Synthetic aperture inversion

This paper considers synthetic aperture radar and other synthetic aperture imaging systems in which a backscattered wave is measured from a variety of locations. The paper begins with a (linearized) mathematical model, based on the wave equation, that includes the effects of limited bandwidth and the antenna beam pattern. The model includes antennas with poor directionality, such as are needed in the problem of foliage-penetrating radar, and can also accommodate other effects such as antenna motion and steering. For this mathematical model, we use the tools of microlocal analysis to develop and analyse a three-dimensional imaging algorithm that applies to measurements made on a two-dimensional surface. The analysis shows that simple backprojection should result in an image of the singularities in the scattering region. This image can be improved by following the backprojection with a spatially variable filter that includes not only the antenna beam pattern and source waveform but also a certain geometrical scaling factor called the Beylkin determinant. Moreover, we show how to combine the backprojection and filtering in one step. The resulting algorithm places singularities in the correct locations, with the correct orientations and strengths. The algorithm is analysed to determine which information about the scattering region is reconstructed and to determine the resolution. We introduce a notion of directional resolution to treat the reconstruction of walls and other directional elements. We also determine the fineness with which the data must be sampled in order for the theoretical analysis to apply. Finally, we relate the present analysis to previous work and discuss briefly implications for the case of a single flight track. 1 Corresponding address: Mathematical Sciences, Research Institute, 1000 Centennial Drive, #5070, Berkeley, CA 94720-5070, USA. After 14 December 2001, Contact at: Math Sciences, Amos Eaton Hall, RPI, 110 8th Street Troy, NY 12180, USA. 0266-5611/02/010221+15$30.00 © 2002 IOP Publishing Ltd Printed in the UK 221 222 C J Nolan and M Cheney