Near-Complete Sampling of Forest Structure from High-Density Drone Lidar Demonstrated by Ray Tracing

Abstract

Drone lidar has the potential to provide detailed measurements of vertical forest structure throughout large areas, but a systematic evaluation of unsampled forest structure in comparison to independent reference data has not been performed. Here, we used ray tracing on a high-resolution voxel grid to quantify sampling variation in a temperate mountain forest in the southwest Czech Republic. We decoupled the impact of pulse density and scan-angle range on the likelihood of generating a return using spatially and temporally coincident TLS data. We show three ways that a return can fail to be generated in the presence of vegetation: first, voxels could be searched without producing a return, even when vegetation is present; second, voxels could be shadowed (occluded) by other material in the beam path, preventing a pulse from searching a given voxel; and third, some voxels were unsearched because no pulse was fired in that direction. We found that all three types existed, and that the proportion of each of them varied with pulse density and scan-angle range throughout the canopy height profile. Across the entire data set, 98.1% of voxels known to contain vegetation from a combination of coincident drone lidar and TLS data were searched by high-density drone lidar, and 81.8% of voxels that were occupied by vegetation generated at least one return. By decoupling the impacts of pulse density and scan angle range, we found that sampling completeness was more sensitive to pulse density than to scan-angle range. There are important differences in the causes of sampling variation that change with pulse density, scan-angle range, and canopy height. Our findings demonstrate the value of ray tracing to quantifying sampling completeness in drone lidar.

1. Introduction

Forests are important terrestrial ecosystems covering 31% of the terrestrial land surface and accounting for 75% of terrestrial gross primary production [1,2,3]. Lidar observations of forests are essential for deriving key traits (e.g., leaf area index), depicting tree morphology, and monitoring spatiotemporal changes [4,5,6,7,8,9,10,11,12]. Detailed lidar observations can also serve as independent estimates to calibrate and validate global data products from spaceborne lidar and radar missions [13,14,15,16,17]. The application of lidar observations in these studies has advanced our understanding of the functioning of forest ecosystems and their roles in the carbon and water cycles under changing climates [18,19,20].

Terrestrial laser scanning (TLS), with its high point density, captures forest structure in great detail [21,22,23]. It is therefore promising for accurately estimating aboveground biomass based on structural metrics such as diameter at breast height (DBH), various crown dimensions, and wood volume on as fine a level as the individual tree [24,25,26,27,28,29]. Ground-based scanning tends to prevent laser pulses from reaching canopy tops, especially during leaf-on seasons in dense forests [15,30]. Multiple scans from different locations can reduce occlusion, but the data acquisition for large areas can be labor-intensive and time-consuming and requires careful planning [31]. Some regions may be inaccessible for acquiring extra scans, and scans may be influenced by wind effects [32]. Airborne laser scanning (ALS) generates more detailed measurements at upper canopies than at lower canopies [30,33], usually with much lower point density compared with TLS [34]. ALS observations or products are essential for scaling between field measurements and satellite data products [35,36,37]. Personal laser scanning (PLS), such as backpack and handheld laser scanning [38,39,40], has become popular due to its lower costs and greater mobility in comparison to TLS [41]. However, it is still used mostly within small areas, since large-area mapping with the system is labor-intensive, and most regions remain inaccessible. Mobile laser scanning (MLS), such as car-based systems or lidar onboard all-terrain vehicles [42], can map larger areas with less labor than PLS [43] but has limited applications in forests [39,44].

High-density drone lidar can produce point clouds with sampling densities that are orders of magnitude greater than traditional ALS [22,45,46,47,48,49] and can scan regions that are inaccessible on the ground [50,51]. Typical point densities for ALS, drone lidar, and TLS vary over about five orders of magnitude, with ALS sensors in the 1–100 points/m² range, drone lidar in the 1000–10,000 points/m² range, and TLS sensors producing point densities close to 10,000 points/m². These numbers depend on flight altitude and scanner configuration [30,45,47,52,53]. Because sampling densities and scan-angle ranges are greater, drone lidar may be able to resolve understory structure that traditional ALS cannot. Overlapping flight lines from ALS or drones, analogous to multiple TLS scans from different locations, can significantly increase the observed canopy volume due to greater scan-angle ranges and pulse densities [6,45,49,52]. However, it remains unknown how scan-angle ranges and pulse densities interact to influence the sampling properties of lidar data from airborne sensors throughout the canopy height profile. Airborne sensors tend to acquire relatively few returns near the ground, but it remains unclear whether measurement characteristics can be modified to produce a more complete sample within the canopy volume. For example, footprint size and pulse energy have an influence on canopy penetration and the distribution of returns [54]. Scan-angle range can impact metrics of canopy and understory volume [55].

