Regularization and L-curves in ice sheet inverse models, a case study in the Filchner-Ronne catchment

Abstract. Over the past three decades, inversions for ice sheet basal drag have become commonplace in glaciological modeling. Such inversions require regularization to prevent over-fitting and ensure that the structure they recover is a robust inference from the observations, confidence which is required if they are to be used to draw conclusions about processes and properties of the ice base. While L-curve analysis can be used to select the optimal regularization level, the treatment of L-curve analysis in glaciological inverse modeling has been highly variable. Building on the history of glaciological inverse modeling, we demonstrate general best practices for regularizing glaciological inverse problems, using a domain in the Filchner-Ronne catchment of Antarctica as our test bed. We show a step by step approach to cost function normalization and L-curve analysis. We show that the optimal regularization level converges towards a finite non-zero limit in the continuous problem, associated with a best knowable basal drag field. We find that, when inversion results are judged by a metric that accounts for both the variance of the result and the quality of the fit, then they support nonlinear as opposed to linear sliding laws. We also find that geometry-based approximations for effective pressure degrade inversion performance, but that an actual hydrology model may marginally improve performance in some cases. Our results with 3D inversions suggest that the additional model complexity is not justified by the 2D nature of the velocity data, but we are hopeful that inversions of 3D models may be better situated to take advantage of new constraints in the future. We conclude with recommendations for best practices in glaciological inversions moving forward.