Bayesian inference of subglacial topography using mass conservation

We develop a Bayesian model for estimating ice thickness given sparse observations coupled with estimates of surface mass balance, surface elevation change, and surface velocity. These fields are related through mass conservation. We use the Metropolis-Hastings algorithm to sample from the posterior probability distribution of ice thickness for three cases: a synthetic mountain glacier, Storglaciären, and Jakobshavn Isbræ. Use of continuity in interpolation improves thickness estimates where relative velocity and surface mass balance errors are small, a condition difficult to maintain in regions of slow flow and surface mass balance near zero. Estimates of thickness uncertainty depend sensitively on spatial correlation. When this structure is known, we suggest a thickness measurement spacing of one to two times the correlation length to take best advantage of continuity based interpolation techniques. To determine ideal measurement spacing, the structure of spatial correlation must be better quantified.