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# Footprint Size Design of Large-Footprint Full-Waveform LiDAR for Forest and Topography Applications: A Theoretical Study

Xuebo Yang, Cheng Wang, Xiaohuan Xi, Yingjie Wang, Ying Zhang, Guoqing Zhou

► To cite this version:

Xuebo Yang, Cheng Wang, Xiaohuan Xi, Yingjie Wang, Ying Zhang, et al.. Footprint Size Design of Large-Footprint Full-Waveform LiDAR for Forest and Topography Applications: A Theoretical Study. IEEE Transactions on Geoscience and Remote Sensing, 2021, 59 (11), pp.9745-9757.
10.1109/TGRS.2021.3054324. hal-04643533

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# Footprint Size Design of Large-Footprint Full-Waveform LiDAR for Forest and Topography Applications: A Theoretical Study

Xuebo Yang<sup>D</sup>, Cheng Wang, Xiaohuan Xi<sup>D</sup>, Yingjie Wang, Ying Zhang, and Guoqing Zhou

**Abstract**—LiDAR footprint, defined as the illumination area of LiDAR sensor on the ground, is the fundamental unit that the sensor collects information from. The design of footprint size crucially influences the acquired LiDAR signals. For large-footprint full-waveform LiDAR, a well-designed footprint size is indispensable to acquire accurate and complete vertical profiles of scene targets. The methods that design the footprint size are increasingly needed to satisfy various application requirements. In this study, an analytical method to designing the footprint size is proposed for forest and topography applications. It is established based on a mixture Gaussian model and the designed footprint size ensures the signals of vegetation and ground can be completely extracted. Experiment results with our method show that the footprint size is preferably in the range of 10.6–25.0 m for forest application, while it is less than 32.3 m for topography application. The intersection of the two sets satisfies both applications. Furthermore, a series of sensibility studies were performed to analyze the influence of multiple key parameters to the optimal footprint size, including the scene characteristics, instrumental configurations, and application requirements. This study provides a theoretical basis for the design of future large-footprint full-waveform laser altimeters.

**Index Terms**—Analytical model, footprint size, forest application, full-waveform LiDAR, topography application.

## NOMENCLATURE

$A_v, A_g$ Amplitudes of vegetation and ground subwaveforms, respectively.

$c$ Speed of light.

Manuscript received March 6, 2020; revised December 24, 2020; accepted January 20, 2021. This work was supported in part by the National Key Research and Development Program of China under Grant 2017YFA0603002, and in part by the Guangxi Natural Science Fund for Innovation Research Team under Grant 2019JJF50001. (Corresponding author: Cheng Wang.)

Xuebo Yang, Cheng Wang, and Ying Zhang are with the Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100094, China, and also with the College of Resources and Environment, University of the Chinese Academy of Sciences, Beijing 100049, China (e-mail: yangxb@radi.ac.cn; wangcheng@radi.ac.cn).

Xiaohuan Xi is with the Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing 100094, China.

Yingjie Wang is with the Centre d'Etudes Spatiales de la Biosphère (CESBIO)—UPS, CNES, CNRS, IRD, University of Toulouse, 31401 Toulouse, France.

Guoqing Zhou is with the College of Geomatics and Geoinformation, Guilin University of Technology, Guilin 541004, China.

Color versions of one or more figures in this article are available at https://doi.org/10.1109/TGRS.2021.3054324.

Digital Object Identifier 10.1109/TGRS.2021.3054324

$D$ Footprint diameter.

$d$ Average tree space.

$H_v$ Canopy (tree) height.

$H_g$ Ground vertical extent.

$L_{crown}$ Crown width.

$m\%, n\%$ Percentages of extracted vegetation and ground signals.

$N$ Number of trees per unit area.

$P_{gap}$ Gap probability.

$r_{crown}$ Ratio of crown height to tree height.

$\text{std}_v, \text{std}_g$ Standard deviations of tree height and ground elevation, respectively.

$x, y$ Time delay and intensity of LiDAR waveform, respectively.

$x_c$ Time delay of critical point.

$Z$ Platform altitude.

$z_v, z_g$ Elevations of gravity centers of vegetation and ground, respectively.

$\beta_v, \beta_g$ Ratio of the distance of critical point to gravity center and the standard deviation in vegetation and ground subwaveforms, respectively.

$\theta_g$ Terrain slope.

$\mu_v, \mu_g$ Gravity center locations of subwaveforms of vegetation and ground, respectively.

$\rho_v, \rho_g$ Vegetation and ground reflectance.

$\sigma_E$ Emitted pulsewidth.

$\sigma_R$ Pulsewidth caused by receiving device.

$\sigma_v, \sigma_g$ Pulsewidths of vegetation and ground subwaveforms, respectively.

## I. INTRODUCTION

LiDAR shows great advantages in quickly and accurately acquiring 3-D information of Earth's surface due to the high-intensity pulse and high-frequency sampling [1], [2]. It has been widely used in forest and topography applications. For forest canopy, a high-intensity laser pulse can penetrate the tree crowns and the received signal reveals the canopy vertical distribution [3]. It explains that the LiDAR signals are used in forest investigations to derive the canopy structural characteristics [3], such as the canopy height [5], leaf area index (LAI) [4], and biomass [5], [6]. For surface topography, LiDAR can directly acquire the surface elevation [7], which can be further used to estimate surface geometry characteristics, e.g., terrain slope [8], [9], aspect [10], and roughness [11].
In more complex situations, such as the mountainous forest area, LiDAR can provide 3-D information for both forest and topography by decoupling the mixed signals [12], [13].

