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Azimuth moveout is a partial-prestack migration operator that can be efficiently applied to 3-D prestack data to transform their effective offset and azimuth Biondi et al. (1998); Chemingui (1999). The cascade operation of any imaging operator with its inverse produces a partial-prestack imaging operator. To obtain the converted-wave azimuth moveout operator (PS-AMO) I use the converted-wave dip moveout operator, presented in Chapter 2, as the initial imaging operator.

PS Azimuth Moveout (PS-AMO) shares the same main property as the conventional AMO operator, that is to transform the offset and azimuth of an arbitrary trace in a prestack data cube into equivalent data with a new offset and azimuth. For converted-wave data, the trace is transformed from the CMP domain to the CRP domain, where the new offset and azimuth are computed, and then back from the CRP domain to the CMP domain.

PS-AMO has several potential applications for 3-D multicomponent processing. One is geometry regularization, through which PS-AMO helps to fill in the acquisition gaps using the information of surrounding traces. Another is data-reduction through partial stacking, which combines PS-AMO and partial stacking to reduce the computational cost of 3-D prestack depth imaging. A third application is the interpolation of unevenly sampled traces, which differs from the first application in the sense that PS-AMO is the main interpolation operator. For this thesis, I combine the geometry regularization application with the data-reduction application to illustrate the converted-wave azimuth moveout operator.

To solve the problem of reorganizing irregular geometries, there are two distinct approaches: 1) data regularization before migration Duijndam et al. (2000), 2) irregular-geometry correction during migration Albertin et al. (1999); Audebert (2000); Bloor et al. (1999); Duquet et al. (1998); Nemeth et al. (1999); Prucha and Biondi (2002). Biondi and Vlad 2002 combine the advantages of the previous two approaches. Their methodology regularizes the data geometry before migration, filling in the acquisition gaps with an AMO operator that preserves the amplitudes in the frequency-wavenumber log-stretch domain. Clapp (2005) extends the work of Biondi and Vlad 2002 by formulating an inverse problem using the AMO operator to create a full regularized 5-D cube with dimensions ($t, {\rm CMP}_x, {\rm CMP}_y,h_x,h_y$).

Clapp (2006) uses the AMO operator to reduce the dimensions of the full regularized 5-D cube. The AMO operator maps the data that are along the crossline offset direction ($h_y \neq 0$) into zero crossline offset (hy = 0). The final result is a common-azimuth cube with dimensions ($t, {\rm CMP}_x, {\rm CMP}_y,h_x$) to be used with common-azimuth migration.

To illustrate the applications of the PS-AMO operator I extend the work of Clapp (2006); Clapp (2005); Clapp (2005) to handle converted-wave data. I combine the geometry regularization with the data reduction applications of the PS-AMO operator in a single least-squares inverse problem; the data space for this inverse problem consists of an irregular 5-D cube with dimensions ($t, {\rm CMP}_x, {\rm CMP}_y,h_x,h_y$); the model space consists of a regular 4-D cube with dimensions ($t, {\rm CMP}_x, {\rm CMP}_y,h_x$). I present two approximations for the solution of this inverse problem, first the adjoint solution; second, the weighted adjoint solution, where I approximate the Hessian of this inverse problem with a diagonal matrix computed using a reference model Rickett (2001).

The PS component of the 3-D OBS dataset acquired above the Alba reservoir in the North Sea illustrates the applications for the PS-AMO operator. I compare the two results for this inverse problem with the conventional method result, that is using simple normal moveout plus stacking along the crossline direction. The final solutions for this chapter become the input data for the converted-wave common-azimuth migration operator in Chapter 5.

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