Geology Reference
In-Depth Information
input spikes and scaled according to the individual spike
amplitudes. Since any transient wave can be represented
as a series of spike functions (Fig. 2.11(e)), the general
form of a filtered output (Fig. 2.11(f )) can be regarded as
the summation of a set of impulse responses related to a
succession of spikes simulating the overall shape of the
input wave.
The mathematical implementation of convolution
involves time inversion (or folding) of one of the func-
tions and its progressive sliding past the other function,
the individual terms in the convolved output being de-
rived by summation of the cross-multiplication products
over the overlapping parts of the two functions. In gen-
eral, if g i ( i = 1,2,..., m ) is an input function and f j ( j =
1, 2, . . . , n ) is a convolution operator, then the convolu-
tion output function y k is given by
Amplitude
Input
Input
displacement
Output
Output
displacement
W
Time
Fig. 2.9 The principle of filtering illustrated by the perturbation
of a suspended weight system.
m
 1
y
=
g f
k
=
12
,,...,
m
+
n
-
1
(2.6)
(
)
k
i
ki
-
response which is defined as the output of the filter when
the input is a spike function (Fig. 2.10).The impulse re-
sponse is a waveform in the time domain, but may be
transformed into the frequency domain as for any other
waveform. The Fourier transform of the impulse re-
sponse is known as the transfer function and this specifies
the amplitude and phase response of the filter, thus
defining its operation completely.The effect of a filter is
described mathematically by a convolution operation such
that, if the input signal g ( t ) to the filter is convolved with
the impulse response f ( t ) of the filter, known as the con-
volution operator, the filtered output y ( t ) is obtained:
i
=
In Fig. 2.12 the individual steps in the convolution
process are shown for two digital functions, a double
spike function given by g i = g 1 , g 2 , g 3 = 2, 0, 1 and an im-
pulse response function given by f i = f 1 , f 2 , f 3 , f 4 = 4, 3, 2,
1, where the numbers refer to discrete amplitude values
at the sampling points of the two functions. From Fig.
2.11 it can be seen that the convolved output y i = y 1 , y 2 ,
y 3 , y 4 , y 5 , y 6 = 8, 6, 8, 5, 2, 1. Note that the convolved
output is longer than the input waveforms; if the func-
tions to be convolved have lengths of m and n , the con-
volved output has a length of ( m + n - 1).
The convolution of two functions in the time domain
becomes increasingly laborious as the functions become
longer. Typical geophysical applications may have func-
tions which are each from 250 to a few thousand samples
long.The same mathematical result may be obtained by
transforming the functions to the frequency domain,
then multiplying together equivalent frequency terms of
their amplitude spectra and adding terms of their phase
spectra.The resulting output amplitude and phase spec-
tra can then be transformed back to the time domain.
Thus, digital filtering can be enacted in either the time
yt
() =
gt
()
f t
()
(2.5)
*
where the asterisk denotes the convolution operation.
Figure 2.11(a) shows a spike function input to a filter
whose impulse response is given in Fig. 2.11(b). Clearly
the latter is also the filtered output since, by definition,
the impulse response represents the output for a spike
input. Figure 2.11(c) shows an input comprising two
separate spike functions and the filtered output (Fig.
2.11(d)) is now the superposition of the two impulse re-
sponse functions offset in time by the separation of the
Spike input
Output = Impulse response
Filter
Fig. 2.10 The impulse response of a filter.
 
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