LiDAR footprint is the fundamental unit that the sensor collects information from, which is defined as the illumination area of LiDAR sensor on the ground [14], [15]. Footprint diameter (D) is a crucial parameter of LiDAR instrument. In general, the LiDAR is divided into small-footprint (D ≤ 5 m) and large-footprint (D > 5 m) systems [15]. The small-footprint LiDARs usually work together with airborne or terrestrial scanning systems to capture fine 3-D structures of surface elements [Fig. 1(a)] [16], [17]. In contrast, most large-footprint LiDARs are designed as the spaceborne and airborne laser altimeters, and place extra emphasis on obtaining surface structures on a large scale [Fig. 1(b)]. In addition, LiDAR signals are commonly divided into three types according to their returns: full-waveform, point cloud, and photon counting [18]. The full-waveform LiDAR records the whole vertical profile of returned pulse while the other two record only a few effective discrete points or photons of returned pulse. The full-waveform that contains more complete information is potentially to extract more comprehensive characteristics of objects [19]. These explain that multiple Earth observation LiDAR missions were designed as large-footprint full-waveform laser altimeters, such as the airborne systems Laser Vegetation Imaging Sensor (LVIS), Scanning Lidar Imager of Canopies by Echo Recovery (SLICER), Laser Vegetation Detecting Sensor (LVDS), and the spaceborne systems Ice, Cloud, and land Elevation Satellite (ICESat)/Geoscience Laser Altimeter System (GLAS), Global Ecosystems Dynamics Investigation (GEDI), Gaofen-7, etc. Among them, the footprint size of airborne systems can be adjusted by changing the platform altitude and beam divergence. For example, the LVIS operates at altitudes up to 10 km and generates the footprint with 25 m diameter [20]. The SLICER flies at a nominal altitude of 5 km and yields a 10 m footprint [21]. The footprint size of LVDS is approximately 15 m at altitudes up to 3 km [22]. In contrast, the instrumental parameters of spaceborne systems keep approximately constant. For example, the ICE-Sat/GLAS operated during 2003–2009 generates the footprints with ~70 m diameter at a global scale [23]. Subsequently, the GEDI and Gaofen-7 both equipped with full-waveform laser altimeters were launched in 2018 and 2019, respectively [24], [25]. The GEDI aiming to observe global forest structures is designed with ~25 m footprint diameter. The Gaofen-7 aiming to measure the Earth’s surface elevations generates the footprints with ~30 m diameter.

Many previous studies have proved the footprint size significantly influences the LiDAR waveform [26]–[29]. An appropriate footprint size helps not only improve the accuracy of signal interpretation but also reduce the difficulty of signal processing. For small-footprint LiDAR, the instruments are developing toward smaller footprints and denser pulses to meet the requirements of acquiring detailed 3-D data. In contrast, for large-footprint LiDAR, the relationship between the optimal footprint size and the application requirement is not so straightforward, and it depends on the scene characteristics and instrumental configurations. Specifically, if the footprint size

is too small to cover an entire plant crown, the received signal would be unable to present canopy vertical structures. If the size is too large, the sloped terrain might make vegetation and ground signals severely mixed. Some researchers have attempted to explore the optimal footprint size of large-footprint full-waveform LiDAR. For instance, Pang et al. [26] found that footprint diameters of 25–30 m were optimal for canopy height estimates. However, the height estimation requires only geometric information while the inversions of many other scene properties (e.g., LAI, gap fraction, and biomass) require both geometric and intensity information, i.e., complete target signal [4], [30], [31]. In this case, an appropriate footprint size should be able to acquire distinguishable target signals so as to meet different requirements of applications. However, there is no rigorous design procedure of this kind of footprint size yet.

Therefore, this article aims to provide a novel method for the design of the footprint size of large-footprint full-waveform LiDAR. This method can derive the optimal footprint size based on the scene characteristics, instrumental configurations, and application requirements. The derived footprint size ensures the vertical profile of each scene object can be completely extracted. To fulfill this objective, four-phased goals are realized: 1) establishing a LiDAR waveform model and quantifying the impact of footprint size on the waveform; 2) defining the conditions of extracting complete target signals from LiDAR waveform; 3) determining the appropriate footprint size range in the function of scene characteristics and instrumental configurations; and 4) designing an optimal footprint size for universal forest and topography applications.

## II. METHODS

### A. LiDAR Waveform Model

LiDAR waveform can be regarded as the convolution of the emitted pulse and the response function of surface objects [32], [33]. Since the emitted pulse follows a 1-D Gaussian distribution in the time domain, the received signal is usually represented by the Gaussian mixture model [34], [35]. Specifically, for a forest canopy with n horizontal layers, the LiDAR waveform can be expressed as a linear superposition of n vegetation subwaveforms and ground subwaveform, as in the following equation:

$$y = \sum_{i=1}^{n} A_{v_i} \exp \left[ -\frac{(x - \mu_{v_i})^2}{2\sigma_{v_i}^2} \right] + A_g \exp \left[ -\frac{(x - \mu_g)^2}{2\sigma_g^2} \right] \quad (1)$$

where the waveform intensity y is a function of the time delay x. A is the amplitude, μ is the gravity center location, σ is the standard deviation (pulsewidth), the subscript v_i and g represent the ith vegetation Gaussian component and the ground Gaussian component, respectively.

Assuming the vegetation within footprint is a one-layer forest, the waveform can be simplified as the sum of the two Gaussian functions, i.e., the vegetation and ground components, as in the following equation:

$$y = A_v \exp \left[ -\frac{(x - \mu_v)^2}{2\sigma_v^2} \right] + A_g \exp \left[ -\frac{(x - \mu_g)^2}{2\sigma_g^2} \right]. \quad (2)$$
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Small-footprint LiDAR waveform and (c) large-footprint LiDAR waveform acquired from (b) flat scene with one-layer ellipsoid forest canopy.</figcaption></figure>

