Foliage randomness and light interception in 3-D digitized trees: an analysis from multiscale discretization of the canopy



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    1. UMR PIAF INRA-UBP, Site de Crouelle, 234 Avenue du Brézet, 63039 Clermont-Ferrand Cedex 2, France,
      H. Sinoquet. E-mail:
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    1. UMR PIAF INRA-UBP, Site de Crouelle, 234 Avenue du Brézet, 63039 Clermont-Ferrand Cedex 2, France,
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    1. UMR PIAF INRA-UBP, Site de Crouelle, 234 Avenue du Brézet, 63039 Clermont-Ferrand Cedex 2, France,
    2. Kasetsart University, Faculty of Science, Department of Botany, Bangkok, Thailand and
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    1. INRIA, UMR AMAP, TA40/PSII, Boulevard de la Lironde, 34398 Montpellier Cedex 5, France
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H. Sinoquet. E-mail:


Light models for vegetation canopies based on the turbid medium analogy are usually limited by the basic assumption of random foliage dispersion in the canopy space. The objective of this paper was to assess the effect of three possible sources of non-randomness in tree canopies on light interception properties. For this purpose, four three-dimensional (3-D) digitized trees and four theoretical canopies – one random and three built from fractal rules – were used to compute canopy structure parameters and light interception, namely the sky-vault averaged STAR (Silhouette to Total Area Ratio). STAR values were computed from (1) images of the 3-D plants, and (2) from a 3-D turbid medium model using space discretization at different scales. For all trees, departure from randomness was mainly due to the spatial variations in leaf area density within the canopy volume. Indeed STAR estimations, based on turbid medium assumption, using the finest space discretization were very close to STAR values computed from the plant images. At this finest scale, foliage dispersion was slightly clumped, except one theoretical fractal canopy, which showed a marked regular dispersion. Taking into account a non-infinitely small leaf size, whose effect is theoretically to shorten self-shading, had a minor effect on STAR computations. STAR values computed from the 3-D turbid medium were very sensitive to plant lacunarity, a parameter introduced in the context of fractal studies to characterize the distribution of gaps in porous media at different scales. This study shows that 3-D turbid medium models based on space discretization are able to give correct estimation of light interception by 3-D isolated trees, provided that the 3-D grid is properly defined, that is, discretization maximizes plant lacunarity.


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 P0 of a horizontally homogeneous canopy in a given direction Ω:


Where GΩ 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 (m2 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.


3-D plants

Eight 3-D plants were included in the study. Four real trees were 3-D digitized in the field, while four additional plants were generated from theoretical assumptions.

One 3-year-old-hybrid walnut tree (NG38 × RA) and two 2-year-old-mango trees (cv. Nam Nok Mai) were 3-D digitized at leaf scale, according to Sinoquet et al. (1998) method, in August 1998 and November 1997, respectively. The walnut tree was grown in an experimental plot in Clermont-Ferrand INRA research centre, France, while the mango trees were grown in a commercial farm in Ban Bung, 150 km south-east of Bangkok, Thailand. The location and orientation of each leaf was recorded with a magnetic digitizer (Fastrak 3Space; Polhemus, Vermont, USA) while leaf length and width were measured with a ruler. A sample of leaves was harvested on similar trees to establish an allometric relationship between individual leaf area and the product of leaf length and width. The individual area of sampled leaves was measured with a leaf area meter (Li-cor 3100: Li-Cor Inc., Lincoln, NE, USA). The data sets therefore consisted of a collection of leaves, the size, the orientation and the location of which have been measured in the field.

One 4-year peach tree (cv. August Red) was digitized in May 2001 in CTIFL Center, Nîmes, South of France, at current-year shoot scale, 1 month after bud break. Given the high number of leaves (˜14 000), digitizing at leaf scale was impossible. The magnetic digitizing device was therefore used to record the spatial co-ordinates of the bottom and top of each leafy shoot. Thirty shoots were digitized at leaf scale in order to derive (1) leaf angle distribution; (2) allometric relationships between number of leaves, shoot leaf area and shoot length. Leaves of each shoot were then generated from (1) allometric relationships; (2) sampling in leaf angle distribution; and (3) additional assumptions, namely constant internode length and leaf size within a shoot (see Sonohat et al. 2004).

