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and
H
(2)
y
(
y
,
z
=
0) on the Earth's surface. The uniqueness theorem for the magnetic
field
H
y
(
y
0) is proved.
Both the methods, MT and MV soundings, have a common mathematical basis.
The 2D conductivity distribution is uniquely determined from exact values of TE
impedances as well as from exact values of tippers or transverse horizontal magnetic
fields given on the infinitely long transverse profile in the entire frequency range.
,
z
=
10.3.3 On the Instability of the Inverse Problem
Inverse problems of magnetotellurics are unstable. The set
, characterized by
small misfits of the impedance tensor and tipper, can contain equivalent solutions
that strongly differ from one another and from the exact model solution.
We illustrate this property of the inverse problem by the example of the 1D inver-
sion. The analysis is based on the theorem of stability of the
S
-distribution proven
by Dmitriev in (Berdichevsky and Dmitriev, 1991, 2002).
Recall that the
S-distribution
stands for a function
z
S
(
z
)
=
(
z
)
dz
(10
.
51)
0
determining the conductance of the Earth on the interval [0
,
z
] The conductivity
is
connected with the conductance
S
through the differential relation
(
z
)
=
dS
(
z
)
/
dz
.
The theorem of stability of the
S
-distribution consists of two statements.
1. The admittance
Y
(
) measured at the Earth's surface depends
continuously on
S
(
z
) . Thus, the condition
)
=
Y
(
z
=
0
,
S
(1)
(
)
C
≤
S
(2)
(
−
.
)
(10
52)
implies that
Y
(1)
(
)
L
2
≤
Y
(2)
(
)
−
(
)
,
(10
.
53)
where
0.
2. The conductance
S
(
z
) is stably determined from the admittance
Y
(
→
0at
→
=
=
)
Y
(
z
,
0
) measured at the Earth's surface. Thus,
S
(1)
(
)
C
→
S
(2)
(
)
−
0
(10
.
54)
if
Y
(1)
(
)
L
2
→
Y
(2)
(
)
−
0
.
(10
.
55)
