Using a stand-level model to predict light absorption in stands with vertically and horizontally heterogeneous canopies
© Forrester et al.; licensee Springer. 2014
Received: 13 May 2014
Accepted: 22 August 2014
Published: 27 September 2014
Forest ecosystem functioning is strongly influenced by the absorption of photosynthetically active radiation (APAR), and therefore, accurate predictions of APAR are critical for many process-based forest growth models. The Lambert-Beer law can be applied to estimate APAR for simple homogeneous canopies composed of one layer, one species, and no canopy gaps. However, the vertical and horizontal structure of forest canopies is rarely homogeneous. Detailed tree-level models can account for this heterogeneity but these often have high input and computational demands and work on finer temporal and spatial resolutions than required by stand-level growth models. The aim of this study was to test a stand-level light absorption model that can estimate APAR by individual species in mixed-species and multi-layered stands with any degree of canopy openness including open-grown trees to closed canopies.
The stand-level model was compared with a detailed tree-level model that has already been tested in mixed-species stands using empirical data. Both models were parameterised for five different forests, including a wide range of species compositions, species proportions, stand densities, crown architectures and canopy structures.
The stand-level model performed well in all stands except in the stand where extinction coefficients were unusually variable and it appears unlikely that APAR could be predicted in such stands using (tree- or stand-level) models that do not allow individuals of a given species to have different extinction coefficients, leaf-area density or analogous parameters.
This model is parameterised with species-specific information about extinction coefficients and mean crown length, diameter, height and leaf area. It could be used to examine light dynamics in complex canopies and in stand-level growth models.
KeywordsComplex forests Mixed-species Stand structure Extinction coefficient Lambert-Beer law Light absorption
where k is the light extinction coefficient for the considered growth period, and L is the leaf area index (m2 m−2). However, most forests and plantations do not have homogeneous canopies; they may consist of multiple species or the canopies may contain gaps, such as in young stands or resulting from thinning and following natural disturbances. Tree-level light absorption models have been developed to deal with this canopy heterogeneity and some have been shown to give comparable predictions to field measurements of APAR (Norman and Welles ; Oker-Blom et al. ; Wang and Jarvis ; Bartelink ; Brunner ; Canham et al. ; Bartelink ; Courbaud et al. ; Gersonde et al. ; Abraha and Savage ; Ligot et al. [2014b]). Inputs for these models may be the leaf area of each tree, vertical and horizontal leaf area distributions, leaf angle distribution, leaf and soil optical properties, and x and y coordinates to indicate the tree positions. Generally, the accuracy of these tree-level models increases with the level of detail used to describe the tree crowns and canopy structure (Brunner ; Sinoquet et al. ; Parveaud et al. ).
Some of these models, such as the Maestra (or Maestro) model (Grace et al. ; Wang and Jarvis ; Medlyn ), have been shown to provide realistic predictions of absorbed light in mixed-species stands (le Maire et al. ; Charbonnier et al. ). However, these tree-level models often require extensive input data and can have high computational demands. In contrast, several equations based on Equation 1 have been developed to predict APAR at the stand level using less information for parameterisation, and models using these can run faster than tree-level models (Duursma and Mäkelä ; Charbonnier et al. ; Forrester ). Such models are therefore useful for stand-level process-based models that need to be easy to use and parameterise and quick to run at large temporal and spatial scales (Ligot et al. [2014a]).
Stand-level light absorption models, such as Equation 1 or modifications of it, are often used in process-based forest growth models but they are rarely tested for the heterogeneous canopies being modelled, even when the same growth models are thoroughly tested for their predictions of growth, transpiration, carbon partitioning and nutrient availability. This project had three aims. The first was to test the APAR predictions of a stand-level light absorption model, similar to that described by Forrester (), for a wide range of canopy structures. The model used by Forrester () has been tested against a detailed tree-level model but not using data from real forests. The second aim was to compare this stand-level model to several other approaches that have been used to predict APAR by mixed-species canopies. The third aim was to determine which parameters this model is most sensitive to.
