Geology Reference
In-Depth Information
2.
that the composite waveform
w
k
for an impulsive
source is minimum delay (i.e. that its contained energy
is concentrated at the front end of the pulse; see
Chapter 2).
From assumption (1) it follows that the autocorrela-
tion function of the seismic trace represents the autocor-
relation function of the composite waveform
w
k
.From
assumption (2) it follows that the autocorrelation func-
tion can be used to define the shape of the waveform, the
necessary phase information coming from the
minimum-delay assumption.
Such an approach allows prediction of the shape of the
composite waveform for use in Wiener filtering. A par-
ticular case of Wiener filtering in seismic deconvolution
is that for which the desired output is a spike function.
This is the basis of
spiking deconvolution
, also known as
whitening deconvolution
because a spike has the amplitude
spectrum of
white noise
(i.e. all frequency components
have the same amplitude).
A wide variety of deconvolution operators can be de-
signed for inverse filtering of real seismic data, facilitat-
ing the suppression of multiples (dereverberation and
deghosting) and the compression of reflected pulses.The
presence of short-period reverberation in a seismogram
is revealed by an autocorrelation function with a series of
decaying waveforms (Fig. 4.21(a)). Long-period rever-
berations appear in the autocorrelation function as a
series of separate side lobes (Fig. 4.21(b)), the lobes
occurring at lag values for which the primary reflection
aligns with a multiple reflection.Thus the spacing of the
side lobes represents the periodicity of the reverberation
pattern.The first multiple is phase-reversed with respect
to the primary reflection, due to reflection at the ground
surface or the base of the weathered layer. Thus the
first side lobe has a negative peak resulting from cross-
correlation of the out-of-phase signals.The second mul-
tiple undergoes a further phase reversal so that it is in
phase with the primary reflection and therefore gives rise
to a second side lobe with a positive peak (see Fig.
4.21(b)). Autocorrelation functions such as those shown
in Fig. 4.21 form the basis of predictive deconvolution
operators for removing reverberation events from
seismograms.
*
Input waveform
Filter operator
=
Filtered output
Desired output
Fig. 4.20
The principle of Wiener filtering.
(a)
φ
xx
(
)
τ
α
(b)
φ
xx
(
τ
)
Fig. 4.21
Autocorrelation functions of
seismic traces containing reverberations.
(a) A gradually decaying function
indicative of short-period reverberation.
(b) A function with separate side lobes
indicative of long-period reverberation.
α
