Mathematical models of elastic waves and their inverse problems

弹性波数学模型及其反问题

• 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

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.

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典型的条件并制造个人计算机的软件，以展示显示屏中波的传播。此外，使用这些结果，我们已经解决了逆问题，从瑞利波（Rayleigh Wave）之日起获取边界的信息。但是，我们无法完全完成我们最初的意图。（iii）：我们获得了似乎适用于反问题的最佳形状问题的数值算法。我们制作了一个个人计算机软件，以展示显示器中波浪的传播。这将开发给一个以显示逆问题各种环境的情况。

项目成果

Scattering theory for the elastic wave equations in perturbed half-spaces

扰动半空间中弹性波方程的散射理论

• 发表时间: 2006
• 期刊: Trans. Amer. Math. Soc (掲載予定、レフェリー済み)
• 作者: 川下美潮;川下和日子;曽我日出夫
• 通讯作者: 曽我日出夫