Generalized oscillatory integrals

The concept of wave fronts is essential in remote sensing problems associated with wave propagation phenomena (that is, inverse scattering problems); wave fronts contain much of the information necessary for addressing such problems. The field of microlocal analysis characterizes the concept of wave fronts with the so-called wave front set. The wave front set provides the position of the singularities of the wave field as well as their high-frequency orientations (directions of propagation). For smooth media, inverse scattering problems can be conveniently formulated with the help of Fourier integral operators (FIOs), which provide geometrical insights into the propagation of the singularities in the wavefield, i.e., the evolution of the wave front set. Real media, however, are more faithfully described by (highly) irregular models, e.g., with discontinuous or fractal-like variations of parameters. Hence, it is desirable to rigorously develop FIOs in the more general case of non-smooth media. For smooth media, kernels of FIOs have oscillatory integral (OI) representations. The analysis of OIs yields many of the microlocal properties of FIOs, the foremost of which is how they govern the evolution of wave front sets. To extend FIOs to non-smooth media we propose, as a first step, the systematic study of OIs in a more general framework that takes the non-smoothness of the medium into account. The general framework will be the Colombeau theory of generalized functions, which is particularly well adapted to solving wave equations in non- smooth media. Particular attention will be paid to pseudodifferential operators as a special case of FIOs. These operators are essential microlocal tools (for instance, in characterizing wave front sets) and are a current research topic in the field of generalized functions. In this project, analyzing OIs and wave front sets will lead naturally to the study of the microlocal properties of Hamiltonian flows in Colombeau theory. Hamiltonian flows provide the proper geometrical tools for following the propagation of singularities in wave phenomena. The rigorous development of FIOs for non-smooth media and the subsequent formulation of inverse scattering problems could have applications in medical and space imaging, ocean acoustics, and geophysics.