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REVIEW 4 major objections 6 minor 36 references

OkadaTorch: A Differentiable Programming of Okada Model to Calculate Displacements and Strains from Fault Parameters

T0 review · 4 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read OkadaTorch makes the Okada dislocation model differentiable, with exact gradients and Hessians of displacements and strains available by automatic differentiation.

desk verdict A useful, well-engineered PyTorch AD port of Okada's model with public code, but the central correctness claim rests on qualitative checks and a self-referential inversion; add quantitative validation before relying on it. read the letter →

arxiv 2507.17126 v1 pith:CU2L3BL4 submitted 2025-07-23 physics.geo-ph cs.LG

Masayoshi Someya , Taisuke Yamada , Tomohisa Okazaki This is my paper · ORCID
classification physics.geo-phcs.LG
keywords Okadamodelautomaticdifferentiationfaultparameterinversioncoseismicdeformationelastichalf-spacedislocationstraincomputationdifferentiableprogramming
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

OkadaTorch gives the standard Okada dislocation solution for earthquake displacements and strains a differentiable form: every subroutine is written as tensor operations in an automatic-differentiation library, so a single forward call returns not only displacements and strains but also exact gradients and Hessians with respect to any station coordinate or fault parameter. The authors' claim is that this removes the need to derive, symbolically or by hand, the sensitivity formulas that fault-parameter inversion has traditionally required. If the claim holds, gradient-based inversion, Bayesian sampling, and physics-informed machine-learning models can all use the same forward program and get their derivatives for free. The paper demonstrates the idea with a synthetic inversion in which nine fault parameters are recovered from noisy surface displacements by minimizing a misfit loss with a standard gradient-based optimizer.

What carries the argument

The machinery is automatic differentiation applied to a line-by-line translation of Okada's four subroutines. SPOINT and SRECTF handle surface deformation for point and rectangular sources, DC3D0 and DC3D handle internal deformation; the translation turns the original scalar arithmetic into tensor operations while keeping its structure, including the return flag that reports whether a computation succeeded. On top of that, the OkadaWrapper class provides compute, gradient, and hessian methods, using forward-mode Jacobian evaluation to differentiate outputs with respect to any single argument or pair of arguments from the same category. Two added flags, compute_strain and is_degree, control whether the nine strain components are evaluated and whether angular parameters are read in degrees.

What would settle it

Run the translated routines on parameter sets that exercise each branch—point source and rectangular fault, surface and subsurface stations, dip angles of 0, 45, and 90 degrees, and stations near fault edges—and compare the displacements, strains, first derivatives, and second derivatives against the original program's outputs and against high-precision finite differences; any mismatch beyond floating-point round-off in any branch would falsify the claim.

Watch

Extended reading notes

Core claim

The central claim is that the Okada model—point and rectangular dislocation sources in a 3D elastic half-space, at the surface and below—survives direct translation into a differentiable tensor program without losing its analytical character. The package translates the four core routines, SPOINT, SRECTF, DC3D0, and DC3D, into vectorized tensor form, and wraps them in an OkadaWrapper class whose three methods compute forward displacements and strains, first derivatives with respect to a chosen coordinate or fault parameter, and second derivatives with respect to two inputs. Because the forward program records every elementary operation, the chain rule supplies exact derivatives automatically, which the authors argue makes hand-coded gradients obsolete. The accompanying synthetic experiment shows the resulting derivatives are ready for off-the-shelf gradient-based parameter estimation.

Load-bearing premise

The load-bearing premise is that the step-by-step rewrite of the original subroutines preserves their exact numerical behavior in every branch—surface and subsurface, point and rectangular source—so that the displacements, strains, and all automatic derivatives are correct wherever the original code is correct.

Editorial extensions

If this is right

  • Fault-parameter inversion becomes a standard gradient loop: define a misfit loss on the computed displacements, call the automatic backward pass, and update parameters with an off-the-shelf optimizer.
  • Exact first and second derivatives with respect to coordinates or fault parameters are available on demand, enabling sensitivity analysis and curvature-based uncertainty estimation.
  • All observation stations are processed in one vectorized call rather than a loop, which the paper argues improves efficiency and allows GPU acceleration.
  • Because the whole model is differentiable, it can be composed with other differentiable components, such as tsunami solvers or neural-network models, to build end-to-end differentiable chains from fault mechanics to surface deformation.
  • The same forward code supports gradient-informed Bayesian inference, since exact model derivatives replace finite-difference approximations inside sampling algorithms.

