# Accounting for Wood, Foliage Properties, and Laser Effective Footprint in Estimations of Leaf Area Density from Multiview-LiDAR Data

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}) distribution for individual plants and forest plots [2]. The approach can be applied to a variety of volumes of interest, assuming random distribution of vegetation inside. These volumes can be either horizontal layers to estimate LAD profiles [3,4,5,6,7], individual tree crowns [8], or voxels to estimate the tridimensional distribution of LAD [9,10,11,12,13,14,15]. In these different approaches, a traversal algorithm is applied to each volume of interest to compute gap fractions, hits, and for some approaches, “free paths” (i.e., distances travelled without interception, in m), in order to derive different metrics to estimate the quantity of interest [3,4,5,6,7,8,9,10,11,12,13,14,15].

## 2. Background and Limitations of Existing Methods

#### 2.1. The Theoretically Bias-Corrected Estimator (TBC-MLE)

^{−1}) is an estimator of the attenuation coefficient, $G$ is the dimensionless leaf projection factor, and H is a dimensionless correction factor that accounts for the laser effective footprint in clumped vegetation [14]. Observations suggest that H decreases with the distance to the scanner to compensate for the increase in effective footprint caused by beam divergence and variation in return detection, which induces an increase of the apparent area of vegetation elements [14,18]. Also, H increases with the voxel size to compensate for the effect of vegetation clumping inside voxels, which causes discrepancies to the theoretically random distribution of vegetation elements, as a consequence of Jensen’s convexity inequality [12,14,18,20]. It also depends on the scanner, and to a lesser extent, on foliage morphological differences between species [14], although the element size and shape can at least partially be accounted for through the notion of “effective” free path ${z}_{e}$ (in m, see a previous study [15] or Equation (3) below and Appendix A). The dimensionless projection function G can be separately estimated [9,21].

^{−1}, of a single element of vegetation (see Appendix A for an estimation of ${\lambda}_{1}$ for cylindrical needles or elliptical flat leaves). Obviously, ${z}_{e}\approx z$ when ${\lambda}_{1}$ is very small (i.e., the turbid medium assumption).

#### 2.2. Theoretical Variance and 68% Confidence Interval of the TBC-MLE

#### 2.3. Accounting for Wood Returns

#### 2.4. Multiview Estimators

## 3. Generalized Maximum-Likelihood Estimation for LAD from Multiview-LiDAR Data

^{3}) of the voxel V (in m

^{3}) occupied by wood elements (Figure 2). Within a voxel volume V, we assume that small leaf elements are randomly distributed in the sub-volume $V-{V}_{w}$ of $V$, which is not occupied by the wood. This sub-volume containing the leaf elements has a (dimensionless) volume fraction $\alpha $ equal to:

^{−1}(i.e., not of the attenuation coefficient $\lambda $), which cancels the first derivative of log-likelihood [16] of the LAD and find (Equation (B6)):

## 4. Numerical Experiments

#### 4.1. Comparison between Formulations to Account for Wood Returns and Volumes

^{−1}. In this simple context, the volume fraction $\alpha $:

#### 4.2. Comparison between Multiview Formulations

_{ref}in a 10-m tri-dimensional mesh grid corresponding to plausible features in terms of LAI, clump size and vertical distribution [23]. Voxel size was equal to 0.1 m, and the cubic vegetation scene had a 10-m lateral extension (and a 10-m height). LAD

_{ref}corresponded to a clumped spatial distribution simulated from RandomFields R package, which was parameterized to correspond to realistic features of natural vegetation (cover fraction of 70% and LAI of about 3.8). The mean clump size, which was representative of the tree crown diameter, was 4 m. Additional clumping (~1 m) occurred to represent branch scale heterogeneity. The LAD vertical profile exhibited a peak around 7 m in height (Figure A1a).

## 5. Discussion

^{−3}, which corresponds to volume fraction to the order of 0.02 [30]. However, such a correction is likely to be necessary when trunks or large branches intersect the voxel, otherwise leading to LAD overestimation, even if the leaf fraction F is correctly estimated. In this context, tree models derived from LiDAR data [22] can provide the appropriate information.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Estimation of ${\mathsf{\lambda}}_{\mathbf{1}}$ for Simple Vegetation Element Shapes

## Appendix B. Optimized Multiview Estimator in a Voxel of Interest

## Appendix C. A Numerical Experiment to Compare Different MULTIVIEW Formulations

_{ref}in a mesh grid, with voxels of size equal to 0.1 m, corresponding to a cubic vegetation scene with a 10-m lateral extension and a 10-m height. LAD

_{ref}corresponded to a clumped spatial distribution simulated from RandomFields R package, which was parameterized to correspond to realistic features of natural vegetation. The mean clump size, which was representative of the tree crown diameter, was 4 m, whereas typical LAD vertical profiles, as well as a projection function, were implemented. In order to get a more realistic reference field, the random field LAD

_{ref}was modified as follows. We multiplied it by a realistic vertical profile to get limited vegetation under 3 m, and a peak in LAD around 7 m height (Figure A1a). Also, the first decile of LAD

_{ref}values was set equal to 0 in order to generate actual gaps between crowns. Finally, random variations were also introduced to simulate the occurrence of small gaps (~1 m), representative of branch-scale heterogeneity inside tree crowns. These settings led to a clumped vegetation scene with a 70% cover fraction (Figure A1b) and a vertical structure (Figure A1a). The LAI of the virtual scene was about 3.8, which corresponds to a mean LAD

_{ref}of 0.38 m-1 (the scene vertical extent was 10 m). Maximal LAD

_{ref}values reached 3.8 m

^{−1}.

