# From Multidirectional-Vector-Based Seismic Reverse Time Migration and Angle-Domain Common-Image Gathers to Full Waveform Inversion Using Phase-Modified and Deconvolved Images in Acoustic and Elastic Media

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## Abstract

Angle-domain common-image gathers (ADCIGs) are an important product from reverse time migration (RTM). Using the Poynting vector (PV) to calculate propagation angles is efficient but suffers from instability problems. First, the PV can give only a single direction per grid point per time step, and thus it fails to give the multiple directions at wavefield overlaps. Second, the current PV formula is only kinematically correct, which leads to an undefined propagation angle at the magnitude peak of the wavefield. Third, the receiver wavefield reconstructed from the observed data is often not as stable as the source wavefield simulated from the synthetic source. We address the first two issues by proposing a dynamically-correct multidirectional PV (MPV) that decomposes the wavefield into several vector bins in the frequency-wavenumber (ω-k) domain and then uses PV to calculate the propagation directions of each decomposed wavefield in the time-space (t-x) domain. We also provide an improved flow to calculate the ADCIGs by using the source wavefield propagation direction and the reflector normal in the k domain.

We propose an improved system for the elastic RTM, which involves three parts. For the P/S wave mode separation, we put forward a scheme to relax the assumption of the (locally) constant shear modulus caused by the Helmholtz theorem. We also give the elastic imaging conditions based on multidirectional vectors, which can give the correct polarities for PP, PS, SP, and SS images without using the reflector normal. For the ADCIG calculation, we give two methods to calculate the multiple propagation directions.

For full waveform inversion (FWI), we propose a new scheme that provides a self-contained and physically-intuitive derivation which establishes a natural connection between the amplitude-preserved RTM, the Zoeppritz equations (the amplitude versus [reflection] angle [AVA] inversion) and the reflectivity-to-impedance inversion and combines them into a single framework to produce a preconditioned inversion formula. The formula also works for inverting only the velocity. For impedance inversion, we propose using rock-physics information to separate the impedance into velocity and density for wavefield extrapolation. Because of the complexity of the rock-physics relationship in the real world, we also suggest combining Machine Learning with this scheme for future development.