The inverse water wave problem of bathymetry detection

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.

A three-dimensional Dirichlet-to-Neumann operator for water waves over topography

“…The authors studied the possibility of reconstructing the sea-bed topography based on surface-wave measurements. Our version of the AFM * formulation is linear and in three dimensions, as opposed to that of Vasan & Deconinck [27] which was nonlinear and in two dimensions. Our goal is to take one step forward by modeling complex (not necessarily smooth) topographies that involve large and rapid variations and, as mentioned, perform simulations with fully three dimensional configurations.…”

“…As mentioned in the introduction, this non-local formulation was proposed in the work of Milewski [22], and more recently by Craig et al [9] and Ablowitz et al [1]. The work of Vasan & Deconinck [27] was one of the first articles in which the AFM formulation was used with bathymetry variations. The authors studied the possibility of reconstructing the sea-bed topography based on surface-wave measurements.…”

“…In this representation formula (33), the complex coefficients X k have to be determined from the bottom boundary condition (25). Notice that if X k vanished for all k, then we would recover (27), the solution of the flat bottom case. Therefore X k contains information related to the geometry of the topography.…”

“…In most cases the typical number of Fourier modes is about 12 for the bottom and around 60 for the wave. As discussed in detail in Vasan & Deconinck [27], the use of a large number of Fourier modes can become an issue, in particular in the presence of a topography and of moderate values of µ.…”

Comparison of Five Methods of Computing the Dirichlet–Neumann Operator for the Water Wave Problem

“…These four methods are the operator expansion method of Craig & Sulem (CS) [10], the transformed field expansion method (TFE) of Bruno & Reitich [7] and Nicholls & Reitich [23,24], the nonlocal implicit formulation of the DNO given by Ablowitz, Fokas & Musslimani (AFM) [2], and a dual version to the AFM method, derived by Ablowitz & Haut [1], which we denote by AFM * . Each of these methods has had remarkable theoretical utility in deriving reduced models for water waves in various physical regimes [11], in deriving conserved quantities [2], and also in providing the theoretical framework to pose some inverse problems [25,28]. Additionally, each of these methods readily generalizes to the case of both varying bottom boundaries and three dimensional fluids.…”