  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. We present a superposition of Gaussian beams over an
arbitrary bounded set of dimension $m$ in phase space, and show that the tools
recently developed in [ H. Liu, O. Runborg, and N. M. Tanushev, Math. Comp.,
82: 919--952, 2013] can be applied to obtain the propagation error of order
$k^{1- \frac{N}{2}- \frac{d-m}{4}}$, where $N$ is the order of beams and $d$ is
the spatial dimension. Moreover, we study the sharpness of this estimate in
examples.
