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REVIEW 3 major objections 6 minor 78 references

A discrete adjoint method for deterministic and probabilistic eikonal-equation-based inversion of traveltime for velocity and source location

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper derives gradient formulas for velocity and source location directly from the discretized fast-marching eikonal solver, giving a single adjoint framework that works for both L-BFGS inversion and Hamiltonian Monte Carlo sampling.

desk verdict Clean derivation, public code, and a real extension to source-location gradients and HMC, but the discrete adjoint is only proved for fixed FMM stencils and the missing finite-difference check leaves the central consistency claim unverified. read the letter →

arxiv 2501.13532 v1 pith:ERVSK4IU submitted 2025-01-23 physics.geo-ph

Andrea Zunino , Scott Keating , Andreas Fichtner This is my paper · ORCID
classification physics.geo-ph
keywords traveltimetomographyeikonalequationdiscreteadjointmethodfastmarchingvelocityinversionsourcelocationHamiltonianMonteCarlouncertaintyquantification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's central proposition is that traveltime tomography based on the eikonal equation can be turned into a coherent inverse problem by deriving gradients directly from the discretized forward solver, rather than from a separately discretized continuous adjoint equation. Working from a second-order fast-marching method, the authors write the discrete eikonal equation as an implicit system, differentiate it to obtain an adjoint equation, and use the same adjoint variable for both the velocity gradient and the source-location gradient. The resulting gradients are, by construction, consistent with the discrete forward model, and the fast-marching update order turns the adjoint system into a triangular one that is cheap to solve. This makes it practical to run joint inversions for velocity and source position, and because the gradients are cheap and accurate, they can feed both deterministic optimization (L-BFGS) and gradient-based Bayesian sampling (Hamiltonian Monte Carlo). The paper demonstrates the approach on synthetic 2D and 3D experiments, including one where Bayesian sampling reveals a bimodal posterior that a deterministic point estimate would miss.

What carries the argument

The load-bearing object is the discrete implicit form of the eikonal equation, $f_i = (\sum_j D^x_{ij} u_j)^2 + (\sum_j D^y_{ij} u_j)^2 + (\sum_j D^z_{ij} u_j)^2 - 1/v_i^2 = 0$, where the $D$ matrices carry the second-order upwind finite-difference stencils selected by the fast-marching solver. The adjoint state $\lambda$ is defined by $(\partial f / \partial u)^T \lambda = -(\partial \psi / \partial u)^T$, and the same $\lambda$ is reused in the velocity-gradient and source-location-gradient chain rules. The structural trick is that ordering grid points by the fast-marching wavefront arrival makes $\partial f / \partial u$ lower triangular, so the transposed adjoint system is upper triangular and solvable in a single pass; the stencils are sparse, with only two or three nonzeros per row. For the refined source-region grid, a second adjoint system is solved on the fine grid with the coarse-grid traveltime acting as the adjoint source, and the chain-rule factor $N_{qa}$ maps the velocity interpolation back to the coarse grid.

What would settle it

Run central-difference checks of the discrete-adjoint gradients against the same FMM code: perturb each velocity voxel and each source coordinate by a small epsilon, recompute traveltimes, and compare the ratio (psi(theta+epsilon)-psi(theta-epsilon))/(2 epsilon) with the adjoint gradient at models near stencil switches and ties in the max operation. If pointwise relative errors are far above round-off where the solver's stencil selection changes, the claim that the gradient is exactly consistent with the implemented forward map is refuted; if the errors stay near machine precision everywhere, the claim is supported.

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Extended reading notes

Core claim

The contribution the paper is trying to establish is that all the pieces of eikonal traveltime tomography—velocity update, source-location update, local grid refinement around the source, and arbitrary source and receiver placement—can be handled by one discrete adjoint formalism built on the same second-order fast-marching forward model. The forward solver is written as an implicit discrete equation; differentiating it gives the adjoint equation whose solution is a single field lambda. The same lambda is then folded into the chain rules for both velocity parameters and source coordinates, with the case of a refined source-region grid handled by solving a second, analogous adjoint problem on the fine grid. Because the fast-marching method solves grid points in increasing traveltime order, the matrix in the adjoint system is triangular, so the gradient computation costs roughly two forward solves and is independent of the number of receivers. The paper claims this yields gradients that correspond exactly to the discretized forward model and demonstrates that these gradients drive successful deterministic L-BFGS reconstructions and Hamiltonian Monte Carlo posteriors in 2D and 3D.

Load-bearing premise

The whole gradient-consistency claim rests on treating the fast-marching solver's selection of neighboring grid points as a fixed smooth configuration when differentiating, although the solver actually switches stencils through max operations and order fallbacks.

