Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal reconstruction of water-waves from noisy pressure measurements at the seabed

Joris Labarbe, Alexandre Vieira, Didier Clamond

Abstract

We consider the problem of recovering the surface wave profile from noisy bottom pressure measurements with (\textit{a priori} unknown) arbitrary pressure at the surface. Without noise, the direct approach developed in \cite{clamond2023steady} provides an effective way to recover the sea surface. However, the assumption of analyticity for the measurement renders this method inefficient in the presence of noise. Therefore, we introduce an optimisation procedure based on the minimisation of a distance between a recovered bottom pressure and its measurement. Such method proves to be well-designed to handle perturbed signals. We illustrate the effectiveness of this approach in the recovery of gravity-capillary waves from unfiltered noisy data.

Optimal reconstruction of water-waves from noisy pressure measurements at the seabed

Abstract

We consider the problem of recovering the surface wave profile from noisy bottom pressure measurements with (\textit{a priori} unknown) arbitrary pressure at the surface. Without noise, the direct approach developed in \cite{clamond2023steady} provides an effective way to recover the sea surface. However, the assumption of analyticity for the measurement renders this method inefficient in the presence of noise. Therefore, we introduce an optimisation procedure based on the minimisation of a distance between a recovered bottom pressure and its measurement. Such method proves to be well-designed to handle perturbed signals. We illustrate the effectiveness of this approach in the recovery of gravity-capillary waves from unfiltered noisy data.
Paper Structure (16 sections, 24 equations, 4 figures, 1 table)

This paper contains 16 sections, 24 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Numerical procedure: direct approach
  4. Cauchy integral formula
  5. Expressions for computing steady surface waves and bottom pressure
  6. Holomorphic functions and recovery formula
  7. The Levenberg--Marquardt procedure
  8. Numerical procedure: optimisation approach
  9. An optimisation formulation
  10. The augmented Lagrangian approach
  11. Numerical experiments
  12. Recovery of capillary waves
  13. Addition of noise in the measurement
  14. Discussion
  15. Comments on the noise addition
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Sketch of the procedure used for computing the bottom pressure from surface information.
  • Figure 2: Relative errors versus the number of Fourier modes $N$ for (a): $B$, (b): $\eta$ and (c): ${p}_\mathrm{s}$. We give the running time (in hours) in panel (d). The circled red lines correspond to the direct approach. The crossed blue (squared magenta) lines represent the solutions from the optimisation method with a low (high) limit on the number of function evaluations.
  • Figure 3: (a): Bottom pressure of reference, computed from expression (\ref{['pb']}). The dashed-dotted black line represents the hydrostatic value $gd$. (b,c): Recovered surface pressure and surface elevation from direct approach (red crosses) and optimisation approach (blue circles). The reference values are displayed in dark green lines. All computations were made with $N=512$.
  • Figure 4: (a): Pressure at the seabed. Ground truth value (computed from (\ref{['pb']})) is depicted in blue line. The signal with $5\%$ added noise is represented in dotted red line. (b,c): Recovered surface pressure and surface elevation without (with) knowledge of the physics at the surface are represented in dotted (dashed) red lines. The reference values are displayed with blue lines. All computations were made with $N=128$.