  We consider the tomography problem of recovering a covector field on a simple
Riemannian manifold based on its weighted Doppler transformation over a family
of curves $\Gamma$. This is a generalization of the attenuated Doppler
transform. Uniqueness is proven for a generic set of weights and families of
curves under a condition on the weight function. This condition is satisfied in
particular if the weight function is never zero, and its derivatives along the
curves in $\Gamma$ is never zero.
