Geoscience Reference
In-Depth Information
Nevertheless, taking into account that the thickness of the atmosphere is much
less than the Earth's radius is, in a number of problems the atmosphere could
be considered by convention as a plane limited with two infinite boundaries:
the bottom - a ground surface and the top - a level, above which the inter-
action between radiation and atmosphere could be neglected. Further, we are
considering only
the plane-parallel atmosphere approximation
.Thegrounds
of the approximation for the specific problems are given in Sect. 1.3. Then the
position of the element
dS
could be characterized with Cartesian coordinates
(
x
,
y
,
z
) choosing the altitude as axis
z
(to put the
z
axis perpendicular to the
top and bottom planes from the bottom to the top). Thus, in a general case
the radiance in the atmosphere could be written as
I
ϑ
ϕ
,
t
). Under the
natural radiation sources (in particular - the solar one) we could neglect the
behavior of the radiance in the time domain comparing with the time scales
considered in the concrete problems (e. g. comparing with the instrument reg-
istration time). The radiation field under such conditions is called a
stationary
one. Further, it is possible to ignore the influence of the horizontal hetero-
geneity of the atmosphere on the radiation field comparing with the vertical
one, i. e. don't consider the dependence of the radiance upon axes
x
and
y
.This
radiation field is called a
horizontally homogeneous
one. Further, we are consid-
ering only stationary and horizontally homogeneous radiation fields. Besides,
following the traditions (Sobolev 1972; Hulst 1980; Minin 1988) the subscript
(
x
,
y
,
z
,
,
λ
λ
is omitted at the monochromatic values if the obvious wavelength dependence
is not mentioned. Taking into account the above-mentioned assumptions, the
formula linking the radiance and irradiance (1.4) is written as:
π
π
2
=
ϕ
ϑ
ϕ
ϑ
ϑ
ϑ
F
(
z
)
d
I
(
z
,
,
) cos
sin
d
.
(1.5)
0
0
ϑ
It is natural to count off the angle
from the selected direction
z
in the at-
mosphere. This angle is called the zenith incident angle (it characterizes the
inclination of incident radiation from the zenith). The angle
ϑ
is equal to zero
π
if radiation comes from the zenith, and it is equal to
if the radiation comes
from nadir. As before we are counting off the azimuth angle from an arbitrary
direction on the plane, parallel to the boundaries of the atmosphere. Then the
integral (1.5) could be written as a sum of two integrals: over upper and lower
hemisphere:
=
F
↓
(
z
)+
F
↑
(
z
),
F
(
z
)
π|
π
2
2
F
↓
(
z
)
=
ϕ
ϑ
ϕ
ϑ
ϑ
ϑ
d
I
(
z
,
,
) cos
sin
d
,
(1.6)
0
0
π
π
2
F
↑
(
z
)
=
ϕ
ϑ
ϕ
ϑ
ϑ
ϑ
d
I
(
z
,
,
) cos
sin
d
.
0
π|
2
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