  The nature of the superconducting order parameter in {Sr}$_2${RuO}$_4$ has
generated intense interest in recent years. Since the superconducting gap is
very small, high resolution methods such as scanning tunneling spectroscopy
might be the best chance to directly resolve the gap symmetry. Recently, a
Bogoliubov quasiparticle interference imaging (BQPI) experiment has suggested
that the $d_{x^2-y^2}$ gap symmetry is appropriate for {Sr}$_2${RuO}$_4$. In
this work, we use a material-specific theoretical approach based on Wannier
functions of the surface of {Sr}$_2${RuO}$_4$ to calculate the continuum
density of states as detected in scanning tunneling microscopy experiments. We
examine several different proposed gap order parameters, and calculate the
expected BQPI pattern for each case. Comparing to the available experimental
data, our results suggest that a $s'+id_{xy}$ gap order parameter is the most
probable state, but the measured BQPI patterns still display features
unaccounted for by the theory for any of the states currently under discussion.