The intensity of each component is proportional to the covered area and reflectance of corresponding object (i.e., vegetation and ground). In case of a gap probability of $P_{gap}$ (ratio of bare ground area to total footprint area), vegetation reflectance of $\rho_v$ and ground reflectance of $\rho_g$, the scene average reflectance is $(1 - P_{gap})\rho_v + P_{gap}\rho_g$ with the contributions of vegetation $(1 - P_{gap})\rho_v$ and ground $P_{gap}\rho_g$. Hence, the normalized amplitudes in (2) can be expressed in the following equations, respectively:

$$A_v = \frac{1}{\sigma_v \sqrt{2\pi}} \cdot \frac{(1 - P_{gap})\rho_v}{(1 - P_{gap})\rho_v + P_{gap}\rho_g} \quad (3)$$

$$A_g = \frac{1}{\sigma_g \sqrt{2\pi}} \cdot \frac{P_{gap}\rho_g}{(1 - P_{gap})\rho_v + P_{gap}\rho_g} \quad (4)$$

The gravity center locations and standard deviations of the vegetation and ground components are determined by the vertical distributions of corresponding objects. In case of a nadir pulse penetrating into vegetation canopy over a flat terrain, the LiDAR signal is shaped as the two separated Gaussian waveforms (Fig. 2). The gravity center locations of vegetation and ground components are derived from the gravity center elevations of corresponding objects, as in the following equations:

$$\mu_g = \frac{2(Z - z_g)}{c} = \frac{2Z}{c} \quad (5)$$

$$\mu_v = \frac{2(Z - z_v)}{c} = \frac{2[Z - H_v \cdot (1 - r_{crown}/2)]}{c} \quad (6)$$

where Z is the platform altitude, $z_g$ and $z_v$ are the ground and canopy gravity center elevations, respectively, $z_g$ is set to 0 in this study, $H_v$ is the tree height, $r_{crown}$ is the ratio of crown height to tree height, and c is the speed of light. Note that the model neglects the tree trunk and assumes the gravity center elevation of forest canopy is located at the crown center.

The standard deviation that quantifies the pulsewidth of the returned waveform is affected by multiple factors, such as the emitted pulse, the receiver bias, and the target vertical extent. We suppose that the influence of each factor on the

received waveform follows the Gaussian distribution. The overall variance is the sum of the variance of each factor as proved in Appendix I. For the flat terrain, the ground vertical extent is 0, which would not increase the width of ground subwaveform. In contrast, the canopy vertical extent, i.e., crown length, would broaden the vegetation subwaveform. Also, the ground roughness would simultaneously broaden the widths of two components and the tree height variance only broadens the vegetation signal. Hence, the standard deviations of vegetation and ground subwaveforms on a flat terrain can be expressed in the following equations, respectively:

$$\sigma_v^2 = \sigma_E^2 + \sigma_R^2 + \frac{(H_v r_{crown})^2}{c^2} + \frac{4\text{std}_v^2}{c^2} + \frac{4\text{std}_g^2}{c^2} \quad (7)$$

$$\sigma_g^2 = \sigma_E^2 + \sigma_R^2 + \frac{4\text{std}_g^2}{c^2} \quad (8)$$

where $\sigma_E$ is the emitted pulsewidth, $\sigma_R$ is the pulsewidth caused by receiving device, $\text{std}_v$ and $\text{std}_g$ are the standard deviations of tree height and ground elevation, respectively.

In case of the sloped terrain, the slope would magnify the ground and canopy vertical extents, thus broadening the waveform widths, as in Fig. 2. The standard deviations of vegetation and ground subwaveforms are expressed in the following equations, respectively:

$$\sigma_v^2 = \sigma_E^2 + \sigma_R^2 + \frac{(H_v R_{crown})^2}{c^2} + \frac{H_g^2}{c^2} + \frac{4\text{std}_v^2}{c^2} + \frac{4\text{std}_g^2}{c^2} \quad (9)$$

$$\sigma_g^2 = \sigma_E^2 + \sigma_R^2 + \frac{H_g^2}{c^2} + \frac{4\text{std}_g^2}{c^2} \quad (10)$$

where $H_g$ is the ground vertical extent due to the terrain slope. It is a function of terrain slope ($\theta_g$) and footprint diameter (D), as in the following equation:

$$H_g = D \cdot \tan \theta_g \quad (11)$$

In the above LiDAR waveform model, it is seen that the footprint size determines the ground vertical extent within the footprint, thus affecting the pulsewidths and normalized intensities of vegetation and ground signals. Additionally, note
<figure><img src="image_4.png" /><figcaption>Fig. 2. Normalized LiDAR waveforms over a flat terrain (black solid line) and a slope terrain (black dashed line). (a) Starting location of vegetation signals. (b) Gravity center of vegetation signals. (c) End location of vegetation signals. (d) Starting location of ground signal. (e) Gravity center of ground signals. (f) End location of ground signals.</figcaption></figure>

that the influence of off-nadir pointing angle is not considered in this study since most laser altimeters are designed to measure on nadir target.

## B. Footprint Size Design for Forest Applications

An appropriate footprint size for forest application must ensure two conditions: 1) the received vegetation signal is completely distinguishable and 2) the footprint area is large enough to cover a tree crown. To extract the vegetation signals completely, the following equation should be verified:

$$A_v \cdot \exp\left[-\frac{(x_c - \mu_v)^2}{2\sigma_v^2}\right] \geq A_g \cdot \exp\left[-\frac{(x_c - \mu_g)^2}{2\sigma_g^2}\right] \quad (12)$$

where $x_c$ is the critical point that ensures m% required vegetation signal can be extracted, with

$$m\% = \frac{\int_{-\infty}^{x_c} A_v \cdot \exp\left[-\frac{(x_c - \mu_v)^2}{2\sigma_v^2}\right] dx}{\int_{-\infty}^{+\infty} A_v \cdot \exp\left[-\frac{(x_c - \mu_v)^2}{2\sigma_v^2}\right] dx} \quad (13)$$

