Tree Physiology 20, 333–345
© 2000 Heron Publishing—Victoria, Canada
Modeling the influence of temperature on monthly gross primary
productivity of sugar maple stands
FRÉDÉRIC RAULIER, PIERRE Y. BERNIER and CHHUN-HUOR UNG
Natural Resources Canada, Canadian Forest Service, Laurentian Forestry Centre, 1055 du P.E.P.S., P.O. Box 3800, Sainte-Foy, Quebec, G1V 4C7,
Canada
Received October 22, 1998
Summary A bottom-up and a top-down model were used to
estimate the effect of temperature on monthly gross primary
productivity (GPP) of sugar maple (Acer saccharum Marsh.).
The bottom-up model computed canopy photosynthesis at an
hourly time step from detailed physiological sub-models of
leaf photosynthesis and stomatal conductance. Leaf mass per
area was used as a covariable to integrate photosynthesis
through the canopy. The top-down model used a radiation-use
efficiency coefficient to relate canopy gross photosynthesis to
absorbed photosynthetically active radiation at a monthly time
step. The parameters of the top-down model were estimated
from simulations with the bottom-up model. Forty single-year
simulations were made using records of daily maximum and
minimum temperatures from weather stations selected within
the natural range of sugar maple in the province of Québec,
Canada. Leaf area index was randomly varied between 4 and
10. Within a broad range of values, temperature had a minor effect on predicted monthly canopy-level GPP and its contribution to explaining the variability of GPP was low, both through
its direct effect on photosynthetic processes (1.1%), and indirectly through the effect of relative humidity on stomatal
conductance (4.0%). This result was unchanged when key parameters relating photosynthesis to temperature and stomatal
conductance to atmospheric humidity were changed in the bottom-up model. An increase in time step from hourly to monthly
resulted in a downward shift in the optimum temperature range
for photosynthesis, from 30 °C for a leaf at saturating irradiance to 22 °C for the canopy at a monthly time scale.
Keywords: Acer saccharum, canopy photosynthesis, gross
primary productivity, leaf mass per area, radiation use efficiency, temperature.
Introduction
The linear relationship between net primary productivity
(NPP) and absorbed photosynthetically active radiation (ϕa )
provides a simple and robust approach for estimating forest
productivity by top-down models (Landsberg et al. 1996).
Such models lack the extreme sensitivity to inputs typical of
bottom-up approaches (Jarvis 1993), because their predictions
are constrained within an expected NPP range. Robustness is
an essential property of any model designed for operational
use (Battaglia and Sands 1998).
A major disadvantage of top-down models, however, is that
they are defined at spatial and temporal scales (canopy, month
or year) that differ greatly from those at which environmental
variables influence NPP (leaf, second or hour). Multiplier
functions yielding values between 0 and 1 are usually included
in top-down approaches, e.g., the radiation-use efficiency
(RUE) models, to account for major external effects on gross
primary productivity (GPP) and NPP (Landsberg 1986,
Runyon et al. 1994, McMurtrie et al. 1994, Landsberg and
Gower 1997). Their inclusion improves NPP predictions
(Runyon et al. 1994), but the shape of these functions and the
values chosen for their parameters are seldom justified in detail (e.g., McMurtrie et al. 1994, Waring et al. 1995).
Although some lumped-parameter RUE models take temperature into account (Running and Coughlan 1988, Aber et
al. 1996), most RUE models do not (McMurtrie et al. 1994,
Runyon et al. 1994, Waring et al. 1995, Landsberg and Waring
1997). It seems likely that temperature makes an important
contribution to explaining the variability in leaf photosynthesis and GPP (Farquhar et al. 1980, Harley and Baldocchi
1995, Leuning et al. 1995, Williams et al. 1996), both through
its direct effect on biochemical processes involved in photosynthesis and through its indirect effect on stomatal conductance through atmospheric humidity. It also seems likely that
the representation of the temperature–photosynthesis interaction is conditioned by the temporal resolution used in its representation. This is an important consideration, because most
RUE models operate at relatively coarse time steps of a month
or more.
The objective of this study was to quantify the effects of
temperature and atmospheric humidity on GPP at the monthly
scale, which is a common scale in RUE models. The effect of
temperature on respiration costs (leaf and other tree components) was not considered in this analysis. Because the effects
of temperature on GPP are usually difficult to separate from
the effects of atmospheric humidity (e.g., Hollinger et al.
1994, Leuning 1995), both effects were considered simultaneously. To do so, we calibrated a bottom-up GPP model from
net photosynthesis light response curves measured throughout
the canopy of a sugar maple stand (Acer saccharum Marsh.)
and used it to estimate the modifiers that should account for
334
RAULIER, BERNIER AND UNG
the influence of temperature and atmospheric humidity on
monthly RUE.
Collection of field data
Study site
Field data were collected in one stand dominated by mature
sugar maple (90% of basal area) in the province of Québec,
Canada. The site (46°26′30′′ N, 71°25′00′′ W, elevation
171 m) is located about 50 km south of Québec City, on the
Saint-Gilles-de-Beaurivage forest property of Daishowa Inc.
The site lies within the sugar maple–basswood–yellow birch
climatic domain, ecological region 2C (Thibault 1985), with
1660 to 1780 degree-days above 5 °C. The site is underlain by
a medium to fine sand deposit of alluvial origin. The stand
rests on a gentle rise in the land, above poorly drained, forested
wetlands. The stand originates from a clear-cut dating from the
1930s and is naturally regenerated. The stand is mostly
even-aged and forms a closed, homogeneous canopy with few
gaps. Dominant and co-dominant tree height ranges from
about 22 to 27 m. Live crown depth is about 5 m. Sugar maple
regeneration forms a nearly continuous single-leaf cover at a
height of about 50 cm.
In summer 1996, a 25-m canopy access tower was erected to
measure leaf gas exchange in canopies of two adjacent sugar
maple trees. A meteorological tower was also installed and
equipped with various sensors to monitor above- and below-canopy environmental variables. All sensors were read
once a minute and the hourly mean was recorded by a data logger.
Leaf gas exchange
Leaf gas exchange was measured in situ on 5 days during summer 1997 (July 8, 10, 21, August 27 and September 19), with
an LCA4 portable gas exchange measurement system (Analytical Development Corp., Hoddesdon, U.K.). Depending on
light conditions and the type of leaves selected, light response
measurements were performed with natural or artificial light.
