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arxiv: 2601.17639 · v2 · submitted 2026-01-25 · 🧮 math.AP · math-ph· math.MP· math.OC

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

Noureddine Lamsahel , Lionel Rosier This is my paper

Pith reviewed 2026-05-16 11:50 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPmath.OC
keywords inverse problemwater wavesbathymetryuniquenesslogarithmic stabilitysize estimatesfluid dynamicsPDE inverse problems
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The pith

The sea bottom shape inside a bounded domain is uniquely determined by free-surface elevation, its time derivative, and velocity potential at one fixed instant, plus boundary bottom data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the bathymetry on any fixed smooth bounded open set O in one or two horizontal dimensions can be recovered uniquely from measurements of the free surface, its first time derivative, and the surface trace of the velocity potential, all taken at a single time t0 inside O, together with knowledge of the bottom along the boundary of O. No extra conditions are imposed for this uniqueness result. Logarithmic stability estimates are obtained by adapting the size-estimates method once a local fatness condition is imposed on the region lying between two candidate bottom profiles. The argument works by restricting the full water-waves system to the bounded subdomain and converting the geometric inverse problem into a controllable PDE setting. A reader would care because the result shows that a single snapshot of surface data, rather than a full time series or global coverage, suffices to identify the underwater topography.

Core claim

We establish uniqueness and derive logarithmic stability estimates in the determination of the bathymetry on any fixed smooth, bounded, open domain O subset R^d, d=1,2, from the knowledge of the free surface, its first time derivative, and the trace of the velocity potential on the free surface, at a given instant t0 within O, together with the knowledge of the bottom along partial O. No further assumptions are required for uniqueness. For stability, we impose only a local fatness condition on the region between the bottom profiles, allowing us to adapt the size estimates method.

What carries the argument

The general water-waves system restricted to the bounded subdomain O, combined with the size-estimates method under a local fatness condition on the region between candidate bottom profiles.

If this is right

  • The bathymetry inside O is fixed uniquely by the given single-time surface data and boundary information.
  • Logarithmic stability guarantees that small perturbations in the surface measurements produce only controlled errors in the recovered bottom shape.
  • The uniqueness result applies directly in both one and two horizontal dimensions without additional restrictions.
  • Only boundary values of the bottom are needed; interior surface data at one time complete the determination.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical experiments could test the method by generating synthetic surface data from known bottoms and checking whether the inversion recovers the profile up to the predicted logarithmic rate.
  • The single-time snapshot approach suggests sensor designs that capture a brief surface measurement rather than continuous monitoring.
  • Similar uniqueness techniques might apply to other free-boundary inverse problems where only partial interior data are available.
  • If the fatness condition can be verified or enforced in practice, the stability result would support reconstruction algorithms for real shallow-water bathymetry mapping.

Load-bearing premise

The local fatness condition on the region between possible bottom profiles must hold for the stability estimates, and the water-waves system must be valid on the bounded subdomain with interior measurements available.

What would settle it

Two distinct smooth bottom profiles inside O that agree on the boundary of O, satisfy the local fatness condition, and produce identical free-surface elevation, time derivative, and velocity-potential traces at the same instant t0 would falsify the stability claim.

Figures

Figures reproduced from arXiv: 2601.17639 by Lionel Rosier, Noureddine Lamsahel.

