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µ
Tab l e 2 . 1 .
Values of escape function
K
0
(
) for cloud layers (0. 65
≤
g
≤
0. 9)
µ
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
µ
K
0
(
)
1.271
1.193
1.114
1.034
0.952
0.869
0.782
0.690
0.591
0.476
µ
µ
The approximation for function
K
0
(
)withtheerror3%for
>
0.4 has
µ
=
µ
been proposed in the topic by Sobolev (1972):
K
0
(
.Inthebook
by Yanovitskij (1997) and in the paper by Dlugach and Yanovitskij (1974) the
results of escape function
K
(
)
0.5+0.75
µ
)havebeenpresentedforthesetofvaluesof
phase function parameter
g
and single scattering albedo
ω
0
. The analysis of
these numerical results yields the following approximation for function
K
0
(
µ
)
with taking into account the phase function dependence:
µ
=
µ
K
0
(
)
(0.7678 + 0.0875
g
)
+ 0.5020 − 0.0840
g
.
(2.32)
The correlation coefficient of the formulas is about 0.99-0.93 depending on
parameter
g
.
In the topic by Minin (1988) it has been proposed to present the function
K
2
(
µ
µ
=
µ
µ
µ
) with the expression
K
2
(
)
n
2
K
0
(
)
w
(
), auxiliary function
w
(
)is
specified with the table.
The numerical analysis in Melnikova (1992) of the table presentation of
escape function
K
(
µ
) according to the paper Dlugach and Yanovitskij (1974)
gives the analytical approximation of function
K
2
(
µ
):
µ
=
µ
µ
=
µ
2
+ 0.1) .
K
2
(
)
n
2
K
0
(
)
w
(
)
1.667
n
2
(
(2.33)
µ
This approximation after the integration with respect of variable
yields value
n
2
with an error less than 0.02%.
In the study by Yanovitskji (1995) the rigorous expression for the function
a
2
(
µ
µ
)accountingfor
the formula from the topic by Minin (1988) (4.55, p. 155) has been deduced
(Melnikova 1992):
) has been derived, and the simple approximation for
a
3
(
)
3
−0.9)+4
q
,
µ
=
µ
µ
a
2
(
)
3
K
0
(
1+
g
(1.271
(2.34)
)
4.5
g
−
)
.
1.6
1+
g
−3−
n
2
w
(
µ
=
µ
µ
a
3
(
)
4
K
0
(
µ
µ
The integration of the expressions for functions
a
2
(
)and
a
3
(
)withrespect
µ
to
leads to values
9
1+
g
(1.271
q
−0.9)
=
12
q
+
=
12
q
+ 0.007
a
2
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