Deriving a Fast P/S Decoupled Operator From 3-D Anisotropic Christoffel Equation With Its Application in Elastic Reverse Time Migration and Angle-Domain Common-Image Gathers

A large-scale computation usually hinders the production of the existing 3D anisotropic P/S wave-mode decomposition methods, such as low-rank approximation, LU factorization or local Fourier transformation. To tackle this problem, we develop a fast decoupled operator in 3D vertical transverse isotropic (VTI) media for P/S wave-mode decomposition and apply this operator to elastic reverse time migration (ERTM) and angle-domain common-image gathers (ADCIGs). To start with, we construct the 3D VTI Christoffel equation in the wavenumber-domain and obtain three elliptical solutions that represent the P-, SV- and SH-polarization direction. We project the original elastic wavefields to these polarization vectors and derive the decoupled formulations of P-, SV-, and SH-waves. To improve the efficiency, these decoupled formulations are then converted to the space domain and depend on the model parameters and phase angles. Further removing the phase angle term from such space-domain decoupled formulations, the approximate P-, SV- and SH-wavefield components are finally obtained. These approximate wave components are proved to compute an accurate phase angle as that of exact P-, SV-, and SH-waves and can be used for ERTM imaging and ADCIGs extraction. A fast decoupled operator is thusly developed and only requires gradient operations and once Fast Fourier transformation (for SV-waves). Both simple and complex synthetic examples demonstrate the effectiveness and feasibility of our approaches.