Annals of Botany 114: 689– 694, 2014 doi:10.1093/aob/mct310, available online at


Quantitative characterization of clumping in Scots pine crowns Pauline Stenberg1,*, Matti Mo˜ttus2, Miina Rautiainen1 and Risto Sieva¨nen3 1

Department of Forest Sciences, PO Box 27, FI-00014 University of Helsinki, Finland, 2Department of Geosciences and Geography, PO Box 64, FI-00014 University of Helsinki, Finland and 3Finnish Forest Research Institute, PO Box 18, FI-01301 Vantaa, Finland * For correspondence. E-mail [email protected] Received: 21 October 2013 Returned for revision: 20 November 2013 Accepted: 13 December 2013 Published electronically: 15 January 2014

Key words: Functional–structural modelling, Pinus sylvestris, radiation, clumping, STAR, crown shape, crown structure.

IN T RO DU C T IO N The spatial aggregation of plant elements in a canopy, also referred to as ‘clumping’, is an adaptive strategy of individual plants and plant communities. The spatial distribution of foliage controls the interaction of radiation with vegetation, and thus indirectly also plant growth and reproduction. Clumping can occur at various hierarchical scales from microscale (shoots in coniferous canopies) to macroscale (spatial tree patterns in forest stands). Clumping causes a decrease in the fraction of sunlit leaf area and a downward shift in the vertical distribution of the sunlit leaf area. From the production ecological point of view, these effects may be translated into a decrease in the fraction of absorbed photosynthetically active radiation (fPAR) but a more even distribution of irradiance on the canopy leaf area and, thus, higher light use efficiency. For canopies carrying a high leaf area index (LAI), in particular, the combined effect may be beneficial for the photosynthetic production (Stenberg, 1998). Proper characterization of the clumped structure of forests allowing, for example, calculation of scattered and absorbed radiation regimes and photosynthetic production in different parts of the canopy, has proved to be a challenging task. This concerns in particular statistical canopy radiation models, where probability density functions are used to describe the spatial distribution and orientation of foliage, and the transfer of radiation is

described using a stochastic variable – the gap probability. Another option is to use deterministic three-dimensional (3-D) structural models, where the plant elements have exact locations and the radiation reaching any specific point in the canopy can be determined by ray tracing. Limitations of such models, however, are their reliance on detailed descriptions of canopy architecture and lack of generality. Statistical models would be preferred for many larger scale applications, given that the models were realistic enough to produce reasonable results. Uncertainties in statistical models depend on how well the within-crown structure is characterized by the models which typically are based on oversimplifications. In this study, we focus on the degree and quantitative characterization of crown-level clumping in Scots pine (Pinus sylvestris). We define clumping with the help of a measure related to the penetration of radiation in plant canopies, namely the spherically averaged silhouette to total area ratio (STAR; Oker-Blom and Smolander, 1988). We start by writing out the general equations for the gap probability describing the penetration of radiation in a forest canopy. Using purely mathematical considerations, we split the effect of clumping on the gap probability into the contributions of spatial distribution of tree crowns and clumping of foliage within the tree crown. Finally, we demonstrate how the crown-level clumping is related to STAR and investigate its variation using a set of modelled Scots pine trees. The specific aims are (1) to study the dependency of crown-level

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† Background and Aims Proper characterization of the clumped structure of forests is needed for calculation of the absorbed radiation and photosynthetic production by a canopy. This study examined the dependency of crown-level clumping on tree size and growth conditions in Scots pine (Pinus sylvestris), and determined the ability of statistical canopy radiation models to quantify the degree of self-shading within crowns as a result of the clumping effect. † Methods Twelve 3-D Scots pine trees were generated using an application of the LIGNUM model, and the crownlevel clumping as quantified by the crown silhouette to total needle area ratio (STARcrown) was calculated. The results were compared with those produced by the stochastic approach of modelling tree crowns as geometric shapes filled with a random medium. † Key Results Crown clumping was independent of tree height, needle area and growth conditions. The results supported the capability of the stochastic approach in characterizing clumping in crowns given that the outer shell of the tree crown is well represented. † Conclusions Variation in the whole-stand clumping index is induced by differences in the spatial pattern of trees as a function of, for example, stand age rather than by changes in the degree of self-shading within individual crowns as they grow bigger.


