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[
W
]
=
[0
W
zy
]
,
H
z
=
W
zy
H
y
,
(4
.
4)
where
J
H
1
z
W
zy
=
.
1
+
J
H
1
y
Clearly in this case the Wiese-Parkinson matrix contains only one component
oriented across the strike.
The similar relation is characteristic of a three-dimensional axisymmetric model.
In coordinates aligned with the radial and tangential directions, the Wiese-
Parkinson matrix has only one component oriented radially.
In closing we follow Zhang et al. (1993) and Ritter and Banks (1998) and
show that the Wiese-Parkinson matrix can be expressed in terms of the impedance
tensor. On eliminating
H
x
o
,
H
y
o
from (1.13
a,b
) and substituting them in (4.1
c
),
we obtain:
H
z
=
U
zx
E
x
+
U
zy
E
y
,
where
J
H
2
z
J
E
1
y
J
H
1
z
(
J
E
2
y
−
−
Z
N
)
U
zx
=
J
E
2
x
J
E
1
y
−
(
J
E
1
x
+
Z
N
)(
J
E
2
−
Z
N
)
y
J
H
1
z
J
E
2
x
−
J
H
2
z
(
J
E
1
x
+
Z
N
)
U
zy
=
Z
N
)
.
J
E
2
x
J
E
1
y
−
(
J
E
1
x
+
Z
N
)(
J
E
2
y
−
In matrix notation
H
z
=
[
U
]
E
τ
=
[
U
][
Z
]
H
τ
=
[
W
]
H
τ
,
where
=
U
zx
U
zy
[
U
]
Z
xx
Z
xy
=
U
zx
U
zy
[
W
]
=
[
U
][
Z
]
Z
yx
Z
yy
=
U
zx
Z
xx
+
U
zy
Z
yy
,
U
zy
Z
yx
U
zx
Z
xy
+
from which
