Pugnaire F.I. Valladares F. Functional Plant Ecology

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between gas molecules leaving and entering stomata (Leuning 1983) are also considered, and

is expressed as

¼ g

þ a

 A= (c

 G) (1 þ D

)½(21:19)

with a

and D

the empirical coefficients, G the CO

compensation point, and g

the stomatal

conductance when A approaches zero; D

and c

are humidity deficit and CO

concentration

at the leaf surface, respectively. Tuzet et al. (2003) further link stomatal conductance to leaf

water potential, which in turn is linked to soil water potential in the rooting zone.

Uncoupled Models

In uncoupled models of stomatal conductance, factors considered to influence g

include

incident PFD, leaf temperature, water vapor mole fraction difference, air humidity, leaf water

potential, and soil water potential (Jarvis 1976, Whitehead et al. 1981, Caldwell et al. 1986,

Jones and Higgs 1989, Beyschlag et al. 1990, Lloyd 1991, Ryel et al. 1993, 2002). Regression

equations often relate g

to one or more of these environmental factors. Uncoupled conduct-

ance models can be easier to parameterize than coupled models, and often give good

correspondence to measured data.

WHOLE-PLANT AND CANOPY MODELS

Models for light attenuation through canopies and interception by foliage are linked with

single-leaf photosynthesis models and form the basis of whole-plant and canopy photosyn-

thesis models. Although other environmental factors affecting photosynthesis throughout the

canopy can be included in model developments, modeling of canopy light relations is the most

important aspect of the canopy photosynthesis models. In this section, we illustrate the

approach commonly used to model light attenuation through plant canopies. Models for

attenuation and interception of direct beam, sky diffuse, and leaf reflected and transmitted

diffuse within canopies with varying complexity are presented, with incident PFD calculated

for both sunlit and shaded leaves.

Uniform Monotypic Plant Canopies

Models for single-species plant communities with relatively homogeneous distribution of

foliage are constructed by dividing the canopy into layers of relatively homogeneous foliage

density and orientation, and interception of radiation is calculated for points at the center of

these layers (Figure 21.2). A random distribution of foliage elements is typically assumed in

these models (but see, e.g., Caldwell et al. 1986). The thickness of layers is determined by the

foliage density distribution in the canopy, and ideally layers have leaf-area index less than

0:5m

foliage area m

2

ground area to facilitate attenuation of leaf-diffuse (Norman 1979).

Foliage areas of leaves and stems (or branches) are often separated as they typically have

different optical properties, photosynthetic rates, and orientations. Leaf inclination, which is

the angle of the major axis of the foliage element from horizontal, and azimuth angle, which

is the directional alignment, comprise the components of foliage orientation. Leaf inclination

is most simply defined as a constant for a layer, but can have a defined distribution (Norman

1979, Campbell 1986). Azimuth orientation of foliage elements may be considered random or

nonrandom in distribution (Lemeur 1973, Caldwell et al. 1986). The model development

presented here considers constant inclination and random azimuth orientation of foliage in

each layer, which is applicable to most uniform canopies.

With randomly distributed foliage within a layer, the penetration of direct beam PFD

declines exponentially with passage through increasing amounts of foliage. The relative area

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640 Functional Plant Ecology

or fraction of sunlit foliage (P

) for the sample point within in layer i (bottom layer ¼1, top

layer ¼n)is

¼ exp

m¼i

þ S

, (21:20)

where L

and S

are leaf and stem densities (m

3

) in layer i, respectively, and l

is the path

length (m) of PFD through layer i. Kl

and Ks

are the light extinction coefficients for leaves

and stems, respectively, and for fixed leaf inclination (a

) and random azimuth can be

calculated (Duncan et al. 1967) for inclination of sun above horizon (b)as

¼ cos a

sin b (21:21)

for a

less than b, and otherwise as

sin a

cos b sin u

þ 1 



cos a

sin b, (21:22)

where u

is the angle (0–908) that satisfies cos u

¼ cot a

tan b.

The flux of direct beam PFD does not decline with attenuation through foliage (Figure

21.7), but the flux incident on foliage is a function of the orientation of the leaf surface relative

to the sun. Incident flux of PFD can hit either the upper or the lower surface of the leaf

surface, and the flux for the upper leaf surface can be expressed (Burt and Luther 1979) as

¼ B( cos a

sin b  sin a

cos b cos d

)(21:23)

and for the lower surface as

¼ B( cos (180  a

) sin b  sin (180  a

) cos b cos d

), (21:24)

FIGURE 21.7 Attenuation of direct beam (left) and sky diffuse (right) radiation through foliage. The

fraction of foliage illuminated by direct beam sunlight declines with interception by foliage, but

the radiation flux does not change in the sunlit portion. In contrast, radiation flux for sky diffuse

(and for scattered diffuse) declines with passage through foliage, but all foliage receives similar flux.

