributions—ultimately resulting in lower FCH estimation accuracy for the rh model compared to the RH model at the DSNY site.

5. Conclusions

To improve the FCH estimation accuracy, this study proposes the IDTW algorithm for spaceborne LiDAR echo waveforms to address the issues of waveform broadening and waveform overlap caused by NHWTLP. Additionally, the ground response function obtained through IDTW processing, combined with terrain slope parameters, is used to address the issues of waveform broadening and waveform overlap caused by non-flat terrain. The algorithm proposed in this study has been validated in FCH estimation under different conditions, such as GEDI full-power beams and coverage beams, varying terrain slopes, different forest canopy densities, and different sites. Based on the research results, the following conclusions can be drawn:

(1)

The FCH estimation accuracy when using the rh model is higher than that with the RH model under conditions such as GEDI full-power beams and coverage beams, varying terrain slopes, different forest canopy densities, and different sites. This indicates that IDTW can effectively address the issues of waveform broadening and waveform overlap caused by NHWTLP.

(2)

The rh model, after the incorporation of terrain slope parameters, shows an improvement in FCH estimation accuracy under conditions such as GEDI full-power beams and coverage beams, varying terrain slopes, different forest canopy densities, and different sites. This indicates that the ground response function still contains terrain slope information, and the incorporation of terrain slope parameters into the rh model can reduce waveform broadening and overlap caused by non-flat terrain.

(3)

The optimal parameter values of the rh model are lower than those of the RH model under conditions such as GEDI full-power beams and coverage beams, different terrain slopes, varying forest canopy densities, and different sites. The evidence indicates that IDTW can reduce the overlap of ground object information in the GEDI echo waveform, accurately reflecting the ground object projection area within the footprint. The findings also validate the applicability of the GEDI ground response function in FCH estimation from the perspective of the waveform energy distribution.

(4)

The rh model shows a greater improvement in the FCH estimation accuracy compared to the RH model for GEDI coverage beams, indicating that IDTW is more effective in denoising the echo waveforms of GEDI coverage beams than those of GEDI full-power beams.