A wave equation method tracing wavefield propagation path is proposed with ray theory. A windowed Fourier frame expansion technique is employed for angle decomposition of wavefields of source-point and target-point. Directional illumination is calculated. An angle coherence technique is applied to directional illumination to acquire propagation path between source-point and target-point. The method takes into account focusing and defocusing effects of the wavefield propagation caused by inhomogeneous media.

In the wave equation depth migration of common shot gathers,the windowed Fourier frame expansion of seismic wavefield is used to obtain the receiver and source wavefields in the local angle domain.We propose a wave equation migration method in angle domain for common shot gathers seismic data.With migration in angle domain,the image matrix in local angle domain,the migration sections with different common-dip angles and reflection angles at imaging points are obtained.It provides fundamental data for processing approaches based on migration,such as analysis of amplitude versus angle,extraction of common image gather in angle domain and velocity analysis in angle domain. Numerical examples in a steep dip model and an international standard Marmousi model show that the wave equation migration method in angle domain is correct and effective.

A frame expansion method of operator is used to the phase shift operator of split-step Fourier propagator.A local split-step Fourier propagator for wavefield propagation is presented and used in a wave equation prestack depth migration.The local split-step Fourier propagator consists of a phase shift operator in phase-space-frequency domain and a windowed time shift operator in space-frequency domain.Different from the global phase shift operator in wavenumber-frequency domain,the phase shift operator in phase-space-frequency domain shows good space localization.A numerical test of wave equation prestack depth migration on the international standard SEG-EAGE 2D salt dome model shows that the local split-step Fourier propagator is stable and is applicable to the media with velocity strong lateral variation.