Jump to Main Content
Estimators and confidence intervals for plant area density at voxel scale with T-LiDAR
- Pimont, François, Allard, Denis, Soma, Maxime, Dupuy, Jean-Luc
- Remote sensing of environment 2018 v.215 pp. 343-370
- confidence interval, equations, leaf area, leaves, lidar, mathematical models, remote sensing, scanners, statistical analysis, transmittance, vegetation types
- 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 random effects. When LAD estimates are averaged over several small voxels -typically to determine a vertical profile at plot scale or to compute the total leaf area of a single plant-, confidence intervals are typically on the order of ±5 to 10% with bias-corrected estimators, which is twice as small as with usual estimators.Our study provides some new ready-to-use estimators and confidence intervals for attenuation coefficients, which are unbiased and efficient within a fairly large range of parameter values. The unbiasedness is achieved for a fairly low beam number, which is promising for application to airborne LiDAR data. They permit to raise the level of understanding and confidence on LAD estimation. Among other applications, their usage should help determine the most suitable voxel size, for given vegetation types and scanning density, whereas existing guidelines are highly variable among studies, probably because of differences in vegetation, scanning design and estimators. The impact of other sources of biases and errors, such as vegetation heterogeneity inside voxels or TLS specifications are not addressed in the present manuscript and would require further investigations.