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In-Depth Information
1.3.3 The Three-Dimensional Impedance Tensor
In the 3D-model the conductivity varies in all three directions, that is, along the
x
,
y
,
z
-axes. Here
D
=
3 and
n
=
8.
Generally
Z
xx
Re
Z
xx
i
Im
Z
xx
Re
Z
xy
Im
Z
xy
Z
xy
[
Z
]
=
=
+
.
(1
.
63)
Re
Z
yx
Re
Z
yy
Im
Z
yx
Im
Z
yy
Z
yx
Z
yy
0. This indicates that the three-dimensional medium
has no vertical plane of mirror symmetry.
From the variety of 3D-models we set off the symmetric models with a vertical
plane of mirror symmetry. If the observation point is on this symmetry plane, then,
in line with (1.62),
ske
Let
ske
w
=
0 and
ske
w
=
S
B
0. Here we have the impedance tensor
[Z] that can be reduced to the antidiagonal form.
Among three-dimensional symmetric models we identify the axisymmetric mod-
els, with a vertical axis of symmetry. Here
ske
w
S
=
0 and
ske
w
B
=
w
B
are everywhere equal
to zero, and the tensor [
Z
] can be reduced to the antidiagonal form at any obser-
vation point. Considering an axisymmetric model, we establish two characteristic
directions: the
radial direction
(notation “
r
”) and the
tangential direction
(notation
“
t”
). Let the
x
-axis run in the radial direction. Then we have a tensor [
Z
] with
Z
xx
w
S
and
ske
0 where
Z
r
and
Z
t
are the radial and
tangential impedances which yield the radial and tangential apparent resistivities
and phases:
=
0
,
Z
xy
=
Z
r
,
Z
yx
=−
Z
t
,
Z
yy
=
2
2
r
=
|
Z
r
|
t
=
|
Z
t
|
o
o
(1
.
64)
r
=
arg
Z
r
t
=
arg
Z
t
.
We see that (1.62) is also the necessary condition for the axisymmetric three-
dimensionality.
In the special case that observation point lies on the axis of symmetry, the
impedance tensor assumes the same form as in the 1D-model:
01
01
01
=
=
+
,
.
[
Z
]
Z
c
Re
Z
c
i
Im
Z
c
(1
65)
−
10
−
10
−
10
where scalar
Z
c
is the
central impedance
.
We have examined some simple effects concerned with symmetry of the 2D and
3D-medium. It is believed that in the 3D-models we may observe more complicated
effects, for instance, the
quasi-symmetry effect
indicated by zero
ske
w
S
and nonzero
ske
w
B
:
ske
w
=
0
,
ske
w
=
0
.
(1
.
66)
S
B
