J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131 123
r
af
= α
w
[wuh]
12
4α L
1 − exp−α
w
2 16
w is the leaf width, α
and α
w
are constants respec- tively equal to 0.005 m s
− 12
and 2.5 dimensionless, uh
is the wind speed at canopy height calculated as uh =
u
a
{ ln[h − dz
] − 9
m
hL} {
ln[z
r
− dz
] − 9
m
z
r
L} 17
The net radiation reaching the soil surface R
ns
is calculated using a Beer’s law of the form
R
ns
= R
n
exp−α
r
L 18
where α
r
is the extinction coefficient. Shuttleworth and Wallace 1985 and Shuttleworth and Gurney
1990 arbitrarily prescribed a value of 0.7 for α
r
in sparse crops, whereas Kustas and Norman 1997 as-
signed a value of 0.45, considered as quote “midway between its likely limits of 0.3–0.6”. A compromise
has been retained in our study by fixing an arbitrary value of 0.5 following Choudhury 1989. This value
is used both in the experimental validation and the numerical simulations. When not available by direct
measurement, the soil heat flux G can be calculated as a given fraction α
g
≈0.3 of the net radiation reaching the substrate: G=α
g
R
ns
. 2.3. Determination of component temperatures
Component temperatures T
f
and T
s
can be di- rectly measured using an IRT sensor with a narrow
acceptance angle to avoid the contamination of fo- liage temperature by substrate temperature. They can
also be determined by calculation from directional measurements of canopy surface temperature i.e.
made at two different angles. A brief outline is given hereafter. In a two-layer representation of the whole
canopy the radiometric surface temperature T
r
is ex- pressed as a function of the component temperatures
as Norman et al., 1995 T
r
= h
f ϕT
4 f
+ 1 − f ϕT
4 s
i
14
19 where f ϕ is the fraction of the field of view of the
radiometer occupied by vegetation a function of the view zenith angle ϕ and where the temperatures are
expressed in Kelvin. If one assumes foliage elements to be randomly distributed above the soil surface, the
fractional vegetation area viewed by a radiometer from a zenith angle ϕ is
f ϕ = 1 − exp
−α
a
L cos ϕ
20 where L
is the leaf area index and α
a
is a coefficient which depends upon the angular distribution of foliage
elements Choudhury, 1989. For uniform leaf-angle distribution α
a
is equal to 0.5. By assuming that the difference between the two component temperatures
T
f
and T
s
expressed in Kelvin is relatively small, the radiometric surface temperature given by 19 can be
approximated by the area-weighted mean of foliage and substrate temperatures Choudhury, 1989; Nor-
man et al., 1995
T
r
= f ϕT
f
+ [1 − f ϕ]T
s
21 If two radiometric surface temperatures T
r1
and T
r2
are measured with two different angles of view ϕ
1
and ϕ
2
, T
f
and T
s
can be retrieved by solving al- gebraically either the set of two equations with two
unknowns 19 Kustas and Norman, 1997, Eqs. 7 and 8 or, with a lesser accuracy, their linearized
forms 21. The problem of retrieving substrate and foliage temperatures from two directional measure-
ments of surface temperature has been analyzed with more details by François et al. 1997.
