  In this work, we describe an approach to stably simulate the 3D isotropic
elastic wave propagation using finite difference discretization on staggered
grids with nonconforming interfaces. Specifically, we consider simulation
domains composed of layers of uniform grids with different grid spacings,
separated by planar interfaces. This discretization setting is motivated by the
observation that wave speeds of earth media tend to increase with depth due to
sedimentation and consolidation processes. We demonstrate that the layer-wise
finite difference discretization approach has the potential to significantly
reduce the simulation cost, compared to its counterpart that uses holistically
uniform grids. Such discretizations are enabled by summation-by-parts finite
difference operators, which are standard finite difference operators with
special adaptations near boundaries or interfaces, and simultaneous
approximation terms, which are penalty terms appended to the discretized system
to weakly impose boundary or interface conditions. Combined with specially
designed interpolation operators, the discretized system is shown to preserve
the energy-conserving property of the continuous elastic wave equation, and a
fortiori ensure the stability of the simulation. Numerical examples are
presented to corroborate these analytical developments.
