  We consider the wave equation with uncertain initial data and medium, when
the wavelength $\varepsilon$ of the solution is short compared to the distance
traveled by the wave. We are interested in the statistics for quantities of
interest (QoI), defined as functionals of the wave solution, given the
probability distributions of the uncertain parameters in the wave equation.
Fast methods to compute this statistics require considerable smoothness in the
mapping from parameters to the QoI, which is typically not present in the high
frequency case, as the oscillations on the $\varepsilon$ scale in the wave
field is inherited by the QoIs. The main contribution of this work is to
identify certain non-oscillatory quadratic QoIs and show
$\varepsilon$-independent estimates for the derivatives of the QoI with respect
to the parameters, when the wave solution is replaced by a Gaussian beam
approximation.