Three theoretical plants were generated from 3-D iterated function systems (IFS; Barnsley 1988) using the PGL library in AMAPmod (Boudon et al. 2001). Each IFS corresponds to a transform made up of a contraction by a factor c (c < 1) followed by n duplications of the contracted object. When applied recursively to an initial object, the IFS successively generates a family of forms that converges towards a fractal object called its attractor. In practice, after only a few iterations a fractal-like object is obtained (Fig. 1). If the duplications of the IFS do not overlap, the theoretical fractal dimension of the IFS attractor is

Figure 1.

Generation of a fractal plant. From left to right: four consecutive generations of an IFS transform consisting of nine duplications of an initial object (a big horizontal leaf) contracted by a factor 3. Theoretical fractal dimension of the attractor is DT = 2. The third and fourth plant (from the left end-side) are the fractal plants, no. 1 and no. 2, respectively, considered in this study.


A first IFS was designed to simulate a 3-D Cantor dust, which can be regarded as an ideal clumped plant structure at different scales. It was made up of a contraction by a factor 1/3 and eight translations to the eight corners of a cube. The first fractal plant corresponds to the fourth iteration of this IFS applied to an initial cube of size 27 m3.

A second IFS was designed to simulate a self-similar plant-like canopy with (n = 9, c = 1/3). The IFS was applied three consecutive times to an initial horizontal disc leading to a first fractal plant (third form in Fig. 1). A further iteration (depth 5) was used to derive a second fractal plant (fourth form in Fig. 1).

Finally a random isolated canopy was generated by randomly locating 1000 leaves within a 1.2-m3 cube. Virtual leaves were discs. Leaf diameter was set to 10 cm, while leaf orientation was sampled in a spherical distribution (Ross 1981).

Multiscale description of canopy structure

For each 3-D plant, a rectangular bounding box was built from the spatial co-ordinates of phytoelements. The diagonal of the bounding box was thus defined by co-ordinates (xmin, ymin, zmin) and (xmax,ymax, zmax), where, for example, xmin is the minimum x-value shown by a leaf point in the tree foliage


where LTree is the number of leaves in the tree canopy.

The bounding box was divided into voxels at different scales n (n = 1, . . . N). At scale 1, the bounding box was represented by a single voxel. At scale n, the bounding box was divided into n parts along each box dimension, making a set of n3 voxels of size dx = (xmax − xmin)/n, dy = (ymax −ymin)/n, dz = (zmax − zmin)/n. As the bounding box was rectangular, voxels resulting from discretization at any scale were also rectangular, namely, in general dx ≠ dy ≠ dz.

Canopy structure parameters computed from the 3-D digitized datasets were the total leaf area and the tree volume abstracted by the volume of the bounding box. Average LAD of the tree canopy was then calculated as the ratio of total leaf area to bounding box volume.

At each scale n, the area of each leaf was affected to voxels according to the spatial co-ordinates of five leaf points, namely proximal, mid and distal points of the midrib, and lamina left and right border points at half leaf length. The co-ordinates of the five leaf points were computed from those of proximal point of the midrib (i.e. the junction point between petiole and lamina) and leaf orientation characterized by the Euler angles (see, e.g. Sinoquet et al. 1998). One-fifth of leaf area was affected to each leaf point, and therefore to each voxel including a leaf point. The number of vegetated voxels Vn was counted and the intervoxel variance of LAD was computed as


Plant lacunarity – a scale-dependent measure of heterogeneity of an object, originally proposed to analyse fractal objects (Mandelbrot 1983) – was then derived from LAD computations (Allain & Cloitre 1991; Plotnick et al. 1996)


Note that this quantity is defined for a given scale n. When the foliage structure is homogeneous at scale n, the lacunarity is close to 1. When the foliage shows a heterogeneous clumped structure at scale n, the lacunarity should be largely greater than 1. If clumps appear at a single scale, lacunarity should be high at this scale and low at other scales. If clumps are observed at different scales, lacunarity should remain high throughout scale.

