Theoretical Framework for Phase-space Local-Angle-Domain Waveform Inversion in Anisotropic Elastic Media

Abstract

We present a phase-space formulation of elastic full-waveform inversion (FWI) based on local-angle-domain (LAD) representations of seismic scattering. Instead of minimizing acquisition-domain waveform misfits, the method minimizes image-domain inconsistencies in a phase-space domain parameterized by position, propagation direction, scattering angle, azimuth, and time. The formulation is motivated by the observation that velocity and impedance perturbations occupy largely distinct regions of phase space. For anisotropic elastic media, we derive a generalized objective function together with the corresponding adjoint-state gradient and Hessian operators. Representing LAD images through directional and opening wavenumbers provides a unified description of propagation and scattering effects. The approach is independent of the forward-modeling engine and can be implemented with finite-difference, finite-element, spectral-element, ray-based, or beam-based methods. Under Born linearization, ray/beam formulations allow explicit Hessian computation. Owing to the near-orthogonality of directional and opening-angle sensitivities, the Hessian is more diagonally dominant than in conventional FWI, improving parameter separation, reducing velocity-reflectivity crosstalk, and enhancing inversion stability. By organizing seismic information in a physically meaningful phase-space representation, LAD-FWI supports adaptive, resolution-driven inversion and may reduce cycle skipping through attributes that vary more smoothly than waveform residuals. The framework naturally integrates directional focusing, reflector-consistency analysis, anisotropic parameter estimation, converted-wave imaging, and diffraction-based characterization, providing a unified workflow for elastic FWI, migration-velocity analysis, impedance inversion, and high-resolution subsurface imaging.