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# Using a stand-level model to predict light absorption in stands with vertically and horizontally heterogeneous canopies

*Forest Ecosystems***volume 1**, Article number: 17 (2014)

## Abstract

### Background

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.

### Methods

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.

### Results

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.

### Conclusion

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.

## 1Background

The absorption of photosynthetically active radiation (APAR) by trees is an important determinant of their growth and accurate estimates of APAR are often critical for process-based growth models. The fraction of PAR (*f*) that is absorbed by homogeneous canopies can be estimated using the Lambert-Beer equation (Monsi and Saeki [1953]; or in English as Monsi and Saeki [2005])

where *k* is the light extinction coefficient for the considered growth period, and *L* is the leaf area index (m^{2} 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 [1983]; Oker-Blom et al. [1989]; Wang and Jarvis [1990]; Bartelink [1998]; Brunner [1998]; Canham et al. [1999]; Bartelink [1998]; Courbaud et al. [2003]; Gersonde et al. [2004]; Abraha and Savage [2010]; 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 [1998]; Sinoquet et al. [2000]; Parveaud et al. [2008]).

Some of these models, such as the Maestra (or Maestro) model (Grace et al. [1987]; Wang and Jarvis [1990]; Medlyn [2004]), have been shown to provide realistic predictions of absorbed light in mixed-species stands (le Maire et al. [2013]; Charbonnier et al. [2013]). 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 [2007]; Charbonnier et al. [2013]; Forrester [2014]). 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 ([2014]), for a wide range of canopy structures. The model used by Forrester ([2014]) 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. [1987]; Wang and Jarvis [1990]; Medlyn [2004]), 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.

## 2Methods

### 2.1 Description of study sites

The stands in this study were composed of monocultures or mixtures of up to four species and all but one of the five data sets contained different tree spacing treatments, resulting in different vertical and horizontal complexity of the canopies (Table 1).

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

The stand-level light model is based on the model described by Forrester ([2014]) and calculates APAR using monthly PAR, and species (or age class, or dominance class)-specific information including the number of trees per ha and mean crown characteristics such as live-crown length (LCL, m), crown diameter (m), tree height (m), tree leaf area (*L*
_{
A
}, m^{2}), and crown shapes, which are approximated by simple geometric forms like ellipsoids, half-ellipsoids, spheres or cones, depending on the respective species. Table 2 contains a list of parameters.

When a stand contains more than one species the canopy is divided into horizontal layers that do not overlap vertically (Figure 1). A given layer may consist of a single species, or several species whose crowns do overlap vertically. A given species is only present in a single layer and, as shown in Figure 1, the crowns are not split between different layers. However, an exception to this is when there are two age classes of a given species between which there is no vertical overlap. In that case, each age class is treated as a different species.

Equation 2 is used to calculate the APAR by each layer, starting with the highest layer, and then reducing the available PAR for each layer by the APAR of all higher layers.

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. [2013]; Bryars et al. [2013]). Duursma and Mäkelä ([2007]) 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, m^{2} m^{−3}), leaf angle distribution, leaf reflectance and leaf transmittance. In such stands the *k*
_{
H
} is independent of trees per ha.

The ${\mathrm{\lambda}}_{{v}_{i}}$ 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 ${\mathrm{\lambda}}_{{h}_{j}}$ 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 ([2007]) and used in the stand-level mixed-species model by Forrester ([2014]). 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.

can be calculated for any species and stand structure using Equation 3 (Forrester [2014]).

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

Adjusted

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 $\frac{{k}_{H,i}{L}_{i}}{{\displaystyle {\sum}_{i=1}^{n}}{k}_{H,i}{L}_{i}}$ 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, [2014]).

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 (m^{3}) of all crowns within a given layer within one hectare divided by the total volume of that layer (m^{3}) over one hectare. The total volume of a layer is (height of the tallest species – minimum *H*
_{
b
}) × 100 m × 100 m.

The *v*
_{
frac
} parameter is used to quantify canopy openness and therefore the potential inter-tree shading. Inter-tree shading increases as the mean midday zenith angle increases due to season and latitude. The zenith angle used to calculate *λ*
_{
h
} is an adjusted value, *z*
_{
adj
}, because while mean midday solar zenith angle is a sine-shaped function of Julian day, at latitudes of about < 23°, the sine shape is distorted when solar zenith angles decline to 0 and increase again, instead of continuing to decline below zero, which would maintain the sine shape (see Appendix B). The *z*
_{
adj
} allows that part of the curve to be negative (× −1). The *λ*
_{
h
} is calculated using two empirical equations (5a and 5b) because *z*
_{
adj
} does not influence *λ*
_{
h
} until it is greater than 30°. These equations are general and can be applied to any species and stand structure.

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 ([2014]). 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*.