Stenberg et al. — Quantitative characterization of clumping in Scots pine crowns

clumping on crown size and needle area, and (2) to examine how well statistical models are able to characterize the effect of clumping within a crown on radiation interception. Using a statistical canopy model, we first present the clumping concept and show how the clumping index is related to STAR of shoots and crowns. We use a series of computer-generated 3-D Scots pine trees (Lintunen et al., 2011) to assess the exact crown silhouette areas and amount of self-shading. These exact values are compared with results produced by the statistical model where the crown shape was represented by a simple geometric volume and a random distribution of shoots within the crowns was assumed.

Gap probability and clumping index

In statistical canopy radiation models, the transfer of radiation within a canopy is expressed in terms of the probability that a beam of radiation incident from the direction (u,f ) (u ¼ zenith angle, f ¼ azimuth angle) will pass through the canopy to reach a given point inside or below the canopy. It is normally assumed that the foliage is azimuthally randomly oriented, in which case the gap probability (P) becomes independent of the azimuth angle. In the following, therefore, the azimuth angle ( f ) of the beam (or view) direction is suppressed from the formulations. In the simplest statistical models, the canopy is treated as a horizontally homogeneous, optically turbid medium (Ross, 1981), where leaves are randomly dispersed with uniform density function in the canopy (i.e. following the Poisson distribution). A purely random leaf dispersion leads to an exponential attenuation of radiation with the downward cumulative LAI. Clumping refers to a more aggregated dispersion of leaves than the purely random (Poisson) distribution (Nilson, 1971). A general expression for the gap probability in clumped canopies can be formulated as: P(u) = exp[−G(u)G(u)L/ cos u]


where L denotes the downward cumulative LAI, G is the mean projection of unit foliage area, and G is the clumping index needed to correct for deviations in the relationship between gap probability and LAI from that of a Poisson canopy composed of randomly distributed leaves. Although G also includes a correction for the presence of woody area (not included in L), it is commonly referred to as a directional clumping index, and GL is referred to as the ‘effective leaf area index’. Since G(u) may vary with u, consistent definition of the effective LAI requires specification of the assumed directional distribution of incoming radiation. A logical choice is to define the effective LAI (Le) with respect to an isotropic radiation field, as originally proposed by Black et al. (1991): p/2 Le = −2

ln[P(u)] cos u sin udu G(u)G(u)L sin udu


G (u)G(u) sin u du




As shown by Miller (1967), for planar leaves the hemispherically averaged value of G is exactly 0.5, and the same result applies to conifer needles as long as they are of convex shape (Lang, 1991; Stenberg, 2006). Thus, in a Poisson canopy with G(u) ¼ 1 for all u, Le coincides with L [Eqn (2)] and the hemispherical clumping index (G) [Eqn (3)] equals one. We note further that Eqn (2) coincides with the formula used in the inversion from measured angular gap fractions [i.e. P(u)] to LAI by devices such as the LAI-2000 Plant Canopy Analyzer (Welles, 1990). Inverted values of LAI should thus be divided by G [Eqn (3)] to yield the true, clumping-corrected LAI.

