Geoscience Reference
In-Depth Information
In the context of the surface energy balance, we are interested in the total amount
of radiation relected from a surface. Therefore, the BRDF can be integrated over all
values of
θ
out
and
ϕ
out
:
π π
/
2
2
=
∫∫
0
r
(, ,
λθϕ
)
r
(, ,
λθϕθϕ ϕθ
,
,
)
dd
(2.21)
in
in
in
in
out
out
out
out
0
But we cannot simplify further. Hence, the relected radiation becomes:
(2.22)
K
↑
(, ,
λθϕ
)
=
K
↓
(, ,
λθϕ λθϕ
)(, ,
r
)
in
in
in
in
in
in
From this equation it is clear that the relected radiation depends not only on wave-
length (of the incoming radiation, and the spectral properties of the surface), but also
on the directional composition of the incoming radiation.
Although in the context of the surface energy balance, the direction of the upwell-
ing radiation is irrelevant (leading to the simpliied Eq. (
2.22
)), in the context of
remote sensing of the surface relectance the full BRDF has to be taken into account
(see
Figure 2.8
). In that application, the sensor on board of the satellite views the radi-
ation relected by Earth's surface into a particular direction (
θ
out
and
ϕ
out
). This direc-
tion depends on the location of the satellite at that particular time. For a cloud-free
situation, most of the radiation is direct radiation, coming from a direction
θ
in
and
ϕ
in
.
Finally, remote sensing sensors usually are observing in narrow wavelength bands.
Thus, such a remote sensing observation of the surface relectance in fact is only a
small part of the entire BRDF.
Albedo
In many applications, we are not interested in (or have no information on) the spec-
tral and directional composition of the incoming radiation, and therefore we deine a
broadband (all wavelengths in the shortwave range), hemispheric (all incident angles)
albedo:
λ
π
/
2
2
π
2
∫∫
(, ,
∫
↓
K
λθϕ θϕ λϕθ
)(, ,
r
)
dd d
in
in
in
in
in
in
↑
K
K
0
0
λ
r
≡=
(2.23)
1
λ
↓
π
/
2
2
π
2
∫
∫
∫
↓
K
(, ,
λθϕλϕθ
)
dd d
in
in
in
in
0
0
λ
1
The main purpose of showing this ratio of two threefold integrals is to clarify that
the broadband hemispheric albedo (which is generally referred to as simply 'albedo')
depends not only on the properties of the surface (as we would like to), but also on
the characteristics of the incoming radiation. This is due to the fact that the two occur-
rences of
K
↓
(within the integral) in the numerator and denominator of Eq. (
2.23
) do
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