  We characterize the range of the attenuated and non-attenuated $X$-ray
transform of compactly supported vector fields in the plane. The
characterization is in terms of a Hilbert transform associated with the
$A$-analytic functions \`{a} la Bukhgeim. As an application we determine
necessary and sufficient conditions for the attenuated Doppler and $X$-ray data
to be mistaken for each other.
