Simple equations to estimate light interception by isolated trees from canopy structure features: assessment with three-dimensional digitized apple trees


Author for correspondence: H. Sinoquet
Tel: +33 4 73 62 43 61
Fax: +33 4 73 62 44 54


  • • Simple models of light interception are useful to identify the key structural parameters involved in light capture. We developed such models for isolated trees and tested them with virtual experiments. Light interception was decomposed into the projection of the crown envelope and the crown porosity. The latter was related to tree structure parameters.
  • • Virtual experiments were conducted with three-dimensional (3-D) digitized apple trees grown in Lebanon and Switzerland, with different cultivars and training. The digitized trees allowed actual values of canopy structure (total leaf area, crown volume, foliage inclination angle, variance of leaf area density) and light interception properties (projected leaf area, silhouette to total area ratio, porosity, dispersion parameters) to be computed, and relationships between structure and interception variables to be derived.
  • • The projected envelope area was related to crown volume with a power function of exponent 2/3. Crown porosity was a negative exponential function of mean optical density, that is, the ratio between total leaf area and the projected envelope area. The leaf dispersion parameter was a negative linear function of the relative variance of leaf area density in the crown volume.
  • • The resulting models were expressed as two single equations. After calibration, model outputs were very close to values computed from the 3-D digitized databases.


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, m2 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):

ɛimax(1 – exp(–K× LAI)  (Eqn 1)

i, light interception efficiency of the canopy; ɛmax, maximum ɛi, 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, m2 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.

Materials and Methods

Model description

The model is aimed at simulating light interception by isolated trees, in terms of projected leaf area (PLA, m2) and silhouette to total area ratio (STAR, Carter & Smith, 1985). PLA characterizes the tree's ability to capture light in terms of intercepting leaf area, while STAR corresponds to average leaf irradiance related to incident radiation. Both variables are related as follows:


(TLA (m2), total leaf area of the plant; PLAΩ and STARΩ values depend on light direction Ω). Here we deal with sky-integrated values – notably PLA and STAR, which globally characterize the interception properties of the trees (Delagrange et al., 2006), for example:

image(Eqn 3)

Ω, weight associated to direction Ω, namely the fraction of incident radiation coming from direction Ω). Here weights ωΩ are computed according to the Standard OverCast sky radiance distribution, which is a satisfactory way to approximate radiation balance at a daily scale (Sinoquet et al., 2004). From this point, all terms in the equations are defined as Ω-integrated values.

In the model, the tree is regarded as a porous envelope. PLA and STAR are therefore formally expressed from the product between the projected area of the crown envelope (PEA, m2) assumed to be opaque, and the interception probability within PEA in relation to crown porosity P0.

PLA = PEA(1 –P0) and STAR = PEA(1 –P0)/TLA  (Eqn 4)

P0 is expressed as:

image(Eqn 5)

(P0〈HOM〉, porosity of the corresponding homogeneous tree canopy, that is, with the same volume and total leaf area as the actual tree, but with uniform distribution of leaf area density within the crown envelope; µ, a leaf dispersion parameter (Nilson, 1971), accounting for the effect of nonrandom and nonuniform distribution of LAD in the crown volume). Indeed, departure from random dispersion (µ = 1) generally occurs in actual tree canopies, which generally show leaf clumping (µ < 1). This leads to higher crown porosity than that of random and uniform canopies (Chen et al., 1993; Cohen et al., 1995; Casella & Sinoquet, in press)

Combining Eqns 4 and 5 leads to, for example, for STAR:

image(Eqn 6)

Eqn 6 is the basis for a simple light interception model. The next step is to relate the Ω-integrated terms PEA, P0<HOM> and µ to canopy structure variables, by using relationships established from virtual experiments on 3-D digitized trees.

PEA is defined as a function of canopy volume (V, m3). The rationale behind this assumption is: (i) Cauchy's theorem (1832; see also Lang, 1991), stating that the Ω-averaged projection of any convex envelope is equal to one-quarter of the envelope area, and (ii) allometry between envelope area and volume.

