Lensing magnification: gravitational waves from coalescing stellar-mass
  binary black holes
  Gravitational waves (GWs) may be magnified or de-magnified due to lensing.
This phenomenon will bias the distance estimation based on the matched
filtering technique. Via the multi-sphere ray-tracing technique, we study the
GW magnification effect and selection effect with particular attention to the
stellar-mass binary black holes (BBHs). We find that, for the observed
luminosity distance $\lesssim 3~\mathrm{Gpc}$, which is the aLIGO/Virgo
observational horizon limit, the average magnification keeps as unity, namely
unbiased estimation, with the relative distance uncertainty
$\sigma(\hat{d})/\hat{d}\simeq0.5\%\sim1\%$. Beyond this observational horizon,
the estimation bias can not be ignored, and with the scatters
$\sigma(\hat{d})/\hat{d} = 1\%\sim 15\%$. Furthermore, we forecast these
numbers for Einstein Telescope. We find that the average magnification keeps
closely as unity for the observed luminosity distance $\lesssim
90~\mathrm{Gpc}$. The luminosity distance estimation error due to lensing for
Einstein Telescope is about $\sigma(\hat{d})/\hat{d} \simeq 10\%$ for the
luminosity distance $\gtrsim 25~\mathrm{Gpc}$. Unlike the aLIGO/Virgo case,
this sizable error is not due to the selection effect. It purely comes from the
unavoidably accumulated lensing magnification. Moreover, we investigated the
effects of the orientation angle and the BH mass distribution models. We found
that the results are strongly dependent on these two components.