Ray tracing in voxel space, originally used for radiative transfer modeling in three dimensions [56,57,58,59,60], is widely used to analyze TLS data [61,62,63,64,65,66,67,68]. Several studies have applied ray tracing to ALS or drone lidar data [55,69,70,71]. Here we use the term to describe the process of tracing the trajectories of individual returns [49,52], which is different from the ray-tracing used to simulate the pulse energy transfer in DART or Helios++ [72,73,74]. When employed, ray-tracing tracks the path of each laser pulse as it travels through voxels [75]. A return is generated when a laser pulse is reflected by vegetation or other material. In forests, voxels that contain a return are known to be occupied by vegetation. However, voxels that do not contain a return may be unoccupied and therefore, no return can be generated, or they can be occupied but not detected. Voxels that are occupied and not detected can exist in a lidar data set for a variety of reasons: they can be unsearched by laser pulses, for example, because pulse density is insufficient, occluded due to the presence of other occupied voxels within the beam path, or searched but not detected. A voxel can be searched and not detected when a laser pulse passes through empty space within an occupied voxel, or not enough laser energy is reflected to be recognized as a discrete return by post-processing algorithms.

We define sampling completeness, $p$, as the number of detected voxels divided by the number of voxels that are occupied. Sampling completeness is a ratio between 0 and 1:

$$p={\displaystyle \frac{n_{detected}}{n_{occupied}}}$$

(1)

Its complement, sampling incompleteness, is $1-p$. Sampling incompleteness can be represented by different types of undetected voxels. Most existing studies have used detected voxels to evaluate sampling completeness or quantified occluded volume to investigate sampling incompleteness [49,52,64,76], and few studies have investigated the voxels that were unsearched by laser pulses [67]. However, occluded and unsearched voxels are not the only kind of voxels that do not trigger a return. Existing ALS or drone lidar studies that employ ray tracing do not provide a comprehensive understanding of how the proportions of various kinds of undetected voxels vary under different sampling conditions throughout the canopy height profile.

Here, we used ray tracing to quantify sampling completeness using data from high-density drone lidar in a temperate mountain forest in the southwest Czech Republic. By using ultra-high-density measurements collected under a wide scan-angle range, we generated simulated datasets using subsampling to evaluate the consequences of different pulse densities and scan-angle ranges on sampling completeness. We benchmarked understory-to-near-ground measurements from drone lidar using TLS data that were collected on the same day. Using a combination of drone lidar and TLS data provided a high-quality reference to assess the sampling completeness of drone lidar. A previous study demonstrated that the combination of TLS and drone lidar reduced occlusion to 2% in a temperate mixed beech forest and a tropical rainforest in Borneo [76].

The objectives of our study were to (1) investigate whether sampling completeness was more sensitive to pulse density than to scan-angle range; (2) quantify how different types of undetected voxels covaried with pulse density and scan-angle range; (3) quantify how sampling completeness varied from the canopy top to the ground; and (4) identify what kinds of undetected voxels were responsible for unsampled forest structure.

2. Materials and Methods

2.1. Drone Lidar

We collected lidar data over a temperate mountain forest in the southern Czech Republic using a high-density drone [45]. The site contained the Zofin forest dynamics plot, a 25 ha permanent-inventory plot in which all free-standing woody plants with DBH greater than 1 cm have been mapped and monitored since 2012 [77,78]. The forest is dominated by old-growth European beech (Fagus sylvatica) and Norway spruce (Picea abies), with occasional silver fir (Abies alba) [78,79]. Field inventory data were used to select locations for this analysis and not in later analyses (

Table 1. Stem density and DBH within each plot used in the study.

Plot	1 ha	100 m2
		1	2	3	4
Mean point density (points/m2)	3354	3870	3881	3828	2953
Stem density (stems/m2)	0.3	0.2	0.5	0.03	0.2
Mean DBH (cm)	5.2	9.3	2.4	47.7	9.6
Maximum DBH (cm)	134.2	60.0	37.0	61.5	73.0
Minimum DBH (cm)	1.0	1.1	1.0	39.4	1.0

).

