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absorption bands indicates the correctness of the computer code. On the other
hand, the essential numerical difference between the observations and calcu-
lations shows the susceptibility of the observed irradiances to the atmospheric
parameters (which have differed from the model values during the observa-
tions).
5.2
Calculation of Derivative from Values of Solar Irradiance
In addition to the irradiance calculations, the derivatives of the irradiances
with respect to all retrieved parameters are necessary for the inverse problem
solving using the methods presented in Chap. 4.
The derivatives computing in the Monte-Carlo method is based on the
differentiation of formal Neumann series (2.22) (Marchuk 1988), i. e. of the
general form of the solution of the equation of radiative transfer (Lenoble
1985; Marchuk 1988). We will keep the same specifications as in Sect. 2.1. Let
the derivative of direct problem (2.22) solution to be inferred:
Ψ
=
Ψ
Ψ
Ψ
K
2
q
+
Ψ
K
3
q
+
...
B
q
+
Kq
+
∂
Ψ
a
B
a
)
|∂
with respect to a certain parameter
a
,i.e.value
a
where the sub-
script of the integral operators symbolizes their dependence of the parameter.
Differentiate Neumann series (2.22):
(
∂
∂
∂
∂
Ψ
a
q
a
)+
∂
Ψ
a
K
a
q
a
)+...+
∂
Ψ
a
B
a
)
=
Ψ
a
K
a
q
a
)+... (5.10)
a
(
a
(
a
(
a
(
∂
∂
Use for brevity of the derivative specification the following form:
∂
∂
Ψ
a
B
a
)
≡
Ψ
a
B
a
)
(
a
(
(5.11)
and assuming the symbolic writing of integral operators (2.20) obtain the
following for the right part of series (5.10)
Ψ
a
(
u
)
B
a
(
u
)
du
Ψ
a
(
u
)
B
a
(
u
)
Ψ
a
(
u
)
du
Ψ
a
(
u
)
+
B
a
(
u
)
=
B
a
(
u
)
(5.12)
=
Ψ
a
(
u
)
B
a
(
u
)
W
a
(
u
)
du
,
where
=
Ψ
a
(
u
)
Ψ
a
(
u
)
+
B
a
(
u
)
W
a
(
u
)
.
(5.13)
B
a
(
u
)
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