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 I-p,I-l (Delta, Lambda),where Lambda is associated to a strong artifact. At the and of the article, we show how to microlocally reduce the strength of the artifact.