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simulation of photon trajectories or of the calculation of the expectation and
variance are depending on the desired value, and hence they wouldn't change.
The difference is concerning the procedure of writing the values to the coun-
ters. Encircle the cone with small solid angle
∆Ω
µ
ϕ
µ
ϕ
).
We will be writing to the counter all photons, which have reached level
z
and
havecometotheconeforradiance
I
according to the equation analogous
to (2.12). Moreover, in this case value 1
(
,
)arounddirection(
,
|
|
µ
|
hastobewrittentothecounter
instead of unity in the case of the irradiance calculation to satisfy the link of
the radiance and irradiance (1.4). Pass further from the above-described (but
not realized) scheme of the direct modeling of the radiances to the schemes
of the
weight modeling
and
local estimation
. Let the photon have coordinates
(
ϕ
). According to the definition of the phase function as a density of the
probability of the scattering (Sect. 1.2), the probability of the photon coming
to solid angle
τ
,
µ
,
τ
is equal to the integral of the
phase function over the angle intervals defined by (1.17) (i. e.
∆Ω
µ
ϕ
(
,
) after scattering at level
∆Ω
and scatter-
µ
,
ϕ
)(
µ
ϕ
|
π
ing angle
(
,
)) with taking into account normalizing factor 1
4
.Let
∆Ω
value
decrease toward zero. Then we are revealing that the density of the
probability of the photon to reach direction (
µ
ϕ
,
) coincides with the value of
χ
=
µ
,
ϕ
)(
µ
ϕ
the phase function for argument
)), which is computed
with (1.46). This probability is necessary to multiply by factor
cos(
(
,
ψ
defined with
τ
(2.18), i. e. by the probability of the photon to reach level
(
z
). Finally, the local
estimation for the radiance is obtained according to the results of the topics by
(Kargin 1984, Marchuk et al. 1980).
χ
)exp
−
τ
τ
w
−
τ
,
ψ
=
π
µ
x
(
µ
4
(2.19)
χ
=
µµ
+
(1 −
µ
µ
2
) cos(
ϕ
ϕ
).
2
)(1 −
−
Thus, the considered algorithm of the radiance computation according to the
Monte-Carlo method differs from the irradiance computation algorithm just
with the other equation for the local estimation (2.19) instead of (2.18) andwith
otherequationsforthecounters:forradianceoversingletrajectory
ξ
µ
ϕ
(
z
,
,
),
ξ
1
(
z
,
µ
ϕ
ξ
2
(
z
,
µ
ϕ
for expectation
). Both
algorithms (for radiance and irradiance) couldbe carriedout on computerwith
one computer code. It is pointed out that the condition of the clear atmosphere
(the small optical thickness) has not been assumed so theMonte-Carlomethod
algorithms can be also applied for the cloudy atmosphere.
In conclusion, illustrate that the considered algorithms actually correspond
to the solution of the equation of radiative transfer (1.47).
The desired radiation characteristic (radiance, irradiance) could be written
in the operator form according to expressions of the radiance through the
source function (1.52), and as per the link of the irradiance and the radiance
(1.4):
,
)andforthesquareoftheexpectation
,
Ψ
=
Ψ
B
(
u
)
B
(
u
)
du
,
(2.20)
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