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and reduce the numerator to zero. Thus, (6.11), (6.13) and (6.14) become
inappropriate and another formulas are necessary to use. The closeness of the
numerator to zero is defined by the expression
mnlK
(
µ
0
) exp(−2
k
τ
)
τ
−→
τ
→∞
C
exp(−2
k
)
1−
ll
exp(−2
k
τ
)
τ
0
equal to 100. The optical thickness is preliminarily
estimated approximately while assuming the conservative scattering as has
been proposed for example in the work by King (1987) and Kokhanovsky et al.
(2003). Then, if
that is about 0.02 for
τ
0
≥
100, the quadratic equations with respect to parameter
s
2
are derived using the expression of
a
(
µ
0
)and
ρ
∞
(
µ
µ
0
) (2.30) taken with the
,
items proportional to
s
2
:
µ
0
)
s
2
−4
K
0
(
µ
0
)
s
+1−
F
↑
(
µ
0
)
=
a
2
(
0
µ
0
)
a
2
(
µ
a
2
(
)
s
2
−4
K
0
(
µ
0
)
K
0
(
µ
ρ
0
(
µ
µ
0
,
ϕ
ρ
=
)
s
+[
,
)−
]
0
12
q
Its solution is trivial:
µ
0
)−
4
K
0
(
µ
0
)
1−
F
↑
(
µ
0
)
µ
0
)
2
−
a
2
(
2
K
0
(
=
s
.
(6.15)
µ
0
)
a
2
(
And the similar expression for case of the reflected radiance:
µ
0
)−
4[
K
0
(
)]
2
−
a
2
(
µ
0
)
a
2
(
µ
)
µ
µ
0
)
K
0
(
µ
ρ
0
(
µ
µ
0
,
ϕ
ρ
2
K
0
(
)
K
0
(
[
,
)−
]
12
q
=
s
.
µ
µ
a
2
(
0
)
a
2
(
)
12
q
(6.16)
Problem of choosing the sign before the radicals is the consequence of the
ambiguity of the inverse problem solution, and it needs the special analysis of
the concrete data. It is easy todemonstrate that justminus has tobe chosenhere.
Indeed, in the case of the conservative scattering the equalities
ρ
=
ρ
0
(
µ
µ
0
,
ϕ
,
)
and
s
2
=
0 are satisfied only with minus before the radical.
In the case of using the transmitted radiance, the corresponding equation
for the values of parameter
s
2
and scaled optical thickness
τ
are similar to
(6.12):
σ
1
K
0
(
µ
1
)
−1
µ
2
)
σ
2
K
0
(
1
s
2
=
,
(6.17)
K
2
(
µ
1
)
−
K
2
(
µ
1
)
2
)
K
0
(
µ
K
0
(
µ
2
)
4
⎡
⎤
µ
0
)
2
ll
+
m
2
K
(
σ
τ
µ
1,2
,
µ
1,2
)
2
K
(
µ
0
)
2
+
m K
(
µ
1,2
)
K
(
µ
0
)
(
,
τ
=
⎣
⎦
,
s
−1
ln
µ
0
)
ll
σ
τ
µ
1,2
,
2
(
,
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