Information Technology Reference
In-Depth Information
where
s
is the root-mean-square (standard) deviation of
. Introduce a parameter
, which is equal to the ratio of variances
D
for the
F
and
:
s
D
F
D
F
s
2
=
=
.
(10
.
85)
Then
1
1
2
s
(
)
=
ln
s
F
s
−
F
(
)
,
(10
.
86)
2
where
2
F
2
+
−
0
(
)
=
−
F
{}
.
(10
.
87)
In this case the condition for maximum likelihood (10.78) reduces to equation
(˜
)
=
inf
∈
M
(
)
(10
.
88)
implying the minimization of Tikhonov's functional
(
) as in the regularization
method. Here the functional
(
) consists of the misfit functional I and the stabi-
lizing functional
with the regularization parameter
:
(
)
=
I
+
F
s
2
2
F
2
s
=
−
0
I
=
−
F
{}
=
.
The problem is conditionally correct and can be solved directly by the regular-
ization method, using (10.71).
We see that the maximum likelihood method leads to the same algorithms as
in the deterministic approach. A similar conclusion can be made for the
Bayesian
inversion
based on the Bayes theorem of hypotheses (Zhdanov, 2002). If we limit
our consideration to the inversion of MT and MV data, the advantage of probabilistic
approaches is not obvious. However, these approaches using the powerful methods
of the modern probability theory and statistics do give a simple and convenient tool
for analyzing a solution obtained.
10.5 Comparison Criteria
To compare magnetotelluric and magnetovariational response functions and deter-
mine their misfits, we use some special criteria taking into account the pequliarities
of the inverse problems of magnetotellurics. The comparison criteria must be con-
structed so that the contributions of data with a similar amount of information will
be the same. The point is that periods,
T
, as well as apparent resistivities,
A
, and
|
|
moduli of impedances,
, can vary over several orders of magnitude. By virtue of
the principle of similitude, the ranges with similar relative variations of these values
Z
