Solving the 3D Acoustic Wave-equation on Generalized Structured Meshes: A FDTD Approach

Abstract:

The key computational kernel of most advanced 3D seismic imaging and inversion algorithms involves calculating solutions of the 3D acoustic wave equation, most commonly with a finite-difference time-domain (FDTD) methodology. While well suited for regularly sampled rectilinear computational domains, FDTD methods seemingly have limited applicability in scenarios involving irregular 3D domain boundary surfaces and mesh interiors that are best described by non-Cartesian geometry (e.g., surface topography). Using coordinate mapping relationships and differential geometry, a FDTD approach can be developed for generating solutions to the 3D acoustic wave equation that is applicable to generalized 3D coordinate systems and (quadrilateral-faced hexahedral) structured meshes. The developed numerical implementation is similar to established Cartesian approaches, save for a necessary introduction of weighted first- and mixed second-order partial-derivative operators that account for spatially varying geometry. The approach is validated on three different types of computational meshes: (1) an ``internal boundary'' mesh conforming to a dipping water bottom; (2) analytical ``semi-orthogonal cylindrical" coordinates; and (3) analytic semi-orthogonal and numerically specified ``topographic" coordinate meshes. Impulse response tests and numerical analysis demonstrate the viability of the approach for kernel computations for 3D seismic imaging and inversion experiments for non-Cartesian geometry scenarios.