3. Model predictions
3.1. Comparison of model outputs with experimental data
The data used to test the model are taken from the HAPEX–Sahel experiment which took place in Niger
in 1992 Goutorbe et al., 1994. The micrometeo- rological data needed to implement the model were
measured on a fallow savannah of the east–central supersite EC near the village of Banizoumbou
13
◦
31
′
N, 2
◦
39
′
E, alt: 240 m. Fallow savannah is a composite vegetation made up of woody shrubs of
Guiera senegalensis scattered above a sparse herba-
ceous substrate. The characteristics of the vegetation and the details of the instrumentation have already
been described in Lhomme et al. 1994 and Monteny et al. 1997. The Guiera shrubs had the following
124 J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131
characteristics: canopy height h=3.5 m, leaf area in- dex L
= 0.5, leaf width w=0.02 m. Bush and substrate
temperatures were measured by two nadir-looking infra-red thermometers Everest Interscience IRTs,
one mounted at 9 m over the grass area in fact a mix- ture of bare soil and senesced foliage and the other
at about 1 m above a Guiera shrub. Wind velocity, air temperature and water vapour pressure were measured
at a reference height of 12 m above the soil surface. The net radiometer REBS, Q6, Campbell, UK was
also installed at a height of 12 m. Soil heat flux was measured by flux plates buried at a depth of 3 cm. All
the micrometeorological measurements were logged as 20 min average values on a data acquisition sys-
tem. Stomatal resistance was not measured on this site. However, measurements of stomatal resistance
were available on a nearby fallow savannah of the west–central supersite WC centred on the village of
Fandou Beri, 10 km West of Banizoumbou and at the same altitude Hanan and Prince, 1997. These data
are used to test the outputs of the model. Resistance measurements of the lower abaxial surface of the
leaf were made with a transient porometer Mark II, Delta-T Devices, Cambridge, UK on 12 Guiera
shrubs selected at random on a line transect across the site and point resistance measurements were averaged
into hour mean values representing an estimate of the bulk average stomatal resistance of the canopy. More
details on the acquisition of stomatal resistance data are given in Hanan and Prince 1997, p. 539. The
structural characteristics of the two fallow savannah sites were roughly similar with shrub density a little
lower on the WC site LAI of 0.32 instead of 0.5. Soil characteristics are also very comparable D’Herbès
and Valentin, 1997. Consequently stomatal resistance values should be very similar for a same day on the
two sites and the comparison between the outputs of the model obtained at the EC supersite and the mea-
surements made at the WC supersite appears to be fully jusifiable. The data of 2 days of October 1992
10 and 15 were used. They are the only days when simultaneous measurements were made on both sites.
The month of October corresponds to the start of the dry season the last rainfall on the site occured on 14
September. The herbaceous canopy is already nearly dry with a mean height of about 0.5 m. Only the
data corresponding to the midday hours, from 9:00 to 15:00 hours, with net radiation generally higher
Fig. 2. For 2 days of October 1992 10 and 15, diurnal course of stomatal resistance of Guiera senegalensis between 9:00 and
15:00 hours, as predicted by the model on a 20 min basis and as measured by a porometer on an hourly basis data taken from
Hanan and Prince 1997.
than 400 W m
− 2
, were selected for these 2 days. The results are shown in Figs. 2 and 3 for the 2 days
retained in the comparison. Fig. 2a and b shows the diurnal course of stomatal resistance as calculated by
the model Eq. 8 from the component temperatures T
s
, T
f
and the micrometeorological data R
n
, G, T
a
, D
a
, u
a
recorded on the EC site and as measured the same day on the WC site. The measured data are
directly taken from Hanan and Prince 1997, Fig. 5a and b, after converting conductance values into re-
sistance, and the values of the coefficients involved in the model are those specified in the theoretical
development, without any adjustment. The agreement
J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131 125
Fig. 3. Predicted values of stomatal resistance vs. measured values on an hourly basis for the 2 days retained in the comparison 10 and 15 October 1992.
between experimental and modeled values is fairly good. Stomatal resistances are higher on 15 October
than on 10 October certainly due to an increasing soil water deficit, as no rain has occurred between these
two dates; and the diurnal trend exhibits a typical in- crease of stomatal resistance during the diurnal part of
the day. Additionally, Fig. 3 shows a plot of predicted versus measured hourly values of stomatal resistance.
The scatter is clearly about the 1:1 line, which indi- cates acceptable agreement. Consequently, two main
statements can be made from the comparison. Firstly, the values predicted by the model globally coincide
with the values measured during these two contrasted days. Secondly, the model accounts well for the char-
acteristic rise of stomatal resistance during the central hours of the day.