The porosity term P0<HOM> is defined as a function of the mean optical density (MOD) in the tree volume. MOD can be expressed as the product of LAD and the average length &#x004c;̄ of beams crossing the tree crown. &#x004c;̄ can be simply written:

=V/PEA (Eqn 7)


MOD = LAD × = (TLA/V)(V/PEA) = TLA/PEA  (Eqn 8)

MOD is therefore similar to the LAI of an isolated tree, defined as the ratio between TLA and the Ω-averaged crown envelope projection, PEA.

Finally parameter µ is defined in two ways. In model 1, µ is a simple parameter which is not related to further canopy structure attributes. In model 2, µ is related to the relative variance of LAD (ξ) in the tree crown (discussed later). This assumption is based on previous results showing that departure from random behaviour in tree canopies is mainly the result of the spatial variations in leaf area density within the tree crown (Sinoquet et al., 2005).

The final form of the model equations is given in the Results section, that is, once relationships between light interception terms and canopy structure parameters have been demonstrated and set up.

Plant material

Apple trees in Lebanon  Trees of apple (Malus domestica Borkh.) cultivars belonging to contrasted ideotypes (Lespinasse, 1992) were planted in 1999 in the Bekaa valley in the American University of Beirut research field (Haouch Sneid; 33°95 N; 36°02E, 900 m asl): Scarletspur Delicious (Type II), Golden Delicious (Type III) and Granny Smith (Type IV). The experiment was a randomized block design and consisted of three two-tree plots for each cultivar, one plot per row. Trees were grafted on M7 rootstock. The orchard layout was 4 × 3.5 m with a south-north row orientation. Agricultural practices included irrigation with mini-sprinklers, standard fertilization and spraying.

After plantation, trees were trained to obtain a central leader (Heinicke, 1975). Training consisted of heading the leader in order to obtain a stronger trunk and heading back one-third of the 1-yr-old lateral shoots. For each cultivar, six healthy trees (two trees per row) were selected for measurement of canopy structure. This made a set of 18 trees. In each pair, one tree was trained as previously described to a central leader while the centrifugal training system (Willaume et al., 2004) was applied to the other one from 2004. Centrifugal training consisted of removing in 2004 all the flowering shoots along the trunk and on the underside of the branches in order to make a light well.

Apple trees in Switzerland  Golden Delicious apple trees were planted in 1991 in Agroscope RAC, Centre des Fougères, Switzerland (46°14′N; 7°18′E, 500 m asl), in a randomized block design. Trees were grafted on M9 rootstock. Three training systems were used: vertical axis, including a central axis bearing fruiting branches; drilling made of three scaffolds; and ycare made of two scaffolds. Canopy heights were, respectively, 3.0, 2.5 and 2.0 m. Row orientation was north-south. Inter-row distance was 4 m, with interplant distance on the row equal to 1.25 and 1 m, for vertical axis and drilling, and ycare systems, respectively. For each system, three trees were selected in two blocks for measurement of canopy structure. This made a set of 18 trees.

Three-dimensional plant digitizing and mock-up reconstruction

The 3-D canopy structure of both the Lebanon and Switzerland trees was measured in summer 2004 and 2005 with a 3-D digitizing technique (Sinoquet & Rivet, 1997). The device – called a 3-D digitizer 3Space Fastrak (Polhemus, Colchester, VT, USA) – consists of a magnetic source making magnetic fields at the plant location, and a pointer containing magnetic coils. When the pointer is submitted to the magnetic fields emitted by the source, that is, in a given volume around the source, currents are induced in the pointer coils The values of induced current depend on the pointer location and orientation with regard to the magnetic source, making it possible to record the spatial co-ordinates and the orientation angles at the pointer location (Polhemus, 1993). In this study, the spatial coordinates of the proximal and distal tips of all leafy shoots of the current year were measured with this device associated with software Pol95 (Adam, 1999). This allowed shoot length and orientation to be computed.

