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where number
k
is defined from condition
z
k
+1
<z
≤
z
k
. The table of the
optical thickness simply appeared from (2.1):
i
−1
1
2
[
τ
i
≡
τ
=
α
α
(
z
i
)
(
z
j
)+
(
z
j
+1
)](
z
j
−
z
j
+1
),
(2.2)
=
j
1
τ
N
=
τ
0
is the optical thickness of the
atmosphere defined by (1.40). We are using the linear interpolation in accor-
dance with the trapezoidal quadrature here and further. Then function
=
τ
1
=
where
i
1,...,
N
,moreover
0,
α
(
z
)in
(2.1) is expressed as
(
z
k
)
z
−
z
k
+1
z
k
−
z
z
k
−
z
k
+1
α
=
α
α
(
z
)
z
k
−
z
k
+1
+
(
z
k
+1
)
,
(2.3)
which with taking into account (2.2) gives the polynomial of the power equal
to two
2
α
α
(
z
k
)(
z
k
−
z
)+
1
(
z
k
+1
)−
(
z
k
)
z
k
−
z
k
+1
τ
=
τ
k
+
α
(
z
k
−
z
)
2
.
(
z
)
(2.4)
τ
After obtaining function
(
z
)accordingto(2.2)and(2.4)itispossibletousethe
altitude, as a coordinate because it is more appropriate in practice. The input
tables of the initial atmospheric parameters are directly converted to
α
τ
i
),
(
ω
0
(
τ
i
)and
x
(
τ
i
,
χ
i
),
i
=
1,...,
N
, and for obtaining the intermediate values, for
ω
τ
example
0
(
), it is possible to use either the linear interpolation directly over
τ
τ
) and to interpolate over altitude
z
in
(2.3) that is more correct. Inverse function
z
(
or to find the altitude as function
z
(
τ
) from (2.4) could be written:
τ
k
)−
α
z
k
+
α
τ
k
)+2
∆
k
(
τ
τ
k
)
2
(
(
−
τ
=
z
(
)
,
(2.5)
∆
k
τ
k
≤
τ
τ
k
+1
,and
where number
k
is deduced from condition
<
∆
k
=
[
α
τ
k
)−
α
τ
k
+1
)
]
|
(
(
[
z
k
−
z
k
+1
].
We should mention that as the procedure for the coordination of the altitude
and optical depth is not linked with the specific of the Monte-Carlo method at
all, it is possible to apply it in other numerical methods of radiative transfer
theory.
Here we will give an account of the Monte-Carlo method. It is based on
the modeling of radiative transfer in the atmosphere as a random process: the
motion of the conditional particle of light called the “photon”, the simulation
of the process on computer, and the calculation of the desired characteris-
tics as a mathematical expectation of random numbers appearing during the
simulation (Kargin 1984; Marchuk et al. 1980).
For the statistical simulation on computer, it is necessary to reproduce
a process that will play the role of the random event. Such an algorithm, called
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