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arxiv: 2510.16413 · v2 · submitted 2025-10-18 · 🧮 math.NA · cs.NA

A multilayer level-set method for eikonal-based traveltime tomography

Wenbin Li , Ken K.T. Hung , Shingyu Leung This is my paper

Pith reviewed 2026-05-18 06:30 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords multilayer level-set methodeikonal equationtraveltime tomographyfirst-arrival traveltimesdiscontinuous slownessinverse problemsregularizationadjoint state method
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The pith

A single level-set function with multiple i_n-level sets represents arbitrarily many interfaces for eikonal traveltime tomography.

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

The paper develops a multilayer level-set method to invert first-arrival traveltimes for slowness models that contain many discontinuities. Instead of tracking interfaces separately, the approach uses one function whose values at a sequence of i_n thresholds each behave locally like a signed-distance function. This lets the same formulation describe any number of phases and subregions inside an Eulerian grid. Traveltimes are computed from the eikonal equation as viscosity solutions, sensitivities come from the adjoint-state method, and three regularizers keep the inversion stable. Experiments recover complex discontinuous models with multiple interfaces.

Core claim

The multilayer level-set method represents multiple phases through a sequence of i_n-level sets, with the function designed to behave like a local signed-distance function near each i_n-level set. This single formulation captures arbitrarily many interfaces and subregions. First-arrival traveltimes are obtained as viscosity solutions of the eikonal equation, Fréchet derivatives of the misfit are computed by the adjoint state method, and the inversion is stabilized by multilayer reinitialization, arc-length penalization, and Sobolev smoothing.

What carries the argument

Multilayer level-set representation in which one function uses a sequence of i_n-level sets, each locally resembling a signed-distance function.

If this is right

  • Complex discontinuous slowness models containing multiple phases and interfaces can be recovered from first-arrival traveltimes without explicit interface tracking.
  • First-arrival traveltimes are obtained inside the Eulerian framework as viscosity solutions of the eikonal equation.
  • Fréchet derivatives needed for the inversion are supplied by the adjoint-state method.
  • Multilayer reinitialization combined with arc-length penalization and Sobolev smoothing keeps the reconstructed interfaces sharp and the inversion stable.
  • An illumination-based error measure supplies an additional diagnostic for reconstruction quality.

Where Pith is reading between the lines

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

  • The same single-function representation could be reused for other interface-rich inverse problems once the eikonal solver and adjoint are replaced.
  • The illumination-based diagnostic might be combined with standard misfit norms to decide when to stop iterations or adapt regularization weights.
  • Because the method stays Eulerian, it may extend directly to three-dimensional domains or to problems that include multiple wave types.

Load-bearing premise

Near each i_n-level set the function behaves like a local signed-distance function.

What would settle it

A controlled numerical experiment on a known multi-interface slowness model in which the recovered interfaces remain visibly offset or merged after the proposed regularizations are applied.

Figures

Figures reproduced from arXiv: 2510.16413 by Ken K.T. Hung, Shingyu Leung, Wenbin Li.

Figure 1
Figure 1. Figure 1: Two examples of the multilayer level-set function. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Section 2.4.1 Test 1) The inner circle expands while the outer circle shrinks. The average [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Section 2.4.1 Test 2) The upper-right curve evolves according to motion in the normal direction, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (Section 2.4.1 Test 2) The upper-right curve evolves according to motion in the normal direction, [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (Section 2.4.2, Test 1) Time evolution of the mean radii of two circles, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Section 2.4.2, Test 2) The outer curve evolves according to mean curvature, while the inner [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Initial guess for Examples 1, 2 and 3. (a) Initial guess of the multilayer level-set function; (b) [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example 1: true model and recovered solution. The initial guess is shown in Figure 7. (a) [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Example 2: true model and recovered solution. The initial guess is shown in Figure 7. (a) [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example 3: true model Strue. 4.1.3. Example 3 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Example 3: recovered solutions with γϕ = 0 and γϕ = 0.01, respectively. The initial guess is shown in [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Initial guess for Example 4. (a) Initial guess of the multilayer level-set function; (b) initial [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example 4: true model and recovered solution. The initial guess is shown in Figure 12. (a) [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Initial guess for Example 5, Example 6, and Example 7. (a) Initial guess of the multilayer [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Example 5: recovering both domains and slowness parameters; true model and recovered [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Example 6: recovering both domains and slowness parameters; true model and recovered [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 14
Figure 14. Figure 14: The penalization on ϕ with a weight of γϕ = 0.01 is applied in this example [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 17
Figure 17. Figure 17: Example 7: recovering both domains and slowness parameters. The initial guess is shown in [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
read the original abstract

We present a novel multilayer level-set method (MLSM) for eikonal-based first-arrival traveltime tomography. Unlike classical level-set approaches that rely solely on the zero-level set, the MLSM represents multiple phases through a sequence of $i_n$-level sets ($n = 0, 1, 2, \cdots$). Near each $i_n$-level set, the function is designed to behave like a local signed-distance function, enabling a single level-set formulation to capture arbitrarily many interfaces and subregions. Within this Eulerian framework, first-arrival traveltimes are computed as viscosity solutions of the eikonal equation, and Fr\'{e}chet derivatives of the misfit are obtained via the adjoint state method. To stabilize the inversion, we incorporate several regularization strategies, including multilayer reinitialization, arc-length penalization, and Sobolev smoothing of model parameters. In addition, we introduce an illumination-based error measure to assess reconstruction quality. Numerical experiments demonstrate that the proposed MLSM efficiently recovers complex discontinuous slowness models with multiple phases and interfaces.