STAR computations

Light interception at tree scale was characterized by STAR values, namely the ratio of silhouette to total leaf area (Oker-Blom & Smolander 1988). As STAR is a directional feature, values were computed for a set of Ω = 1, . . . 46 directions over the sky vault, according to the turtle sky discretization proposed by Den Dulk (1989). Sky-integrated STAR values were then computed by averaging STARΩ values after weighing by standard (SOC; Moon & Spencer 1942) overcast sky radiance distribution, for example


where inline image is the relative contribution of direction Ω to incident radiation in SOC conditions. inline image is thus an estimator of the whole tree ability for light capture. inline image was normalized to get  inline image = 1 for a canopy made of a single isolated horizontal leaf. In order to quantify the contribution of each source of foliage non-randomness, STARΩ values were computed in different ways.

Firstly, STARΩ was computed from virtual images of the 3-D tree mock-ups (Sinoquet et al. 1998), by using software VegeSTAR v.3 (Adam, Donès & Sinoquet 2002). The virtual images were created by using a virtual orthographic camera, namely a camera with parallel rays, in all 46 directions Ω. Image processing simply consisted of counting plant pixels seen on the picture, scaling measured areas according to pixel size and ratioing to total leaf area.  After  summing  up  over  the  sky  vault  (Eqn  6),  the

resulting value inline image (where index PMU refers to plant mock-up) was assumed to be the real STAR value, since it was computed without any assumptions about leaf dispersion.

Secondly, STARΩ was computed from the radiation transfer model included in the RATP model (Sinoquet et al. 2001). In this model, light interception is computed from the turbid medium analogy in a grid of voxels. For each direction Ω, a set of regularly spaced beams is cast in the bounding box, and beam attenuation is computed from the combination of gap fractions inline image in the sequence of voxels k intersected by beam b


where Ab is the cross-section area of a beam, Atree the total leaf area of the plant, and (k = 1, . . . Kb) is the sequence of intersected voxels. Note that inline image is gap fraction in the beam cross-section area Ab, due to leaf area in voxel k. In all simulation runs, beam spacing was 2 cm. Gap fraction inline image was computed in two ways. Firstly, Beer's law was used as in the original version of the RATP model.


where lbk is the length of beam b in voxel k. As previously mentioned, Eqn 8 assumes that leaf size is infinitely small with regard to voxel size.

Secondly inline image was computed by explicitly taking into account the effect of leaf size. For any beam b, inline image was therefore calculated from the product of gap fractions produced by each individual leaf l in voxel k.


where Lk is the number of leaves in voxel k, and abl is the portion of area of leaf l projected onto the beam cross-section area Ab. Given the assumption of uniform distribution of leaves within the voxel volume, abl should be proportional to the volume Vbk associated with beam path in voxel k.


where Al is the projected area of leaf l to a plane perpendicular to beam direction, and V is the voxel volume (V = dx dy dz). We numerically checked that regular and dense beam sampling within the canopy ensured

the  normalization condition inline image and consequently

inline image to be verified.



According to leaf size with regard to voxel volume, the term inline image may become negative, that is, in the case of a big leaf in a small voxel. This property was used to define the lower limit of possible voxel size.

Equation 11 rigorously deals with the case where each leaf is entirely located in a single voxel. As mentioned above, here leaves were cut into five pieces in order to distribute leaf area into voxels. Therefore Eqn 11 was changed as follows in order to simultaneously take into account size of the whole leaf and possible distribution of individual leaf area in several voxels


where Pk is the number of leaf pieces in voxel k, and Al is still the whole projected area of leaf l associated with leaf piece p. One can check that if voxel k includes whole leaves, Eqns 11 & 12 are the same. Otherwise one could remark that inline image tends to inline image if leaf size tends to 0, that is, if Lk tends to ∞ and all Al tend to 0.

