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Fig. 6.1
The
vertical-interface model
The normal fields are defined as
H
N
x
H
N
x
H
x
o
e
ik
z
H
x
o
e
ik
z
=
,
=
.
(6
.
2)
Here
k
=
√
i
k
=
√
i
H
N
H
N
o
/
,
o
/
,
Im
k
>
0 and
H
x
o
=
x
(0)
=
x
(0)
=
2
H
p
(0), where
H
p
(0) is the primary magnetic field on the Earth's surface
z
=
0.
The anomalous fields meet the equations
2
H
A
x
2
H
A
x
2
H
A
x
2
H
A
x
+
+
(
k
)
2
H
x
(
k
)
2
H
x
+
=
0
,
+
=
0 6
.
3)
z
2
y
2
z
2
y
2
H
x
(0)
H
x
(0)
with the boundary conditions
=
=
0 on the Earth's surface and the
conditions
H
x
(
z
H
x
(
z
>
0)
→
0
,
>
0)
→
0
at infinity.
y
→−∞
y
→∞
Solving these equations by the method of separation of variables, we get
∞
∞
H
x
a
m
e
y
sin
mzdm
H
x
a
m
e
−
y
sin
mzdm
=
,
=
,
(6
.
4)
0
0
where
m
2
m
2
=
=
−
(
k
)
2
,
−
(
k
)
2
,
Re
>
0
.
The constants
a
m
and
a
m
can be defined from the boundary conditions at the
vertical interface
y
=
0. It follows from the continuity of
H
x
and
E
z
=−
H
x
/
y
that
y
=
0
H
x
H
x
H
x
H
x
−
=
−
.
(6
.
5)
H
x
H
x
=
y
y
