Geoscience Reference
In-Depth Information
The spatial spectrum of
b
(
i
y
(
x
)is
b
(
i
A
(
k
)=
A
0
exp(
−
kδ
i
)+
···
if
k>
0
,
.
(8.16)
0
if
k<
0
.
i
√
2
πb
(
i
0
δ
i
. The Fourier transform of the logarithmic and sub-
sequent terms in series (8.15) is expressed by generalized functions vanishing
at
k<
0. We do not present here these spectrums, restricting ourselves just
to the final results in the coordinate representation for the logarithmic term.
Since the function
S
(
k
)=0at
k<
0
,
then the coordinate dependencies of the
field components is found using the inverse Fourier transformations along the
positive semi-axis of wavenumbers
k
.
where
A
0
=
−
High Conductive Ground
k
k
Ratio
ζ
4
/ζ
3
(see (7.113), (7.114)) reduces to
ζ
4
/ζ
3
≈−
i
|
|
coth
|
|
for the high
conductive ground. For the wavenumber
k
k
A
and
k
τ
K
=
k
0
hX
K
from (7.128), we have
k
sinh
exp
i
k
∆
(0)
SK
(
k
)
k
≈−
.
(8.17)
By neglecting
Z
(
m
)
in (8.14) for the high conductivity (
σ
g
→∞
) , we obtain
g
+
∞
b
(
i
A
k
exp
k
(
ix
1)
dk.
Φ
3
A
=
−
−
(8.18)
−∞
Finally, substituting (8.16) into (8.18), we have
δ
i
x
+
i
(
h
+
δ
i
)
.
(2
π
)
1
/
2
b
(
i
)
0
Φ
3
A
=
−
(8.19)
Let us consider the situation of the high enough conductivity of the near-
surface ground layer in which the wavelength
λ
is much more than the skin
depth
k
d
g
−
1
=
h/d
g
. Here
d
g
is the skin depth in the ground. Substi-
tuting the expressions (8.19) for
Φ
3
A
into (8.12), (8.13), we find
Σ
P
b
(
i
0
sin
I
,
2
Σ
H
δ
i
b
(
g
)
x
x
+
i
(
δ
i
+
h
)
+
Ck
y
ln [
k
y
x
+
ik
y
(
δ
i
+
h
)] +
≈−
···
(8.20)
2
ik
y
Σ
H
Σ
P
b
(
i
0
δ
i
sin
I
ln [
k
y
x
+
ik
y
(
δ
i
+
h
)] +
b
(
g
)
y
≈
···
,
(8.21)
sin
I
δ
i
,
Ck
y
x
+
i
(
δ
i
+
h
)
2
iZ
h
k
0
Σ
H
Σ
P
b
(
i
)
b
(
g
)
z
≈−
[
x
+
i
(
δ
i
+
h
)]
2
−
(8.22)
0
E
(
g
)
x
Z
h
b
(
g
y
,
(
g
)
y
Z
h
b
(
g
x
,
≈−
≈
(8.23)
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