Turbid medium STARΩ values were computed from Eqn 7 by using both Eqns 8 and 12 at each scale n of bounding box discretization. After summing up on sky directions,

the   resulting   values   were   noted   inline image and

inline image, respectively. Scales n were varied from 1 to N, where N refers to the finest space discretization. N is the largest integer allowing each leaf to be included in one single voxel, that is, the term inline image in Eqn 12 to keep positive.

Computed values inline image, inline image and inline image were compared for a detailed analysis of   foliage   non-randomness.   Comparisons   between

inline image– or inline image– at different tree discretization n show the multiscale effect of foliage clumping, namely accounted for by spatial variations of LAD within the  tree  canopy.  Discrepancies  between  inline image  or inline image and inline image show the effect of foliage non-randomness at a local scale, e.g. leaf arrangement  around the  shoot. Finally  differences between inline image and inline image account for the effect of leaf size on light capture.


Canopy structure

Figure 2 shows virtual images of the 3-D plant mock-ups. Images of digitized plants look like real plants of the same species, and visual comparison between the virtual image and an actual tree photograph in case of mango tree no. 1 and walnut tree shows that 3-D digitizing allows one to get a detailed and accurate description of the real spatial distribution of the tree foliage. Images in Fig. 2 also show the variety of tree canopy geometry used in this study. Random canopy appears like the theoretical random mock-ups proposed by De Castro & Fetcher 1999) and Pinty et al. (2001). The Cantor dust shows a much organized structure based on a ternary space division at consecutive scales. Fractals plants no. 1 and no. 2 look very similar since they were built from the same pattern and only differ by one depth order (Fig. 1).

Figure 2.

Images of the tree canopies used in this study. Images were synthesized from VegeSTAR v.3 software. Actual pictures of mango no. 1 and walnut tree canopies are shown on the left for visual comparison.

Table 1 shows canopy structure parameters computed at tree scale. Actual trees showed large variations in geometry parameters, especially with regard to the number of leaves, from 1559 for mango tree no. 1 to 14260 for the peach tree; total leaf area, from 6.5 to 28 m2; and bounding box volume, from 5 to 25 m3. As a result, mean LAD showed a two-fold range: from 0.66 to 1.32 m2 m−3. Inclusion of theoretical plants increased the range of canopy structure parameters, especially number of leaves (only 730 leaves for fractal plant no. 1), individual leaf size, bounding box volume and mean LAD.

Table 1.  Canopy structure parameters of the 3-D plants
leaf area
Total leaf
Bounding box
voxel volume
Mean LAD
(m2 m−3)
Walnut2.6 155947.27.3511.093.290.660.44
Mango11.8 163639.66.484.982.881.30.36
Mango21.6 245833.
Fractal13.7  72986.26.289.801.990.640.44
Fractal23.6 6561
Cantor3 409619.27.8427.000.760.290.72
Random1.2 100078.57.851.731.734.550.27

The minimum voxel volume, computed as the smallest voxel volume able to include every individual leaf, also showed a large variation. As the minimum voxel size resulted from a combination between leaf dimensions and orientation, the correlation between minimum voxel size and mean leaf area was low (r2 = 0.25, data not shown).

In the range of studied scales, the number of vegetated voxels in the actual tree canopies was closely related to voxel size by a power law, as a log–log relationship was quite linear (r2 > 0.99 for all trees, Fig. 3a and Table 2). The slope of the line was between 2 and 3, that is, values for fractal objects in-between two-dimensional (2-D) surfaces and 3-D volumes. The slope showed inter-tree range, that is, from 2.33 for peach tree to approximately 2.52 for mango trees (Table 2). Theoretical trees also showed this kind of fractal behaviour (Fig. 3b), but the r2 coefficient was lower, especially for the fractal plants and the Cantor dust, although they were built from fractal rules. Indeed, for the fractal plants – mainly the Cantor dust – some points with a smaller voxel size showed a smaller number of vegetated voxels (see the lines on Fig. 3b). This results from mismatch between space discretization by the 3-D grid and space occupation by the plant. As an example, all voxels at scale 4 (i.e. dx = (xmax − xmin)/4) in the Cantor dust were vegetated because it was based on space division by 3n. In contrast, at several scales, the 3-D grid exactly coincided with the space occupation by the plant (e.g. scales 1, 3, 9, 27 for the Cantor dust, see bottom points on the log–log curve Fig. 3b). Otherwise the slope of the log–log line in the random canopy was close to 3, which is the expected value for objects uniformly filling the 3-D space.