Additional 3-D digitizing measurements at leaf scale were made for a set of 30 shoots per shoot type, in order to set rules for foliage reconstruction in shoots according to Sonohat et al.'s (2006) method. These measurements were used to derive allometric relationships, namely the relationships to infer the number of shoot leaves and the shoot leaf area from shoot length. Three-dimensional data at leaf scale also allowed the distributions of leaf angles, namely Euler angles, to be set, by placing the pointer sole parallel to leaf lamina and midrib (Sinoquet et al., 1998). Additional assumptions were used for leaf reconstruction in shoots: constant petiole length, constant internode length and equal area for all leaves in a shoot. The outcome of the foliage reconstruction method was a database of 72 tree mock-ups, where all geometrical parameters at leaf scale were included, namely leaf length and width, leaf location, leaf orientation angles, that is, midrib azimuth, midrib inclination and lamina rolling angle around the midrib. The reconstruction method was fully presented in Sonohat et al. (2006), who also assessed the reconstruction quality in terms of light interception properties at several scales. Figure 1(a) shows examples of canopy foliage reconstruction for two contrasted trees.

Figure 1.

Apple (Malus domestica) tree mock-ups. (a) Actual tree with leaves and fruits (black circular symbols); (b) crown envelope; (c) homogeneous tree canopy with leaves only. Left panels, Swiss Golden tree trained in drilling system in 2004; right panels, Lebanese Scarletspur tree trained in central leader system in 2005. Image size is 4 m and 2.5 m, for Golden and Scarletspur, respectively. Voxel size is 0.2 m.

Three-dimensional tree mock-ups were first used to define the tree canopy envelope. For this purpose, a bounding box was computed around the tree according to the minimum and maximum values of leaf co-ordinates along axes X, Y and Z. The bounding box was then divided into cubic volume elements called voxels. Canopy volume and shape were finally approximated as the cumulated volume of vegetated voxels, that is, voxels including at least one leaf. The voxel method allowed us to define any crown shape, that is, not only simple geometrical shapes like ellipsoids or cones. Computations were made with two voxel sizes: 0.1 and 0.2 m. Examples of canopy envelope reconstruction are shown in Fig. 1(b).

Secondly, 3-D mock-ups were used to generate tree canopies with uniform distribution of leaf area density within the canopy volume. This means that these homogeneous trees had the same total leaf area and the same canopy shape and volume as those given by the voxel method, but leaf area density was about the same in each voxel. Such homogeneous canopies were created by assigning leaves to voxels in a systematic way, that is, leaf 1 to voxel 1, and so on, until the last voxel was filled with one leaf, at which point a second leaf was assigned to voxel 1, and so on, until all leaves were assigned to all voxels. Note, however, that this way of making homogeneous canopies was only approximate, because of variability in leaf size and because the number of leaves was not necessarily a multiple number of the number of voxels. Figure 1(c) shows examples of such homogeneous canopies.

Light interception by the 3-D plant mock-ups

Actual light interception properties of the trees were computed from the 3-D plant mock-ups, and used as virtual measurements. PLA and STAR values at the tree scale were computed using VegeSTAR software (Adam et al., 2002). The principle of VegeSTAR consists of computing the projected leaf area by processing 3-D tree mock-up images (Sinoquet et al., 1998). Indeed, leaf area seen on a plant image taken in a given direction – for example, the direction of the sun – is the leaf area lit in this direction. By using false colours in plant images, it was therefore possible to compute directional PLA by simply counting coloured pixels in the image. Here all leaves were coloured green.

For directional integration over the sky hemisphere, the sky was discretized in 46 solid angle sectors of equal area, according to the Turtle sky proposed by Den Dulk, 1989). Directional PLA and STAR values were computed for the central direction of each solid angle sector. Directional values were summed up over the sky hemisphere by using weighting coefficients derived from the Standard OverCast distribution of sky radiance (Moon & Spencer, 1942).