To achieve these aims, stand-level APAR predictions were compared with those of a detailed tree-level model, Maestra (Grace et al. ; Wang and Jarvis ; Medlyn ), which was run using data collected from five experiments: (i) thinned and unthinned Eucalyptus nitens plantations in southeastern Australia, (ii) monospecific and mixed-species plantations of Eucalyptus grandis and Acacia mangium in the tropics of Brazil, (iii) mixed-species plantations of Hopea odorata and Acacia hybrid (A. mangium × A. auriculiformis) in Vietnam, (iv) mixed-species forests and Cunninghamia lanceolata plantations in the subtropics of China, and (v) temperate mixed-species forests composed of Abies alba, Picea abies and Fagus sylvatica at six sites in southwestern Germany, each of which was managed according to a range of thinning treatments.
2.1 Description of study sites
Site characteristics, species compositions, experimental designs and mean tree dimensions (standard deviations in parentheses) for the five stands used
Eucalyptus grandis / Acacia mangium
Acacia hybrid (A. mangium x A. auriculiformis) / Hopea odorata
Castanopsis eyrei / Castanopsis sclerophylla / Cunninghamia lanceolata / Cyclobalanopsis glauca / Liquidambar formosana
Abies alba / Fagus sylvatica / Picea abies
Carrajung, Victoria, Australia (38°23′S, 146°41′E)
Itatinga experimental station, southern Brazil (23°2′S, 48°38′W)
Phu Loc, Thua Thien Hue Province, Vietnam (16°18'N, 107°42'E)
Shitai county, Anhui Province, China (29°59′-30°24′N, 117°12′-117°59′ E )
Black Forest, Swabian-Franconian Forest, and south-western Swabian Alp, Germany (47°44′N to 48°56′N, 7°58′E to 9°34′E)
Site and stand characteristics
Mean annual precipitation 1124 mm. Minor slope (<10%). Examined from age 3.4 to 8.1 years, leaf area index 1.5 to 5.1, 291 to 935 trees per ha.
Mean annual precipitation 1360 mm. Minor slope (<3%). Examined for a rotation, between 1 and 6 years, leaf area index up to about 4.7, from 1111 to 2222 trees per ha.
Mean annual precipitation > 3500 mm. Minor slope (<5%). Examined when Acacia hybrid were 3 years old and H. odorata were 1 to 2 years old. Leaf area index of 4.6 (strip design) and 3.4 (circular design).
Mean annual precipitation is about 1200 mm. Level to very steep slopes (up to 107%). Leaf area index of 1.3 to 10.2, from 88 to 2829 trees per ha. Tree ages were 20–31 / 16–49 / 15–25 / 20 / 16–36 years.
Mean annual precipitation from 941 to 1850 mm, Level to slopes of 51%. About 100 years old, leaf area index of 1.4 to 7.7, 184 to 757 trees per ha.
Monospecific plantation with unthinned (about 900 trees per ha) plots and plots thinned down to 300 trees per ha at age 3.2 years. These were unfertilised and fertilised with 300 kg N per ha at age 3.2 years. Plots of about 23 m x 32 m surrounded by buffers of about 5 m with the same treatment.
Monospecific plots of each species (1111 trees per ha) and (additive) mixtures containing 1111 E. grandis trees per ha + 277, 555 or 1111 A. mangium trees per ha and (replacement) mixtures containing 555 trees per ha of each species. 18 m x 18 m plots surrounded by 6 m buffers of the same treatment.
Three mixed-species plots. Two strip plantings where H. odorata seedlings were planted in strip gaps (5- or 7.5-m wide) between rows of Acacia hybrid. In the third experiment, a circular gap 22-m in diameter was made in 3 year old Acacia hybrid that was planted at a spacing of 2 m x 2 m. Within this circular gap (and 1 m away from the gap under the Acacia hybrid canopy) H. odorata seedlings were planted at a spacing of 2 m x 2 m.
49 plots with a radius of 10 m were established in a native forest and adjacent C. lanceolata plantations. Some plots contained additional species but these contributed a maximum (by trees per ha or basal area) of 10%.
Mixed-species forest plots on 6 sites, each containing four thinning treatments. Plot sizes were 50 m x 50 m or 60 m x 60 m. The outer 10 m of these plots were buffers.
