  Propagation characteristics of a wave are defined by the dispersion
relationship, from which the governing partial differential equation (PDE) can
be recovered. PDEs are commonly solved numerically using the finite-difference
(FD) method, with stencils constructed from truncated Taylor series expansions
which, whilst typically providing good approximation of the PDE in the
space-time domain, often differ considerably from the original partial
differential in the wavenumber-frequency domain where the dispersion
relationship is defined. Consequentially, stable, high-order FD schemes may not
necessarily result in realistic wave behavior, commonly exhibiting numerical
dispersion: lagging high-frequency components as a product of discretization. A
method for optimizing FD stencil weightings via constrained minimization to
better approximate the partial derivative in the wavenumber domain is proposed,
allowing for accurate propagation with coarser grids than would be otherwise
possible. This was applied to second derivatives on a standard grid and first
derivatives on a staggered grid. To evaluate the efficacy of the method, a pair
of numerical simulations were devised to compare spatially-optimized stencils
with conventional formulations of equivalent extent. A spatially-optimized
formulation of the 1D acoustic wave equation with Dirichlet boundary conditions
is presented, evaluating performance at a range of grid spacings, examining the
interval between the theoretical maximum grid spacings for the conventional and
optimized schemes in finer detail. The optimized scheme was found to offer
superior performance for undersampled wavefields and heavily oversampled
wavefields. Staggered-grid first derivative stencils were then applied to the
P-SV elastic wave formulation, simulating seismic wave propagation for a
two-layer, water-over-rock model.
