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In-Depth Information
We obtained the known problem (1.40) for the impedance of a 1D medium with the
conductivity
D
. The function
Z
(
z
)in
the model under consideration evidently represents the normal impedance
Z
N
(
z
).
Setting
Z
(
z
)=
Z
N
(
z
) and taking into account (10.43), (10.44) and (10.45), we find
the far-zone asymptotics
N
(
z
)
,
0
≤
z
≤
D
and
D
=
const
,
z
>
|
y
−
y
S
|
>>
d
=−
|
y
−
y
S
|
>>
d
(10
H
A
H
o
z
(
y
)
H
o
y
(
y
)
i
o
(
y
−
y
S
)
o
z
(
y
)
i
o
(
y
−
y
S
)
Z
N
(0)
=−
2
H
A
2
o
y
(
y
)
46)
that coincides with the known expression for a remote infinitely long linear current
(Vanyan, 1965). The normal impedance
Z
N
(0) is connected with the ratio of the
components
.
H
A
H
A
0
y
of the anomalous magnetic field, which can be determined
from values of the tipper
W
zy
known at all points of the
y-
axis from -
0
z
and
∞
to
∞
.To
H
A
find
0
y
, we solve the integral equation (5.80)
∞
H
A
o
y
(
y
o
)
1
W
zy
(
y
)
H
A
o
y
(
y
)
+
dy
o
=−
W
zy
(
y
)
.
y
−
y
o
−∞
Then we compute
H
A
o
z
H
A
=
W
zy
(1
+
o
y
)
(10
.
47)
.
H
A
0
z
and calculate the normal impedance
Z
N
from the far-zone asymptotics. With known
H
A
Knowing
W
zy
, we synthesize the normalized anomalous magnetic field
0
y
,
H
A
H
A
0
z
and
Z
N
, we integrate the second Maxwell equation (the Faraday law) and
continue the longitudinal impedance
Z
0
y
,
to the entire
y
-axis:
⎧
⎨
⎫
⎬
⎭
.
y
E
x
(
y
)
H
y
(
y
)
=
1
H
A
Z
(
y
)
=
Z
N
−
i
o
o
z
(
y
)
dy
(10
.
48)
H
A
o
y
⎩
1
+
−∞
Thus, we find
Z
from
W
zy
. A one-to-one correspondence exists between
Z
and
W
zy
. Therefore, we can apply the Gusarov theorem (1981), stating that inversion of
Z
has a unique solution, and extend this result to inversion of
W
zy
. The uniqueness
theorem for 2D MT inversion (the TE-mode) gives rise to that for 2D MV inversion.
Moreover these two theorems can be supplemented by the uniqueness theorem for
the horizontal magnetic field.
IV. Return to a 2D model shown in Fig. 10.3. Let the longitudinal impedance
Z
(
y
)
Z
(
y
=
,
z
=
0)
=
E
x
(
y
,
z
=
0)
/
H
y
(
y
,
z
=
0) be known at all
−∞
∞
points of the
y
-axis from
to
in the entire range of frequencies
∞
from 0 to
.
