Focusing Waves in Unknown Media by Modified Time Reversal Iteration
Focusing Waves in Unknown Media by Modified Time Reversal Iteration
复制标题
通过改进的时间反转迭代聚焦未知介质中的波
DOI：
10.1137/070705192
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发表时间：
2009
影响因子：
2.2
通讯作者：
Dahl M
中科院分区：
文献类型：
--
作者：
Dahl M
研究了紧致黎曼流形上有界区域Minor上的波动方程.假设我们不知道波动方程的系数，而只给出双曲罗宾-狄利克雷映射，它对应于边界上的一部分物理测量。在本文中，我们表明，在一个固定的时间波可以被切断以外的一个适当的集合。也就是说，如果是一个球在旅行时间度量的联盟，其中心在边界处，那么我们可以修改波的给定Robin边界值，使得在时间处，修改后的波在N内任意接近原始波，在N外任意小。此外，在时间的修改波的时间导数是任意小的所有M。我们应用这个结果来构造一个Robin边界值序列，使得在某个时刻，相应的波收敛到Delta分布，而波的时间导数收敛到零。这样的边界值是通过测量的迭代序列产生的。在每一步迭代中，我们应用时间反演和其他简单的算子来测量数据，并计算下一步迭代的边界值。这一结果的一个关键特征是，它不需要知道波动方程中的系数，即介质内部的材料参数。然而，我们假设波的焦点在走时坐标中是已知的，并且满足一定的几何条件。
We study the wave equation on a bounded domainMinor on a compact Riemannian manifold with boundary. Assume that we do not know the coefficients of the wave equation but are given only the hyperbolic Robin-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. In this paper we show that at a fixed timea wave can be cut off outside a suitable set. That is, ifis a union of balls in the travel time metric having centers at the boundary, then we can modify a given Robin boundary value of a wave such that at timethe modified wave is arbitrarily close to the original wave insideNand arbitrarily small outsideN. Also, at timethe time derivative of the modified wave is arbitrarily small in all ofM. We apply this result to construct a sequence of Robin boundary values so that at a timethe corresponding waves converge to a delta distributionwhile the time derivative of the waves converge to zero. Such boundary values are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the pointwhere the wave focuses is known in travel time coordinates, andsatisfies a certain geometrical condition.
影响因子：
2.1
作者：
M. Klibanov;A. Timonov
通讯作者：
A. Timonov