Weighted average of ${k}_{H}\frac{{L}_{A}}{{S}_{A}}$ in Equation 5 =

The monospecific data contained simulated monospecific stands with crown characteristics that varied in terms of LAD (0.27 to 2.62 m^{2} m^{−3}), monthly *k*
_{
H
} (0.04 to 1.48), live-crown lengths (2 to 18 m), *L*
_{
A
} (10 to 390 m^{2}), 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 ([1983]). Transmission and absorption of diffuse radiation is modelled using the method of Norman ([1979]), 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. [2013]). 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. ([2013]) 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. [2013]). 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

Previous studies have already used Maestra to calculate APAR for four of the data sets used in this study, and in the other stand the crown architecture has been examined in detail (Table 1). Therefore, the same Maestra parameters were used in this study (Table 3). Input data to the Maestra model included individual tree *x* and *y* coordinates, and individual tree characteristics including crown diameter, LCL and *L*
_{
A
}. These were estimated for each tree or were predicted using site- and species-specific allometric equations described in the studies cited in Table 1. One exception was Stand 5, where the *L*
_{
A
} allometric equations were not specific to the sites. As APAR of trees within the plot could still be influenced by trees beyond the buffers, which were up to 10 m wide (Table 1), a 25-m wide buffer with mean stand characteristics was created around the outside of the plot to avoid potential estimation bias. This buffer had the mean characteristics of the plot. Daily PAR data was estimated from solar radiation measured at the specific sites or calculated from other climatic data collected from the sites or nearby. Maestra parameters that differed between stands and species are provided in Table 3. Other parameters were constant for all stands. These included the number of individual tree crown layers to integrate over: nolay (6), pplay (12), nzen (5), naz (11); the distribution of diffuse radiation incident from the sky (difsky; 0); the number of time steps per day (khrsperday) was 24; crown shape was a half-ellipse (cshape = ELIP); there was only one age considered for the beta distribution of LAD; there was no clumping of foliage into shoots. Further details about the meaning and significance of these parameters/settings can be found in the Maestra manual (see Medlyn [2004]).

The PAR data used for the stand-level model were monthly totals of the daily data used for Maestra. The stand-level model was parameterised using the *k*
_{
H
}, mean height, mean LCL, mean crown diameter and mean *L*
_{
A
} for all trees of a given species in each plot. The actual number of trees per ha was also used. That is, plot specific means were used. For each species, *k*
_{
H
} was estimated using Maestra and Equation 7, where *f*
_{
H
} is the fraction of PAR that is absorbed by a homogeneous canopy

The *k*
_{
H
} used for each species is shown in Table 4 and were calculated using Maestra. As explained above, *k*
_{
H
} is calculated for a given species by simulating a stand with a homogeneous canopy such that all trees have the same height, live-crown length, LAD, leaf angle distribution, and leaf optical properties, and have box-shaped crowns that fit perfectly together (no space between crowns). The varying *k*
_{
H
} values for the *Acacia* hybrid (Stand 3) result from the different LAD and leaf angles that were found in each of the experiments. In the same stand the high *k*
_{
H
} for *H. odorata* is related to the relatively high LAD and low to medium leaf inclination angle (Tables 1 and 3).

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

The predictions of *f* from the stand-level model were compared with those from the Maestra model using criteria such as the relative average error (average bias, *e*%, Equation 8), the relative mean absolute error (MAE%, Equation 9), and the mean square error (MSE, Equation 10) (Janssen and Heuberger [1995]; Vanclay and Skovsgaard [1997]).

where *O* is the observed calculations from Maestra and *P* is the predicted values from the stand-level model, and $\tilde{O}$ and $\tilde{P}$ are the means.

The accuracy test described by Freese ([1960]) 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 [2013]).

The APAR predictions of the stand-level model based on Equation 2, were also compared with three other approaches. The first was to simply use Equations 7 and 11.

where ζ is the fraction of ground covered by the canopy. Both of these equations ignore any vertical heterogeneity within the canopy and inter-specific differences in shading abilities. Equation 7 also ignores horizontal heterogeneity in terms of gaps between trees, while Equation 11 takes this into account by considering the fraction of the ground that is actually covered by the canopy of that species. The second approach was to use Equation 12, which ignores vertical or horizontal heterogeneity (in terms of canopy gaps) but considers the inter-specific differences in shading abilities.

In Equation 12, instead of using the empirical parameter *λ*
_{
v
} to partition the PAR absorbed by a given canopy layer to each species within that layer, the contribution made by each of n species is weighted based on their *L* and *k* (Rimmington [1984]; Sinoquet and Bonhomme [1991]; Sinoquet et al. [2000]). The third approach was taken from some forest gap models, which assume that all of the leaf area of a species (or age-class) is at the top of its crown (Figure 2b; Bugmann [2001]), as opposed to the stand-level model of this study where the leaf area is distributed between the top and bottom of the crowns (Figure 2a; Equation 2). A potential problem with this gap model approach is shown in Figure 2. For example, species 1 is taller than species 2, but species 2 probably shades species 1 a lot more than vice versa. In the five stands in this study, all species had different heights and so for this third approach, each species was assumed to occupy its own canopy layer, even if it was only a few centimetres taller than another species, and Equation 2 was used without the *λ*
_{
v
} parameter.

