The detection and parameter estimation challenges of Type-II lensed binary black hole signals

 Strong lensing can produce three types of images denoted as Type-I, Type-II and Type-III corresponding to the minimum, saddle and maximum of the total arrival time of the lensed GWs. The Type II images, in particular, receive a non-trivial phase shift of $\pi/2$ (for positive frequencies). This phase shift can introduce (additional) distortions in the strains produced by the Type-II image of the binary black hole (BBH) signals depending on the morphology of the signals, e.g., when they have contributions from higher harmonics (HMs) and/or precession. The optical depth to the Type-II images is nearly the same as the strong lensing (from galaxies) and thus are very likely to be observed in the near future.  In this work, we investigate the potential applicability of these distortions in helping to identify the Type-II signals individually and also the systematic biases that could arise in the inferences of their parameters if they are unknowingly recovered with the unlensed GW templates. We show that at high SNRs ($\gtrsim 100$), and thus mostly relevant for 3G detectors and beyond, most Type-II images would be identifiable. These identified images will, however, have significantly biased estimates of the parameters (such as, sky location, distance, inclination, etc.) and in many cases fully biased. Thus, in the future, using Type-II lensed template for such signals would be necessary to avoid the potentially wrong astrophysical implications otherwise.