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first scattering event. Then, the value of the upward irradiance at the top as per
(2.13) is to be equal to the value of the downward irradiance due to the law of
energy conservation (there is no absorption), and this condition wouldn't be
broken only if the writing to the counters according to (2.12) does not depend
on value
µ .
In result, the following algorithm of the irradiance calculation is obtained:
τ =
µ = µ 0 ,
1. In the beginning of every trajectory it can be written
0,
ϕ =
0.
2. Further, the photon free path is simulated according to (2.7) with the
writing to the counters (2.12).
3. If the photon is going out of the atmosphere (
τ 0), its trajectory will
finishandthetrajectoryofthefollowingphotonwillbesimulated.
4. If the photon reaches the surface (
τ τ
0 ), its interaction (reflection or
absorption)withthesurfacewillbesimulated.
5. The absorption means the end of the trajectory and the trajectory of the
following photon is simulated.
6. The reflection from the surface gives the new direction of the trajectory
according to (2.11) and then the photon recurrent free path is simulated.
7.Ifthephotonstaysintheatmosphere(0 <
τ 0 ), its interaction
(scattering or absorption) with the atmosphere is simulated.
8. The absorption leads to the end of the trajectory.
9. In the case of the scattering, thenewdirectionof thephoton is determined
using (2.9) and (2.10) and the photon following free path is simulated.
The desired values of the irradiances are foundwith (2.13) after sufficient
numbers K of the trajectories.
τ
<
ξ 1 ( z )
ξ 1 ( z )
|
|
Mark that ratios
K are the expectations (the arithmetic
means) of photon numbers written to the counters as the result of the simu-
lation of a single photon trajectory. Introduce counters
K and
ξ ( z )with
zeroth energies in the beginning of every trajectory and write a relation anal-
ogous to (2.12) for a single trajectory. Then, (2.12) transforms to:
ξ ( z )and
ξ 1 ( z ):
= ξ 1 ( z )+
ξ 1 ( z ):
= ξ 1 ( z )+
ξ ( z ),
ξ ( z ) ,
(2.14)
moreover, the writing to the counters as per (2.14) is carried out at the end of
every trajectory.
Thus, the problem of the irradiance calculation by the Monte-Carlo method
is reduced in fact to the calculation of the expectations of random values
ξ ( z )
ξ ( z ) (the number of photons written to the counters) over the finite sample
from K trajectories by (2.13) and (2.14). It is possible to calculate not only the
expectation but also other statistical estimations of random values
and
ξ ( z )and
ξ ( z ). Further, obtain their variance. For that we introduce the counters of
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