Geoscience Reference
In-Depth Information
for the upward and downward semispherical solar fluxes:
τ
0
−
τ
δ
µ
0
)
(1 −
A
)[3(1 −
g
)(
)+1.5
−2]+4
A
F
↑
(
τ
µ
0
)
=
K
0
(
,
τ
0
+3
δ
(1 −
A
)[3(1 −
g
)
]+4
A
(2.49)
τ
0
−
τ
δ
µ
0
)
(1 −
A
)[3(1 −
g
)(
)+1.5
+2]+4
A
F
↓
(
τ
µ
0
)
=
,
K
0
(
.
τ
0
+3
δ
(1 −
A
)[3(1 −
g
)
]+4
A
It is possible to apply the formulas of the radiative characteristics in the case of
conservative scattering for a rough estimation even for very weak absorption
but the computational errors increase fast when the absorption grows and it is
necessary to use the equations for the absorption medium to reach a certain
accuracy.
2.2.6
Case of the Cloud Layer of an Arbitrary Optical Thickness
The optical thickness of certain cloud layers is not sufficient in some cases
for making use of the above-presented equations and their application leads
to significant errors and it causes the necessity of different approaches. We
would like to mention the two-flux Eddington and delta-Eddington methods
among all analytical approaches (Josef et al. 1980). These methods are no-
table for the simple expressions and they provide sufficient accuracy of the
calculations, however, they are approximations. In addition, they are awkward
enough and hence, are not convenient for the inverse problem transforming.
A mathematically rigorous method has been developed for the calculation of
the irradiances at the boundaries of the layer of arbitrary optical thickness in
Yanovitskij (1991,1997) and Dlugach and Yanovitskij (1983). The restrictions
tothetrueabsorptionaremorerigorousthanaboveandtheopticalthickness
is accepted in the range 0.1
<
τ
0
<
5.0 The irradiance outgoing from the layer
is described with the following:
F
↑
=
µ
τ
0
)ch
k
τ
0
−
v
(
µ
τ
0
)] ,
1−
f
[
u
(
,
,
F
↓
=
µ
τ
0
)−
v
(
µ
τ
0
)ch
k
τ
0
],
f
[
u
(
,
,
(2.50)
4
s
sh
k
=
f
.
τ
0
τ
0
and ch
k
τ
0
specify the hyperbolic sine and cosine, functions
Functions sh
k
τ
0
) are defined in several studies (Dlugach and Yanovitskij
1983; Yanovitskij 1991, 1997) and they are similar to the escape function.
Here we are not adducing these definitions. It should be emphasized only that
they depend on the optical thickness as well. Besides, these functions depend
inexplicitly on the phase function. The tables containing the values of functions
u
(
µ
0
,
τ
0
)and
v
(
µ
0
,
u
(
τ
0
)forthewidesetoftheargumentsandseveralvaluesof
phase function parameter
g
have been calculated and presented in Yanovitskij
(1991) and Dlugach and Yanovitskij (1983)
µ
0
,
τ
0
)and
v
(
µ
0
,
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