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follows (Melnikova 1992; Melnikova et al. 2000):
=
ϕ
µ
ϕ
µ
µ
µ
µ
µ
(
)
(
0
)+
g
[4.8
−3.0(
0
+
) + 1.9]
0
ρ
0
(
µ
µ
0
)
,
.
(2.39)
µ
0
+
µ
4(
)
In the case of the Henyey-Greenstein phase function the high harmonics are
close to zero (
ρ
m
(
µ
µ
0
)
≈
0,
m>
0) if either of zenith angle cosines
µ
µ
0
,
and
µ
limit
distinguish for different harmonics
and they are shown in Tables 2.3 and 2.4. The approximation by (2.37) with
coefficients
a
m
,
b
m
and
c
m
in Tables 2.3-2.4 gives an acceptable presentation
for all the harmonics of the reflection function considered here. The errors
of this approximation have been shown to depend on the values of the zenith
solar and viewing angles cosines, on the number of the harmonic
m
,andon
phase function parameter
g
(Melnikova et al. 2000). Some details of the error
analysis will be presented in Sect. 2.4.
The presented totality of rigorous asymptotic formulas (2.24)-(2.28), ex-
pansions (2.29)-(2.31) and approximations (2.32)-(2.35) allows computing
the reflected and transmitted radiance and irradiance together with the radia-
tive flux divergence for the cloud layer if the layer properties and the geometry
of the problem are known. The considered model has to satisfy the applica-
bility ranges of the presented formulas:
large optical thickness
and
weak true
absorption
. These ranges will be analyzed in Sect. 2.4 in detail. However, it is
necessary to point out that for the application of (2.24)-(2.28) the large optical
thickness is a condition with known asymptotic functions and constants. The
using of expansions (2.29)-(2.31) needs the weak absorption condition.
Wewouldliketomentionthattheapproximationformulaforthereflection
function,forwhichneedstobeknownthewholephasefunction,hasalsobeen
proposed in the study by Konovalov (1997).
µ
limit
.Thevaluesof
are greater than
2.2.4
Diffused Radiation Field Within the Cloud Layer
τ
N
−1
>>
1and
τ
1
>>
1) is described with formulas different from those presented above. The
correspondent analysis could be found in Minin (1988) and Ivanov (1976).
Here we are offering the results useful for further consideration.
The diffused radiance in energetic units in the diffusion domain at optical
depth
τ
0
−
Radiation within the cloud layer (in the diffusion domain:
τ
τ
τ
0
−
τ
>>
1 and is expressed with the
equation, derived in the topic by Minin (1988)
satisfied conditions
>>
1,
τ
µ
µ
τ
=
µ
µ
τ
I
(
,
,
0
,
0
)
SK
(
0
)
0
exp(−
k
)
µ
τ
0
−
τ
|
µ
)
l
exp(−
k
(
τ
0
−
τ
i
(
)exp(
k
(
))
i
(−
))
(2.40)
×
,
1−
ll
exp(−2
k
τ
0
)
µ
where
S
is the solar constant, function
i
(
) characterizes the angular depen-
dence of the radiance in deep levels of the semi-infinite atmosphere. The
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