Fig. 3 gives an example, if the intensity of the vegetation signal at the critical point $x_c$ is larger than that of the ground signal (such as the ground subwaveform of which center is located at $\mu_{g2}$), m% of vegetation signal can be extracted and the mixed ground signal can be considered negligible. Otherwise, too much ground signal wound mix into the vegetation signal, such as the ground subwaveform of which center is located at $\mu_{g1}$. We define $x_c = \mu_v + \beta_v \sigma_v$ for extracting the vegetation subwaveform, so that the ratio m% can be directly inquired according to $\beta_v = (x_c - \mu_v)/\sigma_v$ in the truncated standard normal distribution table [36]. Some examples of standard normal distribution table are shown in Appendix II. By further simplifying, condition (12) can be

<figure><img src="image_5.png" /><figcaption>Fig. 3. Critical condition for extracting complete vegetation waveform.</figcaption></figure>

expressed by the distance between the gravity centers of two components, as in the following equation:

$$\mu_v - \mu_g \geq \sigma_g \cdot \sqrt{2 \ln \frac{A_g \sigma_v}{A_v \sigma_g} + \beta_v^2 + \sigma_v \cdot \beta_v} \quad (14)$$

The upper boundary of footprint size can be derived by substituting (3)–(6), (9)–(11) into (14). Note that (14) is always verified for flat terrain, it only gives the upper boundary of footprint size in case of sloped terrain.
<figure><img src="image_6.png" /><figcaption>Fig. 4. Critical minimal value of footprint size in a regular-distributed forest. The gray filled circle represents a single tree in the forest. The blue hollow circle represents the LiDAR footprint area.</figcaption></figure>

Besides, the footprint size should be large enough to cover an entire tree crown. For the forest with different tree spatial distributions, the regular pattern has the largest average tree space and it is most difficult that the footprint can cover an entire tree at any location. Suppose the regularly distributed trees have average crown width of $L_{\text{cover}}$. The average tree space (d) is the square root of reciprocal of tree density (N) (i.e., number of trees per unit area), as in (15). The location that is the most difficult to detect an entire crown is located at the circumcircle center of the four adjacent crown tops, as in Fig. 4. Only when footprint size satisfies (16), can the footprint detect the treetop and cover over an entire tree crown. Condition (16) gives the lower boundary of footprint size

$$d = \sqrt{\frac{1}{N}} = \sqrt{\frac{\pi (L_{\text{crown}}/2)^2}{1 - P_{\text{gap}}}} \quad (15)$$

$$D \ge \sqrt{2d} = \sqrt{\frac{\pi L_{\text{crown}}^2}{2 \cdot (1 - P_{\text{gap}})}} \quad (16)$$

### C. Footprint Size Design for Topography Application

For topography applications, we assume that n% of ground waveform is required to be extracted from LiDAR signal, with

$$n\% = \frac{\int_{x_c}^{+\infty} A_g \cdot \exp[-\frac{(x_c - \mu_g)^2}{2\sigma_g^2}] dx}{\int_{-\infty}^{+\infty} A_g \cdot \exp[-\frac{(x_c - \mu_g)^2}{2\sigma_g^2}] dx} \quad (17)$$

The critical point $x_c$ of ground subwaveform is located at $\mu_g - \beta_g \sigma_g$ with $\beta_g = (\mu_g - x_c)/\sigma_g$. As in Fig. 5, if the gravity center of vegetation signal is located at $\mu_{v2}$, the truncated ground component below the critical point contains overmuch vegetation signal. Only if the intensity of ground subwaveform at the critical point is lower than that of vegetation subwaveform, can the mixed vegetation signal in the ground signal be ignored. Hence, similar with (12), the condition to extract n%

<figure><img src="image_7.png" /><figcaption>Fig. 5. Critical condition for extracting complete ground waveform.</figcaption></figure>

<table><caption>TABLE I
VALUES OF FACTORS INFLUENCING THE FOOTPRINT SIZE DESIGN</caption><thead><tr><th>Factor</th><th>Unit</th><th>Default</th><th>Minimum</th><th>Maximum</th></tr></thead><tbody><tr><td>β<sub>v</sub></td><td>-</td><td>0.84</td><td>1</td><td>2.5</td></tr><tr><td>β<sub>g</sub></td><td>-</td><td>0.84</td><td>1</td><td>2.5</td></tr><tr><td>θ<sub>g</sub></td><td>degree</td><td>30</td><td>0</td><td>50</td></tr><tr><td>H<sub>v</sub></td><td>m</td><td>20</td><td>5</td><td>80</td></tr><tr><td>P<sub>gap</sub></td><td>-</td><td>0.5</td><td>0.01</td><td>0.99</td></tr><tr><td>ρ<sub>v</sub> / ρ<sub>g</sub></td><td>-</td><td>1</td><td>0.05</td><td>20</td></tr><tr><td>σ<sub>E</sub></td><td>ns</td><td>4</td><td>0.2</td><td>10</td></tr><tr><td>r<sub>crown</sub></td><td>-</td><td>0.5</td><td>0.01</td><td>0.99</td></tr><tr><td>L<sub>crown</sub></td><td>m</td><td>6</td><td>0.5</td><td>15</td></tr><tr><td>std<sub>g</sub></td><td>m</td><td>0</td><td>0</td><td>10</td></tr><tr><td>std<sub>v</sub></td><td>m</td><td>0</td><td>0</td><td>10</td></tr></tbody></table>

of ground signal is expressed in the following equation:

$$A_v \cdot \exp\left[-\frac{(x_c - \mu_v)^2}{2\sigma_v^2}\right] \le A_g \cdot \exp\left[-\frac{(x_c - \mu_g)^2}{2\sigma_g^2}\right] \quad (18)$$

Condition (18) is further expressed by the distance between the gravity center locations of two components, as in (19). By substituting (3)–(6), (9)–(11) into (19), the upper boundary of footprint size for extracting complete ground signal can be derived

$$\mu_v - \mu_g \ge \sigma_v \cdot \sqrt{2 \ln \frac{A_v \sigma_g}{A_g \sigma_v} + \beta_g^2 + \sigma_g \cdot \beta_g} \quad (19)$$

Note that there is no minimum limit on the footprint size in the topography application. This is because the smaller the footprint, the easier it is to extract the ground signal.
<figure><img src="image_8.png" /><figcaption>Fig. 6. Relationship among critical maximal footprint size, canopy height, and terrain slope for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Relationship among critical maximal footprint size, canopy height, and gap probability for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

## D. Footprint Size Design for Forest and Topography Applications

In some cases, the vegetation and ground signals are required to be extracted simultaneously. For example, LAI is usually estimated by building a semiempirical relationship with the ratio of vegetation and ground signals [4], [37]. It is necessary to design the appropriate footprint that meets the demands of extracting vegetation and ground signals simultaneously. For these requirements, the footprint size should verify the conditions (14), (16), and (19).

