Ice-Sheet Models Face a Hidden Trade-Off: New Theory Reveals When Glaciers Give Up Their Secrets

Deep beneath the surface of Antarctica and Greenland, two hidden properties quietly control how fast the world’s ice flows toward the sea: how slippery the bed is, and how viscous the ice itself is. For more than three decades, the scientists who build large-scale ice-sheet models have disagreed about whether satellite measurements of surface velocity can reveal both of these hidden fields at the same time. Some modelling teams routinely estimate them together; others insist the problem is fundamentally underdetermined and restrict their inversions to basal slipperiness alone. Now a new theoretical study by Camilla A. O. Schelpe and G. Hilmar Gudmundsson of Northumbria University, published in The Cryosphere, has tackled the dispute head-on, and its verdict is striking: the joint retrieval is possible in principle, but only under specific geometric conditions that the real world does not always provide.

The stakes could hardly be higher. Every projection of how much sea level will rise this century depends on initialising a computational ice-flow model so that its simulated velocities match the velocities observed from space. The inversion step that performs this initialisation dates back to Douglas MacAyeal’s pioneering control-methods work in 1992, and it now sits inside virtually every major ice-sheet model, including Úa, BISICLES, ISSM, MALI, STREAMICE, Elmer/Ice and WAVI. Yet the design choices vary wildly between these codes. Úa, WAVI and MALI perform a joint inversion for ice stiffness and basal slipperiness over any domain containing grounded ice. STREAMICE and ISSM invert solely for basal slipperiness over grounded ice, on the belief that viscosity cannot be extracted simultaneously. The deep-learning model IGM restricts its inversion to a linear combination of the two fields amid claims that simultaneous inference leads to non-unique solutions. Until now, there has been little theoretical work to arbitrate between these positions.

Schelpe and Gudmundsson approached the problem by linearising the shallow ice stream approximation, the momentum-conservation framework used in vertically integrated ice-sheet models, into a set of transfer functions. These functions describe, in Fourier space, how small perturbations in each hidden field are transmitted to the surface velocity fields. The key insight comes from inspecting where viscosity actually appears in the governing equations: it multiplies only the derivatives of the horizontal velocity fields. In the absence of velocity gradients, the viscosity has no visible effect on the surface to first order. Horizontal gradients of the velocity field are therefore essential for retrieving the viscosity field in any ice-flow model, a point the authors stress as fundamental.

To build a background state with such gradients, the authors considered an idealised but physically meaningful scenario: a uniform slab of ice flowing down an inclined plane, laterally confined within a channel whose side walls exert drag, much like an Antarctic ice stream. In this domain, lateral drag, which is proportional to viscosity, competes with basal drag, which is proportional to slipperiness, to determine the flow. The velocity profile has a simple analytical solution, a hyperbolic cosine shape that slows toward the channel margins, and from it the authors derived four transfer functions: two describing how viscosity perturbations reach the along-flow and transverse surface velocities, and two doing the same for slipperiness perturbations.

The frequency profiles of these transfer functions turn out to be distinct. The viscosity transfer functions drop to zero wherever the background velocity gradients vanish, and they are maximal at the lateral edges of the channel, where horizontal stresses in the ice are greatest. The slipperiness transfer functions, by contrast, are maximal at the centre of the flow, where the velocity is greatest. At very short wavelengths, all transfer functions tend to zero, meaning fine-scale features in either field simply cannot be retrieved through inverse methods because they leave no visible imprint on the surface. Crucially, the two sets of functions are not linear scalings of one another in general, so the hidden fields have distinct frequency profiles and do not mix. In principle, the spatial distribution of viscosity can be extracted uniquely at the same time as the slipperiness distribution.

