Chapters 4 and 5 present the 3-D contribution of this thesis. Chapter 4 describes the converted-wave azimuth moveout operator, whereas Chapter 5 discusses the converted-wave common-azimuth migration operator. The 3-D OBS dataset from the Alba oil field illustrates the application of these two operators.

The PS-AMO operator transforms an input data trace
with an arbitrary offset and azimuth position
into an equivalent trace with a new offset and
azimuth position.
The PS-AMO operator has two main characteristics:
1) it preserves the resolution of the dipping events, and
2) it internally performs the CMP to CRP correction.
Therefore, *a priori* CRP binning is not necessary before
applying azimuth moveout to converted-wave data.

To illustrate the applications of the PS-AMO operator I combined the geometry regularization with the data-dimensionality reduction problems as an inverse problem that uses the cascade operation of the linear interpolation operator and the PS azimuth moveout operator. The model space for this inverse problem consists of an uniformly sampled common offset-azimuth cube. I presented two solutions for this inverse problem, the adjoint solution and the weighted adjoint soluiton. These two solutions were compared with the result for the conventional method, using normal moveout plus stacking.

These three solutions were analyzed using a subset for the real 3-D OBS seismic dataset from the Alba oil field in the North Sea. The results showed that both the adjoint solution and the weighted adjoint solution produced a regularized common-azimuth cube of PS data that honored the input data and the characteristics of converted-wave data. These attributes were not observed in the conventional method result. The weighted adjoint solution produced a final model with an even distribution of the energy along the model cube. This is result of the approximation of the Hessian using a diagonal operator.

12/14/2006