Crown-level clumping

In the multiscale modelling approach (Da Silva et al., 2008), the plant organization is decomposed into a collection of components or clusters of leaves at different scales. The components are represented by porous envelopes generated as convex hulls containing components at a finer scale. The most commonly used clustering levels in radiation models for coniferous forests are clumping of needles into shoots and of shoots into tree crowns (Fig. 1) (e.g. Norman and Jarvis, 1975; Oker-Blom and Kelloma¨ki, 1983; Nilson, 1999). Here, we are interested in the degree of self-shading within individual tree crowns and thus want to separate the total clumping index [G(u)] into clumping at stand level [Gstand(u)] and crown level [Gcrown(u)], i.e. G(u) ¼ Gstand(u)Gcrown(u). We define the crown clumping index Gcrown(u) so that in the case of Poisson distributed trees (i.e. no clumping at stand level), it equals the total clumping index. Oker-Blom and Kelloma¨ki (1983) have demonstrated that in the case of Poisson distributed trees [i.e. when Gstand(u) ¼ 1, for all u], the gap probability depends only on the stand density N (number of trees per unit ground area) and the orthogonal projection area (i.e. silhouette area) of an individual average crown S(u): P(u) = exp[−NS(u)L/cos u]


Thus, Eqn (1) takes the form: P(u) = exp[−Gstand (u)Gcrown G(u)L/cos u] = exp[−Gstand (u)NS(u)/cos u]


From Eqn (5) we obtain the following mathematical relationship between

Gcrown (u) =

0 p/2




NS(u) S(u) = G(u)L G(u)lC


where lC ¼ L/N is the hemisurface leaf area of a single average tree.

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The total hemispherical clumping index is then defined as G ¼ Le/L:

Stenberg et al. — Quantitative characterization of clumping in Scots pine crowns


Shoot Forest stand



F I G . 1. The different structural levels used in describing the structure of a coniferous stand.


Gcrown = 2

Gcrown (u)G(u) sin udu



and the hemispherical stand-level clumping index as Gstand ¼ G/Gcrown. With these definitions, the total hemispherical clumping index [Eqn (3)] can be decomposed as: p/2


Gstand (u)Gcrown (u)G(u) sin udu = Gstand (u)Gcrown 0

(8) Relationship between Gcrown and STAR at shoot and crown level

We may extend the mathematical framework for describing canopy clumping to finer levels than the crown by decomposing the crown clumping index as Gcrown(u) ¼ Gs(u)Gn(u), where Gs and Gn quantify the reduction in crown silhouette area from overlapping of shoots within the crown and from overlapping of needles within the shoot, respectively. More precisely, Gn(u) is defined as the sum of the silhouette areas of all shoots in the crown [SSAtot (u)], when projected in the direction u, divided by the quantity G(u)lC which corresponds to the sum of the silhouette areas of all needles when projected in the same direction and retaining the natural orientation but assumed not to shade each other: Gn(u) ¼ SSAtot (u)/[G(u)lC]. From Eqn (6) follows that Gs(u) is the ratio of crown silhouette area to the sum of shoot silhouette areas: Gs(u) ¼ Gcrown(u)/Gn(u) ¼ S(u)/ SSAtot (u). We define the mean directional shoot silhouette to total needle area ratio, STARshoot (u) (Oker-Blom and Smolander, 1988), of all shoots in the crown as STARshoot (u) ¼ SSAtot (u)/(2lC). The quantity Gn(u)G(u) ¼ 2STARshoot (u) is thus the ratio of shoot silhouette area to hemisurface needle area, i.e. the mean projection of unit shoot area. In case shoots and needles have no preferred

orientation (G ¼ 0.5), parameter Gn(u) becomes independent of direction and is related to the spherically averaged shoot silhouette to total needle area ratio as Gn(u) ¼ 4 × STARshoot, where the factor 4 arrives from the fact that the total (i.e. allsided) needle area which is used as the denominator in STARshoot is four times larger than the spherically projected needle area. We may similarly define the STAR at crown level (STARcrown) as the spherically averaged ratio of crown silhouette area [S(u)] to the total all-sided needle area of the crown (2lC): p/2 STARcrown = 0

S(u) sin udu 2lC


From Eqns (6), (7) and (9) follows that the hemispherical crown clumping index and crown-level STAR are related to each other as Gcrown ¼ 4 × STARcrown. We further define the hemispherical shoot clumping index (Gn) as Gn ¼ 4 × STARshoot (note that this definition does not presume a spherical orientation of shoots and needles). Gcrown can now be expressed as the product of two indices: Gn ¼ 4 × STARshoot and Gs ¼ STARcrown/STARshoot, which in the following will be used to quantify the clumping of needles in the shoots and clumping of shoots in the crown.

