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UBC Theses and Dissertations

A generalized Kirchhoff-WKBJ depth migration theory for multi-offset seismic reflection data : reflectivity model construction by wavefield imaging and amplitude estimation Lumley, David Edward


This thesis embodies a mathematical, physical, and quantitative investigation into the imaging and amplitude estimation of subsurface earth reflectivity structure within the framework of pre stack wave-equation depth migration of multi-offset seismic reflection data. Analysis is performed on five prestack depth migration reflectivity "imaging conditions" with respect to image quality and quantitative accuracy of recovered reflectivity amplitudes. A new computationally efficient and stable prestack depth migration imaging method is proposed which is based upon a geometric approximation to the theoretically correct, but unstable, "dynamic" imaging condition. The "geometric" imaging condition has the desirable property of true-amplitude reflectivity recovery in regions of both 1-D and 2-D velocity variation, while fully retaining and optimizing the favorable imaging characteristics of current, non-true amplitude formulations. The currently predominant "crosscorrelation" and "excitation-time" migration imaging methods are shown to possess significantly less accurate imaging and amplitude-recovery characteristics relative to the proposed geometric migration. The respective signal-to-noise recovery of their imaged amplitudes deteriorates approximately linearly (excitation-time) and quadratically (crosscorrelation) with depth. As a necessary prerequisite to the imaging analysis, a true-amplitude prestack depth migration equation is derived which appears to be new to the literature. This result is obtained in the form of a 2.5-D farfield Kirchhoff integral solution to the acoustic wave equation, after the application of a dynamic imaging condition to the reconstructed upgoing and downgoing wavefields. This solution is in harmony with zeroth order asymptotic ray theory (ART) assumptions, and depends upon WKBJ Green's functions which can be numerically evaluated for arbitrary migration models by raytracing methods. A new and "generalized" Kirchhoff prestack depth migration equation is subsequently obtained by the introduction of a weighting function into the true-amplitude migration integral. The weight is a function of both the reconstructed upgoing and downgoing wavefields, and is determined analytically by a mathematical application of each specific reflectivity imaging condition. This generalized equation is significant in that it provides a common mathematical, physical and computational basis for the comparative analytical and quantitative analysis of reflectivity image quality and amplitude recovery among current prestack migration philosophies and variants of those migration themes. In addition, three ancillary research objectives are achieved. The first achievement is the development of a Kirchhoff prestack depth migration computer algorithm to implement the generalized imaging of surface-recorded seismic reflection data. This algorithm can be readily modified to perform seismic wavefield imaging for other recording geometries such as cross-borehole or vertical seismic profiling, and may be suitable to non-seismic applications such as the imaging of electromagnetic wavefields and satellite-acquired synthetic aperture radar data. The second result is the development of a fast two-point raytracing computer algorithm which provides accurate computation of a subsurface grid of traveltimes and 1.5-D zeroth order ART amplitudes in a 1-D acoustic medium. This algorithm is useful for subsurface wavefield reconstruction and imaging, and for inversion applications such as geotomography. The third objective is the detailed quantitative examination of migration imaging quality and true relative-amplitude normal-incidence reflectivity recovery from numerically migrated depth images. This is achieved successfully in an extensive 2.5-D synthetic data analysis, using a challenging 2-D structural model, synthetic multifold reflection seismogram shot gathers, and the numerical imaging and modelling algorithms developed as part of this thesis research.

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