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det[
]
=
xx
yy
−
xy
yx
,
(3
.
76)
=
xx
+
xy
+
yx
+
yy
.
Substituting [
] from (3.72) into (3.71), we obtain:
2
2
2
1
F
2
=
−
1
)
,
(3
.
77)
2
1
sin
2
(
2
−
1
)
+
2
cos
2
(
where
1
= −
.
This is the equation of an ellipse in polar coordinates
F
,
.
The major and minor semiaxes of the ellipse make the angles
=
1
and
=
1
+
/
2 with the
x
-axis. They are equal to the principal values
1
and
2
of the phase
tensor, which determine the principal phases
1
=−
arctg
1
and
2
=−
arctg
2
lying in the IV quadrant.
The phase-tensor ellipse can be constructed using (3.73), (3.74), and (3.75). It
is defined by four independent parameters
that are free of distortions
caused by near-surface inhomogeneities. These parameters characterize the regional
background and fill all four degrees of freedom possessed by the matrix [
1
,
2
,
,
].
The phase-tensor ellipses for a three-dimensional asymmetric and a two-
dimensional regional backgrounds are exemplified in Fig. 3.13.
Let us consider the angles
and
, which control the orientation of the phase-
tensor ellipse.
The angle
is defined by (3.73). It is curious that the Caldwell-Bibby-Brown
formula (3.73), deduced for a three-dimensional asymmetric background, coincides
with the basic Bahr formula (3.8), which yields the strike of the two-dimensional
regional background. In fact, with due regard for (3.59),
1
2
arctan
xy
+
yx
=
xx
−
yy
2
arctan
Re
Z
yy
Im
Z
xy
−
Re
Z
xy
Im
Z
yy
+
Re
Z
xx
Im
Z
yx
−
Re
Z
yx
Im
Z
xx
1
=
Re
Z
yy
Im
Z
xx
−
Re
Z
xx
Im
Z
yy
+
Re
Z
yx
Im
Z
xy
−
Re
Z
xy
Im
Z
yx
2
arctan
Im(
Z
yx
Z
xx
+
Z
xy
Z
yy
)
1
=
Z
xy
Z
yx
)
.
Im(
Z
xx
Z
yy
+
.
(3
78)
is defined by (3.74). This angular rotationally invariant parame-
ter indicates the regional asymmetry. It is referred to as the
Caldwell-Bibby-Brown
skew
:
The angle
.
1
2
arctan
xy
−
yx
xx
+
yy
ske
w
= || =
(3
.
79)
CBB
w
,
/
.
The Caldwell-Bibby-Brown
ske
CBB
is taken in the range [0
2]
In models with
w
1D, 2D and axially symmetric 3D regional conductivity,
ske
CBB
= 0. Departure
