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This problem coincides with the magnetotelluric problem on the horizontally uni-
form magnetic field in a horizontally layered medium (Berdichevsky and Dmitriev,
2002). Thus, we can introduce the normal one-dimensional impedance
Z
N
satisfying
the Riccati equation
Z
N
z
−
N
Z
N
=
i
o
(5
.
10)
and express
Z
N
as
1
d
dz
.
Z
N
=−
(5
.
11)
N
Summing up (5.6), (5.8) and (5.11), we write
(
z
)
Z
N
(
z
)
U
o
(
x
,
y
)
(
z
)
Z
N
(
z
)
U
o
(
x
,
y
)
E
x
E
y
E
z
=
,
=−
,
=
0
,
y
x
(
z
)
U
o
(
x
,
y
)
(
z
)
U
o
(
x
,
y
)
H
x
H
y
=
,
=
,
x
y
=−
(
z
)
Z
N
(
z
)
i
2
U
o
(
x
,
y
)
+
2
U
o
(
x
,
y
)
H
z
.
o
x
2
y
2
(5
.
12)
,
Thus, determination of the normal field reduces to differentiating
U
o
(
x
y
).
>>
r
i
,
>>
Next
we come bac
k to Fig. 5.1 and assume that
R
o
R
o
r
s
, where
x
o
+
R
o
=
y
o
+
z
o
is a distance between the centre O(
x
o
,
y
o
,
z
o
) of the source
domain I and the centre M
o
(0
0) of the observation domain S, while
r
i
and
r
s
are maximum radii of the domains I and S. Then we define
U
o
(
x
,
0
,
,
y
)as
U
o
(
x
,
y
)
≈
U
o
(
R
)
,
R
≈
R
o
,
(5
.
13)
=
(
x
where
R
−
x
o
)
2
+
(
y
−
y
o
)
2
+
z
o
is a distance between the observation site
M(
x
,
y
,
z
=
0) and the centre O(
x
o
,
y
o
,
z
o
) of the source domain I. So, we obtain:
R
=
R
o
=
U
o
(
x
,
y
)
dU
o
(
R
)
dR
R
x
o
)
1
R
dU
o
(
R
)
dR
x
o
)
1
R
dU
o
(
R
)
dR
≈
x
=
(
x
−
≈
(
x
−
C
o
(
x
−
x
o
)
,
x
R
=
R
o
=
U
o
(
x
,
y
)
dU
o
(
R
)
dR
R
y
o
)
1
R
dU
o
(
R
)
dR
y
o
)
1
R
dU
o
(
R
)
dR
=
y
=
(
y
−
≈
(
y
−
C
o
(
y
−
y
o
)
,
y
R
=
R
o
=
2
U
o
(
x
2
U
o
(
x
,
y
)
+
,
y
)
1
R
dU
o
(
R
)
dR
≈
C
o
,
x
2
y
2
.
(5
14)
where
