Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T06:34:18.429Z Has data issue: false hasContentIssue false
  • English
  • Français

The inverse water wave problem of bathymetry detection

Published online by Cambridge University Press:  02 January 2013

Vishal Vasan  and
Bernard Deconinck
Vishal Vasan*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Bernard Deconinck
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The inverse water wave problem of bathymetry detection is the problem of deducing the bottom topography of the seabed from measurements of the water wave surface. In this paper, we present a fully nonlinear method to address this problem in the context of the Euler equations for inviscid irrotational fluid flow with no further approximation. Given the water wave height and its first two time derivatives, we demonstrate that the bottom topography may be reconstructed from the numerical solution of a set of two coupled non-local equations. Owing to the presence of growing hyperbolic functions in these equations, their numerical solution is increasingly difficult if the length scales involved are such that the water is sufficiently deep. This reflects the ill-posed nature of the inverse problem. A new method for the solution of the forward problem of determining the water wave surface at any time, given the bathymetry, is also presented.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 714 , 10 January 2013 , pp. 562 - 590
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ablowitz, M. J., Fokas, A. S. & Musslimani, Z. H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.CrossRefGoogle Scholar
Ablowitz, M. J. & Haut, T. S. 2008 Spectral formulation of the two fluid Euler equations with a free interface and long wave reductions. Anal. Appl. 6, 323–348.Google Scholar
Collins, M. D. & Kuperman, W. A. 1994 Inverse problems in ocean acoustics. Inverse Problems 10, 1023–1040.Google Scholar
Craig, W., Guyenne, P., Nicholls, D. P. & Sulem, C. 2005 Hamiltonian long-wave expansions for water waves over a rough bottom. Proc. R. Soc. A 461, 839–873.Google Scholar
Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83.Google Scholar
Deconinck, B. & Oliveras, K. 2011 The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167.Google Scholar
Evans, L. 1998 Partial Differential Equations. American Mathematical Society.Google Scholar
Grilli, S 1998 Depth inversion in shallow water based on nonlinear properties of shoaling periodic waves. Coast. Engng 35, 185–209.CrossRefGoogle Scholar
Guenther, R. B. & Lee, J. W. 1996 Partial Differential Equations of Mathematical Physics and Integral Equations. Dover.Google Scholar
Lannes, D. 2005 Well-posedness of the water-wave equations. J. Am. Math. Soc. 18, 605–654.Google Scholar
Nicholls, D. P. & Taber, M. 2008 Joint analyticity and analytic continuation of Dirichlet–Neumann operators on doubly perturbed domains. J. Math. Fluid Mech. 10, 238–271.CrossRefGoogle Scholar
Paley, R. C. & Wiener, N. 1934 Fourier Transforms in the Complex Domain, Colloquium Publications , vol. 19. American Mathematical Society.Google Scholar
Piotrowski, C. & Dugan, J. 2002 Accuracy of bathymetry and current retrievals from air-borne optical time-series imaging of shoaling waves. IEEE Trans. Geosci. Remote Sens. 40, 2606–2618.CrossRefGoogle Scholar
Taroudakis, M. I. & Makrakis, G. 2001 Inverse Problems in Underwater Acoustics. Springer.CrossRefGoogle Scholar
Wu, S. 2011 Global wellposedness of the 3-D full water wave problem. Invent. Math. 184, 125–220.Google Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Tekh. Fiz. 8, 86–94.Google Scholar