We used neutral density filters to provide a range of low irradiances for the generation of light response curves. Target irradiances were at or near points of physiological interest
according to the recommendations of Hanson et al. (1987). We
aimed for irradiances at the compensation point, a point in the
steepest portion of the light response curve, one point nearer to
the zone of curvature, including saturation (above 1200 µmol
m –2 s –1) and total darkness (0 µmol m –2 s –1).
On each sampling day, light response curves were obtained
for three to seven leaves, depending on climatic conditions and
other measurements to be performed. The leaves were selected
from three canopy positions: direct sun exposure at the top of
the crown, an intermediate canopy position, and either the
base of the living crown or the maple regeneration layer on the
forest floor. All measurements were carried out between 1000
and 1500 h. A total of 24 response curves were obtained during the summer, at temperatures varying between 21 and 27 °C
(mean of 24.7 °C). Equilibration times during the field
measurements were variable, but were generally between 10
and 15 min. In accordance with the manufacturer’s recommendations, we used the stabilization of the internal CO2
concentration (ci ) as the criterion for the attainment of equilibrium. This variable is sensitive to both CO2 and H2O fluxes because it is computed from photosynthetic rates and stomatal
conductance values.
Temperature response curves were obtained in the field on
five sugar maple leaves chosen to cover the whole range of
leaf mass per unit area (LMA). For two sun leaves, net photosynthesis was measured at saturating irradiance at temperatures of 15, 20, 25 and 30 °C. For three shade leaves, net
photosynthesis was measured below and above saturating
irradiance in a non-systematic way at 10, 15 and 20 °C.
After the measurements were completed, the leaves were
tagged, detached, placed in a leaf press and left to dry. In late
September, after the last field measurements, we scanned each
leaf (Scanjet IIcx/T, Hewlett Packard, Minneapolis, MN) and
determined its area, excluding any holes, to the nearest
0.1 mm 2 with image analysis software (NIH Image v1.61, National Institutes of Health, Bethesda, MD). Dry mass of each
leaf was determined to the nearest 0.1 mg. Leaf mass per area
(LMA) of individual leaves was computed as the ratio of dry
mass to leaf area (g m –2). All measurements excluded the petiole.
Characterization of leaf population
Destructive measurements were performed within a single
1000-m 2 circular plot in the same stand to determine the canopy-level relationship between leaf mass and area. Within the
plot, trees were divided into three social classes (dominant +
co-dominant, intermediate and suppressed). Two to three trees
from each class were felled. Leaf area was then estimated by
randomized branch sampling (Gregoire et al. 1995), after dividing the crown in two at the level of maximum crown width,
to provide a measure of the frequency distribution of leaf area
per LMA class for each felled tree. A weighted mean of these
distributions was then calculated at the plot level using the
basal area of individual trees as weight. The detailed protocol
is given in Raulier et al. (1999).
Bottom-up model: derivation and integration
The bottom-up model is a spatially inexplicit, multilayer
model that integrates through the canopy a coupled model of
net photosynthesis and stomatal conductance in which the effects of temperature and atmospheric humidity are taken into
account. Integration from leaf to canopy is achieved with leaf
mass per area as a covariable to describe leaf properties and
leaf area distribution in the canopy and radiative environment.
The work on this model was carried out in two parts. The
first part was the derivation of the model operating at an
hourly time step and the leaf scale. The second was the numerical integration of the photosynthesis and the stomatal conductance submodels to produce coefficients for the top-down
RUE model operating at a monthly time step and the canopy
scale.
TREE PHYSIOLOGY VOLUME 20, 2000
INFLUENCE OF TEMPERATURE ON MONTHLY CUMULATED GPP
Leaf photosynthesis and stomatal conductance submodels
The coupled models of Farquhar et al. (1980) and Ball et al.
(1987) were used to model leaf photosynthesis and stomatal
conductance. At a constant atmospheric CO2 pressure, the
Farquhar et al. (1980) model describes the response of photosynthesis to light by a two-segment curve. In the lower segment, photosynthesis is limited by the light-limited rate of
electron transport (carboxylation rate Wj ). In the upper segment, photosynthesis is limited by the CO2-limited
carboxylation rate linked to the Rubisco activity
(carboxylation rate Wc ). The model of Farquhar et al. (1980)
can be summarized as follows (Harley et al. 1992, Harley and
Baldocchi 1995):
 Γ
An = 1 −  min(Wc , Wj ) − Rd ,
 ci 
(1)
where
Wc = Vcmaxc i / ( c i + K c (1 + O / K o))
Wj = Jc i / ( 4( c i + 2Γ))
J = αI a / 1 + (αI a / Jmax) 2
Symbols (see Table 1): An is net photosynthesis (µmol m –2
s –1), Γ is the CO2 compensation point in the absence of day
respiration (Pa), ci is the intercellular concentration of CO2
(Pa), Wc and Wj are carboxylation rates (µmol m –2 s –1), Rd is
leaf dark respiration rate (µmol m –2 s –1), Vcmax is the maximum
carboxylation rate allowed by Rubisco activity (µmol m –2 s –1),
Kc and Ko are the Michaelis-Menten constants for carboxylation and oxygenation (Pa), O is the partial atmospheric pressure of O2 (Pa), J and Jmax are the electron transport and the
potential electron transport rates (µmol m –2 s –1), α is the leaf
photosynthetic quantum efficiency (mol electron (mol photon) –1), and Ia is the photosynthetically active radiation (PAR)
(µmol (m 2 leaf) –1 s –1) incident on the leaf surface. The form
presented in Harley et al. (1992) was chosen because it is the
form used by Wullschleger (1993) for a comparative study of
109 plant species. At constant temperature, the calibration of
Equation 1 is reduced to the estimation of three parameters
(Jmax, Vcmax and Rd), all other parameters being considered constant for C3 plants. Values of all parameters, except for Jmax,
Vcmax and Rd, are taken from Harley and Baldocchi (1995).
Under a constant atmospheric CO2 pressure, stomatal conductance is well described by the Ball et al. (1987) model. This
model was simplified by Harley and Baldocchi (1995) by assuming a strong coupling between the leaf and the atmosphere
(Williams et al. 1996). Stomatal conductance is assumed to be
linearly related to the product of net photosynthesis and relative humidity:
g s = g 0 + g 1 An
h
,
ca
(2)
where gs is stomatal conductance to water (mmol m –2 s –1), h is
relative humidity (decimal fraction), ca is atmospheric CO2
335
pressure (expressed as µmol mol –1), and g0 and g1 are adjusted
parameters. The algorithm presented by Sellers et al. (1992)
was used to solve simultaneously for net photosynthesis,
stomatal conductance and intercellular CO2 concentration.
