Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness and stability in bottom detection through surface measurements of water waves

Noureddine Lamsahel, Lionel Rosier

TL;DR

This work studies the problem of recovering a bounded portion of the seafloor ${b}$ from surface measurements within a truncated domain using the general water-waves system. The authors connect bathymetry to surface data via the Dirichlet-to-Neumann operator ${G(\zeta,b)}$ and prove that the bottom geometry is locally identifiable from time-${t_0}$ surface measurements together with ${b}$ on ${\partial\mathcal{O}}$. They establish $L^1$-log–log and logarithmic stability under a relaxed local fatness condition using a size-estimates framework, extending previous unbounded-domain results to a bounded, Lipschitz setting and removing several restrictive assumptions. These results enable robust bathymetry detection from surface data and pave the way for optimization-based and numerically efficient bathymetry reconstruction methods on truncated domains.

Abstract

This paper investigates the geometric inverse problem of recovering the bottom shape from surface measurements of water waves. Using the general water-waves system on a bounded subdomain of the fluid domain, we address this inverse problem, focusing on the identifiability and the stability issues. We establish uniqueness and derive logarithmic stability estimates in the determination of the bathymetry on any fixed smooth, bounded, open domain ${\mathcal O}\subset {\mathbb R} ^d$, $d=1,2$, from the knowledge of the free surface, its first time derivative, the velocity potential at a given instant $t_0$ within $\mathcal O $, and the knowledge of the bottom along $\partial \mathcal O$. No further assumptions are required for uniqueness. For stability, we impose only a \textit{local fatness} condition on the region between the bottom profiles, allowing us to adapt the size estimates method.

Uniqueness and stability in bottom detection through surface measurements of water waves

TL;DR

This work studies the problem of recovering a bounded portion of the seafloor from surface measurements within a truncated domain using the general water-waves system. The authors connect bathymetry to surface data via the Dirichlet-to-Neumann operator and prove that the bottom geometry is locally identifiable from time- surface measurements together with on . They establish -log–log and logarithmic stability under a relaxed local fatness condition using a size-estimates framework, extending previous unbounded-domain results to a bounded, Lipschitz setting and removing several restrictive assumptions. These results enable robust bathymetry detection from surface data and pave the way for optimization-based and numerically efficient bathymetry reconstruction methods on truncated domains.

Abstract

This paper investigates the geometric inverse problem of recovering the bottom shape from surface measurements of water waves. Using the general water-waves system on a bounded subdomain of the fluid domain, we address this inverse problem, focusing on the identifiability and the stability issues. We establish uniqueness and derive logarithmic stability estimates in the determination of the bathymetry on any fixed smooth, bounded, open domain , , from the knowledge of the free surface, its first time derivative, the velocity potential at a given instant within , and the knowledge of the bottom along . No further assumptions are required for uniqueness. For stability, we impose only a \textit{local fatness} condition on the region between the bottom profiles, allowing us to adapt the size estimates method.
Paper Structure (11 sections, 10 theorems, 147 equations, 2 figures)

This paper contains 11 sections, 10 theorems, 147 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the paper
  4. Problem setting and preliminaries
  5. The geometric inverse problem
  6. Preliminaries on Elliptic problems
  7. Uniqueness
  8. Stability estimates
  9. Conclusions, further comments, and future work
  10. Proof of Lemma \ref{['uppergradfortheupper']}
  11. Proof of Proposition \ref{['lemmaaboutupperboundofinte']}

Key Result

Theorem 5

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^{d+1}$ ($d\ge 1$) with constants $r_0$ and $M_0$ according to Definition lipsdef, and let $u\in H^{1}(\Omega)$ be a solution of (Nuemannproblem) such that $v \in L^2(\partial\Omega)$. Then $u\in H^1(\partial \Omega)$ and there exist two posit where $\mu_{d+1}$ denotes the Lebesgue measure in $\mathbb R ^{d+1}$.

Figures (2)

  • Figure 1: Scheme for the inverse problem when $\mathcal{O}=(a_1,a_2)\subset \mathbb R$ and $d=1$.
  • Figure 2: Graphs of the functions $b_0$, $b$, $\zeta _0$ and $\zeta$ for $d=1$ and $\mathcal{O}=(a_1,a_2)$.

Theorems & Definitions (26)

  • Remark 1
  • Remark 2
  • Definition 3: alessandrini2003size, Lipschitz regularity
  • Theorem 5
  • Theorem 6: alessandrini2000optimal, Lipschitz propagation of smallness
  • Theorem 7: choulli2020new, log-log stability
  • Remark 8
  • Lemma 9
  • proof
  • Proposition 10
  • ...and 16 more