Physics-aware Machine Learning to Estimate Ice Thickness of Glaciers in West Svalbard
Authors: Steidl Viola , Bamber Jonathan , Zhu Xiao Xiang 
Published: March 2024

Glacier ice thickness is a fundamental variable required for modelling flow and mass balance. However, direct measurements of ice thickness are scarce. Physics-based and data-driven approaches aim to reconstruct glacier ice thicknesses from the limited in-situ data. Farinotti et al. compared 17 models and found that their ice thickness estimates differ considerably on test glaciers.[1] Following these results, Farinotti et al. created an ensemble of models to develop the so-called consensus estimate of the ice thickness for the world&#8217;s glaciers in 2019.[2] Later, Millan et al. derived ice thickness estimates for the world&#8217;s glaciers using ice motion as the primary constraint. However, these results differ considerably from existing estimates and the 2019 consensus estimates.[3] It is evident, therefore, that significant uncertainty remains in ice thickness estimates.Deep learning approaches are flexible and adapt well to complex structures and non-linear behaviour. However, they do not guarantee physical correctness of the predicted quantities. Therefore, we employ a physics-informed neural network (PINN), which integrates physical laws into their training process and is not purely data-driven. We include, for example, the conservation of mass in the loss function and estimate the depth-averaged flow velocity. Teisberg et al. also employed a mass-conserving PINN to interpolate the ice thickness of the well-studied Byrd glacier in Antarctica.[4] In this work, we extend the methodology by integrating the ratio between slope and surface flow velocities in estimating the depth-averaged flow velocity and mapping the coordinate variables to higher dimensional Fourier Features.[5] This allows to encompass glaciers in western Svalbard, addressing challenges posed by basal sliding, surface melting, and complex glacier geometries. Using surface velocity data from Millan et al. and topographical data from Copernicus DEM GLO-90[6] gathered through OGGM[7],&#160; the model predicts ice thickness on glaciers with limited measurements. We are extending it to perform as a predictor of thickness for glaciers with no observations. Here, we present the machine learning pipeline, including the physical constraints employed and preliminary results for glaciers in western Svalbard.
[1] Daniel Farinotti et al., &#8216;How Accurate Are Estimates of Glacier Ice Thickness? Results from ITMIX, the Ice Thickness Models Intercomparison eXperiment&#8217;, The Cryosphere 11, no. 2 (April 2017): 949&#8211;70, https://doi.org/10.5194/tc-11-949-2017.

[2] Daniel Farinotti et al., &#8216;A Consensus Estimate for the Ice Thickness Distribution of All Glaciers on Earth&#8217;, Nature Geoscience 12, no. 3 (March 2019): 168&#8211;73, https://doi.org/10.1038/s41561-019-0300-3.

[3] Romain Millan et al., &#8216;Ice Velocity and Thickness of the World&#8217;s Glaciers&#8217;, Nature Geoscience 15, no. 2 (February 2022): 124&#8211;29, https://doi.org/10.1038/s41561-021-00885-z.

[4] Thomas O. Teisberg, Dustin M. Schroeder, and Emma J. MacKie, &#8216;A Machine Learning Approach to Mass-Conserving Ice Thickness Interpolation&#8217;, in 2021 IEEE International Geoscience and Remote Sensing Symposium IGARSS, 2021, 8664&#8211;67, https://doi.org/10.1109/IGARSS47720.2021.9555002.

[5] Matthew Tancik et al., &#8216;Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains&#8217;, (arXiv, 18 June 2020), https://doi.org/10.48550/arXiv.2006.10739.

[6] {https://doi.org/10.5270/ESA-c5d3d65}

[7] Fabien Maussion et al., &#8216;The Open Global Glacier Model (OGGM) v1.1&#8217;, Geoscientific Model Development 12, no. 3 (March 2019): 909&#8211;31, https://doi.org/10.5194/gmd-12-909-2019.