A Mass Conservative Technique for Interpolation and Smoothing of Ice Surface Velocity Observations

Jesse V. Johnson
Department of Computer Science
University of Montana
Missoula, Montana, USA


The velocity of ice flow can be measured using interferometric techniques on satellite images. Such data have proven remarkably useful for critical investigations of the mass balance of ice sheets, ice sheet response to perturbation, and as a constraint on ice sheet models. When used for mass balance, an area is defined at the mouths of major outlet glaciers and the velocity times that area provides the flux of ice moving into the ocean. Measurements of this sort produce the lowest uncertainty we currently have on the mass balance of Antarctica and Greenland. Time series of surface velocity measurements reveal that changes in the position of the calving front and seasonal surface melting are strongly correlated with increases in the surface velocity. These correlations, in turn, allow investigators to form hypotheses on the key mechanisms responsible for ice dynamical changes. Used as a constraint for ice sheet models, model output is brought into agreement with surface velocity observation, often using adjoint based assimilation methods. The agreement between model and observation is achieved through manipulation of the basal traction field, an unknown of considerable interest because it reveals something about the interaction between ice, rock, sediment, and basally generated melt water.

Given the high importance of these data, it is critical that they be presented in the best way possible. However, as they now exist, there are numerous visible artifacts and large gaps in the coverage. The artifacts arise from different tracks being followed by satellites, and slightly different processing occurring within each track. Sharp discontinuities appear between the tracks. Gaps in the data emerge in regions of high snow accumulation or high surface slope, where interferometric analysis struggles to lock on key features for comparison. The problems in the data are most critical to the model assimilation work, where surface artifacts produce artifacts in basal traction.

Here, I present an optimization technique which preserves the essential nature of the data while smoothing over artifacts and filling in gaps in the coverage. The technique involves solving the differential form of the continuity equation for ice to determine the ice speed. Placed in an adjoint framework for optimization, I minimize the differences between computed and observed speeds by using the snowfall driven accumulation of ice as a control variable. Observed ice flow directions are assumed to be correct, and the guessed to be downhill in regions where there are no observations. The results are 1) within the stated uncertainties of the velocity observations, 2) continuous over the entire domain, 3) consistent with the conservation of mass, and 4) providing reasonable estimates of the velocity in regions of no coverage. They should be of assistance to all those involved with the interpretation of interferometrically derived surface velocity data. The values of the control variable, accumulation, are also instructive, as they reveal inaccuracies in other data used, in particular the thickness of ice.