Reading between the lines

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

  • If the translation is exact in every branch, then existing hand-derived or symbolically generated gradient code for the Okada model becomes redundant, and future modifications to the forward model would update all derivatives automatically.
  • The paper does not quantitatively validate its outputs or derivatives against the original program, finite differences, or published benchmarks; adding such tests would convert the correctness claim from a structural one into a verified one.
  • Automatic differentiation through the original branch structure gives derivatives only where the forward function is smooth; near fault-edge singularities and at the free surface the gradients may be undefined or unstable, so users should treat derivative values in those regions with caution.
  • The same translation recipe could be applied to other analytical geophysical Green's-function solutions, such as layered half-space or viscoelastic dislocation models, to give each one an automatically differentiable version.
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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

4 major / 6 minor

Summary. The paper presents OkadaTorch, a PyTorch implementation of Okada's analytical solutions for displacements and strains caused by point and rectangular dislocation sources in an elastic half-space. The core is a direct translation of the original FORTRAN subroutines SPOINT, SRECTF, DC3D0, and DC3D into vectorized PyTorch tensor operations, together with a wrapper class OkadaWrapper that exposes compute, gradient, and hessian methods. The authors demonstrate forward modeling, derivative fields, diagonal Hessians, and a synthetic fault-parameter inversion using Adam. The central claim is that the entire code is differentiable, so gradients and Hessians with respect to station coordinates or fault parameters can be obtained by automatic differentiation without hand-derived formulas.

Significance. If the implementation is numerically faithful to the original Okada solution, this is a genuinely useful software contribution: it removes the need to derive and code gradients by hand, enables GPU-accelerated vectorized forward modeling, and plugs directly into PyTorch-based optimization, Bayesian inference, and scientific machine learning workflows. The public repository, the explicit permission from NIED to publish the translated code, and the clean wrapper interface are concrete strengths. However, the significance is contingent on correctness of the translation, and the manuscript currently offers no quantitative evidence for that correctness: the central claim is unsupported by numerical validation.

major comments (4)
  1. The central claim that OkadaTorch is a direct translation of the original Okada subroutines is not quantitatively validated. Figures 1 and 2 are described only as 'physically consistent deformation patterns' from visual inspection, and no comparison is made against the original FORTRAN code [11], an independent implementation [13-19], or published benchmark values. Because the paper's core value proposition is exactness of the translation, a quantitative comparison table is load-bearing: for a set of representative configurations, report maximum absolute and relative errors for all displacement components and all nine strain components, and the IRET status, comparing OkadaTorch against the original FORTRAN or a trusted reference over both surface (z=0) and subsurface (z<0) points. Without this, a translation error in any branch would invalidate the computed displacements, strains, and all derived gradients.
  2. Gradients and Hessians are never checked against finite differences or analytic derivatives. The code uses PyTorch's jacfwd, so the returned derivatives are exact derivatives of the implemented program, but they are exact derivatives of the physical model only if the program itself is correct. The manuscript should include a finite-difference convergence test, for example comparing gradient and hessian outputs to central differences with decreasing epsilon for at least one rectangular-fault configuration, and report the achieved agreement for derivatives with respect to both coordinates and fault parameters. This is particularly important because the is_degree flag changes derivative scaling, and because the second derivative with respect to slip being zero (Figure 4) is a trivial linearity property that does not test the implementation.
  3. The inversion example is self-referential as a validation: the synthetic observations are produced by the same OkadaTorch forward model, so the successful recovery of parameters can detect optimizer or gradient bugs, but it cannot detect a systematic error shared by the forward and inverse calculations. The manuscript should explicitly state this limitation. In addition, the final parameters in Table 1 are not particularly close to the true values: depth is recovered as 0.604 km versus 0.100 km and slip as 6.14 m versus 5.62 m, and no convergence criterion or uncertainty measure is given, so the statement that 'all estimated fault parameters are reasonably close to their true values' is overstated. This example is fine as a software demonstration, but it should not be presented as evidence of physical correctness.
  4. The manuscript does not address how vectorized tensor operations handle the scalar conditional branches and singular limits present in the original FORTRAN subroutines. Specific edge cases that need explicit tests include dip=90 degrees, strike at 0 or 180 degrees, stations on the free surface z=0, stations near fault edges and corners, negative z, and the point-source limit with potency-based slip. The difference between scalar control flow and PyTorch tensor operations can change behavior in exactly these branches, so a dedicated set of edge-case validation tests, with numerical comparisons to the original implementation or known analytical limits, is required to support the claim that the translation preserves exact numerical behavior in every branch.
minor comments (6)
  1. The affiliation contains a typo: 'Earthquake Reseach Institute' should be 'Earthquake Research Institute'.
  2. The units of strain are given as 'm/km' in Figure 2, while displacements are in m and station coordinates are not assigned explicit units in the text. Please state the assumed unit convention for coordinates and clarify how the strain components scale if coordinates are provided in kilometers or meters.
  3. When is_degree=True (the default), gradients with respect to strike, dip, and rake are per degree rather than per radian, and Hessians inherit the corresponding mixed scaling. This should be documented explicitly, because it affects numerical values used in optimization and uncertainty quantification.
  4. The manuscript says that IRET has the same shape as the input coordinate tensors, but the original subroutines return a scalar status flag. Please clarify how the per-element IRET is defined for vectorized calls and which values indicate failure in the tensor case.
  5. For point sources, the parameter 'slip' is said to represent potency, but potency is never defined and its units are not given. Please define it and state the expected units to avoid confusion with the rectangular-fault slip.
  6. The caption 'Units are km from x_fault to width' is confusing; it should be rephrased to list the units for each column separately. Also, the source of the true parameters, 'model 10 of Table S1 in [32]', is not self-contained in the manuscript; please provide the values in a way the reader can verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the paper is an implementation of an external analytical solution, and the only self-referential test is explicitly labeled as illustrative.

