## 1 Introduction

[2] Realistic modeling of the Antarctic Ice Sheet is essential to improve projections of its past, present, and future contributions to sea level rise in a warming climate [*IPCC-AR4*, 2007]. Boundary conditions are required inputs for ice sheet numerical models. Among these boundary conditions, basal friction is one of the main controls of ice sheet mechanics and it is also one of the most poorly known variables because it cannot be observed directly. Inverse methods that combine ice sheet modeling and surface observations provide a viable alternative to constrain basal conditions. This approach has been applied to simplified two-dimensional ice sheet models [*MacAyeal*, 1992] and extended to higher-order and full-Stokes models [*Morlighem et al.*, 2010; *Seroussi et al.*, 2011; *Jay-Allemand et al.*, 2011]. *Larour et al.* [2012] and *Gillet-Chaulet et al.* [2012] applied this approach to the Greenland Ice Sheet using different ice flow models, but inversion of basal friction has never been attempted at the scale of the Antarctic continent, which is 7 times larger than Greenland. *Pollard and DeConto* [2012] recently used a simplified approach to infer basal friction beneath the Antarctic Ice Sheet at a resolution of 40 km by tuning basal friction to best match observations of ice sheet surface elevation.

[3] Here, we present and apply an inverse method to the entire Antarctic Ice Sheet using a three-dimensional, thermomechanical, higher-order, ice flow model combined with high-resolution (300 m) ice motion data. To apply this method to the entire continent, the approach needs to be scalable and the cost function must accommodate flow regimes spanning from near stagnant ice in the interior (cm/yr) to fast-flowing ice along the periphery (km/yr), almost 6 orders of magnitude difference in speed.

[4] Inverting for basal friction requires the construction of an adjoint model. A common approximation is to neglect the nonlinearity of ice viscosity (e.g., *MacAyeal* [1992]). The impact of this incomplete adjoint approximation on the performance of the inversion has not been fully established. *Goldberg and Sergienko* [2011] showed that for a hybrid model [*Schoof and Hindmarsh*, 2010; *Goldberg*, 2011], the exact adjoint may be advantageous in some cases to minimize the cost function. Here, we address this issue by deriving the exact solution of the adjoint model and by comparing the results to those obtained with the incomplete adjoint. We also compare the performance of two descent algorithms. Finally, we analyze and discuss the inferred pattern of basal friction in Antarctica and the implications of the results for ice sheet modeling.