Solving the tensorial 3D acoustic wave equation: A mimetic finite-difference time-domain approach

Generating accurate numerical solutions of the acoustic wave equation (AWE) is a key computational kernel for many seismic imaging and inversion problems. Although finite-difference time-domain (FDTD) approaches for generating full-wavefield solutions are well-developed for Cartesian computational domains, several challenges remain when applying FDTD approaches to scenarios arguably best described by more generalized geometry. In particular, how best to generate accurate and stable FDTD solutions for scenarios involving grids conforming to complex topography or internal surfaces. We address these issues by developing a mimetic FDTD (MFDTD) approach that combines four key components: a tensorial 3D AWE, mimetic finite-difference (MFD) operators, fully staggered grids (FSGs), and MFD Robin boundary conditions (RBC). The tensorial formulation of the 3D AWE permits wave propagation to be specified on (semi-) analytically defined coordinate meshes designed to conform to complex domain boundaries. MFD operators allow for higher order FD stencils to be applied throughout the model domain, including the boundary region where implementing centered FD stencils can be problematic. The FSG approach combines wavefield information propagated on four complementary subgrids to ensure the existence of all wavefield gradients required for computing the tensorial Laplacian operator, and thereby avoids interpolation approximations. The RBCs are implemented with a flux-preserving mimetic boundary operator that forestalls introduction of nonphysical energy into the grid by enforcing underlying flux-conservation laws. After validating the 3D MFDTD scheme on a sheared Cartesian mesh, we generate 3D wavefield simulation examples for internal boundary (IB) and topographic coordinate systems. The numerical examples demonstrate that the MFDTD scheme is capable of providing accurate and low-dispersion impulse responses for scenarios involving distorted IB meshes conforming to water-bottom surfaces and topographic coordinate systems exhibiting 2.5 km of topographic relief and including steep (65°) slope angles.