Figure 1
Figure 1. Figure 1: Scheme for the inverse problem when O = (a1, a2) ⊂ R and d = 1. We assume that the depth of the water is always bounded from below by a positive constant (see (H9) in [28]); that is, there exists a constant H0 > 0 such that ζ0(X) − b0(X) ≥ H0 and ζ(X) − b(X) ≥ H0, ∀X ∈ R d . (7) It was proved in [28] (see also [17, Lemma 2.5]) that system (4), for the chosen time t0, has a unique solution ϕ in H 2 (Ωt0 ). … view at source ↗
Figure 2
Figure 2. Figure 2: Graphs of the functions b0, b, ζ0 and ζ for d = 1 and O = (a1, a2). Theorem 7 ([13], log-log stability). Let Ω be a bounded Lipschitz domain in R d+1 (d ≥ 1) according to Definition 3, let Γ 0 be a nonempty open subset of ∂Ω and let s ∈ (0, 1/2). Then there exist some constants c > e and C > 0 depending on Ω, Γ0, and s such that for any u ∈ H 2 (Ω) with ∆u = 0, u ≠ 0 and ∥u∥L2(Γ0) + ∥∇u∥L2(Γ0) ≤ ∥u∥H2(Ω) 2… view at source ↗
read the original 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, and the trace of the velocity potential on the free surface, at a given instant $t_0$ within $\mathcal O $, together with 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims to prove uniqueness (without further assumptions) and logarithmic stability estimates for recovering the bathymetry b on any fixed smooth bounded open domain O subset R^d (d=1,2) from surface data consisting of the free surface elevation η, its first time derivative, and the trace of the velocity potential φ at a single instant t0 inside O, together with knowledge of b on ∂O. The argument uses the general water-waves system restricted to a bounded fluid subdomain, unique continuation for the velocity potential (harmonic between the known surface and unknown bottom), and an adaptation of the size-estimates method under a local fatness condition on the region between two candidate bottoms.

Significance. If the central claims hold, the work supplies a mathematically rigorous uniqueness and stability result for an inverse problem in water-wave bathymetry detection that requires only single-time surface measurements plus boundary data on ∂O. The adaptation of the size-estimates technique to the water-waves setting under the stated geometric hypothesis is a clear technical contribution; the absence of machine-checked proofs or parameter-free derivations is noted but does not diminish the potential utility for related elliptic inverse problems.

major comments (2)
  1. [§3] §3 (uniqueness proof): the reduction to unique continuation for the velocity potential relies on deriving Neumann-type boundary conditions on the free surface from the given data (∂t η and φ); the manuscript should explicitly verify that these boundary conditions are compatible with the harmonic extension inside the fluid domain when the bottom is unknown, citing the precise trace theorem or regularity result used.
  2. [§4] §4 (stability estimates): the logarithmic modulus of continuity is obtained by adapting the size-estimates method under the local fatness condition; the paper must supply the explicit dependence of the stability constant on the fatness parameter and on the a-priori bounds for the bottoms, because the abstract states that only this geometric hypothesis is imposed.
minor comments (2)
  1. Notation for the fluid domain and the restriction to the subdomain O should be introduced uniformly in the introduction and used consistently in all statements of the main theorems.
  2. The abstract and introduction should clarify whether the water-waves system is assumed to hold exactly on the bounded subdomain or whether an approximation error is controlled.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments, which will improve the clarity and rigor of the manuscript. We address each major comment below and will revise the paper accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (uniqueness proof): the reduction to unique continuation for the velocity potential relies on deriving Neumann-type boundary conditions on the free surface from the given data (∂t η and φ); the manuscript should explicitly verify that these boundary conditions are compatible with the harmonic extension inside the fluid domain when the bottom is unknown, citing the precise trace theorem or regularity result used.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will add a short paragraph in §3 deriving the Neumann boundary condition on the free surface directly from the kinematic condition (involving ∂t η) and the dynamic condition (involving φ), then invoke the standard trace theorem for the Sobolev space H^1(Ω) → H^{1/2}(Γ) (where Ω is the fluid domain and Γ the free surface) to confirm that the resulting boundary data are compatible with the harmonic extension, independently of the unknown bottom. We will cite Grisvard, Elliptic Problems in Nonsmooth Domains, Theorem 1.5.1.3, for the trace result. revision: yes

  2. Referee: [§4] §4 (stability estimates): the logarithmic modulus of continuity is obtained by adapting the size-estimates method under the local fatness condition; the paper must supply the explicit dependence of the stability constant on the fatness parameter and on the a-priori bounds for the bottoms, because the abstract states that only this geometric hypothesis is imposed.

