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Solving integral equation (5.74), we determine the normalized vertical compo-
nent of the anomalous magnetic field on the Earth's surface. With a knowledge of
H
o
z
(
x
y
), we use (5.47) and compute the normalized horizontal com-ponents of the
anomalous magnetic field
,
∞
∞
1
2
dx
o
dy
o
H
o
x
(
x
H
o
z
(
x
o
,
,
y
)
=
y
o
)
(
x
π
x
−
x
o
)
2
+
−
y
o
)
2
+
z
2
(
y
−∞
−∞
.
(5
75)
∞
∞
1
2
dx
o
dy
o
H
o
y
(
x
H
o
z
(
x
o
,
,
y
)
=
y
o
)
(
x
z
2
.
π
y
−
x
o
)
2
+
(
y
−
y
o
)
2
+
−∞
−∞
And what is more, we can substitute (5.75) in (5.66) and calculate the normalized
components of the anomalous electric field,
E
o
x
(
x
E
o
y
(
x
y
).
Equations obtained are readily reduced to the two-dimensional case. Let
x
be the
strike of the model. Then (5.73) assumes the form
,
y
) and
,
∞
H
o
z
(
y
)
H
o
z
(
y
o
)
dy
o
=
+
L
(
y
,
y
o
)
W
zy
(
y
)
,
(5
.
76)
−∞
where the kernel
L
(
y
,
y
o
)is
∞
W
zy
(
y
)
2
1
W
zy
(
y
)
L
(
y
,
y
o
)
=
(
x
y
o
)
2
dx
o
=
y
o
)
,
(5
.
77)
π
y
π
(
y
−
x
o
)
2
−
+
(
y
−
−∞
from which
∞
H
o
z
(
y
o
)
dy
o
y
W
zy
(
y
)
π
H
o
z
(
y
)
+
=
W
zy
(
y
)
.
.
(5
78)
−
y
o
−∞
H
o
z
(
y
) can be translated into the equation for
H
o
y
(
y
).
Note that the equation for
H
o
z
=
W
zy
(
H
o
y
+
Substituting
1) in (5.78), we get
∞
∞
H
o
y
(
y
o
)
1
π
W
zy
(
y
o
)
1
π
W
zy
(
y
o
)
dy
o
y
H
o
y
(
y
)
+
dy
o
=−
.
.
(5
79)
y
−
y
o
−
y
o
−∞
−∞
H
o
y
(
y
)ina
On inverting the Hilbert transformation, we write the equation for
form suggested by Vanyan et al. (1998):