Data were collected using a RIEGL VUX-1 laser scanner coupled to an Oxford Technical Solutions (OxTS) Survey + 2 GPS-IMU. The VUX-1 could record up to 500,000 measurements per second at a wavelength of 1550 nm. The sensor recorded the return time of emitted laser pulses and processed the recorded waveform to identify discrete reflective surfaces (returns) within every emitted laser pulse. All returns were included in later analyses. The OxTS GPS-IMU was a survey-grade instrument designed for aerial mapping applications. The nominal roll and pitch accuracy were 0.03°, and the nominal heading accuracy in dual-antenna mode was 0.05°. The standard deviation of roll and pitch angles were 0.942° and 0.791°, respectively, indicating that stability of the aircraft during flight had minimal impact on our analysis. During flight operations, we collected an independent Global Navigation Satellite System (GNSS) data stream using a Novatel FlexPak 6 Triple Frequency + L-band GNSS receiver. We used this data stream to differentially correct the OxTS GPS-IMU measurements in post processing. A previous analysis demonstrated that post-processed range accuracy was less than 5 cm (one sigma) on a heterogeneous target (a fallen dead tree) [45].

The data were collected in two sets of orthogonal flight lines on 6 June 2018 under leaf-on conditions. There were 45 flight lines in the NW–SE direction, and 45 flight lines in the NE–SW direction (Figure 1A). We used two sets of flight lines over the same area to ensure high-density point coverage of stem and branch structure. The nominal flight altitude was 110 m above ground, and the nominal flight speed was 6 m/s. The maximum scan-angle range was ±60°. Mean point density in the leaf-on point cloud was 2166 points/m². We classified ground returns within the calibrated and georeferenced point cloud using a progressive morphological filter [80] and excluded ground returns from the subsequent analyses (Figure 2).

2.2. Terrestrial Laser Scanning

We collected TLS data within a 1 ha area at 22 scanning positions using a Leica P20 ScanStation on 6 June 2018 (Figure 1B). All TLS data were co-registered using ground-based reflective targets. The mean absolute co-registration error was 1 mm [81]. We classified ground returns in the calibrated and georeferenced point cloud using the terrain-from-octree algorithm in 3D Forest [82]. We then manually co-registered TLS data to the drone lidar point cloud. To estimate the registration error between TLS and drone lidar data, we measured the distance between 20 pairs of TLS and drone lidar points in each of the four 100 m² plots (80 pairs in total) and calculated the arithmetic mean. The mean distance was 0.25 m, and the standard deviation was 0.2 m.

2.3. Ray Tracing and Voxel-Traversal

We assessed the impact of pulse density and scan-angle range on detected and different types of undetected voxels, using a grid with 0.5 m voxels. For each voxel, we determined whether the voxel was occupied by vegetation or not, based on the union of the TLS and drone lidar points. Occupied voxels were voxels containing at least one return from TLS or drone lidar data. Part of our analysis was to investigate under what pulse density and scan-angle range the occupied voxels would be detected by simulating different pulse densities and scan-angle ranges. Our analysis identified three types of undetected voxels (Figure 3). First, we describe the reconstruction of pulse trajectories and the determination of different undetected voxels with ray tracing. In Section 2.4, we then document how different pulse densities and scan-angle ranges were simulated.

2.3.1. Reconstruction of Pulse Trajectories

We considered pulse trajectories as line segments with a starting point and an ending point. The location of the ending point was determined by the coordinates of a lidar return. To denote the starting point, we used the aircraft location at the time a return was recorded. For each recorded return, we determined the aircraft’s location in latitude (Y), longitude (X), and elevation (Z) using linear interpolation of the 100 Hz GPS trajectory. After determining starting points of the pulse trajectories, we identified multiple returns associated with the same pulse using the recorded return number and the GPS time.

2.3.2. Voxel Traversal

We calculated the searching distances of the pulse trajectories within each voxel to identify different types of undetected voxels. Before calculating the searching distances, we determined the intersection points of each pulse trajectory with each voxel, based on Equations (2)–(4). First, we assessed whether the pulse was parallel to any plane of a voxel using

$$\overrightarrow{u}·\overrightarrow{np}\ne 0,$$

(2)

where $\overrightarrow{u}$ represents the direction vector of a pulse trajectory, and $\overrightarrow{np}$ represents the normal vector of a plane coinciding with one of the six voxel surfaces. When the dot product of the two vectors was equal to 0, the pulse was parallel to the plane of the voxel and could not intersect that plane. The purpose of this parallel detection step was to refine the search scope, ensuring that only pulses likely to intersect with a voxel plane were considered in the subsequent analyses.

