120 J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131
satellite-based infrared sensors, foliage temperature is often difficult to separate from the soil component,
which makes the CWSI poorly reliable. To overcome this difficulty, Moran et al. 1994 developed the veg-
etation index-temperature VIT trapezoid method, based on a combination of spectral vegetation indices
and composite surface temperature measurements. Jones 1999 modified the calculation of the CWSI
by replacing theoretical estimates of the upper bound and baseline temperatures by measured temperatures
of appropriate reference surfaces.
Concurrently, attempts
have been
made to
directly determine canopy resistance from the surface temperature of the foliage, as measured by infrared
thermometry, in conjunction with standard meteoro- logical measurements above the canopy. When there
is complete canopy cover, the single-layer approach underlying the Penman–Monteith equation Monteith,
1965 provides an adequate basis for this determina- tion Jackson et al., 1981; Hatfield, 1985. For sparse
vegetation, the problem is not so straightforward. Smith et al. 1988 developed and tested a two-layer
approach based on the Shuttleworth–Wallace model 1985 to calculate the mean stomatal resistance of
a sparse crop from infrared measurements of foliage temperature. However, a major drawback of their
approach is that the theoretical expression of foliage resistance is a function of soil evaporation that is a pri-
ori unknown. Their assumption of using equilibrium evaporation as an estimate of soil evaporation limits
the applicability of their method because evidently it does not hold when soil dries out. In a similar man-
ner, Shuttleworth and Gurney 1990 investigated the performance of the Shuttleworth–Wallace model to
calculate the canopy resistance from measurements of foliage temperature. But the exacting assumption ap-
pearing in Smith et al. 1988 still exists in the form of a fixed and arbitrary value of soil surface resistance.
They recognize quote that: “In practical applica- tions a better specification of substrate interaction is
identified as the primary problem in using this theory in very sparse canopies”, adding that “a measurement
of substrate temperature used in conjunction with [basic equations] represents an attractive and simple
alternative” Shuttleworth and Gurney, 1990, p. 517.
The present study follows the lines drawn by these previous papers in the sense that it aims at present-
ing and evaluating the theoretical basis needed to infer stomatal resistance in sparse vegetation from measure-
ments of radiometric surface temperature. It is shown that stomatal resistance can be calculated in a more re-
liable manner than in the previous models, i.e. without major assumption concerning the substrate interaction,
when foliage and substrate temperatures are simul- taneously used in the calculation. Consequently, the
model presented constitutes an upgrading of the pre- vious models of Smith et al. 1988 and Shuttleworth
and Gurney 1990. The paper is divided into two main sections. The first section infers the theoretical expres-
sion relating stomatal resistance to foliage and sub- strate temperatures. The second section presents the
practical implementation of the model with an experi- mental validation, numerical simulations and a sensi-
tivity analysis.
2. Theoretical development
The basic equations used are those of the one-dimensional two-layer model originally devised
by Shuttleworth and Wallace 1985 and updated by Choudhury and Monteith 1988 and Shuttleworth
and Gurney 1990 Fig. 1. This model is based on diffusion theory K-theory even though turbulent
transport within canopies is not a strict diffusive pro- cess. In the recent years, news theories have been
Fig. 1. Schematic diagram and potential-resistance network for the one-dimensional two-layer model. The definitions of the symbols
are given in Section 5.
J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131 121
developed to describe the turbulent transport of scalar within the canopy such as the ‘higher-order-closure’
approach Meyers and Paw U, 1987 or the La- grangian approach Raupach, 1989. Unfortunately,
these new approaches, which are supposed to replace the traditional K-theory, are generally too complex
to be easily implemented in simple canopy models for practical applications. In the Lagrangian approach
the profile of scalar concentration is described as the sum of a far-field contribution predicted by diffu-
sion theory and a near-field contribution, for which the diffusion equation fails. Van den Hurk and Mc-
Naughton 1995 showed that the non-diffusive part of the transport process in a two-layer resistance
model could be accounted for by adding a near-field resistance in series with the bulk boundary-layer
resistance of the upper-layer foliage. Although sur- prising, this additional resistance has been found to
be small in comparison with other aerodynamic resis- tances with a small effect on canopy microclimate
and does not call into question the basic principles of the Shuttleworth–Wallace model. McNaughton
and van den Hurk 1995 completed the revision of the two-layer model by re-evaluating the other re-
sistances. Their conclusion is quote “K-theory in the guise of far-field theory is an adequate basis for
constructing two-layer models”, which justifies the approach used in the following development.