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 manuscript proposes a multilayer level-set method (MLSM) for eikonal-based first-arrival traveltime tomography. It represents multiple phases and interfaces via a sequence of i_n-level sets within a single level-set function that is designed to behave locally as a signed-distance function near each level. First-arrival traveltimes are obtained as viscosity solutions of the eikonal equation, Fréchet derivatives via the adjoint-state method, and inversion is stabilized by multilayer reinitialization, arc-length penalization, and Sobolev smoothing. An illumination-based error measure is introduced, and numerical experiments are presented to illustrate recovery of complex discontinuous slowness models with multiple phases and interfaces.

Significance. If the local signed-distance property is reliably maintained, the MLSM would provide a compact Eulerian representation for models containing arbitrarily many interfaces, which is a useful advance over classical single-interface level-set formulations in traveltime tomography. The reliance on established eikonal solvers and adjoint-state derivatives is a methodological strength, and the illumination-based error measure offers a practical diagnostic. The overall significance hinges on whether the numerical results remain robust when the reinitialization is applied across multiple levels during adjoint-driven updates.

major comments (2)
  1. [§3] §3 (Multilayer reinitialization): The procedure is stated to enforce local signed-distance behavior near each i_n-level set, yet no analysis or diagnostic is supplied showing that |∇φ| remains close to 1 simultaneously at all levels after each adjoint update. Because the central claim that a single formulation captures arbitrarily many interfaces rests on this property, its preservation must be demonstrated explicitly.
  2. [§5.2] §5.2 (Numerical experiments, complex model test): The illumination-based error measure reports low misfit in illuminated zones, but the experiments do not include a quantitative check (e.g., max | |∇φ| − 1 | near each i_n) after convergence. Without this, it is unclear whether interface locations remain accurate when multiple phases are present.
minor comments (2)
  1. [§2] The notation i_n is introduced without an explicit equation defining the level values; adding a short definition and a schematic in §2 would improve readability.
  2. [Figure 4] Figure captions for the reconstruction results should state the regularization weights used, as these are free parameters listed in the method.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of the multilayer reinitialization and its verification. We address each major comment below and will revise the manuscript accordingly to strengthen the supporting evidence for the local signed-distance property.

read point-by-point responses

  1. Referee: [§3] §3 (Multilayer reinitialization): The procedure is stated to enforce local signed-distance behavior near each i_n-level set, yet no analysis or diagnostic is supplied showing that |∇φ| remains close to 1 simultaneously at all levels after each adjoint update. Because the central claim that a single formulation captures arbitrarily many interfaces rests on this property, its preservation must be demonstrated explicitly.

    Authors: We agree that explicit verification of the preservation of |∇φ| ≈ 1 near all i_n-level sets after adjoint updates is necessary to substantiate the claim of representing arbitrarily many interfaces with a single function. The multilayer reinitialization is constructed to act locally and sequentially around each level set, which we expect to maintain the signed-distance property due to its localized application and the viscosity solution framework. In the revised manuscript, we will add a short analysis of the reinitialization operator together with quantitative diagnostics (e.g., evolution of max |∇φ| − 1| near each level during iterations) to demonstrate this preservation explicitly. revision: yes

  2. Referee: [§5.2] §5.2 (Numerical experiments, complex model test): The illumination-based error measure reports low misfit in illuminated zones, but the experiments do not include a quantitative check (e.g., max | |∇φ| − 1 | near each i_n) after convergence. Without this, it is unclear whether interface locations remain accurate when multiple phases are present.

    Authors: We acknowledge that a post-convergence quantitative check on |∇φ| near the level sets would provide direct confirmation of interface accuracy for the multi-phase case. While the reported experiments already show faithful recovery of the discontinuous slowness model, we will augment §5.2 with the requested diagnostic: computation and reporting of max | |∇φ| − 1 | in small neighborhoods of each i_n-level set at convergence. This addition will explicitly link the maintained signed-distance property to the observed interface fidelity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; multilayer formulation and adjoint derivatives are independent of fitted inputs

full rationale

The paper introduces a multilayer level-set representation with i_n-level sets designed to act as local signed-distance functions, computes traveltimes via standard viscosity solutions to the eikonal equation, and obtains Fréchet derivatives through the adjoint-state method. Regularization terms (multilayer reinitialization, arc-length penalization, Sobolev smoothing) and the illumination-based error measure are presented as stabilization and assessment tools rather than quantities that reduce the central reconstruction result to its own inputs by construction. No derivation step equates a claimed prediction or uniqueness result to a fitted parameter or self-citation chain; the numerical experiments on discontinuous slowness models supply independent validation outside the formulation itself. The method is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The claim rests on the new MLSM construction, standard viscosity solutions for the eikonal equation, and several regularization strategies whose specific forms and parameter values are not detailed beyond naming.

free parameters (1)
  • regularization weights for multilayer reinitialization, arc-length penalization, and Sobolev smoothing
    These are introduced to stabilize the inversion and are expected to require tuning for different models.
axioms (1)
  • standard math First-arrival traveltimes are viscosity solutions of the eikonal equation.
    Invoked as the forward modeling framework within the Eulerian level-set setting.
invented entities (1)
  • Multilayer level-set method (MLSM) with i_n-level sets no independent evidence
    purpose: To represent arbitrarily many interfaces and subregions through a single function with local signed-distance behavior near each level set.
    New entity introduced to overcome limitations of classical zero-level-set approaches.

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