Tensorial Elastodynamics for Anisotropic Media

Elastic wavefield solutions computed by finite-difference (FD) methods in complex anisotropic media are essential elements of elastic reverse-time migration and full waveform inversion analyses. Cartesian formulations of such solution methods, though, face practical challenges when aiming to represent curved interfaces (including free-surface topography) with rectilinear elements. To forestall such issues, we propose a general strategy for generating solutions of tensorial elastodynamics for anisotropic media (i.e., tilted transversely isotropic (TTI) or even lower symmetry) in non-Cartesian computational domains. For the specific problem of handling free-surface topography, we define an unstretched coordinate mapping that transforms an irregular physical domain to a regular computational grid on which FD solutions of the modified equations of elastodynamics are straightforward to calculate. Our fully staggered grid with a mimetic finite-difference (FSG+MFD) approach solves the velocity-stress formulation of the tensorial elastic wave equation where we compute the stress-strain constitutive relationship in Cartesian coordinates and then transform the resulting stress tensor to generalized coordinates to solve the equations of motion. The resulting FSG+MFD numerical method has a computational complexity comparable to Cartesian scenarios using a similar FSG+MFD numerical approach. Numerical examples demonstrate that the proposed solution can simulate anisotropic elastodynamic field solutions on irregular geometries and is thus a reliable tool for anisotropic elastic modeling, imaging and inversion applications in generalized computational domains including handling free-surface topography.