  We develop a paradigm using microlocal analysis that allows one to
characterize the visible and added singularities in a broad range of incomplete
data tomography problems. We give precise characterizations for photo- and
thermoacoustic tomography and Sonar, and provide artifact reduction strategies.
In particular, our theorems show that it is better to arrange Sonar detectors
so that the boundary of the set of detectors does not have corners and is
smooth. To illustrate our results, we provide reconstructions from synthetic
spherical mean data as well as from experimental photoacoustic data.