2.1. Formulation of stomatal resistance Energy balance equations for the vegetation layer
and the substrate layer can be detailed respectively as a function of foliage temperature T
f
and substrate
temperature T
s
in the following manner Shuttleworth and Wallace, 1985; Choudhury and Monteith, 1988
A − A
s
= ρc
p
γ [e
∗
T
f
− e ]
r
sf
+ r
af
+ ρc
p
T
f
− T
r
af
1 A
s
= ρc
p
γ [e
∗
T
s
− e ]
r
ss
+ r
as
+ ρc
p
T
s
− T
r
as
2 where A=R
n
− G
is the total available energy, with R
n
the net radiation and G the soil heat flux; A
s
= R
ns
− G
the available energy of the substrate level, with R
ns
the net radiation reaching the substrate. T and e
are the air temperature and the air vapour pressure at canopy source height, respectively. e
∗
T denotes the saturated vapor pressure at temperature T. r
as
denotes the aerodynamic resistance between the substrate and
the canopy source height and r
af
is the bulk bound- ary layer resistance of the canopy foliage. r
ss
is the substrate resistance, r
sf
the foliage resistance to wa- ter vapor transfer, γ the psychrometric constant, ρ the
air density and c
p
is the specific heat of air at con- stant pressure. These basic equations can be solved
for substrate and foliage resistances by means of the following manipulations. Linearizing the difference of
saturated pressure between the foliage and the canopy source height leads to
e
∗
T
f
− e =
D +
sT
f
− T
3 where D
is the water vapor pressure deficit at canopy source height and s the slope of the saturated vapor
pressure curve at the temperature of the air. The sen- sible heat flux H written between the canopy source
height and the reference height gives
T =
T
a
+ Hr
a
ρc
p
4 A similar equation can be written for the saturation
deficit Shuttleworth and Wallace, 1985, Eq. 8 D
= D
a
+ [H s + γ − γ A]r
a
ρc
p
5 Combining Eqs. 3–5 with Eq. 1 leads to
r
sf
= D
a
γ − [Ar
a
+ A − A
s
r
af
]ρc
p
+ 1 + sγ T
f
− T
s
A − A
s
ρc
p
+ [Hr
a
ρc
p
− T
f
− T
a
]r
af
6 Writing that the total flux of sensible heat is
the sum of the contributions from each layer H=ρc
p
[T
f
− T
r
af
+ T
s
− T
r
as
] and taking into account Eq. 4 leads to
H = ρc
p
[T
s
r
as
+ T
f
r
af
− T
a
1r
as
+ 1r
af
] [1 + r
a
1r
as
+ 1r
af
] 7
Replacing H in Eq. 6 by Eq. 7 yields the following expression of foliage resistance
122 J.P. Lhomme, B. Monteny Agricultural and Forest Meteorology 104 2000 119–131
r
sf
= D
a
γ − [Ar
a
+ A − A
s
r
af
]ρc
p
+ 1 + sγ T
f
− T
a
A − A
s
ρc
p
+ r
a
T
s
− T
f
+ r
as
T
a
− T
f
r
a
r
af
+ r
as
+ r
as
r
af
8 The mean stomatal resistance r
st
per unit leaf area is calculated as r
st
= r
sf
L , where L
is the leaf area index Lhomme, 1991. We have to note that a similar
calculation carried out on the basis of substrate energy balance 2 leads to a similar expression for substrate
resistance
r
ss
= D
a
γ − Ar
a
+ A
s
r
as
ρc
p
+ 1 + sγ T
s
− T
a
A
s
ρc
p
+ r
a
T
f
− T
s
+ r
af
T
a
− T
s
r
a
r
as
+ r
af
+ r
af
r
as
9 Eq. 8 should be valid for the limit case of a closed
canopy. When there is no substrate interaction, A
s
= 0 and the resistance r
as
is infinite in Eq. 8, which means that the expression of r
sf
transforms into
r
sf
= D
a
γ −Ar
a
+ r
af
ρc
p
+ 1 + sγ T
f
− T
a
Aρc
p
+ T
a
− T
f
r
a
+ r
af
10 This equation is directly obtained from the one-layer
description of the vegetation–atmosphere interac- tion following the simpler Penman–Monteith format
Shuttleworth and Gurney, 1990, Eq. 4.
2.2. Specification of air resistances and available energy partition
The aerodynamic resistance r
a
assumed to be the same for heat and water vapor is calculated by means
of the classical formula that takes into account the stability correction functions for wind and temperature
Brutsaert, 1982
r
a
= 1
k
2
u
a
ln z
r
− d
z −
9
h
z
r
L .
× ln
z
r
− d
z −
9
m
z
r
L 11
u
a
is the wind speed at the reference height z
r
, k
is von Karman’s constant, 9
h
and 9
m
are the integral diabatic correction functions respec-
tively for heat and momentum given by Paulson 1970. L is the Monin–Obukhov length defined by
L=−ρc
p
T
a
u
3 ∗
kgH, where u
∗
is the friction velocity, g
is the acceleration of gravity, T
a
is the air tempera- ture expressed in Kelvin and H is the sensible heat flux
counted positively upwards. The zero plane displace- ment height d and the roughness length for momentum
z are determined following Choudhury and Monteith
1988, who fitted simple functions to the curves ob- tained by Shaw and Pereira 1982 from second-order
closure theory d =
1.1h ln1 + X
14
X = c
d
L 12
z =
z
0s
+ 0.3hX
12
0 X 0.2 0.3h1 − d h
0.2 X 1.5 13
where c
d
= 0.2 is the mean drag coefficient assumed
to be uniform within the canopy, h is the height of the canopy and z
0s
is the roughness length of the sub- strate. For bare soil z
0s
is commonly taken as 0.01 m Shuttleworth and Wallace, 1985. For vegetated sub-
strate, z
0s
is calculated as one-tenth of the height of the lower vegetation Lhomme et al., 1997. The
formulations of the resistances within the canopy are essentially those based on K-theory proposed by
Choudhury and Monteith 1988. The aerodynamic resistance between the substrate and the source height
of the whole canopy d+z
is defined as the integral of the reciprocal of eddy diffusivity over the height
range [z
0s
, d+z ]
r
as
= h
expα
w
α
w
Kh exp
−α
w
z
0s
h −
exp −α
w
d + z h
14 K
h is the value of eddy diffusivity at canopy height given by
Kh = k
2
u
a
h − d {
ln[z
r
− dz
] − 9
m
z
r
L} 15
The bulk boundary-layer resistance of the canopy is calculated by integrating the leaf boundary-layer
conductance over the canopy height, assuming leaf area index to be uniformly distributed with height
Choudhury and Monteith, 1988
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
