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λ
χ
χ
λ
χ
λ
lengths
and cosines of scattering angles
.Coefficients
a
i
(
,
),
b
i
(
,
),
χ
λ
λ
c
i
(
values are derived using linear interpolation. Sub-
script
i
means the dependence of the model upon altitudinal zone. The at-
mosphere has been divided into three zones: 0-4 km (the surface layer as
per Krekov and Rakhimov 1986), 4-11 km (the troposphere) and higher than
11 km (the stratosphere and upper atmosphere). Factor
C
is introduced to the
denominator of (5.9) to satisfy normalizing condition (1.18).
The SNR for model (5.9) is minimal over all considered approximation
although it exceeds unity. Thus, (5.9) will be used further for the phase function
parameterization in the applied algorithm. The evident shortcoming of this
parameterization (5.9) is its explicit referencing to the a priori aerosol model
[through coefficients
a
i
(
,
) for intermediate
χ
λ
χ
λ
χ
λ
)], however, the problem of the
adequacy of our a priori notions about the reality is the general difficulty
of all inverse problems of atmospheric optics, especially of the retrieval of
aerosol parameters (Zuev and Naats 1990). Although, dividing the atmosphere
looks like an artificial adjustment to the initial model structure by Krekov
and Rakhimov (1986), it could be grounded. Actually, according to studies
by Vasilyev A and Ivlev (1995, 1996) in the general case coefficients
a
i
(
,
),
b
i
(
,
),
c
i
(
,
χ
λ
,
),
χ
λ
χ
λ
b
i
(
) depend on the type of the aerosol substance only, and it is
accounted for during the selection of the altitude zones in (5.9).
Using model (5.9), where the phase function is uniquely defined with the
aerosol scattering coefficient means the excluding of the parameters describ-
ing the phase function shape and, hence, the rejection of the retrieval of all
phase function characteristics. The only parameters to obtain are the volume
coefficients of scattering and absorption. It seems to contradict the above con-
firmation about the strong relationship between the irradiances and the phase
function shape. However, we are using not real phase functions but the an-
alytical approximations similar to the real ones in shape. For the real phase
function, a strong correlation either with the scattering coefficient or with
the aerosol substance is observed (Barteneva et al. 1967, 1978; Gorchakov and
Isakov 1974; Gorchakov et al. 1976; Vasilyev A and Ivlev 1995, 1996). Thus, the
parameters of the phase function shape are hardly retrieved in the limits of the
concrete a priori model (Vasilyev O and Vasilyev A 1989a, 1989b, 1994a).
The weak point of parameterization (5.9) is a tabulated relationship between
phase function and scattering angle. Certainly, the analytical parameterization
is preferable. Nevertheless, all our attempts to find the phase function analyti-
cal presentation even for the calculations of the semispherical irradiances have
failed. The rude numerical estimation indicates the needed accuracy of the
phase function approximation about 5-10% for computing the upwelling irra-
diance above the dark (water) surface with the accuracy about 1%. It is a very
rigorous demand because the accuracy of the field observations of the phase
function is the same.
Note lastly, that all described approaches of the estimation of uncertainty
have been implemented while computing the irradiances above the surfaces
with small albedo (water surfaces) because the scattered radiation yields max-
imal solar upwelling irradiance. The link between the upwelling irradiance
calculation and phase function parameterization is essentially weaker for the
,
),
c
i
(
,
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