Geology Reference
In-Depth Information
Frequency domain
Time domain
(a)
(b)
Sinc
function
f
c
f
t
(c)
(d)
Filter
operator
Fig. 2.16
Design of a digital low-pass
filter.
f
c
f
t
(BR) in terms of their frequency response. Frequency
filters are employed when the signal and noise compo-
nents of a waveform have different frequency character-
istics and can therefore be separated on this basis.
Analogue frequency filtering is still in widespread use
and analogue antialias (LP) filters are an essential compo-
nent of analogue-to-digital conversion systems (see Sec-
tion 2.2). Nevertheless, digital frequency filtering by
computer offers much greater flexibility of filter design
and facilitates filtering of much higher performance than
can be obtained with analogue filters. To illustrate the
design of a digital frequency filter, consider the case of a
LP filter whose cut-off frequency is
f
c
. The desired out-
put characteristics of the ideal LP filter are represented by
the amplitude spectrum shown in Fig. 2.16(a).The spec-
trum has a constant unit amplitude between 0 and
f
c
and
zero amplitude outside this range: the filter would there-
fore pass all frequencies between 0 and
f
c
without atten-
uation and would totally suppress frequencies above
f
c
.
This amplitude spectrum represents the transfer func-
tion of the ideal LP filter.
Inverse Fourier transformation of the transfer func-
tion into the time domain yields the impulse response of
the ideal LP filter (see Fig. 2.16(b)). However, this im-
pulse response (a sinc function) is infinitely long and
must therefore be truncated for practical use as a convo-
lution operator in a digital filter. Figure 2.16(c) repre-
sents the frequency response of a practically realizable LP
filter operator of finite length (Fig. 2.16(d)). Convolu-
tion of the input waveform with the latter will result in
LP filtering with a ramped cut-off (Fig. 2.16(c)) rather
than the instantaneous cut-off of the ideal LP filter.
HP, BP and BR time-domain filters can be designed
in a similar way by specifying a particular transfer func-
tion in the frequency domain and using this to design a
finite-length impulse response function in the time do-
main.As with analogue filtering, digital frequency filter-
ing generally alters the phase spectrum of the waveform
and this effect may be undesirable. However,
zero phase
filters
can be designed that facilitate digital filtering with-
out altering the phase spectrum of the filtered signal.
2.5.2 Inverse (deconvolution) filters
The main applications of inverse filtering to remove the
adverse effects of a previous filtering operation lie in the
field of seismic data processing. A discussion of inverse
filtering in the context of deconvolving seismic records
is given in Chapter 4.
2.6 Imaging and modelling
Once the geophysical waveforms have been processed
to maximize the signal content, that content must be
extracted for geological interpretation. Imaging and
modelling are two different strategies for this work.
As the name implies, in imaging the measured wave-
