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Conclusions
Seismic noise attenuation is an important part of the processing
workflow before a migrated image can be obtained
and interpreted. This thesis tackles the noise problem with inversion
to preserve any signal present in the data. In particular,
spiky and coherent noise are attenuated by looking at the statistics
(i.e., PDF and pattern) of the undesirable events.
For spiky events, Chapter
introduces
the Huber norm. The Huber norm behaves
like the
norm for large residuals
and like the
norm for small residuals.
Because it is continuous and differentiable everywhere, the
Huber norm is minimized with a quasi-Newton method
called L-BFGS. This technique maintains most
of the convergence properties of the standard BFGS method for
non-linear problems while keeping the memory requirements very low.
This last point is particularly important for seismic processing where
the volume of data can be quite large.
In this thesis, the combination Huber norm/L-BFGS solver proves being very versatile.
For instance, in Chapter
, velocity scans are
derived from data with bad traces and noise bursts. In Chapter
,
small geological features at the bottom of the Sea of Galilee are unraveled from extremely noisy
data. In Chapter
, multiples and primaries are
better separated by estimating matching filters with the Huber norm.
This list of applications is not exhaustive.
It is my belief that many more geophysical problems
could benefit from using the Huber norm with the L-BFGS solver.
For coherent noise attenuation, I introduce in Chapter
two strategies based on
the observation that coherent noise generally results from an
approximate modeling of the seismic data.
These techniques achieve an important goal of
least-squares inversion: to have IID residuals.
One strategy approximates the inverse data covariance
operator with multidimensional prediction-error filters (PEFs) by weighting
the data residual. The other strategy incorporates a coherent noise modeling term
inside the inversion. Compared to the weighting approach, the modeling technique
yields the smallest residual with the best convergence properties.
One advantage of the weighting approach with PEFs, however, is its
ability to separate non-stationary noise and signal according to their
pattern.
This pattern-based approach is very effective at separating primaries and
multiples as illustrated in Chapters
and
. In particular, I demonstrate that the
pattern-based approach with 3-D PEFs is more robust to modeling
uncertainty than adaptive subtraction, an industry standard for this
task.
Therefore, the main contributions of this thesis are four fold:
- 1.
- It presents a particular implementation of a robust solver for
the inversion of data contaminated with spiky noise. This
implementation is based on the Huber norm and the L-BFGS method
(Chapters
and
).
- 2.
- It develops processing techniques for coherent noise
attenuation based on the need to have IID residuals
(Chapters
and
).
- 3.
- It establishes multidimensional prediction-error filters with
helical boundary conditions as an extremely flexible tool for noise
filtering and/or modeling (Chapters
,
, and
).
- 4.
- It highlights the limitations of the standard adaptive
subtraction technique for the multiple attenuation problem
by introducing two viable alternatives:
(1) matching filters estimation with the Huber norm (Chapter
), and (2)
multiple subtraction with a pattern-based approach (Chapters
and
).
One important problem omitted in this thesis that directly links to my
work, however, is how to build stable inverse non-stationary PEFs. In practice,
patching solves it but in a very crude manner. More elegant solutions
to that problem should be developed. Valcarce et al. (2000)
give conditions for building stable inverse filters, but
their approach seems to be difficult to apply in
practice. Beylkin (1995) proposes transforming infinite
impulse response filters (IIR) into finite response filters (FIR).
This approach could impact our way of doing non-stationary deconvolution.
If stable non-stationary PEFs can be computed, then the modeling approach,
which has very good signal/noise separation properties, could be
also used for non-stationary coherent noise removal.
There, inverse non-stationary PEFs would play the role of modeling
operators, as illustrated in Chapter
for the stationary case.
Next: Algorithm for minimizing the
Up: Multidimensional seismic noise attenuation
Previous: Acknowledgments
Stanford Exploration Project
5/5/2005