论文标题
弹性散射具有随机电势
Inverse Elastic Scattering for a Random Potential
论文作者
论文摘要
本文涉及具有随机电势的时谐波弹性波方程的反向散射问题。被解释为一个分布，假定电势为微局部的普遍高斯随机场，其协方差算子由经典的伪差异操作员描述。目的是从在有界域中测得的散射波的主要符号确定与电势域有正距离的散射波。对于如此粗糙的潜力，通过研究等效的lippmann--schinginger积分方程来确定分布意义上直接散射问题的良好性。对于反散射问题，概率表明，协方差算子的主要符号可以通过从随机电位的单个实现中平均在频带上平均的散射波的振幅来确定。该分析采用高频出生的近似值，弹性波方程的绿色张量的渐近分析以及傅立叶积分算子的微局部分析。
This paper is concerned with an inverse scattering problem for the time-harmonic elastic wave equation with a random potential. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic generalized Gaussian random field with the covariance operator being described by a classical pseudo-differential operator. The goal is to determine the principal symbol of the covariance operator from the scattered wave measured in a bounded domain which has a positive distance from the domain of the potential. For such a rough potential, the well-posedness of the direct scattering problem in the distribution sense is established by studying an equivalent Lippmann--Schwinger integral equation. For the inverse scattering problem, it is shown with probability one that the principal symbol of the covariance operator can be uniquely determined by the amplitude of the scattered waves averaged over the frequency band from a single realization of the random potential. The analysis employs the Born approximation in high frequency, asymptotics of the Green tensor for the elastic wave equation, and microlocal analysis for the Fourier integral operators.
