Correct analysis of amplitude information requires prestack imaging to determine the location and extent of reflectivity anomalies. Kirchhoff techniques have been widely used for imaging 3D data because of their I/O flexibility and computational efficiency. They evade the question of total cost by being target oriented. The mathematical derivations of integral operators assume continuous wavefields. In generating the seismic model, each trace is modeled as a weighted sum of image sources on a reflector. Therefore, Kirchhoff imaging of seismic data involves a weighted sum of filtered surface-recorded traces where the weights are deduced from the theory of asymptotic inversion. In practice, the resulting imaging algorithms are applied to discretely sampled seismic data and their implementation reduces to a matrix-vector multiplication. Due to irregular coverage, the matrix is often ill-conditioned, and the linear system that need to be solved is badly scaled.
In Chapter 4, I present a new processing sequence for irregularly sampled 3D prestack data. The method focuses on both algorithmic accuracy and proper handling of irregular geometry. Therefore it allows for reliable AVO analysis on migrated data. It employs the AMO operator to organize the data into regularly gridded common-azimuth (CA) and common-offset (CO) subsets. The regular CA/CO cubes are then ready for imaging by efficient migration algorithms.
To correct for the irregular surface coverage and the varying subsurface illumination, I propose two new developments for Kirchhoff operators: a data-space formulation based on row scaling of pull (sum) operators, and a model-space normalization based on column scaling of push (spray) operators. In both approaches, the final image is normalized by a reference model that is the operator's response to an input vector with all components equal to one. A more general approach is to properly scale both row and columns of the operator. As will be shown in Chapter 5, this weighting is suitable for the preconditioning of Kirchhoff matrices for iterative linear solvers.
Results from applying the new processing flow to a 3D land survey showed that using AMO to regularize the coverage of the data by partial stacking improves the quality of the final image. The normalization technique has proved to equalize the AMO operator for the effects of the varying illumination. The advantage of applying the AMO transformation before migration is threefold: (1) reduction of the size of prestack data subsets (2) interpolation to a regular grid before imaging (3) common-azimuth common-offset processing of 3D surveys.