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Re
Z
yy
Im
Z
xy
−
Re
Z
xy
Im
Z
yy
Re
Z
yy
Im
Z
xy
−
Re
Z
xy
Im
Z
yy
xy
=
Re
Z
xy
Re
Z
yx
=
Re
Z
xy
Re
Z
yx
,
Re
Z
xx
Re
Z
yy
−
Re
Z
xx
Re
Z
yy
−
Re
Z
xx
Im
Z
yx
−
Re
Z
yx
Im
Z
xx
Re
Z
xx
Im
Z
yx
−
Re
Z
yx
Im
Z
xx
yx
=
Re
Z
xy
Re
Z
yx
=
Re
Z
xy
Re
Z
yx
,
Re
Z
xx
Re
Z
yy
−
Re
Z
xx
Re
Z
yy
−
Re
Z
xx
Im
Z
yy
−
Re
Z
yx
Im
Z
xy
Re
Z
xx
Im
Z
yy
−
Re
Z
yx
Im
Z
xy
yy
=
Re
Z
xy
Re
Z
yx
=
Re
Z
xy
Re
Z
yx
.
(3
Re
Z
xx
Re
Z
yy
−
Re
Z
xx
Re
Z
yy
−
.
59)
It is seen that [
] is independent of local distortions. Without any assumptions
of regional dimensionality we may derive the tensor [
] immediately from the
superimposition tensor [
Z
S
] as a true performance of the regional (one-dimensional,
two-dimensional or three-dimensional) impedance [
Z
R
].
Rotating the tensor [
] clockwise trough an angle
, we get
)]
−
1
,
.
[
(
)]
=
[
R
(
)] [
][
R
(
(3
60)
from which
xx
(
)
=
+
3
sin 2
+
4
cos 2
2
xy
(
)
=
+
3
cos 2
−
4
sin 2
1
(3
.
61)
yx
(
)
=−
+
3
cos 2
−
4
sin 2
1
yy
(
)
=
−
3
sin 2
−
4
cos 2
,
2
where
−
+
=
=
xy
yx
xx
yy
,
1
2
2
2
(3
.
62)
=
xy
+
yx
2
=
xx
−
yy
2
.
3
4
The rotational invariants of the tensor [
]are
J
14
=
=
2
=
xx
+
yy
,
tr[
]
2
J
15
=
=
xx
yy
−
xy
yx
,
J
16
=
xy
−
yx
,
J
17
=
=
det[
]
(3
.
63)
xx
+
xy
+
yx
+
yy
.
Take a model with a horizontally homogeneous (1D) medium containing local
three-dimensional near-surface inhomogeneities. In this model,
Z
xx
=
Z
yy
=
0 and
Z
xy
=−
Z
yx
Z
R
, where
Z
R
=
is the regional one-dimensional impedance. Then,
according to (3.60),
