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

On the Range Characterization of the two dimensional attenuated Doppler Transform