Species proportions by trees per ha
0% to 100%
87% / 13%
58% / 42%
0% to 100%
14 % to 100% / 0% to 56% / 0% to 83%
Mean number of live stems per ha at age of simulation
870 / 511
927 / 139
1267 / 916
90 / 802 / 1197 / 353 / 207
121 / 94 / 85
Mean height (m)
5.4 (0.7) to 24.4 (3.1) / 2.5 (0.7) to 11.3 (3.8)
13.5 (1.43) / 1.5 (0.22)
14.6 (1.56) / 1.4 (0.33)
8.3 (1.7) / 10.2 (1.9) / 10.4 (1.7) / 8.8 (2.0) / 13.1 (1.5)
27.1 (8.7) / 23.3 (6.0) / 32.2 (2.9)
Mean height to crown base (m)
0.9 (0.001) to 18.4 (2.27) / 0.1 (0.02) to 5.5 (1.42)
4.5 (1.54) / 0.2 (0.03)
10.7 (1.49) / 0.2 (0.03)
2.2 (0.7) / 4.2 (0.8) / 4.6 (1.0) / 3.1 (1.7) / 5.0 (1.5)
15.4 (5.4) / 10.3 (2.5) / 14.5 (1.9)
Mean live-crown length (m)
4.5 (0.7) to 6.0 (1.5) / 2.4 (0.7) to 5.8 (2.6)
8.9 (1.6) / 1.2 (0.22)
4 (1.02) / 1.1 (0.34)
6.1 (0.9) / 6.0 (1.6) / 5.9 (1.8) / 5.8 (0.9) / 8.0 (1.6)
11.6 (3.9) / 13.2 (3.5) / 17.6 (2.6)
Mean stem diameter at 1.3 m (cm)
5.1 (0.8) to 16.5 (2.8) / 2.4 (0.8) to 10.9 (5.0)
10.6 (2.12) / 1.2 (0.15)**
9.6 (2.07) / 1.2 (0.27)**
10.2 (3.9) / 19.0 (5.0) / 15.2 (3.6) / 11.5 (1.8) / 18.1 (5.1)
41.7 (15.3) / 25.5 (8.5) / 45.9 (8.8)
Mean crown radius (m)
1.3 (0.1) to 1.5 (0.3) / 0.8 (0.4) to 1.9 (0.5)
1.6 (0.36) / 0.5 (0.08)
1.1 (0.25) / 0.4 (0.12)
2.3 (0.6) / 3.1 (0.7) / 1.8 (0.2) / 2.2 (0.3) / 4.1 (1.1)
2.8 (0.61) / 2.4 (0.31) / 2.8 (0.43)
Mean tree leaf area (m2)
33.0 (10.8) to 35.6 (13.5) / 3.0 (1.5) to 26.6 (18.0)
49.8 (24.56) / 0.8 (0.24)
26.3 (19.76) / 0.7 (0.71)
40.8 (13.3) / 28.4 (10.5) / 34.6 (21.6) / 30.3 (7.7) / 109.3 (50.3)
209 (105) / 122 (65) / 231 (70.9)
Mean leaf-area density (m2 m−3)
1.92 (0.29) to 1.30 (0.30) / 0.79 (0.01) to 0.54 (0.03)
1.03 / 1.38
2.77 / 1.73
0.63 (0.09) / 0.27 (0.09) / 0.84 (0.12) / 0.53 (−)/ 0.40 (0.08)
1.05 (0.24) / 0.79 (0.14) / 0.77 (0.11)
Forrester et al. 
le Maire et al. 
Forrester and Albrecht  (only used the 4th growing period)
In most of the mixed-species stands the mixing of species was relatively uniform; i.e. tree-by-tree mixtures rather than a spatial clustering of species. However, the mixtures of Acacia hybrid and H. odorata in Vietnam (Stand 3, Table 1) included two different nurse crop designs. One of these consisted of rows of H. odorata (about 1.5 m tall) planted between rows of much taller Acacia hybrid (about 14 m tall). The second design was a 22-m diameter circular gap inside an Acacia hybrid plantation that had been planted with H. odorata trees. The stand-level model assumes that species mixtures are intimate. Therefore, this planting design allows for testing of possible errors that can result when this assumption is not met. Details about each stand are provided in Table 1.
