  Computation of high frequency solutions to wave equations is important in
many applications, and notoriously difficult in resolving wave oscillations.
Gaussian beams are asymptotically valid high frequency solutions concentrated
on a single curve through the physical domain, and superposition of Gaussian
beams provides a powerful tool to generate more general high frequency
solutions to PDEs. An alternative way to compute Gaussian beam components such
as phase, amplitude and Hessian of the phase, is to capture them in phase space
by solving Liouville type equations on uniform grids. In this work we review
and extend recent constructions of asymptotic high frequency wave fields from
computations in phase space. We give a new level set method of computing the
Hessian and higher derivatives of the phase. Moreover, we prove that the
$k^{th}$ order phase space based Gaussian beam superposition converges to the
original wave field in $L^2$ at the rate of $\ep^{\frac{k}{2}-\frac{n}{4}}$ in
dimension $n$.