3.2. Operational expression of stomatal resistance and numerical simulations
The rest of the paper aims at developing an op- erational expression of stomatal resistance in terms
of component temperatures T
f
and T
s
to examine its general behavior and to assess its sensibility. Net ra-
diation and soil heat flux measurements are easy to make and can be directly used as input to the model as
shown in the previous section. However, when avail- able energy is not measured, R
n
and G can be pa- rameterized in terms of standard meteorological data
and canopy characteristics as described below. R
n
is expressed as a function of long-wave radiation bud-
get R
l
and incoming solar radiation S a meteorolog- ical variable commonly measured in the following
way: R
n
= 1 − a
e
S + R
l
R
l
= εL ↓ −σ T
4 r0
22 where a
e
is the effective surface albedo, ε the sur- face emissivity assumed to be constant during the
day, σ the Stefan–Boltzmann constant and T
r0
the nadir radiometric surface temperature. Surface tem-
perature being necessarily an input to the model, T
r0
can be either directly measured or calculated from component temperatures T
f
and T
s
by means of Eq. 19 or 21 with ϕ=0. L↓ is the incom-
ing long-wave radiation which can be calculated by means of the following empirical formula Brutsaert,
1982
L ↓= ε
a
σ T
4 a
ε
a
= 0.552e
17 a
23 where T
a
is the air temperature in Kelvin and e
a
the air vapor pressure expressed in hPa. It is possible to
use other formulae for the apparent emissivity of the atmosphere ε
a
without altering the generality of the results obtained. The long-wave radiation balance can
be rewritten as R
l
= εσ
[ε
a
T
4 a
− f
T
4 f
− 1 − f
T
4 s
] 24
where f =
1−exp−α
a
L is the shielding factor in
the sense defined by Deardorff 1978 i.e. the relative
126 J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131
area of the substrate covered by the vegetation. It is obtained from Eq. 20 with ϕ=0. Denoting foliage
and substrate albedos respectively by a
f
and a
s
, the effective albedo of the two-layer interface is
a
e
= f
a
f
+ 1 − f
2
a
s ∞
X
i=
f a
f
a
s i
= f
a
f
+ a
s
1 − f
2
1 − f a
f
a
s
25 This formula is obtained by adding the multiple reflec-
tions between the substrate and the canopy Taconet et al., 1986. Taking into account the formulae derived
above, the stomatal resistance formulation Eq. 8 transforms into
r
st
= D
a
γ + 1 + sγ T
f
− T
a
+ c
1
S + c
2
R
l
ρc
p
c
3
S + c
4
R
l
ρc
p
+ c
5
T
s
− T
f
+ c
6
T
a
− T
f
L 26
Coefficients c
i
i=1, . . . , 4 are expressed as c
1
= − 1 − a
e
[1 − α
g
exp−α
r
L ]r
a
+ [1 − exp−α
r
L ]r
af
} 27
c
2
= c
1
1 − a
e
28 c
3
= 1 − a
e
[1 − exp−α
r
L ]
29 c
4
= 1 − exp−α
r
L 30
Coefficients c
5
and c
6
are those of Eq. 8. Eq. 26 involves two types of input data: driving variables and
canopy characteristics. The driving variables are the standard weather data air temperature T
a
, air humid- ity D
a
and e
a
, wind velocity u
a
, solar radiation S and the component temperatures T
f
and T
s
obtained by infrared thermometry. The canopy is characterized
by structural data canopy height h, LAI L , leaf
width w, roughness length of the substrate z
0s
, radiative data component albedos a
f
and a
s
, surface emissivity ε and pseudo-constant coefficients c
d
, α
, α
a
, α
g
, α
r
, α
w
related to the type of canopy. Nu- merical simulations have been performed with Eq. 26
using the values of the parameters given in Table 1. Fig. 4 shows the relationship between stomatal resis-
tance and foliage temperature for different substrate temperatures in a given scenario with high solar ra-
diation and high air temperature. As expected r
st
is
Table 1 Standard values of model parameters used in the numerical sim-
ulations Symbol
Equation Value
w 16
0.02
a
z
0s
13, 14 0.01
a
z
r
11, 15, 17 5
a
a
f
25 0.20
a
s
25 0.20
ǫ 22, 24
0.97 c
d
12 0.2
α 16
0.005 α
a
20 0.5
α
g
27 0.3
α
r
18 0.5
α
w
14, 16 2.5
a
The values are measured in meters.
