Estimators and confidence intervals for plant area density at voxel scale with T-LiDAR

• Published in: Remote sensing of environment 2018-09, Vol.215, p.343-370
• Main Authors: Pimont, François, Allard, Denis, Soma, Maxime, Dupuy, Jean-Luc
• Format: Article
• Language: English
• Subjects: Attenuation coefficients; Beams (radiation); Bias; Computer simulation; Confidence intervals; Data processing; Density; Efficiency; Element size; Estimators; Heterogeneity; LAD; LAI; Lasers; Leaf area; Leaves; Lidar; Life Sciences; Maximum likelihood estimation; Measurement techniques; Numerical simulations; Optical analysis; Optical scanners; Planting density; Remote sensing; Scanning; Terrestrial environments; Terrestrial LiDAR; Theoretical analysis; TLS; Variability; Vegetal Biology; Vegetation
• ISSN: 0034-4257; 1879-0704
• DOI: 10.1016/j.rse.2018.06.024

Estimating leaf and plant area density with Terrestrial Laser Scanners (TLS) continues to be more and more popular, as tridimensional point clouds appear as an appealing measurement technique for heterogeneous environments. Some approaches implement a discretization of the point cloud in a grid (referred to as “voxel-based”) to 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 unbiasedness (zero bias) and efficiency (smallest error) have not been evaluated. Also, confidence intervals are almost never provided. In the present paper, we assumed that the vegetation elements were randomly distributed within voxels and that TLS beams were infinitely thin, in order to focus on the remaining sources of biases and errors. In this simplified context, we both solve the transmittance equation and use the Maximum Likelihood Estimation (MLE), to derive some new estimators for the attenuation coefficient, which is proportional to leaf area density at voxel scale in this idealized context. These estimators include bias corrections and confidence intervals, and account for the number of beams crossing the voxel (beam number), the inequality of path lengths in voxel, the size of vegetation elements, as well as for the variability of element positions between vegetation samples. These theoretical derivations are complemented by numerous numerical simulations for the evaluation of estimator bias and efficiency, as well as the assessment of the coverage probabilities of confidence intervals. Our simulations reveal that the usual estimators are biased and exhibit 95% confidence intervals on the order of ±100% of the estimate, when the beam number is smaller than 30. Second, our bias-corrected estimators -especially the bias-corrected MLE- are truly unbiased and efficient in a wider range of validity than the usual ones, even for beam number as low as 5. Third, we found that the confidence intervals can be as high as ≈ ± 50% when the projected area of a single element was on the order of 10% of voxel cross-sectional area and vegetation was dense (optical depth of the voxel equal to 2), even for a beam number larger than 1000. This is explained by the variability of element positions between vegetation samples, which implies that a significant part of residual error is caused by