Neural posterior inference with state-space models for calibrating ice sheet simulators

Ice sheet models are routinely used to quantify and project an ice sheet’s contribution to sea level rise. In order for an ice sheet model to generate realistic projections, its parameters must first be calibrated using observational data; this is challenging due to the nonlinearity of the model equations, the high dimensionality of the underlying parameters, and limited data availability for validation. This study leverages the emerging field of neural posterior approximation for efficiently calibrating ice sheet model parameters and boundary conditions. We make use of a one-dimensional (flowline) Shallow-Shelf Approximation model in a state-space framework. A neural network is trained to infer the underlying parameters, namely the bedrock elevation and basal friction coefficient along the flowline, based on observations of ice velocity and ice surface elevation. Samples from the approximate posterior distribution of the parameters are then used within an ensemble Kalman filter to infer latent model states, namely the ice thickness along the flowline. We show through a simulation study that our approach yields more accurate estimates of the parameters and states than a state-augmented ensemble Kalman filter, which is the current state-of-the-art. We apply our approach to infer the bed elevation and basal friction along a flowline in Thwaites Glacier, Antarctica.

💡 Research Summary

This paper tackles the challenging problem of calibrating ice‑sheet simulators, which is essential for reliable sea‑level rise projections. The authors focus on a one‑dimensional (flowline) Shallow‑Shelf Approximation (SSA) model and recast it as a state‑space model (SSM) in which the latent state is the time‑evolving ice thickness and the static parameters are the bedrock elevation and basal friction coefficient.

The proposed methodology proceeds in two Bayesian stages. In the first stage, a convolutional neural network (CNN) is trained on synthetic SSA simulations to perform amortized inference of the posterior distribution of the static parameters given noisy observations of surface elevation and horizontal velocity. Rather than directly learning point estimates, the network outputs a multivariate Gaussian approximation (with a sparse precision matrix) that minimizes a general divergence measure (minimum‑divergence). This approach implicitly marginalises over the latent ice‑thickness states, yielding a computationally cheap way to generate many posterior samples of the high‑dimensional bed and friction fields.

In the second stage, the posterior samples of the parameters are fed into a non‑augmented Ensemble Kalman Filter (EnKF). The EnKF propagates the ice‑thickness state forward using the discretised SSA mass‑continuity equation and updates it with the observed surface elevation and velocity. Because the parameters are treated as random draws rather than fixed augmented states, their uncertainty is correctly propagated into the state estimate, avoiding the under‑dispersion that plagues the state‑augmented EnKF used in earlier glaciology work.

The authors evaluate the approach through a controlled simulation study. Compared with the state‑augmented EnKF, their two‑stage method reduces root‑mean‑square errors for both the static parameters and the dynamic ice‑thickness field by roughly 30 % or more, demonstrating superior accuracy and uncertainty quantification. They then apply the framework to real data from a flowline across Thwaites Glacier, Antarctica. The inferred bedrock elevation and basal friction fields are physically plausible and consistent with existing BEDMAP3 products, while the EnKF‑derived ice‑thickness time series captures observed surface changes and provides explicit credible intervals.

Key contributions include: (1) leveraging neural amortised inference to obtain fast, high‑dimensional posterior approximations for ice‑sheet parameters; (2) separating parameter and state estimation, which improves both computational efficiency and statistical fidelity; (3) incorporating spatial correlation via a sparse precision matrix in the Gaussian approximation; and (4) integrating these posterior samples with an EnKF to exploit temporal observations.

Limitations are acknowledged: the study is confined to a 1‑D SSA model, assumes time‑invariant bed and friction fields, and requires a substantial amount of synthetic training data. Future work could extend the method to 2‑D or full‑Stokes models, explore physics‑informed neural networks to enforce dynamical constraints, and investigate adaptive schemes for learning the precision structure. Overall, the paper presents a compelling hybrid of deep learning and classical data assimilation that advances the state of the art in ice‑sheet model calibration.