When the pulse was not parallel to a voxel plane, a line (not a line segment) sharing the same direction as the pulse trajectory intersected the infinite plane that coincided with the voxel plane. The intersection point between the line and the infinite plane is denoted by $P$. Solving for $P$ requires the distance from the aircraft location, $u_0$, to $P$, denoted using the scalar $t$. We solve for $P$ and $t$ using Equations (3) and (4):

$$P=u_0+t·\overrightarrow{u},$$

(3)

$$(P-P_0)·\overrightarrow{np}=0,$$

(4)

where $P_0$ represents a random point on the plane. Only one random point is sufficient to solve for $P$ and $t$. We used one vertex of the plane as the random point because coordinates of the vertices of the voxel planes were known from voxelization.

After determining all intersection points based on the line and the infinite planes of a voxel, we excluded intersection points outside the voxel. Each voxel had at most two intersection points. We denominated the distances from the aircraft location to the intersection points D_min and D_max according to the relative quantity of the distances, where D_min represented the distance from the starting point of the pulse trajectory to the entry point into the voxel, and D_max represented the distance to the departure point. The distance from the aircraft location to the lidar return point was D. If, for a given voxel, D < D_min or D > D_max, the lidar return was generated in a different voxel.

The searched distance of a pulse trajectory concerning a given voxel was calculated under four scenarios (Figure 3A). In scenario #1 (Figure 3A: c, d, g, h), there were no intersection points for a given voxel, meaning that the searched distance of pulses inside the voxel was 0 because the laser was not fired in the direction of the voxel. In scenario #2 (Figure 3A: f, i), D < D_min. Here, the searched distance within the voxel was 0 because the lidar return was triggered in a voxel closer to the sensor along the beam path. Scenario #3 (Figure 3A: e) illustrated a searched voxel where D_min < D < D_max. Here the searched distance was D − D_min and a lidar return was generated inside the voxel. Lastly, in scenario #4 (Figure 3A: a, b), D > D_max, representing a voxel searched without producing a lidar return. The searched distance was D_max − D_min. The total distance searched for a given voxel was the sum of the searched distances produced by all pulse trajectories within the voxel.

When a voxel could have been traversed by a pulse but was not due to interception by other material along the beam path, this voxel was an occluded voxel. Voxels could be partially occluded or completely occluded. A voxel was partially occluded when non-occluded pulses also traversed it. A voxel was completely occluded when it was never traversed by a pulse, despite individual pulses being fired in the direction of the voxel.

2.4. Quantifying Sampling Completeness under Simulated Conditions

We quantified sampling completeness and the proportions of three types of undetected voxels (Figure 3B; undetected and searched voxels, undetected and unobserved voxels, and undetected and completely occluded voxels) under simulated pulse densities and scan-angle ranges. Sampling completeness was quantified using the proportion of occupied voxels that were detected. The proportions of each type of undetected voxel were the ratios of those voxels relative to the total number of occupied voxels. Sampling completeness was quantified in both a 1 ha plot and four 100 m² plots, while proportions of different types of undetected voxels were quantified in the 100 m² plots (Figure 1C, Table 1). Note that for the 100 m² plots, only returns within each plot were considered. The undetected voxels were studied only in the 100 m² plots because the three types of undetected voxels needed to be determined by tracing pulse trajectories at each pulse density and scan-angle range, which was computationally demanding for the 1 ha plot.

To quantify sampling completeness, we simulated different conditions by changing pulse density and scan-angle range. We considered pulse densities of 1 pulse/m², 10 pulses/m², 50 pulses/m², 100 pulses/m², 200 pulses/m², 300 pulses/m², 400 pulses/m², 500 pulses/m², 1000 pulses/m², and 1500 pulses/m². The nominal scan-angle range in our data was ±60°. We therefore considered scan-angle ranges of ±5° to ±60° in increments of ±5° (±5°, ±10°, ±15°, ±20°, ±25°, ±30°, ±35°, ±40°, ±45°, ±50°, ±55°, ±60°). For example, a scan-angle range of ±5° meant that the observations were acquired within a scan-angle range of −5° to 5°; a scan-angle range of ±30° meant that the observations were acquired within a scan-angle range of −30° to 30°. For each combination of pulse density and scan-angle range, we excluded returns outside the specific scan-angle range, randomly sampled pulses from drone lidar according to a certain pulse density, identified voxel types according to the sampled pulse trajectories and relevant returns, and calculated proportions of different types of voxels. Note that we examined multiple returns when sampling pulses, because sampling the pulse trajectory of any return sampled all returns associated with that pulse. We repeated this process 10 times for each combination of pulse density and scan-angle range. The proportions were the arithmetic mean over these 10 simulations. Each combination of pulse density and scan-angle range was generated within tiles to reduce computation time, and results from tiles were then aggregated to derive results. Each tile in the 1 ha plot was 50 m², and each tile in a 100 m² plot was 4 m². Note that in the simulations based on tiles, the number of pulses under a simulated scan-angle range might be smaller than the pulse density to be simulated. Thus, there were missing simulated combinations for certain scan-angle ranges and pulse densities. We used linear regression to quantitatively assess the contribution of pulse density and scan-angle range to sampling completeness. The proportion of detected voxels was the response variable, and predictor variables were pulse density and scan-angle range. We compared three models: (1) a pulse density model; (2) a scan-angle range model; (3) a model including both pulse density and scan-angle range.