## E. Appropriate Footprint Size in Various Cases

The above method takes into accounts terrain slope, canopy height, crown height ratio, crown width, gap probability, canopy and ground reflectance, emitted pulsewidth, bias from receiving device, vegetation height variation, and ground height variation. Once these parameters are configured, the appropriate footprint size range can be derived. Table I lists the reference values of these parameters. The default values are considered as the attributes of the main observed

objects, which is used to derive a common range of footprint size. Additionally, this study conducts a series of sensitivity experiments to explore the impact of the factors on the optimal footprint size. The bias from receiving device is not discussed since it is insignificant and usually stable. According to the GLAS system, the width caused by the receiving device is about 1/12⁰.⁵ ns. In addition, due to the normalized intensity, the reflectance ratio of vegetation and ground instead of their absolute reflectance is used for impact analysis. Note that the vegetation reflectance here is not the mixed value of vegetation and understory ground reflectance but the ideal reflectance of vegetation without underlying. The default value of β is set as 0.84, which meets the application requirement to extract 80% of waveform component.

# III. EXPERIMENTAL RESULTS

## A. Appropriate Footprint Size in Default Case

In the default scene, the forest canopy with 20 m height grows on the sloped terrain with 30 gradient. In order to extract over 80% of vegetation signals, the footprint diameter should
<figure><img src="image_10.png" /><figcaption>Fig. 8. Relationship among critical minimal footprint size, crown width, and gap probability in forest application.</figcaption></figure>

be designed less than 25.0 m. Also, the diameter is required over 10.6 m to cover at least one plant with 6-m crown width. For topography application, the footprint diameter should be designed less than 32.3 m so that 80% of ground signals are extracted. Overall, the footprint size range that meets the requirement of both applications is 10.6–25.0 m.

For airborne large-footprint LiDAR system, the footprint size is preferably designed close to the minimal value in the appropriate range. In this way, the LiDAR systems can be used in areas with more severe terrain reliefs (slope > 30°), and acquire more high-quality data. In contrast, the footprint size of satellite LiDARs should be designed close to the critical maximal value. This is because the large footprint can collect more representative data on a global scale.

## B. Footprint Size Under Various Tree Height

Table II shows the critical maximal value of footprint size in cases of various tree heights and terrain slopes. Once the footprint diameter is lower than the critical value in Table II, the relatively complete vegetation and ground signals can be extracted at the corresponding slope. For instance, for 30° terrain slope, 25-m footprint can acquire data of which vegetation and ground components are easily separated in the forest with 20 m height. The vegetation with 15 m height requires the footprint with less than 18.8 m diameter.

Besides Table II, Figs. 6–14 display the relationships among critical maximal footprint size, tree height, and other factors. Keeping other factors constant, the critical maximal footprint size increases with the tree height. This can be explained that tall trees have a significant elevation difference with the ground. Aliasing of vegetation and ground components only occurs in case of a large footprint.

## C. Footprint Size Under Various Terrain Slopes

Fig. 6 displays the critical maximal footprint size under various terrain slopes. Results indicate that the critical maximal size decreases with the terrain slope. In theory, the terrain slope broadens both the vegetation and ground subwaveforms. As the terrain slope increases, the canopy vertical extent mixes

more with the ground extent. With the same tree height, a small footprint is required to extract the object components on the high-slope terrain. In addition, in default case that the energy returned by vegetation is equal to that of ground, the vegetation component is wider than the ground component. Hence, the footprint size required to extract the vegetation signal is smaller than that of extracting the same ground signal. This leads that the footprint size which can extract the complete vegetation signal is able to separate the two components. That is, the result in Fig. 6(c) is consistent with that in Fig. 6(a).

## D. Footprint Size Under Various Gap Probability

Fig. 7 shows the critical maximal footprint size under various gap probability. We observe that the results for extracting vegetation and ground components are exactly opposite. For forest application, the vegetation signal of any footprint can be extracted once the gap probability is less than 0.38 [Fig. 7(a)]. Above the threshold, the critical footprint size for forest application decreases with the gap probability. In contrast, when the gap probability exceeds 0.6, the ground signal is strong enough to be extracted [Fig. 7(b)]. Below the threshold, the critical footprint size for topography application increases with the gap probability. The footprint size for both applications in Fig. 7(c) is the intersection of the results in Fig. 7(a) and (b). Specifically, if the gap probability is less than about 0.46 [minimum point of the curves in Fig. 7(c)], the footprint size which can extract the ground signal must be able to separate two components. For the gap probability over 0.46, the footprint depends on the value derived from forest application.

Theoretically, the gap probability affects the relative intensity of vegetation and ground signals. Large gap probability makes it hard to extract weak vegetation signal from LiDAR returns while small gap probability corresponds to the inability to extract the ground signal. In these cases, the extracted component could be distinguished only when there is a narrow width for both canopy and ground components, which requires a small footprint.

In addition, the gap probability determines the lower limit of footprint size. Fig. 8 shows the critical minimal footprint size under various gap probability and crown width. Results indicate that the footprint size increases with the crown width and decreases with the gap probability. Actually, either larger crown width or lower gap probability would result in lower plant density and larger average plant spacing. In this case, a large footprint is required to cover more than one tree.

