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In-Depth Information
1
1
f
=
f
=
1
1
o
S
1
h
2
,
o
S
1
h
2
,
−
i
−
i
and
Z
N
Z
N
are normal impedances of side and central segments defined in the thin-
sheet approximation:
,
i
o
h
i
o
h
Z
N
Z
N
=−
o
S
1
h
2
,
=−
h
2
.
o
S
1
1
−
i
1
−
i
Solutions of these equations are
⎧
⎨
Z
N
Ae
−
g
1
y
/
f
+
≥
v
+
y
q
Z
N
+
Be
−
g
2
y
/
f
Ce
g
2
y
/
f
+
v
+
q
≥
y
≥
v
Z
⊥
(
y
)
=
(9
.
4)
⎩
D
cos h
g
f
Z
N
+
≤
≤
v.
y
0
y
S
1
(
y
)
Z
⊥
(
y
)
Constants
A
,
B
,
C
,
D
are
found
from
conditions
that
and
2
(
y
)
dS
1
(
y
)
Z
⊥
(
y
)
dy
=
ν
=
ν
+
q
.
The first condition ensures the horizontal component of the current density
j
y
to
be continuous at the Earth's surface:
are continuous at
y
and
y
h
1
1
(
y
)
E
y
(
y
,
0)
h
1
H
x
S
1
(
y
)
Z
⊥
(
y
)
=−
=−
j
y
(
y
,
0)
.
H
x
The second condition ensures the vertical component of the electric field
E
z
to
be continuous at the floor of sediments:
2
(
y
)
dS
1
(
y
)
Z
⊥
(
y
)
dy
d
[
H
x
=−
2
(
y
)
H
x
dS
1
(
y
)
E
y
(
y
,
0)
=
2
(
y
)
H
x
−
H
x
(
y
,
h
1
)]
dy
dy
=−
2
(
y
)
H
x
dH
x
(
y
,
h
1
)
E
z
(
y
,
h
1
)
=
.
dy
H
x
On cumbersome mathematics we get
2
e
[
g
1
(
w
+
q
)
−
g
2
q
]
f
−
1
2
1
ke
−
2
g
2
q
f
1
ke
−
2
g
2
q
f
coth
k
)
η
g
g
2
g
f
v
Z
N
(1
A
=
+
2
−
+
+
η
−
2
e
g
2
w
f
−
1
2
1
+
ke
−
2
g
2
q
f
1
−
ke
−
2
g
2
q
f
coth
Z
N
η
g
g
2
g
f
v
B
=
2
−
+
η
e
−
g
2
(
w
+
2
q
)
f
−
1
2
1
+
ke
−
2
g
2
q
f
1
−
ke
−
2
g
2
q
f
coth
Z
N
η
g
g
2
g
f
v
2
2
C
=
−
+
η
g
η
−
1
−
1
ke
−
2
g
2
q
f
2
1
+
ke
−
2
g
2
q
f
1
−
ke
−
2
g
2
q
f
coth
2
2
−
g
g
2
g
f
v
Z
N
D
=
+
η
,
g
f
v
η
g
2
sinh
(9
.
5)
where
