Geoscience Reference
In-Depth Information
π
S
from the
Define these functions to the direction of the incoming irradiance
µ
, 0) and assume this back-standing geometry completely similar to
thecaseoftheilluminationfromthetopthen
bottom (
µ
τ
0
in this
>
0andconsider
case as a top of the atmosphere.
The symmetry relations are the most important property of the reflection
and transmission functions:
ρ
µ
µ
ϕ
=
ρ
µ
µ
ϕ
(
,
0
,
)
(
0
,
,
),
ρ
µ
µ
0
,
ϕ
=
ρ
µ
0
,
µ
ϕ
˜
(
,
)
˜
(
,
),
(1.78)
σ
µ
µ
0
,
ϕ
=
σ
µ
0
,
µ
ϕ
(
,
)
˜
(
,
).
In general case, the proof of (1.78) is complicated and presented e. g. in the
topic by Yanovitskij (1997). Specify the analogous functions for the system
“atmosphere plus surface”
¯
ρ
µ
µ
0
,
ϕ
σ
µ
µ
0
,
ϕ
).
It is possible to exclude the azimuthal dependence of the reflection and
transmission functions presenting them as expansions over the azimuthal
harmonics as follows:
(
,
)and
¯
(
,
N
ρ
µ
µ
0
,
ϕ
=
ρ
0
(
µ
µ
0
)+2
1
ρ
m
(
µ
µ
0
) cos
m
ϕ
(
,
)
,
,
,
(1.79)
=
m
σ
µ
µ
0
,
ϕ
ρ
µ
µ
0
,
ϕ
and the analogous expressions for the functions
(
,
),
˜
(
,
)etc.
ρ
m
(
µ
µ
0
,)
=
ρ
m
(
µ
0
,
µ
Every harmonic satisfies relations of the symmetry (
,
)
etc.).
Now consider the simplest but widespread case of the orthotropic surface
with albedo
A
. It is easy to demonstrate (Sobolev 1972) that the consideration
of the only zeroth harmonics for the isotropic reflection is enough. Actually, if
non-zeroth harmonics (1.79) varied it would mean the azimuthal dependence
of the reflected radiation as per definitions (1.76)-(1.77) that contradicts the
assumption about the orthotropness of the reflection.
Write the explicit form of the integral operators from (1.75). According to
the definition of
T
operator (1.58) and to the expression for extraterrestrial
radiance
I
0
(1.41) we are getting the following:
π
2
1
ϕ
η
T
(
µ
,
ϕ
)
µ
−
ϕ
−0)
=
µ
ϕ
π
δ
µ
0
)
δ
=
π
µ
µ
0
,
ϕ
TI
0
d
d
,
,
S
(
(
ST
(
,
,0)
0
0
comparing it with (1.76) and taking into account only the zeroth harmonics
the following is inferred:
=
µ
0
µ
µ
,
ϕ
ϕ
)
π
ρ
0
(
µ
µ
0
)
T
(
,
,
,
for the top of the atmosphere
=
µ
0
µ
µ
,
ϕ
ϕ
)
π
σ
0
(
µ
µ
0
)
T
(
,
,
,
for the bottom of the atmosphere
(
µ
≡
µ
0
ϕ
≡
0) .
and
Search WWH ::
Custom Search