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In-Depth Information
W
zx
=
U
zx
Z
xx
+
U
zy
Z
yx
(4
.
5)
W
zy
=
U
zx
Z
xy
+
U
zy
Z
yy
.
There is good reason to believe that generally the impedance tensor
[
Z
]
reflects
vertical and horizontal variations in the Earth's conductivity and that its components
tend to zero as
0. In view of (4.5) we can assume that the same properties are
sha-red by the Wiese-Parkinson matrix [
W
].
In a two-dimensional model with strike along the
x
→
−
axis the relation (4.5)
reduces to
W
zx
=
0
(4
.
6)
W
zy
=
U
zx
Z
xy
,
where
Z
xy
is the longitudinal impedance. Let the anomalous magnetic field be
caused by a near-surface body, whose dimensions are far less than the skin depth.
Then we can ignore the induction effect and take
U
zx
as a real-valued factor. In that
event
W
zy
and
Z
xy
are in-phase.
4.1.1 Rotation of the Wiese-Parkinson Matrix
How do the components of the Wiese-Parkinson matrix change as the
x
,
y
-axes
rotate about the
z
-axis? Let
be a clockwise rotation angle. Then
)]
−
1
[
R
(
H
z
=
[
W
]
H
τ
=
[
W
][
R
(
)]
H
τ
=
[
W
(
)]
H
τ
(
)
,
where
)]
−
1
[
W
(
)]
=
[
W
][
R
(
H
τ
(
)
=
[
R
(
)]
H
τ
.
Thus,
W
zx
(
)
=−
W
zx
(
±
)
=∓
W
zy
(
±
/
2)
=
W
zx
cos
+
W
zy
sin
(4
.
7)
W
zy
(
)
=−
W
zy
(
±
)
=±
W
zx
(
±
/
2)
=−
W
zx
sin
+
W
zy
cos
.
As is easy to see, in two-dimensional and axisymmetric three-dimensional mod-
els both the components,
W
zx
(
) and
W
zy
(
), are in-phase or anti-phase. Really,
according to (4.4) and (4.7), we have
W
zx
(
)
=
W
zy
(
)tan
,
whence