Figure 3.

Log–log relationship between voxel size and number of vegetated voxels in the tree canopy. Voxel size is defined as the cube-root of voxel volume since voxel size is different according to X-, Y- and Z-axis. Regression lines are not shown for sake of figure clarity, but r2 coefficients are given in Table 2. (a) digitized real trees; (b) theoretical canopies.

Table 2.  Determination coefficients and absolute values of slopes of linear regression analysis between log(voxel size) as X-variable, and (1) log(number of vegetated voxels) and (2) log(plant lacunarity)
 Log(no. vegetated voxels)Log (plant lacunarity)
  1. Voxel size is defined as the cube-root of voxel volume since voxel size is different according to X-, Y- and Z-axis. For plant lacunarity, data of the 3 largest voxel sizes were discarded from computation because they do not belong to the linear portion of the line. See Figs 3 and 4 for data points.

Real trees
 Mango no. 10.9962.540.9890.48
 Mango no. 20.9982.510.9740.58
Theoretical trees
 Fractal no. 10.9902.320.9431.01
 Fractal no. 20.9912.220.9771.01

For real trees, the variance of LAD as represented by plant lacunarity (Eqn 5) monotonously increased with decreasing voxel size as a power law: the smaller voxel size, the larger variance of LAD taken into account by space division. The log–log relationship was quite linear (Fig. 4a and Table 2), with r2 coefficients above 0.97. Theoretical canopies did not show the same behaviour (Fig. 4b). The two fractal trees showed almost the same increase of plant lacunarity with smaller voxel size, but at some points, a smaller voxel size resulted in a smaller variance in LAD. The Cantor dust showed large oscillations of plant lacunarity with voxel size, again resulting from obvious mismatch between space occupation by the plant and space division by the voxel grid. As a result, the log–log relationship was more scattered, with lower r2 coefficients (Table 2), especially for fractal plant no. 1 (r2 = 0.94) and the Cantor dust (r2 = 0.78). Otherwise the lacunarity line for the random canopy was monotonous but highly concave and therefore non-linear, as reported by Plotnick et al. (1996) for random sets.

Figure 4.

Log–log relationship between voxel size and plant lacunarity. Voxel size is defined as the cube-root of voxel volume since voxel size is different according to X-, Y- and Z-axis. Regression lines are not shown for sake of figure clarity, but r2 coefficients are given in Table 2. (a) digitized real trees; (b) theoretical canopies.

The maximum variance of LAD taken into account by space discretization showed a large range among real trees. As computed from Eqn 5, this corresponds to coefficient of variation of voxel LAD ranging between 3.6 and 18.5 for walnut and peach tree, respectively. Theoretical plants showed maximum values of LAD variance included in this range.

In contrast with the effect of voxel size, variance of LAD showed small variations in response to the 3-D grid translation and rotation (Fig. 5). For walnut tree at a fine scale (n = 10), fluctuations of LAD variance were very low, with a coefficient of variation of 0.65%. For mango no. 1 tree at an intermediate scale (n = 6), fluctuations of LAD variance were slightly larger (coefficient of variation of 1.9%). Other trees showed the same trends (data not shown).

Figure 5.

Changes in LAD variance of walnut and mango no. 1 trees as a function of grid rotation around Z-axis (a) and translation along X-axis (b). The bounding box was divided at scale 10 and 6 for walnut and mango trees, respectively.

STAR computations

Values of inline image of digitized trees, i.e. real sky vault-integrated STAR values computed from VegeSTAR, showed  a  two-fold  range  (Table 1,  last  column):  the  peach

tree had the lowest inline image (0.22) whereas the walnut tree with the lowest LAD had the highest score (0.44).