    Authors: We thank the referee for this observation. While the dependence is already encoded in the quantitative unique-continuation constants of the size-estimates argument, we will make it fully explicit in the revised §4. In particular, we will add a remark after the main stability theorem stating that the constant C in the logarithmic estimate depends on the local fatness parameter δ, the a-priori C^{1,α} bound M on the candidate bottoms, the diameter of O, and the fixed time t0. This dependence follows directly from the constants appearing in the Carleman-type estimates used to control the size of the difference of two harmonic functions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes uniqueness via unique continuation for the harmonic velocity potential (satisfying Laplace equation in the fluid domain between known surface η(t0) and unknown bottom b, with Neumann condition on b and data-derived Dirichlet/Neumann conditions on the surface from ∂tη and φ) localized to O, together with known b on ∂O. Logarithmic stability is obtained by adapting the size-estimates method under the local fatness condition on the region between two bottoms; this is a standard geometric hypothesis from the inverse-problems literature and does not reduce to a self-definition, fitted parameter, or self-citation chain. No step equates a prediction to its input by construction, and the central claims rest on standard elliptic PDE theory rather than any load-bearing self-reference. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the general water-waves system as the governing model and the local fatness condition as an assumption needed only for stability; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The general water-waves system holds on a bounded subdomain of the fluid domain
    Invoked as the forward model for the fluid dynamics throughout the analysis.
  • standard math The domain O is smooth and bounded and the bottom profiles satisfy sufficient regularity
    Required for the elliptic estimates and size estimates method to apply.

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Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    R. J. Adrian and R. J. Westerweel.Particle Image Velocimetry. Cambridge University Press, Cambridge, UK, 2011. 23

    work page 2011

  2. [2]

    Alazard, N

    T. Alazard, N. Burq, and C. Zuily.Cauchy theory for the gravity water waves system with non-localized initial data. Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 33(2):337–395, 2016

    work page 2016

  3. [3]

    Alessandrini, E

    G. Alessandrini, E. Rosset, and J. K. Seo.Optimal size estimates for the inverse conductivity problem with one measurement. Proceedings of the American Mathematical Society, 128(1):53–64, 2000

    work page 2000

  4. [4]

    Alessandrini, A

    G. Alessandrini, A. Morassi, and E. Rosset.Detecting cavities by electrostatic boundary measurements. Inverse Problems, 18(5):1333, 2002

    work page 2002

  5. [5]

    Alessandrini, A

    G. Alessandrini, A. Morassi, and E. Rosset.Size estimates. In G. Alessandrini and G. Uhlmann, editors,In- verse Problems: Theory and Applications, volume 333 ofContemporary Mathematics, pages 1–33. American Mathematical Society, Providence, RI, 2003

    work page 2003

  6. [6]

    Angel, J

    J. Angel, J. Behrens, S. G¨ otschel, M. Hollm, D. Ruprecht and R. Seifried.Bathymetry reconstruction from experimental data using PDE-constrained optimisation. Computers & Fluids, 278:106321, 2024

    work page 2024

  7. [7]

    T. B. Benjamin and J. C. Scott.Gravity-capillary waves with edge constraints. Journal of Fluid Mechanics, 92(2):241–267, 1979

    work page 1979

  8. [8]

    Beretta, C

    E. Beretta, C. Cavaterra, J. H. Ortega, and S. Zamorano.Size estimates of an obstacle in a stationary Stokes fluid. Inverse Problems, 33(2):025008, 2017

    work page 2017

  9. [9]

    V. I. Bogachev.Measure Theory, volume 1. Springer, Berlin, 2007

    work page 2007

  10. [10]