Editorial extensions

If this is right

  • Gradients produced by the discrete adjoint are consistent with the FMM forward model by construction, removing the forward/adjoint discretization mismatch that can arise when the continuous adjoint equation is discretized separately.
  • The same single adjoint field supplies both velocity and source-location gradients, so joint inversion becomes a natural extension of the single-parameter case rather than a separate two-step procedure.
  • Ordering grid points along the fast-marching wavefront makes the adjoint matrix triangular, so gradient computation is efficient and the cost is nearly independent of the number of receivers.
  • The formalism accommodates arbitrary source and receiver positions plus a refined grid around the source, which reduces the dominant traveltime errors near the source without dropping those terms from the gradient.
  • Because gradients are cheap and accurate, the same framework feeds both L-BFGS deterministic inversion and HMC probabilistic sampling, yielding uncertainty quantification that can expose multimodality in the posterior.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same discrete-adjoint construction should extend to other grid-based eikonal solvers, such as factored eikonal forms or higher-order stencils, by replacing the derivative matrices in the forward relation; the chain-rule structure of the derivation is solver-agnostic.
  • A practical validation step would be a finite-difference check of the gradients near points where the forward solver switches stencils or ties in the max selection; that would quantify how often the fixed-stencil differentiability assumption matters in practice.
  • For field applications, the probabilistic branch could be used to produce posterior covariance maps for both velocity and hypocenter locations, which would feed directly into seismic hazard or event-relocation studies.
  • The triangular structure of the adjoint system suggests the gradient cost scales like the forward solve, so large 3D regional models with many sources and receivers are the natural next target; the paper's 3D examples use relatively small grids.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper develops a discrete adjoint method for traveltime tomography based on the eikonal equation, with second-order fast marching used as the forward solver. Gradients are derived with respect to both velocity structure and source location, including a treatment of grid refinement around the source, and are then embedded in deterministic (L-BFGS) and probabilistic (HMC/NUTS) inversion frameworks. Synthetic 2D and 3D examples are presented for both approaches. The central claim is that the discrete adjoint derivation yields gradients that are exactly consistent with the discretized FMM forward model, enabling efficient joint inversion for velocity and source position.

Significance. If the consistency claim holds, the paper would be a useful contribution: it unifies velocity and source-location gradients in one discrete adjoint framework, allows arbitrary source/receiver positions, handles local grid refinement, and demonstrates both deterministic and probabilistic inversion, with open-source code provided. The derivations are self-contained and the numerical experiments illustrate the intended use. However, the exact-consistency claim is not numerically verified against the actual piecewise-differentiable FMM forward map, and this gap is load-bearing for both the L-BFGS and HMC applications. The paper is therefore promising but requires additional validation and careful qualification before the central claim can be accepted.

major comments (3)
  1. [Section 3.1.1, Eqs. (9)-(22)] The adjoint derivation differentiates the implicit equation (9) while treating the stencil matrices D^x, D^y, D^z as fixed. In the implemented FMM (Section 2.1, Eq. (5)), the stencils are selected at runtime by the max(·,0) upwind rule, with fallback between second-order (Eq. (3)) and first-order (Eq. (4)) stencils depending on the availability of upwind values. The stencil configuration is a piecewise-constant function of the traveltime field, so Eq. (20) gives a branch-local derivative, not the derivative of the full implemented forward map. At stencil switches or ties, the gradient in Eq. (22) does not correspond to a unique linearization of the forward code. This is load-bearing because the paper's stated advantage over continuous adjoint methods is exactly discrete consistency, and the mismatch is never quantified.
  2. [Section 4 (numerical experiments)] No finite-difference validation of the adjoint gradients is presented in any of the deterministic or probabilistic experiments. Given the nondifferentiable stencil selection in the forward solver, a direct comparison of the adjoint gradient against finite differences of the actual FMM implementation is essential to support the claim of exact consistency stated in the Introduction, Section 5, and the abstract. Without such a test, the magnitude and frequency of any gradient error remain unknown, which also affects the HMC/NUTS results because a biased gradient biases the stationary distribution.
  3. [Section 3.2, Eqs. (59)-(60)] The gradient with respect to source location differentiates Eq. (2) while assuming a fixed set S of nodes surrounding the source. As the source moves across a grid-cell boundary, the set of enclosing nodes changes discontinuously, making the source-location forward map piecewise differentiable. The derivation does not address this, and no numerical test checks the behavior of dψ/ds_r at such boundaries. Since joint inversion for source location is a central contribution, this branch-dependence should be discussed and ideally validated numerically.
minor comments (6)
  1. [Introduction, Section 5] The phrases “exactly the counterparts of the forward model” and “machine precision” overstate the result given the piecewise-differentiable stencil selection; the text should be qualified to refer to differentiability within fixed stencil configurations.
  2. [Section 2.1, Eq. (2)] The velocity v_s in Eq. (2) is not uniquely defined when the four (or eight) surrounding grid points have different velocities; the paper should specify how v_s is chosen.
  3. [Section 4.1, Figure 4 caption] The caption states that the inversion ran for 150 iterations, while the text reports 80 iterations; this discrepancy should be corrected.
  4. [Section 4.2, first example] The text reports 50000 NUTS iterations but then states that 150000 models were saved after burn-in; the relationship between these numbers should be clarified.
  5. [Section 4.2, Figure 7 caption] The caption refers to a random model “after 1454 iterations,” while the text says 10000 iterations were run; the reported iteration counts should be reconciled.
  6. [Section 3.1.4, Eq. (44)] The right-hand side written as ∂u/∂τ is ambiguous; it should be written as ∂û_h/∂τ_p to match the notation introduced in Eq. (42).

