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In-Depth Information
Take the set of conductivity distributions obtained from the inversion of 1D
admittance:
(
z
):
(
z
)]
L
2
≤
Y
}
,
Y
(
∈
={
)
−
Y
[
,
(10
.
56)
where
Y
(
) is the measured admittance,
Y
[
,
(
z
)] is the operator calculating the
admittance from a given distribution
Y
is the error in the admittance. The
theorem of stability of the
S
-distribution implies that, for any
(
z
), and
(1)
(2)
(
z
) and
(
z
)
from the set
, the following condition is valid:
C
≤
z
z
(1)
(2)
(
z
)
dz
−
(
z
)
dz
(
Y
)
,
(10
.
57)
0
0
(1)
(2)
where
(
z
) meet condition (10.57), they are
equivalent, i.e., they are characterized by closely related
S
-distributions and cannot
be resolved by MT observations performed with an error
→
0as
Y
→
0. If
(
z
) and
Y
. Such
−
distribu-
tions are called
S
-
equivalent distributions
. We say that
is the set of
S
-equivalent
distributions of the conductivity. In the framework of one-dimensional magnetotel-
lurics, we can formulate the following generalized principle of
S
-equivalence: the
conductance
S
characterizes the whole set
of equivalent solutions of the inverse
problem. To specify the entire set
it is sufficient to know its
S
-distribution.
Differentiating the conductance
S
(
z
), one intends to find the conductivity
(
z
).
However, the immediate numerical differentiation of
S
(
z
) is an unstable operation
generating a scatter in the distribution
)
is evidently an ill-posed problem. It is easy to show, that there exist essentially
different distributions
(
z
). The determination of
(
z
)from
Y
(
(2)
(
z
) corresponding to close distributions
S
(1)
(
z
)
and
S
(2)
(
z
), and thereby to close distributions
Y
(1)
(
(1)
(
z
) and
) and
Y
(2)
(
).
As an example, consider a model with an infinite homogeneous basement at a
depth
h
.Let
0
[
z
,
z
+
for
z
∈
h
]
(1)
(
z
)
(2)
(
z
)
−
=
√
(10
.
58)
[
z
,
z
+
c
/
h
for
z
∈
h
]
,
where
z
+
h
<
h
, while
c
and
h
are arbitrary positive constants. Then
z
S
(1)
(
z
)
S
(2)
(
z
)
(1)
(
z
)
(2)
(
z
)]
dz
−
=
[
−
0
⎧
⎨
z
0
for
0
≤
z
≤
(10
.
59)
z
)
c
(
z
−
z
≤
z
+
=
√
for
z
≤
h
⎩
h
c
√
z
+
h
for
h
≤
z
≤
h
.
