This paper develops a method for forming a synthetic-aperture image of a flat surface seen through a homogeneous layer of a material that is dispersive, i.e., its wave speed varies with frequency. We outline first a simplified scalar model for electromagnetic wave propagation in a dispersive medium; the resulting equation could also be used for acoustics. We show that the backscattered signal can be viewed as a Fourier integral operator applied to the ground reflectivity function. The reconstruction method, which is based on backprojection, can be used for arbitrary sensor paths and corrects for the radiated beam pattern, the source waveform and geometrical spreading factors. The method correctly reconstructs the singularities (such as edges) that are visible from the sensor.