Inverse Problems, Layer Stripping and the Riesz Transform

首席研究员和合作者开发了一种数学上严谨且高度稳定的反散射算法，用于无损分层介质（一维亥姆霍兹方程），该算法已成功地处理了高噪声水平的实验遥感数据。该方法是一种基于非线性Riesz变换的层剥离方法，而不是跟踪公式。它具有一个非线性Plancherel等式和一些非线性Payley-Weiner定理。与Riesz变换一起，这些使我们能够开发出相当完整的傅立叶分析，即使在存在强烈的多重反射的情况下。在这个项目中，我们计划研究将这些方法扩展到更复杂的（例如有损和非分层）媒体。要了解更详细的概述，请参阅“www.math.washington.edu/~sylvest/”。科学的基本任务是探索我们周围的世界。实现这一目标最有力的方法是将能量以波的形式引导到一个物体上，并在波与该物体相互作用后观察它们。例如，传统的照片是通过将闪光灯的光波引导到被研究的物体上，并将反射波形成的图像记录在胶片上而产生的。虽然人类科学家很容易理解这样一个实验的结果（我们可以看一下图片），但使用更深入研究介质的波（例如微波、x射线和一些声波）进行的类似实验的结果就没有那么直接的意义了。能深入介质的波使我们有机会看到表面以下的东西。然而，我们得到的图像是一堆图像的顶部表面和它下面的各种层，所有叠加在同一张“照片”。此外，还有更多的叠加图像，由层之间的内部反射形成（多重反射）。该项目继续发展一种数学方法，将由叠加图像堆叠而成的“微波照片”转化为一堆单独的照片，每个照片包含单层介质的图像。从这些单独的照片中，我们看到了潜在媒介的真实结构。

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.