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ξ
2
(
z
)withzerothinitialvaluesandwritetogetherwith
(2.14) the following:
ξ
2
(
z
)and
squares
ξ
2
(
z
):
=
ξ
2
(
z
)+[
ξ
2
(
z
):
=
ξ
2
(
z
)+[
ξ
↓
(
z
)]
2
,
ξ
↑
(
z
)]
2
.
(2.15)
ξ
=
ξ
ξ
2
)−
M
2
(
Using the known expression for variance
D
(
)
M
(
), where
M
(...)
is expectation, we obtain:
K
ξ
2
(
z
)−
1
K
ξ
1
2
K
ξ
2
(
z
)−
1
K
ξ
1
2
1
1
ξ
↓
)
ξ
↑
)
=
=
D
(
,
D
(
.
(2.16)
ξ
↑
(
z
)isunknown.
However, the distribution of its expectations according to the central limit the-
orem tends to the normal distribution as
K
ξ
↓
(
z
)and
Thebehaviorofthedistributionofrandomvalues
.Hence,desiredirradiances
(2.13), which are also considered as random values, have the distributions
asymptotically close to the normal distribution. It is known that the normal
distribution is characterized with
the expectation
and
the variance
expressed
by (2.16). For
the standard deviation
(
SD
)(
s
(...)
→∞
=
√
D
(...))oftheirradiances
in accordance with the study by Marchuk et al. (1980) with taking into account
the known rule for the variances addition the following is obtained:
µ
0
D
(
µ
0
D
(
s
(
F
↓
(
z
))
=
ξ
↓
)
|
s
(
F
↑
(
z
))
=
ξ
↑
)
|
F
0
K
,
F
0
K
.
(2.17)
As follows from (2.17), the increasing of the number of trajectories
K
leads
to the minimization of the standard deviation (SD), i. e. of the random error
of the irradiances calculation. Evaluating the SD with (2.15)-(2.17) is of great
practical interest because it allows accomplishment of the calculations with
the accuracy fixed in advance. Actually, the calculation of the SD gives the
possibility of estimating the necessary number of photon trajectories and as
soon as the SD is less than the fixed value, the simulating is finished.
The above-considered scheme of the simulating of photon trajectories is
called
“direct modeling”
(Kargin 1984) as it directly reflects our implication
concerning photon motion throughout the atmosphere. However, direct mod-
eling is not enough for accelerating the calculation according to the algorithm
of the Monte-Carlomethod or for the radiance calculation (Kargin 1984). Con-
sider two approaches to increase the calculation effectiveness that we have
applied. It is possible to find detailed descriptions of other approaches in the
topics by Kargin (1984), and Marchuk et al. (1980).
The basis of optimizing the calculation with the Monte-Carlo method is an
idea of decreasing the spread in the values written to the counters. Then the
variance expressed by (2.16) decreases too and fewer trajectories are necessary
for reaching the fixed accuracy according to (2.17).
Assume that the photon could be divided into parts (as it is a mathematical
object and not a real quantum here). Then a part of the photon equal to
1−
ω
0
(
τ
) is absorbed at every interaction with the atmosphere and the rest
ω
0
(
τ
) is scattered and, then, continues themotion. During the interactionwith
the surface these parts are equal to 1 −
A
and to
A
(
A
is the surface albedo)
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