In this study, Ω-integrated values of PLA and STAR were computed for the actual trees, their crown envelopes and the corresponding homogeneous canopies. Finally, porosity terms were computed from PLA values of both actual and homogeneous trees and that of the crown envelope, that is:

P0= 1 – PLA/PEA and P0<HOM>= 1 – PLA<HOM>/PEA (Eqn 9)

Canopy structure parameters

Canopy structure parameters were computed from the information available in the 3-D digitized plant databases. TLA was computed as the sum of individual leaf areas. V was computed as the sum of vegetated voxel volumes. The relative variance of LAD (ξ) – which has previously been proposed as contributing to foliage clumping – was computed from values of LAD in voxels:

image(Eqn 10)

(Nv, number of vegetated voxels; LADv, LAD in voxel v; LAD, mean LAD in the tree crown (LAD = TLA/V)).

Mean leaf inclination angle α was computed from Euler angles of individual leaves, as digitized at leaf scale in 2004. The α angles were 31° and 39° for Lebanese and Swiss trees, respectively.

Parameter µ was computed from the inversion of Eqn 5, namely

µ= ln(P0)/ln(P0<HOM>) (Eqn 11)

where P0 and P0<HOM> were computed from Eqn 9.

Model calibration and validation

Calibration consisted of finding the model parameter values which allowed the smallest discrepancy between STAR values measured from the 3-D tree mock-ups and those simulated with the simple model. The deviation between the measurements and model outputs was assessed by the root mean square error of prediction (RMSE), which is the root mean square difference between measured and modelled values.

The model was calibrated separately for Swiss and Lebanese trees. For each group of trees, a subset of trees was selected for calibration purposes. For Swiss trees, we chose trees digitized in 2004 in block A, that is, a set of nine trees. For Lebanese trees, we selected all trees digitized in 2004 in order to account for cultivar variations, that is, a set of 18 trees.

Free calibration was first made by allowing all model parameters to change. Then constrained calibration consisted of setting some parameters to constant values, in order to test the generality of the model parameterization.


PLA and STAR variations

Close relationships with r2 coefficients ranging between 0.90 and 0.98 were found between PLA and TLA for each cultivar in each country (Fig. 2). The relationships were not the same for all cultivars and countries. Swiss Golden trees were the most efficient in PLA at a given TLA, while Lebanese Golden and Scarletspur trees were the least efficient.

Figure 2.

Projected leaf area (PLA) as a function of total tree leaf area. Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Consequently, Swiss Golden trees showed the highest values of STAR (0.33–0.48), while Lebanese Scarletspur trees had the lowest values (0.18–0.41), although they bore less leaf area (Fig. 3). Lebanese Golden and Granny trees showed a smaller range of STAR values (0.25–0.32 and 0.26–0.39, respectively). The relationship between STAR and TLA was not as close as that between PLA and TLA, with r2 coefficients between 0.52 and 0.74. This confirms that TLA is not the only determinant of light capture in isolated trees, as has already been suggested by Delagrange et al. (2006).

Figure 3.

Silhouette to total area ratio (STAR) as a function of total tree leaf area. Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Model parameterization

For both voxel sizes, PEA was closely related to V for all cultivars in both countries (Fig. 4). A power function was chosen because of the expected allometry between PEA and V. Coefficients r2 ranged between 0.85 and 0.96. At a given canopy volume, Swiss Golden trees and Lebanese Granny trees were the most efficient in displaying the crown envelope area. However, for all cultivars, the exponent in the power function was close to 2/3, that is, the value corresponding to allometry between area and volume of geometrical objects. The only exception was Lebanese Granny with a voxel size of 0.2 m, where the exponent was 0.77. However, discarding one very high point (in brackets in Fig. 4a) made the exponent value for Granny 0.70, that is, also closer to 2/3. Because of the fractal nature of tree canopies (Zeide & Pfeifer, 1991), using the smaller voxel size led to smaller V and PEA (Fig. 4b). Consequently, the parameters of the relationship were different. Moreover, the relationships were slightly closer for the larger voxel size.

Figure 4.

Projected area of the crown envelope as a function of tree crown volume. (a) Voxel size = 0.2 m; (b) voxel size = 0.1 m. Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Porosity of the tree canopy (P0) was related to MOD with a negative exponential function similar to Beer-Lambert's law (Ross, 1981) (Fig. 5 for a voxel size of 0.2 m). Coefficients r2 ranged between 0.80 and 0.96. Golden and Granny trees showed similar extinction coefficients (0.30–0.37), while the value for Scarletspur trees was much lower (0.15). Consequently, the latter trees showed greater crown porosity, although optical density was slightly higher. The results with a voxel size of 0.1 m showed the same trends (data not shown).