Parameter estimation of the bottom-up model
The parameters of the leaf photosynthesis model were estimated from our light response curves as described by
Niinemets and Tenhunen (1997; their Appendix C). In the first
step, the model parameters of Hanson et al. (1987), which include photosynthetic capacity, dark respiration and light compensation point, were estimated leaf by leaf by nonlinear
ordinary least squares (OLS) (PROC NLIN, SAS Institute
1996, Cary, NC). For each leaf, the equation was solved for
saturating irradiance (irradiance required to reach 95% of
photosynthetic capacity). Photosynthesis values beyond this
point were used to estimate a mean carboxylation rate Vcmax.
The value of Jmax was determined as the junction point between the two segments of Equation 1 (Wc = Wj ). Because of
the type of hyperbola used, the equation could be solved to obtain the values of Jmax for only 12 of the 24 leaves. The values
of Jmax for the other 12 leaves were estimated by a modification of Wullschleger’s (1993) model:
Jmax = ω 1Vcmaxeω 2Vc max ,
(3)
To calibrate the model, we used the observations of Harley et
al. (1992), Harley and Baldocchi (1995) and Wang et al.
(1996) in addition to Wullschleger’s data set. A logarithmic
transformation was applied to model heteroscedasticity. The
values of Jmax determined with the procedure of Niinemets and
Tenhunen (1997) were not significantly different from those
estimated with Equation 3 (tobs = 1.26, t[0.05,11] = 2.20). Consequently, Jmax was estimated with Equation 3 for the remaining
light response curves.
The second step was the integration of leaf photosynthesis
over the canopy. Leaf morphology and physiology of sugar
maple change throughout the canopy (Hagen and Chabot
1986) and leaf mass per area of sugar maple varies concurrently with the properties of leaf photosynthesis (Ellsworth
and Reich 1993). We therefore used leaf mass per area as a
covariable and looked for dependencies of Vcmax and Rd with
LMA. Our data revealed that Vcmax was linearly related to
LMA. A linear relationship between Rd and Vcmax was first
shown by Farquhar et al. (1980). The terms Vcmax, Jmax and Rd
in Equation 1 were then replaced with their respective
parameterized equations:
Vcmax = α 1LMA
(4a)
Rd = α 2Vcmax
(4b)
Jmax = ω 1Vcmax exp(ω 2Vcmax)
(4c)
where LMA is leaf mass per area (g m –2), α1 and α2 are the parameters to be estimated and ω1 and ω2 are those estimated for
Equation 3. This form of Equation 1 will henceforth be referred to as Equation 4 (Table 2).
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336
RAULIER, BERNIER AND UNG
Table 1. List of the main symbols used. Secondary symbols are explained in the text.
Ac
An
ca
ci
fb
fϕ, fT, fh, fL
gs
g0–1, g11–12
h
H , H0
HaV, HaJ, HaR
Ia, Ia.b, Ia..d
J, Jmax
KC , KO
L*
O
Rd
Vcmax
Wc , Wj
α
α1−2
ε
Γ
µ1–2
ϕa
ω1−2
Canopy net photosynthesis (µmol (m –2 ground) –1 s –1)
Leaf net photosynthesis (µmol (m 2 leaf) –1 s –1)
Partial atmospheric pressure of CO2 (µmol mol –1)
Intercellular concentration of CO2 (Pa)
Fraction of sunlit leaf area
RUE modifiers for PAR, temperature, relative humidity and LAI
Stomatal conductance (mmol m –2 s –1)
Parameters of the Ball et al. (1987) model (Equations 2 and 6)
Relative humidity, expressed in decimal fractions
Daily PAR radiation averaged over the growing season, incident on a horizontal surface at the top of the canopy (mol (m –2
ground) –1 day –1)
Parameters of the temperature dependence of photosynthesis (Equation 5) (kJ mol –1)
Instantaneous PAR irradiance incident on a unit leaf area (µmol (m –2 leaf) –1 s –1). Ia.b and Ia.d are direct and diffuse irradiances absorbed per unit leaf area
Electron transport rate and potential electron transport rate (µmol m –2 s –1)
Michaelis-Menten constants for carboxylation and oxygenation (Equation 1) (Pa)
Leaf area index (m 2 leaf (m –2 ground) –1)
Partial atmospheric pressure of O2 (Pa)
Leaf dark respiration rate (µmol (m 2 leaf) –1 s –1)
Maximum carboxylation rate allowed by Rubisco activity (µmol m –2 s –1)
Carboxylation rates limited either by Rubisco activity or RuBP regeneration (µmol m –2 s –1)
Leaf photosynthetic quantum efficiency (mol electron (mol photon) –1) (Equation 1)
Parameters of the leaf photosynthesis model (Equation 4)
Radiation use efficiency (RUE) (mol C (mol photon) –1)
CO2 compensation point in the absence of day respiration (Pa)
Parameters of the relationship between LMA and relative irradiance (Equation 7)
Monthly PAR radiation absorbed by the canopy (mol (m 2 ground) –1 month –1)
Parameters of the Wullschleger model (Equation 3)
Our light response curves and the measurements used by
Wullschleger (1993) for the calibration of his model were both
obtained at a similar temperature (24.7 and 25.3 °C, respectively). The estimation of Vcmax, Jmax and Rd with Equations
4a–c is thus valid for a temperature of 25 °C. The sensitivity of
Rd to temperature was obtained directly from the function proposed by Harley and Baldocchi (1995, their Equation 1). The
sensitivities of Vcmax and Jmax to temperature were computed
with Equation 2 of Harley and Baldocchi (1995):
 T − 298) H aR 
Rd( T k) = Rd298 exp k
 ,
 RT k 298 
(5a)
Vc max( T k) =
Vcopt
H dV exp((1 / ToptV − 1 / T k) H aV / R)
,
H dV − H aV(1 − exp((1 / VoptV −1 / T k) H aV / R))
(5b)
Jmax( T k) =
Jopt
H dJ exp((1 / ToptJ − 1 / T k) H aJ / R)
,
H dJ − H aJ (1 − exp((1 / ToptJ −1 / T k) H aJ / R))
(5c)
where Tk is temperature (°K), R is the gas constant (0.00831 kJ
°K –1 mol –1), and Ha and Hd are the activation and deactivation
energies (kJ mol –1) for the corresponding parameter. The val-
ues of Vcmax and Jmax estimated with Equations 4a and 4c at
25 °C were used to estimate the corresponding values at the
optimum temperature (Vcopt and Jopt) with Equations 5b and 5c.
