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3.therelativeangledistributionoftheradiancedoesnotdependonthe
optical depth (Sobolev 1972).
The name “diffusion” appears because the equation of radiative transfer is
transformed to the diffusion equation in that case (Hulst 1980). In the scattering
layer of a large optical thickness the analytical solution of the transfer equation
is possible and it is expressed through the asymptotic formulas of the theory
of radiative transfer (Sobolev 1972; Minin 1988), moreover the existence and
uniqueness of the solution have been proved (Germogenova 1961). According
to the topics by Sobolev (1972), Hulst (1980), and Minin (1988), the solution
of the transfer equation, expressed through reflection
ρ
σ
and transmission
functions, is the following:
)−
mlK
(
µ
µ
0
)exp(−2
k
τ
0
)
)
K
(
ρ
µ
µ
0
,
ϕ
=
ρ
∞
(
µ
µ
0
,
ϕ
(0,
,
)
,
1−
ll
exp(−2
k
τ
0
)
(2.24)
m K
(
µ
0
) exp(−
k
τ
0
)
1−
ll
exp(−2
k
µ
)
K
(
σ
τ
0
,
µ
µ
0
)
=
(
,
.
τ
0
)
ρ
∞
(
µ
µ
0
,
ϕ
In these equations
,
) is the reflection function for a semi-infinite at-
µ
mosphere;
K
(
) is the escape function, which describes an angular dependence
of the reflected and transmitted radiance;
m
,
l
,
k
are the constants, depending
on the cloud optical properties, the formulas for its computing are presented
below;
K
(
)and
l
depends on ground albedo
A
as well. The following ex-
pressions are taking into account the ground surface reflection according to
Sobolev (1972), Ivanov (1976) and Minin (1988):
µ
Amn
2
1−
Aa
∞
µ
)+
Aa
(
)
n
1−
A
l
=
K
(
µ
=
µ
l
−
,
)
K
(
.
(2.25)
)istheplanealbedoand
a
∞
is the spherical albedo of
a semi-infinite atmosphere (the atmosphere of the infinite optical thickness).
µ
In these expressions
a
(
n
1−
Aa
∞
K
(
µ
=
µ
)+
A Qa
(
µ
=
l
=
l
−
Am QQ
)
K
(
),
n
,
(2.26)
µ
),
a
∞
,andvalue
n
are defined by the integrals:
where
a
(
1
1
µ
=
ρ
µ
µ
0
)
µ
0
d
µ
0
,
a
∞
=
µ
µ
µ
a
(
)
2
(
,
2
a
(
)
d
0
0
1
1
=
µ
µ
µ
=
K
(
µ
µ
µ
n
2
K
(
)
d
,
n
2
)
d
,
0
0
µ
It is seen that (2.24) are the asymmetric formulas relatively to variables
and
µ
0
, which are input with escape functions
K
(
µ
)and
K
(
µ
). It links with different
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