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Canopy Photosynthesis Modeling 641

where B is the PFD flux on a surface normal to the solar beam and d is the azimuth angle of

the major axis of the leaf surface relative to the sun.

Attenuation of sky-diffuse PFD through the canopy is considered from the midpoint of

concentric bands of equal width or area (skybands) that divide the hemisphere of sky above

the canopy. Unlike direct beam PFD flux, the flux of sky diffuse PFD declines with passage

through increasing amounts of foliage (Figure 21.7), and can be expressed as

i,w

¼ D

sky

exp

m¼i

þ S

, (21:25)

where w is the skyband (e.g., w ¼ 1, 2,..., 9 for inclination bands ¼ 08–108,108–208,...,

808–908), D

sky

is the sky diffuse PFD flux on a horizontal surface, and A

(view factor, see

Duncan et al. 1967 for formulations) is the fractional portion of the sky hemisphere within

band w.

Diffuse radiation reflected and transmitted by foliage is difficult to accurately portray in

canopy models. Although leaf optical properties, foliage density, and characteristics of

adjacent foliage contribute to scattering, the process of scattering is complex. Simplistic

approaches have been proposed with one approach to assume that radiation striking the

upper foliage surface reflects upward or transmits downward, with the opposite occurring for

radiation striking the lower surface. Using this approach (see Norman 1979, Ryel et al. 1990),

downward PFD flux from the midpoint of layer i would be

¼t

(Qu

þ Du

) þ r

(Ql

þ Dl

) þ Tu

i1

þ Td

iþ1

þ Td

iþ1

(1  L

 S

(21:26)

whereas the upward flux would be

¼r

(Qu

þ Du

) þ t

(Ql

þ Dl

) þ Td

iþ1

þ Tu

i1

þTu

i1

(1  L

 S

(21:27)

where r

and t

are leaf reflectance and transmittance for PFD, respectively.

Sunlit leaves have incident PFD fluxes from direct beam, sky diffuse, and reflected and

transmitted diffuse. The total PFD incident on both sides of a sunlit leaf in layer i is

¼ Qu

þ Ql

þ Du

þ Dl

þ Tu

i1

þ Td

iþ1

: (21:28)

Incident fluxes for shaded leaves are calculated similarly, but without direct beam compon-

ents Qu

and Ql

Uniform Multispecies Plant Canopies

The model structure for uniform multispecies canopies is a generalization of the single-species

models. Equations describing interception of PFD can be extended from those for monotypic

plant stands. The sunlit fraction of foliage in layer i for species x is simply

x,i

¼ exp

m¼i

y¼x

y,m

þ S

y,m

)

, (21:29)

where N is the number of species in the canopy and corresponds to Equation 21.20. Equation

21.25 through Equation 21.27 are similarly extended to apply to multiple species (Ryel

et al. 1990).

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642 Functional Plant Ecology

Inhomogeneous Canopies

The cylinder model of Ryel et al. (1993) is used as an example of models for complex canopy

structure as it contains the spectrum of complexity found in these models.

Individual plants within the canopy are fitted to concentric cylinders and horizontal

layers of relatively uniform foliage density and orientation (Figure 21.4), a process analog-

ous to the layer divisions in uniform canopies. A three-dimensional array of points

(Figure 21.4) is used to sample the plant canopy for calculation of light interception and

gas exchange. Consistent calculations of flux rates throughout the course of a day may

require 1000 or more points within an individual plant crown (Ryel et al. 1993, Falge et al.

1997). Values for points within the canopy are appropriately weighted, averaged, and

summed to generate whole-plant flux rates. Simulations may be conducted for all plants

within the canopy (e.g., Grace et al. 1987b), for individuals representing plants of similar

structure and stature (e.g., Falge et al. 1997), or for one individual when all plants are

assumed to be similar in structure and relatively uniformly distributed (e.g., Beyschlag et al.

1994, Ryel et al. 1994). Whole canopy rates can be calculated as the sum of individual

plants and expressed per ground area if desired for comparisons between vegetation

communities or plots.

Many of the model equations are analogous to the homogeneous canopy model and are

simply generalizations of those formulations. The fraction of sunlit foliage (analogous to

Equation 21.20) is calculated for each sample point k in plant x as

x,k

¼ exp

y¼1

y,l

i¼1

y,c

m¼1

y,i,m

þ S

y,i,m



y,i,m

! !

, (21:30)

where N

is the total number of plants in the canopy, and n

y,l

and n

y,c

are the number of

layers and subcylinders in plant individual y, respectively. The flux of sky diffuse PFD is

calculated for both sky bands, and azimuth directions to account for differential placement

of neighboring plants. Sky diffuse flux (analogous equation to Equation 21.25) is calcu-

lated as

x,p,w,a

¼ D

sky

w,a

exp

y¼1

i¼1

m¼1

y,i,m

þ S

y,i,m



y,i,m

! !

