直接优化法同时反演声波方程的两个系数

Simultaneous reconstruction of the acoustic equation via a direct optimization approach

摘要: 本文在时域范围内考虑一个层状半无限空间的声波方程两个系数同时反演的问题。通过构造一个偏差的平方和形式的目标泛函,声波方程系数的反演问题可以转化为一个由双曲型偏微分方程控制的优化问题。在本文中,一种直接优化的技术被用来解这个优化问题。通过寻找目标泛函的极小点,声波方程的两个系数可以方便地得到同时反演。由于直接优化法只使用了目标泛函的函数值而不用它的梯度信息,因此这种算法具有稳定、精确的优点。文中同时还给出了这种算法的稳定性定理以及数值解的例子。从数值解的情况上看,这种直接优化法反演声波方程系数的结果是令人满意的。

Abstract: The acoustic inverse problem on a stratified half-space is considered in the time domain.By constructing the objective functional,the inverse problem is thereby reduced to an optimization problem where the profiles of the velocity and the density minimizing the functional are to be found.A direct optimization algorithm is employed to solve the optimization problem and reconstruct the profiles of the velocity and the density simultaneously.The stability theorem and numerical examples are given.The results of reconstruction excellently agree with the true profiles