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the basis for various other approaches of the calculation optimization in the
Monte-Carlo method (Kargin 1984; Marchuk et al. 1980), e. g. the computing
of the derivatives of the irradiances that will be considered in Chap. 5. As
has been shown using these methods, the same transfer equation (1.47) is
solved with different versions of operators K and
Ψ
simulating. In practice,
it is appropriate to use the following procedure. Assume that the probability
density of transition K isalwaysdeterminedbytheconcreteschemeofthe
photons trajectories simulating, and operator
Ψ
is determined by the concrete
writing to the counters (in other words, K is responsible for radiative transfer
and
Ψ
answers for the model of its “observation”).
2.2
Analytical Method for Radiation Field Calculation in a Cloudy Atmosphere
Let us consider the model of an extended and horizontally homogeneous cloud
of large optical thickness
τ 0 >> 1 as Fig. 2.1 illustrates. At the first stage, the
cloud layer is assumed vertically homogeneous as well and the influence of
the clear atmosphere layers above and below the cloud layer is not taken into
account. The volume coefficients of scattering
α
κ
and absorption
,linkedwith
κ
α τ
|∆
α ω
τ
|∆
κ τ
ω
|∆
the cloud characteristics as
z ,
are used for the cloud description. The optical properties of the cloud are
described by the following parameters: single scattering albedo
+
z ,
z ,
0 (1 −
0 )
0
0
0
ω 0 ;optical
τ
thickness
, and mean cosine of the scattering angle g , which characterizes
a phase function. From the bottom the cloud layer adjoins the ground surface
anditsreflectanceisdescribedbygroundalbedo A . Theunderlying atmosphere
could be taken into account if albedo A is implying as an albedo of the system
“surface+atmosphere under the cloud”. Parallel solar flux
π
S is falling on the
µ 0 . The reflected and transmitted radiance
is observed at viewing angle arccos
cloud top at incident angle arccos
µ
. The reflected radiance (in the units
π
µ 0 ) is expressed with reflection function
of incident extraterrestrial flux
S
ρ
µ 0 ) and the transmitted radiance (in the same units) is expressed with
transmission function
τ 0 ,
µ
(
,
σ
τ 0 ,
µ
µ 0 ).
(
,
2.2.1
The Basic Formulas
At a sufficiently large optical depth within the cloud layer far enough from the
top and bottom boundaries the asymptotic or diffusion regime set in owing
to the multiple scattering. This regime permits a rather simple mathematical
description (Sobolev 1972; Hulst 1980). The region within the cloud layer is
called a diffusion domain. The physical meaning yields the following specific
features of the diffusion domain:
1. the role of the direct radiation (transferred without scattering) is negli-
gibly small compared to the role of the diffused radiation;
2. the radiance within the diffusion domain does not depend on the az-
imuth;
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