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![]() | Interpolation of near offsets using multiples and prediction-error filters | ![]() |
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Autocorrelation can be used to extract synthetic active source data
from incoming waves that reflect at the free surface and return within
the recording array
(Schuster, 2001; Claerbout, 1968; Cole, 1995; , ,).
This has traditionally been thought of in a passive context, where
random noise is assumed to be arriving from all locations. The
reflecting waves from active-source experiments can be treated in the
same manner, where the primary reflections correlate with the
free-surface multiples (Shan, 2003; Berkhout and Verschuur, 2003; Reiter et al., 1991; Berkhout and Verschuur, 1994). Autocorrelating data, ,
(implemented as multiplication of complex-conjugates in the
domain) for two receiver points,
and
, both for a single
shot,
, gives the pseudo-primaries,
,
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sigsbee1shot
Figure 2. Crosscorrelation of a single shot. (a): Original fully-sampled split-spread shot; (b): The same shot recreated from crosscorrelating the traces in (a) using equation 1. (c): correlations are made for ![]() ![]() ![]() |
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Figure 2a shows an example of a fully-sampled
split-spread shot from the Sigsbee2B data set, while Figure
2b shows a slice of the pseudo-primary output of
autocorrelating the zero-offset trace with all of the other traces
within that single shot. Clearly, the original shot and the
pseudo-primaries generated from the autocorrelation are different.
The water-bottom reflection at zero offset is present as are the first
group of diffractors, but they are not present at the more distant
offsets. The many other correlations between events produce undesired
noise in the output. Primaries can correlate with other primaries,
multiples with multiples, and noise with other noise or signal. Much
of these undesired correlations, however, vary as a function of source
position, , so when the correlations for the same receiver pair
and
are performed for many different source locations,
, and are then summed, the unwanted events destructively interfere,
while the correct pseudo-primaries constructively interfere.
As shown in Figure 2c, where sources were
used, this summing over multiple source positions thus can greatly
improve the pseudo-primary signal. The range of usable offsets is
greatly improved, as is the signal-to-noise ratio. The
pseudo-primaries now look more similar in character to the original
data.
Pseudo-primaries generated from recorded multiples are interesting in
part because they have different illumination than do the recorded
primaries. Figure 3a illustrates a desired
near-offset primary ray-path that is not recorded because of the gap
between the source and the nearest receiver. In Figure
3b, the source is positioned such that the
raypath first travels from the source to the water-bottom and back up
to the water surface within the recording array. This ray then
reflects back into the subsurface and eventually returns into the
recording array at another receiver. This four-segment raypath is a
free-surface multiple, which can be reconsidered as two distinct
events: the first, a recorded primary event from the source to the
first receiver
and the second, another primary from the virtual
source
to another receiver
. Crosscorrelating the multiple
recorded at
with the primary recorded at
produces a
pseudo-primary trace with a virtual source at
and a receiver at
. Comparing Figures 3a and
3b, we see that we can transform multiples
that we record into pseudo-primaries with virtual source locations
where we did not originally record data. This is most useful at
unrecorded near offsets.
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nearoffsetcartoon
Figure 3. Raypaths of a primary and multiple reflection. (a): an unrecorded primary reflection that hits the surface at near offset. (b): a recorded multiple that first reflects at the surface within the recording array, then returns within the recording array. ![]() |
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The generated pseudo-primaries differ from the recorded data in several ways. First, the crosscorrelation of the data squares the amplitude spectrum of the pseudo-primaries relative to the original data, meaning that the wavelet of the pseudo-primaries will become zero-phase, different than that of the input data. Second, the amplitudes of the pseudo-primaries will differ from the amplitudes of the actual primaries, as the pseudo-primaries are a correlation and summation of different events. This amplitude difference will be both on a global scale difference between the pseudo-primaries and the primaries, as well as different relative amplitudes within each data set. Third, spurious events from other correlations may still exist in the data as the number of sources is limited.
Because of these differences between the pseudo-primaries and primaries, a direct substitution of the created near-offset pseudo-primaries is not adequate. The pseudo-primaries, however, can be used as training data for a nonstationary prediction-error filter, as the PEF is insensitive to the phase of the training data. This PEF is then used to interpolate the missing near offsets, so that the negative aspects of the pseudoprimaries, such as the incorrect wavelet and extra noise, do not affect the PEF, while the positive aspects of the pseudo-primaries, contained in the autocorrelation of these data, are used.
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![]() | Interpolation of near offsets using multiples and prediction-error filters | ![]() |
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