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arxiv: 1906.08209 · v2 · pith:D6GYWJXWnew · submitted 2019-06-19 · ⚛️ physics.comp-ph

Parameter sensitivity analysis of dynamic ice sheet models-Numerical computations

Gong Cheng , Per L\"otstedt This is my paper

Pith reviewed 2026-05-25 19:42 UTC · model grok-4.3

classification ⚛️ physics.comp-ph
keywords ice sheet modelsadjoint equationssensitivity analysisStokes equationsshallow shelf approximationparameter perturbationsnumerical computationsdynamic models
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The pith

Adjoint equations of the stress and height equations supply weights to quantify how perturbations in friction and base topography affect surface velocity and height in ice sheet models.

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

The paper develops numerical solutions to adjoint equations that turn perturbations in the friction coefficient and base topography into sensitivity weights for surface velocity and height. The approach is applied to both stationary and time-dependent ice sheets using either the full Stokes equations or the shallow shelf approximation. Examples are solved in two dimensions and checked against analytical solutions for simpler problems. A reader would care because these weights show which base parameters most influence observable surface quantities, allowing focused use of measurements when the forward models are expensive to run repeatedly.

Core claim

The sensitivity to the perturbations of the velocity and the height at the surface is quantified by solving the adjoint equations of the stress and height equations providing weights for the perturbed data. The adjoint equations are solved numerically and the sensitivity is computed in several examples in two dimensions. Comparisons are made with analytical solutions to simplified problems for both the full Stokes model and the shallow shelf approximation applied to stationary and dynamic ice sheets.

What carries the argument

The adjoint equations of the stress and height equations, solved numerically to produce weights that measure the effect of changes in friction coefficient and base topography on surface observations.

Load-bearing premise

The adjoint equations remain accurate representations of the sensitivity when they are discretized and solved numerically for the full dynamic ice sheet models.

What would settle it

Direct finite-difference recomputation of surface velocity and height after small changes in friction or topography, compared against the adjoint-derived weights on the same two-dimensional mesh, would disagree if the discretized adjoints fail to capture the true sensitivity.

Figures

Figures reproduced from arXiv: 1906.08209 by Gong Cheng, Per L\"otstedt.

