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ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002 
  
  
Figure 1. LIDAR sensing geometry (red stars represent LIDAR 
measurements; blue lines represent pulse vectors 
composing the scan space T; y, represents a single 
range measurement along pulse vector f). 
The distribution of tree crowns over the entire scene is then 
modeled as an object configuration, x. If individual plants were 
actually solid objects (e.g. ellipsoids, spheres, etc.) in object 
space U, these LIDAR measurements, y, would represent the 
location where each vector f intercepted the surface of the 
object. In the terminology introduced above, these 
measurements would represent the signal, or the deterministic 
influence of the actual configuration of objects x on the series of 
LIDAR range measurements that are observed. A more realistic 
approach, however, would need to account for the fact that 
plants are not solid geometric objects, and LIDAR pulses 
actually penetrate a certain distance into the canopy through 
foliage gaps. 
This would incorporate a stochastic element to the LIDAR 
measurements, y, due to the irregular spatial distribution of 
foliage elements (leaves, branches, etc.) in the pathway of a 
laser pulse as it intersects a tree crown. Therefore, again using 
the notation introduced above, the conditional distribution of 
individual LIDAR measurements, given the signal, is given by a 
family of densities g(y |x). In the context of LIDAR 
measurement of tree crowns, these probability densities will be 
related to the laser attenuation function, which in the case of 
discrete LIDAR systems is directly related to the probability of 
reflection. If the values of individual LIDAR measurements 
along a pulse vector can be considered conditionally 
independent, given the true configuration of objects, x, the 
likelihood function, representing the joint probability of the 
data, is given by: 
I(y|x) » [26,12 Q) 
teT 
3.3.1 Modelling the distribution of foliage in complex forest 
scenes: Previous studies of laser transmission through the 
forest canopy have utilized three-dimensional grid models 
populated with generalized geometric forms that represent 
individual plants (Sun and Ranson, 2000). 
In this study, a three-dimensional array, with 0.91-m cell size, 
was used to model the distribution of foliage density throughout 
a forested area. The spatial distribution of foliage is a function 
of individual tree locations, sizes, crown forms, and an average 
leaf area density (LAD). Crown forms were represented as 
generalized ellipsoids following Sheng et al.. (2001), where the 
space occupied by the foliage within an individual tree crown is 
determined by four parameters: crown width (cw), crown height 
(ch), crown curvature (cc), tree height (hf), and the 2-D 
coordinate of the crown top (X, Y; ) (see Figure 2). 
Ap 
a) ht:47, cw:11, ch:14, cc:1.45 b) At:30, cw:8, ch:18, cc:0.75 
Figure 2. Generalized ellipsoid crown models. 
The surface of a tree crown is then given by the following 
mathematical expression: 
cc/2 
(Z+ch+2,, Y. i (de. +(Y-%,, ’) 
eh ert 
zd. (3) 
  
where the elevation of the crown top, Z,,, is determined by 
adding the tree height to the elevation of the base as determined 
from the DTM. Values for LAD were obtained from previous 
research findings (Webb and Ungs, 1993). 
3.3.2 Modelling laser-canopy foliage interaction: The 
analysis of data acquired from active remote sensing 
technologies requires an understanding of the interactions 
between the emitted radiation and the physical properties of the 
target. 
In our model, where the laser footprint (0.4 m) is significantly 
less than the cell size (0.91 m), the probability that a direct light 
beam that enters a cell exits from the cell without being 
intercepted is calculated as a function of the leaf area density 
and the off-nadir angle (0) of the laser pulse. The model then 
calculates the probability of penetration to the center of any cell 
by direct laser energy originating from outside the cell. 
Specifically, the probability of a laser beam that enters a canopy 
cell, z;, at a specific off-nadir angle, 6,, reflects from this cell 
with foliage density LAD; is given by the following function 
(Vanderbilt, 1990): 
p(reflection) = [1 —exp-1/cos0 - LAD, -G(0,,2;)- dz) | (4) 
where LAD; is the leaf area density (m°/ m?) within cell z;, G(0, 
Z;) is the projection of the vegetation within cell z; in the 
direction of 0, and dz; is the depth of cell z;. Often, a spherical 
leaf angle distribution can be assumed, in which case G(6, z;) is