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

摘要

A pragmatic decomposition of a vector wavefield into P- andS-waves is based on the Helmholtz theory and the Christoffelequation. It is applicable to VTI media when the plane-wave po-larization is continuous in the vicinity of a given wavenumberand is uniquely defined by that wavenumber, except for the kisssingularities on the VTI symmetry axis. Unlike divergence andcurl, which separate the wavefield into a scalar and a vector field,the decomposed P- and S-wavefields are both vector fields, withcorrect amplitude, phase, and physical units. If the vector compo-nents of decomposed wavefields are added, they reconstructthose of the original input wavefield. Wavefield propagation inany portions of a VTI medium that have the same polarizationdistribution (i.e., the same eigenvector) in the wavenumber do-main have the same decomposition operators and can be recon-structed with a single 3D Fourier transform for each operator(e.g., one for P-waves and one for S-waves).This applies to iso-tropic wavefields and to VTI anisotropic wavefields, if the polar-ization distribution is constant, regardless of changes in the ve-locity. Because the anisotropic phase polarization is local, notglobal, the wavefield decomposition for inhomogeneous aniso-tropic media needs to be done separately for each region that hasa different polarization distribution. The complete decomposedvector wavefield is constructed by combining the P-, SV-, andSH-wavefields in each region into the corresponding compositeP-, SV-, and SH-wavefields that span the model. Potential practi-cal applications include extraction of separate images for differ-ent wave types in prestack reverse time migration, inversion, ormigration velocity analysis, and calculation of wave-propaga-tion directions for common-angle gathers.