## Introduction

Simulation models of light interception depend on the way canopy structure is described (Ross, 1981). Simple models are useful to identify the key structural parameters involved in light capture. Target users are plant scientists, who need to compare species, cultivars or training systems for their ability to capture light. Simple models allow them to determine the structural parameters to be measured in the field and to obtain a synthetic view of their effects on light capture.

In the case of horizontally homogeneous canopies, simple description of canopy structure based on leaf area index (LAI, m^{2} m^{−2}) combined with Beer-Lambert's law of light attenuation (e.g. see Ross, 1981) has been proposed, e.g. (Gosse *et al*., 1986):

_{i}=ɛ

_{max}(1 – exp(–

*K*× LAI) (Eqn 1)

(ɛ_{i}, light interception efficiency of the canopy; ɛ_{max}, maximum ɛ_{i}, namely when LAI → ∞; *K*, extinction coefficient, which depends of the leaf inclination angle distribution).

In the case of isolated trees, making simple models is more difficult. This is because the spatial distribution of leaf area is not uniform: first, the tree foliage occupies a limited canopy volume, the shape of which may be complex; and second, the leaf area density (LAD, m^{2} m^{−3}) within the canopy space is not uniformly distributed (Whitehead *et al*., 1990; Sinoquet & Rivet, 1997). Two types of light model have been proposed to take into account the complexity of tree canopy structure. The first, for trees, is based on the turbid medium analogy, that is Beer's law, as in horizontally homogeneous canopies. Canopy structure is described as a tree shape filled with leaf area. Authors have proposed simple tree shapes (ellipsoids, frustrums: Thorpe *et al*., 1978; Norman & Welles, 1983; Boudon, 2004) and more sophisticated shapes (convex envelopes: Cluzeau *et al*., 1995; Boudon, 2004; complex parametric shapes: Cescatti, 1997). In contrast to simple shapes, convex envelopes and Cescatti's (1997) model offer more flexibility to fit actual canopy shapes, but also require more input parameters. In turbid medium models applied to isolated trees, another way to describe tree shape is to approximate the canopy volume by a set of three-dimensional (3-D) cells called voxels (Kimes & Kirchner, 1982; Myneni, 1991; Sinoquet *et al*., 2001; Gastellu-Etchegorry *et al*., 2004). The voxel approach additionally allows one to take into account the spatial variations of LAD within the tree crown, namely by assigning different values of LAD in each voxel. However, this method is very demanding in terms of input parameters, and methods of obtaining LAD values at the voxel scale are scarce and tedious (e.g. 2-D or 3-D leaf clipping method: Cohen & Fuchs, 1987; 3-D vegetation digitizing: Sinoquet & Rivet, 1997). Another way to take into account the nonuniform distribution of leaf area in the canopy envelope is to use a multiscale approach based on botanical principles. For example, Oker-Blom & Kellomäki (1983) proposed to a two-scale approach for conifers where needles were assumed to be randomly distributed in the shoot envelopes, while the latter were randomly distributed in the crown envelope. This method allowed foliage clumping to be dealt with at several scales (Norman & Jarvis, 1975). For all these light models based on the turbid medium analogy, beams are traced in the tree canopy and beam extinction is computed according to the Beer-Lambert's law relating light transmission to the optical density of the canopy.

The second type of light model for trees is based on the use of explicit mock-ups of trees, where the information about canopy structure is dramatically detailed (Room *et al*., 1996), namely a database in which the shape, size, location and orientation of each plant organ is included. Such databases can be generated from measurements in the field, for example, by using 3-D digitizing devices at the leaf scale (Lang, 1973; Sinoquet *et al*., 1998). This kind of measurement is very tedious and cannot be used for large numbers of trees or large trees, although methods combining partial 3-D digitizing and reconstruction of the nondigitized elements have been proposed (Sonohat *et al*., 2006). In the case of 3-D tree mock-ups, another way to obtain the canopy structure is to simulate tree architecture from botanical rules (Prusinkiewicz & Lindenmayer, 1990) to simulate of the interactions between tree structure and function (Allen *et al*., 2005). Light models based on 3-D tree mock-ups may use projection methods for computing light interception without scattering (Meyer *et al*., 1984). Scattering can be computed using Monte-Carlo ray tracing (Ross & Marshak, 1988; Govaerts & Verstraete, 1998) or radiosity methods (Chelle & Andrieu, 1998).

From our knowledge, the only simple light model for trees has been proposed by Jackson & Palmer (1979) for orchards. In a first step, the model assumes the canopy as a solid, that is, a nontransmitting and nonreflecting shape, as in a previous model proposed by Jackson & Palmer (1972). In a second step, the fraction of intercepted radiation by the solid shape is weighted by a term similar to a shape porosity factor and which is expressed with a Beer's law formalism.

The objective of this study is to propose two simple equations for light interception by an isolated tree crown, involving the minimum number of canopy structure parameters. The modelling approach is based on Jackson & Palmer (1979), as interception is decomposed into two terms: (i) the projection of the canopy envelope, and (ii) the foliage porosity in the projection envelope. The canopy envelope is treated with a voxel approach, while the parameterization of the crown porosity term involves the mean leaf angle inclination and explicitly takes into account foliage clumping within the canopy. The two models were tested with virtual experiments involving a large database of 3-D digitized apple trees, which allowed both canopy structure parameters and light interception properties to be computed.