The purpose of this work is to present results about the composition of Fourier integral operators with certain singularities, for which the composition is not again a Fourier integral operator. The singularities considered here are folds and blowdowns. We prove that for such operators, the Schwartz kernel of F * F belongs to a class of distributions associated to two cleanly intersection Lagrangians. Such Fourier integral operators appear in integral geometry, inverse acoustic scattering theory and Synthetic Aperture Radar imaging, where the composition calculus can be used as a tool for finding approximate inversion formulas and for recovering images.

In this article, we analyze the microlocal properties of the linearized forward scattering operator F and the normal operator F * F (where F * is the L 2 adjoint of F ) which arises in Synthetic Aperture Radar imaging for the common midpoint acquisition geometry. When F * is applied to the scattered data, artifacts appear. We show that F * F can be decomposed as a sum of four operators, each belonging to a class of distributions associated to two cleanly intersecting Lagrangians, I p,l (Λ 0 , Λ 1 ), thereby explaining the latter artifacts.

We consider a linearized inverse problem arising in offshore seismic imaging. Following Nolan and Symes [28], one wishes to determine a singular perturbation of a smooth background soundspeed in the Earth from measurements made at the surface resulting from various seismic experiments; the overdetermined data set considered here corresponds to marine seismic exploration. In the presence of only fold caustics for the background, we identify the geometry of the canonical relation underlying the linearized forward scattering operator F , which is a Fourier integral operator. We then establish a composition calculus for general FIOs associated with similar canonical relations, which we call folded cross caps, sufficient for identifying the normal operator F * F . In contrast to the case of a single source experiment, treated by Nolan [25] and Felea[5], the resulting artifact is 1 2 order smoother than the main pseudodifferential part of F * F .

We prove microlocal properties of a generalized Radon transform that integrates over lines in R 3 with directions parallel to a fairly arbitrary curve on the sphere. This transform is the model for problems in slant-hole SPECT and conical-tilt electron microscopy, and our results characterize the microlocal mapping properties of the SPECT reconstruction operator developed and tested by Quinto, Bakhos, and Chung. We show that, in general, the added singularities (or artifacts) are increased as much as the singularities of the function we want to image. Using our microlocal results, we construct a differential operator such that the added singularities are, relatively, less strong than the singularities we want to image.

We analyse further inverse problems related to synthetic aperture radar imaging considered by Nolan and Cheney (2002 Inverse Problems 18 221). Under a nonzero curvature assumption, it is proved that the forward operator F is associated with a two-sided fold, C. To reconstruct the singularities in the wave speed, we form the normal operator F*F. In Felea (2005 Comm. Partial Diff. Eqns 30 1717) and Nolan (2000 SIAM J. Appl. Math. 61 659), it was shown that F*F ∊ I2m,0(Δ, C1), where C1 is another two-sided fold. In this case, the artefact on C1 has the same strength as the initial singularities on Δ and cannot be removed. By working away from the fold points, we construct recursively operators Qi which, when applied to F*F, migrate the primary artefact. One part is lower order, has less strength and is smoother than the image to be reconstructed. The other part is as strong as the original artefact, but is spatially separated from the scene.