• Bayesian methods in glaciology

      Brinkerhoff, Douglas; Truffer, Martin; Aschwanden, Andy; Tape, Carl; Bueler, Ed (2017-12)
      The problem of inferring the value of unobservable model parameters given a set of observations is ubiquitous in glaciology, as are large measurement errors. Bayes' theorem provides a unified framework for addressing such problems in a rigorous and robust way through Monte Carlo sampling of posterior distributions, which provides not only the optimal solution for a given inverse problem, but also the uncertainty. We apply these methods to three glaciological problems. First, we use Markov Chain Monte Carlo sampling to infer the importance of different glacier hydrological processes from observations of terminus water flux and surface speed. We find that the opening of sub-glacial cavities due to sliding over asperities at the glacier bed is of a similar magnitude to the opening of channels due to turbulent melt during periods of large input flux, but also that the processes of turbulent melting is the greatest source of uncertainty in hydrological modelling. Storage of water in both englacial void spaces and exchange of water between the englacial and subglacial systems are both necessary to explain observations. We next use Markov Chain Monte Carlo sampling to determine distributed glacier thickness from dense observations of surface velocity and mass balance coupled with sparse direct observations of thickness. These three variables are related through the principle of mass conservation. We develop a new framework for modelling observational uncertainty, then apply the method to three test cases. We find a strong relationship between measurement uncertainty, measurement spacing, and the resulting uncertainty in thickness estimates. We also find that in order to minimize uncertainty, measurement spacing should be 1-2 times the characteristic length scale of variations in subglacial topography. Finally, we apply the method of particle filtering to compute robust estimates of ice surface velocity and uncertainty from oblique time-lapse photos for the rapidly retreating Columbia Glacier. The resulting velocity fields, when averaged over suitable time scales, agree well with velocity measurements derived from satellites. At higher temporal resolution, our results suggest that seasonal evolution of the subglacial drainage system is responsible for observed changes in ice velocity at seasonal scales, and that this changing configuration produces varying degrees of glacier flow sensitivity to changes in external water input.
    • Exact and numerical solutions for stokes flow in glaciers

      Mitchell, William H. (2012-08)
      We begin with an overview of the fluid mechanics governing ice flow. We review a 1985 result due to Balise and Raymond giving exact solutions for a glaciologically-relevant Stokes problem. We extend this result by giving exact formulas for the pressure and for the basal stress. This leads to a theorem giving a necessary condition on the basal velocity of a gravity-induced flow in a rectangular geometry. We describe the finite element method for solving the same problem numerically. We present a concise implementation using FEniCS, a freely-available software package, and discuss the convergence of the numerical method to the exact solution. We describe how to fix an error in a recent published model.
    • Modeling supraglacial lake drainage and its effects on the seasonal evolution of the subglacial drainage system in a tributary glacier setting

      Franco, Nevil Arley; Truffer, Martin; Wackerbauer, Renate; Delamere, Peter (2021-08)
      This work aims to gain a better understanding of the relationship between glacier motion and water distributed through subglacial drainage systems. A numerical scheme (GlaDS) is used to model both inefficient and efficient drainage systems to see which dominates after the draining of a supraglacial lake on a synthetic glacier that is made up of an outline that features a main branch and a tributary. The geometry is based on the surgetype Black Rapids Glacier (Ahtna Athabascan name: Da lu'itsaa'den) in Alaska, where a lake develops in the higher ablation zone, and drains rapidly early in the melt season. It has also been observed that this lake drainage causes a twofold or threefold speed-up of the main branch, with some acceleration of the lower-lying Loket tributary. This speed-up can be considered a surrogate for a surge, which also initiates in the main branch, while, during times of quiescence, the ice flow on the tributary is dominant. We investigate the effects of varying timing and volume inputs of lake drainage with a focus on its effects beneath the tributary. We find that the response of the glacier depends on the seasonal timing, the amount of water from the draining lake, and its location on or near the margins of the glacier. Results show that an inefficient drainage system is the cause of the glacier speed-up, both when the lake drains rapidly or when there is an extended time in drainage, at any time of the season. The speed signals vary throughout the glacier depending on the location of the lake relative to that of an evolved efficient drainage system.