Parameter Rd298 corresponds to dark respiration at 25 °C. With
the calibration method used by Harley and Baldocchi (1995),
only the activation energy is allowed to vary for the three parameters (HaV, HaJ and HaR for Vcmax, Jmax and Rd, respectively).
Therefore, we used the values of HdV, HdJ, ToptV and ToptJ given
by Harley et al. (1992). For parameters Γ, Kc and Ko, the temperature dependencies given by Harley and Baldocchi (1995)
were used. Parameters HaV, HaJ and HaR were estimated by
nonlinear OLS by including Equations 5a–c in Equation 4
with the temperature response curves obtained for five leaves
(Table 2, Equation 4′). The values obtained for the parameters
of Equation 4′ are comparable to those of Harley and
Baldocchi (1995), although the standard deviation for HaR is
abnormally high (Table 2, Equation 4′). In a statistical sense,
this indicates that the temperature dependencies used with the
Farquhar et al. (1980) model are overparameterized for our
data. Harley et al. (1992) also encountered this problem.
The parameters of the stomatal conductance model were estimated by OLS from the 24 light response curves used to calibrate Equation 4. An analysis of the residuals of Equation 2
showed that parameter g1 is a function of LMA (Table 2,
Equation 6):
g s = g 0 + ( g 11 + g 12LMA) An
TREE PHYSIOLOGY VOLUME 20, 2000
h
,
ca
(6)
INFLUENCE OF TEMPERATURE ON MONTHLY CUMULATED GPP
337
Table 2. Bottom-up model statistics. Number of observations (n), mean square error (MSE), adjusted coefficient of determination (Radj2), parameter estimates, standard error (SE) and correlation matrix.
Model
n
MSE
Radj2
Parameters
Estimated value
SE
Correlation matrix
Equation 3
183
0.0455
0.885
Equation 4
125
1.013
0.909
Equation 4′1
14
0.462
0.972
ω1
ω2
α1
α2
Ha.V
Ha.R
Ha.J
g0
g11
g12
µ1
µ2
2.716
–3.20 × 10 –3
0.654
0.015
77.74
69.44
54.38
20.86
2396.95
53.30
0.078
70.07
0.085
4 × 10 –4
0.193
0.003
8.36
216
8.49
2.69
789.95
10.90
0.007
1.464
1
–0.86
1
0.46
1
0.83
0.76
1
–0.57
0.33
1
–0.47
Equation 6
Equation 7 2
1
2
119
56
351
0.782
0.0110
0.893
1
1
1
0.90
1
–0.92
1
Equation 4′ is Equation 4 modified to add the sensitivity of leaf photosynthesis to temperature (Equation 5).
From Raulier et al. (1999).
Six out of 125 observations were rejected as outliers with
t-tests.
Integration through the canopy
Because LMA of sugar maple leaves is related to the time-averaged irradiance that impinges on the leaves (Ellsworth and
Reich 1993), it can be used as a covariable to estimate the
irradiance needed for the leaf photosynthesis model. However, the use of a mean irradiance will tend to overestimate
photosynthesis (Norman 1993). Moreover, irradiance needs to
be separated into its diffuse and direct components. Three elements must therefore be specified to integrate leaf photosynthesis through the canopy with LMA: the relationship between
LMA and the time-averaged irradiance, the distribution within
the canopy of leaf area by LMA classes and, within each LMA
class, the proportion of leaves subjected to direct and diffuse
radiation.
An analysis of available data for sugar maple (Ellsworth and
Reich 1993, Tjoelker et al. 1995, Burton and Bazzaz 1995,
Ellsworth, unpublished data) showed that a general relationship between LMA and relative irradiance could be considered for this species (Raulier et al. 1999):
H
1
,
=
1
+
exp(
−
µ
H0
1( LMA − µ 2)
1  H
L *sh = − ln
,
k  H0
(8)
where k is a mean light extinction coefficient (Appendix 2).
The L* included within a specified LMA class was determined
as the difference between the shading L* calculated for that
class and that calculated for the next LMA class. With Equation 8, stand L* determines the minimum LMA that should be
observed in the considered stand. We compared the predicted
frequency distributions of L* per LMA class with the observed
distribution at St-Gilles. For this purpose, the frequency distribution of L* was simulated for LMA classes (5, 10 and 20 g
m –2) and a visual comparison was made between the observed
and predicted distributions (Figure 1).
Finally, the shading L* estimated for an LMA class was used
to approximate the proportion of direct and diffuse irradiance
reaching the specified LMA class. Following Spitters (1986),
the fraction of sunlit leaf area equals the fraction of direct
irradiance reaching the considered class. Also, the fraction of
direct or diffuse irradiance absorbed by the leaves in a given
class is equal to the product of the irradiance impinging on the
leaves and the corresponding extinction coefficient (Spitters
1986, Thornley and Johnson 1990). Canopy net photosynthesis was estimated as:
(7)
Ac =
∑L
*
[ fbi An( I ab + I ad) + (1 − fbi ) An( I ad)] ,
i
(9)
i
where H and H0 are daily PAR, averaged over the growing
season, incident on a horizontal surface (mol (m 2 ground) –1
day –1) at the level where LMA is measured and at the top of
the canopy, respectively, and µ1 and µ2 are parameters to be estimated by OLS (Table 2, Equation 7). The time-averaged relative irradiance estimated with Equation 7 was then used to
estimate the leaf area index shading the leaves of the considered class (L*sh) with the Beer-Lambert law (e.g., Thornley
and Johnson 1990):
where Ac is net canopy photosynthesis (mol (m 2 ground) –1
s –1), An(Ia) is net photosynthesis (Equation 1) corresponding to
absorbed irradiance Ia, L*i is the L* of a given LMA class i, fbi
corresponds to the fraction of sunlit leaf area for an LMA class
i (exp( −kL* shi )), and Iab and Iad are the direct and diffuse irradiances (µmol (m 2 leaf) –1 s –1) absorbed by the leaves. Their estimation requires the use of appropriate extinction coefficients
(Appendix 2). Scattering of PAR was not considered (Baldocchi 1993).
