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where functions
K
0
(
µ
)and
K
2
(
µ
) are definedwith formulas (2.35). The positive
valueofthesquarerootischosen,owingtothedemandofthelogarithm
argument positiveness.
Any of the values of
σ
1
or
σ
2
(
ρ
1
or
ρ
2
) corresponding to cosines of the
µ
2
could be substituted to the expressions of the scaled
optical thickness. However, for better accuracy we recommend the use of the
observations for all available viewing angles and then to average the retrieved
values. We should mention that if the data of radiation measured in arbitrary
unitsisenoughfortheparameter
s
2
retrieval it will be necessary to use these
data in relative units of the incident solar flux at the top of the atmosphere for
the scaled optical thickness retrieval.
Itisnecessarytopointoutthattherigorousdemandofthecloudfieldstabil-
ity is suggested in the case of the approach applied to the transmitted irradiance
observationsbecausethisapproachneedscarryingoutthemeasurementsat
several time moments. Using different pixels of the satellite images [as per
(6.14)] needs the horizontal homogeneity of the cloud field, which is checked
out at the initial stage of the approximate retrieval of the optical thickness with
assumption of the conservative scattering. The likewise demand is advanced,
while using the transmitted radiance at different viewing angles, where the
verification of the horizontal homogeneity is provided with the observations
at several azimuth angles.
µ
1
or
viewing angles
6.1.4
InverseProblemSolutionintheCaseoftheCloudLayer
of Arbitrary Optical Thickness
The case of the cloudiness with arbitrary optical thickness (not very thick
clouds) is described by the formulas derived in the study by Dlugach and
Yanovitskij (1974) and cited in Sect. 2 [(2.50)]. Applying the above-mentioned
transformations to (2.50), we deduce the inverse formulas of the optical thick-
ness and parameter
s
2
. The following is obtained for the nonreflecting surface:
(1 −
F
↑
)
2
−
F
↓
2
16[
u
2
−
v
2
]
s
2
=
,
(6.18)
±
(
u
2
−
v
2
)(
t
2
−1)
u
+
tv
1−
F
↑
F
↓
s
−1
ln
tu
+
v
τ
0
=
=
3(1 −
g
)
, w e
t
.
The expression in the numerator of the first formula is the difference of squares
ofthenetfluxesatthetopandbottomofthecloudlayerinunitsofthesolar
incident flux at the top, and value
t
is the ratio of the same net fluxes. The
account of the surface reflection with albedo
A
transforms the functions and
values in (6.18) as follows:
=
u
−
AF
↓
(
p
−1),
=
v
+
AF
↓
p
,
u
v
1−
F
↑
(1 −
A
)
F
↓
(6.19)
F
↓
is changed to (1 −
A
)
F
↓
and
t
is changed to
t
=
.
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