Superconducting gap symmetry from Bogoliubov quasiparticle interference analysis on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Sr</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>RuO</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math>

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.