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338
RAULIER, BERNIER AND UNG
cord of daily minimum and maximum temperatures was selected at random (Canadian Daily Climate Data: Temperature
and Precipitation, Eastern Canada 1996, Environment Canada
1998, Downsview, ON). For the 40 stations, degree-days
above 5 °C vary between 1200 and 2200 °C (mean of 1720 °C)
and mean yearly temperature between 1.6 and 6.6 °C (mean of
4.1 °C). Medlyn (1998) showed that RUE depends on L*. Consequently, the chosen climatic records were randomly assigned to four groups of L* values of 4, 6, 8 and 10, a range
typical of sugar maple stands. The growth season was arbitrarily defined as extending from May 1 to October 31.
Top-down model: derivation
The top-down model takes the form of a simple function in
which the monthly absorbed photosynthetically active radiation (ϕa; mol (m 2 ground) –1 month –1) is multiplied by a mean
radiation-use efficiency (RUEav) to obtain monthly GPP (mol
(m 2 ground) –1 month –1). This relationship can be improved by
a series of multipliers describing the influence of various external variables. The monthly relationship between canopy
gross photosynthesis and ϕa was expressed as:
GPP = RUE avϕ a .
(10)
Monthly ϕa was calculated with the Beer-Lambert law from L*
and a mean extinction coefficient that depends on L* (Appendix 2). No distinction was made between direct and diffuse
components of irradiance. Monthly GPP was calculated as the
cumulated hourly canopy net photosynthesis plus respiration
obtained from the 40 single-year simulations carried out with
the bottom-up model. The value of RUEav was obtained by fitting Equation 10 to these data as a first step in finding an expression for total RUE.
Estimation of RUE modifiers
Figure 1. Observed and predicted (Equation 6) frequency distributions of L* by LMA classes of 20 (a), 10 (b) and 5 g m –2 of amplitude
(c), at Saint-Gilles.
The influence of temperature and other environmental variables on monthly GPP was tested in a three-step procedure.
Variables were first selected through an evaluation of their
correlation with the residuals of Equation 10. The modifiers
for all selected variables were then computed with a standard
quadratic function in which the variable centered around its
mean:
Simulations with the bottom-up model
The variables required to use the bottom-up model are stand
leaf area index, diffuse and direct irradiances above the canopy, air temperature and atmospheric humidity. Hourly temperature, relative humidity and radiation were derived from
daily minimum and maximum temperatures (Appendix 1).
Hourly values of predicted GPP were cumulated at a monthly
scale to provide the monthly canopy GPP data needed to derive the parameters of the top-down model.
Simulations with the bottom-up model were run for various
climatic conditions observed at 40 permanent meteorological
stations within the natural range of sugar maple in the province
of Québec, Canada. For each selected station, one year’s re-
2
x −x
x x
 + β qx  −  ,
fx = 1 + β lx 
 x 
 x 
(11)
where fx is the RUE modifier for the variable x, βlx and βqx are
the linear and quadratic effects of the variable on RUE and x is
its mean. Finally, the modifiers were included in a stepwise
procedure in Equation 10, on the basis of maximum reduction
of the mean square error (MSE) of the model. The regressions
were estimated with nonlinear OLS (NLIN, SAS Institute,
Cary, NC). Because two parameters were introduced at each
step, the significance of entry or exit of a modifier was tested
with a likelihood ratio (Bates and Watts 1988).
TREE PHYSIOLOGY VOLUME 20, 2000
INFLUENCE OF TEMPERATURE ON MONTHLY CUMULATED GPP
RUE sensitivity to key parameters
We tested the sensitivity of the GPP–temperature relationship
to the initial values of two sets of key parameters used in the
bottom-up model and to the effect of spatial and temporal integration. The key parameters examined in this analysis were
derived from our leaf photosynthesis measurements and define the sensitivity of leaf photosynthesis and stomatal conductance to temperature (HaV, HaJ and HaR) and relative
humidity (g11 and g12). The sensitivity analysis was conducted
by performing the simulations of the bottom-up model with
parameter values augmented or reduced by 25%, and by re-estimating the parameters of the top-down model. The analysis
was conducted on the relative change in monthly GPP.
Sensitivity of GPP to temperature is dependent on both spatial and temporal scales. Effect of scale was investigated by
analyzing simulations made with the bottom-up model at three
combinations of spatial and temporal scales: the mean leaf at
the scale of the second and the canopy at the scales of the second and the day. Sensitivity was estimated as the relative
change in GPP divided by the relative change in temperature.
The response of leaf gross photosynthesis to temperature was
estimated at six values of LMA (20 to 120 g m –2), six temperatures (5 to 30 °C) and six irradiances (200 to 1200 µmol m –2
s –1). Relative humidity was kept constant at 72.2%. The response curve for the mean leaf by irradiance classes was then
estimated by weighting the response curves calculated by
LMA classes, with weights equal to the frequency distribution
of L* at Saint-Gilles. For the canopy sensitivity at the second
scale, the bottom-up model was used to simulate canopy photosynthesis for four values of L* (4, 6, 8 and 10), six temperatures (5 to 30 °C), and six irradiances (200 to 1200 µmol m –2
s –1), again at a constant relative humidity of 72.2%. The mean
response of canopy gross photosynthesis to temperature was
estimated by averaging the response curves calculated for
each L* class. Finally, for the canopy sensitivity at the day
scale, all daily GPP predictions made with the bottom-up
model were averaged by classes of temperature (class amplitude of 5 °C) and daily mean irradiance (class amplitude of
200 µmol m –2 s –1). The mean GPP sensitivity to temperature
by irradiance class was estimated by averaging the sensitivities estimated in steps of 5 °C. To show the interaction between irradiance and GPP sensitivity to relative humidity, the
daily GPP predictions made with the bottom-up model were
averaged by classes of relative humidity (class amplitude of
0.1) and daily mean irradiance (class amplitude of 200 µmol
m –2 s–1).
