Direct, Nonlinear Inversion Algorithm for Hyperbolic Problems via Projection-Based Model Reduction
Direct, Nonlinear Inversion Algorithm for Hyperbolic Problems via Projection-Based Model Reduction
复制标题
通过基于投影的模型简化解决双曲问题的直接非线性反演算法
DOI：
10.1137/15m1039432
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
M. Zaslavsky
中科院分区：
文献类型：
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作者：
V. Druskin;A. Mamonov;Andrew E. Thaler;M. Zaslavsky
我们通过构造一个与离散时域数据相匹配的降阶模型（ROM），从边界测量中估计声波方程中的波速。ROM的状态变量表示可以等效地看作是时域解的快照所张成的Krylov子空间上的Galerkin投影。我们的算法的成功取决于数据驱动的Gram-施密特正交化的快照，抑制多次反射，可以被看作是一个离散形式的Marchenko-Gel'fand-Levitan-Krein算法。特别是，正交化的快照是本地化的功能，（平方）规范的基本上是加权平均的波速。平方正交快照的质心为我们提供了重建速度的网格。该网格弱依赖于走时坐标中的波速，因此网格点可以通过已知参考速度生成的类似平方正交快照集的质量中心来近似。我们提出了一维和二维合成模型的反演实验结果。
We estimate the wave speed in the acoustic wave equation from boundary measurements by constructing a reduced-order model (ROM) matching discrete time-domain data. The state-variable representation of the ROM can be equivalently viewed as a Galerkin projection onto the Krylov subspace spanned by the snapshots of the time-domain solution. The success of our algorithm hinges on the data-driven Gram--Schmidt orthogonalization of the snapshots that suppresses multiple reflections and can be viewed as a discrete form of the Marchenko--Gel'fand--Levitan--Krein algorithm. In particular, the orthogonalized snapshots are localized functions, the (squared) norms of which are essentially weighted averages of the wave speed. The centers of mass of the squared orthogonalized snapshots provide us with the grid on which we reconstruct the velocity. This grid is weakly dependent on the wave speed in traveltime coordinates, so the grid points may be approximated by the centers of mass of the analogous set of squared orthogonalized snapshots generated by a known reference velocity. We present results of inversion experiments for one- and two-dimensional synthetic models.