Abstract 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.

中文翻译：

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T-LiDAR 体素尺度植物面积密度的估计值和置信区间

摘要 使用地面激光扫描仪 (TLS) 估计叶子和植物面积密度继续变得越来越流行，因为三维点云似乎是一种用于异构环境的有吸引力的测量技术。一些方法在网格中实现点云的离散化（称为“基于体素”）以解释这种植被异质性，并且在这个最近的研究领域已经完成了重要的工作，但没有通用的理论分析可用。尽管已经提出了估计量并且已经确定了一些偏差的原因，但是它们的无偏性（零偏差）和效率（最小误差）还没有得到评估。此外，几乎从不提供置信区间。在本文中，我们假设植被元素随机分布在体素内，并且 TLS 光束无限薄，以便关注剩余的偏差和错误来源。在这种简化的上下文中，我们既求解透射率方程，又使用最大似然估计 (MLE) 来推导出衰减系数的一些新估计量，在这种理想化的上下文中，衰减系数与体素尺度的叶面积密度成正比。这些估计量包括偏差校正和置信区间，并考虑穿过体素的光束数量（光束数）、体素中路径长度的不等式、植被元素的大小以及植被样本之间元素位置的可变性. 这些理论推导得到了大量数值模拟的补充，用于评估估计量偏差和效率，以及评估置信区间的覆盖概率。我们的模拟表明，当光束数小于 30 时，通常的估计量是有偏差的，并且表现出 95% 的置信区间约为估计值的 ±100%。 其次，我们的偏差校正估计量——尤其是偏差校正的 MLE——真正无偏和有效，在更广泛的有效性范围内比通常的有效，即使对于低至 5 的光束数。 第三，我们发现当单个元素的投影面积时，置信区间可以高达 ≈ ± 50%大约为体素横截面积的 10%，植被密集（体素的光学深度等于 2），即使光束数大于 1000。这是由植被样本之间元素位置的可变性来解释的，这意味着很大一部分残差是由随机效应引起的。当 LAD 估计在几个小体素上平均时——通常是为了确定地块规模的垂直剖面或计算单个植物的总叶面积——，置信区间通常在 ±5% 到 10% 的数量级，并使用偏差校正估计器，这是通常估计量的两倍。我们的研究为衰减系数提供了一些新的即用型估计器和置信区间，它们在相当大的参数值范围内是无偏和高效的。在相当低的波束数下实现了无偏性，这有望应用于机载 LiDAR 数据。它们允许提高对 LAD 估计的理解和信心水平。在其他应用中，对于给定的植被类型和扫描密度，它们的使用应该有助于确定最合适的体素大小，而现有的指南在研究之间变化很大，可能是因为植被、扫描设计和估算器的差异。本手稿未涉及其他偏差和错误来源的影响，例如体素内的植被异质性或 TLS 规范，需要进一步调查。扫描设计和估算器。本手稿未涉及其他偏差和错误来源的影响，例如体素内的植被异质性或 TLS 规范，需要进一步调查。扫描设计和估算器。本手稿未涉及其他偏差和错误来源的影响，例如体素内的植被异质性或 TLS 规范，需要进一步调查。