Although our flight plan was designed to minimize sampling variation from different observation angles, the observation angle may have an impact on sampling completeness. Therefore, we sampled combinations of four azimuth-angle ranges crossed by pulse density. The azimuth angle of each pulse was defined as the angle from 0° to the projected pulse trajectory on the horizontal plane. We considered four azimuth-angle ranges: 0°–90°, 90°–180°, 180°–270°, and 270°–360°.

To gain a deeper understanding of the sampling capability of the high-density drone lidar, we investigated voxels at different heights by quantifying the different types of voxels throughout the canopy height profile. We obtained the ground elevation beneath each voxel using a digital terrain model (DTM) generated from ground returns. The aboveground height for each voxel was calculated by subtracting the ground elevation from the voxel elevation. We classified voxels based on their heights above ground. Voxel heights above ground used in this study ranged from 0 to 46 m, with increments of 0.5 m (e.g., 0 refers to the 0–0.5 m height class, 0.5 refers to the 0.5–1 m height class, and so forth). For each height increment, we quantified proportions of detected and different undetected voxels in the 100 m² plots under each combination of simulated pulse density and scan-angle range.

3. Results

Results from all four 100 m² plots were broadly consistent, so we report results from Plot 1 in the main text. Summaries for the remaining plots are in Appendix A.

3.1. Sampling Completeness under Different Pulse Densities and Scan-Angle Ranges

Sampling completeness was more sensitive to pulse density than to scan-angle range (Figure 4). Although a strong relationship existed between sampling completeness and scan-angle range (Figure 4B), decoupling scan-angle range from pulse density revealed that most of the variation in sampling completeness was driven by pulse density (Figure 4A). For example, even with a scan-angle range as large as ±60°, a pulse density smaller than 100 pulses/m² could not achieve sampling completeness > 30% (Figure 4A). Results from the linear regression showed that pulse density explained about ten times more variation in sampling completeness than scan-angle range (R² = 0.650 for pulse density and 0.061 for scan-angle range;

Table A1. Linear regressions show that pulse density ($pd$) explains more of the variation in sampling completeness than scan-angle range ($sar$).

Model	R2
$samp. completeness=0.0006\times pd+0.0006\times sar+0.2544$	0.652
$samp. completeness=0.0006\times pd+0.2741$	0.650
$samp. completeness=0.0039\times sar+0.3101$	0.061

). All regressions were statistically significant (p < 0.001 for models with both variables in combination and for the model with pulse density, and p = 0.011 for the model with scan-angle range). Including both pulse density and scan-angle range did not meaningfully increase the amount of variance explained (R² = 0.652).

3.2. Undetected Voxels under Different Pulse Densities and Scan-Angle Ranges

Combining data from drone lidar and TLS, we identified 12,823 occupied voxels in the 100 m² study area (plot 1). Before subsampling to limit pulse density and scan-angle range, the drone lidar searched 12,577 of the occupied voxels (98.1%) and detected 10,486 of the occupied voxels (81.8%), while 2091 of the searched voxels (16.6%) remained undetected. There were no undetected and unobserved voxels in the original data, meaning that all the undetected and unsearched voxels were unsearched due to occlusion, and were therefore regarded as undetected and completely occluded voxels.

