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

We are interested in a linearized inverse scattering problem considered by Symes and Nolan [10] for the marine data acquisition geometry. This arises as follows: a vessel carries an acoustic source (airgun) and recording instruments (hydrophones). The airgun sends acoustic waves through the ocean to the subsurface and the reflections occur when the sound waves encounter singularities in the material of the subsurface. The reflected rays are received by hydrophones towed behind the vessel. The vessel makes repeated passes along parallel paths. The pressure field at the surface is used to reconstruct an image of the subsurface. Our eventual goal is to invert the Fourier integral operator F , the linearized operator which maps singularities in the sound speed in the subsurface to the singularities in the resulting pressure field at the surface. Standard techniques tell us to consider the normal operator F ∗ ◦ F .