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2
+
4
cos 2
−
4
sin 2
[
(
)]
=
,
(3
.
68)
−
4
sin 2
2
−
4
cos 2
where
R
R
1
2
(tan arg
sin(arg
1
+
arg
2
)
1
+
2
)
2
=
tan arg
=
2
,
1
cos arg
2cos arg
R
R
1
2
(tan arg
sin(arg
1
−
arg
2
)
R
R
4
=
1
−
tan arg
2
)
=
2
.
1
cos arg
2cos arg
In arbitrary coordinates, the two-dimensional phase tensor [
] is symmetric. The
similar properties of [
] are observed in models with axially symmetric three-
dimensional regional background. If the three-dimensional regional background is
asymmetric, the symmetry of the phase tensor is violated.
3.5.2 Polar Diagrams of the Phase Tensor
The dependence of the components of the phase tensor upon the direction can be
displayed by means of polar diagrams.
Let the phase tensor [
] be defined on measurement axes
x
,
y
. Introduce new
axes
x
,
y
rotated through a clockwise angle
. In accordance with (3.61),
=
|
|
=
|
|
,
xx
(
)
arctan
xx
(
)
arctan(
+
3
sin2
+
4
cos2
)
2
arctan
)
= |
(3
.
69)
xy
(
)
=
xy
(
arctan(
+
3
cos2
−
4
sin2
)
|
.
1
Plotting these quantities on the
x
-axis and changing
, we trace the
polar diagrams of the phase tensor. They take the form of regular or irregular ovals
with more or less narrow waist and may consist of four petals.
The
from0to2
xy
-diagrams are exemplified in Fig. 3.12. They clearly indicate
the dimensionality of regional structures.
In the event of horizontally homogeneous (1D) regional background, the
xx
- and
xy
-
diagram contracts to a point and vanishes, while the
xx
-diagram is a circle of radius
arg
Z
R
where
Z
R
is the regional one-dimensional impedance.
In the event of a two-dimensional or axially symmetric three-dimensional
regional background, we have
arctan (
)
,
+
yy
sin
2
xx
=
xx
cos
2
(3
.
70)
arctan
,
xy
=
{
(
xx
−
yy
)sin
cos
}
xx
,
yy
are tangents of the phases of the longitudinal and transverse (or
tangential and radial) regional impedances. The
where
xx
-diagram assumes the form of
