Inverse Source and Inverse Scattering Problems

遥感的基本特征是，所有信息都必须通过(电磁波或声波)通过中间介质传输。我们只观察散射场，即波穿过中间介质(可能是空的空间)后的场。最简单的数学模型是标量波动方程，它通过傅里叶变换分解成每个波数的标量亥姆霍兹方程。消逝现象限制了分辨率，因为只允许有限维波族以未严重衰减的幅度到达传感器。该族的维度提供了有关波源大小的重要信息，而表示不同基数的波所需的非零系数的数量(以不同点为中心的汉克尔函数展开)携带了有关一个或多个源的位置的信息。我们正在开发利用这些信息的方法。由于震源不是由散射波唯一确定的，这一点变得复杂起来。我们通过寻找支持(携带)辐射散射场的源的最小集合来解决这个问题，并且必须是辐射相同场的任何其他源的一部分。例子表明，一般不存在这样的集合，但存在一个良好分离的凸集的最小并，它满足这两个准则。这个项目的一个主要目标是找到一种算法来找到这种良好分离的凸集的并，并给出可靠的估计，不同大小的源必须相距多远才能保证单个源在数据中可见。这一分析与许多其他逆散射问题有关。一个特别重要的应用是理解在何种情况下，Born或单次散射近似是完全散射问题的可靠替代。虽然所有逆散射技术的实践者都知道，两个散射体之间的距离越远，它们相互作用就越少，因此Born近似的效果越好，但当前的数学理论似乎正好相反。这种明显的悖论是由于我们用来测量和比较海浪的标准。大多数数学上易于处理的规范都有一些错误地反映了一些物理原理的特征。我们继续努力制定标准，这些标准足够简单，可以进行数学分析，但又能忠实地反映物理现实。遥感实验收集关于远处物体的信息，而不需要任何直接接触。水下声学传感器阵列试图根据潜艇发动机辐射的噪声来定位潜艇(源)，这是被动遥感系统的一个例子。而声纳阵列发射自己的波，以便根据回波的特性定位潜艇，是主动遥感系统的一个例子。在这种情况下，阵列而不是潜艇是主要的信号源，潜艇被称为散射体。逆源和散射问题是遥感的重要组成部分，寻找小点状信号源和散射体的算法对包括天线设计在内的技术发展产生了重大影响。该项目旨在通过重新表述分析，更准确地反映该算法对有限数据集的工作方式，来弥合有效算法和理论之间的差距。

The essential feature of remote sensing is that all information must be transported through an intervening medium by (electromagnetic or acoustic) waves. We observe only the scattered field, the wave after it has passed through the intervening medium (which may be empty space). The simplest mathematical model is the scalar wave equation, which decomposes, via Fourier transform, into the scalar Helmholtz equation at each wavenumber. The phenomenon of evanescence limits resolution by allowing only a finite dimensional family of waves to reach the sensors with an amplitude that has not been severely attenuated. The dimension of that family gives important information about the size of the source of the wave, and the number of nonzero coefficients needed to represent the wave in different bases (Hankel function expansions centered at different points) carries information about the location of the source or sources. We are developing methods to use this information. This is complicated by the fact that the source is not uniquely determined by the scattered wave. We resolve this by seeking the smallest set that supports (carries) a source that radiates that scattered field, and must have been part of any other source that radiated the same field. Examples show that there is no such set in general, but there is a smallest union of well separated convex sets that satisfies both criteria. A major goal of this project is to find an algorithm to find this union of well separated convex sets, and give reliable estimates on how far apart sources of various sizes must be in order to guarantee that the individual sources will be visible in the data. This analysis is relevant to many other inverse scattering problems. A particularly important application is understanding the circumstances in which the Born, or single scattering, approximation is a reliable substitute for the full scattering problem. While all practitioners of inverse scattering techniques know that the farther apart two scatterers are, the less they interact, and therefore the better the Born approximation works, current mathematical theory seems to say the opposite. The apparent paradox is due to the norms we use to measure and compare the waves. Most mathematically tractable norms have features which misrepresent some of the physics. We continue to work to formulate norms which are simple enough to allow mathematical analysis and yet faithfully reflect physical reality.A remote sensing experiment gathers information about an object from a distance, without any direct contact. An underwater array of acoustic sensors that seeks to locate a submarine (the source) based on the noise radiated from the submarine's engine, is an example of passive remote sensing system. While a sonar array that transmits its own wave in order to locate the submarine based on the properties of the echo, is an example of active remote sensing system. In this case the array, and not the submarine, is the primary source, and the submarine is referred to as the scatterer.The inverse source and scattering problems are an essential part of remote sensing, and algorithms for finding small point-like sources and scatterers have had major impact on technological development, including antenna design. The project aims to bridge the gap between effective algorithms and theory by reformulating the analysis in a way that more accurately reflects the way this algorithms work with finite data sets.