Inversion from Time Domain Backscattering Data for the Wave Equation

研究人员建议研究三维空间中等离子体波动方程(波算符加零阶项)的形式确定的反问题。目标是从放置在球边界上的重合的源-接收器对产生的依赖时间的数据中恢复支撑在球中的势。这个问题是对从后向散射数据中恢复电势的频域问题在时间域上的模拟。研究人员将研究向数据发送势的映射的解析性，函数处于该映射范围内的必要条件，势被限制为角度变量中的解析函数时的唯一性，以及从数据中恢复势的奇异性。研究人员提出的问题的解决和解决这些问题的方法将在地球物理、石油勘探、医学成像、国防等利用声波或电磁波探测隐藏对象的领域中有用。

The investigator proposes studying a formally determined inverse problem for the plasma wave equation (wave operator plus a zeroth order term) in three space dimensions. The goal is the recovery of a potential, supported in a ball, from time dependent data generated by coincident source-receiver pairs placed on the boundary of the ball. This problem is an imitation, in the time domain, of the frequency domain problem of recovering a potential from back-scattering data. The investigator will study, the analyticity of the map sending potentials to the data, necessary conditions for a function to be in the range of this map, uniqueness when the potentials are restricted to functions which are analytic in the angular variables, and the recovery of the singularities of the potential from the data.The solution of the problems proposed by the investigator and the techniques introduced to solve them would be useful in geophysics, oil exploration, medical imaging, defense and other fields where hidden objects are probed by acoustic or electromagnetic waves.