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In-Depth Information
τ
0
=
Σ
τ
i
−
τ
i
−1
),
Suppose the optical thickness of the cloud is equal to
(
τ
i
is the optical deepness of the
i
-th observational level. The values of
where
ε
i
=
α
i
are the volume coefficients of extinction,
absorption, and scattering, where
z
i
is the geometrical deepness of the
i
-th
level. The value of
τ
i
−
τ
i
−1
)
|
ε
i
=
κ
i
+
(
(
z
i
−
z
i
−1
),
ω
0,
i
=
α
i
|ε
i
isthesinglescatteringalbedointhe
i
-th sublayer
(situated between
i
−1
st
and
i
-th levels). In the short-wave spectral range one
can assume 1−
ω
0,
i
<<
1. The phase function is characterized withmean cosine
of scattering angle
γ
=
γ
:
g
i
cos
.Letthenumberoftheanalyzedsublayersbe
equal to
N
.
We should mention the main points of deriving the formulas defining pa-
rameters
s
2
and
τ
from
the irradiances observations
within the cloud. The
cloud sublayer adjusting the cloud top is described by the first group of the
formulas, the inner sublayers are described by the second group and adjusting
the cloud bottom sublayer is described by the third group. Consider (2.26) to-
gether with (2.42) and expansions (2.29) and (2.30) to infer the corresponding
formulas. The equations linear in value
s
2
canbeobtainedafterthealgebraic
manipulation while keeping the terms proportional to
s
2
. The results are con-
sidered in detail elsewhere (Melnikova 1998; Melnikova and Fedorova 1996;
Melnikova and Mikhailov 2001). The formulas for the optical thickness of the
sublayer (
τ
i
−1
) between measurement levels
i
−1and
i
are deduced from
the combination of (2.26) and (2.42). The following expressions are obtained:
- for the cloud top sublayer (level numbers are 0, 1)
τ
i
−
τ
1
)
2
F
(0)
2
−
F
(
s
1
=
(6.3)
µ
0
)
2
−4(
F
1
+
F
1
)
2
+
F
(
τ
1
)
2
(2
n
2
−9
q
2
)−2
F
(0)
a
2
(
µ
0
)
16
K
0
(
ln
(
F
(
τ
1
)−4
F
1
s
(1−2
s
))(
F
(0) +
s
(4
K
0
(
µ
0
)+
a
2
(
µ
0
)
s
))
1
2
s
τ
1
=
3(1 −
g
)
τ
1
)+4
F
1
s
(1−2
s
))(
F
(0) −
s
(4
K
0
(
µ
0
)−
a
2
(
µ
0
)
s
))
(
F
(
- for the internal sublayers (level numbers are
i
−1,
i
)
τ
i
−1
)
2
−
F
(
τ
i
)
2
]
F
2
(
τ
i
−1
)
[
F
(
s
i
=
(6.4)
τ
i
)
2
F
↓
(
τ
i
−1
)
F
↑
(
τ
i
−1
)−
F
(
τ
i
−1
)
2
F
↓
(
τ
i
)
F
↑
(
τ
i
)]
16[
F
(
ln
(
F
(
τ
i
−1
)+4
F
i
−1
s
(1−2
s
))(
F
(
τ
i
)−4
F
i
s
(1−2
s
))
1
2
s
τ
i
−1
−
τ
i
)
=
3(1 −
g
)(
τ
i
−1
)−4
F
i
−1
s
(1−2
s
))(
F
(
τ
i
)+4
F
i
s
(1−2
s
))
(
F
(
- for the cloud bottom sublayer (level numbers are
N
−1,
N
)
τ
N
−1
)
2
−
F
(
τ
N
)
2
F
(
s
2
N
=
4(
F
N
−1
+
F
N
−1
)
2
−16
F
↑
N
−
F
(
τ
N
−1
)
2
(2
n
2
−9
q
2
)−24
q
F
N
F
(
τ
N
)
τ
N
−
τ
N
−1
=
τ
N
−
τ
N
−1
)
=
3(1 −
g
)(
(6.5)
ln
l
(
F
(
τ
N
−1
)+4
F
N
−1
s
(1−2
s
))(
F
(
τ
0
)−4
F
N
s
(1+3
q
s
))
1
2
s
=
τ
N
−1
)−4
F
N
−1
s
(1−2
s
))(
F
(
τ
0
)+4
F
N
s
(1−3
q
s
))
(
F
(
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