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A Direct Approach for Detection of Bottom Topography in Shallow Water

Lamsahel Noureddine, Carole Rosier

TL;DR

The paper tackles the problem of reconstructing underwater bottom topography from surface measurements in a 1D open-channel flow. It develops a direct inverse approach derived from the shallow-water equations, requiring the free surface and its first two time derivatives at a single time, plus upstream discharge and bed information, and solves for the bed via a second-order finite-difference discretization with a Lipschitz stability guarantee. The method yields analytic solutions in steady flows and demonstrates high accuracy and noise robustness across subcritical, transcritical, and supercritical regimes in extensive numerical tests. This work provides a fast, data-efficient tool for bathymetry estimation with rigorous stability analysis, potentially extendable to deeper-water models and no-stagnation settings.

Abstract

We propose a fast, stable, and direct analytic method to detect underwater channel topography from surface wave measurements, based on one-dimensional shallow water equations. The technique requires knowledge of the free surface and its first two time derivatives at a single instant $t^{\star}$ above the fixed, bounded open segment of the domain. We first restructure the forward shallow water equations to obtain an inverse model in which the bottom profile is the only unknown, and then discretize this model using a second-order finite-difference scheme to infer the floor topography. We demonstrate that the approach satisfies a Lipschitz stability and is independent of the initial conditions of the forward problem. The well-posedness of this inverse model requires that, at the chosen measurement time $t^{\star}$, the discharge be strictly positive across the fixed portion of the open channel, which is automatically satisfied for steady and supercritical flows. For unsteady subcritical and transcritical flows, we derive two empirically validated sufficient conditions ensuring strict positivity after a sufficiently large time. The proposed methodology is tested on a range of scenarios, including classical benchmarks and different types of inlet discharges and bathymetries. We find that this analytic approach yields high approximation accuracy and that the bed profile reconstruction is stable under noise. In addition, the sufficient conditions are met across all tests.

A Direct Approach for Detection of Bottom Topography in Shallow Water

TL;DR

The paper tackles the problem of reconstructing underwater bottom topography from surface measurements in a 1D open-channel flow. It develops a direct inverse approach derived from the shallow-water equations, requiring the free surface and its first two time derivatives at a single time, plus upstream discharge and bed information, and solves for the bed via a second-order finite-difference discretization with a Lipschitz stability guarantee. The method yields analytic solutions in steady flows and demonstrates high accuracy and noise robustness across subcritical, transcritical, and supercritical regimes in extensive numerical tests. This work provides a fast, data-efficient tool for bathymetry estimation with rigorous stability analysis, potentially extendable to deeper-water models and no-stagnation settings.

Abstract

We propose a fast, stable, and direct analytic method to detect underwater channel topography from surface wave measurements, based on one-dimensional shallow water equations. The technique requires knowledge of the free surface and its first two time derivatives at a single instant above the fixed, bounded open segment of the domain. We first restructure the forward shallow water equations to obtain an inverse model in which the bottom profile is the only unknown, and then discretize this model using a second-order finite-difference scheme to infer the floor topography. We demonstrate that the approach satisfies a Lipschitz stability and is independent of the initial conditions of the forward problem. The well-posedness of this inverse model requires that, at the chosen measurement time , the discharge be strictly positive across the fixed portion of the open channel, which is automatically satisfied for steady and supercritical flows. For unsteady subcritical and transcritical flows, we derive two empirically validated sufficient conditions ensuring strict positivity after a sufficiently large time. The proposed methodology is tested on a range of scenarios, including classical benchmarks and different types of inlet discharges and bathymetries. We find that this analytic approach yields high approximation accuracy and that the bed profile reconstruction is stable under noise. In addition, the sufficient conditions are met across all tests.

Paper Structure

This paper contains 15 sections, 3 theorems, 86 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline of the paper
  4. The Forward and the inverse problems
  5. Governing equations
  6. The inverse model
  7. Sufficient Conditions and Stability
  8. Nondegeneracy Sufficient Conditions
  9. Sensitivity analysis
  10. Numerical results
  11. Discretization of the forward system
  12. Discretization of the inverse system
  13. Numerical tests
  14. Conclusions, further comments, and future work
  15. Proof of stability

Key Result

Theorem 3.1

Let $\varepsilon>0$ such that $h(t,x)\geq \varepsilon,\; \forall (t,x)\in(0,t_f)\times I$. Assume that the bed profile $b\in C^1(I)$ and that the fluid flow satisfies the hypothesis subcriticalflow. If there exists $\rho >0$, such that for any $t\in \left(\frac{a_2-a_1}{c_0},t_f\right)$, where $c_0= where $(\gamma_i)_i$ are such that $b'(\gamma_i)=0$ and $\bigcup_{i\geq 1} (\gamma_i,\gamma_{i+1})=

Figures (16)

  • Figure 1: Scheme for the inverse problem
  • Figure 2: Comparison of the forward and reconstructed discharge profiles.
  • Figure 3: Comparison of the analytic bed topography and true bathymetry.
  • Figure 4: Comparison of the reconstructed bed topography and true bathymetry.
  • Figure 5: Comparison of the forward and reconstructed discharge profiles.
  • ...and 11 more figures

Theorems & Definitions (14)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 3.1
  • proof
  • Remark 3.5
  • ...and 4 more