Sensitivity of the mass conservation method to the regularisation scheme

While being one of the most important variables for predicting the future of the ice sheets, observations of ice thickness are only available along flight tracks, separated by a few to a few tens of kilometres. For many applications, these observations need to be interpolated on grids at a much higher resolution than the actual average spacing between tracks.

The mass conservation method is an inverse method that combines the sparse ice thickness data with high resolution surface velocity observations to obtain a high-resolution map of ice thickness that conserves mass and minimizes the departure from observations.  As with any inverse method, the problem is ill-posed and requires some regularisation. The classical approach is to use a Tikhonov regularisation that penalizes the spatial derivatives of the ice thickness and therefore favours smooth solutions with implicit spatial correlation structures. In a Bayesian framework, regularization can be seen as an implicit assumption for the prior probability distribution of the inverted parameter. Other popular geostatistical interpolation algorithms, such as kriging, usually require to parameterize the spatial correlation of the interpolated field using standard correlation functions (e.g., gaussian, exponential, Matèrn).

Here we replace the Tikhonov regularisation term in the mass conservation method  by a term that penalises the departure from a prior, where the error statistics are parametrized with the same standard correlation functions. This makes the regularisation independent from grid spacing and regularisation weights do not need to be adjusted. We present and discuss the sensitivity of the mass conservation method to the regularisation scheme using a suite of synthetic and “true” bed from deglaciated areas and show that prescribing the correct regularisation always provides the most accurate solution.