Is a curved flight path in SAR better than a straight one?

In the plane, we study the transform $R_{\gamma }f$ of integrating a unknown function $f$ over circles centered at a given curve $\gamma$. This is a simplified model of SAR, when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering the wave front set $\text{WF}(f)$. If the visible singularities of $f$ hit $\gamma$ once, we show that the “artifacts” cannot be resolved. If $\gamma$ is a closed curve, we show that this is still true. On the other hand, if $f$ is known a priori to have singularities in a compact set, then we show that one can recover $\text{WF}(f)$, and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation.