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wavelength (5.4), the wavelength remains the fixed one while modeling every
single trajectory. Hence, it is enough to consider the monochromatic case only
during the differentiation and the derivative of integral (5.4) will be obtained
automatically. It should be emphasized also that the optical thickness itself is
the function of differentiated parameters. Thus, the atmospheric pressure is
to be used as a vertical coordinate, while computing the derivatives. Nothing
changesintherealmodelingbutforthederivationof(2.8)thephotonfreepath
probability from altitude level P 1 (in the pressure scale) to level P is written as:
1
µ
P
α
( P ) dP
,
1−exp
P 1
α
where
( P ) is the extinction coefficient, then probability density (2.8) trans-
forms to the following:
1
µ
P
α
( P )
| µ |
ρ
=
α
( P ) dP
.
( P )
exp
(5.17)
P 1
It is just (5.17), which is to be used as a probability density of the photon free
path, while differentiating.
Now apply the algorithm of the irradiance calculation, described in Sect. 2.1
to the algorithm for the calculating of derivatives while taking into account the
explicit form of the functions in (5.16).
Counters W a are introduced for the whole set of parameters. Starting every
trajectory of the counter W a :
=
0 is assumed.Whilemodeling every photon free
path, the following value is assigned to the counter while taking into account
(5.17):
1
1
| µ |
=
α
∆τ ( P 1 , P 2 )) ,
W a :
W a +
a (
( P 2 )) −
a (
(5.18)
α
( P 2 )
∆τ ( P 1 , P 2 ) is the photon free path from level P 1 to level P 2 (2.7). If the
photon reaches the surface, then the itemwith value
where
α
( P 2 )willbeabsent.While
modeling every act of the interaction between the photon and atmosphere, i. e.
while multiplying the photon weight by
ω 0 (
τ ),thefollowingvalueiswritten
to the counter:
1
ω 0 ( P )
=
ω 0 ( P )) ,
W a :
W a +
a (
(5.19)
where P is the current coordinate (in the atmospheric pressure scale) corre-
sponding to optical thickness
τ . Analogously, the values for the interaction of
thephotonwiththesurfaceiswrittentothecounterinaccordancewith(2.23):
W a + 1
A
=
W a :
a ( A ) .
(5.20)
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