69
Comparative Analysis of Surface Energy Balance Models
where T c is the canopy temperature (in Kelvin), T s is the soil temperature (in Kelvin),
r s is the soil resistance to heat transfer (in seconds per meter) (Sauer et al. 1995), and
r x is the resistance of the canopy boundary layer (in seconds per meter; McNaughton
and van den Hurk 1995).
An estimate of the vegetation directional fractional cover, f θ , (dimensionless) is
used to estimate T c and T s from T r using the following equation:
T
f T
f T
r
c
s
=
+ −
θ
θ
4
4
1 4
1
(
)
,
/
(4.9)
whereas r s is computed from a relatively simple formulation predicting the wind speed
close to the soil surface (Goudriaan 1977; Norman et al. 1995; Kustas and Norman
1999a,b), and r x is derived assuming a parameterization suggested by Grace (1981).
With the additional use of the Priestley–Taylor formulation (Priestley and Taylor
1972) for estimating canopy transpiration and, consequently, T c , the closure of the set of
available Equations 4.8 and 4.9 is achieved. In the TSEB model, since Priestly–Taylor
formulation is appropriate for well-watered grass surface, when the canopy is in water
stress, the result is an overestimation of canopy transpiration, which in turn results in
a condensation on the soil surface based on the energy balance principle. The latter
condition is not physically realistic during daytime and is overridden by searching for a
new solution by iteratively reducing the Priestley–Taylor coefficient (Kustas et al. 2004).
For both models, once the spatial distributions of R n , G 0 , and H are obtained, the
spatial distribution of the instantaneous λET (in watts per square meter) is computed
using Equation 4.1. Additionally, the fluxes can be used to derive the evaporative
fraction, Λ (Menenti and Choudhury 1993):
Λ =
−
λET
R G
n
0
.
(4.10)
Different from the SEBAL and TSEB models, S-SEBI is a simplified approach; first
introduced by Roerink et al. (2000), it allows the direct computation of instantaneous
evaporative fraction from an analysis of the correlation between the surface albedo
and T r . It has been observed that T r and α are correlated over an area characterized
by constant atmospheric forcing and their relationship can be applied to determine
the effective land surface properties (Menenti et al. 1989). A simple representation
of the S-SEBI basic principle is given in Figure 4.2.
Basically, the α–T r scatterplots are bounded by two lines representing minimum
and maximum T r values for all albedo conditions (as shown in Figure 4.2). These two
lines correspond to the maximum sensible heat flux (H max ) and, subsequently, low
ET, and to the maximum latent heat flux (λET max ) and, therefore, the potential ET.
With these assumptions, the evaporative fraction can be determined as
Λ =
+ −
−
+
−
a
b T
a
a
b b
H
H
r
H
E
H
E
α
α
(
)
(
)
,
(4.11)
Comparative Analysis of Surface Energy Balance Models
where T c is the canopy temperature (in Kelvin), T s is the soil temperature (in Kelvin),
r s is the soil resistance to heat transfer (in seconds per meter) (Sauer et al. 1995), and
r x is the resistance of the canopy boundary layer (in seconds per meter; McNaughton
and van den Hurk 1995).
An estimate of the vegetation directional fractional cover, f θ , (dimensionless) is
used to estimate T c and T s from T r using the following equation:
T
f T
f T
r
c
s
=
+ −
θ
θ
4
4
1 4
1
(
)
,
/
(4.9)
whereas r s is computed from a relatively simple formulation predicting the wind speed
close to the soil surface (Goudriaan 1977; Norman et al. 1995; Kustas and Norman
1999a,b), and r x is derived assuming a parameterization suggested by Grace (1981).
With the additional use of the Priestley–Taylor formulation (Priestley and Taylor
1972) for estimating canopy transpiration and, consequently, T c , the closure of the set of
available Equations 4.8 and 4.9 is achieved. In the TSEB model, since Priestly–Taylor
formulation is appropriate for well-watered grass surface, when the canopy is in water
stress, the result is an overestimation of canopy transpiration, which in turn results in
a condensation on the soil surface based on the energy balance principle. The latter
condition is not physically realistic during daytime and is overridden by searching for a
new solution by iteratively reducing the Priestley–Taylor coefficient (Kustas et al. 2004).
For both models, once the spatial distributions of R n , G 0 , and H are obtained, the
spatial distribution of the instantaneous λET (in watts per square meter) is computed
using Equation 4.1. Additionally, the fluxes can be used to derive the evaporative
fraction, Λ (Menenti and Choudhury 1993):
Λ =
−
λET
R G
n
0
.
(4.10)
Different from the SEBAL and TSEB models, S-SEBI is a simplified approach; first
introduced by Roerink et al. (2000), it allows the direct computation of instantaneous
evaporative fraction from an analysis of the correlation between the surface albedo
and T r . It has been observed that T r and α are correlated over an area characterized
by constant atmospheric forcing and their relationship can be applied to determine
the effective land surface properties (Menenti et al. 1989). A simple representation
of the S-SEBI basic principle is given in Figure 4.2.
Basically, the α–T r scatterplots are bounded by two lines representing minimum
and maximum T r values for all albedo conditions (as shown in Figure 4.2). These two
lines correspond to the maximum sensible heat flux (H max ) and, subsequently, low
ET, and to the maximum latent heat flux (λET max ) and, therefore, the potential ET.
With these assumptions, the evaporative fraction can be determined as
Λ =
+ −
−
+
−
a
b T
a
a
b b
H
H
r
H
E
H
E
α
α
(
)
(
)
,
(4.11)