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(1998) where it is proposed to classify the inverse problems coming from the
type of known and desired parameters. We will return to the topic of choice
in Sect. 4.3, and now let us assume that the concrete parameters contained in
vector
X
are specified. Equation (4.1) could now be rewritten as:
Y
(
X
)
=
G
(
X
,
U
\
X
),
(4.2)
where
U
X
is the set of vector
U
components not included in vector
X
,i.e.
the known parameters of the direct problem. Thus:
G
(
U
)
\
=
\
X
), i. e.
solution of the direct problem is not to depend on which parameters are to be
retrieved.
The inverse problem could be formulated as a determination of vector
X
from the equation:
G
(
X
,
U
=
G
(
X
,
U
\
X
)
Y
.
(4.3)
However, in a general case system (4.3) may have no solution. Indeed, as has
been shown above, the operator of the direct problem
G
is just an approxi-
mation of reality. Hence, even if we supposed that it reflected reality exactly,
vector
Y
wouldnotbeadequatetorealitybecauseofthesystematicandran-
dom observational uncertainties. Thus, a set of possible solutions of the direct
problem
Y
(
X
) could disagree with a set of possible values of the observational
results
Y
. In addition, the case of the nonexistence of the solution for (4.3) is
quite a likely one, even in the simplest variant of the linear operator
G
.Itiscon-
nected with the general properties of the abstract linear operators (Tikhonov
and Aresnin 1986; Kolmogorov and Fomin 1989). However in our version of
the inverse problem statement it is evident: if observations
are linearly
independent and their quantity exceeds the quantity of the parameters under
retrieval (
M>K
), the system of the linear equations will be unsolved. There-
fore, generally the inverse problem of atmospheric optics can be formulated
as follows: to find a set of parameters of the direct problem so that its solution
would be as close as possible to the observational results. In mathematical
wording given in the topic by Tikhonov and Aresnin (1986), it means to find
value
X
, for which the minimum is reached:
{
y
i
}
X
∈
T
ρ
(
Y
,
Y
(
X
))
=
X
∈
T
ρ
min
min
(
Y
,
G
(
X
,
U
\
X
)) ,
(4.4)
ρ
where
T
is the set of possible solutions,
(. . .) is the certain measure in space
of the observational vectors, i. e. the metrics (more details are in the topic by
Kolmogorov and Fomin 1989). Note that in particular cases the minimum in
question could be equal to zero, i. e. the equality in relation
Y
=
\
G
(
X
,
U
X
)is
possible.
The essential factor, which is to be accounted for while solving the inverse
problem, is the observational uncertainty. These questions will be considered
in further detail and here we only mention that unknown parameters
X
are
determined with the uncertainty as well. Hence, accounting for the uncertainty
is an alienable and important stage of the inverse problem solving in atmo-
spheric optics. Besides, as the base of the inverse problem solving consists of
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