Circularity Check

0 steps flagged · score 0.0 of 10

The discrete adjoint derivation is self-contained; cited self-work only supplies HMC implementation details and is not load-bearing.

full rationale

The paper derives its gradients from its own discrete forward model and objective function: eq. (9) defines the implicit discrete eikonal forward model, eqs. (18)-(24) obtain the velocity gradient via the discrete adjoint equation, and eqs. (56)-(60) do the same for source location by differentiating eq. (2). No parameter is fitted to data and then relabeled as a prediction, and no external uniqueness theorem or ansatz is imported from the authors' prior work. Self-citations (Fichtner et al. 2019, Zunino et al. 2023) are used only to reference the HMC sampler and the code framework; the adjoint derivation does not depend on them. The numerical experiments are synthetic demonstrations rather than confirmations of a physical theory. A separate correctness risk exists: the FMM solver's stencil selection (eqs. 3-5) is piecewise constant, so the gradient in eq. (20) is conditional on a fixed stencil, and no finite-difference validation is reported; that concern bears on the accuracy of the computed gradients, not on the circularity of the derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The method derives gradients from the discrete forward model and introduces no fitted physical parameters or new physical entities. It relies on standard implicit differentiation and on properties of the FMM solver. The interpolation choices for off-grid sources and for fine-grid velocities are user-defined modeling assumptions. No new particles, forces, or dimensions are postulated.

assumptions (5)
  • standard math The discrete forward operator (9) is differentiable and its Jacobian is invertible at the solution, so implicit differentiation applies (eqs. 16-19).
    The adjoint derivation uses the implicit function theorem on the discrete eikonal equation. Nonsingularity of the Jacobian is assumed but not proven; FMM solutions typically satisfy this away from kinks.
  • domain assumption The FMM computes the viscosity solution of the eikonal equation, and the discrete forward map is exactly described by eqs. (3)-(5).
    Section 2.1; the paper relies on standard properties of fast marching methods without re-deriving them.
  • domain assumption Traveltimes at grid nodes surrounding an off-grid source are accurately given by the analytic formula (2), which assumes locally homogeneous velocity in the source cell.
    Eq. (2) in Section 2.1; this approximation is part of the forward model and affects source-region derivatives, but its accuracy is not investigated beyond citing previous practice.
  • domain assumption With grid refinement, the fine-grid velocity is obtained by linear interpolation (eq. 37), and the sampling operator H extracts coarse-grid collocated values (eq. 42).
    Section 3.1.4; these interpolation choices are user-defined modeling assumptions that enter the gradient calculation.
  • domain assumption Ordering unknowns by increasing FMM traveltime makes the matrix A in eq. (25) lower triangular, enabling an efficient triangular solve of the adjoint system.
    Section 3.1.1 after eq. (25); the property is asserted for the FMM ordering but not proven in the paper.

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Cite this review

Pith. "Pith review of A discrete adjoint method for deterministic and probabilistic eikonal-equation-based inversion of traveltime for velocity and source location." pith.science (2026). https://pith.science/paper/ERVSK4IU

@misc{pith2026250113532,
  author       = {Pith},
  title        = {Pith review of: A discrete adjoint method for deterministic and probabilistic eikonal-equation-based inversion of traveltime for velocity and source location},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ERVSK4IU}},
  note         = {Machine review of arXiv:2501.13532}
}
read the original abstract

Seismic traveltime tomography represents a popular and useful tool for unravelling the structure of the subsurface across the scales. In this work we address the case where the forward model is represented by the eikonal equation and derive a formalism to solve the inverse problem where gradients are calculated efficiently using the discrete adjoint state method. Our approach provides gradients with respect to both velocity structure and source locations, allowing us to perform a consistent joint inversion. The forward problem is solved using a second-order fast-marching method, which provides a strategy to efficiently solve the adjoint problem. Our approach allows for arbitrary positions of both sources and receivers and for a refined grid around the source region to reduce errors in computed traveltimes. We show how gradients computed using the discrete adjoint method can be employed to perform either deterministic inversion, i.e., solving an optimization problem, or for a probabilistic (Bayesian) approach, i.e., obtaining a posterior probability density function. We show applications of our methodology on a set of synthetic examples both in 2D and 3D using the L-BFGS algorithm for the deterministic case and the Hamiltonian Monte Carlo algorithm for the probabilistic case.

Figures

Figures reproduced from arXiv: 2501.13532 by the authors.

Figure 1
Figure 1. An example of the grid used for the FMM scheme in 2D, with arbitrary position of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. An example of 2D grid without (a) and with (b) grid refinement around the source. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Deterministic inversion for a 2D velocity model. a) Ground-truth model from which the [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Deterministic inversion for both a 2D velocity model and source locations. a) Ground [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Deterministic inversion for a 3D velocity model. a) Ground-truth model from which the [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Probabilistic inversion for both a 2D velocity model and source locations. a) Ground [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Probabilistic inversion for a 3D velocity model. a) Ground-truth model from which [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.