2.2 Description of the stand-level model
A list of the abbreviations and symbols used in the text and in equations
Absorption of photosynthetically active radiation (PAR) (MJ m−2 month−1)
Relative average error
Fraction of ground area covered by the canopy
Fraction of PAR that is absorbed by a homogeneous canopy
Fraction of PAR that is absorbed by species i
Indicates the i th species of n species
Indicates the j th canopy layer of n layers
Light extinction coefficient
Light extinction coefficient of a homogeneous canopy
Height to crown base (m)
The mid-crown-height (m) of a layer or species; given as (height – H b )/2 + H b
Empirical parameter to account for the effects of horizontal heterogeneity within the j th canopy layer
Empirical parameter to quantify the vertical structure of a canopy layer and to partition layer APAR to each species within the layer
Leaf area index (m2 m−2)
Individual tree leaf area (m2)
Leaf-area density in terms of leaf area per crown volume (m2 m−3)
Live-crown length (m)
Relative mean absolute error
Mean square error
Observed calculations from the tree-level model (Maestra)
Mean of O
Predicted values from the stand-level model
Mean of P
Crown surface area of individual trees (m2)
Sum of the crown volume (m3) of all crowns within a given layer within one hectare divided by the total volume of that layer
Empirical parameter to quantify the horizontal structure of a canopy layer (see Appendix A)
An adjusted mean midday solar zenith angle (°) (see Appendix B)
The fraction of PAR intercepted by the whole canopy layer is the term inside the square brackets. k H is defined as an extinction coefficient for a homogeneous canopy. Typical long-term (monthly, annual) extinction coefficients, k (Equation 1) vary for a given species as the crown architecture and canopy structures (e.g. the space between neighbouring crowns) change with factors such as age and resource availability (Binkley et al. ; Bryars et al. ). Duursma and Mäkelä () showed that this variability in k could be accounted for by first replacing k in Equation 1 with an extinction coefficient for a homogeneous canopy, k H . This canopy is characterized as being composed of trees of the same height, with the same live-crown length, having box-shaped crowns that fit together perfectly (no space between crowns), and of the same leaf-area density (LAD, leaf area per crown volume, m2 m−3), leaf angle distribution, leaf reflectance and leaf transmittance. In such stands the k H is independent of trees per ha.
The is an empirical parameter that partitions the absorbed light between each of the species (i) within a given layer based on the vertical structure of the layer as well as the extinction coefficients and leaf area. The is used to account for the effects of horizontal heterogeneity within the j th canopy layer. This replaces an empirical parameter (ϕ, Appendix A) that was proposed by Duursma and Mäkelä () and used in the stand-level mixed-species model by Forrester (). However, preliminary analyses indicated that ϕ can be difficult to calculate for some crown architectures and it is occasionally not calculable for others due to its asymptotic form.
where h m,i is the mid-crown-height of species i, h m is the mid-crown-height of the layer in which species i belongs. For Equation 3 R 2 = 0.75 and P < 0.0001. The mid-crown-height is the height of a point half way between tree height and the height to crown base (H b ), where the H b is the height where the lowest live branch joins the stem. Thus, the mid-crown-height of a given species is calculated as (height – H b )/2 + H b , and the mid-crown-height of a given layer is calculated using the height of the tallest species in that layer, and the minimum H b of all species in that layer. The sum of λ v for all species in a given layer is then normalized to unity such that
This ensures that the λ v values only change the partitioning of individual species APAR and not the total layer (or stand) APAR. Equation 3 was established using canopy layers with up to eight species. In canopy layers containing > 8 species the values would probably be lower, on average, than those in the data set used to fit Equation 3, and therefore, Equation 3 should be used cautiously for layers with > 8 species, although the whole canopy can contain more than eight species if they are in multiple layers (Forrester, ).
The horizontal heterogeneity λ h is calculated as a function of mean midday solar zenith angle, k H , the ratio of mean tree leaf area (L A ) to mean tree crown surface area (S A ) and v frac , which is the sum of the crown volume (m3) of all crowns within a given layer within one hectare divided by the total volume of that layer (m3) over one hectare. The total volume of a layer is (height of the tallest species – minimum H b ) × 100 m × 100 m.
