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I. Tikhonov (1965) proved the uniqueness theorem for 1D MT inversion in
the class of piecewise-analytical functions
(
z
). In our topic we present a sim-
plified proof of the Tikhonov theorem for the case of a homogeneously layered
model.
Let
(
z
) be a piecewise-constant function of the depth
z
:
=
m
(
z
)
at
z
m
−
1
<
z
<
z
m
,
m
and
h
m
are the conductivity and thickness of the mth layer and
z
m
is the depth of its
lower boundary. At the depth
z
m
∈
[1
,
M
]
,
z
o
=
0
,
z
M
=∞
,
h
m
=
z
m
−
z
m
−
1
, where
=
z
M
−
1
, the model rests on an infinite homogeneous
basement of conductivity
=
const
. The admittance
Y
(
z
,
) in this homoge-
M
neously layered model satisfies the Riccati equation
dY
(
z
,
)
o
Y
2
(
z
+
i
,
)
=−
(
z
)
,
z
∈
[0
,
z
M
−
1
]
,
∈
[0
.
∞
]
( 0
.
30)
dz
with the boundary conditions
i
)
M
[
Y
(
z
,
)]
S
=
0
,
Y
(
z
M
−
1
,
)
=
(1
+
o
.
2
Using (10.30), we can easily derive a recurrent formula expressing
Y
m
−
1
=
Y
(
z
m
−
1
,
) through
Y
m
=
Y
(
z
m
,
):
m
+
−
m
−
Y
m
)
e
2
ik
m
h
m
(
Y
m
)
(
Y
m
−
1
=
m
Y
m
)
e
2
ik
m
h
m
,
.
(10
31)
(
m
+
Y
m
)
+
(
m
−
where
k
m
is the wavenumber of the mth layer:
i
)
o
m
2
k
m
=
(1
+
and
i
)
k
m
o
=
m
m
=
(1
+
o
.
2
Inverse of (10.31) yields a formula determining
Y
m
through
Y
m
−
1
(converting the
admittance from the upper boundary of the mth layer to its lower boundary):
Y
m
−
1
)
e
2
ik
m
h
m
(
m
+
Y
m
−
1
)
−
(
m
−
Y
m
=
m
Y
m
−
1
)
e
2
ik
m
h
m
.
(10
.
32)
(
m
+
Y
m
−
1
)
+
(
m
−
