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+
∆
)(
∆
m
+
∆
n
+
∆
∆τ
0
τ
0
=
∆
l
µ
m
n
K
∆
µ
)−
F
↑
)+
l
(
∆
∆
F
↑
)
s
+
mnK
(
K
)+
l
(
a
(
a
−
l
µ
µ
)−
F
↑
)
s
mnK
(
)+
l
(
a
(
1
τ
0
+
∆
∆
F
↑
a
−
.
µ
a
(
)−
F
↑
And in case of using the transmitted irradiance:
∆
∆
∆
∆
∆
µ
1
)
+
∆
µ
1
)+
∆
K
0
F
↓
K
0
F
↓
s
2
w
+
n
2
FK
0
+
FK
0
(
=
µ
2
))
Q
2
+
,
µ
1
)−
w
(
F
1
K
0
(
µ
2
)−
F
2
K
0
(
2
F
2
K
0
(
s
(
w
(
µ
1
)
(6.47)
⎡
⎣
∆
⎤
⎦
ll
∆
r
2
l
+
∆
∆
l
l
+
2
r
∆τ
0
τ
0
=
∆
s
r
r
1
τ
0
+
+
1+
r
2
ll
+1
2
1+
r
2
,
s
ll
where
l
∆
=
∆
+
∆
+
∆
+
∆
+
∆
+
∆
r
r
F
F
m
m
n
n
K
K
l
.
l
l
∆τ
0
|τ
0
∼
10% for
reflected irradiance and for transmitted irradiance - 6% and 10% correspond-
ingly, if the observational uncertainty is about 2%. In general, the irradiances
data allow obtaining the optical parameters within the cloud more accurately
than the radiances do, according to the study by McCormick and Leathers
(1996).
∆
|
∼
The error analysis as per (6.46)-(6.47) gives
s
s
8% and
6.3.2
The Applicability Region
As has been mentioned in Sect. 2.4, the main lower bound connecting with the
diffusion domain is set on the optical thickness. The restriction on the true
absorption arises due to expansions over the small parameter for the asymp-
totic constants. The applicability region of the inverse expressions for values
s
and
τ
have been studied in several studies (Melnikova 1992, 1998; Melnikova
et al. 2000) for the wide set of parameters. Calculation of the direct problem
has been accomplished with the doubling and adding method, and the ob-
tained radiative characteristics have served as measured values (Demyanikov
and Melnikova 1986). The retrieved parameters have been compared with the
model parameters of the direct problem for estimating the relative error. About
50 numerical models have been analyzed in total. The values of the relative un-
certainties of 1−
τ
0
with fixed phase function parameter
g
are presented
in Figs. 6.5 and 6.6 versus the single scattering albedo and optical thickness
correspondingly. We should point out that the only uncertainties caused by the
break of the applicability region have been studied in the above-mentioned
ω
0
and
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