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The norms of deviations (10.58) and (10.59) are determined as
⎧
⎨
⎫
⎬
1
/
2
h
N
=
(2)
(
z
)
L
2
=
(1)
(
z
)
(1)
(
z
)
(2)
(
z
)]
2
dz
−
[
−
=
c
⎩
⎭
0
(10
.
60)
⎧
⎨
⎩
⎫
⎬
⎭
1
/
2
h
N
S
=
S
(1)
(
z
)
S
(2)
(
z
)
L
3
=
[
S
(1)
(
z
)
S
(2)
(
z
)]
2
dz
−
−
0
c
=
h
(
h
−
z
−
2
h
/
3)
.
h
, the deviation
N
can always be done arbitrar-
ily large, and the deviation
N
S
arbitrarily small. Consequently, arbitrarily differing
conductivities can correspond to close conductances and close admittances.
Let the medium contain a thin layer, whose conductance
S
o
is much smaller
than the conductance
S
of the overlying layers. The conductivity of the layer can
vary within wide limits constrained by the condition
S
o
<<
S
, but these variations
scarcely affect the admittance measured on the Earth's surface.
The one-dimensional inverse problem is unstable. Evidently, we have every rea-
son to extend this conclusion to the two-dimensional and three-dimensional inverse
problems. Compare, for example, a 2D or 3D-model with a slowly varying boundary
between two deep layers and a model in which this boundary rapidly fluctuates
around its slow variation. Their MT and MV response functions observed on the
Earth's surface will virtually coincide, although these models are largely different.
Inverse problems of magnetotellurics are unstable. An arbitrarily small error in
initial MT and MV data can lead to an arbitrarily large error in the inversion of
these data, i.e., in the conductivity distribution. Using the terminology of Hadamard,
we state that the inverse problems of magnetotellurics are ill-posed. An immediate
solution of an ill-posed (unstable) problem is generally meaningless, because it can
yield results far from reality.
Choosing large
c
and small
10.4 In the Light of the Theory of Ill-Posed Problems
...
The cornerstone of the MT and MV data interpretation is the theory of ill-posed
problems. Its basic principles were formulated by Tikhonov (1963). Presently, meth-
ods of this theory have been developed rather comprehensively and are widely used
in practice (Tikhonov and Arsenin, 1977; Lavrentyev et al., 1980; Glasko, 1984;
Tikhonov and Goncharsky, 1987; Zhdanov, 2002) The Russian mathematical school
headed by Tikhonov gave rise to a new doctrine of physical experiment encompass-
ing various fields of science and technology.
Following (Berdichevsky and Dmitriev, 1991, 2002; Zhdanov, 2002), we
consider inverse problems of magnetotellurics in light of Tikhonov's theory of
regularization, which provides a basis for developing the strategy of MT and MV
inversions.