Figure 5.

Porosity of the tree canopy as a function of mean tree optical density (voxel size = 0.2 m). Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Porosity of the homogenous tree canopy (P0<HOM>) was also related to MOD with a negative exponential function (Fig. 6 for voxel size of 0.2 m). However, a single relationship fitted the data points for all cultivars in the two countries with a high r2 coefficient (0.99). The extinction coefficient (0.50) was greater than that involved in the relationship between actual porosity P0 and MOD (Fig. 5). This is because foliage clumping in actual trees makes P0 higher than that of the homogeneous canopy. The same trends were found with voxel size of 0.1 m, with an extinction coefficient of 0.45 and r2 value of 0.98 (data not shown).

Figure 6.

Porosity of the homogeneous tree canopy as a function of mean tree optical density (voxel size = 0.2 m). Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

The leaf dispersion parameter (µ) ranged between 0.45 and 0.75, and between 0.57 and 0.86, for voxel sizes of 0.2 and 0.1 m, respectively (Fig. 7). Such values (µ < 1) are related to foliage clumping within tree crowns. The highest leaf clumping was found in Lebanese Scarletspur trees, while foliage of Swiss Golden trees showed the slightest clumping. For both voxel sizes, µ was closely related to the relative variance of LAD (ξ), with a single linear relationship for all trees (Fig. 7). The r2 coefficient was greater than 0.8, a high value confirming the importance of spatial variations of LAD in leaf dispersion behaviour (Sinoquet et al., 2005). Scarletspur trees obviously also showed the greatest values of ξ. Voxel size affected the range of values of ξ and the parameters of the relationship. Smaller voxel size consistently led to higher variance of LAD. The intercept of the regression line was also closer to 1 for voxels of 0.1 m, showing that smaller voxels allowed more spatial variations of LAD to be taken into account.

Figure 7.

Leaf dispersion parameter µ as a function of the relative variance of leaf area density (LAD), ξ. (a) Voxel size = 0.2 m; (b) voxel size = 0.1 m. Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Final model equations

The results shown in Figs 4–7 were used to set up the general model (Eqn 6). However, the relationships shown in the figures were not directly used. Instead, some parameters were fixed at values ensuring model consistency and parsimony.

PEA was modelled as follows:

PEA =aV2/3(Eqn 12)

where a is a parameter. The exponent value was fixed at 2/3, according to the expected allometry between envelope area and volume (Fig. 4).

As MOD is equal to TLA/PEA (Eqn 8), porosity P0 of the actual tree crown was expressed as:

P0= exp(–K(α)µTLA/(aV2/3)) (Eqn 13)

(K(α), extinction coefficient as a result of mean leaf inclination angle α in the tree crown). K(α) accounts for the extinction of hemispherical radiation in a random canopy. It was computed according to the formula proposed by Sinoquet et al. (2000):

K(α) = 0.988 cos(α/2)2.4(Eqn 14)

Equation 13 is similar to the classical gap fraction equation proposed by Nilson (1971). The proportion parameter was set to 1, so that porosity equals 1 when optical density is 0. In a first model (model 1), µ was regarded as a bulk dispersion parameter and was not related to any canopy structure parameter. In a second model (model 2), leaf dispersion µ was modelled as:

µ=µ0(1 –bξ) (Eqn 15)

where µ0 and b were parameters. The constant in Eqn 15 was set to 1, so that µ = µ0 when ξ = 0, that is, in the case of homogeneous canopies. The rationale behind Eqn 15 is the distinction between foliage clumping as a result of spatial variations in LAD, and nonrandomness of foliage distribution at the local – voxel – scale. The latter was taken into account by parameter µ0.

