### Introduction

- Top of page
- Summary
- Introduction
- Materials and Methods
- Results
- Discussion
- Acknowledgements
- References

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):

- ɛ
_{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.

### Discussion

- Top of page
- Summary
- Introduction
- Materials and Methods
- Results
- Discussion
- Acknowledgements
- References

Simple models of light interception allow one to identify the main structural determinants involved in light capture. Indeed, it is quite impossible to identify the role of structural parameters if one uses a sophisticated model, for example, a ray tracing model. This is because canopy structure is described using a huge number of parameters (e.g. the location, orientation and size of each leaf), so that the user cannot have a synthetic view of the relationships between canopy structure and light interception properties. Sophisticated models can, however, be used to derive simple models.

The simple model proposed in this paper is based on the general idea developed by Jackson & Palmer (1979), where light interception by the tree canopy is decomposed into two components: (i) light interception by the canopy envelope assumed to be opaque; and (ii) porosity of the crown envelope. In comparison with Jackson & Palmer's (1979) model, the equations presented in this study have several modifications and improvements.

First, our simple equations deal with hemispherical (i.e. Ω-integrated) radiation, while the basic equations were demonstrated for directional fluxes (Ross, 1981). The assumption that relationships well established for directional fluxes are still valid for hemispherical radiation is a hypothesis which has previously been used in several light models. This is especially the case in models based on the Kubelka–Munk equations (Kubelka & Munk, 1931; Bonhomme & Varlet-Grancher, 1977). In this study, Ω-integrated variables were successfully used for light interception properties (PLA and STAR) and its components (crown porosity, leaf dispersion parameters).

Secondly, the light interception components were expressed as a function of structural properties of the canopy. In Jackson & Palmer's (1979) model, PEA was geometrically computed by abstracting the tree as a simple geometrical shape. This holds for training systems where tree shape is strongly constrained, but does not hold in the case of free systems such as centrifugal training (Lauri, 2002). In this study, canopy shape was abstracted as an array of voxels of fixed size. Using a single size for all voxels and all trees allowed us to cope with the fractal nature of plants, namely using the same scale (i.e. voxel size) to estimate the volume of fractal objects was necessary to compare crown volumes. PEA was modelled as a function of canopy volume and the computation did not need any assumption about tree shape. The rationale behind the relationship between PEA and *V* was: (i) Cauchy's theorem (Cauchy, 1832), which formally relates the Ω-integrated PEA to the area of the canopy envelope surface, and (ii) allometry between canopy envelope area and volume. Cauchy's theorem theoretically holds only for convex envelopes. Although crown envelopes are not all convex, Fig. 4 shows that the relationship between PEA and canopy volume was valid, although it was different between cultivars. The relationships were, however, closer in the case of a voxel size of 0.2 m, because larger voxel size decreased with crown concavities. Finally, the proportion coefficient between PEA and canopy volume exponent 2/3 was about constant for all studied apple tree populations (Fig. 4), although tree shapes may be markedly different (Fig. 1).

In this study, crown porosity was related in a single equation to the main structural parameters involved in the light interception process: TLA, angle α (see Eqn 14) and foliage dispersion. Moreover, the porosity equation led to a definition of LAI for isolated trees (Eqn 8):

- LAI = TLA/PEA = TLA/(
*aV*^{2/3}) (Eqn 18)

With this definition of LAI, the STAR expression given in Eqns 16 and 17 is the same as that of a horizontally homogeneous canopy. In our opinion, this property confirms that the proposed definition of LAI for isolated trees is valid, and gives an answer to the question of how much ground area is occupied by an isolated tree.

Only a few light models have dealt with leaf dispersion (Kuusk & Nilson, 2000). Here leaf dispersion was explicitly taken into account in the simple model 2 (Eqn 17). According to a previous study (Sinoquet *et al*., 2005), leaf dispersion was decomposed into two sources: (i) local leaf dispersion (µ_{0}), which expressed departure from the random dispersion at the voxel scale; and (ii) spatial variations of LAD (ξ). To our knowledge, the present study is the first one which experimentally shows a close relationship between Ω-integrated leaf dispersion and ξ (Fig. 7). Note, however, that this relationship is consistent with previous proposals. On the one hand, Nilson (1971) demonstrated that leaf dispersion in the direction Ω is related to the variance of number of contacts between phytoelements and a needle inserted in the canopy in direction Ω. On the other hand, Warren-Wilson (1960) showed the relationship between the number of contacts and projected LAI in the direction Ω. Combination of the two results leads to a consistent relationship between variance of LAI and directional leaf dispersion. In our work, we further showed that such a relationship also held for Ω-averaged values. Finally, note that the relationship between µ and ξ held for a large set of contrasted apple tree shapes (Fig. 7).

As already demonstrated by Sinoquet *et al*. (2005), the leaf dispersion parameter depended on the chosen voxel size. Indeed, the smaller the voxel size, the greater the spatial variations in LAD taken into account by the voxel grid. As a result, local leaf dispersion with a voxel size of 0.2 m was the combination between local leaf dispersion in voxels of 0.1 m and the LAD variance of the 0.1 m voxels within the 0.2 m voxels. Consequently, as LAD variance resulted in foliage clumping, values of µ_{0} for the smaller voxels were systematically higher than values for larger voxels (Fig. 7).

Thirdly, the simple model was tested with a large database of 3-D apple tree mock-ups, namely 36 trees sampled during 2 yr, in two locations, three cultivars and five training systems. The reference values of light interception by the tree mock-ups were computed with a projection method, which did not involve any assumption. The only uncertainty source was the geometrical structure of the 3-D mock-ups, owing to the reconstruction rules. However, Sonohat *et al*. (2006) provided a careful assessment of the reconstruction method, and showed that the method is especially accurate for light computations at tree scale.

Methods to get the input parameters requested by the simple model are in existence. Canopy volume can be derived from measurements of transmitted light to an array of light sensors located on the ground surface (Giuliani *et al*., 2000), or from a set of photographs taken around the isolated tree (Shlyakhter *et al*., 2001; Phattaralerphong & Sinoquet, 2005). Total leaf area can be estimated from inversion methods of gap fraction applied to isolated plants, for example, LAI-2000 plant analyser (Villalobos *et al*., 1995), or photo method (Phattaralerphong *et al*., 2006). By contrast, there is no published method to estimate the LAD variance. However, the next step in our photo method implemented in Tree Analyser software (available at http://www2.clermont.inra.fr/piaf/eng/download/download.php) should be the computation of the 3-D distribution of leaf area density in a grid of voxels. Preliminary unpublished results show that the method works if the canopy is not too dense and if voxels are large enough. Note also that the effect of LAD variance on STAR computation was almost important for Lebanese Scarletspur trees, that is, trees showing the highest LAD variance.

From a methodological point of view, this study showed the usefulness of virtual plants to carry out virtual experiments (Room *et al*., 1996; Godin & Sinoquet, 2005). Here, virtual plants allowed the computation of both structural and light interception properties for isolated trees – while the trees were not isolated in the orchards. Virtual plants also allowed us to create homogeneous tree canopies that are useful to study leaf dispersion experimentally (Fig. 7), although these canopies do not exist in the real world.