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

Abstract

Over the past 3 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 explore the spatial and spectral characteristics of the solution as a function of regularization, and we test the sensitivity of L-curve analysis and regularization to model resolution, effective pressure, sliding nonlinearity, and the flow equation. We find that the optimal regularization level converges towards a finite non-zero limit in the continuous problem, associated with a best knowable basal drag field. Nonlinear sliding laws outperform linear sliding in our analysis, with both a lower total variance and a more sharply cornered L-curve. By contrast, geometry-based approximations for effective pressure degrade inversion performance when added to a sliding law, but an actual hydrology model may marginally improve performance in some cases. Our results with 3D inversions suggest that the additional model complexity may not be justified by the 2D nature of the surface velocity data. We conclude with recommendations for best practices in future glaciological inversions.