Figure 1
Figure 1. Figure 1: The initial ice geometry with height h (blue), ice base b (orange), and ocean bathymetry (black). The domains in Eq.(2) are the ice domain Ω between the blue and orange curves, the upper surface Γs in blue, the lower boundary on the bedrock Γb and on water Γw in orange, Γu at x = 0 and Γd at x = L = 1.6 × 106 m. steady state solution by C(x) = C0kuk −2/3 . The resulting friction law becomes Cf(u) = C(x) wh… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the weights Tu·Tv in Eq. (10) for perturbations δC at different observation points x∗ = 0.25×106 ,0.5×106 ,0.7× 106 and 0.9 × 106 (blue, orange, green, and pink).Upper panels: transient simulations; lower panels: steady states. Left panels: wuC with pointwise u response; right panels: whC with pointwise h response. The adjoint solutions v1 at Γb of all the four cases are concentrated at the o… view at source ↗
Figure 3
Figure 3. Figure 3: The response at Γs with different wavelengths λ in the perturbation of C in Eq. (28). Left panel: δu1(x∗,λ)/δu1,∞(x∗); right panel: δh(x∗,λ)/δh∞(x∗). We perform a pair of experiments to compare the results from perturbing the forward equation and the prediction by the adjoint solutions. A relative 1% perturbation δC(x) is added at x ∈ [0.9,1.0]×106 m to the friction coefficient C(x). The differences betwee… view at source ↗
Figure 4
Figure 4. Figure 4: The changes on the horizontal velocity u1 (upper panel) and surface elevation h (lower panel) after one year with 1% perturbation on C(x) at x ∈ [0.9,1.0] × 106 m. Solid lines are the differences between the steady state and perturbed transient solutions in Eq. (5). Red dots are the estimated perturbation using Eq. (10). 3.2 SSA The same MISMIP benchmark experiment as in Sect. 3.1 is solved by the SSA on a… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the steady state numerical solutions of the SSA velocity u and the thickness H in Eq. (17) (orange) and the analytical solutions in Eq. (A1) (blue). The weight functions wuC and whC in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the weights wuC and whC in Eq. (19) for perturbations δC with m = 1 at different observation points x∗ = 0.25 × 106 ,0.5 × 106 ,0.7 × 106 and 0.9 × 106 (blue, orange, green, and pink). The black dashed line in the lower panels are wuC and whC computed from the analytical solutions of u in Eq. (A1) and v in Eq. (A2) and Eq. (A4) at x∗ = 0.7 × 106 . Upper panels: transient simulations; lower pa… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the weights wub and whb in Eq. (19) for perturbations δb at different observation points x∗ = 0.25 × 106 ,0.5 × 106 ,0.7 × 106 and 0.9 × 106 (blue, orange, green, and pink). The black dashed line in the lower panels are the weights of δb in Eq. (A3) and Eq. (A5) at x∗ = 0.7×106 . Upper panels: transient simulations; lower panels: steady states. Left panels: wub for pointwise u response; right… view at source ↗
Figure 8
Figure 8. Figure 8: A close-up view of the steady state weights in the lower panels of [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The singular values of the transfer matrices WuC , WhC , Wub and Whb. The corresponding comparisons for the steady state problem are made in Figs. 12 and 13 with the weights in the lower panels of Figures 6 and 7. The analytical solutions of the steady state perturbations from (A3) and (A5) are shown with black dashed lines in these two figures. The rapid change of δh in Figs. 10 and 11 is explained by the… view at source ↗
Figure 10
Figure 10. Figure 10: The changes in the horizontal velocity u (upper panel) and surface elevation h (lower panel) after one year with 1% perturbation of C(x) in x ∈ [0.9,1.0] × 106 m. Solid lines are the differences between the steady state and the perturbed solutions in Eq. (13). Red dots represent the estimated perturbation using Eq. (15). perturbations by the simplified adjoint SSA systems in [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 11
Figure 11. Figure 11: The changes in the horizontal velocity u (upper panel) and surface elevation h (lower panel) after one year with 0.01 m pertur￾bation of b(x) in x ∈ [0.9,1.0] ×106 m. Solid lines are the differences between the steady state and the perturbed solutions in Eq. (13). Red dots represent the estimated perturbation using Eq. (15). We solve the FS adjoint problem only one step backward in time to verify the nume… view at source ↗
Figure 12
Figure 12. Figure 12: The changes in the horizontal velocity u (upper panel) and surface elevation h (lower panel) after 15000 years (close to the steady state) with 1% perturbation of C(x) in x ∈ [0.9,1.0] × 106 . Solid lines are the differences between the steady state and perturbed solutions in Eq. (13). Red dots represent the estimated perturbation using Eq. (15). The transfer relation WuC between small perturbations of th… view at source ↗
Figure 13
Figure 13. Figure 13: The changes in the horizontal velocity u (upper panel) and surface elevation h (lower panel) after 15000 years (close to the steady state) with 0.01 m perturbation of b(x) in x ∈ [0.9,1.0]×106 . Solid lines are the differences between the steady state and perturbed solutions in Eq. (13). Red dots represent the estimated perturbation using Eq. (15). solve N + 1 forward problems to compute WuC . In the inve… view at source ↗
Figure 14
Figure 14. Figure 14: The changes in the horizontal velocity u after one year with 1% perturbation of C(x) in x ∈ [0.9,1.0] × 106 m. Solid lines are the differences between the steady state and the perturbed solutions in Eq. (13). Red dots represent the estimated perturbation using Eq. (15). Upper panel: forward viscosity. Lower panel: without advection equation. 0 200 400 600 800 1000 Singular values 10−10 10−8 10−6 10−4 Σe u… view at source ↗
Figure 15
Figure 15. Figure 15: The singular values of the transfer matrices with simplifications from MacAyeal (1993). Σe uC corresponds to the forward viscosity case and Σb uC is from the adjoint SSA without coupling to the ψ equation. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
read the original abstract