The standard errors of parameter values in Equations 5a and 5b are in parentheses below the estimate and P < 0.0001. These equations were fitted using the monospecific data set described by Forrester (). Monocultures were used so that the effects of horizontal heterogeneity could be quantified in the absence of any vertical heterogeneity that occurs in mixtures due to inter-specific differences in the vertical distribution of LAD, even if the height and height to crown base does not vary between species. When Equations 5a and 5b are used for mixed-species layers the k H × L A /S A is a weighted mean of all species within the given layer. The mean is weighted by the contribution that each species makes to the sum of k H × L.
The monospecific data contained simulated monospecific stands with crown characteristics that varied in terms of LAD (0.27 to 2.62 m2 m−3), monthly k H (0.04 to 1.48), live-crown lengths (2 to 18 m), L A (10 to 390 m2), crown diameters (2.7 to 7.6 m), L A /S A (0.16 to 1.94), mean leaf angles (20° to 70° from horizontal), crown shapes (cones, ellipses, and half-ellipses), and densities ranged from 10 to 5300 trees per ha. About 548 stands were replicated at five latitudes between 0 and 65° and f was calculated for each month using Maestra. The k H of these stands was calculated using simulations of homogeneous canopies consisting of box-shaped crowns.
2.3 Description of detailed tree-level light model
Maestra is a three dimensional tree-level model that calculates APAR by individual trees based on their crown architectures and any shading from neighbouring trees. To account for shading from neighbours the canopy is represented as an array of tree crowns (shaped as spheres, cones, ellipses, half-ellipses) whose positions are defined by x and y coordinates. Each crown is divided into horizontal layers and each layer is divided into several points, and for each, LAD, leaf angle distributions and leaf optical properties are used to calculate APAR. The penetration of radiation through the canopy is calculated using the radiative transfer model of Norman and Welles (). Transmission and absorption of diffuse radiation is modelled using the method of Norman (), while the non-intercepted radiation reaching a canopy point is calculated in the sun direction for direct beam according to the hourly zenith and azimuth angles of the sun and of various azimuth and zenith directions for the diffuse sky radiation.
Maestra has been tested against field measurements in several stands, and performed well. For example, Maestra calculations of diffuse radiation were compared with field measurements in agroforestry plantations of widely-spaced Erythrina poeppigiana trees and a dense understorey of Coffea arabica plants (Charbonnier et al. ). Transmittance through the overstorey layer as well as through both layers, was compared and the goodness of fit (R 2 ) was > 0.75 in all cases. Maestra slightly underestimated values at the high end of the range of diffuse transmittance and Charbonnier et al. () suggested that this may have resulted from errors relating to 1) field data collection, 2) estimates of Maestra parameters, and 3) assumptions used by Maestra, such as the simplification of crown architecture descriptions.
Maestra was also tested in an experiment containing monocultures and mixtures of Eucalyptus grandis and Acacia mangium (Stand 2 in Table 1; le Maire et al. ). Gap fractions were slightly overestimated by Maestra in the monocultures and underestimated in the mixtures, which resulted in underestimates of APAR of 3.4% for A. mangium monocultures, 4.5% for E. grandis monocultures and overestimates of 4.6% for 1:1 mixtures.
