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arxiv: 2606.09020 · v1 · pith:T7444IZDnew · submitted 2026-06-08 · 🧮 math-ph · math.MP

On an n-Dimensional Travel Time Tomography Problem

Michael V. Klibanov This is my paper

Pith reviewed 2026-06-27 15:00 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords travel time tomographyeikonal equationCarleman estimateLipschitz stabilityuniqueness theoremsemi-discrete methodinverse problemn-dimensional
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The pith

A Lipschitz stability estimate holds for the semi-discrete n-dimensional travel time tomography problem under a truncated Fourier series assumption, implying uniqueness.

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

The n-dimensional travel time tomography problem with formally determined incomplete data has remained open for stability and uniqueness since its 1D solution over a century ago. This work examines a semi-discrete formulation where the governing eikonal PDE is discretized in finite differences for n-1 variables. It assumes the solution takes the form of a truncated Fourier-like series in an orthonormal basis that depends solely on the source position. Under these conditions, the paper establishes a Lipschitz stability estimate for the unknown speed function. The estimate directly yields uniqueness, achieved through a novel Carleman estimate in weighted spaces.

Core claim

Under the semi-discrete finite-difference formulation of the n-D TTTP and the assumption that the solution is represented by a truncated Fourier-like series in a special orthonormal basis depending only on source position, a Lipschitz stability estimate holds, implying uniqueness. This is proven using a new Carleman estimate.

What carries the argument

New Carleman estimate in Carleman weighted spaces applied to the semi-discrete eikonal equation with truncated series solution representation.

If this is right

  • The Lipschitz stability implies uniqueness of the solution.
  • Numerical methods for the TTTP with formally determined data gain theoretical justification.
  • The approach extends classical 1D results to higher dimensions in the semi-discrete setting.
  • Carleman estimates become applicable to stability questions in travel time tomography.

Where Pith is reading between the lines

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

  • If the series truncation can be controlled, this might inform fully discrete numerical schemes.
  • The basis choice depending only on source position could be tested in specific geometries like layered media.
  • Similar Carleman techniques might apply to other nonlinear inverse problems governed by eikonal-type equations.

Load-bearing premise

The solution of the semi-discrete PDE generated by the eikonal equation must be representable as a truncated Fourier-like series in a special orthonormal basis depending only on the source position.

What would settle it

A numerical or analytical example in which two distinct velocity functions produce identical travel time data in the semi-discrete model, but where the solution does not admit the required truncated series representation, would falsify the stability claim.

read the original abstract

In their seminal works Herglotz (1905) and Wiechert and Zoeppritz (1907) have solved the so-called Travel Time Tomography Problem (TTTP) in the 1-D case. However, the question about stability estimates and uniqueness theorems for an n-D n>= 2 TTTP with formally determined incomplete input data still mostly stands open after more than one hundred years period. \textquotedblleft Formally determined input data" means that the number p of free variables in the input data equals the number $n$ of free variables in the unknown right hand side of the governing nonliniear eikonal PDE, p=n. Some previous publications demonstrate that it is possible to develop well performed numerical methods for the TTTP with formally determined input data, which indicates the importance of such data for practical applications. This is the first publication in which the above question is addressed. More precisely, we consider a semi-discrete case, in which a PDE generated by the eikonal equation is written in finite differences with respect to n-1 variables. In addition, it is assumed that the solution of that semi-discrete PDE is represented via a truncated Fourier-like series with respect to a special orthonormal basis of functions, which depend only on the position of the point source. Under these conditions, Lipschitz stability estimate is proven, and this estimate implies uniqueness. An important tool of this paper is a new Carleman estimate. Carleman Weighted Spaces are introduced. Carleman estimates were not applied previously to address questions about stability estimates and uniqueness theorems for the TTTP.

