A fast anisotropic decoupled operator of elastic wave propagation in the space domain for P-/S-wave-mode decomposition and angle calculation, with its application to elastic reverse time migration in a TI medium

Abstract

The conventional anisotropic P-/S-wave-mode decomposition and angle calculation usually are performed in the wavenumber domain and require multiple Fourier transforms and rely on strong model assumptions. Therefore, it is difficult to use in production due to the prohibitively high computational cost. To tackle this problem, we develop a fast anisotropic P-/S-wave-mode decomposition approach in the space domain and further apply this decomposition to the vertical transversely isotropic (VTI)/tilted transversely isotropic (TTI) elastic reverse time migration (ERTM) and the associated angle-domain common-image gathers (ADCIGs). To start with, we solve the Christoffel equation of the VTI media and obtain an anisotropic-Helmholtz (AH) operator that represents the P- and S-wave polarizations in the wavenumber domain. The decoupled operators for the P-/S-wave decomposition are then derived in terms of the AH operator. To improve the efficiency and model adaptability, we convert the decoupled operators from the wavenumber domain to the space domain as a function of the model parameters and phase angle. By further eliminating the phase-angle term, such space-domain decoupled operators lead to the approximate P- and S-wave components which are proven to achieve the same effect as the accurate P and S wavefields applied in the angle calculation and ERTM images. A fast space-domain AH decomposition approach is thus obtained and can be used for P-/S-wave-mode decomposition, ADCIGs, and ERTM. In addition, we extend our approach to the TTI media by using the coordinate transform. Three examples are used to demonstrate the effectiveness and feasibility of our method.