Elastic Wave-equation Depth Migration of Seismic Data for Isotropic and Azimuthally Anisotropic Media
Richard Andrew Bale
Two separate seismic processing steps for multicomponent data, shear-wave splitting correction and migration, are brought together in this thesis. The framework for doing this is the theory of elastic anisotropic propagator matrices, combined with generalized phase-shift plus interpolation (PSPI). The resulting extrapolation can be written as matrix pseudodifferential operators acting on vector wavefields. Each extrapolation step includes a decomposition based on eigenvectors of the Kelvin-Christoffel equation, a phase-shift step based on eigenvalues of the Kelvin-Christoffel equation and a recomposition step. It is the decomposition and recomposition operations which enable the shear-wave splitting correction within migration.
Practical implementation of this approach is achieved by a new method of adaptive spatial windowing, designed to minimize the number of Fourier transforms which are required to represent the pseudodifferential operator. The windows are made up of elementary Gaussian functions, and are selected based on minimizing a phase error criterion over a range of phase angles. adaptive windowing (PSPAW).
Another algorithm is derived for the special case of isotropic elastic media. This algorithm is closer in spirit to the original scalar PSPI algorithm, and relies upon approximate separation of the dependence on P-wave and S-wave velocities. Both PSPAW and PSPI algorithms are tested on examples and compared with a reference, which is computed using the full pseudodifferential operator.
For both methods, a prestack depth migration scheme is constructed by the addition of an appropriate imaging condition. The PSPAW algorithm is used to apply prestack depth migration on a synthetic modelled dataset with a faulted HTI (transversely isotropic with horizontal symmetry axis) layer, and compared with isotropic migration on the same data.
Finally, the PSPI algorithm is tested on a new elastic version of the Marmousi dataset, known as Marmousi-2. The results reveal both the potential benefits of the method and areas where challenges remain.