an increasing function of T
f
. When substrate temperature increases with foliage temperature kept
constant, stomatal resistance decreases. This is due to the fact that the sensible heat flux emanating from the
substrate contributes to increasing foliage tempera- ture as stomatal closure does. Since this contribution
increases with substrate temperature, the relative con- tribution of stomatal closure to the heating of the
foliage at a given foliage temperature is lower for a higher substrate temperature, which means that r
st
Fig. 4. Stomatal resistance vs. foliage temperature for differ- ent substrate temperatures in a given scenario: h=0.5 m, LAI=1,
S= 1000 W m
− 2
, T
a
= 35
◦
C, RH
a
= 60, u
a
= 2 m s
− 1
.
J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131 127
Fig. 5. Stomatal resistance vs. foliage temperature: a for different values of LAI; b for different values of vegetation height h m.
S= 800 W m
− 2
, T
a
= 30
◦
C, RH
a
= 60, u
a
= 2 m s
− 1
and T
s
= 40
◦
C.
should be lower. The influence of the main structural characteristics of the canopy LAI and vegetation
height on the relation r
st
= f
T
f
is examined in Fig. 5a and b. All other conditions being equal, the
stomatal resistance increases with an increased LAI for a given foliage temperature. The same behavior
occurs with canopy height: The higher the canopy, the greater the stomatal resistance. Fig. 6 illustrates
the role of wind velocity. All other conditions being equal, the greater the wind velocity, the greater the
stomatal resistance. This result is logical since greater wind velocity leads to increased energy dissipation
by convection, which tends to lower the foliage tem- perature. Hence, at a given foliage temperature, stom-
atal resistance should be greater for a greater wind velocity.
Fig. 6. Stomatal resistance vs. foliage temperature for different values 1, 2, 4 of wind velocity u
a
m s
− 1
. The scenario simulated is: h=0.5 m, LAI=1, S=800 W m
− 2
, T
a
= 30
◦
C, RH
a
= 60 and
T
s
= 40
◦
C.
3.3. Sensitivity analysis In the calculation of stomatal resistance from
Eq. 26 the major cause of uncertainty certainly comes from the component temperatures T
f
and T
s
, either directly measured or estimated from two direc-
tional measurements of radiometric surface tempera- ture. We have calculated the relative error made in r
st
due to an error in foliage temperature T
f
and substrate temperature T
s
, all other quantities being assumed to be correctly measured or estimated. Differentiating
Eq. 26 leads to δr
st
r
st
= δ
N N−δDD,where N and
D are respectively the numerator and denominator of
Eq. 26. Calculating δN and δD as a function of δT
f
and δT
s
and regrouping the terms in δT
f
and δT
s
yields δr
st
r
st
= 1
N ∂N
∂T
f
− 1
D ∂D
∂T
f
δT
f
+ 1
N ∂N
∂T
s
− 1
D ∂D
∂T
s
δT
s
31 where the derivatives of N and D with respect to T
f
and T
s
are detailed in Appendix A. In a given scenario, the variations of δr
st
r
st
corresponding to a given error in T
f
and T
s
δT
f
=± 1
◦
C and δT
s
=± 1
◦
C have been calculated as a function of foliage temperature T
f
. One should concede nevertheless that, when component
temperatures are derived from radiometric tempera- ture observations at two viewing angles, the error in
substrate and foliage temperatures will likely be larger
128 J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131
Fig. 7. Relative error in stomatal resistance estimate vs. foliage temperature for the range of foliage temperature corresponding ap-
proximately to 50r
st
500 s m
− 1
: a for a fixed error |dT
f
|=1 in foliage temperature T
f
; b for a fixed error |dT
s
|=1 in substrate temperature T
s
. The scenario simulated is character- ized by: h=0.5 m, LAI=1, S=800 W m
− 2
, T
a
= 30
◦
C, RH
a
= 60,
u
a
= 2 m s
− 1
.
