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µ,
µ
0
µ
µ
0
µ,
µ
0
(µ′,
µ′
0
µ,
µ
0
µ
Fig. 2.1.
Model of the atmosphere
the top of the plane parallel horizontally homogeneous atmosphere
z
∞
.The
base of the atmosphere
z
=
0 is the orthotropic surface with albedo
A
(note
that the condition of the orthotropic reflection is not essential for the Monte-
Carlomethod. The anisotropic reflectionmodel will be considered in Chap. 5).
The initial optical parameters of the atmosphere are provided as look-up
tables over the altitude grid: volume extinction coefficient
α
(
z
i
), probability
ω
0
(
z
i
), phase function as a table over altitudes and
cosines of the scattering angles
x
(
z
i
,
of the quantum surviving
χ
j
),
j
=
χ
1
=
χ
M
=
1,...,
M
,
1,
−1, where
=
=
=
i
0.Thephysicalatmosphericmodel(thevertical
profiles of the temperature, pressure, concentrations of the absorbing gases
and the aerosol model described in Sect. 1.2) defines all these parameters.
It is necessary to find the numerical values of the semispherical fluxes - the
downward one
F
↓
(
z
)andupwardone
F
↑
(
z
) - at arbitrary altitude 0
1,...,
N
,
z
1
z
∞
,
z
N
≤
z
≤
µ
ϕ
µ
ϕ
z
∞
or (and) radiance
I
(
z
,
). All mentioned
parameters and values are monochromatic for the chosen wavelength.
Let us express the optical thickness as a function of altitude by means
of (1.39) before presenting the Monte-Carlo method. Using the trapezoidal
quadrature, we obtain:
,
) for arbitrary direction (
,
k
−1
1
2
[
(
z
j
+1
)](
z
j
−
z
j
+1
)+
1
τ
=
α
α
α
α
(
z
)
(
z
j
)+
2
[
(
z
)+
(
z
k
)](
z
k
−
z
)
(2.1)
=
j
1
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