  As the interferometers detecting gravitational waves are upgraded, improving
their sensitivity, the probability of observing strong lensing increases. Once
a detection is made, it will be critical to gain as much information as
possible about the lensing object from these observations. In this work, we
present a methodology to rapidly perform model selection between differing mass
density profiles for strongly lensed gravitational wave signals, using the
results of the fast strong lensing analysis pipeline GOLUM. We demonstrate the
validity of this methodology using some illustrative examples adopting the
idealised singular isothermal sphere and point mass lens models. We take
several simulated lensed signals, analyse them with GOLUM and subject them to
our methodology to recover both the model and its parameters. To demonstrate
the methodology's stability, we show how the result varies with the number of
samples used for a subset of these injections. In addition to the analysis of
simulations, we also apply our methodology to the gravitational wave event pair
GW191230--LGW200104, two events with similar frequency evolutions and sky
locations, which was analysed in detail as a potential lensing candidate but
ultimately discarded when considering the full population and the uncertain
nature of the second event. We find a preference for the singular isothermal
sphere model over the point mass, though our posteriors are much wider than for
the lensed injections, in line with the expectations for a non-lensed event.
The methodology developed in this work is made available as part of the
Gravelamps package of software.
