  Theories of gravity extending or modifying general relativity typically allow
for black hole solutions different from the Schwarzschild/Kerr geometries.
Electromagnetic observations have been used to place constraints on
parametrized deviations from the Schwarzschild/Kerr metrics, in an effort to
gain insight on the underlying gravitational theory. In this work, we show that
observations of the gravitational quasi-normal modes by existing and future
interferometers can be used to bound the same parametrized black hole metrics
that are constrained by electromagnetic observations (e.g. by the Event Horizon
Telescope). We argue that our technique is most sensitive to changes in the
background black hole metric near the circular photon orbit, and that it is
robust against the changes that a gravitational theory differing from general
relativity necessarily introduces in the equations for the gravitational
perturbations. We demonstrate our approach by reconstructing the background
metric from a set of simulated observations using a Bayesian approach.
