A pragmatic decomposition of a vector wavefield into P-and S-waves is based on the Helmholtz theory and the Christoffel equation. It is applicable to VTI media when the plane-wave polarization is continuous in the vicinity of a given wavenumber and is uniquely defined by that wavenumber, except for the kiss singularities on the VTI symmetry axis. Unlike divergence and curl, which separate the wavefield into a scalar and a vector field, the decomposed P-and S-wavefields are both vector fields, with correct amplitude, phase, and physical units. If the vector components of decomposed wavefields are added, they reconstruct those of the original input wavefield. Wavefield propagation in any portions of a VTI medium that have the same polarization distribution ͑i.e., the same eigenvector͒ in the wavenumber domain have the same decomposition operators and can be reconstructed with a single 3D Fourier transform for each operator ͑e.g., one for P-waves and one for S-waves͒.This applies to isotropic wavefields and to VTI anisotropic wavefields, if the polarization distribution is constant, regardless of changes in the velocity. Because the anisotropic phase polarization is local, not global, the wavefield decomposition for inhomogeneous anisotropic media needs to be done separately for each region that has a different polarization distribution. The complete decomposed vector wavefield is constructed by combining the P-, SV-, and SH-wavefields in each region into the corresponding composite P-, SV-, and SH-wavefields that span the model. Potential practical applications include extraction of separate images for different wave types in prestack reverse time migration, inversion, or migration velocity analysis, and calculation of wave-propagation directions for common-angle gathers.

We have developed an alternative (new) method to produce common-image gathers in the incident-angle domain by calculating wavenumbers directly from the P-wave polarization rather than using the dominant wavenumber as the normal to the source wavefront. In isotropic acoustic media, the wave propagation direction can be directly calculated as the spatial gradient direction of the acoustic wavefield, which is parallel to the wavenumber direction (the normal to the wavefront). Instantaneous wavenumber, obtained via a novel Hilbert transform approach, is used to calculate the local normal to the reflectors in the migrated image. The local incident angle is produced as the difference between the propagation direction and the normal to the reflector. By reordering the migrated images (over all common-source gathers) with incident angle, common-image gathers are produced in the incident-angle domain. Instantaneous wavenumber takes the place of the normal to the reflector in the migrated image. P- and S-wave separations allow both PP and PS common-image gathers to be calculated in the angle domain. Unlike the space-shift image condition for calculating the common-image gather in angle domain, we use the crosscorrelation image condition, which is substantially more efficient. This is a direct method, and is less dependent on the data quality than the space-shift method. The concepts were successfully implemented and tested with 2D synthetic acoustic and elastic examples, including a complicated (Marmousi2) model that illustrates effects of multipathing in angle-domain common-image gathers.

P- and S-wavefield separation is necessary to extract PP and PS images from prestack elastic reverse time migrations. Unlike traditional separation methods that use curl and divergence operators, which do not preserve the wavefield vector component information, we did P and S vector decomposition, which preserves the same vector components that exist in the input elastic wavefield. The amplitude and phase information was automatically preserved, so no amplitude or phase corrections were required. We considered two methods to realize P and S vector decomposition: selective attenuation and decoupled propagation. Selective attenuation uses viscoelastic extrapolation, in which the Q-values are used as processing parameters, to remove either the P-waves or the S-waves. Decoupled propagation rewrites the stress and particle velocity formulation of the elastic equations into separate P- and S-wave components. In both methods, the decomposition is realized during the extrapolation of an elastic wavefield. These algorithms could also perform P and S decomposition in [Formula: see text] gather data by extrapolating the data downward from the receivers, during which the decomposition is performed, and then back upward to record the decomposed P- and S-waves at the receivers. Comparisons of the two methods in terms of efficiency, accuracy, and memory showed that both could separate P- and S-waves in the vector domain. The decoupled propagation is preferable in terms of speed and memory cost, but was applicable only to elastic propagation.

Reverse time migration (RTM) was implemented with a modified crosscorrelation imaging condition for data from 2D elastic vertically transversely isotropy (VTI) media. The computation cost was reduced because scalar qP- and qS-wavefield separations are performed in VTI media, for the source and receiver wavefields only at the RTM imaging time, to calculate the migrated qP and qS images. Angle-domain common-image gathers (CIGs) were extracted from qPqP and qPqS common-source RTM images. The local incident angle was produced as the difference between the qP-wave phase angle, obtained directly from the source wavefield polarization, and the normal to the reflector, calculated as the instantaneous wavenumber direction via a directional Hilbert transform of the stacked image. Angle-domain CIGs were extracted by reordering the prestack-migrated images by local incident phase angle, source by source. Vector decomposition of the source qP-wavefield was required to calculate the qP-wave phase polarization direction for each image point at its imaging time. RTM and angle-domain CIG extraction were successfully implemented and illustrated with a synthetic 2D elastic VTI example.

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