An Inverse Problem in Elastodynamics: Uniqueness of the Wave Speeds in the Interior
An Inverse Problem in Elastodynamics: Uniqueness of the Wave Speeds in the Interior
复制标题
弹性动力学反问题：内部波速的唯一性
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
Lizabeth V. Rachele
中科院分区：
文献类型：
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作者：
Lizabeth V. Rachele
摘要 我们考虑通过在表面进行的测量来独特地确定非均匀各向同性弹性物体的内部特性。 3 维物体通过弹性动力学线性双曲方程组的解进行建模，其（主）系数对应于物体的内部属性（其密度和弹性）。我们通过有限时间间隔内的狄利克雷-诺依曼映射对表面测量进行建模。作者在之前的一篇论文中表明，物体表面的密度和弹性属性是由狄利克雷到诺依曼映射唯一确定的。在这里，我们应用该结果得出结论，物体内部的某些属性（波速）也被确定。然后我们观察到弹性动力极化是由狄利克雷到诺依曼图在物体外部确定的。我们得出结论，在波速恒定的情况下，偏振数据不能确定内部的密度。这个问题及其研究中使用的技术与地震学和医学成像等领域密切相关。不过，这里使用的技术（来自几何光学、积分几何和微局部分析）可以解决这个完全三维的问题。
Abstract We consider the unique determination of internal properties of a nonhomogeneous, isotropic elastic object from measurements made at the surface. The 3-dimensional object is modelled by solutions of the linear hyperbolic system of equations for elastodynamics, whose (leading) coefficients correspond to the internal properties of the object (its density and elasticity). We model surface measurements by the Dirichlet-to-Neumann map on a finite time interval. In a previous paper the author has shown that the density and elastic properties of the surface of the object are uniquely determined by the Dirichlet-to-Neumann map. Here we apply that result to conclude that certain properties of the interior of the object (the wave speeds) are also determined. We then observe that the elastodynamic polarization is determined outside the object by the Dirichlet-to-Neumann map. We conclude, in the case that the wave speeds are constant, that the polarization data do not determine the density in the interior. This problem and techniques used in its study are closely related to those in, for example, seismology and medical imaging. The techniques used here, though (from geometric optics, integral geometry, and microlocal analysis) lead to the solution of this fully three-dimensional problem.