M AT E RI AL S A ND M E T HO DS The computer-generated trees

We used an application of the LIGNUM model (Lintunen et al., 2011) that generates 3-D Scots pine trees on the basis of empirical equations, with tree height, diameter and values of a few competition indices as input. The generated Scots pine trees have been shown to reproduce the observed distribution of foliage inside the crown (Lintunen et al., 2011). The application predicts the needle area, needle length and needle angle of the shoots, thus all necessary variables for assessing the shoot silhouette area (SSA) and directional STAR [STARshoot (u)] of the shoots. We used the application to generate 12 pine trees with heights of 3, 6, 9, 12, 15 and 18 m. This span of heights corresponds to the validity range of empirical models in Lintunen et al. (2011). Two needle area configurations were used when generating the trees: ‘dense’ and ‘sparse’. We assigned the values of competition indices to correspond to high basal area (dense) and low basal area (sparse) on the basis of Finnish recommendations for Scots pine management (see Lintunen et al., 2011). Total (all-sided) needle areas for the generated trees varied between 15 and 200 m2 (Table 1).

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The stand-level clumping index [Gstand(u)] depends on the spatial pattern of trees and attains values larger (smaller) than one if the spatial pattern is more regular (aggregated) than the Poisson. It should be noted, however, that Gstand(u) is not uniquely determined by the tree pattern (except for the Poisson case) but for a given spatial pattern its values in different directions also vary with crown-level properties (Nilson, 1999). Finally, we define the hemispherical crown clumping index (Gcrown) similarly to the total hemispherical clumping index G [Eqn (3)] as:


Stenberg et al. — Quantitative characterization of clumping in Scots pine crowns

TA B L E 1. Basic characteristics of LIGNUM-generated trees Tree height (m) Needle area (sparse) (m2) Needle area (dense) (m2) Crown STAR (sparse) Crown STAR (dense)

3.0 15.9

6.0 35.5

9.0 69.4

12.0 101.6

15.0 117.2

18.0 164.6



















where t(u, p) is the transmittance through the crown to the point p on the plane perpendicular to the direction u, and Ac is a region on this plane that contains the silhouette area of the crown. We approximated the integral in Eqn (10) with a sum: we calculated the transmittance values t(u, p) for each point p in a grid of 3 × 3 cm covering the crown silhouette area. There were at least 5000 calculation points in the grid. The transmittance through the crown to point p is:    t u, p = m (u) (11) j[P(u,p) j

We tested the suitability of different simple geometric crown shape models for scaling between the canopy structural levels. We divided the crowns of computer-generated pine trees (Fig. 2, left) into segments with a thickness of 30 cm, calculated the maximum extent of the crown in each segment, and represented the tree crown as a stack of cylindrical segments (Fig. 2, middle). To test the suitability of an even simpler crown shape model, we modelled the crown as an ellipsoid of rotation with the same height and volume as the cylinder stack (Fig. 2, right). The approach where the crown envelope is modelled using maximum values of crown radii corresponds to the simple max mockup as described by Da Silva et al. (2011). Finally, tests were made where the diameter of the ellipsoid was altered (reduced or increased by a constant factor) while preserving its length (i.e. crown height).