Results and discussion
Estimation of RUE modifiers
Analysis of the relationship between residuals of the top-down
model (Equation 10) and environmental variables yielded five
variables that appeared to be related to RUE: monthly mean
temperature, monthly mean relative humidity, leaf area index,
monthly cumulated PAR (mol m –2 month –1) and monthly
variance of the daily temperature amplitude. Monthly mean
temperature was computed as the mean of monthly mean min-
339
imum and maximum temperatures (Running et al. 1987, Aber
and Federer 1992). Monthly mean relative humidity was estimated using the monthly mean and minimum temperatures
(Aber and Federer 1992). The stepwise analysis revealed that,
of these five variables, only the monthly variance of the daily
temperature amplitude had no effect on RUE. The other four
variables all significantly contributed to explaining the variability in GPP predicted with the bottom-up model (Table 3,
Figure 2):
GPP = RUE av fh fL fϕ fT ϕ a ,
(12)
where fh, fL, fϕ and fT are RUE modifiers (Equation 11) for
monthly relative humidity, leaf area index, monthly cumulated PAR and monthly mean temperature, respectively, that,
together with RUEav, define the total RUE. As defined in
Equation 11, the modifiers can have a value > 1. The value of
RUEav estimated from Equation 12 is 0.031 mol C (mol photon) –1 (Table 3). Landsberg and Waring (1997) used a value of
0.031 mol C (mol photon) –1, based on the results of
McMurtrie et al. (1994), Waring et al. (1995) and Williams et
al. (1996). Goetz and Prince (1998) estimated a RUEav of
0.035 mol C (mol photon) –1 for boreal stands of Picea
mariana B.S.P. and Populus tremuloides Michx. in Minnesota.
On the annual scale, cumulated GPP predicted with the
top-down model (Equation 12) is not significantly different
from that predicted with the bottom-up model (Fobs = 0.10,
F(0.05, 2, 38) = 3.24) (Figure 3). The predicted annual GPP varies
between 113 and 195 mol C (m 2 ground) –1 year –1, with a mean
of 172 mol C (m 2 ground) –1 year –1.
The ϕa explained most of the variability in GPP predicted by
the bottom-up model (92.6%, Table 3) (cf. Goetz and Prince
1998). Relative humidity and temperature accounted for only
5.2% of the variability in GPP (Table 3). Temperature by itself, after the stepwise inclusion of relative humidity, L* and
PAR modifiers, explained only 1.1% of GPP variability (Table 3).
In general, all of the modifiers had a linear effect on total
RUE within the observed ranges of relative humidity, L*, PAR
and temperature (Figure 2). Re-estimation of the parameters
of Equation 12 without considering the quadratic term of the
modifiers only slightly changed the coefficient of determination of the relationship (from 99.2 to 98.9%) and the linear
terms remained mostly unaffected (results not shown). When
ranking the modifiers by order of linear sensitivity (parameters βlx, Table 3), RUE was most sensitive to atmospheric humidity, but an augmentation of 10% in mean monthly relative
humidity will increase total RUE by only 5.5% (Table 3). An
augmentation of 10% in mean monthly temperature will cause
only a 1.2% increase in total RUE (Table 3). A slope of 1 indicates a proportional influence. When the slope is less than 1,
the dependent variable is said to be inelastic to variation of the
independent variable (Soares et al. 1995). Thus, RUE and GPP
are inelastic to changes in any of the considered variables.
Importance of temperature and relative humidity
Temperature had only a slight influence on RUE and monthly
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340
RAULIER, BERNIER AND UNG
Table 3. Monthly RUE model statistics. Order of entry of the modifiers in the stepwise regression procedure, variable for which the model included an RUE modifier (Equation 11), parameter estimates, standard error (SE) and contribution to the adjusted coefficient of determination
(based on 240 observations).
Order of entry
Variable
1
ϕa
2
Mean relative humidity
3
Mean
Parameter Estimated value
SE
Contribution
ε
0.0307
0.00010
92.6%
0.722
βl.h
βq.h
0.557
0.671
0.0352
0.3066
4.0%
L*
7
βl.L
βq.L
0.0847
–0.0854
0.0051
0.0186
1.2%
4
PAR
994 mol m –2 month –1
βl.ϕ
βq.ϕ
–0.201
0.188
0.0115
0.0382
0.3%
5
Mean temperature
13.9 °C
βl.T
βq.T
0.125
–0.113
0.0065
0.0176
1.1%
GPP (Figure 2) for the June–September monthly mean temperatures ranging from 10 to 20 °C. The minor importance of
temperature on RUE and monthly GPP seems counterintuitive, given the important and well-defined effect of temperature on instantaneous net photosynthesis at the leaf level.
However, two factors combine to reduce the importance of the
temperature effect. The first is related to the naturally high correlation between irradiance and temperature, a correlation that
exists at all time scales, but that increases when moving from
the hourly, to the daily, and to the monthly scales. Inclusion of
irradiance in the model therefore leaves less for temperature to
explain. The second is a canopy-level property that emerges
from the integration of the leaf-level response. A strong dependence of GPP on temperature only holds at saturating
irradiance. At lower irradiances, the sensitivity of net leaf pho-
tosynthesis to temperature decreases (Figure 4a) (Farquhar et
al. 1980, their Figure 9, Harley et al. 1985, their Figure 11).
This loss of sensitivity is further marked when considering
canopy photosynthesis at the scale of seconds (Figure 4b) and
days (Figure 4c). For a PAR of 650 µmol m –2 s –1, corresponding to the mean for our simulations and within a range of
temperatures between 5 and 30 °C, the mean sensitivity of
photosynthesis to temperature changes from 0.57 at the leaf
level to 0.27 at the canopy level and drops to 0.19 when passing from the second to the day time step.
Canopy sensitivity to temperature has not often been measured (Ruimy et al. 1995, p. 37), but Goulden et al. (1996) observed a similar insensitivity in an oak and maple stand in the
Harvard Forest over a 5-year period with the eddy-covariance
technique. In a sensitivity analysis of the model PnET, a
Figure 2. Relationships between monthly mean relative
humidity, PAR, temperature
and L* with its RUE modifier.
The observed values correspond to the GPP predicted
with the bottom-up model and
divided by the GPP predicted
with the top-down model
(Equation 12), but without
considering the modifier in
question.
TREE PHYSIOLOGY VOLUME 20, 2000
INFLUENCE OF TEMPERATURE ON MONTHLY CUMULATED GPP
341
Figure 3. Relationship between cumulated annual GPPs predicted
with the bottom-up and top-down models.
model calibrated for Appalachian forests relatively close to
our own site (Aber and Federer 1992), gross carbon exchange
was inelastic to changes in the parameters of the temperature
modifier applied to gross photosynthesis (Aber et al. 1996,
their Table 3). These results support our conclusion that direct
GPP dependency on temperature is of minor importance in
models operating at the canopy level at a time step of a month
or more.