than 1
◦
C in practice mainly because of some uncer- tainty in the fractional cover estimate. The results of
the calculation are shown in Fig. 7. It appears that r
st
is very sensitive to an error in foliage temperature since
an error of ±1
◦
C leads to a relative error in stomatal resistance of not less than 30. Conversely, r
st
is very little sensitive to an error in substrate temperature. An
error of ±1
◦
C in T
s
leads to a relative error much lower than 5 in stomatal resistance. This means that
foliage temperature should be measured or determined with a greater care than substrate temperature because
its impact on stomatal resistance estimate is greater. Among the canopy characteristics required as inputs to
the model, leaf area index is certainly the most difficult to determine accurately, especially if evaluated with
remote sensing techniques Sellers and Hall, 1993. It is the reason why the model sensitivity to an uncer-
tainty in LAI has been studied in the way illustrated in Fig. 8. The relative error δr
st
r
st
in stomatal resis- tance resulting from a ±25 variation in LAI is plot-
ted against foliage temperature for a given scenario. The effect on r
st
appears to be relatively small: the relative error is much lower than 5 over the largest
part of the temperature range. The sensitivity of stom- atal resistance expression to the main coefficients in-
volved in its formulation has also been assessed. Only the two coefficients related to the partitioning of avail-
Fig. 8. Relative error in stomatal resistance estimate resulting from a given error ±25 in LAI for the same scenario as in Fig. 7
and for the range of foliage temperature corresponding roughly to 50r
st
500 s m
− 1
.
able energy have been retained, namely α
r
, which de- scribes the exponential decay in net radiation within
the canopy, and α
g
, which determines the soil heat flux from the net radiation reaching the substrate. The
results are shown in Figs. 9 and 10. The magnitude of the response to changes in the other coefficients those
involved in the resistances of the air: c
d
, α , α
w
has been thoroughly examined by Shuttleworth and Wal-
lace 1990 in relation with their proper formulation of stomatal resistance and will not be re-examined here.
In sparse and clumped vegetation, the use of Beer’s law Eq. 18 for estimating net radiation divergence
Fig. 9. Relative error in stomatal resistance estimate due to a given error ±0.2 in coefficient α
r
involved in the partitioning of available energy with the preferred value α
r
= 0.5. Same scenario
and same conditions as in Fig. 7.
J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131 129
Fig. 10. Relative error in stomatal resistance estimate due to a given error ±0.1 in coefficient α
g
involved in soil heat flux estimation with the preferred value α
g
= 0.3. Same scenario and
same conditions as in Fig. 7.
is certainly questionable unless the extinction coeffi- cient α
r
is properly adjusted. The problem is that the magnitude of this coefficient is not easily estimated a
priori. In Fig. 9, for a given scenario, the relative error made in stomatal resistance δr
st
r
st
is plotted versus foliage temperature when α
r
is set to its prescribed value ±0.2 a realistic uncertainty in the case of sparse
vegetation. In Fig. 10, the same type of graph is drawn with α
g
Eq. 27 set to its prescribed value ±0.1 also a realistic uncertainty. The sensitivity of r
st
to α
r
appears to be much greater than to α
g
. For the spe- cific range 30
◦
CT
f
37
◦
C, a variation of +0.2 and –0.2 in the prescribed value of α
r
0.5 leads to a rel- ative error a little bit less than −40 and greater than
+ 70 respectively. On the other hand, a variation of
± 0.1 in the prescribed value of α
g
0.3 does not lead to a relative error greater than 8 it is lower than
5 on most of the range. Consequently, an accurate determination of Beer’s law coefficient α
r
appears to be determinant in the implementation of this stomatal
resistance model, whereas the role of α
g
is of minor importance.
4. Conclusion