where w is the acute angle between the directions of the ray of light and the axis of the shoot cylinder, K is the directional extinction coefficient within a shoot, Af is the needle area of the shoot, Vf is the volume of the shoot cylinder and d is the distance the ray of light has travelled in the cylinder. The extinction coefficient (K) was derived based on empirical data by Oker-Blom and Smolander (1988) and it also accounts for the shading of the shoot’s woody twig inside the shoot cylinder. In addition to foliage shading, we checked whether the ray hits the needleless parts of the crown; in such a case m ¼ 0. We calculated the crown silhouette area [Eqn (10)] at 158 steps from the horizontal direction to the zenith. For each zenith angle, the silhouette area was averaged over three rotational angles of the crown. Using appropriate weights for the directions, we calculated the spherically averaged STARcrown as defined by Eqn (9) (Table 1). Crown silhouette areas and STARcrown for the crown models using geometric shapes were similarly calculated by Eqn (10) but modelling the crown transmittance (t) assuming a Poisson distribution of shoots (exponential attenuation) within the crown volumes. The extinction coefficient (mean projection of unit shoot area) then equals Gn(u)G(u) ¼ 2STARshoot (u), and the transmittance through the crown is obtained as:      (13) t u, p = exp −2STARshoot (u)rs p

Calculation of crown STAR

where r denotes the needle area density within the crown volume and s( p) is the path length of the ray within the crown before reaching point p.

F I G . 2. A sample computer-generated tree (height 9 m, sparse) and the corresponding geometric crown shapes.

Geometric shapes of tree crowns

The crown silhouette area of the computer-generated trees was calculated using the surface integral:     1 − t u, p da (10) S(u) = AC

R E SULT S A ND D IS CUS SIO N The crown STAR was rather independent of tree height, needle area or growth conditions (Fig. 3). Although a slightly decreasing

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where P(u, p) is the set of shading elements (shoots with or without needles) along the path starting from point p with direction u. The shoots were modelled in this calculation as cylinders, the dimensions of which are determined by the length and diameter of the woody twig, and the needle length and needle angle. Whether the path intersected a shoot cylinder, and the length of the path within the cylinder if it did, was determined using analytical geometry as described in Perttunen et al. (1998). The transmission through an intercepted cylinder was evaluated on the basis of Oker-Blom and Smolander (1988) as:

   Af d (12) m = exp −K w Vf

Stenberg et al. — Quantitative characterization of clumping in Scots pine crowns 0·10



0·08 0·06 0·04 LIGNUM Cylinder stack Ellipsoid

0·02 0

150 50 100 Tree all-sided needle area (m2)


Ellipsoid STARcrown



0·05 0·05

0·06 LIGNUM STARcrown


F I G . 3. (A) STARcrown values calculated using the three crown models presented in Fig. 2 plotted against crown all-sided needle area. (B) STARcrown from the computer-generated crown compared with that from the ellipsoid model with diameter shrunk by 20 %.

trend with needle area may be noted, a near-constant value of STARcrown ¼ 0.06 or Gcrown ¼ 0.24 was predicted using the computer-generated crowns. The mean value of STARshoot calculated with the same directional weights as STARcrown was around 0.14 (Gn ¼ 0.56), yielding Gs ¼ STARcrown/STARshoot ¼ 0.43. The overlapping of shoots in the crown thus caused a slightly larger reduction in the crown silhouette area than the overlapping of needles in the shoots. The model based on geometric crown shapes filled with a random medium of shoots initially overestimated STARcrown. The values were slightly above 0.08 when the crown was approximated by a stack of cylinders, and slightly below 0.08 when the crown was modelled as an ellipsoid of rotation with the same volume (Fig. 3A). Similarly to results for the computergenerated trees, however, there was no dependency of modelled STARcrown on crown size. Furthermore, when crown diameter was artificially reduced by 20 %, there was close to perfect agreement between STARcrown values calculated using computergenerated and geometric tree crowns (Fig. 3B). The initial overestimation of STARcrown by the statistical model seems to suggest a more aggregated distribution of shoots in the crown than the completely random (Poisson) distribution. However, rather than invalidating the stochastic approach (modelling tree crowns as geometric shapes filled with a random

