A Bayesian hierarchical model for glacial dynamics based on the shallow ice approximation and its evaluation using analytical solutions

Giri Gopalan,Finnur Pálsson,Birgir Hrafnkelsson,Guðfinna Aðalgeirsdóttir,Alexander H Jarosch

doi:10.5194/tc-12-2229-2018

Abstract

Abstract. Bayesian hierarchical modeling can assist the study of glacial dynamics and ice flow properties. This approach will allow glaciologists to make fully probabilistic predictions for the thickness of a glacier at unobserved spatiotemporal coordinates, and it will also allow for the derivation of posterior probability distributions for key physical parameters such as ice viscosity and basal sliding. The goal of this paper is to develop a proof of concept for a Bayesian hierarchical model constructed, which uses exact analytical solutions for the shallow ice approximation (SIA) introduced by Bueler et al. (2005). A suite of test simulations utilizing these exact solutions suggests that this approach is able to adequately model numerical errors and produce useful physical parameter posterior distributions and predictions. A byproduct of the development of the Bayesian hierarchical model is the derivation of a novel finite difference method for solving the SIA partial differential equation (PDE). An additional novelty of this work is the correction of numerical errors induced through a numerical solution using a statistical model. This error-correcting process models numerical errors that accumulate forward in time and spatial variation of numerical errors between the dome, interior, and margin of a glacier.

Highlights

The shallow ice approximation (SIA) is a nonlinear partial differential equation (PDE) that describes ice flow when glacier thickness is relatively small compared to the horizontal dimensions
The primary objective of this paper is to develop a Bayesian hierarchical model (BHM) for glacier flow utilizing the framework espoused by Wikle (2016) and Cressie and Wikle (2015), which allows one to (1) infer ice viscosity and basal sliding parameters and (2) make probabilistic predictions for glacial thickness at unobserved spatiotemporal coordinates
As opposed to using other benchmarks such as the EISMINT experiment (Payne et al, 2000), which itself is based on numerical modeling and subject to numerical errors, the benchmark solutions provided in this work can be treated as ground truth to compare to. (This is in the sense that these are exact solutions of the SIA, but it must be stressed that the SIA is an approximation of the true physical dynamics governing a glacier.) These analytical solutions serve as a basis www.the-cryosphere.net/12/2229/2018/

Summary

Introduction

The shallow ice approximation (SIA) is a nonlinear partial differential equation (PDE) that describes ice flow when glacier thickness is relatively small compared to the horizontal dimensions. The primary objective of this paper is to develop a Bayesian hierarchical model (BHM) for glacier flow utilizing the framework espoused by Wikle (2016) and Cressie and Wikle (2015), which allows one to (1) infer ice viscosity and basal sliding parameters and (2) make probabilistic predictions for glacial thickness at unobserved spatiotemporal coordinates. This BHM relies upon a finite difference scheme for solving the SIA that is inspired by the Lax–Wendroff method (Hudson, 1998). This BHM is verified and diagnosed through a combination of assessments of posterior probability intervals, checks of predictive accuracy for glacial thickness prediction, and a comparison between observed and expected errors due to the numerical solution of the SIA

Objectives

Methods

Results