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In-Depth Information
The electric field
E
x
(
y
,
z
) in the air is a solution of the problem
2
E
x
(
y
2
E
x
(
y
,
z
)
+
,
z
)
k
o
E
x
(
y
+
,
z
)
=
0
−∞
<
y
<
∞
0
≥
z
>
−∞
(10
y
2
z
2
.
49)
with boundary condition on the Earth's surface
z
=
0
Z
(
y
)
i
E
x
(
y
,
z
)
Z
(
y
)
H
y
(
y
E
x
(
y
,
z
=
0)
=
,
z
=
0)
=
o
z
and absorption condition in the air
0 s
y
2
E
x
(
y
E
o
e
ik
o
z
→
,
z
)
−
+
z
2
→∞
,
were
k
o
is the air wavenumber, Im
k
o
>
0 , and
E
o
is the amplitude of the incident
wave. It is well known that a problem of this kind has a unique solution continuously
depending on the coefficient
Z
(
y
) in the boundary condition. Consequently, to the
different impedances
Z
(1)
(
y
) and
Z
(2)
(
y
) , the different electric fields
E
(1)
(
y
,
z
) and
x
E
(2
x
(
y
z
) correspond.
Does it mean that to different impedances there correspond different magnetic
fields on the Earth's surface?
Let us give the proof by contradiction. The boundary problem for the electric
field can be rewritten as
,
2
E
x
(
y
2
E
x
(
y
,
z
)
+
,
z
)
k
o
E
x
(
y
+
,
z
)
=
0
−∞
<
y
<
∞
,
0
≥
z
>
−∞
y
2
z
2
z
=
0
=
,
E
x
(
y
z
)
o
H
y
(
y
,
=
i
z
0)
z
0 s
y
2
E
x
(
y
E
o
e
ik
o
z
→
,
z
)
−
+
z
2
→∞
.
(10
.
50)
Solution of this problem exists and is unique. Hence, to identical magnetic fields
H
(1)
y
H
(2)
y
0) identical electric fields
E
(1)
x
E
(2)
x
(
y
,
z
=
0)
≡
(
y
,
z
=
(
y
,
z
)
≡
(
y
,
z
)
correspond.
Assume that to the different impedances
Z
(1)
(
y
) and
Z
(2)
(
y
,
,
) , the identical
magnetic fields
H
(1)
y
H
(2)
y
0) correspond. But it follows from
(10.50) follows that in this case the identical electric fields
E
(1)
x
(
y
,
z
=
0)
≡
(
y
,
z
=
E
(2)
x
(
y
,
z
)
≡
(
y
,
z
)
also correspond to the different impedances
Z
(1)
(
y
) and
Z
(2)
(
y
) , which con-
tradicts the statement derived from (10.49). So, we say that to different impedances
Z
(1)
(
y
,
,
) and
Z
(2)
(
y
=
0) correspond. And taking into account the Gusarov uniqueness theorem for the lon-
gitudinal impedance
Z
(
y
,
,
) different magnetic fields
H
(1)
y
,
=
0) and
H
(2)
y
,
(
y
z
(
y
z
,
) , we state that to different conductivity distributions
(1)
(
y
,
z
) and
(2)
(
y
,
z
) there correspond different magnetic fields
H
(1)
y
(
y
,
z
=
0)