Mango trees had intermediate inline image values although their mean LAD was larger. This underlines that the trees sampled for this study showed a range of light interception responses. inline image values of theoretical trees were included in the same range, except the Cantor dust where inline image was higher (0.72), due to its low LAD. Note that the random canopy had a inline image similar to that of the peach tree, although mean LAD was about four-fold.

Light interception capacity as computed from the turbid

medium model, namely , inline image showed large variations as a function of space discretization scale (Fig. 6). For

Figure 6.

inline image, namely light interception capacity computed from the turbid medium model, as a function of log(voxel size). Voxel size is defined as the cube-root of voxel volume since voxel size is different according to X-, Y- and Z-axis. (a) digitized real trees; (b) theoretical canopies.

real trees, the smaller voxel size, the lower inline image. The STAR value computed at bounding box scale (i.e. inline image) was about +50% of the value computed at finest space discretization (i.e. inline image) for walnut and  mango  trees,  but  it  was  about  two-fold  for  the peach tree. inline image changes showed high correlation with voxel size, with r2 coefficients about 0.92–0.97. For theoretical trees, inline image was also generally lower for smaller voxel size (Fig. 6b). However, the random canopy showed only small changes according to voxel size. Moreover, unlike the real trees, the relationship for fractal objects was not strictly monotonous, especially for the Cantor dust where the line showed large oscillations.

When inline image values were plotted against plant lacunarity, relationships were strictly monotonous and quite linear, for both real and theoretical trees (Fig. 7). Determination coefficients r2 were very close to 1 for all plants (Table 3) but the peach tree (r2˜0.96) and the Cantor dust (r2 = 0.97). Nevertheless inline image was always more correlated to plant lacunarity than to voxel size or number of vegetated voxels (Table 3).

Figure 7.

inline image, namely the light interception capacity computed from the turbid medium model, as a function of log(plant lacunarity), for a range of space divisions (n = 1, . . . N). (a) digitized real trees; (b) theoretical canopies.

Table 3.  Determination coefficients of linear regression analysis between inline image and (i) log(voxel size); (ii) log(number of vegetated voxels) and (iii) log(plant lacunarity)
 Log(voxel size)Log(no. vegetated voxels)Log(plant lacunarity)
  1. Voxel size is defined as the cube-root of voxel volume since voxel size is different according to X-, Y- and Z-axis. See Figs 6 and 7 for data points.

Real trees
 Mango no. 10.9550.9480.996
 Mango no. 20.9630.9600.995
Theoretical trees
 Fractal no. 10.9620.9400.991
 Fractal no. 20.9710.9620.983

The small changes in plant lacunarity due to grid translation and rotation at fixed voxel size resulted in small variations  in  inline image  (Fig. 8,  for  grid  rotation  in mango

Figure 8.

inline image, namely light interception capacity computed from the turbid medium model, as a function of log(plant lacunarity), for a range of grid rotations at a fixed voxel size (scale = 6) for the mango no. 1 tree.

no. 1 tree). Although they were small, changes in inline image were highly correlated to plant lacunarity: the higher plant

lacunarity, the smaller inline image. Finally Figs 7 and 8 show that STAR computations with a turbid medium model are very sensitive to canopy lacunarity included in the model.

Values of STAR computed from the turbid medium model at finest space discretization, namely inline image,

were generally close to reference values, that is, inline image computed from plant images (Fig. 9). Data points below the

Figure 9.

inline image, namely light interception capacity computed from plant images, versus inline image, i.e. light interception capacity computed from a 3-D turbid medium model at finest space discretization.

1 : 1 line indicate  inline imageinline image, namely foliage dispersion at finest scale is clumped. All plants but the Cantor dust showed almost random leaf dispersion at finest scale: peach, mango and fractal trees were slightly clumped, whereas the walnut and the random canopy were quite random. The Cantor dust showed high regular dispersion at local scale due to the regular geometric rules used for its construction.