## E. Footprint Size Under Various Reflectance Ratios of Vegetation and Ground

Fig. 9 displays the critical maximal footprint size under various ratios of vegetation and ground reflectance. Like the gap probability, the reflectance ratio affects the relative intensities of two components, thus affecting the footprint size design. When the vegetation reflectance is over 1.67 times of the ground reflectance, the vegetation signal can be separated unconditionally. Below the threshold, the critical footprint size
<table><caption>TABLE II
CRITICAL MAXIMAL FOOTPRINT SIZE FOR SEPARATING THE VEGETATION AND GROUND SIGNALS UNDER VARIOUS CANOPY HEIGHTS AND TERRAIN SLOPES</caption><thead><tr><th>Critical maximal footprint size [m]</th><th>θ<sub>g</sub> = 10°</th><th>θ<sub>g</sub> = 20°</th><th>θ<sub>g</sub> = 30°</th><th>θ<sub>g</sub> = 40°</th><th>θ<sub>g</sub> = 50°</th></tr></thead><tbody><tr><td>H<sub>v</sub> = 5 m</td><td>19.4</td><td>9.4</td><td>5.9</td><td>4.1</td><td>2.9</td></tr><tr><td>H<sub>v</sub> = 10 m</td><td>40.6</td><td>19.7</td><td>12.4</td><td>8.5</td><td>6.0</td></tr><tr><td>H<sub>v</sub> = 15 m</td><td>61.4</td><td>29.7</td><td>18.8</td><td>12.9</td><td>9.1</td></tr><tr><td>H<sub>v</sub> = 20 m</td><td>82.1</td><td>39.8</td><td>25.0</td><td>17.2</td><td>12.1</td></tr><tr><td>H<sub>v</sub> = 25 m</td><td>102.7</td><td>49.8</td><td>31.4</td><td>21.6</td><td>15.2</td></tr><tr><td>H<sub>v</sub> = 30 m</td><td>123.4</td><td>59.8</td><td>37.7</td><td>25.9</td><td>18.3</td></tr><tr><td>H<sub>v</sub> = 35 m</td><td>144.0</td><td>69.8</td><td>44.0</td><td>30.3</td><td>21.3</td></tr><tr><td>H<sub>v</sub> = 40 m</td><td>164.6</td><td>79.7</td><td>50.3</td><td>34.6</td><td>24.4</td></tr><tr><td>H<sub>v</sub> = 45 m</td><td>185.2</td><td>89.7</td><td>56.6</td><td>38.9</td><td>27.4</td></tr><tr><td>H<sub>v</sub> = 50 m</td><td>205.8</td><td>99.7</td><td>62.9</td><td>43.2</td><td>30.4</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Relationship among critical maximal footprint size, canopy height, and reflectance ratio of vegetation and ground for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Relationship among critical maximal footprint size, canopy height, and crown height ratio for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

for extracting vegetation signal decreases with the reflectance ratio. In contrast, if the vegetation reflectance is less than 0.7 times of ground reflectance, the ground waveform is always sharp in any footprint. Above the threshold, the footprint size for topography application increases with the reflectance ratio.

For the applications that require to simultaneously extract two components, the suitable footprint size is the intersection of the intervals for extracting respective components. Especially when the vegetation reflectance is about 1.17 times of ground, the appropriate footprint size range is the largest.
<figure><img src="image_13.png" /><figcaption>Fig. 11. Relationship among critical maximal footprint size, canopy height, and standard deviation of tree height for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Relationship among critical maximal footprint size, canopy height, and standard deviation of ground elevation for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

## F. Footprint Size Under Various Crown Height Ratios

Fig. 10 displays the critical maximal footprint size under various crown height ratios. A large crown height ratio implies long crown and short trunk. The crown length has a significant stretching effect on the vegetation waveform, thus dropping its center of gravity and weakening its normalized intensity. Hence, a small footprint is required to extract the weak vegetation signal. The results in Fig. 10(a) also show that the footprint size for extracting vegetation signal decreases with the crown height ratio. In contrast, the crown height ratio has an indirect effect on extracting the ground signal. The critical value of footprint size in Fig. 10(b) hardly changes with the crown height ratio.

## G. Footprint Size Under Variances of Tree Height and Ground Elevation

Figs. 11 and 12 display the critical maximal footprint sizes under various standard deviations of tree height and

ground elevation, respectively. Results show that the critical footprint size decreases with the standard deviation of either canopy height or ground elevation for extracting either vegetation or ground signals. In theory, the object height variance broadens the corresponding waveform component, thus requiring a relatively small footprint to narrow the waveform width. Specifically, the canopy extent variation makes the vegetation component wider while the ground extent variation affects the two components. Hence, the standard deviation of tree height more obviously influences the footprint size for extracting the vegetation waveform compared with the ground waveform, as comparison of Fig. 11(a) and (b). Also, the critical footprint size in both applications changes with the standard deviation of ground elevation more significantly than that of tree height by comparing Figs. 11 and 12. Additionally, due to the crown length, it is more difficult to extract vegetation signal with the same proportion as ground signal. The footprint size that meets the requirement of forest application must be able to meet the same demand of ground application. That is, the results
<figure><img src="image_15.png" /><figcaption>Fig. 13. Relationship among critical maximal footprint size, canopy height, and emitted pulsewidth for extracting (a) complete vegetation waveform, (b) complete ground waveform, and (c) both vegetation and ground waveforms.</figcaption></figure>

in Figs. 11(c) and 12(c) are consistent with those in Figs. 11(a) and 12(a), respectively.

## H. Footprint Size Under Various Emitted Pulsewidths

Fig. 13 displays the critical maximal footprint size under various emitted pulsewidths. Results indicate that the critical maximal footprint size slightly decreases with the emitted pulsewidth in both applications. This is caused by the fact that the emitted pulse stretches the widths of two components but its broadening effect is not as significant as those of crown and ground vertical extents. For a large emitted pulsewidth, a relatively small footprint is required to narrow the signal widths and further extract the complete signal.

