## Inference of the Bed Topography Beneath Glaciers by Assimilation of Multi-sensor Data

J. Monnier (University of Toulouse - INSA, Mathematics Institute of Toulouse, France) and J. Zhu (1 year CNES fellowship, 2017-2018).

### Objectives

This research project aims at improving the estimation of the bed topography beneath glaciers, more particularly beneath ice-sheets where no airborne measurements are available (or very few only). The knowledge of the bed topography is a basic step in setting up numerical flow models; moreover, if combined with the surface measurements this provides volumeâ€“mass estimations of the ice-sheets.

In the presence of relatively dense airborne measurements (e.g. CReSIS radar datasets), the bed topography elevation beneath fast sliding ice-streams (~100 m/y and more) can be estimated by inverting a mass conservation equation (see e.g. BedMachine project in Greenland [Morlighem et al. 2017]).

Inferring the bed topography elevation where very few airborne measurements are available remains challenging. The current estimations (both in Antarctica and Greenland) derive from multiple airborne ice thickness surveys plus Kriging interpolation techniques; see [Bamber et al. 2013] for Greenland and [Fretwell et al. 2013] for Antarctica (BedMap2 project, international dataset, map produced by British Antarctic Survey). In East Antarctica Ice Sheet (EAIS) areas that are more than 50 km from direct ice-thickness measurements the estimations uncertainties are large, up to +/- 1000m.

The objective of the present project is to improve these bed topography estimations in interior sectors, in particular in EAIS where the current uncertainties are large.

### Method

The inversions are based on: a) a dedicated physical-based model (mass equation plus momentum equations) natively integrating InSAR data (surface velocities) and altimetry data (surface elevation); b) a purely data-driven model aiming at estimating a dimensionless physical parameter; and c) an assimilation process of the airborne thickness measurements into the flow model a).

Given surface satellites measurements, the key difficulty is to separate the effects of the bed shape from the glacier basal sliding and from the varying vertical ice rate factor profile (thermal vertical gradient). To solve this "source separation" problem, an original reduced uncertainty formulation of a non-isothermal version of the shallow ice flow model is derived; it is combined with state-of-the-art know-hows in Variational Data Assimilation (VDA) and a deep learning model (based on the airborne measurements datasets).

The obtained inferences are valid at large scale only, with minimal wavelength ~10 times the thickness (~30 km in EAIS). However they are valid inland where no airborne measurements are available (where the surface ice flows at ~10-50+ m/y).

### Results (March 2019)

As an example, numerical results obtained in "Ant 1" area of EAIS are presented (see Fig. 1). The algorithms have been performed to the 6 others portions indicated in Fig. 1; all of them respect the surface velocity range a-priori compatible with the method domain of validity.

The "first guesses" in the iterative optimization algorithms are BedMap2 estimations ([Fretwell et al. 2013]) and the climatic source term Racmo2 ([Noel et al., 2018]). All the numerical sensitivities experiments (led on ~8 large areas inland Antarctica and Greenland) have demonstrated a good robustness of the inversion method. The investigated sensitivities have been with respect to the first guess values, the surface data smoothing length scales, the data-driven model choice, the grid size, and with respect to the flight tracks density and/or locations. The inference relies on the surface observations and the estimation of a dimensionless physical parameter obtained by a (global) data driven model. As a consequence, the method remains relatively robust with respect to the flight tracks locations or density.

The obtained estimations in Ant1 area are presented in Fig. 2. Far from the *in-situ* (airborne) measurements (i.e. farer than 2 minimal wave lengths, ~50 km), the method provides much less regular patterns than the Kriging model employed to obtain the Bedmap2 estimations.

The next step of the study consists to investigate more in detail the obtained differences of estimations, in particular at upstream of fast flows.

### References

J. Monnier and J. Zhu. "Inference of the bottom topography in non-isothermal mildly-sheared shallow ice flows". Comput. Meth. Applied Meth. Eng., 2019.

J. Monnier and P.-E. des Boscs. "Inference of the Bottom Properties in Shallow Ice Approximation Models". Inverse Problems, vol. 33 (9) 2017.

J. Monnier and J. Zhu. "Some estimations of bed topography elevation inland East Antarctica Ice Sheet". Submitted.

[Dass] DassFlow computational software (Data Assimilation for Free Surface flows). Available online upon request since 2005. University of Toulouse & INSA, France.