Assimilating sparse data in glaciological inverse problems

Most of the existing work on solving inverse problems in glaciology has assumed that the observational data used to constrain the model are spatially dense. This assumption is very convenient because it means that the model-data misfit term in the objective functional can be written as an integral. In many scenarios, however, the computational mesh can locally be much finer than the observational grid, or the observations can have large patches of missing data. Moreover, pretending as if the observations are a globally-defined continuous field obscures valuable information about the number of independent measurements we have. It is then impossible to apply a posteriori sanity checks on the expected model-data misfit from regression theory. Here we'll describe some recent work we've done on assimilating sparse point data into ice flow models and how this allows us to be more rigorous about the statistical interpretation of our results. For now we are focusing on the kinds of inverse problems that have been solved in the glaciology literature for a long time -- inferring rheology and basal friction from surface velocities. But these developments open up the possibility of assimilating new sources of data, such as measurements from strain gauges or ice cores.