Geoscience Reference
In-Depth Information
Figure 6.2 The filter function,
Eq. (6.12)
, and transfer function,
Eq. (6.11)
,ofa
one-dimensional, low-pass, wave-cutoff filter.
T
and
G
for a one-dimensional, low-pass wave-cutoff filter are plotted in
From
Eq. (6.6)
low-pass wave-cutoff filtering gives in physical space
∞
∞
x
)
sin
[
κ
c
(x
−
]
f
r
(x)
x
)f (x
)dx
=
f(x
)dx
.
=
G(x
−
(6.13)
x
)
π(x
−
−∞
−∞
Figure 6.2
shows that this filter function
G
is negative in some regions. This
is required in order that
the filter have a sharp cutoff in the wavenumber
domain.
The
high-pass
wave-cutoff filter rejects Fourier components of smaller wave-
number and passes those of larger wavenumbers:
T(κ)
=
0
,
κ
≤
κ
c
;
T(κ)
=
1
,
κ>
c
.
(6.14)
6.2.4 The Gaussian filter
Both the running-mean filter,
Figure 6.1
,
and the wave-cutoff filter,
Figure 6.2
,
have
negative values in physical or wavenumber space. One that does not is the Gaussian
filter,
1
x
2
2
σ
2
.
√
2
πσ
e
−
G(x)
=
(6.15)
Its transfer function is also Gaussian
(Problem 6.25)
.