When the pulse density and the scan-angle range were at their maximum subsampled value (1500 pulses/m², ±60°), 3065 (23.9%), 10 (0.1%), and 465 (3.6%) undetected voxels were identified as undetected and searched, undetected and unobserved, and undetected and completely occluded, respectively (Figure 5). However, when we simulated smaller pulse densities and narrower scan-angle ranges, proportions of different undetected voxels varied substantially (Figure 5 and Figure 6). For example, with a scan-angle range of ±10° and a pulse density of 1 pulse/m², the proportions of undetected and searched, undetected and unobserved, and undetected and completely occluded voxels were 8.7%, 76.7%, and 13.7%, respectively (Figure 5). At 300 pulses/m² with the same scan-angle range, these values changed to 33.6%, 4.4%, and 20.3%. Holding the pulse density at 300 pulses/m² and increasing the scan-angle range to ±60° resulted in values of 40.9%, 0.9%, and 9.0%, respectively (Figure 5).

The types of undetected voxels depended strongly on pulse density and weakly on scan-angle range (Figure 5 and Figure 6). Increasing pulse density or scan-angle range always reduced the proportion of undetected and unsearched or more specifically, undetected and unobserved voxels (Figure 6A,C). However, this was not always the case for undetected and searched voxels or undetected and completely occluded voxels; whether the numbers of these two types of undetected voxels decreased or increased depended on the pulse density and the scan-angle range. As the pulse density increased, the numbers of undetected and searched voxels and undetected and completely occluded voxels first increased and then decreased. For example, the proportion of undetected and searched voxels increased when the pulse density increased from 1 to 50 pulses m² (Figure 6B), as it did for undetected and completely occluded voxels when the pulse density increased from 1 to 10 pulses/m² (Figure 6D). When the scan-angle range increased, the proportion of undetected and searched voxels increased (Figure 6B), while the proportions of the other three types of voxels decreased (Figure 6A,C,D). For all three types of undetected voxels, when the pulse density was as low as 1 pulse/m², the scan-angle range had little impact.

3.3. Vertical Variability of Proportions of Different Types of Voxels

High-density drone lidar detected voxels throughout the canopy height profile. There was a greater likelihood of detection near the top of the canopy (Figure 7 and Figure 8). For example, at 30–30.5 m above ground, 91.3% of occupied voxels were detected when the pulse density was 1000 pulses / m² and the scan-angle range was ±60°. These percentages decreased to 57.7%, 52.4%, and 29.9% at 20–20.5 m, 11–11.5 m, and 3–3.5 m above ground, respectively (Figure 7). In general, almost no voxels were undetected and completely occluded near the top of the canopy. Both types of undetected and unsearched voxels (i.e., undetected and unobserved voxels and undetected and completely occluded voxels) were more common at lower heights and at narrower scan-angle ranges. For example, in the upper canopy (30–30.5 m in Figure 7), there were no undetected and completely occluded voxels and all voxels were searched when the scan-angle range exceeded ±20° and the pulse density was greater than 100 pulses/m². At 10–11.5 m above ground, some voxels were consistently occluded, regardless of the scan-angle range and the pulse density (Figure 7 and Figure 8).

At different heights, the proportion of undetected voxels varied with pulse densities and scan-angle ranges differently (Figure 7). For example, undetected and unobserved voxels decreased as the pulse density increased at both lower and higher heights. However, undetected and searched voxels increased with pulse density at lower heights (e.g., scan-angle range ±20°, heights 3–3.5 m, Figure 7 and Figure 9) and decreased with pulse density at higher heights (e.g., scan-angle range ±20°, heights 30–30.5 m, Figure 7 and Figure 9). At lower heights (Figure 7: 3–3.5 m, 11–11.5 m), undetected and completely occluded voxels first increased then decreased as the pulse density increased at most scan-angle ranges, while at higher heights they decreased with increasing pulse density (Figure 10).

The types of undetected voxels that were mainly responsible for unsampled forest structure changed with pulse density and height above ground. Closer to the ground (3–3.5 m and 11–11.5 m in Figure 7), undetected voxels were mostly completely occluded. As pulse density increased, undetected voxels were more likely to have been searched but undetected. For example, at 11–11.5 m above ground, when the pulse density was 100 pulses/m² and the scan-angle range was ±10°, 66% of the occupied voxels were undetected and unsearched (8.9% were undetected and unobserved, and 57.1% were undetected and completely occluded), 26.8% were undetected and searched, and the detected voxels accounted for 7.2%. At the same height, when the pulse density was 1500 pulses/m² and the scan-angle range was ±60°, the proportion of undetected and searched voxels increased to 38.1%. Closer to the canopy (20–20.5 m and 30–30.5 m in Figure 7), the proportion of undetected and searched voxels was greater than that of undetected and completely occluded voxels at most of the pulse densities examined. There were undetected and searched voxels even at a high pulse density of 1500 pulses/m² (Figure 7).

