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(
y
,
z
) are piecewise-analytical. An infinite homogeneous basement of conductiv-
ity
const
occurs at a depth
D
. The model is excited by the plane
E
-polarized
electromagnetic wave incident vertically on the Earth's surface
z
D
=
0.
The Dmitriev theorem states that the piecewise-analytical distribution of conduc-
tivity
=
N
(
z
)
M
∈
S
(
M
)
=
N
(
z
)
+
(
y
,
z
)
M
∈
S
is uniquely determined by exact values of the tipper
H
z
(
y
,
z
=
0)
W
zy
(
y
)
=
0)
,
−∞
<
y
<
∞
,
0
≤
<
∞
,
H
y
(
y
,
z
=
given on the Earth's surface
z
=
0 at all points of the
y
-axis from
−∞
to
∞
in the
entire range of frequencies
.
The uniqueness theorem is proven in two stages. First we derive the asymptotics
of the tipper
W
zy
(
y
) at a great distance from the inhomogeneity S and show that
it determines the normal conductivity
from 0 to
∞
N
(
z
). Then, with the known conductivity
N
(
z
), we prove that the tipper uniquely determines the longitudinal impedance of
the inhomogeneous medium.
The anomalous magnetic field
H
A
on the Earth's surface can be represented as a
field produced in a horizontally homogeneous layered medium by excess current of
density
j
x
distributed in the domain S. Normalizing
H
A
, we write:
H
y
(
y
,
z
=
0)
H
A
o
y
(
y
)
=
=
j
x
(
M
o
)
h
y
(
y
,
M
o
)
dS
H
y
(
z
=
0)
S
(10
.
35)
H
z
(
y
,
z
=
0)
H
A
o
z
(
y
)
=
=
j
x
(
M
o
)
h
z
(
y
,
M
o
)
dS
,
H
y
(
z
=
0)
S
where
h
y
(
y
M
o
) are magnetic fields produced at the surface of a hori-
zontally homogeneous medium by an infinitely long linear current of the unit den-
sity flowing at the point
M
o
(
y
o
,
,
M
o
)
,
h
z
(
y
,
z
o
)
∈
S
in the
x
-direction. The functions
h
y
(
y
,
M
o
)
and
h
z
(
y
,
M
o
) assume the form (Dmitriev, 1969; Berdichevsky and Zhdanov, 1984)
∞
i
o
y
o
)
e
λ
z
U
(
h
y
(
y
,
M
o
)
=
lim
z
cos
λ
(
y
−
λ,
z
=
0
,
z
o
)
λ
d
λ
→
0
0
(10
.
36)
∞
i
o
y
o
)
e
λ
z
U
(
h
z
(
y
,
M
o
)
=−
lim
z
→
0
sin
λ
(
y
−
λ,
z
=
0
,
z
o
)
λ
d
λ,
0
where the factor
e
λ
z
relates to the upper half-space
z
≤
λ,
,
0 and the function
U
(
z
z
o
)
is the solution of the boundary problem
