This paper presents an analytic method for synthetic-aperture inversion when the measurements are corrupted with noise and clutter. We use microlocal analysis in a statistical setting to develop filtered-backprojection-type reconstruction methods. The inversion method is applicable in non-ideal scenarios, such as those involving arbitrary source trajectories or variable antenna beam patterns. We show that the backprojection preserves the location and orientation of the singularities of the first- and second-order statistics of the target scene. We derive backprojection filters with respect to different statistical criteria. In particular, if we use a criterion based on first-order statistics, the resulting image can be interpreted as approximately unbiased. Alternatively, if we use a criterion based on second-order statistics to design the backprojection filter, such as a minimum-mean-square error criterion, the strength of the singularities due to noise and clutter is suppressed in the resulting image. Although we have developed our approach specifically for synthetic-aperture radar application, the method is also applicable to other inversion problems in which microlocal techniques are relevant, such as geophysics and x-ray tomography.