Composition of Fourier Integral Operators with Fold and Blowdown Singularities

Abstract

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.

Microlocal Analysis of SAR Imaging of a Dynamic Reflectivity Function

“…In this case, we show that both projections π L and π R have singularities called blowdowns (see section 2). Such canonical relations were studied before in [12] and in [7], [4] (where only one projection has blowdown singularities).…”

“…We can find the strength of the artifacts by finding the order of F * F on Λ \ ∆. It is shown in section 3 that these artifacts are stronger (case 2) or have equal strength with ∆ (cases 3,4).…”

An FIO Calculus for Marine Seismic Imaging: Folds and Cross Caps

“…In the case of the single source model, with only fold caustics appearing, Nolan [25] showed that F is an FIO associated to a folding canonical relation in the sense of [21] (also called a two-sided fold), and stated that the Schwartz kernel of the operator F * F belongs to a class of distributions associated to two cleanly intersecting Lagrangians in (T * Y \ 0) × (T * Y \ 0). This was fully proved in [5]. The corresponding canonical relations are the diagonal ∆ and a folding canonical relation, different from the original one, which lies in T * X × T * Y .…”

“…Following an idea from [12], [5], we now make a singular change of variables, T : R n → R n−1 , T (θ ′′ , z n−1 , z n ) = (ξ ′′ , ξ n−1 ), given by:…”

“…Composition of FIOs under other singular geometries arising in integral geometry and inverse problems has been previously studied in, e.g., [13], [9], [11], [12], [25] and [5].…”