Mathematical models of elastic waves and their inverse problems

在这个研究项目中，我们最初的目标是(i)研究表面波（如瑞利波等）的散射，并研究从这些波的数据中提取表面情况的反问题；(2)建立与地震探头相对应的数学模型，研究利用人工入射波反射波资料了解地层形状的反问题；（三）在典型条件下对上述反问题进行数值求解，并制作个人计算机软件，用于显示波在显示器中的传播情况。关于(i)：我们构造了瑞利波的散射理论及其对解反问题有用的渐近解，以及表示边界情况的散射核的表示。此外，利用这些结果，我们还解决了从瑞利波数据中获取边界信息的逆问题。关于（ii）：我们对地震波反问题作了适当的数学设置，并在一些假设下发展了适用于该问题的方法。但是我们还没有完全完成我们最初的计划。关于（iii）：我们得到了一种最优形状问题的数值算法，它似乎适用于逆问题。我们制作了一个个人电脑软件，用于显示显示波的传播。这将发展为展示反问题的各种设置的情况。

In this research project we aimed initially(i) to examine scattering of surface waves (e.g. the Rayleigh wave, etc) and investigate inverse problems to extract the situations of the surface from the data of those waves ;(ii) to set up some mathematical model corresponding the seismic probe and study inverse problems to know the shape of stratums from the data of reflected waves for artificial incident waves ;(iii) to solve the above inverse problems numerically under typical conditions and make softwares of personal computers for exhibition of the propagation of the waves in the display.About (i) : We have constructed a scattering theory of the Rayleigh wave and its asymptotic solution useful for solving the inverse problem, and a representation of the scattering kernel expressing the situations of the boundary. Furthermore, using these results, we have solved the inverse problem to get the information of the boundary from the date of the Rayleigh wave.About (ii) : We have made an appropriate mathematical setup of the inverse problem of seismic waves and have developed the methods applied to it under some assumptions. But we have not been able to accomplish completely what we intended initially.About (iii) : We have obtained a numerical algorism of optimal shape problems which seems applicable to the inverse problem. We have made a software of personal computers for exhibition of the propagation of the waves in the display. This will be developed to the one to show the situations of various setting of the inverse problem.