The modeling results also show that irradiance influences
the temperature optimum of net photosynthesis in addition to
reducing its temperature dependence. At saturating irradiance,
and using the model of Harley and Baldocchi (1995), leaf photosynthesis is optimal around 30 °C. At lower irradiances, the
temperature optimum for leaf net photosynthesis shifts to
lower temperatures (Figure 4a). This temperature shift is accentuated with increase in the temporal scale over which the
temperature is averaged. Figure 2d shows that the modeled
monthly mean temperature optimum for canopy GPP is
around 22 °C. As the time step increases, the optimum range
of mean temperature for photosynthesis shifts downwards.
About 2.5 °C of this downward shift is caused simply by the
inclusion of nighttime temperatures in the mean temperature.
The rest of this shift, about 5.5 °C, is caused by the interaction
between variations in irradiance and temperature over time,
and the averaging of this interaction when passing from the
scale of the second to that of the day (Figure 4c).
The low contribution of relative humidity to the determination coefficient of Equation 12 (Table 3) was in part due to the
narrow range of monthly relative humidity that we used
(mostly between 0.60 to 0.80, Figure 2). As with temperature,
irradiance also interacts with the sensitivity of net photosynthesis to relative humidity (Figure 5). Both factors can be used
to explain the low sensitivity of total RUE to relative humidity
(Aber et al. 1996, their Table 3).
The results remain mostly unchanged when key parameters
of the bottom-up model are changed by ± 25%. Values of GPP
predicted by the bottom-up model are affected at most by
± 4%. The contribution of temperature and relative humidity
Figure 4. Temperature response curves and temperature optima (䉬)
of instantaneous gross leaf photosynthesis (a), instantaneous canopy
gross photosynthesis (b), and daily canopy gross photosynthesis as a
function of six irradiances (µmol m –2 s –1) (c). (Refer to “RUE sensitivity to key parameters” for detailed explanations).
to the determination coefficient of Equation 12 varies only between 4.3 and 6.4%. When the parameters describing the photosynthesis sensitivity to temperature (HaV, HaJ and HaR,
Equation 5) are augmented by 25%, mean RUE diminishes
from 0.0307 to 0.0294 mol C (mol photon) –1 and linear RUE
sensitivity to temperature (parameter βlT, Table 3) is raised
from 0.06 to 0.24. In effect, raising the temperature sensitivity
of GPP reduces GPP for months with lower temperatures.
When parameters HaV, HaJ and HaR are lowered by 25%, the reverse occurs with the mean RUE rising to 0.032 and its linear
sensitivity to temperature decreasing slightly to 0.03. When
TREE PHYSIOLOGY ON-LINE at http://www.heronpublishing.com
342
RAULIER, BERNIER AND UNG
Figure 5. Relative humidity response curves of daily gross photosynthesis as a function of six irradiances (µmol m –2 s –1). The daily GPP
predicted by the bottom-up model were averaged by classes of relative humidity (classes of 0.1) and mean daily PAR (classes of 200
µmol m –2 s –1).
key parameters (parameters g11 and g12, Equation 6) of the bottom-up model describing photosynthesis sensitivity to relative
humidity are augmented by 25%, RUE sensitivity (parameters
βlh and βqh, Table 3) is unaffected, but mean RUE is slightly
enhanced (0.0315), whereas it is reduced (0.0295) when the
parameters are reduced by 25%.
Effects of light regime and L* on total RUE
Modeled total RUE increased with a drop in monthly PAR
(Figure 2). During the months of September and October, the
daily potential radiation is on average 40% lower than during
the other months of the growing season. At these low irradiances, canopy photosynthesis is less often saturated and is therefore more efficient per unit of ϕa. This influence of light
climate on RUE at the monthly time step was also noted by
Medlyn (1998, her Figure 2) when cumulating NPP with
MAESTRO.
The dependency of RUE on L* is related to the increase in
RUE with decreasing irradiance. Unlike the effect of monthly
PAR on RUE, the effect of L* on RUE depends on the interaction between the Beer-Lambert light distribution and
leaf-level light response within the canopy. In the bottom-up
model, L* is always augmented by adding more efficient (unsaturated) shade leaves at the bottom of the canopy, resulting
in a gradual increase of RUE with increasing L* (Figure 2). A
similar dependency of RUE on L* has been observed by
Medlyn (1998, her Figure 3).
Conclusions
Two temperature-related properties clearly emerge from scaling up from the leaf to the canopy, and from a short time step
(one hour or less) to a long time step (a day or a month). The
first property is that, within a relatively broad range of values,
temperature has a minor effect on predicted monthly canopy-level GPP both directly on the photosynthetic process,
and indirectly on stomatal conductance through relative hu-
midity. A primary reason for this is the dependence of temperature response on irradiance, which is low when, as is usually
the case, mean irradiance within the canopy is below saturation. The second property is that, as the time step increases, the
optimum range of mean temperature for photosynthesis shifts
downward. Both of these temperature-related properties need
to be considered in any top-down model.
The exercise of scaling up physiologically based models to
larger spatial and temporal scales necessarily entails simplifications and loss of process representation in order to maintain
or augment model robustness. The simpler scaled-up models
are thus easier to calibrate than detailed bottom-up models
(Jarvis 1993), but the simplification is done at the cost of generalization. As it is simplified, the scaled-up model becomes
increasingly dependent on the database to derive the form of
the modifiers and estimate the correct values for their parameters. It is therefore important to recognize this limitation when
developing scaled-up models for operational use, and either to
specify clearly the domain of applicability, or to broaden the
database used in model development.
Acknowledgments
We thank Sébastien Dagnault, Adrien Forgues, Valérie Hudon,
Gérard Laroche, Roger Mongrain, Robert Saint-Laurent and René
Turcotte for their dedicated work in getting the site established and
for their help in obtaining the various field data. We are also grateful
to Pamela Cheers for her editorial comments. Postdoctoral support
for F.R. was provided by Natural Resources Canada through the Natural Sciences and Engineering Research Council of Canada. This
work is part of the ECOLEAP project, an effort dedicated to the modeling of forest productivity in Canada over large areas but with a spatial resolution pertinent to forest management applications (Bernier et
al. 1999).
