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Im
Z
R
Re
Z
R
Im
Z
R
Re
Z
R
tan arg
Z
R
tan arg
Z
R
xx
=
=
,
xy
=
0
,
yx
=
0
,
yy
=
=
,
(3
.
64)
from which
tan arg
Z
R
tan arg
Z
R
tan arg
Z
R
10
01
0
[
]
=
=
.
(3
.
65)
0
] has a scalar matrix with the factor tan arg
Z
R
Here the tensor [
characterizing the
phase of the regional impedance.
Next examine a superimposition model with local three-dimensional near-surface
inhomogeneities placed against a two-dimensional regional background. Let the
x-
axis run along the regional strike. Then
0
R
1
[
Z
R
]
=
,
R
2
−
0
2
are principal values of the regional impedance tensor (the longitu-
dinal and transverse impedances). By virtue of (3.60)
1
where
and
R
2
R
1
Im
Im
R
R
xx
=
2
=
tan arg
2
,
xy
=
0
,
yx
=
0
,
yy
=
1
=
tan arg
1
,
Re
Re
(3
.
66)
from which
tan arg
R
2
0
[
]
=
.
(3
.
67)
R
1
0
tan arg
2
Here the tensor [
] has a diagonal matrix with the components tan arg
and
1
characterizing the phases of the transverse and longitudinal impedances.
Evidently, the longitudinal and transverse directions are principal directions of the
two-dimensional tensor [
tan arg
1
2
], while tan arg
and tan arg
are its principal values.
In accordance with (3.65) and (3.67), the tensor [
] is given the name
phase
tensor
.
Let us rotate the two-dimensional phase tensor [
] clockwise through an angle
. According to (3.61),
xx
=
2
+
4
cos 2
=−
4
sin 2
xy
yx
=−
4
sin 2
yy
=
2
−
4
cos 2
,
from which