, (21:31)

where A

w,a

is the view factor including azimuth orientation (a) of skyband subsections.

As with the uniform canopy model, reflected and transmitted diffuse within diverse

canopies is difficult to model. One approach is to calculate average reflected and transmitted

diffuse for horizontal layers of points within the canopy using formulations similar

to Equation 21.26 and Equation 21.27 (Norman and Welles 1983, Ryel et al. 1993). Cescatti

(1997) used a similar approach, but used a weighted average for the flux at each point within

a layer.

PARAMETERIZATION

A lengthy treatise of data-collection methods is too voluminous for this chapter, but an

overview is provided. The reader is referred to the cited literature for further details. Esti-

mating model parameters is often challenging and time consuming because of the difficulty

involved in measuring flux rates and structural parameters. Since parameter estimates can

affect model outputs and study conclusions, parameter estimation must be done carefully.

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Canopy Photosynthesis Modeling 643

Assessing model sensitivity to parameter estimates (Forrester 1961, Steinhorst et al. 1978)

should be conducted to evaluate the robustness of model outputs.

Model Parameters for Single-Leaf Photosynthesis

Parameters for single-leaf photosynthesis and stomatal models are often derived from gas

exchange measurements (e.g., Harley et al. 1992, Falge et al. 1997). For the model of

Farquhar et al. (1980), the initial slope of the A versus c

relationship characterizes the

activity status of Rubisco, and A at saturating light and CO

relates to the maximum RuBP

regeneration rate of the Calvin cycle (von Caemmerer and Farquhar 1981). The initial slope

of the relationship A versus incident PFD at saturating CO

is an estimate of the maximum

quantum-use efficiency of the light reaction of photosynthesis, and can also be measured with

chlorophyll fluorescence (Schreiber et al. 1994). Biochemical parameters are often obtained

from the literature, but some may vary considerably within or among species (Evans and

Seemann 1984, Keys 1986).

Canopy Structural Parameters

Both destructive and nondestructive methods are used to estimate density and distribution of

foliage. Destructive methods often involve harvesting foliage from portions of the plant

canopy (e.g., Caldwell et al. 1986, Beyschlag et al. 1994) with foliage area estimated with

analytical devices such as the leaf-area meter. Within canopy light measurements are the most

commonly used nondestructive methods for estimating canopy structure (Norman and

Campbell 1989), and are often combined with inverted light extinction equations to estimate

foliage area in many types of canopies (Lang et al. 1985, Perry et al. 1988, Walker et al. 1988).

This estimation procedure has been automated with the LAI-2000 plant canopy analyzer

(LI-COR, Lincoln, NE, USA) that efficiently calculates foliage area and average leaf inclin-

ation despite some limitations (Gower and Norman 1991, Deblonde et al. 1994, Stenberg et al.

1994). High-contrast fish-eye photography has also been used to estimate foliage area

(Anderson 1971, Bonhomme and Chartier 1972). An additional nondestructive within-

canopy method, the linear probe or inclined-point-quadrat method uses the frequency of

contacts of foliage obtained from the repeated movement of a rod oriented at a fixed angle

through the canopy (Warren Wilson 1960, Caldwell et al. 1983). Estimates of leaf area from

outside the canopy are most commonly conducted using the normalized difference vegetation

index (NDVI). This nondestructive method using remotely sensed images of visible and near

infrared reflectances obtained from aircraft or satellite (Field 1991, Wang et al. 2005) is

effective at the community level where more detailed measurements are time consuming.

Ground truthing using the above methods, however, is necessary to calibrate the NDVI to

actual LAI.

Foliage orientation can be measured in situ (Caldwell et al. 1986), with the LAI-2000

analyzer or by linear probe (Warren Wilson 1960, 1963). Leaf reflectance and transmittance

of PFD, total shortwave and longwave radiation are often measured with a spectroradio-

meter (Brunner and Eller 1977) or obtained from suitable literature values (e.g., Sinclair and

Thomas 1970, Ross 1975).

MODEL VALIDATION

Model validation is necessary to assess whether model simulations mimic the dynamics

of the modeled system. Model validation is conducted by comparing model output

with independently measured variables (Forrester 1961, Innis 1974) to evaluate model

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644 Functional Plant Ecology

performance. Comparisons are often made graphically, but statistical evaluation

methods have been proposed (Kitanidis and Bras 1980). Simulations extrapolating beyond

the range of validation run the risk of producing faulty results, but are often performed to

investigate system dynamics in response to hypothetical conditions. However, validations are

conducted under a range of conditions to increase the confidence of extrapolations using the

model.