## I. Footprint Size Under Various Stringent Requirements of Forest and Topography Applications

Fig. 14(a) and (b) display the critical maximal footprint size under different stringent requirements of the two applications. The strict degrees of requirements are quantified by the value of $\beta$, which is related to the proportion of the extracted pure signal to the corresponding component. A strict requirement means that the practical applications need more complete vegetation or ground profiles, which also needs a small footprint to extract the signals. For instance, as in Fig. 14(a), if it is required to extract over 80% of signal of vegetation with 20 m height, the footprint size must be less than 25.0 m. If 90% of vegetation signal is required, only the footprints less than 18.0 m are suitable. Fig. 14(c) displays the application requirements that four footprint sizes (10, 30, 50, and 70 m) can meet. Once the requirements of extracting the vegetation and ground components and canopy height are set, the appropriate footprint size can be easily confirmed.

# IV. DISCUSSION

## A. Design of LiDAR System Parameters

Footprint is the spatial unit of LiDAR sensor. For laser scanning systems, most pulses are fired with off-nadir pointing.

The off-nadir pointing has a distorted effect on the LiDAR waveform in the case of sloped terrain, stretching or compressing [26], [27]. This uncertainty makes laser scanning systems hardly designed as the large-footprint full-waveform LiDAR. In contrast, the laser altimeters are commonly designed with a nadir pointing, which can collect high-quality full-waveform data. For the on-nadir laser altimeters, the footprint size is determined by the flight altitude and beam divergence [14]. Additionally, the flight altitude, flight speed, and pulse frequency jointly affect the center-to-center spacing and footprint density. The design of footprint size straightforwardly promotes the work of determining these instrument parameters. For instance, the spaceborne laser altimeter has constant onboard configurations. The higher the platform altitude designed, the longer the orbit cycle and the denser the footprints along the tracks generated. At the same time, a higher platform altitude represents a larger footprint, which might be unable to acquire high-quality data in the forest area with high terrain reliefs.

## B. Relationship Among Instrumental Configurations, Scene Characteristics, and LiDAR Waveform

LiDAR waveform is a comprehensive expression of laser pulse and Earth scenes. Some researchers have explored the effects of instrumental configurations, topography, and vegetation characteristics on LiDAR waveforms using various methods. For example, Yang *et al.* [27] qualitatively assessed the impact of surface topography and canopy structure on LiDAR waveforms using a geometric optical and radiative transfer model. They concluded that the critical terrain slope on which the vegetation and ground components could be separated was relevant to the footprint size. Pang *et al.* [26] used a 3-D LiDAR radiative transfer model to quantify the optimal footprint size for forest height estimation. Hilbert and Schmullius [29] analyzed the influence of surface topography on forest height estimation and waveform shape by developing a variety of waveform shapes subjectively. Different from them, this study used an analytical LiDAR waveform model
<figure><img src="image_16.png" /><figcaption>Fig. 14. Relationship among critical maximal footprint size, canopy height, and requirement values of (a) forest application, (b) topography application, and (c) forest and topography applications.</figcaption></figure>

to quantitatively express the relationship among instrumental configurations, scene characteristics, and LiDAR signal. In the model, the object vertical distributions affect the waveform widths and the gravity center locations, both of which belong to waveform geometry information. The coverage rate and the reflectances influence the waveform intensity. Another point of view, the waveform geometry information mirrors the vertical distribution of objects, while the waveform intensity is strongly related to the object horizontal distribution and spectral properties. This confirms that LiDAR waveform contains abundant information concerning the 3-D spatial and spectral characteristics of surface objects.

<figure><img src="image_17.png" /><figcaption>Fig. 15. Standard normal distribution.</figcaption></figure>

### C. Conditions for Separating Target Signals

Since the LiDAR waveform conveys abundant information of the surface objects, it should be highlighted how to accurately extract the target signal from the waveform. Until now, few studies quantitatively evaluated whether the extracted signal is pure. In this study, based on the cumulative Gaussian distribution function, we derive the conditions of extracting target components and evaluate the pureness of the extracted signal quantitatively. Specifically, whether the target component can be extracted completely is determined by a combination of the distance to other adjacent components and the spectral and spatial characteristics of the target. This evaluation method for target components would provide inspiration for quantitative waveform processing.

<table><caption>TABLE III
EXAMPLES IN STANDARD NORMAL TABLE</caption><thead><tr><th>β</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th></tr></thead><tbody><tr><th>Area</th><td>0.5</td><td>0.84134</td><td>0.97725</td><td>0.99865</td><td>0.99997</td></tr><tr><th>β</th><td>0.84</td><td>1.04</td><td>1.28</td><td>1.65</td><td>2.33</td></tr><tr><th>Area</th><td>0.79955</td><td>0.85083</td><td>0.89973</td><td>0.95053</td><td>0.99010</td></tr></tbody></table>

### D. Limitations of the Analytical Method

The analytical LiDAR waveform model proposed in this study is established based on very strong simplifications and assumptions. It represents the LiDAR waveform with the Gaussian mixture model that has been widely implemented in waveform processing and applications [35], [38], [39]. However, it is almost impossible that the actual waveform absolutely conforms mixture Gaussian model due to complex surface scene. For example, the analytical model regards the signal of a one-layer vegetation as a symmetric Gaussian function. However, due to the canopy occlusion, the bottom

part of the vertically symmetrical canopy returns fewer signals than the upper part, which causes the vegetation peak is slightly upper-skewed, as in Fig. 1(c). Also, the multiple scattering among scene elements strengthens the waveform intensity to different degrees at various time delays [40]. However, these effects are not considered in the analytical model, which causes the deviation between the analytical Gaussian waveform and the actual LiDAR signal. This deviation directly limits the practicality of the method of footprint size design.