medium) the result stresses the importance of how to choose the geometric shape to best represent the actual crown (Mo˜ttus et al., 2006; Rautiainen et al., 2008; Da Silva et al., 2011). The crown shape determination method presented first was chosen to simulate a visual measurement in a forest where a person performing the measurements would determine the maximum extent of the crown at different heights. By doing so, we (and the measurer) have included a considerable amount of empty space in the crown. As actual tree crowns are not rotationally symmetric and their projected width depends on direction, using the maximum extent overestimates average crown width. Thus, we have decreased the needle area density within the crowns, increased average crown transmittance (i.e. reduced selfshading) and thus overestimated the crown silhouette area. Interestingly, modelling the crown as an ellipsoid yielded better results than the more detailed stack model and, simply by adjusting the diameter of the ellipsoid, the statistical model was able to reproduce the ‘true’ values of STARcrown from the computer-generated crowns. The result is in line with those of Da Silva et al. (2011), who found good agreement between measured and modelled light transmission in forest stands of different species when approximating crowns using simple, rotationally symmetrical geometric shapes and mean values of crown radii (the simple mean mockup). A potential application area of STARcrown is the parameterization of canopy-absorbed radiation using the photon recollision probability p, defined as the probability with which a photon, after having survived a scattering inside the canopy, will interact again with another canopy element (Smolander and Stenberg, 2005). This parameterization possesses great potential for robustly quantifying canopy structure in optical remote-sensing applications (e.g. Knyazikhin et al., 2013). At the level of a shoot, the probability p has been shown to depend on the STAR (Smolander and Stenberg, 2003), and a mathematically similar equation has been derived for the whole canopy (Stenberg, 2007). The results support the capability of the stochastic approach in characterizing clumping in crowns given that the outer shell of the tree crown is well represented, and therefore facilitate the applicability of photon recollision probability at crown level. For quantitatively accurate determination of STARcrown, or, equivalently, within-crown photon recollision probability, a physical method for determining the outer shell of a tree crown has to be developed. Another research question addressed in this study was on the magnitude of the crown-level clumping and its dependency on tree height (crown size) and growth condition (needle area density in the crown). The obtained value of Gcrown ¼ 0.24 can be interpreted such that the silhouette area of an individual crown is reduced by 76 % from overlapping of needles within shoots and between shoots. The result also implies that in the case of Poisson distributed trees (with G ¼ Gcrown ¼ 0.24), optically based indirect estimates of the effective LAI (Le) would correspond to only a quarter of the true LAI. In managed forests, however, trees are typically more regularly spaced (Gstand .1) than the Poisson, and the total clumping factor (G) is thus larger than Gcrown. Against (our) expectations, the crown clumping index for the 12 Scots pine trees did not change with age (tree height) or simulated growth conditions. Our set of computer-generated trees mimics only a few of all crown configurations that can be

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Stenberg et al. — Quantitative characterization of clumping in Scots pine crowns

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found in Scots pine in Finland due to the limitations of the empirical models in Lintunen et al. (2011). The computer-generated trees are random but we did not use multiple repetitions of similar trees to the 12 used because the random variation influenced the results only marginally (not shown). On the other hand, the size differences of our computer-generated trees, from 3 to 18 m in height and from 15 to 200 m2 in needle area should be large enough to reveal the main variation of STARcrown with crown characteristics. We carried out experiments with other trees (grown by LIGNUM, see Sieva¨nen et al. 2008) but the results were similar (not shown). While it is far too preliminary, based on this small case study, to reach the conclusion that the crown clumping index is constant in Scots pine, we suggest that the issue and its relevance from both the ecological and modelling point of view is worth exploring further. Finally, we draw attention to the fact that even if the degree of self-shading within crowns would remain constant, clumping at the whole-stand level would vary with the spatial pattern of trees, which in turn changes with, for example, stand age, growth conditions and management procedures.

Quantitative characterization of clumping in Scots pine crowns.

Proper characterization of the clumped structure of forests is needed for calculation of the absorbed radiation and photosynthetic production by a can...
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