Diurnal course measurements of single-leaf net photosynthesis, transpiration, intercel-

lular CO

, and stomatal conductance for H

O are usually compared with simulated results

for photosynthesis and conductance models (Figure 21.8; see also Tenhunen et al. 1987,

Ryel et al. 1993). Validation of light interception models are conducted by comparing

model outputs to light sensor measurements taken within the canopy (Figure 21.9; see also

Norman and Welles 1983, Beyschlag et al. 1994). Model predictions of whole-plant or

canopy gas fluxes can be compared with several types of field measurements for model

validation including whole-plant gas exchange measured within large gas-exchange cuvettes

(Figure 21.10); sap-flow measurements (Cermak et al. 1973, Ishida et al. 1991, Falge et al.

2000) of a suitable sample of plants within a canopy (Dye and Olbrich 1993), and eddy-

correlation measurements (Baldocchi et al. 1986, Verma 1990, Dugas et al. 1991) of gases

within and above the canopy (Baldocchi and Harley 1995, Aber et al. 1996).

Time (h)

Wheat

Wild oat

8.00

−1

0.20

(Pa)

G (mol m

−2

−1

)

A (µmol m

−2

−1

)

0.40

0.60

10.00 12.00 14.00 16.00

8.00 10.00 12.00 14.00 16.00 18.00

FIGURE 21.8 Model validation conducted for single-leaf photosynthesis model of Tenhunen et al.

(1987) for wheat (Triticum aestirum) and wild oat (Avena fatua). Model output for the course of a single

day was compared with data measured by gas exchange for net photosynthesis (A), stomatal conduct-

ance (G), and intercellular CO

). (From Beyschlag, W., Barnes, P.W., Ryel, R.J., Caldwell, M.M.,

and Flint, S.D., Oecologia, 82, 374, 1990. With permission.)

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Canopy Photosynthesis Modeling 645

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 0.2 0.3 0.4 0.5

Fraction of sensors sunlit

Predicted fraction of

sunlit area

0.6 0.7 0.8 0.9 1.0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

200 400 600 800 1000

Intercepted PFD, measured

(µmol m

–2

–1

)

Intercepted PFD, predicted

(µmol m

–2

–1

)

1200 1400 1600 1800 2000

200

400

600

800

1000

1200

1400

1600

1800

2000

200 400 600 800 1000

Intercepted PFD, measured

(µmol m

–2

–1

)

Intercepted PFD, predicted

(µmol m

–2

–1

)

1200 1400 1600 1800 2000

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 0.2 0.3 0.4 0.5

Fraction of sensors sunlit

Predicted fraction of

sunlit area

0.6 0.7 0.8 0.9 1.0

FIGURE 21.9 Model validation for light flux in uniform canopy as attenuated through mixtures of

wheat and wild oat (upper left) and through monocultures of wheat or wild oat (lower left). Measured

data were collected with photodiodes mounted on a long stick inserted into the canopy at defined

canopy depths. Fluxes below 400 mmol m

1

are for shaded leaves and sensors. The fraction of foliage

sunlit was also compared between model and measured data for mixtures (upper right) and monocul-

tures (lower right). Measured flux data was bimodal in nature and allowed for estimating the portion of

fully sunlit sensors. Data points from measurements made deep in the canopy where individual photo-

diodes may not have been fully sunlit are circled. (From Ryel, R.J., Barnes, P.W., Beyschlag, W.,

Caldwell, M.M., and Flint, S.D., Oecologia, 82, 304, 1990. With permission.)

A. desertorum

(a)

Time (h)

12 16 8 12 16 8 12 16 8 12 16

(b) (c) (d)

P. spicata

(Pa)

(mmol mol

−1

)

WUE

tuss

(mmol m

−2

−1

)

tuss

( mmol m

−2

−1

)

tuss

FIGURE 21.10 Validation for cylinder model of Ryel et al. (1993) conducted for whole-plant net

photosynthesis (A), transpiration (E), water-use efficiency (WUE), and average intercellular CO

con-

centration (C

) for two tussocks of crested wheat grass (Agropyron desertorum a,b), and two of blue-bunch

wheat grass (Pseudoroegneria spicata c,d). Measured data were from whole-plant gas-exchange cuvettes.

(From Ryel, R.J., Beyschlag, W., and Caldwell, M.M., Functional Ecology, 7, 115, 1993. With permission.)

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646 Functional Plant Ecology

ACKNOWLEDGMENTS

Part of this work has been funded by the Deutsche Forschungsgemeinschaft, Bonn (SFB 251,

University of Wu

rzburg) and the Utah Agricultural Experiment Station. We would also like

to thank Mr. H. Weigel, Bielefeld for the artwork in Figure 21.2 through Figure 21.4 and

Figure 21.7.

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