2.4 Model runs and parameterization
Parameters used for the Maestra model in each of the five stands
Parameter name and definition
E. grandis / A. mangium
Acacia hybrid / H. odorata
C. eyrei / C. sclerophylla / C. lanceolata/ C. glauca / L. formosana
A. alba / F. sylvatica / P. abies
Rhosol: soil reflectance in PAR, NIR and thermal
0.10, 0.30, 0.05
0.07, 0.27, 0.05
0.10, 0.30, 0.05
0.11, 0.28, 0.05
0.10, 0.30, 0.05
Atau: leaf transmittance in PAR, NIR and thermal
0.093, 0.34, 0.01
0.034, 0.328, 0.01 / 0.063, 0.296, 0.01
0.063, 0.296, 0.01 / 0.03, 0.32, 0.01
C.lan 0.03, 0.26, 0.01 / Others 0.046, 0.336, 0.017
0.03, 0.26, 0.00 / 0.05, 0.30, 0.05 / 0.03, 0.26, 0.00
Arho: leaf reflectance in PAR, NIR and thermal
0.082, 0.49, 0.05
0.048, 0.247, 0.05 / 0.074, 0.206, 0.05
0.074, 0.206, 0.05 / 0.05, 0.25, 0.05
0.09, 0.33, 0.05 / 0.067, 0.382, 0.05
0.09, 0.33, 0.05 / 0.06, 0.35, 0.05 / 0.09, 0.33, 0.05
Nalpha: number of leaf angle classes from 0 to 90 degrees
Falpha: proportion of leaf area in each angle class
0.007, 0.022, 0.041, 0.064, 0.094, 0.132, 0.176, 0.219, 0.245 / 0.053, 0.130, 0.156, 0.148, 0.129, 0.111, 0.098, 0.090, 0.086
Avgang: mean leaf inclination angle
(36.7 / 31.0)*
42.2 (strip) 39.2 (circle) / 27.8 (strip) 41.5 (circle)
Jleaf: specification of leaf-area density distribution
1 (vertical direction)
1 or 2 (in vertical and horizontal directions)
1 (vertical direction)
1 (vertical direction)
bpt: beta dist. parameters for the vertical (and horizontal if used) leaf area density
1.647, 0.791, −0.057
5.707, 1.296, 0.711, 2.280, 1.218, 1.048 / 2.825, 0.840, 0.340, 0.0, 0.0, 0.0
1.614, 0.072, 0.358 / 2.304, 1.520, −0.050 / 2.035, 1.327, −0.066 / 3.869, 0.966, 0.518 / 2.210, 0.642, 0.185 / 1.060, 0.416, −0.210
3.53, 0.58, 0.78
Homogeneous extinction coefficients ( k H ) for each species and typical extinction coefficients ( k ) for all monospecific stands
k H (sd)
Stand 3 - Circular gap
Stand 3 - Strip gap
Some of the stands contained deciduous species. The months during which leaf fall and leaf development occurred were excluded from the analyses. These were March and October for L. formosana in Stand 4 and May and October for F. sylvatica in Stand 5. During the leafless period, the deciduous species were given L A values of zero, while all other species continued to absorb light.
2.5 Model comparisons and sensitivity analysis
where O is the observed calculations from Maestra and P is the predicted values from the stand-level model, and and are the means.
The accuracy test described by Freese () was used to test whether the calculations from Maestra and those from the stand-level model differed by > 10% with a 95% confidence limit (α = 0.05). All statistical analyses were performed using R 3.0.2 (R Core Team ).
In all cases k H was used instead of k because k could not be determined for all species using this data set (monocultures are required) and it was assumed that k H was a good approximation of k. It is also problematic to determine a k for each species because it varies with stand density and L, unlike k H . To increase the range of stand structures and inter-specific differences that were examined the comparisons for the three different approaches were made using data from Stands 4 and 5, or using the simulated dataset developed by Forrester () that contained 548 monocultures and 495 mixtures replicated at 3–5 latitudes between 0 and 65°. In these stands crown characteristics varied in terms of LAD (0.27 to 2.62 m2 m−3), monthly k H (0.04 to 1.60), live-crown lengths (2 to 18 m), L A (10 to 390 m2), crown diameters (2.7 to 8.0 m), L A /S A (0.16 to 2.05), mean leaf angles (20° to 70° from horizontal), crown shapes (cones, ellipses, and half-ellipses), and densities ranged from 10 to 5300 trees per ha.
Finally, a sensitivity analysis was used to examine how much the APAR predictions using Equation 2 change in response to a 10% change in k H , L A , crown diameter and live-crown length.
3Results and discussion
Stand 1 – Equation 2
Stand 2 – Equation 2
Stand 3 – Equation 2
Stand 4 – Equation 2
Stand 4 – Equation 12
Stand 5 – Equation 2
Stand 5 – Equation 12
Vertical heterogeneity is initially considered in the model by dividing the canopy into layers, each of which contains species with crowns that overlap vertically (Figure 1). Then the vertical heterogeneity within a layer is considered by using the constant λ v (Equations 2 & 3), which is determined by the height of the midpoint of the crowns compared with the midpoint of the given layer. The same constant is used to partition the total layer f to each of the species within that layer based on their k H × L.
