### Abstract:

In this article we consider four particular cases of Synthetic Aperture
Radar imaging with moving objects. In each case, we analyze
the forward operator F and the normal operator F∗F, which appear
in the mathematical expression for the recovered reflectivity function
(i.e. the image). In general, by applying the backprojection operator
F∗ to the scattered waveform (i.e. the data), artifacts appear in the
reconstructed image. In the first case, the full data case, we show
that F∗F is a pseudodifferential operator which implies that there is
no artifact. In the other three cases, which have less data, we show
that F∗F belongs to a class of distributions associated to two cleanly
intersecting Lagrangians Ip,l(Δ; Λ), where Λ is associated to a strong
artifact. At the and of the article, we show how to microlocally reduce
the strength of the artifact.