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where
S
1
(
x
,
y
)
=
2
y
B
x
1
−
M
yy
(
x
B
,
,
y
)
+
[
M
xy
(
x
B
,
y
B
|
x
,
y
)
Z
xx
(
x
B
,
y
B
)
+
M
yy
(
x
B
,
y
B
|
x
,
y
)
Z
xy
(
x
B
,
y
B
)]
S
(
x
B
,
y
B
)
/
2
Z
xy
(
x
,
y
)
S
1
(
x
,
y
)
2
.
1
−
M
xx
(
x
B
,
y
B
|
x
,
y
)
−
[
M
xx
(
x
B
,
y
B
|
x
,
y
)
Z
yx
(
x
B
,
y
B
)
+
M
yx
(
x
B
,
y
B
|
x
,
y
)
Z
yy
(
x
B
,
y
B
)]
S
(
x
B
,
y
B
)
/
2
Z
yx
(
x
=
,
y
)
Here
2
S
1
−
S
1
=
S
1
+
S
1
characterizes the measurement and model errors.
The above technique is a part of more general approach which is referred to as
the
Obukhov Z
int
transformation
. Let us recall this interesting approach which
can be usefull in indentifying the geoelectric structures.
On the surface of the 1D model, the internal magnetic field
H
int
−
τ
=
H
τ
/
2
.
In view
of this, we can introduce an “internal” 1D impedanse
Z
int
as
E
x
H
int
y
E
y
H
int
x
Z
int
=
=−
=
2
Z
(11
.
39)
int
and an “internal” apparent resistivity
as
Z
int
2
2
o
=
|
Z
|
int
=
o
=
.
(11
.
40)
A
4
int
coincide
respectively with the doubled Tikhonov-Cagniard impedance
Z
and the Tikhonov-
Cagniard apparent resistivity
In the 1D model, the impedance
Z
int
and apparent resistivity
A
.
Similarly we can introduce an “internal”-impedance tensor in the 2D and 3D
models. To this end, we turn to (1.13) and replace
H
x
,
H
y
for
H
int
H
in
y
:
,
x
E
x
E
x
H
x
o
J
E2
J
E1
x
E
x
=
+
=
+
H
y
o
(
Z
N
+
)
a
x
E
y
E
y
J
E2
y
H
y
o
J
E1
E
y
=
+
=
H
x
o
(
−
Z
N
+
)
+
b
y
(11
.
41)
H
int
x
5
H
x
H
x
J
H2
x
H
y
o
J
H1
x
=
0
.
+
=
H
x
o
(0
.
5
+
)
+
c
H
int
y
5
H
y
H
y
H
x
o
J
H2
J
H1
y
=
0
.
+
=
+
H
y
o
(0
.
5
+
)
.
d
y
Eliminating
H
x
o
,
H
y
o
from (11.41
c,d
) and substituting these values in (11.41
a,b
),
we establish