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In-Depth Information
impedances are defined from the magnetic field
H
y
measured at a base station and
the filtered electric fields
E
fl
x
measured on the profile.
The low-frequency spatial filtering is performed by means of the Hanning trans-
formation (Bernard and Rader, 1969):
x
0
+
W
/
2
1
2
∞
E
flt
x
e
x
(
k
x
)
h
(
k
x
)
e
−
ik
x
x
0
dk
x
(
x
0
)
=
E
x
(
x
)
H
(
x
−
x
0
)
dx
=
π
x
0
−
W
/
−∞
2
(11
.
10)
∞
1
2
e
fl
x
(
k
x
)
e
−
ik
x
x
0
dk
x
,
=
π
−∞
where
H
(
x
) is the Hanning window of width
W
:
1
⎧
⎨
⎩
1
W
cos
2
π
x
W
2
+
|
x
| ≤
W
H
(
x
)
=
W
2
,
0
|
x
|
>
h
(
k
x
) is its frequency response as a function of the spatial frequency
k
x
:
1
e
ik
x
x
dx
W
/
2
∞
1
W
cos
2
π
x
H
(
x
)
e
ik
x
x
dx
h
(
k
x
)
=
=
+
W
−∞
−
W
/
2
⎛
⎝
⎞
⎠
=
2
4
π
sin
Wk
x
2
Wk
x
2
1
2
1
2
sin
Wk
x
2
Wk
x
2
W
2
k
x
=
1
−
−
1
,
Wk
x
Wk
x
2
2
2
W
2
k
x
−
4
π
1
+
1
−
e
x
(
k
x
) is the spatial spectrum of the observed electric field
E
(
x
):
∞
E
x
(
x
)
e
ik
x
x
dx
e
x
(
k
x
)
=
,
−∞
and
e
fl
x
(
k
x
) is the spatial spectrum of the filtered electric field
E
fl
x
(
x
):
e
flt
x
(
k
x
)
=
h
(
k
x
)e
x
(
k
x
)
.
Figure 11.19 presents the filter using the Hanning windows. Its 3 dB cutoff point
with
h
≈
0
.
7 is located at
Wk
x
≈
4
.
52. With increasing
Wk
x
the amplitude falls as
(
Wk
x
)
3
.At
Wk
x
≈
1
/
10
.
36 we have 20 dB attenuation with
h
=
0
.
1.
The critical question is how to choose the optimum window width
W
. Bostick
(1984) believes that
W
should be proportional to the effective penetration depth
h
eff
:
