Extracting angle-domain common-image gathers in VTI media using ansiotropic-Helmholtz P/S wave-mode decomposition

Common-image gathers are extensively used in amplitude versus angle (AVA) and migration velocity analysis (MVA). The current state of methods for anisotropic angle gathers extraction use slant-stack, local Fourier transform or low-rank approximation, which requires much computation. Based on an anisotropic-Helmholtz P/S wave-mode decomposition method, we propose a novel and efficient approach to produce angle-domain common-image gathers (ADCIGs) in the elastic reverse time migration (ERTM) of VTI media. To start with, we derive an anisotropic-Helmholtz decomposition operator from the Christoffel equation in VTI media, and use this operator to derive the decomposed formulations for anisotropic P/S waves. Second, we employ the first-order Taylor expansion to calculate the normalized term of decomposed formulations and obtain the anisotropic-Helmholtz decomposition method, which generates the separated P/S wavefields with correct amplitudes and phases. Third, we develop a novel way that uses the anisotropic-Helmholtz decomposition operator to define the polarization angles for anisotropic P/S waves and substitute these angles to decomposing formulations. The polarization angles are then calculated directly from the separated vector P- and S-wavefields and converted to the phase angles. The ADCIGs are thusly produced by applying the phase angles to VTI ERTM. In addition, we develop a concise approximate expression of residual moveout (RMO) for PP-reflections of flat reflectors in VTI media, which avoids the complex transformations between the group angles and the phase angles. The approximate RMO curves show a good agreement with the exact solution and can be used as a tool to assess the migration velocity errors. As demonstrated by two selected examples, our ADCIGs not only produce the correct kinematic responses with regards to different velocity pertubatation, but also generate the reliable amplitude responses versus different angle. The final stacking images of ADCIGs data exhibit the identical imaging effect as that of VTI ERTM.