Anelastic and anisotropic direct and adjoint wave propagation with force memory variables and variable topography or bathymetry: Applications to the finite-difference SBP-SAT scheme in the curvilinear coordinate system

High-end seismic imaging, formulated as an inverse problem, requires accounting for anelastic effects and computing the gradient of a data misfit function. When using second-order displacement wave equations with a time-domain finite-difference scheme, memory requirements can be reduced by employing three memory variables, either displacement or force-based, per relaxation mechanism. We present a formulation using force memory variables, which correspond to the divergence of stress memory variables. This approach minimizes storage demands during gradient computation and is therefore advantageous for seismic imaging. However, total stresses and tractions cannot be directly recovered from force memory variables, complicating the implementation of traction-dependent boundary conditions. We then derive a set of boundary conditions that separately incorporate elastic and anelastic (viscous) stresses, based on a comparison of the weak forms of the force and displacement memory variable formulations. We propose a summation-by-parts with simultaneous approximation terms (SBP-SAT) finite-difference scheme to model variable topography and bathymetry, incorporating an acoustic/elastic interface, in a curvilinear coordinate system. Stability is ensured by adding SAT to the time-evolution equations of both the displacement fields and the memory variables. This results in a dual-consistent and stable numerical scheme. We validate the numerical implementation by comparing it with a spectral element method applied to a simple elastic model. We then present a 3D simulation using an anelastic and anisotropic model representing the Mont Blanc region.