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

1 major / 0 minor

Summary. The manuscript claims to prove a Lipschitz stability estimate (implying uniqueness) for the n-dimensional travel time tomography problem (TTTP) with formally determined incomplete data. The setting is semi-discrete: the eikonal PDE is discretized via finite differences in n-1 variables, and the solution is assumed to admit an exact representation as a truncated Fourier-like series in a special orthonormal basis whose elements depend only on source position. The proof employs a new Carleman estimate in Carleman-weighted spaces.

Significance. If the central claim holds under the stated assumptions, the result would constitute the first stability/uniqueness theorem for the n-D TTTP (n≥2) beyond the classical 1-D Herglotz–Wiechert–Zoeppritz solution, addressing a long-standing open question. The application of Carleman estimates to TTTP stability is novel, and the semi-discrete reduction with source-position-dependent basis permits formally determined data. The strength of the contribution is limited by the lack of justification for the representation assumption.

major comments (1)
  1. [Abstract] Abstract (paragraph on the semi-discrete case): The Lipschitz stability estimate is derived only after imposing that 'the solution of that semi-discrete PDE is represented via a truncated Fourier-like series with respect to a special orthonormal basis of functions, which depend only on the position of the point source.' No argument is supplied showing that this basis is complete in the natural function space for solutions of the finite-difference eikonal equation, nor is the truncation error controlled with respect to the Carleman weight. Because this representation is used to close the stability argument, the result applies at present only to an artificially restricted subclass of slowness fields.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of the manuscript and for acknowledging the potential significance of the result if the central claim holds. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph on the semi-discrete case): The Lipschitz stability estimate is derived only after imposing that 'the solution of that semi-discrete PDE is represented via a truncated Fourier-like series with respect to a special orthonormal basis of functions, which depend only on the position of the point source.' No argument is supplied showing that this basis is complete in the natural function space for solutions of the finite-difference eikonal equation, nor is the truncation error controlled with respect to the Carleman weight. Because this representation is used to close the stability argument, the result applies at present only to an artificially restricted subclass of slowness fields.

    Authors: We acknowledge that the manuscript imposes the truncated series representation as an explicit modeling assumption on the solutions of the semi-discrete eikonal equation, without providing a proof of completeness of the basis in the natural function space or an estimate on the truncation error relative to the Carleman weight. The Lipschitz stability and uniqueness are established precisely under this assumption, which is introduced to obtain a formally determined problem (equal number of free variables in data and unknown). The assumption is not claimed to hold for arbitrary solutions of the finite-difference eikonal equation; the result therefore applies to the subclass of slowness fields whose associated solutions admit such a representation. We agree that this restricts the scope and will add an explicit remark in the revised abstract and introduction clarifying that the representation is an assumption whose justification (completeness and error control) lies outside the present work. revision: partial

standing simulated objections not resolved
  • Justification of completeness of the special orthonormal basis in the function space of finite-difference eikonal solutions and control of truncation error with respect to the Carleman weight

Circularity Check

0 steps flagged

No circularity: stability estimate derived from new Carleman estimate under explicit ansatz assumption

full rationale

The paper states an explicit assumption that the semi-discrete eikonal solution admits a truncated Fourier-like series representation in a source-position-dependent orthonormal basis, then proves Lipschitz stability (hence uniqueness) for the formally determined incomplete-data TTTP via a new Carleman estimate in Carleman-weighted spaces. No step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation; the derivation is a direct mathematical argument conditional on the stated ansatz and does not invoke prior author results to close the chain. The result is therefore self-contained as a conditional theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on two modeling assumptions that are not derived from prior literature but introduced to make the semi-discrete problem tractable: finite-difference discretization in n-1 variables and representation by a truncated series in a source-dependent orthonormal basis. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The solution of the semi-discrete PDE is represented via a truncated Fourier-like series with respect to a special orthonormal basis of functions depending only on the position of the point source.
    This representation is the key structural assumption that enables the application of the new Carleman estimate (abstract).

pith-pipeline@v0.9.1-grok · 5818 in / 1370 out tokens · 25951 ms · 2026-06-27T15:00:52.969013+00:00 · methodology

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Reference graph

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