Higher-order pseudospectral methods for the acoustic wave equation

The phase-shift time-stepping equation (PSTS) is a wavefield propagator
that allows two-way in time propagation for the acoustic wave equation.
The PSTS is based an an exact solution to the acoustic wave equation. It
is adapted to variable velocity wave equation by a windowed Fourier
transform where in each window a constant velocity solution is computed.
We consider a correction to the phase-shift time-stepping equation that
corrects the wave propagators for inhomogeneity velocity variations.  The
correction is based on a similar Taylor-series expansion used to derive
the split-step correction for one-way depth steppers or to derive
higher-order in time pseudospectral methods using the modified equation
approach or Lax-Wendroff method. Its computational properties are similar
to higher-order in time pseudospectral methods.