In this article we deal with a class of geometric inverse problem for bottom detection by one single measurement on the free surface in water--waves. We found upper and lower bounds for the size of the region enclosed between two different bottoms, in terms of Neumann and/or Dirichlet data on the free surface. Starting from the general water--waves system in bounded domains with side walls, we manage to formulate the problem in terms of the Dirichlet to Neumann operator and thus, as an elliptic problem in a bounded domain with Neumann homogeneous condition on the rigid boundary. Then we study the properties of the Dirichlet to Neumann map and analyze the called method of size estimation.

中文翻译：

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水波底恢复反问题的稳定性

在本文中，我们通过在水的自由表面上的单次测量来处理一类用于底部检测的几何逆问题——波。根据自由表面上的诺依曼和/或狄利克雷数据，我们找到了包围在两个不同底部之间的区域大小的上限和下限。从具有侧壁的有界域中的一般水波系统开始，我们设法根据 Dirichlet 到 Neumann 算子将问题公式化，因此，作为刚性边界上具有 Neumann 齐次条件的有界域中的椭圆问题。然后我们研究了Dirichlet 到Neumann 映射的性质并分析了称为尺寸估计的方法。