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Appendix 1. Simulation of hourly temperature, relative
humidity and radiation
The bottom-up model runs at the hourly scale. The most frequently collected data in meteorological stations are daily
minimum and maximum temperatures. Daily variation of temperature normally follows the sun course with a small delay
(Parton and Logan 1981, Strandman et al. 1993). We used the
model of Parton and Logan (1981) to simulate the diurnal time
course of temperature, and estimated the parameters of this
model using the hourly measurement of temperature from four
meteorological stations maintained by the ECOLEAP project
since 1996 at Saint-Gilles, Duchesnay (46°26′ N, 71°25′ W),
Forêt Montmorency (47°19′ N, 71°06′ W) and Green River
(47°44′ N, 68°04′ W). A continuous first-order autoregressive
process (Jones and Boadi-Boateng 1991) was also added to
account for autocorrelation (Table A1, Equation A1):
 π(t ij − t srj ) 
T ij = Tnj + ( Txj − Tnj ) sin
 + ε ij ,
 l dj + 2δ T 
(A1)
with
ε ij = ε ij − 1 exp( − φ T (t ij − t ij − 1))
where the indices i and j correspond to the measure i of day j,
Tnj and Txj are the daily minimum and maximum temperatures,
ldj is day length (hours), δ T and φ T are the parameters to be estimated with nonlinear OLS, and δ T equates the time delay
(hours) between the solar midday and the time when maximum temperature is observed. The algorithm for solar elevation of the Astronomical Almanac (Michalsky 1988) was used
to estimate the daily time difference between local and solar
times at solar midday.
The calculation of the hourly relative humidity depends on
the observed temperature and the temperature at dew point.
Relative humidity was calculated with the model of Murray
(1967), assuming that the dew point occurs at the daily minimum temperature (Running et al. 1987).
Daily PAR incident on a horizontal surface was calculated
from the product of a clear-sky potential radiation and a clearness index depending on daily temperature amplitude. Extra-terrestrial short-wave radiation was calculated with the
algorithm of Collares-Pereira and Rabl (1979). The clear-sky
transmission coefficient was estimated with the procedure of
Nikolov and Zeller (1992), fixing to 0 the variable expressing
the importance of cloud cover. A daily mean elevation of the
sun, required by Nikolov and Zeller’s (1992) procedure, was
calculated with the algorithm of Wang and Polglase (1995,
p. 1242), with the help of the sun elevation algorithm of
Collares-Pereira and Rabl (1979). The PAR was estimated as
one half of total shortwave radiation (Ross 1975, Aber et al.
1996). We then calibrated an algorithm to estimate the daily
clearness index from silicon pyranometer measurements
(LI200SB, Li-Cor Inc., Lincoln, NE) done at the top of the
four ECOLEAP meteorological stations:
H = K hH pot
(A2)
where
K hn + β 0( Tx − Tn)
Kh = 
K hx
TREE PHYSIOLOGY VOLUME 20, 2000
if ( Tx − Tn) < ∆Tjoin
otherwise.
INFLUENCE OF TEMPERATURE ON MONTHLY CUMULATED GPP
345
Table A1. Model statistics. Number of observations (n), mean square error (MSE), adjusted coefficient of determination (Radj2), parameter estimates, standard error (SE) and correlation matrix.
Model
n
MSE
Radj2
Parameters
Equation A1
13151
2.409
0.984
δT
φT
Estimated value
SE
Correlation matrix
1.649
0.174
0.022
5.1 × 10 –3
1
–0.34
1
–3
Equation A2
1024
0.421
0.724
β1
β2
0.112
–0.056
4.5 × 10
3.4 × 10 –3
1
–0.96
1
Equation A5
7
6.86 × 10 –5
0.989
κ1
κ2
0.672
–0.079
6.5 × 10 –3
3.7 × 10 –3
1
–0.88
1
∆Tjoin =
K hx − K hn
,
β0
was modeled as (e.g., Jarvis and Leverenz 1983, Chen and
Cihlar 1995):
β 0 = β 1 exp(β 2( T x − T n)).
kb = Ω
Tx and Tn are the monthly mean maximum and minimum temperatures. Equation A2 is a four parameter model (β1, β2, Khn
and Khx) with three intermediate parameters (Kh, ∆Tjoin and β0)
that depend on the previous parameters. It is composed of two
linear segments that join at ∆Tjoin and can be considered a close
variant of the algorithm of Bristow and Campbell (1984, their
Equations 4 and 5). The value of Kh corresponds to the clearness index and should vary between 0 and 1, which are the expected values for Khn and Khx. The three parameters of
Equation A2 were estimated with weighted nonlinear OLS,
with a square root transformation. A simplified version of
Equation A2, with Khn and Khx maintained constant at 0 and 1
respectively (Table A1, Equation A2), was not significantly
different from Equation A2 (Fobs = 0.54, F[0.05, 2, 1020] = 3.00).
Finally, daily PAR was used to estimate hourly radiation
with the algorithm of Wang and Polglase (1995, their Equation A9). The relative importance of diffuse and direct radiation was estimated with the model of Collares-Pereira and
Rabl (1979, their Equation 6), which depends on day length
and the daily clearness index (Equation A2). Sun zenith angle
was estimated with spherical trigonometry (e.g., Michalsky
1988).
Appendix 2. Extinction coefficients
The method used to integrate leaf photosynthesis through the
canopy (Equation 9) requires estimates of the extinction coefficients for direct and diffuse irradiances as the fraction of direct or diffuse irradiance absorbed by the leaves is equal to the
product of the irradiance impinging on the leaves and the corresponding extinction coefficient (Spitters 1986, Thornley and
Johnson 1990). The extinction coefficient for direct irradiance
G (θ)
,
cos θ
(A3)
where Ω is a foliage clumping index, G(θ) is the mean cosine
of the angle between the leaves and the sun and θ is the solar
zenith angle. We used a value of 0.367 for the product ΩG.
This value was derived from the results of Burton et al. (1991),
irrespective of the sun elevation by comparing their measured
relative irradiances with the L* estimates calculated by their
allometric method. For the diffuse part of irradiance, a mean
extinction coefficient was estimated assuming an isotropic sky
and dividing the sky hemisphere into 10 zenithal angle classes:
 10

ln 01
. ∑ exp( − kbi L *) 
 i=1

,
kd = −
*
L
(A4)
where kbi is the extinction coefficient for direct irradiance in
the zenithal angle class i (Equation A3). The value of kd depends on the L* and was therefore estimated for values of L*
between 2 and 10 increments of 2.
A mean extinction coefficient (Equation 8) was also estimated for L* values between 2 and 10 in increments of 2 by
first estimating the irradiance below the considered L* with
Equations A3 and A4, and then summing this irradiance over
the whole growing season to calculate the mean extinction coefficient. Results depended on L* and were used to adjust the
following function:
k = κ 1 + κ 2 ln( L *),
(A5)
where κ1 and κ2 are two parameters estimated by OLS (Table A1, Equation A5).
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