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

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.

A Hierarchical Spatiotemporal Statistical Model Motivated by Glaciology

“…As such, glaciology synthesizes elements from a multitude of scientific disciplines including continuum mechanics, fluid mechanics, hydraulics, chemistry, and meteorology. Bueler et al (2005) introduce analytical solutions for the SIA PDE, a commonly used model for the dynamics of glaciers (Fowler and Larson, 1978;Hutter, 1982;Flowers et al, 2005;Cuffey and Paterson, 2010;van der Veen, 2013;Brinkerhoff et al, 2016;Guan et al, 2016;Gopalan et al, 2018). Based on the principle of conservation of mass, the SIA dictates that glacier flow is in the direction of the (negative) gradient of the glacier surface and is due to gravity and basal sliding (also referred to as friction or drag if in the direction of the positive gradient).…”

“…Based on the principle of conservation of mass, the SIA dictates that glacier flow is in the direction of the (negative) gradient of the glacier surface and is due to gravity and basal sliding (also referred to as friction or drag if in the direction of the positive gradient). While an explanation of the SIA PDE is given in Gopalan et al (2018), our focus is on ice viscosity, B. Intuitively, this parameter controls the softness of the ice.…”

“…Such scenarios can be modeled with a variant of a Bayesian hierarchical spatio-temporal model that was introduced in Gopalan et al (2018) for glacial dynamics, if considered more generally. We delineate three methods to make posterior inference efficient: the first is to utilize bandwidth limited linear-algebraic routines for likelihood evaluation (Rue, 2001), the second is to utilize an embarrassingly parallel approximation to the likelihood, and the third is to use first-order emulators (Hooten et al, 2011) for speeding up computer simulators.…”

“…This approach has been applied in a variety of scientific contexts, including the study of ozone concentrations (Berrocal et al, 2014), sediment loads at the Great Barrier Reef (Pagendam et al, 2014), precipitation in Iceland (Sigurdarson and Hrafnkelsson, 2016), Antarctic contributions to sea level rise (Zammit-Mangion et al, 2014), and tropical ocean surface winds (Wikle et al, 2001) (among many others). In Gopalan et al (2018), the motivating example for the work in this paper, a Bayesian hierarchical model for shallow glaciers based on the shallow ice approximation (SIA) PDE was developed and evaluated. Kennedy and O'Hagan (2001) suggest constructing Bayesian statistical models that incorporate the output of a computer simulator of a physical process, such as a numerical solver for the underlying system of PDEs.…”

“…We must also be clear about what distinguishes this work from its predecessor, Gopalan et al (2018).…”