  Full--waveform inversion (FWI) is a method used to determine properties of
the Earth from information on the surface. We use the squared Wasserstein
distance (squared $W_2$ distance) as an objective function to invert for the
velocity of seismic waves as a function of position in the Earth, and we
discuss its convexity with respect to the velocity parameter. In one dimension,
we consider constant, piecewise increasing, and linearly increasing velocity
models as a function of position, and we show the convexity of the squared
$W_2$ distance with respect to the velocity parameter on the interval from zero
to the true value of the velocity parameter when the source function is a
probability measure. Furthermore, we consider a two--dimensional model where
velocity is linearly increasing as a function of depth and prove the convexity
of the squared $W_2$ distance in the velocity parameter on large regions
containing the true value. We discuss the convexity of the squared $W_2$
distance compared with the convexity of the squared $L^2$ norm, and we discuss
the relationship between frequency and convexity of these respective distances.
We also discuss multiple approaches to optimal transport for non--probability
measures by first converting the wave data into probability measures.
