LSJIMP: Choice of Imaging Operator  

The exact form of the modeling operators, $\bold L_{i,k,m}$, shown in equation () has not yet been discussed. Any candidate prestack imaging operator for multiples must accomplish two tasks: focus the multiples in time/depth and offset/angle at the position of the primary and correct their amplitude to make the multiples directly comparable to the corresponding primary.

The literature contains many multiple modeling techniques which use wavefield extrapolation to ``add a multiple bounce'' to recorded data, and thus transform primaries into a model of the multiples, which is then generally adaptively subtracted from the data. These modeling techniques can roughly be divided into earth-model-based Berryhill and Kim (1986); Lu et al. (1999); Morley (1982); Wiggins (1988) and autoconvolutional Riley and Claerbout (1976); Tsai (1985); Verschuur et al. (1992) approaches. It is possible to reverse the multiple modeling process-in other words, to ``remove a multiple bounce'' from the data and transform multiples into pseudo-primary events, which can then be imaged as primaries Berkhout and Verschuur (2003); Shan (2003).

Existing migration techniques for multiples perform the reverse modeling process either explicitly (i.e., using an earth model) or implicitly. Reiter et al. (1991) imaged pegleg multiples with Kirchhoff prestack depth migration. He and Schuster (2003) present a least-squares joint imaging scheme for multiples that uses poststack Kirchhoff depth migration. Yu and Schuster (2001) and Guitton (2002) migrate peglegs with shot-profile depth migration, while Berkhout and Verschuur (1994) used a similar crosscorrelation technique. Shan (2003) uses source-geophone migration after crosscorrelation at the surface. None of these techniques explicitly addresses the issue of amplitudes, beyond a polarity flip.

In this thesis, I use an earth-model-based multiple modeling strategy to simulate the kinematics and angle-dependent amplitude behavior of pegleg multiples. In section I derive an extension to the normal moveout (NMO) equation for pegleg multiples. In section I introduce HEMNO (Heterogeneous Earth Multiple NMO Operator), an extension of the NMO equation for multiples, which independently images split peglegs in a moderately heterogeneous earth. In Sections - I derive a series of amplitude correction operators to normalize the angle-dependent reflectivity of imaged multiples to be commensurable with their imaged primaries.

I postpone giving the full motivation for my particular choice of multiple imaging operator, as well as many implementation details until section . However, we can at this early stage state some important facts which have bearing on how the problem is regularized and how it is implemented computationally. The prestack multiple image, $\bold m_{i,k,m}$, shown in Figure , is parameterized by zero-offset traveltime, $\tau$, offset, x, and midpoint, y. However, one important feature of the my combined imaging operator is the fact that it operates on a CMP-by-CMP basis. This underscores the fact that HEMNO applies a vertical stretch, and does not move information across midpoint. The regularization schemes I present also do not operate across midpoint. This implementation allows a coarse-grained parallelization scheme, enabling straightforward parallel computation on a Linux cluster.