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Gravitational waves (GWs) can be distorted, similarly to electromagnetic (EM)waves, by massive objects through the phenomenon of gravitational lensing. Theimportance of gravitational lensing for GW astronomy is becoming increasinglyapparent in the GW detection era, in which nearly 100 events have already beendetected. As current ground-based interferometers reach their designsensitivities, it is anticipated that these detectors may observe a few GWsignals that are strongly lensed by the dark halos of intervening galaxies orgalaxy clusters. Analyzing strong lensing effects on GW signals is, thus,becoming important to understand the lens' properties and correctly infer theintrinsic GW source parameters. However, one cannot accurately infer lensparameters for complex lens models with only GW observations because there arestrong degeneracies between the parameters of lensed waveforms. In this paper,we discuss how to conduct parameter estimation of strongly lensed GW signalsand infer the lens parameters with the help of EM observations, such as theshape and the redshift of the lens. We find that for simple sphericallysymmetric lens models, the lens parameters can be well recovered using only GWinformation. On the other hand, recovering the lens parameters requires systemsin which four or more GW images are detected with additional EM observationsfor non-axially symmetric lens models. Combinations of GW and EM observationscan further improve the inference of the lens parameters.

Inferring properties of dark galactic halos using strongly lensed gravitational waves