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.
