Inversion from Time Domain Backscattering Data for the Wave Equation

研究者提出在三维空间中研究等离子体波动方程（波算符加零阶项）的正式确定的逆问题。目标是从放置在球边界上的同步源-接收器对产生的时间相关数据中恢复支持在球中的电位。这个问题是在时域上模仿从后向散射数据中恢复电势的频域问题。研究者将研究，向数据发送电位的图的分析性，函数在该图范围内的必要条件，当电位被限制为在角变量中解析的函数时的唯一性，以及从数据中恢复电位的奇异性。研究者提出的解决问题的方法和所介绍的解决问题的技术，对于利用声波或电磁波探测隐藏物体的地球物理、石油勘探、医学成像、国防等领域具有重要意义。

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.