In this paper, we consider a problem inspired by the real-world need to identify the topographical features of ocean basins. Specifically, we consider the problem of estimating the bottom impermeable boundary to an inviscid, incompressible, irrotational fluid from measurements of the free-surface deviation alone, within the context of dispersive shallow-water wave models. The need to consider the shallow-water regime arises from the ill-posed nature of the problem and is motivated by prior work. Assuming only a relatively inaccurate initial guess for the bottom-boundary, we design an algorithm to accurately deduce the fluid velocities and the true bottom-boundary profile. We achieve this by considering two separate inverse problems: one to deduce the bottom-boundary from velocities and the surface deviation, and another to recover the velocities from the surface deviation and an approximate bottom-boundary. The former is a classic inverse problem that requires the inversion of an ill-conditioned matrix, whereas the latter employs the observer framework. Combining the two inverse problems leads to our reconstruction algorithm. We emphasize the role played by model selection and its impact on algorithm design and the accuracy of the reconstruction.

中文翻译：

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使用浅水波模型的海洋深度测量

在本文中，我们考虑了一个受现实世界需要确定海洋盆地地形特征的启发的问题。具体而言，我们考虑了在弥散浅水波模型的背景下，仅根据自由表面偏差的测量来估计底部不渗透边界到无粘性、不可压缩、无旋流体的问题。考虑浅水区的需要源于问题的不适定性质，并受到先前工作的推动。假设仅对底部边界的初始猜测相对不准确，我们设计了一种算法来准确推导出流体速度和真实的底部边界轮廓。我们通过考虑两个独立的逆问题来实现这一点：一个是从速度和表面偏差推导出底边界，另一个是从表面偏差和近似底部边界恢复速度。前者是一个经典的逆问题，需要对病态矩阵求逆，而后者使用观察者框架。结合这两个逆问题导致我们的重建算法。我们强调模型选择所起的作用及其对算法设计和重建精度的影响。