_{ref}, as well as a two-dimensional horizontal distribution of this vegetation field, are shown in Figure A1a,b. They correspond to a LAI of 3.8 and a cover fraction of 70%.

**Figure A1.**Reference vegetation: (

**a**) vertical profile of $LA{D}_{ref}$; (

**b**) horizontal distribution of $LA{D}_{ref}$ at z = 6 m.

_{ref}for a given scan j depends on leaf projection, leaf fraction, vegetation heterogeneity, and scanner properties (inverting Equation (1)). Let $\left({x}_{j},{y}_{j},{z}_{j}\right)$ be the coordinates of the scanner corresponding to scan j and $\left(x,y,z\right)$ the coordinates of the center of a voxel in the vegetation scene. The effective attenuation coefficient for both leaf and wood for scan j was:

**Figure A2.**Estimated horizontal distribution of ${\tilde{LAD}}^{M}$ at z = 6 m. This distribution could directly be compared to $LA{D}_{ref}$ in Figure A1b. Blank pixels correspond to unexplored voxels, which revealed occluded locations in the canopy.

**Figure A3.**Comparison between predicted and reference LAD, for the three multiview formulations for two classes of beam numbers: (

**a**) ${\tilde{\mathrm{LAD}}}^{\mathrm{Nmax}},$ $N\in \left[5,15\right[$; (

**b**) ${\tilde{\mathrm{LAD}}}^{\mathrm{NW}}$, $N\in \left[5,15\right[$; (

**c**) ${\tilde{\mathrm{LAD}}}^{\mathrm{M}}$, $N\in \left[5,15\right[$; (

**d**) ${\tilde{\mathrm{LAD}}}^{\mathrm{Nmax}}$, $N\in \left[100,500\right[$; (

**e**) ${\tilde{\mathrm{LAD}}}^{\mathrm{NW}}$, $N\in \left[100,500\right[$; (

**f**) ${\tilde{\mathrm{LAD}}}^{\mathrm{M}}$, $N\in \left[100,500\right[$.

## Appendix D. Leaf Fraction Corresponding to the 200 Numerical Simulations Presented in Section 4.1

**Figure A4.**Leaf fraction $F=\frac{{\mathrm{Ni}}^{l}}{\mathrm{Ni}}$ in the 200 simulations presented in Section 4.1.

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**Figure 1.**Scheme of the information provided by the traversal algorithm which is used to compute the MLE of the attenuation coefficient: number of hits Ni (blue dots) and free paths (distances z travelled by the beams; blue lines) in each voxel. The dotted lines represent pulse trajectory.

**Figure 2.**Scheme of the representation of wood volumes ${V}_{w}$ (in dashed blue) in the voxel of volume $V$. We assume that leaf elements are randomly distributed in volume $V-{V}_{w}$, which exhibits a very complex and unknown topology.

**Figure 3.**Scheme of the information provided by the traversal algorithm, which is used to compute the Maximum Likelihood Estimator (MLE) of the Leaf Area Density (LAD) from multiview data from Scan A (in red) and Scan B (in blue): leaf hits (blue and red dots) and free paths (distances z travelled by the beams; blue and red lines) in the voxel. The dotted lines represent pulse trajectories; ${c}_{A}=\frac{{G}_{A}}{{H}_{A}}$ and ${c}_{B}=\frac{{G}_{B}}{{H}_{B}}$ represent the correcting factors for viewpoints A and B, respectively, which differs with distance to scanner and view angle. For simplicity, correction for effective free path (${z}_{e}$; Equation (3)) is ignored. Note that in this framework, no leaf can be distributed within the volume ${V}_{W}$ occupied by wood elements (in brown). Also, and contrary to Figure 1, the hits corresponding to woody elements (e.g., 5th beam of scan 1) are ignored in the hit sum, but the corresponding free paths are accounted for in the free-path sum, in which c

_{A}and c

_{B}are used as multiplicative factors.

