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Multiple reflections have long been treated as noise in the seismic imaging process.
In contrast to many other types of ``noise'', like surface waves, multiply reflected
body waves may still penetrate deep into the earth, and thus have a potential to
aid in imaging the prospect zone. I refer generically to joint imaging with multiples
as any process which creates a ``pseudo-primary'' image from multiples by removing the
propogation effects of body waves through arbitrary multiple layer (generator + free
surface), and which then seeks to integrate the information provided by the primary
and pseudo-primary images.
Reiter et al. (1991) present an early example of imaging multiples directly
using a prestack Kirchhoff scheme. Yu and Schuster (2001) describe a
cross-correlation method for imaging multiples. Berkhout and Verschuur (1994) and
Guitton (2002) apply shot-profile migration for multiples.
The aforementioned approaches produce separate-but-complementary pseudo-primary and primary
images, yet they either do not attempt to, or employ simplistic methods to integrate the information
contained in the two images; either add Reiter et al. (1991) or multiply Yu and Schuster (2001)
them together.
In this paper, I introduce a new methodology for jointly imaging primaries and multiples.
In addition to a desire to correctly image the multiples, my approach is driven by three
primary motivations:
- 1.
- Data Consistency - The primary and pseudo-primary images both should be
maximally consistent with the input data.
- 2.
- Self-consistency - The primary and pseudo-primary images should be consistent
with one another, both kinematically and in terms of amplitudes.
- 3.
- Noise Suppression - In the primary image, all orders of multiples should be
suppressed. In the pseudo-primary image created from, say first-order water-bottom
multiples, contributions from primaries and secord-order or greater multiples should be
suppressed.
Least squares optimization provides an excellent, and perhaps the only viable approach to
address all three requirements.
I adopt an approach similar to Nemeth et al. (1999), which used a least-squares scheme
to jointly image compressional and surface waves, for improved wavefield separation.
Data consistency is effected by minimization of a data residual; self-consistency and noise
suppression through the use of regularization terms which penalize 1) differences between
primary and pseudo-primary images, and 2) attributes which are not characteristic to
true primaries or pseudo-primaries.
In my approach, I use the simplest possible imaging operation, Normal Moveout (NMO). I
derive an NMO equation for water-bottom multiple reflections, which maps these multiples
to the same zero-offset traveltime as their associated primaries, creating a ``pseudo-primary''
section. To account for the amplitude differences between the primary and pseudo-primary
sections, I assume constant seafloor AVO behavior and estimate a single water-bottom
reflection coefficient from the data. To address the AVO differences between primary
and pseudo-primary, I derive an expression - valid only for constant velocity - for the
AVO of the pseudo-primary as a function of the AVO of the primary, and then enforce
this constraint in the inversion via an offset- and time-dependent regularization term.
Next: methodology
Up: Brown : Imaging with
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Stanford Exploration Project
6/10/2002