full rationale

OkadaTorch is a software-implementation paper: the forward model is Okada's published analytical solution, cited to external references [1,2,11] rather than to the authors' own prior work. The code is described as 'a direct translation of the original FORTRAN subroutines' (Section 2), and the paper makes no claim to derive new geophysical predictions or to fit a parameter and then present a dependent quantity as an independent prediction. The gradient and Hessian outputs are exact derivatives of the implemented program by construction of PyTorch's automatic differentiation, which is a property of the computing framework rather than a derived result that could reduce to its own inputs. The inversion demonstration in Section 3.4 generates synthetic observations using the same forward model and then recovers parameters from them; however, the paper explicitly states, 'The purpose of this example is not to propose a robust inversion framework, but rather to illustrate that gradient-based parameter optimization is straightforward when using a fully differentiable implementation.' This makes it a self-consistency check, not a claimed external validation or a prediction that is forced by definition. The absence of quantitative comparisons against the original FORTRAN implementation, finite differences, or published benchmarks is a correctness and validation gap, not circularity, because no step in the paper's derivation chain is equivalent, by construction or by self-citation, to the claim it is used to support. Self-citations such as references [36,37] appear only as examples of possible SciML integrations and are not load-bearing for the central implementation claim. Consequently, no circular step satisfying the quoting requirement is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The model introduces no new physical entities or free parameters; the fault parameters fitted in the synthetic inversion example are an application of the code, not part of the central claim. The central claim rests on the fidelity of the FORTRAN-to-PyTorch translation and on PyTorch autograd being correct for this program.

assumptions (3)
  • domain assumption Okada's analytical solutions correctly describe displacements and strains in a homogeneous elastic half-space.
    The entire implementation is a translation of these formulas; the paper does not re-derive or test the physical assumptions.
  • domain assumption PyTorch's automatic differentiation computes exact derivatives of the translated program.
    The gradient and Hessian methods rely on jacfwd and autograd, but the paper provides no finite-difference or analytic comparison to verify these derivatives.
  • ad hoc to paper The FORTRAN-to-PyTorch translation preserves the original subroutines' numerical behavior in all branches and edge cases.
    Section 2 asserts a direct translation, but no quantitative comparison against the original FORTRAN implementation is provided, leaving translation fidelity as an unverified assumption.

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Cite this review

Pith. "Pith review of OkadaTorch: A Differentiable Programming of Okada Model to Calculate Displacements and Strains from Fault Parameters." pith.science (2026). https://pith.science/paper/CU2L3BL4

@misc{pith2026250717126,
  author       = {Pith},
  title        = {Pith review of: OkadaTorch: A Differentiable Programming of Okada Model to Calculate Displacements and Strains from Fault Parameters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CU2L3BL4}},
  note         = {Machine review of arXiv:2507.17126}
}
read the original abstract

The Okada model is a widely used analytical solution for displacements and strains caused by a point or rectangular dislocation source in a 3D elastic half-space. We present OkadaTorch, a PyTorch implementation of the Okada model, where the entire code is differentiable; gradients with respect to input can be easily computed using automatic differentiation (AD). Our work consists of two components: a direct translation of the original Okada model into PyTorch, and a convenient wrapper interface for efficiently computing gradients and Hessians with respect to either observation station coordinates or fault parameters. This differentiable framework is well suited for fault parameter inversion, including gradient-based optimization, Bayesian inference, and integration with scientific machine learning (SciML) models. Our code is available here: https://github.com/msomeya1/OkadaTorch

Figures

Figures reproduced from arXiv: 2507.17126 by the authors.

Figure 1
Figure 1. Three displacement components calculated by [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Nine strain components calculated by compute method (units: m/km, displacement [m] differentiated with respect to distance [km]). – "strike", "dip", "rake": fault orientation. – "slip": slip amount (for rectangular faults) or potency (for point sources). Optional keys: "length" and "width" for rectangular faults. Other optional arguments are as follows. • compute_strain: whether to compute strain components in addit… view at source ↗
Figure 3
Figure 3. First-order derivatives of the vertical displacement [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Diagonal Hessian of vertical displacement [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Synthetic observation data generated by adding random noise to the forward modeling results. The grid [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Displacement components computed using the initial fault parameters at the start of the optimization. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Estimated displacement components computed using the optimized fault parameters. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Evolution of the loss function over 2000 optimization epochs. [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reference graph

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