2D and 3D elastic wavefield vector decomposition in the wavenumber domain for VTI media

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 eigenvector) 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-waves). 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.