Mathematical models of elastic waves and their inverse problems

在这个研究项目中，我们最初的目的是（i）检查表面波的散射（2）建立相应的数学模型，研究反演问题，从人工入射波的反射波数据中了解地层的形状;（iii）在典型条件下数值求解上述反问题，并制作了在显示器上显示波的传播的个人计算机软件。关于（i）：我们构造了一个对求解反问题有用的瑞利波散射理论及其渐近解，以及一个表示边界情况的散射核的表示。此外，利用这些结果，我们解决了反问题，从瑞利波的数据得到边界的信息。关于（ii）：我们建立了一个适当的数学模型，并在一定的假设下发展了适用于地震波反问题的方法。但我们还没有能够完全完成我们最初打算。关于（iii）：我们得到了一个数值算法的最佳形状问题，这似乎适用于反问题。我们已经做了一个软件的个人电脑的展览中的波的传播显示。这将发展到一个显示的情况下，各种设置的反问题。

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.