The final equations of models 1 and 2 were obtained by replacing variables PEA, P0, and µ in Eqn 6 with their expression given in Eqns 12–15, that is:

Model 1: STAR =aV2/3(1 – exp(–K(α)µTLA/ (aV2/3)))/TLA  (Eqn 16)
Model 2: STAR =aV2/3(1 – exp(–K(α)µ0(1 –bξ)TLA/ (aV2/3)))/TLA (Eqn 17)

Finally model 1 involved two canopy structure variables (V and TLA) and two parameters (a and µ). Model 2 used a third variable (ξ) and three parameters (a, µ0 and b).

Model calibration and validation

Results of model calibration and validation are only shown for a voxel size of 0.2 m, because both voxel sizes led to similar results, with slightly better model prediction in the case of the 0.2 m voxel size.

Model calibration allowed the values of the model parameters to be defined. For model 1, free calibration led to similar values of parameter a around 2 for both Lebanese and Swiss trees (Table 1). By contrast, parameter µ showed some differences, with a low value about 0.6 showing high foliage clumping in the case of Lebanese trees. By contrast, the foliage of Swiss trees was only slightly clumped, with a µ-value of around 0.9. For model 2, parameter a for Lebanese trees was still close to 2, but was larger (2.64) for Swiss trees. Parameter µ0 still showed variations between Swiss and Lebanese trees, indicating that leaf dispersion at the local scale was more clumped for Lebanese trees. Finally, parameter b was not very different between Lebanese and Swiss trees.

Table 1.  Model parameters and root mean square error of prediction (RMSE) after calibration (voxel size = 0.2 m)
 Model 1Model 2
  1. Values in italics were fixed and were not submitted to calibration. RMSE values are those of silhouette to total area ratio (STAR) computed on all apple (Malus domestica) trees.

Free calibration
 Lebanese trees1.9280.6150.0212.1090.7210.0780.018
 Swiss trees2.0620.8960.0152.6360.8900.0900.014
Constrained calibration
 Lebanese trees2.0000.5960.0212.4000.6820.0850.019
 Swiss trees2.0000.9220.0152.4000.9300.0850.014

With these sets of parameters, the deviations (RMSE) between outputs of model 1 and STAR measured from the tree images were 0.021 and 0.015 for Lebanese and Swiss trees, respectively (Table 1). When using model 2, the RMSE of STAR values decreased slightly to 0.018 and 0.014, respectively.

Model calibration with fixed parameters a and b also led to similar results (Table 1), with very low effect on RMSE. This shows that the important parameters in models 1 and 2 were µ and µ0.

Finally model validation with parameters computed from the constrained calibration showed that the model satisfactorily simulated PLA and STAR values for this large range of apple trees. Since regression slopes were very close to the line 1 : 1, both simulated variables were unbiased. Moreover, r2 coefficients were high. For PLA, the two models led to similar performance, with r2 coefficients about 0.98 (Figs 8, 9). For STAR values, model 2 showed slightly higher r2 values (0.91 vs 0.94) (Figs 10, 11). The difference between the two models was mainly due to Lebanese Scarletspur trees, which showed higher variations in ξ.

Figure 8.

Measured vs simulated projected leaf area (PLA) with simple model 1. Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Figure 9.

Measured vs simulated projected leaf area (PLA) with simple model 2 (i.e. including the relative variance of leaf area density (LAD), ξ). Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Figure 10.

Measured vs simulated silhouette to total area ratio (STAR) with simple model 1. Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.

Figure 11.

Measured vs simulated silhouette to total area ratio (STAR) with simple model 2 (i.e. including the relative variance of leaf area density (LAD), ξ). Apple (Malus domestica): GoS, Golden Swiss trees; GoL, Golden Lebanese trees; GrL, Granny Lebanese trees; ScL, Scarletspur Lebanese trees.


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/(aV2/3) (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 (µ0), 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 µ0 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 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.


We are grateful to the American University of Beyrouth, especially Prof. Salma Talhouk and M. Nicolas Haddad, for allowing field work in Lebanon, and to students who helped with digitizing the Lebanese trees. We also thank Anne-Marie Potel for her contribution to digitizing the Swiss trees, and Fruit-Union Suisse (FUS, Zug) for financial support of the Swiss part of the study.