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correspondingly. We specify the value
w
called
the weight of a photon
(Kargin
1984; Marchuk et al. 1980), which it is possible to formally consider as a fourth
coordinate. Assume value
w
=
1 in the beginning of every trajectory and
while writing to the counters, (2.12) will be assigned not unity but value
w
.
Then the simulation of the interaction with the atmosphere is reducing to the
assignment
w
τ
) at every step, and the simulation of the interaction
with the surface is reducing to the assignment
w
=
w
ω
0
(
:
=
w
A
.Nowthephoton
trajectory can't break (the surviving part of the photon always remains), the
break of the trajectory occurs only when the photon is outgoing from the
atmosphere top. Usually for not driving the photon with too small weight
within the atmosphere
parameter of the trajectory break
W
is introduced: the
trajectory is broken if
w
<W
. It is suitable to evaluate value
W
based on the
accuracy needed for the calculation:
W
:
=
δ
,where
s
is the minimal (over all
altitudes
z
for the downward and upward irradiances) needed relative error of
the calculation;
s
δ
δ
=
10
−2
). This approach of
the photon “dividing” is known under the unsuccessful name
“the analytical
averaging of the absorption”
(Kargin 1984) (the words “analytical averaging”
are associated with a certain approximation, which is not used in reality).
Consider a photon at the beginning of the trajectory at the top of the atmo-
sphere. In this case, before the simulation of the first free path (
isthesmallvalue(wehaveused
τ
=
µ
=
µ
0
,
0,
w
=
1) using Beer's Law (1.42) it is possible to account direct radiation, i. e.
radiation reaching level
τ
(
z
) without interaction with the atmosphere. For that
it is necessary to write to all counters
ξ
↓
(
z
)thevaluedependingon
z
instead
unity:
w
exp
−
τ
,
τ
−
ψ
=
(2.18)
µ
and further writing to the counters is not implemented for the first free path
(direct radiation). This approach is easy to extend to other parts of the trajec-
tory: before the writing of the free path to the counter, which the photon can
reach (
ψ
calculated with (2.18) is writing and the further photon flight through the
countersisnotregistering.Notethatasithasbeenshownabovetheexponent
in (2.18) is a probability of the photon started from level
ξ
↓
(
z
), for
µ
τ
≥
τ
,or
ξ
↑
(
z
), for
µ
τ
≤
τ
)value
>
0and
<
0and
τ
τ
.
This general approach of writing to the counter the probability of the photon
to reach the counter is called
“a local estimation”
(Kargin 1984; Marchuk et al.
1980).
The analysis of the above-described algorithmof the irradiances calculation
indicates that the irradiances are not depending on photon azimuth
to reach level
ϕ
.Actu-
ϕ
does not
influence other coordinates and hence, the values written to the counters. Thus,
the “photon azimuth” coordinate is excessive in the task and it could be ex-
cluded for accelerating the calculations (but only in this task of the irradiances
calculations above the orthotropic surface).
Consider the second of the problems described above: the problem of ra-
diance
I
(
z
,
ally, calculated only in two cases with (2.10) and (2.11), azimuth
µ
ϕ
,
) calculation. It is obvious that the procedures either of the
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