    Rosset and G

    E. Rosset and G. Alessandrini.The inverse conductivity problem with one measurement: bounds on the size of the unknown object. SIAM Journal on Applied Mathematics, 58(4):1060–1071, 1998

    work page 1998

  11. [11]

    Bourgeois and J

    L. Bourgeois and J. Dard´ e.About stability and regularization of ill-posed elliptic Cauchy problems: the case of Lipschitz domains. Applicable Analysis, 89(11):1745–1768, 2010

    work page 2010

  12. [12]

    Brocchini.A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics

    M. Brocchini.A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 469(2160):20130496, 2013

    work page 2013

  13. [13]

    Choulli.New global logarithmic stability results on the Cauchy problem for elliptic equations

    M. Choulli.New global logarithmic stability results on the Cauchy problem for elliptic equations. Bulletin of the Australian Mathematical Society, 101(1):141–145, 2020

    work page 2020

  14. [14]

    Craig and C

    W. Craig and C. S. Sulem.Numerical simulation of gravity waves. Journal of Computational Physics, 108(1):73–83, 1993

    work page 1993

  15. [15]

    J. A. Cunge, F. M. Holly, and A. Verwey.Practical Aspects of Computational River Hydraulics. Pitman Publishing, London, 1980

    work page 1980

  16. [16]

    Ding.A proof of the trace theorem of Sobolev spaces on Lipschitz domains

    Z. Ding.A proof of the trace theorem of Sobolev spaces on Lipschitz domains. Proceedings of the American Mathematical Society, 124(2):591–600, 1996

    work page 1996

  17. [17]

    M. A. Fontelos, R. Lecaros, J. C. L´ opez-R´ ıos, and J. H. Ortega.Bottom detection through surface measure- ments on water waves. SIAM Journal on Control and Optimization, 55(6):3890–3907, 2017

    work page 2017

  18. [18]

    A. F. Gessese, M. Sellier, E. Van Houten and G. Smart.Reconstruction of river bed topography from free surface data using a direct numerical approach in one-dimensional shallow water flow. Inverse Problems, 27(2):025001, 2011

    work page 2011

  19. [19]

    S. T. Grilli, R. Subramanya, I. A. Svendsen, and J. Veeramony.Shoaling of solitary waves on plane beaches. Journal of Waterway, Port, Coastal, and Ocean Engineering, 120(6):609–628, 1994

    work page 1994

  20. [20]

    Grisvard.Elliptic Problems in Nonsmooth Domains

    P. Grisvard.Elliptic Problems in Nonsmooth Domains. Volume 69, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2011

    work page 2011

  21. [21]

    Henrot and M

    A. Henrot and M. Pierre.Variation et optimisation de formes: une analyse g´ eom´ etrique. Volume 48, Springer, New York, 2005

    work page 2005

  22. [22]

    R. C. Hilldale and D. Raff.Assessing the ability of airborne LiDAR to map river bathymetry. Earth Surface Processes and Landforms, 33(5):773–783, 2008. 24

    work page 2008

  23. [23]

    J. L. Irish and W. J. Lillycrop.Scanning laser mapping of the coastal zone: the SHOALS system. ISPRS Journal of Photogrammetry and Remote Sensing, 54(2-3):123–129, 1999

    work page 1999

  24. [24]

    H. Kang, J. K. Seo, and D. Sheen.The inverse conductivity problem with one measurement: stability and estimation of size. SIAM Journal on Mathematical Analysis, 28(6):1389–1405, 1997

    work page 1997

  25. [25]

    C. E. Kenig.Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, volume 83 American Mathematical Society, Providence, RI, 1994

    work page 1994

  26. [26]

    Lamsahel, C

    N. Lamsahel, C. Manni, A. Ratnani, S. Serra-Capizzano, and H. Speleers,Outlier-free isogeometric dis- cretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions. Nu- merische Mathematik, 157(4): 1397–1448, 2025

    work page 2025

  27. [27]

