  This work characterizes, analytically and numerically, two major effects of
the quadratic Wasserstein ($W_2$) distance as the measure of data discrepancy
in computational solutions of inverse problems. First, we show, in the
infinite-dimensional setup, that the $W_2$ distance has a smoothing effect on
the inversion process, making it robust against high-frequency noise in the
data but leading to a reduced resolution for the reconstructed objects at a
given noise level. Second, we demonstrate that for some finite-dimensional
problems, the $W_2$ distance leads to optimization problems that have better
convexity than the classical $L^2$ and $H^{-1}$ distances, making it a more
preferred distance to use when solving such inverse matching problems.