Partitioning and consideration of the vertical heterogeneity is particularly important in mixtures because if this is ignored the model will give one species an unrealistic competitive advantage for light absorption and hence growth. Biased estimates can be obtained if this vertical heterogeneity is quantified by only using the mean height of each species as shown in Figure 2b (Sinoquet et al. ). This will exaggerate the asymmetry of competition for light such that if one tree or species is able to overtop another only by a few metres or even centimetres it gains a complete competitive advantage in terms of light because shorter trees can only intercept the light that is transmitted through the crowns of the taller trees. The bias that can result from assuming the canopy structure of Figure 2b is shown in Figure 3. Figure 3e and 3g show the APAR estimates when considering the relative positions of the whole crown length (i.e. Figure 2a), not just the height of each species. In contrast, Figure 3f & 3h assume that all of the leaf area of a given species is positioned at the top of the crown (i.e. Figure 2b). This results in an overestimate of APAR for the taller species, such as P. abies, and a corresponding underestimate for the shorter species, A. alba or F. sylvatica (Figure 3h). Even though there is a difference in height, the majority of the length of the crowns in Stands 4 and 5 overlap in a similar way to species 1 and species 3 in Figure 2 (Table 1). If one species is only slightly shorter than another, but has a similar or shorter LCL, or a higher L A or extinction coefficient then it may even have a greater shading effect on the other species than vice versa. This may not be a problem when all species have very low k H values, but it could lead to biased estimates where the taller species within a given layer have higher leaf areas or higher extinction coefficients. Total layer APAR should be unaffected.
In this study it was assumed that there is only minor variability in k H for a given species, and a single k H value was used for each species. That is, it is assumed that leaf angle distributions or LAD or L A /S A do not change enough with age, resource availability or climatic conditions to significantly influence k H , and instead these variables influence tree APAR via changes in crown length, crown diameter or L A . This is based on the results of other studies that have found that LAD is less sensitive to spacing and resource availability than L A , crown length or crown width (Forrester et al. ; Dong ; Guisasola ) and that inter-specific variability in LAD tends to be greater than intra-specific variability (Ligot et al. [2014a]). However, there was considerable variability in k H in Stands 1 and 2 (Table 4).
In Stand 1 the variability in k H was probably an artifact of assuming a half-ellipsoidal crown shape, when the crowns were not quite half-ellipsoidal. By making this assumption, the LAD estimates for trees in unthinned plots were 0.67 m2 m−3 while those in thinned plots were 0.57 m2 m−3, however more detailed calculations that did not assume any crown shape resulted in no significant thinning effect on LAD (Forrester et al. ). It is not clear what caused the large variability in k H and LAD in Stand 2. This level of variability is also large in comparison to each of the other stands in this study. Stand 2 differs from the others in that it was examined for a whole rotation period, which spanned the age of 1 to 6 years. During this time the mean tree heights were between 2.5 to 24.4 m and the stand L increased to its peak at about age 3 years before stabilizing and finally declining towards the end of the rotation (le Maire et al. ). During this rotation period there may be significant changes in leaf display, however, there was no clear relationship between age and k H for either species.
It may be hard to predict APAR in stands containing species with such variable crowns using any tree- or stand-level model that does not allow individuals (or cohorts) of a given species to vary in terms of k H , LAD or variables that describe the within-crown architecture. It is worth noting that while there was clearly some variability in k H in the five stands examined in this study, this variability, in terms of standard deviations, was only about half that for typical (non-uniform canopy) extinction coefficients, k (Table 4). This is because the crowns of trees within a canopy do not fit together perfectly, which results in some empty space between individual tree crowns and this space will influence the estimates of k. The variability in k H is lower because in a homogeneous canopy there are no spaces and the k H is only influenced by the crown architecture and leaf characteristics.
