Mathematical models of elastic waves and their inverse problems

弹性波数学模型及其反问题

基本信息

批准号：
17540145
负责人：
SOGA Hideo
金额：
$ 1.41万
依托单位：
Ibaraki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540145/
关键词：
mathematical physics inverse problems elastic equations seismic waves scattering theory Rayleigh wave wave equations partial differential equations 偏微分方程

项目摘要

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.

在这个研究项目中，我们最初的目标是（i）检查表面波（例如瑞利波等）的散射并研究反问题，以从这些波的数据中提取表面的情况；（ii）建立一些数学模型建立与地震探头相对应的模型并研究反问题，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，并制作个人计算机软件以显示传播的关于（i）：我们构建了瑞利波的散射理论及其渐近解，可用于解决反演问题，以及表达边界情况的散射核的表示。此外，利用这些结果，我们解决了反演问题，以从瑞利波的日期中获取边界信息。关于（ii）：我们对地震波的反演问题做了适当的数学设置，并开发了在某些假设下应用的方法。但我们还没有能够完全实现我们最初的预期。关于(iii)：我们已经获得了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，用于在显示器上展示波的传播。这将发展到展示逆问题的各种设置的情况。