Information Technology Reference
In-Depth Information
⎧
⎨
)
+
1
−
1
1
+
1
a
2
cos
E
y
(
−
r
cos
for
r
≥
a
r
U
(
r
,
,
)
=
(7
.
1)
1
1
+
1
2
⎩
E
y
(
−
)
r
cos
for
r
≤
a
,
where
E
y
(
) is the normal electric field on the Earth's surface
z
=
0 (Smythe,
1950). The function
U
satisfies the boundary conditions
=+
0
,π
−
0
=
r
=
a
+
0
=
r
=
a
−
0
.
U
1
U
1
1
U
0
U
|
r
=
a
+
0
=
U
|
r
=
a
−
0
r
r
On differentiating
U
, we get the electric field along the
y
-axis:
=
0
=−
U
(
r
,
,
)
E
y
(
y
,
)
=
E
y
(
y
,
z
=
0
,
)
r
⎧
⎨
1
E
y
(
+
1
−
1
1
+
1
a
2
y
2
(7
.
2)
)f r
|
y
| ≥
a
=
1
1
+
1
2
⎩
E
y
(
|
|
≤
)
for
y
a
,
whence
⎧
⎨
1
Z
N
(
+
1
−
1
1
+
1
a
2
y
2
)f r
|
y
| ≥
a
E
y
(
y
,
)
Z
⊥
(
y
,
)
=−
=
(7
.
3)
2
1
+
1
2
⎩
H
x
Z
N
(
)
for
|
y
| ≤
a
,
where
H
x
E
y
/
H
x
is the normal magnetic field on the Earth's surface and
Z
N
=−
is the normal impedance.
Thus,
⎧
⎨
1
2
+
1
−
1
1
+
1
a
2
y
2
Z
⊥
(
y
)
=
N
(
N
(
)
)f r
|
y
| ≥
a
2
,
⊥
(
y
,
)
=
=
2
2
o
⎩
1
1
+
=
N
(
)
N
(
)
for
|
y
| ≤
a
,
.
(7
4)
2
,
where
= |
Z
N
|
/
0
is the normal apparent resistivity and
are the real
N
frequency-independent distortion factors
⎧
⎨
⎫
⎬
⎧
⎨
⎫
⎬
1
2
1
−
1
a
2
y
2
2
=
,
=
1
+
.
(7
.
5)
1
1
⎩
⎭
⎩
⎭
1
+
1
+
1
1