It is important to note, that when calculating k H values using tree-level models, it is critical to use models that do not require all trees of a given species to have the same within-canopy k or LAD. Maestra is an example of a suitable model for this purpose because it does not assume any particular k and even though it allows the vertical and horizontal shapes of the LAD distribution to be defined, the absolute values of the LAD at any given height can vary between individual trees in response to their leaf area and crown sizes.
In most of the stands the trees were roughly evenly spaced. However, in Stand 3 there was a very uneven distribution such that the H. odorata trees were planted within a 22-m wide circular gap within an Acacia hybrid plantation (Dong ). This spatial distribution is taken into account by the detailed tree-level model but not by the stand-level model. As a result the stand-level model assumes that the Acacia hybrid trees are evenly spaced, which would increase APAR per tree because there is less shading from other Acacia hybrid trees. However, in reality Acacia hybrid trees are grouped (and are all outside the gap) so the stand-level model overestimates their APAR (Figure 3) because it underestimates the shading effect. Despite this, the overestimate was minor and if the model is to be used for these designs the bias could be determined and accounted for.
Another simplification in terms of stand structure is that the stand-level model ignores slopes and aspects, whereas the detailed tree-level model takes these into account. At Stands 4 and 5 there were some plots with very steep slopes of > 50% and sometimes > 100% (Table 1). However, the slope and aspect do not appear to have any influence on the f estimates, which is consistent with other studies that have suggested that crown architecture and canopy structure are more important determinants of f than slope (Courbaud et al. ; Duursma et al. ). This may be because if a tree gains an advantage by being upslope of another tree, it is disadvantaged to a similar degree by being downslope of other trees.
The stand-level model provided adequate predictions of APAR for a wide range of crown architectures, species compositions, species proportions and stand densities. It uses two empirical parameters to account for the vertical and horizontal heterogeneity within the canopy and the model could be improved further by improving this part of the model. It can be parameterised with species-specific information about mean crown length, diameter, height and leaf area as well as extinction coefficients for a simulated homogeneous canopy. These extinction coefficients can be estimated using the mean crown length, diameter, height and leaf area information to run a tree-level light model. It should be noted that the stand-level model did not perform well in one of the five stands examined because of the high variability in k H within that stand. The prediction of APAR in such circumstances may require (tree- or stand-level) models that allow different individuals or cohorts of the same species to have different k H , LAD or analogous parameters. The stand-level model could be used to examine light dynamics in complex canopies and in stand-level growth models.
where ϕ is an empirical parameter that depends on the mean solar zenith angle and therefore considers latitude and season. Duursma and Mäkelä () and Forrester () calculated ϕ using empirical equations. However, the equation used by Duursma and Mäkelä () was developed for a single k H and while the equation used by Forrester () included a range of k H , it approached an asymptote and preliminary analyses showed that it could not provide adequate estimates of ϕ for some crown architectures and was occasionally not calculable for others due to its asymptotic form.
This study was part of the Lin2Value project (project number 033 L049) supported by the Federal Ministry of Education and Research (BMBF, Bundesministerium für Bildung und Forschung). Thanks to B. Medlyn and R. Duursma for maintaining the Maestra model. We are also grateful to those who established and maintain the experiments. Hancock Victorian Plantations Pty. Ltd. provided the site for Stand 1 and established the plantation. Stand 2 measurements and model parameterisation were carried out within the ANR (Agence Nationale de la Recherche) SYSTERRA program, ANR-2010-STRA-004 (Intens&fix). N. V. Phan and N. V. Minh provided the site for Stand 3 and established the plantation. We also thank Director An’guo Fan, Mr. Bailing Ding and Miss Yue’e Chu from the Shitai Forest Bureau for useful advice and assistance when organising the fieldwork at Stand 4, and also Mr. Xiaozhu Wang and Mr. Hongxing Ruan for fieldwork support. Plots in Stand 5 are maintained and monitored by the Forest Research Institute of Baden-Württemberg and the Forest Districts where the field sites are located. Thanks to Dr. Lutz Fehrmann, who provided useful comments that improved the manuscript and also helped with data collection and data processing for Stand 4. Lastly, we would like to thank two anonymous reviewers who provided useful comments that improved manuscript.
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