68
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
The estimation of H in Equation 4.2 requires the computation of r ah , which in SEBAL
is based on the single-layer approach (Figure 4.1, left panel) given by
r
z d
z
z L
z d
z
ah
m
m
MO
h
=
−
−
⋅
−
ln
( ,
) ln
0
0
0
0
Ψ
−
⋅
Ψ h
MO
z L
k u
( ,
)
,
2
(4.7)
where k is the von Kármán number (0.41), u is the wind speed at height z (in meters
per second), Ψ h and Ψ m are the two stability correction functions for momentum and
heat transfer, respectively, and L MO is the Monin–Obukhov length (in meters).
The correction functions Ψ h , Ψ m , and L MO depend on H and then on r ah . For this
reason, the solution of Equations 4.2 and 4.7 is calculated by means of an iterative
procedure. In SEBAL, as stressed in the previous paragraph, the empirical adjustment of Equation 4.2 is carried out assuming a linear relationship between T r and
δT = (T 0h – T a ) to be calibrated on the basis of two boundary conditions, including dry
nonevaporating and fully wet surfaces.
In contrast to SEBAL, the TSEB scheme considers the contributions of soil and
canopy separately (Figure 4.1, right panel) and uses a few additional resistances to
retrieve H. In particular, H is expressed as the sum of the contributions of soil, H s ,
and canopy, H c , according to the assumption of an “in-series” resistance network
(Shuttleworth and Wallace 1985).
This allows computing T 0h in Equation 4.2 by using the following expression:
T
T
r
T
r
T
r
r
r r
h
a
ah
c
x
s
s
ah
x
s
0
1 1 1
=
+ +
+ +
,
(4.8)
R n
R n
r ah
λET
λE s
λET c
H s
H
H c
r s
T s
T 0
T a
r x
r a
G 0
G 0
δT
H
T c
FIGURE 4.1 Scheme of the key energy balance variables and “in-series” resistances used
in SEBAL (left panel) and TSEB (right panel) models.
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
The estimation of H in Equation 4.2 requires the computation of r ah , which in SEBAL
is based on the single-layer approach (Figure 4.1, left panel) given by
r
z d
z
z L
z d
z
ah
m
m
MO
h
=
−
−
⋅
−
ln
( ,
) ln
0
0
0
0
Ψ
−
⋅
Ψ h
MO
z L
k u
( ,
)
,
2
(4.7)
where k is the von Kármán number (0.41), u is the wind speed at height z (in meters
per second), Ψ h and Ψ m are the two stability correction functions for momentum and
heat transfer, respectively, and L MO is the Monin–Obukhov length (in meters).
The correction functions Ψ h , Ψ m , and L MO depend on H and then on r ah . For this
reason, the solution of Equations 4.2 and 4.7 is calculated by means of an iterative
procedure. In SEBAL, as stressed in the previous paragraph, the empirical adjustment of Equation 4.2 is carried out assuming a linear relationship between T r and
δT = (T 0h – T a ) to be calibrated on the basis of two boundary conditions, including dry
nonevaporating and fully wet surfaces.
In contrast to SEBAL, the TSEB scheme considers the contributions of soil and
canopy separately (Figure 4.1, right panel) and uses a few additional resistances to
retrieve H. In particular, H is expressed as the sum of the contributions of soil, H s ,
and canopy, H c , according to the assumption of an “in-series” resistance network
(Shuttleworth and Wallace 1985).
This allows computing T 0h in Equation 4.2 by using the following expression:
T
T
r
T
r
T
r
r
r r
h
a
ah
c
x
s
s
ah
x
s
0
1 1 1
=
+ +
+ +
,
(4.8)
R n
R n
r ah
λET
λE s
λET c
H s
H
H c
r s
T s
T 0
T a
r x
r a
G 0
G 0
δT
H
T c
FIGURE 4.1 Scheme of the key energy balance variables and “in-series” resistances used
in SEBAL (left panel) and TSEB (right panel) models.