Geoscience Reference
In-Depth Information
γ
ϕ
The substitutionof the value
C
d
(
,
) from(1.15) to (1.16) gives a
normalization
conditionofthephasefunction:
π
π
2
1
4
ϕ
γ
ϕ
γ
γ
=
d
x
(
,
) sin
d
1 .
(1.17)
π
0
0
γ
ϕ
=
If the scattering is equal over all directions, i. e.
C
d
(
,
)
const
,itiscalled
γ
ϕ
isotropic
and the relation
x
(
≡
,
)
1 follows from the normalization (1.17).
π
Thus, the multiplier 4
is used in (1.16) for convenience. In many cases, (for
example themolecular scattering, the scattering on spherical aerosol particles)
the phase function does not depend on the scattering azimuth. Further, we are
considering only such phase functions. Then the normalization condition
converts to:
π
1
2
γ
γ
γ
=
x
(
) sin
d
1 .
(1.18)
0
The integral from the phase function in limits between zero and scattering
angle
2
0
x
(
γ
1
γ
γ
d
γ
) sin
could be interpreted as a
probability of scattering
γ
to the angle interval
[0,
]. It is easy to test this integral for satisfying all
demands of the notion of the “probability”. Hence the phase function
x
(
γ
)is
γ
the
probability density of radiation scattering to the angle
. Often this assertion
is accepted as a definition of the phase function.
3
The real atmosphere contains different particles interacting with solar ra-
diation: gas molecules, aerosol particles of different size, shape and chemical
composition, and cloud droplets. Therefore, we are interested in the interac-
tion not with the separate particles but with a total combination of them. In
the theory of radiative transfer and in atmospheric optics it is usual to abstract
from the interaction with a separate particle and to consider the atmosphere
as a continuous medium for simplifying the description of the interaction
between solar radiation and all atmospheric components. It is possible to at-
tribute the special characteristics of the interaction between the atmosphere
and radiation to an elementary volume (formally infinitesimal) of this contin-
uous medium.
Scrutinize
the elementary volume
of this continuous medium
dV
=
dSdl
(Fig. 1.5), on which the parallel flux of solar radiation
F
0
incomes normally
to the side
dS
. The interaction of radiation and elementary volume is reduced
to the processes of scattering, absorption and radiation extenuation after ra-
diation transfers through the elementary volume. Specify the radiation flux
3
Point out that the phase function determines scattering only in the case of unpolarized incident
radiation. After the scattering (both molecular and aerosol), light becomes the polarized one and the
consequent scatteringorders (secondary and soon) can't bedescribedonlyby thephase functionnotion.
Thus the theory of scattering, which doesn't take into account the polarization, is an approximation. In
a general case, the accuracy of this approximation is estimated within 5% according to Hulst (1980). In
special cases, it is necessary to test the accuracy that will be done in the following sections.
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