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the optimal ones will be the observations, where the demanded compromise
between the exactness of the parameter retrieval and needed expenditures for
obtaining them is reached, contrary to the observations providing themaximal
exactness. Note that testing the solution of the inverse problem by a compar-
ison with the independent measurements strictly speaking is reasonable for
the direct measurements only. If the parameters for the comparison have been
also obtained from the solution of another inverse problem, it is possible to
discuss the comparing of the instruments and methodics only.
Accounting for the above-mentioned difficulties together with the fact that
there has been no direct simultaneous observations for the considered sound-
ings hereinafter consider the problem of the analysis of the adequacy of the
inverse problem solution with the theoretical means.
Either the observation or the direct problem solution contains systematic
uncertainties. These uncertainties evidently cause the minimum of discrep-
ancy
(
Y
,
Y
(
X
)) reached while the inverse problem solving will not correspond
to the minimum of the discrepancy of true values of the observational data
and direct problem solution. Take into account that the desired parameters
are linearly expressed through the difference of the observations and direct
problem solution in the formulas of Sects. 4.2 and 4.3, i. e.
X
ρ
=
A
(
Y
−
Y
), where
Y
+
Y
,
where
Y
is the true mean value of the measured characteristic,
Y
is the ab-
solutely exact solution of the direct problem,
=
∆
Y
,
Y
=
∆
Y
+
A
is a certain linear “solving” operator. Thenwriting
Y
Y
are the corresponding
systematic uncertainties of the observations and calculations, we are obtaining
X
∆
∆
Y
,
A
(
Y
−
Y
)+
A
(
Y
). The first item is the desired adequate value
X
,
but the second item means its distortion by a random shift. As the random
observational uncertainty causes the obtaining of the vector of parameters
X
either with the random uncertainty, the mentioned systematic shift is to be
estimated from its comparison with the random uncertainty of vector
X
.If
the systematic shift is not less than the random uncertainty is then the result
ignoring this shift will be evidently inauthentic. In practice, it is more conve-
nient to compare not the retrieval errors but the errors of the observation and
direct problem solution (Zuev and Naats 1990).
The systematic uncertainties of the observations are always much more
than the random ones, so value
=
∆
∆
Y
−
Y
is of main interest. The simple receipt
of its accounting is presented in the topic by Zuev and Naats (1990); if it is
essentially less than the random uncertainty is, then a subject to
∆
Y
will not
be needed, otherwise it should be added to the random uncertainty. With this
adding, the observations become less accurate and it causes the corresponding
increase of the random uncertainty, i. e. SD of the retrieved parameters, and
the systematic shift does not cause the escape of parameters vector
X
out of
the admissible range of the confidence interval. Thus, the reliability of the
result is reached by increasing the SD. Quite often this fact is difficult to be
accepted psychologically, particularly, in limits of the “fight for accuracy”
traditional in the observational technology. However, it is obvious: in general
form while solving the inverse problems the measurements provide not only
the instrument readings but the results of their numerical modeling as well, so
both processes influence the accuracy. On the basis of the above arguments, the
∆
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