## INTRODUCTION

Simulation models of light interception by vegetation canopies have been developed for many years for purposes ranging from plant production and ecophysiology to remote sensing (Ross 1981; Myneni, Ross & Asrar 1989; Varlet-Grancher, Bonhomme & Sinoquet 1993). Most light models are based on the turbid medium analogy, namely Beer's law which takes into account the amount of leaf area, and the leaf angle distribution with regard to the direction of incident radiation. The most common application of Beer's law is the computation of gap fraction *P*_{0} of a horizontally homogeneous canopy in a given direction Ω:

Where *G*_{Ω} is the projection coefficient of leaf area on a plane perpendicular to direction Ω, which depends on leaf angle distribution, *L* is the leaf area index (m^{2} m^{−2}) and *h* is the elevation angle of direction Ω.

Theoretical derivation of Eqn 1 was proposed in Nilson's (1971) pioneering work from the following original assumptions: (1) the stand consists of a large number of statistically independent layers; (2) the probability of observing more than one contact within a layer is infinitely small compared with the probability of one contact; (3) the probability of observing a contact within a small layer is equal to the mean number of contacts per layer. When applied to any vegetation canopy, the original assumptions imply three assumptions about canopy structure: (1) leaf size is assumed to be infinitely small; (2) leaves are assumed to be randomly located within the vegetation canopy, that is, the spatial location of one leaf does not depend on that of other leaves; (3) leaf area density (LAD) is assumed to be uniformly distributed within the canopy volume. Interaction exists between foliage randomness and the spatial distribution of leaf area: if leaves are randomly distributed in the canopy space, this should lead to uniform leaf area density.

Equation 1 cannot be directly used for isolated trees because they are not horizontally homogeneous crops. Light interception at the tree canopy scale can be characterized by STAR values, namely the Silhouette to Total Area Ratio (Oker-Blom & Smolander 1988), which depends on incident direction Ω. Sky-integrated STAR expresses the overall light interception of the tree, and is also the average relative leaf irradiance. For isolated trees, assumptions used in Beer's law derivation can be violated in two ways. On one hand, leaf size may be significant with regard to the ground area occupied by the tree, especially for seedlings (e.g. Planchais & Sinoquet 1998). On the other hand, foliage may be aggregated within clumps around the current-year shoots (e.g. Whitehead, Grace & Godfrey 1990; Cohen, Mosoni & Meron 1995).

Solutions have been proposed which relax the classical Beer's law assumptions. Leaf size has been explicitly included by replacing the original Beer's law by a binomial or multinomial law (Fukai & Loomis 1976; Thanisawanyangkura *et al*. 1997). Non-randomness of leaf location has been taken into account by using binomial and Markov models (Nilson 1971; Cohen *et al*. 1995). These models lead to modified forms of Beer's law where one or two additional leaf dispersion parameters have been introduced. Usually the leaf dispersion parameters were not explicitly related to canopy geometry parameters or botanical features, except in Foroutan-Pour, Dutilleul & Smith (2001) where leaf dispersion was characterized by the fractal dimension of the leafless branching system measured on a photograph. In addition recently, Niinemets *et al*. (2004) found some correlation between clumping and petiole length in poplar canopies.

Non-uniform distribution of leaf area has been taken into account in two general ways: firstly, the canopy can be divided into subcanopy envelopes filled with uniform LAD (e.g. Thorpe *et al*. 1978; Norman & Welles 1983). Multiscale applications have been proposed, where needles have been included in shoot envelopes, and shoots in whorls and crowns canopies (Norman & Jarvis 1975; Oker-Blom & Kellomäki 1983). Secondly, the space occupied by the vegetation is discretized in two or three-dimensional rectangular cells (e.g. Kimes & Kirchner 1982; Cohen & Fuchs 1987). In computer graphics jargon, the three-dimensional (3-D) cells are called voxels, which is a nickname for ‘volume element’. In the voxel grid, differences between voxels in foliage density account for the spatial variations of LAD within the canopy, while the original Beer's law is applied within each voxel with assumptions of random and uniform leaf area distribution.

Three-dimensional plant mock-ups built from 3-D digitizing (e.g. Sinoquet & Rivet 1997) or generated from growth simulation rules (e.g. Prusinkiewicz 1998) give an alternative way to compute light interception by isolated plants (e.g. STAR values, Sinoquet *et al*. 1998). Comparison between turbid medium light model outputs and STAR values computed from virtual images of 3-D plant mock-ups has shown the overall effect of clumping on light interception by trees (e.g. Chen *et al*. 1993).

The objective of this study was to quantify the effect of the three assumptions used in Beer's law on light interception computations for a range of isolated tree canopies. For this purpose, actual trees were 3-D digitized and theoretical trees were generated from fractal rules or in order to get a random canopy (Pinty *et al*. 2001). Although any feature of the canopy radiation field is largely affected by foliage clumping, light interception was characterized only by sky-integrated STAR values. They were computed from virtual images of the tree mock-ups, and from a turbid medium model using 3-D discretization of the canopy into voxels at different scales. Virtual images of 3-D digitized plants allowed us to estimate the actual STAR values, while comparison with turbid medium computations at different scales made it possible to separate the effects of the three assumptions. The second objective of the paper was to define the best grid for 3-D models based on the turbid medium analogy, in terms of resolution, rotation and translation.