The friction coefficient and the base topography of a stationary and a dynamic ice sheet are perturbed in two models for the ice: the full Stokes equations and the shallow shelf approximation. The sensitivity to the perturbations of the velocity and the height at the surface is quantified by solving the adjoint equations of the stress and the height equations providing weights for the perturbed data. The adjoint equations are solved numerically and the sensitivity is computed in several examples in two dimensions. Comparisons are made with analytical solutions to simplified problems.

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 develops adjoint-based methods to quantify the sensitivity of surface velocity and height to perturbations in the friction coefficient and base topography for both stationary and time-dependent ice-sheet models. It derives adjoint equations from the full Stokes and shallow-shelf approximation (SSA) systems, solves them numerically, and computes sensitivities in several 2D examples, with comparisons restricted to analytical solutions of simplified problems.

Significance. If the numerical adjoint solutions are shown to be accurate for the dynamic cases, the work would supply an efficient, derivative-based tool for sensitivity analysis that avoids repeated forward solves, with direct relevance to data assimilation and uncertainty quantification in glaciology. The 2D demonstrations and analytic comparisons for simplified cases are a useful starting point, but the absence of cross-validation for the full dynamic systems limits the immediate applicability of the reported weights.

major comments (1)
  1. [Abstract] Abstract: the central claim that the numerically solved adjoint equations of the stress and height equations 'provide weights for the perturbed data' for dynamic models rests on the unverified assumption that discretization and solver errors do not invalidate the sensitivities; the only reported checks are analytic comparisons for simplified (non-dynamic) problems, with no mention of finite-difference perturbation tests on the same meshes or manufactured-solution verification for the coupled time-dependent Stokes/SSA systems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The point raised about verification of the adjoint sensitivities for the dynamic cases is well taken, and we address it directly below with a commitment to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the numerically solved adjoint equations of the stress and height equations 'provide weights for the perturbed data' for dynamic models rests on the unverified assumption that discretization and solver errors do not invalidate the sensitivities; the only reported checks are analytic comparisons for simplified (non-dynamic) problems, with no mention of finite-difference perturbation tests on the same meshes or manufactured-solution verification for the coupled time-dependent Stokes/SSA systems.

    Authors: We agree that the analytic comparisons in the manuscript are limited to simplified problems and that the dynamic Stokes and SSA cases lack explicit cross-validation such as finite-difference perturbation tests or manufactured-solution checks on the same meshes. The numerical results for the time-dependent models are obtained by solving the derived adjoint equations, but without those additional tests the claim that the computed sensitivities provide reliable weights rests on the consistency of the discretization rather than direct verification. To address this, we will add finite-difference perturbation tests for the dynamic cases in the revised manuscript, performed on the same meshes and with the same solvers used for the adjoint computations. This will allow direct comparison of the adjoint-derived sensitivities against finite-difference approximations for the full time-dependent systems. revision: yes

Circularity Check

0 steps flagged

No circularity: adjoint sensitivity follows standard derivation from governing PDEs

full rationale

The paper derives adjoint equations directly from the stress and height equations of the full Stokes and SSA models to compute surface sensitivities to perturbations in friction coefficient and base topography. This is a standard, non-circular application of adjoint methods for PDE sensitivity analysis, with numerical solutions compared to analytical results on simplified problems. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations are present in the abstract or described method. The derivation chain is self-contained and independent of the target sensitivities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the assumption that adjoint equations can be derived and numerically solved for the given ice-sheet PDEs without additional free parameters or new entities being introduced.

axioms (1)
  • domain assumption Adjoint equations derived from the stress and height equations correctly quantify sensitivities to friction and topography perturbations.
    Invoked as the core mechanism for providing weights to the perturbed data.

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