Values of STAR computed with the turbid medium with either Beer's law (Eqn 8, inline image) or by taking into account the finite leaf size (Eqn 9, inline image) were also very close (Fig. 10). The ratio inline image/inline image was always greater than 1, because finite leaf size as taken into account in Eqn 9 means that a given leaf does not shade itself. As expected the ratio increased with smaller voxel size, that is, as a function of the ratio between leaf and voxel size, but the overall effect was small: For real trees, the maximum values of the ratio was between 1.02 and 1.04. The same kind of response was found for the theoretical trees (data not shown).

Figure 10.

Ratio of STAR values computed from a 3-D turbid medium light by taking into account finite leaf size (inline image) or not (inline image), as a function of voxel size, for four digitized real trees.


Although Beer's law has been intensively used in light models for vegetation purposes, it shows a major drawback which is the random dispersion assumption. Usually departure from foliage randomness is assessed from comparison between light interception/transmission as computed from a turbid medium model and as measured in the field (see, e.g. Table 1 in Myneni et al. 1989). In this study we have attempted to identify the contribution of three sources of foliage non-randomness to light interception properties of a range of actual and theoretical ‘deciduous’ trees.

STAR computations with a classical 3-D turbid medium (RATP, Sinoquet et al. 2001) were close to reference STAR values, that is, as computed from plant images, only when the canopy volume was discretized at finest scale allowed by leaf size (Fig. 9). Other potential sources of foliage non-randomness, namely leaf size and non-randomness at local scale had a much lesser effect.

Deviation from foliage randomness may have occurred from finite leaf size. Indeed Beer's law assumptions results in computing self-shading within a given leaf, because the latter is abstracted as an infinite number of small elements randomly distributed in space. Plane leaves of real plants do not show self-shading. Consequently a random set of finite leaves theoretically behaves as a regular canopy (Sinoquet, Varlet-Grancher & Bonhomme 1993). Previous model comparisons on cotton canopies (Thanisawanyangkura et al. 1997) and tree seedlings (Planchais & Sinoquet 1998; Farque, Sinoquet & Colin 2001) have already shown that the effect of leaf size is usually negligible, except in the case of very young plants with only some leaves. Note however, that Fukai & Loomis (1976) explicitly included leaf size in their 2-D light interception model.

Non-randomness at local scale has rarely been directly assessed. In case of coniferous trees, needles are known to be clumped in shoots (Oker-Blom & Kellomäki 1983), so that several authors have proposed to input measured STAR values at shoot scale in the light models (Stenberg, Smolander & Kellomäki 1993; Nilson & Ross 1997). In that case shoots replace needles as basic canopy elements and within-shoot needle clumping is implicitly taken into account. In the case of deciduous trees, foliage randomness at local scale could also be derived from light interception at shoot scale. Takenaka (1994) described the 3-D geometry of shoots and related their light interception properties to petiole length and lamina narrowness. In a similar way, Falster & Westoby (2003) digitized in three dimensions the shoot geometry of a wide range of species. Such datasets could be used to check if foliage randomness at local scale, as found in our study, is a general issue. If not, the only present way to include non-randomness at local scale in light models would be to introduce a local leaf dispersion parameter or replace leaf elements by shoot elements as already mentioned for conifers.

Figure 9 shows that 3-D turbid medium models based on space division in voxels (e.g. Kimes & Kirchner 1982; Myneni 1991; Sinoquet et al. 2001) are able to provide correct estimations of light interception, namely that non-randomness in canopy structure is mainly driven by spatial variations in leaf area density. Note that Lang & Yueqin's method (1986) of leaf area index estimation from canopy transmittance is based on this assumption. However results are dramatically sensitive to voxel size (Fig. 6), which is the way the spatial variations of canopy structure are taken into account. Previous studies have mainly showed strong deviations between actual light interception and light interception in a horizontally homogeneous canopy of the same leaf area index (Chen et al. 1993; Béguéet al. 1996). Here we have shown that computed STARBeer values decreased with voxel size in real digitized trees (Fig. 6). This is in agreement with Knyazikhin et al. (1997)′s results where PAR transmittance was reported to increase with smaller voxel size.  This  is  purely  the  result  of  a  mathematical  property of the negative exponential function, namely the curve

convexity which makes that inline image. This property is reported as Jensen's inequality. Its consequences on light interception in 3-D heterogeneous optical media have been extensively discussed by Davis & Marshak (2004). In our case, this makes trees behave as clumped at canopy (i.e. bounding box) scale, although they show random dispersion at local scale (Whitehead et al. 1990).

