Geology Reference
In-Depth Information
designed using statistical methods.This special approach
to the deconvolution of seismic records, known as pre-
dictive deconvolution, is discussed further in Chapter 4.
2.4.2 Deconvolution
Deconvolution
or
inverse filtering
(Kanasewich 1981) is a
process that counteracts a previous convolution (or
filtering) action. Consider the convolution operation
given in equation (2.5)
2.4.3 Correlation
Cross-correlation
of two digital waveforms involves cross-
multiplication of the individual waveform elements and
summation of the cross-multiplication products over
the common time interval of the waveforms.The cross-
correlation function involves progressively sliding one
waveform past the other and, for each time shift, or
lag
,
summing the cross-multiplication products to derive the
cross-correlation as a function of lag value. The cross-
correlation operation is similar to convolution but does
not involve folding of one of the waveforms. Given two
digital waveforms of finite length,
x
i
and
y
i
(
i
= 1, 2, . . . ,
n
), the cross-correlation function is given by
yt
()
=
gt
f t
()
()
*
y
(
t
) is the filtered output derived by passing the input
waveform
g
(
t
) through a filter of impulse response
f
(
t
)
.
Knowing
y
(
t
) and
f
(
t
), the recovery of
g
(
t
) represents a de-
convolution operation. Suppose that
f
¢(
t
) is the function
that must be convolved with
y
(
t
) to recover
g
(
t
)
gt
( )
=
yt
( )
f t
¢
( )
(2.7)
*
Substituting for
y
(
t
) as given by equation (2.5)
n
-
Â
1
t
gt
(
)
=
gt
f t
f t
¢
(
)
(
)
(
)
(2.8)
**
(
)
=
xy m
-
<
< +
m
ft
(
t
)
(2.11)
xy
i
+
i
t
i
=
Now recall also that
where
t
is the lag and
m
is known as the maximum
lag value of the function. It can be shown that cross-
correlation in the time domain is mathematically
equivalent to multiplication of amplitude spectra and
subtraction of phase spectra in the frequency domain.
Clearly, if two identical non-periodic waveforms are
cross-correlated (Fig. 2.13) all the cross-multiplication
products will sum at zero lag to give a maximum positive
value.When the waveforms are displaced in time, how-
ever, the cross-multiplication products will tend to
cancel out to give small values. The cross-correlation
function therefore peaks at zero lag and reduces to small
values at large time shifts.Two closely similar waveforms
will likewise produce a cross-correlation function that is
strongly peaked at zero lag. On the other hand, if two
dissimilar waveforms are cross-correlated the sum of
cross-multiplication products will always be near to zero
due to the tendency for positive and negative products to
cancel out at all values of lag. In fact, for two waveforms
containing only random noise the cross-correlation
function
f
xy
(
t
) is zero for all non-zero values of
t
.
Thus, the cross-correlation function measures the
degree of similarity of waveforms.
An important application of cross-correlation is in the
detection of weak signals embedded in noise. If a wave-
form contains a known signal concealed in noise at un-
known time, cross-correlation of the waveform with the
signal function will produce a cross-correlation function
gt
(
)
=
gt
t
(2.9)
(
)
*
d
(
)
where
d
(
t
) is a spike function (a unit amplitude spike at
zero time); that is, a time function
g
(
t
) convolved with
a spike function produces an unchanged convolution
output function
g
(
t
). From equations (2.8) and (2.9) it
follows that
ft
f t
¢
(
)
=
t
(2.10)
(
)
d
(
)
*
Thus, provided the impulse response
f
(
t
) is known,
f
¢(
t
)
can be derived for application in equation (2.7) to re-
cover the input signal
g
(
t
). The function
f
¢(
t
) represents
the deconvolution operator.
Deconvolution is an essential aspect of seismic data
processing, being used to improve seismic records by re-
moving the adverse filtering effects encountered by seis-
mic waves during their passage through the ground. In
the seismic case, referring to equation (2.5),
y
(
t
) is the
seismic record resulting from the passage of a seismic
wave
g
(
t
) through a portion of the Earth, which acts as a
filter with an impulse response
f
(
t
)
.
The particular prob-
lem with deconvolving a seismic record is that the input
waveform
g
(
t
) and the impulse response
f
(
t
) of the Earth
filter are in general unknown. Thus the 'deterministic'
approach to deconvolution outlined above cannot be
employed and the deconvolution operator has to be
