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As a final example, consider the problem of separating spatiallyaliased groundroll
from body waves in land data. This a more challenging application of
the algorithm because the body waves have curvature that changes
rapidly with both offset and time so to match it I need small filters in
relatively small
patches. The groundroll, on the other hand, has little global curvature
(although it may have strong local curvature due to aliasing) and matching it
is more successful with large filters in large patches. A refinement to the
method could use different regularization operators or at least different
regularization parameters on the nonstationary filters for the
primaries and the multiples. For the sake of simplicity, I used the same
regularization for both.
Figure 15 shows the original shot as well as the
initial estimates of the body waves and the groundroll.
The groundroll estimate was computed simply by
highcut filtering the data to 24 Hz using a Butterworth filter with six poles.
I allowed significant energy from the body
waves to leak into the estimate of the groundroll to illustrate the problem
described in the previous paragraph. Similarly, the estimate of the body waves
was computed by lowcut filtering the data to 18 Hz also with a Butterworth
filter with 6 poles. Since I don't want to reduce the
low frequency components of the signal too much, I allowed strong groundroll
to leak into the estimate of the body waves. The purpose is to eliminate this
groundroll without hurting the signal and ideally, mapping back some of the
bodywaves from the estimate of the groundroll.


shot1estimates1
Figure 15. Land shot gather with strong groundroll
(a), initial estimate of groundroll (b), and body waves (c).






shot1matchedbw
Figure 16. Estimate of body waves after one
outer iteration (a), after 5 outer iterations (b) and after 10
outer iterations (c). Notice how after the fifth iteration the
groundroll is essentially gone.






shot1matchedgr
Figure 17. Estimate of groundroll after one
outer iteration (a), after 5 outer iterations (b) and after 10
outer iterations (c). Some of the body waves have been removed in
panel (c) but much still remains.




Figure 16 shows the estimate of the bodywaves
after one, five and 10 outer iterations of the proposed algorithm. Even after
just the first iteration, most of the groundroll has been eliminated and
after five iterations it is almost gone. For this example I used just two
patches in time and one in offset. Figure 17 shows
similar results for
the groundroll. Since the patches were so large, the energy of the leaked
bodywaves were only slightly attenuated (see the reflector at about 1.7 secs).
This energy was mapped back to the estimate of the bodywaves.
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Up: Examples with real data
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