4. Discussion

4.1. Sampling Completeness under Different Pulse Densities and Scan-Angle Ranges

Our study showed that the scan-angle range had a limited impact on sampling completeness after it was decoupled from pulse density (Figure 4). We quantified the impact of pulse density and scan-angle range on sampling completeness and found that sampling completeness was more sensitive to pulse density than to scan-angle range. Increasing the pulse density increased the percentage of occupied voxels that were detected among all pulse densities examined, but increasing the scan-angle range did not necessarily increase sampling completeness. Thus, pulse density is the main driver of sampling completeness in high-density lidar data, no matter how large the scan-angle range is.

Although scan-angle range was less important than pulse density for sampling completeness, some voxels were more likely to be observed at wide scan-angle ranges, including voxels occupied by vertical stems (Figure A1, Figure A2 and Figure A3). This finding aligns with previous studies that have demonstrated the importance of a wide scan-angle range or multiple viewing angles in generating stem returns [30,45,55,83]. In contrast to previous studies, this study extends insight into scan-angle range by investigating when the impact of scan-angle range plateaued before and after it was decoupled from pulse density (Figure 4).

4.2. Undetected Voxels under Different Pulse Densities and Scan-Angle Ranges

We observed that undetected voxels exhibited patterns of distinct covariation with pulse density and scan-angle range. Undetected and searched voxels or undetected and completely occluded voxels did not necessarily decrease with greater pulse densities or scan-angle ranges, whereas undetected and unobserved voxels consistently decreased as the pulse density increased. Whether the former two types of undetected voxels decreased or increased with greater pulse densities depended on the specific pulse density and scan-angle range. As the pulse density increased, voxels that were undetected and searched or that were undetected and completely occluded first increased and then decreased (Figure 5 and Figure 6).

This study demonstrated that the undetected and searched voxels were the most common type of undetected voxels under the investigated voxel size (0.5 m), even under the largest possible pulse density and scan-angle range, followed by undetected and completely occluded voxels and undetected and unobserved voxels (Figure 5, when the scan-angle range was ±60°). The existence of undetected and searched voxels underscores that space searched by pulses in the absence of a return is not necessarily unoccupied. The proportion of undetected and searched voxels was negatively related to pulse density (Figure 5 and Figure 6B). This was probably caused by insufficient searching distance within the voxel, although it could also have been linked to voxel size, the number of pulses, and the distribution of vegetation in the voxel. The existence of such voxels could also depend on the amount of vegetation in comparison to other factors (i.e., footprint size and laser power). If there is too little vegetation inside a voxel, the reflected pulse might be insufficient to trigger a return, especially when footprint size is large relative to the amount of vegetation in the voxel. Future work should investigate the relationship between sampling completeness and search effort, including whether the relationship changes in response to voxel size and laser power.

A majority of undetected and searched voxels had a large proportion of occluded pulses (Figure A16). When the pulse density was 1500 pulses/m² and the scan-angle range was ±60°, most of the undetected and searched voxels were occluded by more than half the pulses. The number of undetected and searched voxels with pulses that were occluded >50% of the time increased with pulse density and scan-angle range.

Azimuth-angle range had less impact on detected voxels than undetected voxels (Figure A7, Figure A8, Figure A9 and Figure A10). When the pulse density was <100 pulses/m², the proportions of detected voxels from the four azimuth-angle ranges were similar. Variation in sampling completeness from different azimuth-angle ranges was greater at large pulse densities. This probably reflected local differences in forest structure within the 100 m² plots.

Returns outside a plot may have influenced our results, but we believe the impact was small. Because we tracked all pulse trajectories associated with returns inside the plot, returns outside the plot could have affected the proportions of undetected and unsearched voxels, but not the proportions of detected voxels. By tracing additional pulses that triggered returns outside the plot, some undetected and unobserved voxels might be reclassified as undetected and searched or undetected and completely occluded voxels. Voxels that were undetected and completely occluded might remain the same or be reclassified as undetected and searched voxels in this scenario. Because the proportion of undetected and unsearched voxels was small (Figure 5, especially when the pulse density was greater than 1000 pulses/m²), pulses outside the plot would have had limited impact on our results.

