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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:

Data Consistency - The primary and pseudo-primary images both should be maximally consistent with the input data.
Self-consistency - The primary and pseudo-primary images should be consistent with one another, both kinematically and in terms of amplitudes.
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.

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