**Figure 4.**Comparison between predicted and reference LAD for a variety of formulations to account for wood in estimators (see Table 1 for details): (

**a**) Equation (7); (

**b**) Equation (8); (

**c**) Equation (15), (with $\alpha =1$, ${\mathrm{Ni}}^{l}$>>1, and ${\lambda}_{1}\ll 1$); (

**d**) Equation (7), with $\alpha $ multiplicative factor; (

**e**) Equation (8), with $\alpha $ multiplicative factor; (

**f**) Equation (15) (${\mathrm{Ni}}^{l}$>>1 and ${\lambda}_{1}\ll 1$). Formulations presented in subplots (a–c) ignored wood volumes, contrary to subplots (d–f).

**Figure 5.**Vertical profiles of percentages of voxels with number of beams smaller than 2, 10, 30, and 100, in the numerical experiment described in Appendix C (five different viewpoints located at 1 m above the ground).

**Table 1.**Different estimators used to for numerical experiment described in Section 4.1.

Equation | Simplified for Mulation | Reference |
---|---|---|

Equation (7) | $\left(a\right)\frac{{\mathrm{Ni}}^{l}}{G{\sum}_{\ne wood\text{}hits}z}$ | [9] |

Equation (8) | $\left(b\right)-\frac{\mathrm{log}\left(1-\frac{{\mathrm{Ni}}^{l}}{{\mathrm{N}}^{\ne w}}\right)}{\delta}$ | [17] |

Equation (15) (with $\alpha =1$, ${\mathrm{Ni}}^{l}$>>1 and ${\lambda}_{1}\ll 1$) | $\left(c\right)\frac{{\mathrm{Ni}}^{l}}{G\Sigma \mathrm{z}}$ | This publication |

Equation (7), with $\alpha $ multiplicative factor | $\left(d\right)\frac{\alpha {\mathrm{Ni}}^{l}}{G{\sum}_{\ne wood\text{}hits}z}$ | [9] and this publication |

Equation (8), with $\alpha $ multiplicative factor | $\left(e\right)-\frac{\alpha \mathrm{log}\left(1-\frac{{\mathrm{Ni}}^{l}}{{\mathrm{N}}^{\ne w}}\right)}{\delta}$ | [17] and this publication |

Equation (15) (${\mathrm{Ni}}^{l}$ >> 1 and ${\lambda}_{1}\ll 1$) | $\left(f\right)\frac{\alpha {\mathrm{Ni}}^{l}}{G\Sigma \mathrm{z}}$ | This publication |

**Table 2.**Mean biases (in % of the mean LAD

_{ref}) of the three estimators for three different classes of total beam number N.

Range of Beam Number | $\tilde{\mathbf{LAD}}{}^{\mathbf{Nmax}}$ | $\tilde{\mathbf{LAD}}{}^{\mathbf{NW}}$ | $\tilde{\mathbf{LAD}}{}^{\mathbf{M}}$ |
---|---|---|---|

$\mathrm{N}\ge 2\text{}\mathrm{and}\text{}\mathrm{N}10$ | −6.0% | −15% | +2.2% |

$\mathrm{N}\ge 10\text{}\mathrm{and}\text{}\mathrm{N}15$ | +0.8% | −2.8% | +0.4% |

$\mathrm{N}\ge 15$ | +0.0% | −0.4% | +0.0% |

**Table 3.**Root Mean Square Error (in % of the mean LAD) of the three multiview estimators for six different classes of total beam numbers.

Range of Beam Number | $\tilde{\mathit{LAD}}{}^{\mathit{Nmax}}$ | $\tilde{\mathit{LAD}}{}^{\mathit{NW}}$ | $\tilde{\mathit{LAD}}{}^{\mathit{M}}$ |
---|---|---|---|

$N\ge 2\text{}and\text{}N10$ | 450% | 410% | 416% |

$N\ge 10\text{}and\text{}N15$ | 137% | 234% | 114% |

$N\ge 15\text{}and\text{}N30$ | 99% | 183% | 83% |

$N\ge 30\text{}and\text{}N100$ | 61% | 52% | 51% |

$N\ge 100\text{}and\text{}N1000$ | 37% | 31% | 30% |

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**MDPI and ACS Style**

Pimont, F.; Soma, M.; Dupuy, J.-L.
Accounting for Wood, Foliage Properties, and Laser Effective Footprint in Estimations of Leaf Area Density from Multiview-LiDAR Data. *Remote Sens.* **2019**, *11*, 1580.
https://doi.org/10.3390/rs11131580

**AMA Style**

Pimont F, Soma M, Dupuy J-L.
Accounting for Wood, Foliage Properties, and Laser Effective Footprint in Estimations of Leaf Area Density from Multiview-LiDAR Data. *Remote Sensing*. 2019; 11(13):1580.
https://doi.org/10.3390/rs11131580

**Chicago/Turabian Style**

Pimont, François, Maxime Soma, and Jean-Luc Dupuy.
2019. "Accounting for Wood, Foliage Properties, and Laser Effective Footprint in Estimations of Leaf Area Density from Multiview-LiDAR Data" *Remote Sensing* 11, no. 13: 1580.
https://doi.org/10.3390/rs11131580