## Relative-Amplitude-Preserving Prestack Time Migration by the Equivalent Offset Method

### Hugh D. Geiger

The kinematics of prestack time migration by the equivalent offset method (EOM) are well established as a simple reformulation of the double-square-root equation of seismic imaging. EOM is implemented as a nonrecursive diffraction stack, where samples in the data space are weighted, filtered, and summed to produce samples in the image space. In this dissertation, I determine the exact optimum weighting function that produces an image as a stack of angle-dependent reflectivities, and suggest practical alternatives that are appropriate for imaging using prestack time migrations.

The imaging problem is treated as an inverse problem consisting of an estimation problem and an appraisal problem. As is typical in geophysical inverse problems, a quantitative solution is provided for the estimation problem, and the appraisal problem is replaced by a validation process. A framework for qualitative validation of prestack time migration is described in terms of accuracy of focusing, accuracy of relative positioning, and accuracy of absolute positioning. Quantitative validation is achieved by testing the weighting functions using synthetic seismic data.

The theoretical basis for acoustic wavefield extrapolation is developed from first principles. The Kirchhoff-Helmholtz integral representation, the fundamental equation of wavefield extrapolation and imaging, provides a mathematical description of Huygens' principle, yields simplified formulae for forward and inverse extrapolation from planar and non-planar interfaces, and gives reciprocity relations for Green's functions and acoustic pressure.

Two methods of depth imaging are developed, Kirchhoff-approximate migration and Kirchhoff-approximate migration/inversion. Both rely on the Kirchhoff approximation at the reflecting surface. The second method, determined from Born-approximate inversion, provides exact expressions for constant-wavespeed common-offset migration/inversion required for relative amplitude preserving prestack time migration. The common-shot and common-receiver migration/inversion formulae are shown to produce biased estimates for asymmetric acquisition configurations. Simplifications to the depth imaging formulae are proposed that greatly increase the efficiency of implementation without any significant loss of accuracy.

Relative-amplitude-preserving EOM prestack time migration is tested against conventional processing over a portion of LITHOPROBE SNORCLE line 1. EOM prestack time migration can provide a better image for interpretation because it enhances imaging of steeper dips, and improves relative positioning of reflectors with conflicting dips.