Mathematical Sciences: Waves and Diffusion in Random Media

点击翻译按钮获取中文摘要

The proposed research has three components: (a) The study of direct and inverse problems for acoustic pulse reflection from randomly layered media, (b) the study of the focusing singularity of the nonlinear Schroedinger and related equations, and (c) convection enhanced diffusion. In pulse reflection a statistical theory will be developed for reflected signals generated by pulses emitted by a point source located on the surface or above a randomly layered half space. The range of validity of the theory, based on the asymptotic theory of stochastic equations, will be explored with numerical simulations. For pulsed plane waves incident on a randomly layered half space, a number of statistical inverse problems will be formulated and solved, where properties of the medium are based on the statistics of the observed reflected signals. The work on pulse reflection is of direct interest in exploration seismology and to a lesser extent in connection with the detection of seismic events that are natural or man-made. The work on the nonlinear Schroedinger equation is useful in nonlinear optics for laser surgery, for example, and in laser fusion. The work on convection-diffusion is useful in the study of dispersion of pollutants in various environments such as the ocean and the atmosphere.