4.3. Vertical Variability of Proportions of Different Types of Voxels

Our study demonstrates that, despite a reduction in sampling completeness as pulses penetrated deeper into the canopy (i.e., as the height above ground decreased; Figure 7), the use of drone lidar data with a wide scan-angle range was able to achieve sampling completeness > 50% close to ground level (Figure A14). For example, when the pulse density was 1000 pulses / m², approximately half of the occupied voxels at 5–5.5 m above ground were detected, even with a scan-angle range of ±30° (Figure A14, when the scan-angle range was ±30°). When the pulse density was as high as 1500 pulses/m² and the scan-angle range was ±60°, nearly 60% of the occupied voxels at 10–10.5 m above ground were detected (Figure A14, when the scan-angle range was ±60°).

Nevertheless, undetected voxels persisted at both lower and higher heights, even when the pulse density was large and the scan-angle range was wide (Figure 7). For example, at lower heights, greater pulse densities eliminated undetected and unobserved voxels, but undetected and completely occluded voxels and undetected and searched voxels remained. In the upper canopies, greater pulse densities eliminated undetected and completely occluded voxels and undetected and unobserved voxels, but undetected and searched voxels persisted. The persistence of undetected voxels at lower heights was probably due to the scarcity of pulses at these heights. A similar challenge near the top of the canopy exists in TLS sampling. Previous studies showed that a single scan from the center of a plot failed to detect upper-canopy vegetation and around 40% of the trees [21,84,85,86], and even multiple scans did not guarantee complete sampling of canopy volume or stem structure [21,67]. One study in a temperate broadleaf forest in North America showed that as many as nine TLS scans did not provide complete sampling of the upper canopy structure, where more than half of the voxels were unobserved, and that the proportion of unsampled voxels increased above 10 m in height [67].

4.4. Future Work

Some of our findings are likely to have depended on voxel size. For example, undetected and searched voxels may be more common among larger individual voxel sizes than smaller ones. A future investigation should consider voxels of different sizes, including sizes that approach the diameter of the laser footprint. We anticipate that when the voxel size is smaller, especially when the voxel size is close to the laser footprint size, there will be relatively fewer undetected and searched voxels, and relatively more undetected and unsearched voxels. Note that there are also other factors that may result in variation in sampling completeness across different sites, including flight and atmospheric conditions and properties of the lidar sensor, including wavelength, pulse repetition rate, and laser power [54]. Also note that although pulse trajectories are usually traced as lines, the actual interaction between a laser pulse and an object of interest may be influenced by beam divergence.

The proportion of detected voxels in this study was assessed from a 1 ha area in an old-growth temperate mountain forest, and proportions of undetected voxels were quantified in 100 m² plots. Coincident drone lidar and TLS collected at different locations with different canopy densities can be used to test the generality of these findings. Given the scarcity of research exploring various types of failures to generate lidar returns, the empirical proportions derived from our study could be useful for calibrating sensor outputs, but the generality of our conclusions requires validation.

Our approach to sub-setting lidar data used tiles overlaid on each plot. Generating samples independently within each tile may have influenced our conclusions by preventing spatial heterogeneity in the density of returns. We expect the impact of sampling strategy to be most important when pulse density is small, but future work should consider alternative approaches to sub-sampling.

5. Conclusions

High-resolution drone lidar can acquire detailed measurements of vertical and horizontal forest structure. Drone lidar searched 98.1% of voxels known to contain vegetation in our study area, and generated returns in 81.8% of these voxels. By decoupling the impact of pulse density and scan-angle range on the likelihood of generating a return, this study demonstrates that nearly complete sampling of forest structure is possible using high-density drone lidar in a temperate mountain forest. The ability to generate a return when vegetation was present was more sensitive to pulse density than to scan-angle range. At very large pulse densities (>1000 pulses/m²), failure to generate a return was almost exclusively associated with searched voxels that contained vegetation but did not trigger a return.

This study documents three ways that a return can fail to be generated in the presence of vegetation: (1) the voxel could be searched but no return generated, for example, because the voxel was sparsely populated and contained mostly empty space; (2) the voxel could be unsearched because it was occluded; and (3) the voxel could be unsearched because no pulse was fired in the direction of the voxel. Our findings demonstrated that all types of undetected voxels existed, and the proportion of each type changed with pulse density and scan-angle range. The strength and form of the relationships depended on canopy height, pulse density, and scan-angle range. These findings provide an empirical assessment of sampling completeness using a combination of high-resolution drone lidar and coincident TLS data. Our findings demonstrate the value of ray tracing when considering questions of sampling completeness in drone lidar data.