Inverse Problems, Layer Stripping and the Riesz Transform

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The principal investigator and collaborators have developed a mathematically rigorous and highly stable inverse scattering algorithm for a lossless stratified medium (1-D Helmholtz Equation), which has successfully treated experimental remote sensing data with high noise levels. The method is a layer stripping method, based on a nonlinear Riesz transform, rather than a trace formula. It features a nonlinear Plancherel equality and some nonlinear Payley-Weiner theorems. Together with the Riesz transform, these allow us to develop a rather complete Fourier analysis, even in the presence of strong multiple reflections. In this project, we plan to investigate the extensions of these methods to more complicated (e.g. lossy and non-stratified) media. For a more detailed overview, see "www.math.washington.edu/~sylvest/".The fundamental task of science is to probe the world around us. The most powerful method for accomplishing this goal is to direct energy, in the form of waves, at an object and to observe the waves after they have interacted with that object. For example, a conventional photograph is produced by directing light waves from the flash bulb to the object under study and recording the image formed by the reflected waves on film.While the results of such an experiment can be readily understood by the human scientist (we can just look at the picture), results of analogous experiments using waves which penetrate more deeply into the medium under study (e.g. microwaves, X-rays, and some sound waves) are less directly meaningful.A wave which penetrates deeply into a medium gives us the opportunity to see below the surface. However, the picture we obtain is a stack of images of the top surface and the various layers below it, all superimposed on the same "photograph". In addition, there are even more superimposed images, formed from internal reflections between layers (multiple reflections).This project continues development of a mathematical method for turning a "microwave photograph", made up of a stack of superimposed images, into a stack of individual photographs, each containing the image of a single layer of the medium. From these individual photographs, we see the true structure of the underlying medium.