Estimators and Confidence Intervals for Plant Area Density at Voxel Scale with T-LiDAR
Authors: Pimont Francois , Allard Denis , Soma Maxime , Dupuy Jean-Luc 
Published: September 2017

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. &bull; Although commonly used when the beam number is low, the usual estimators are strongly biased and the 95% confidence intervals can be &asymp;&plusmn;100% of the estimate. &bull; Our unbiased estimators are consistent in a wider range of validity than the usual ones, especially for the unbiased MLE, which is consistent when the beam number is as low as 5. The unbiased MLE is efficient, meaning it reaches the lowest residual errors that can be expected (for an unbiased estimator). Also the unbiased MLE does not require any bias correction when path lengths are unequal. &bull; When elements are small (or voxel is large), 103 beams entering the voxel leads to some confidence intervals &asymp;&plusmn;10%, but when elements are larger (or voxel smaller), it can remain wider than &plusmn;50%, even for a large beam number. This is explained by the variability of element positions between vegetation samples. Such a result shows that a significant part of residual error can be explained by random effects. &bull; Confidence intervals are much smaller (&plusmn;5 to 10%) when LAD estimates are averaged over several small voxels, typically within a horizontal layer or in the crown of individual plants. In this context, our unbiased estimators show a reduction of 50% of the radius of confidence intervals, in comparison to usual estimators. Our study provides some new ready-to-use estimators and confidence intervals for attenuation coefficients, which are consistent and efficient within a fairly large range of parameter values. The consistency is achieved for a low beam number, which is promising for application to airborne LiDAR data. They entail 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.