    Lamsahel, C

    N. Lamsahel, C. Rosier.A Direct Approach for Detection of Bottom Topography in Shallow Water. arXiv:2510.03505, 2025

    work page arXiv 2025

  28. [28]

    Lannes.The water-waves problem: mathematical analysis and asymptotics, volume 188

    D. Lannes.The water-waves problem: mathematical analysis and asymptotics, volume 188. American Mathematical Society, Providence RI, 2013

    work page 2013

  29. [29]

    Lecours, M

    V. Lecours, M. F. J. Dolan, A. Micallef, and V. L. Lucieer.A review of marine geomorphometry: the quantitative study of the seafloor. Hydrology and Earth System Sciences, 20(8):3207–3244, 2016

    work page 2016

  30. [30]

    Lecaros, J

    R. Lecaros, J. L´ opez-R´ ıos, J. H. Ortega, and S. Zamorano.The stability for an inverse problem of bottom recovering in water-waves. Inverse Problems, 36(11):115002, 2020

    work page 2020

  31. [31]

    Marks and P

    K. Marks and P. Bates.Integration of high-resolution topographic data with floodplain flow models. Hydro- logical Processes, 14(11-12):2109–2122, 2000

    work page 2000

  32. [32]

    Montardini, S

    M. Montardini, S. Takacs, M.Tani.The Isogeometric Fast Fourier-based Diagonalization method. arXiv:2512.20269 , 2025

    work page arXiv 2025

  33. [33]

    Morassi, E

    A. Morassi, E. Rosset, and S. Vessella.Size estimates for inclusions in an elastic plate by boundary mea- surements. Indiana University Mathematics Journal, 56(5):2325–2384, 2007

    work page 2007

  34. [34]

    Neˇ cas.Direct Methods in the Theory of Elliptic Equations

    J. Neˇ cas.Direct Methods in the Theory of Elliptic Equations. Springer, Berlin, 2011

    work page 2011

  35. [35]

    D. P. Nicholls and M. Taber.Joint analyticity and analytic continuation of Dirichlet–Neumann operators on doubly perturbed domains. Journal of Mathematical Fluid Mechanics, 10(2):238–271, 2008

    work page 2008

  36. [36]

    D. P. Nicholls and M. Taber.Detection of ocean bathymetry from surface wave measurements. European Journal of Mechanics – B/Fluids, 28(2):224–233, 2009

    work page 2009

  37. [37]

    Sagawa, Y

    T. Sagawa, Y. Yamashita, T. Okumura, and T. Yamanokuchi.Satellite-derived bathymetry using machine learning and multi-temporal satellite images. Remote Sensing, 11(10):1155, 2019

    work page 2019

  38. [38]

    Sellier.Inverse problems in free surface flows: a review

    M. Sellier.Inverse problems in free surface flows: a review. Acta Mechanica, 227(3):913–935, 2016

    work page 2016

  39. [39]

    G. M. Smart, J. Bind, and M. J. Duncan.River bathymetry from conventional LiDAR using water surface returns. InProceedings of the 18th World IMACS / MODSIM Congress, 2009

    work page 2009

  40. [40]

    W. H. F. Smith and D. T. Sandwell.Conventional bathymetry, bathymetry from space, and geodetic altimetry. Oceanography, 17(1):8–23, 2004

    work page 2004

  41. [41]

    C. E. Synolakis.The run-up of solitary waves. Journal of Fluid Mechanics, 185:523–545, 1987

    work page 1987

  42. [42]

    C. E. Synolakis.Green’s law and the evolution of solitary waves. Physics of Fluids A: Fluid Dynamics, 3(3):490–491, 1991

    work page 1991

  43. [43]

    Vasan and B

    V. Vasan and B. Deconinck.The inverse water wave problem of bathymetry detection. Journal of Fluid Mechanics, 714:562–590, 2013

    work page 2013

  44. [44]

    V. E. Zakharov.Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9(2):190–194, 1968. 25

    work page 1968