Currently, some physically-based radiative transfer models have been developed to simulate the LiDAR waveforms accurately [41], [42]. However, due to the complexity of these models, the relationships among the waveform characteristics, instrumental configurations, and scene characteristics are hard to be straightforwardly expressed. It is expected that further work can establish rigorous sensor design procedures based on the more accurate LiDAR waveform models.
## V. CONCLUSION

This study proposes a footprint size design method for large-footprint full-waveform LiDAR based on an analytical waveform model. It defines appropriate footprint size for topography and forest applications so that the signals of vegetation and ground can be extracted completely. For a forest that consists of trees with 20 m canopy height, 6 m crown width, 0.5 crown height ratio, and 0.5 gap probability on a 30° terrain slope, our method shows that the footprint diameter should be less than 32.3 m for extracting over 80% of ground signal, while in the range of 10.6–25.0 m for extracting over 80% of vegetation signal. The intersection of the two sets satisfies both applications. Furthermore, this study explores the relationship between the expected footprint size and multiple relevant factors. Results demonstrate that a small footprint size is expected for large terrain slope, high tree height, large standard deviations of tree height and ground elevation, and strict requirements of extracting pure signal. Also, a large-footprint size is required in forest with large crown width and sparse plants.

In summary, our method and discussions provide a theoretical basis for the design of footprint size in spaceborne and airborne full-waveform laser altimeter missions, and our results provide the reference value of footprint size. The criteria to extract complete target signal also can provide inspiration for quantitative LiDAR waveform processing. However, the method is limited if the LiDAR signal is far from the ideal mixture Gaussian distribution. The future work will focus on exploring the optimal footprint size scene using more accurate LiDAR waveform models.

## APPENDIX I

For the two following Gaussian functions:

$$f_1(x) = e^{-\frac{x^2}{2\sigma_1^2}}, \quad f_2(x) = e^{-\frac{x^2}{2\sigma_2^2}}.$$

$f_3(x)$ is the convolution of the two functions. It is calculated as follows:

$$\begin{aligned}
f_3(x) &= \int_{-\infty}^{+\infty} f_1(t) \cdot f_2(x-t) dt \\
&= \int_{-\infty}^{+\infty} e^{-\frac{t^2}{2\sigma_1^2}} \cdot e^{-\frac{(t-r)^2}{2\sigma_2^2}} dt \\
&= e^{-\frac{x^2}{2(\sigma_1^2+\sigma_2^2)}} \cdot \int_{-\infty}^{+\infty} e^{-\frac{(\sigma_1^2+\sigma_2^2)\left(t-\frac{\sigma_1^2 x}{\sigma_1^2+\sigma_2^2}\right)^2}{2\sigma_1^2\sigma_2^2}} dt \\
&= \sqrt{2\pi} \frac{\sigma_1\sigma_2}{\sqrt{\sigma_1^2 + \sigma_2^2}} \cdot e^{-\frac{x^2}{2(\sigma_1^2+\sigma_2^2)}}.
\end{aligned}$$

Therefore, the convolution of two 1-D Gaussian function is still the Gaussian function. The variance $\sigma_3^2$ of new function $f_3(x)$ is the sum of the variances of two original functions $f_1(x)$ and $f_2(x)$. That is, $\sigma_3^2 = \sigma_1^2 + \sigma_2^2$.

## APPENDIX II

Standard normal table is a mathematical table that displays the values of the cumulative distribution function of normal

distribution. The table is used to find the probability or the area that a statistic is observed below $\beta$ on the standard normal distribution (Fig. 15). Some examples in the standard normal table are listed in Table III.

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<figure><img src="image_18.png" /></figure>

**Xuebo Yang** received the B.E. degree in remote sensing science and technology from Wuhan University, Wuhan, China, in 2016. She is pursuing the joint Ph.D. degree with the Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, China, and the College of Resources of Environment, University of the Chinese Academy of Sciences, Beijing, China.

Her research interests include LiDAR radiative transfer modeling, design of LiDAR sensors, and multifield applications of LiDAR data.

<figure><img src="image_19.png" /></figure>

**Cheng Wang** received the Ph.D. degree in remote sensing from Université Louis Pasteur (ULP), Strasbourg, France, in 2005.

He is a Professor with the Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, China, and also with the College of Resources of Environment, University of the Chinese Academy of Sciences, Beijing. His research interests include LiDAR remote sensing mechanisms and multimode LiDAR data processing.

<figure><img src="image_20.png" /></figure>

**Xiaohuan Xi** received the M.S. degree in environmental geography from Peking University, Beijing, China, in 2000.

She is an Associate Professor with the Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing. Her research interests include forest characteristics retrievals using LiDAR data.

<figure><img src="image_21.png" /></figure>

**Yingjie Wang** received the B.S. degree in geography information science from Peking University, Beijing, China, in 2016, and the M.S. degree in energy system and environment from IMT Atlantique, Nantes, France, in 2018. He is pursuing the Ph.D. degree with the Centre d'Etudes Spatiales de la Biosphère, Toulouse, France.

His research interests include 3-D radiative transfer modeling.

<figure><img src="image_22.png" /></figure>

**Ying Zhang** received the B.E. degree in remote sensing science and technology from Wuhan University, Wuhan, China, in 2016. He is pursuing the joint Ph.D. degree with the Aerospace Information Research Institute, Chinese Academy of Sciences, Beijing, China, and the College of Resources of Environment, University of the Chinese Academy of Sciences, Beijing.

His research interests include monitoring terrain motion using remote sensing technologies.

<figure><img src="image_23.png" /></figure>

**Guoqing Zhou** received the Ph.D. degree in photogrammetry and remote sensing from Wuhan University, Wuhan, China, in 1994.

He is a Professor with the College of Geomatics and Geoinformation, Guilin University of Technology, Guilin, China. His research interests include the designs of advanced laser altimeters and laser sounders.