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 ([2014]) 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 m^{2} m^{−3}), monthly *k*
_{
H
} (0.04 to 1.60), live-crown lengths (2 to 18 m), *L*
_{
A
} (10 to 390 m^{2}), 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

The stand-level light absorption model provided similar predictions of *f* or APAR to the much more detailed tree-level model (Maestra) in four of the five stands (Figure 3, Table 5). In contrast, *f* or APAR was not predicted well in Stand 2 where the crown architectures (e.g. *k*
_{
H
}) were more variable and it appears unlikely that *f*
_{
i
} could be predicted in such stands using tree- or stand-level models that do not allow different trees of the same species to have different *k*
_{
H
} or other architectural variables such as LAD.

The stand-level model based on Equation 2 and the model from which it was developed described in Forrester ([2014]) both consider the horizontal and vertical heterogeneity within the canopy, and gave very similar predictions of *f*
_{
i
} (Figure 4). In terms of horizontal heterogeneity, as *L* declines and canopy openness increases, the total stand APAR declines but the APAR per tree increases because there is less inter-tree shading. The effect of openness or canopy gaps was taken into account by considering the zenith angle of the sun, canopy openness and crown architecture (Equation 5), which are all summarised using the *λ*
_{
h
} parameter (Equation 2). In contrast, the openness could not be sufficiently accounted for by only considering the reduction in *L* (e.g. Equation 1 or 7; Figures 5a and 6a) or the fraction of ground area covered by the canopy by using Equation 11 (Figure 5b). Equation 11 was also applied to mixtures and resulted in very biased predictions of APAR that were similar to those for monocultures in Figure 5b (data not shown).

For some species, the use of Equation 12 to predict APAR was as good as, or even better, than Equation 2 (Table 5), even though it does not consider the vertical or horizontal heterogeneity within the canopy layers. Nevertheless, Equation 2 outperformed Equation 12 more often than not, and APAR predictions from Equation 12 were nearly always more variable, as indicated by the higher MSE in Table 5, Figure 7 and also Figure 3e compared with Figure 6b.

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. [2000]). 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.

The model is more sensitive to changes in *k*
_{
H
} and *L*
_{
A
} than to changes in crown length and width (Figure 8). This sensitivity depends on the mean midday solar zenith angle and therefore varies between months, especially at higher latitudes (data not shown). For a given species and month, the sensitivity tends to increase with declining *L* because at higher stand densities (or *L*) there is already a lot of inter-tree shading and changes in crown parameters have a smaller influence on APAR.

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. [2013]; Dong [2014]; Guisasola [2014]) 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 m^{2} m^{−3} while those in thinned plots were 0.57 m^{2} m^{−3}, however more detailed calculations that did not assume any crown shape resulted in no significant thinning effect on LAD (Forrester et al. [2013]). 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. [2013]). 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 [2014]). 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. [2003]; Duursma et al. [2003]). 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.

## 4Conclusions

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.

## Appendix A

The stand-level model of this study is based on a model described by Forrester ([2014]) where the fraction of PAR (*f*) that is intercepted by species *i* within a given layer is calculated using Equation A1.

where *k*
_{
eff,i
} is an effective extinction coefficient derived by Duursma and Mäkelä ([2007]) and is given as

where *ϕ* is an empirical parameter that depends on the mean solar zenith angle and therefore considers latitude and season. Duursma and Mäkelä ([2007]) and Forrester ([2014]) calculated *ϕ* using empirical equations. However, the equation used by Duursma and Mäkelä ([2007]) was developed for a single *k*
_{
H
} and while the equation used by Forrester ([2014]) 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.

## Appendix B

The relationship between mean midday solar zenith angle and the day of the year for four different latitudes is shown in Figure 9a (southern hemisphere) and Figure 9c (northern hemisphere). At latitudes between about −23° and 23° the sine-shaped function is distorted because the solar zenith angle is never negative. However, a smooth sine-shaped function of solar zenith angles is easier to use when predicting the effect of solar zenith angle on the horizontal heterogeneity parameter *λ*
_{
h
} in Equation 5. Therefore an adjusted zenith angle *z*
_{
adj
} is used, which allows the curve to be negative (× − 1) to retain the sine shape. To determine when the curve is to be multiplied by −1 for a given latitude, it is necessary to calculate the two days of the year where the Zenith angle = 0. The first time during the year when Zenith angle = 0 is given by Equation B1 and the second time during the year when Zenith angle = 0 is given by Equation B2. Note that when developing Equation 5, where *z*
_{
adj
} is used, Forrester ([2014]) incorrectly calculated the solar zenith angles such that those for the northern hemisphere were used for the southern hemisphere and vice versa and this reversal was retained in this study in order to retain Equation 5.

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## Acknowledgements

This study was part of the Lin^{2}Value 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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The authors declare that they have no competing interests.

### Authors’ contributions

All authors carried out the field work and processed the data. DIF developed the stand-level model, ran the light absorption models, analysed the data and drafted the manuscript. All authors jointly revised the manuscript. All authors read and approved the final manuscript.

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### Keywords

- Complex forests
- Mixed-species
- Stand structure
- Extinction coefficient
- Lambert-Beer law
- Light absorption