SEISMIC MIGRATION/INVERSION FOR TRANSVERSELY ISOTROPIC ELASTIC MEDIA

David W.S. Eaton

ABSTRACT

Classical seismic imaging techniques assume an acoustic isotropic medium. In this thesis, methods are developed for isotropic and/or transversely isotropic (TI) elastic media. These new assumptions facilitate the analysis of both mode-converted waves and anisotropic parameters within the earth.

Processing mode-converted seismic data requires special binning techniques, because the lateral position of the conversion point varies with depth. Previously published algorithms for approximate common-conversion-point (CCP) trace sorting are unsuitable for imaging multiple depth zones, and are sometimes prone to periodic binning artifacts. In this thesis, a depth-variant CCP mapping technique is used to overcome these difficulties. The mapping algorithm produces unmigrated P-SV stacked images that are directly comparable to common-midpoint (CMP) stacked sections obtained from conventional ( P-P ) seismic data. An example of single-depth trace sorting for a strongly anisotropic material with a vertical infinite-fold symmetry axis illustrates that, unlike isotropic media, conversion-point shift toward the source is possible for qP-qSV arrivals.

The latter half of this thesis deals with migration and inversion of seismic data based on a least-squares ray-Born formalism. The following assumptions are employed to simplify the problem: the orientation of the infinite-fold anisotropic symmetry axis is known, and coplanar with the sources and receivers; the medium and acquistion geometry are two-dimensional; based on prior information, an accurate and smooth (ray-valid) reference model can be defined; coherent noise has been removed from the data. At least six parameters are required to characterize an elastic medium with TI symmetry. Here, for convenience, the model-parameter set is chosen to be qP and qS velocities, density, and the three Thomsen anisotropy parameters.

In order to implement the migration/inversion strategy, robust and efficient methods for computing high-frequency background Green's functions are required. For this purpose, an existing methodology for finite-difference traveltime and amplitude computation for isotropic media is adapted for use with TI media. The traveltime technique tracks seismic wavefronts by solving the sixth-order anisotropic eikonal equatin on a hexagonal mesh. Differentiation of the computed traveltime field yields estimates of the slowness and polarization vectors. The initial ray parameters are determined by perturbing the source location on the grid, and are then used estimate the 2.5-dimensional geometrical-spreading function. In numerical tests using a Sun Sparcstation 2, accurate traveltimes were computed at a rate of about 50 per CPU second. However, the computed amplitudes contain small oscillatory artifacts.

The migration/inversion problem is posed as a least-squares optimization, which is solved by an iterative algorithm consisting of three steps: filtered backprojection (migration) of the current data residual, application of an approximate inverse-Hessian matrix to yield parameter perturbation estimates, and re-scattering from the new model to update the data-residual vector. Application of this procedure to synthetic crosswell data demonstrates that Thomson's ε parameter can be resolved as well as, or better than, any of the isotropic parameters. Migration/inversion of a ray-traced dataset produced erroneous estimates of density and anisotropic parameters, but illustrates that superior imaging of slope discontinuities in the subsurface is possible when all elastic wave types (ie., qP-qP, qP-qSV, qSV-qSV, etc.) are used.

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