Geology Reference
In-Depth Information
Waveform 1
Waveform 2
lag
Cross-correlation function
- ve lag
+ ve lag
Fig. 2.13
Cross-correlation of two
identical waveforms.
Zero
lag
centred on the time value at which the signal function
and its concealed equivalent in the waveform are in
phase (Fig. 2.14).
A special case of correlation is that in which a wave-
form is cross-correlated with itself, to give the
autocorre-
lation
function
f
xx
(
t
)
.
This function is symmetrical about
a zero lag position, so that
The autocorrelation function contains all the am-
plitude information of the original waveform but none
of the phase information, the original phase relation-
ships being replaced by a zero phase spectrum. In fact,
the autocorrelation function and the square of the am-
plitude spectrum
A
(
f
) can be shown to form a Fourier
pair
(
)
=-
ft f
(
t
)
(2.12)
)
2
xx
xx
(
)
ยด
Af
ft
xx
(
(2.13)
The autocorrelation function of a periodic waveform is
also periodic, with a frequency equal to the repetition
frequency of the waveform.Thus, for example, the auto-
correlation function of a cosine wave is also a cosine
wave. For a transient waveform, the autocorrelation
function decays to small values at large values of lag.
These differing properties of the autocorrelation func-
tion of periodic and transient waveforms determine one
of its main uses in geophysical data processing, namely,
the detection of hidden periodicities in any given wave-
form. Side lobes in the autocorrelation function (Fig.
2.15) are an indication of the existence of periodicities in
the original waveform, and the spacing of the side lobes
defines the repetition period.This property is particular-
ly useful in the detection and suppression of multiple
reflections in seismic records (see Chapter 4).
Since the square of the amplitude represents the power
term (energy contained in the frequency component)
the autocorrelation function can be used to compute the
power spectrum
of a waveform.
2.5 Digital filtering
In waveforms of geophysical interest, it is standard prac-
tice to consider the waveform as a combination of
signal
and
noise
.The signal is that part of the waveform that re-
lates to the geological structures under investigation.The
noise is all other components of the waveform.The noise
can be further subdivided into two components,
random
and
coherent
noise. Random noise is just that, statistically
