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size. In the general case (
ske
0) the 3D,b diagrams
look like a figure eight. It would be difficult to find any regularity in their relative
positions.
To conclude, let us compare Fig. 3.12 with Fig. 1.7. The diagrams of
w
S
=
0
,
ske
w
B
=
0
,
ske
w
CBB
=
xx
and
arg
Z
xy
have similar form, while their orientations differ approximately by
/
2.
3.5.3 The Phase-Tensor Eigenstate Problem
To solve the eigenstate problem, Caldwell and his workmates apply technique,
which has its origin in the method of ellipses employed in the telluric current and
dc-resistivity methods (Leonardon, 1948; Bibby, 1986).
Let us show the construction of the phase-tensor ellipse in some detail. The initial
relation is
F
(
)
=
[
]
1
τ
,
(3
.
71)
where the tensor [
] transforms a circle, described by the horizontal unit vector
1
τ
,
into the ellipse, described by the horizontal vector
F
which makes an angle
with
the
x
-axis. To find the equation of the phase-tensor ellipse, we apply the singular
value decomposition of the phase tensor:
cos(
1
cos(
−
)
−
sin(
−
)
0
+
)sin(
+
)
[
]
=
,
sin(
−
)
cos(
−
)
0
2
−
sin(
+
) cos(
+
)
(3
.
72)
where
1
and
2
are the principal values of the phase tensor,
1
≥
2
>
0.
Solving the matrix equation (3.72), we get
1
2
arctan
xy
+
yx
=
xx
−
yy
,
(3
.
73)
where
∈
[0
,
/
2] when
xy
+
yx
≥
0 and
∈
(
/
2
,
) when
xy
+
yx
<
0,
1
2
arctan
xy
−
yx
.
(3
74)
=
xx
+
yy
,
where
∈
[0
,
/
2] when
xy
−
yx
≥
0 and
∈
(
/
2
,
) when
xy
−
yx
<
0,
1
2
2
2
1
=
+
2
|
det[
]
| +
−
2
|
det[
]
|
,
(3
.
75)
1
2
2
2
|
|
−
|
|
2
=
+
2
det[
]
−
2
det[
]
,
where