The actual trees showed fractal behaviour, namely high log–log correlations between voxel size, number of vegetated voxels and lacunarity (Table 2). It was therefore difficult to distinguish between all these three variables as candidates of clumping parameters. Paradoxically, using objects built from fractal rules allowed us to – slightly – break the correlation between the three variables (Figs 3b & 4b). Indeed regular space discretization in voxels did not match the regular pattern of space occupation by these deterministic fractal objects. Using them consequently showed that plant lacunarity better than voxel size accounts for the effect of spatial variations in leaf area density on STAR computations when grid resolution varies (Fig. 7 and Table 3): the higher plant lacunarity, the more uniform distribution of leaf area density within the voxels, and the smaller effect of Jensen's inequality at local scale. Such a conclusion was already qualitatively drawn by Andrieu & Sinoquet (1993) from directional gap fraction measurements in physical plant mock-ups. Here we showed a quantitative relationship between sky vault-integrated light interception and canopy lacunarity. Although clumping is known to show directional changes (e.g. Chen 1996; Kucharik, Norman & Gower 1999), canopy lacunarity thus appears as a general clumping parameter for direction-integrated light interception.

A practical consequence of this study is that 3-D turbid medium models should be very careful with the space discretization into voxels, although this is seldom discussed in the literature (Knyazikhin et al. 1997). We showed here that plant lacunarity and light interception computations are virtually insensitive to grid translation and rotation (Fig. 8), but very sensitive to voxel size (Fig. 6). As discussed above, the best grid should be that which maximizes plant lacunarity taken into account by space discretization, so that intravoxel variance of LAD is minimal and Beer's law holds at voxel scale. Unfortunately plant lacunarity cannot generally be estimated a priori, as we made here from 3-D digitizing data. Decreasing voxel size generally increases the fraction of canopy lacunarity accounted for by the space division into voxels, because real plants behave as statistical fractal objects (Figs 3a & 4a). There is however, a limit in decreasing voxel size (Knyazikhin et al. 1997), according to leaf size (see Eqn 11). As shown in the range of tree canopies used in this study, the optimal voxel size is the minimal one allowed by leaf size constraints (Fig. 9).


Three-dimension digitized trees were used to assess the relative contribution of three sources of non-randomness of leaf dispersion within the canopy volume. The spatial distribution of leaf area density in the canopy accounted for most of departure from random leaf dispersion at canopy scale, while trees exhibited almost local random leaf dispersion and leaf size had also a minor effect on light interception. STAR computations from a 3-D turbid medium model were very sensitive to voxel size, and were closely related to the variance of LAD taken into account by space discretization.

The next step should be to explicitly include canopy lacunarity in a turbid medium model at canopy (i.e. bounding box) scale, without any 3-D discretization of the canopy volume. On one hand, this has already been proposed for small LAD fluctuations by using the Reynolds decomposition to both the radiation field and the canopy structure parameters in the general equation of radiation transfer (Menzhulin & Anisimov 1991; Anisimov & Fukhansky 1993). On the other hand, Nilson (1971) proposed in his pioneer work to relate leaf dispersion to the variance of number of contacts in the canopy. As the mean number of contacts is related to leaf area index, one can expect explicit relationships between canopy lacunarity, leaf dispersion and light interception by isolated trees.


The authors are grateful to D. Combes (INRA-Lusignan, France) for 3-D digitizing data of walnut tree and contribution to peach tree digitizing, and to P. Kasemsap and S. Thanisawanyangkura (Kasetsart University, Bangkok, Thailand) for contribution to mango tree digitizing. Financial support from PFI programme of INRA was provided for peach digitizing, while peach trees were made available by CTIFL, Balandran.