Ocean‐depth measurement using shallow‐water wave models. Issue 4 (21st June 2021)

• Record Type: Journal Article
• Authors: Vasan, V.; Manisha, ; Auroux, D.
• Is Part Of: Studies in applied mathematics. Volume 147:Issue 4(2021)
• Page Start: 1481
• Page End: 1518
• Publication Date: 2021-06-21
• Subjects: bathymetry -- dispersive equations -- inverse problem -- observers -- water waves; Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12418
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract: 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.