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.
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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
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.
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
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 (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 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 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 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 was computed by explicitly taking into account the effect of leaf size. For any beam b, 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 a′bl 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, a′bl should be proportional to the volume Vbk associated with beam path in voxel k.
where A′l 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 and consequently
to be verified.
According to leaf size with regard to voxel volume, the term 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 A′l 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 tends to if leaf size tends to 0, that is, if Lk tends to ∞ and all A′l 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 and
, 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 in Eqn 12 to keep positive.