Terrestrial LiDAR becomes more and more popular to estimate leaf and plant area density.Voxel-based approaches account for this vegetation heterogeneity and significant work has been done in this recent research field, but no general theoretical analysis is available.Although estimators have been proposed and several causes of biases have been identified, their consistency and efficiency have not been evaluated. Also, confidence intervals are almost never provided.In the present paper, we solve the transmittance equation and use the Maximum Likelihood Estimation (MLE), to derive unbiased estimators and confidence intervals for the attenuation coefficient, which is proportional to leaf area density. The new estimators and confidence intervals are defined at voxel scale, and account for the number of beams crossing the voxel, the inequality of path lengths in voxel, the size of vegetation elements, as well as for the variability of element positions between vegetation samples. They are completed by numerous numerical simulations for the evaluation of estimator consistency and efficiency, as well as the assessment of the coverage probabilities of confidence intervals.• Although commonly used when the beam number is low, the usual estimators are strongly biased and the 95% confidence intervals can be ≈ ±100% of the estimate.
Reliable measurements of the 3D distribution of Leaf Area Density (LAD) in forest canopy are crucial for describing and modelling microclimatic and eco-physiological processes involved in forest ecosystems functioning. To overcome the obvious limitations of direct measurements, several indirect methods have been developed, including methods based on Terrestrial LiDAR scanning (TLS). This work focused on various LAD estimators used in voxel-based approaches. LAD estimates were compared to reference measurements at branch scale in laboratory, which offered the opportunity to investigate in controlled conditions the sensitivity of estimations to various factors such as voxel size, distance to scanner, leaf morphology (species), type of scanner and type of estimator. We found that all approaches to retrieve LAD estimates were highly sensitive to voxel size whatever the species or scanner and to distance to the FARO scanner. We provided evidence that these biases were caused by vegetation heterogeneity and variations in the effective footprint of the scanner. We were able to identify calibration functions that could be readily applied when vegetation and scanner are similar to those of the present study. For different vegetation and scanner, we recommend replicating our method, which can be applied at reasonable cost. While acknowledging that the test conditions in the laboratory were very different from those of the measurements taken in the forest (especially in terms of occlusion), this study revealed existence of strong biases, including spatial biases. Because the distance between scanner and vegetation varies in field scanning, these biases should occur in a similar manner in the field and should be accounted for in voxel-based methods but also in gap-fraction methods.
The spatial distribution of Leaf Area Density (LAD) in a tree canopy has fundamental functions in ecosystems. It can be measured through a variety of methods, including voxel-based methods applied to LiDAR point clouds. A theoretical study recently compared the numerical errors of these methods and showed that the bias-corrected Maximum Likelihood Estimator was the most efficient. However, it ignored (i) wood volumes, (ii) vegetation sub-grid clumping, (iii) the instrument effective footprint, and (iv) was limited to a single viewpoint. In practice, retrieving LAD is not straightforward, because vegetation is not randomly distributed in sub-grids, beams are divergent, and forestry plots are sampled from more than one viewpoint to mitigate occlusion. In the present article, we extend the previous formulation to (i) account for both wood volumes and hits, (ii) rigorously include correction terms for vegetation and instrument characteristics, and (iii) integrate multiview data. Two numerical experiments showed that the new approach entailed reduction of bias and errors, especially in the presence of wood volumes or when multiview data are available for poorly-explored volumes. In addition to its conciseness, completeness, and efficiency, this new formulation can be applied to multiview TLS—and also potentially to UAV LiDAR scanning—to reduce errors in LAD estimation.
9Terrestrial LiDAR becomes more and more popular to estimate leaf and plant area density. 10Voxel-based approaches account for this vegetation heterogeneity and significant work has 11 been done in this recent research field, but no general theoretical analysis is available.
2• Our unbiased estimators are consistent in a wider range of validity than the usual ones, 26 especially for the unbiased MLE, which is consistent when the beam number is as low as 5. 27The unbiased MLE is efficient, meaning it reaches the lowest residual errors that can be 28 expected (for an unbiased estimator). Also the unbiased MLE does not require any bias 29 correction when path lengths are unequal. 30• When elements are small (or voxel is large), 10 3 beams entering the voxel leads to some 31 confidence intervals ≈ ±10%, but when elements are larger (or voxel smaller), it can remain 32 wider than ±50%, even for a large beam number. This is explained by the variability of 33 element positions between vegetation samples. Such a result shows that a significant part of 34 residual error can be explained by random effects. 35• Confidence intervals are much smaller (±5 to 10%) when LAD estimates are averaged over 36 several small voxels, typically within a horizontal layer or in the crown of individual plants. 37In this context, our unbiased estimators show a reduction of 50% of the radius of confidence 38 intervals, in comparison to usual estimators. 39Our study provides some new ready-to-use estimators and confidence intervals for attenuation 40 coefficients, which are consistent and efficient within a fairly large range of parameter values. 41The consistency is achieved for a low beam number, which is promising for application to 42 airborne LiDAR data. They entail to raise the level of understanding and confidence on LAD Highlights: 52• Voxel-based estimations of LAD/PAD may lack of consistency and efficiency 53• We propose new estimators based on theoretical derivation and numerical simulations 54• Estimators for confidence intervals are also provided 55• New estimators should help determine the most appropriate voxel resolution 56 57
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