The authors then framed the inversion as a Gaussian maximum a-posteriori estimation with explicitly stated priors, a Bayesian formulation equivalent to the velocity-misfit minimisation and Tikhonov regularisation used in operational ice-sheet models. In this framework, the retrieved estimate has one component coming from the priors, scaled by the errors in that estimate, and another component that updates the priors using the observed surface measurements, scaled by their measurement errors. The prior covariance matrix, which can be chosen to mimic Tikhonov regularisation, suppresses high-frequency components in the retrieval, a smoothing that proves essential because fine-scale perturbations are poorly transmitted to the surface.

The synthetic experiments revealed a sharp dependence on measurement accuracy and position along the flow. With almost perfect data, errors of ten to the minus four percent, both fields could be perfectly retrieved regardless of position. But even with very small data errors of 0.005 percent, the viscosity retrieval near the centre of the flow became poor, contaminated by significant high-frequency noise, because surface measurements there do not constrain spurious fluctuations from appearing. Applying a prior improved the retrieval dramatically. With realistic errors of one percent, regularisation became essential for retrieving both fields anywhere along the flow. Slipperiness was then reasonably well retrieved all along the flow, while the viscosity retrieval struggled even at the lateral margins. Near the centre of the flow, the trade-off between suppressing noise and recovering the true signal proved impossible to satisfy, and the underlying viscosity signal could not be extracted from the noise.

Importantly, the authors found no evidence of mixing between the viscosity and slipperiness fields. In regions of small background strain rates, the surface data simply provides little additional information about ice viscosity beyond what the prior already encodes, but this does not translate into less accurate estimates of basal slipperiness in those regions. To confirm that the analytical conclusions survive non-linear finite-amplitude effects, the authors repeated the experiments with a full numerical inversion in the Úa ice-sheet model, using adjoint gradients and a direct-adjoint Hessian. With perfect data, the numerical inversion achieved near-perfect retrieval of both fields, and with noise and regularisation it followed the same pattern: viscosity best resolved toward the lateral margins, slipperiness best toward the centre, and no mixing between the fields. In regions of poor transference, the numerical inversion stayed close to the initial field values rather than producing noisy estimates.

The study carries a clear recommendation for the modelling community: always invert for basal slipperiness and viscosity together over grounded ice areas in ice-sheet models, and explicitly recognise prior knowledge of the hidden fields through appropriate priors and regularisation. Models that ignore spatial variation in viscosity during inversion are, whether they admit it or not, prescribing a prior on the viscosity field with one hundred percent confidence. Making that choice explicit, by including viscosity as an unknown field with strong confidence in its prior, or equivalently large regularisation parameters, is the honest approach. The authors also caution that their experiments assumed linear bed sliding and rheology, perfect knowledge of surface elevation, and uncorrelated measurement errors; relaxing these assumptions, or exploring non-linear flow laws with exponents greater than one, could make separating the effects of slipperiness and viscosity harder, although they expect non-linearity would likely differentiate the surface transference of the two fields even further.

For a field in which the degree of regularisation chosen by the modeller has been identified as one of the largest contributors to uncertainty in Antarctic sea level rise projections, the theoretical clarity offered by this study is a significant step. It suggests that the long-running disagreement between modelling groups is not a dispute about mathematical possibility but about how much confidence to place in priors where the ice itself refuses to cooperate. Where lateral drag bends the flow and velocity gradients are strong, the glaciers give up both of their secrets. Where the flow is uniform and wide, the viscosity remains hidden, and only an explicit, well-chosen prior can fill the gap.

Subject of Research: Earth Sciences

Article Title: On the theoretical limitations of joint inversion for basal slipperiness and effective viscosity in ice-flow models

Article References: Schelpe, C. A. O., & Gudmundsson, G. H. (2026). On the theoretical limitations of joint inversion for basal slipperiness and effective viscosity in ice-flow models. The Cryosphere, 20(10), 6003-6020. https://doi.org/10.5194/tc-20-6003-2026

Image Credits: AI Generated

DOI: 10.5194/tc-20-6003-2026

Keywords: glaciology, ice-sheet models, inversion methods, sea level rise, The Cryosphere, theoretical, limitations, joint, inversion, basal, slipperiness, effective