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A Review of Bayesian Modelling in Glaciology

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Statistical Modeling Using Bayesian Latent Gaussian Models

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Abstract

Bayesian methods for modelling and inference are being increasingly used in the cryospheric sciences and glaciology in particular. Here, we present a review of recent works in glaciology that adopt a Bayesian approach when conducting an analysis. We organise the chapter into three categories: (i) Gaussian–Gaussian models, (ii) Bayesian hierarchical models, and (iii) Bayesian calibration approaches. In addition, we present two detailed case studies that involve the application of Bayesian hierarchical models in glaciology. The first case study is on the spatial prediction of surface mass balance across the Icelandic mountain glacier Langjökull, and the second is on the prediction of sea-level rise contributions from the Antarctic ice sheet. This chapter is presented in such a way that it is accessible to both statisticians and Earth scientists.

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Notes

  1. 1.

    https://www.globalmass.eu.

  2. 2.

    https://arcsaef.com/.

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Acknowledgements

Zammit-Mangion was supported by the Australian Research Council (ARC) Discovery Early Career Research Award (DECRA) DE180100203. McCormack was supported by the ARC DECRA DE210101433. Zammit-Mangion and McCormack were also supported by the ARC Special Research Initiative in Excellence in Antarctic Science (SRIEAS) Grant SR200100005 Securing Antarctica’s Environmental Future. The authors would like to thank Bao Vu for help with editing the manuscript, as well as Haakon Bakka for providing a review of an earlier version of this manuscript.

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Authors and Affiliations

  1. Statistical Sciences Group, Los Alamos National Laboratory, Los Alamos, NM, USA

    Giri Gopalan

  2. School of Mathematics and Applied Statistics and Securing Antarctica’s Environmental Future, University of Wollongong, Wollongong, NSW, Australia

    Andrew Zammit-Mangion

  3. Securing Antarctica’s Environmental Future, School of Earth, Atmosphere & Environment, Monash University, Clayton, VIC, Australia

    Felicity McCormack

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  1. Giri Gopalan
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  1. University of Iceland, Reykjavik, Iceland

    Birgir Hrafnkelsson

Appendix: Governing Equations

Appendix: Governing Equations

For ease of exposition, in this appendix, we omit notation that establishes dependence of a variable or parameter on space and time. All variables and parameters should be assumed to be a function of space and time unless otherwise indicated.

Modelling ice-sheet flow relies on the classical laws of conservation of momentum, mass, and energy. For incompressible ice flow, conservation of momentum is described by the full Stokes equations:

$$\displaystyle \begin{aligned} \nabla\cdot\boldsymbol{\sigma} + \rho \boldsymbol{g} &= \boldsymbol{0}, \end{aligned} $$
(11)
$$\displaystyle \begin{aligned} \text{Tr}(\boldsymbol{\dot{\varepsilon}})&=0, \end{aligned} $$
(12)

where \(\nabla \cdot \boldsymbol {\sigma }\) is the divergence vector of the stress tensor \(\boldsymbol {\sigma }\), \(\rho \) is the constant ice density, \(\boldsymbol {g}\) is the constant gravitational acceleration, \(\boldsymbol {\dot {\varepsilon }}\) is the strain rate tensor, and \(\text{Tr}\) is the trace operator. The stress and strain rates are related by the material constitutive relation:

$$\displaystyle \begin{aligned} \boldsymbol{\sigma'}=2\eta\boldsymbol{\dot{\varepsilon}}, \end{aligned} $$
(13)

where \(\boldsymbol {\sigma '}=\boldsymbol {\sigma }+p\boldsymbol {I}\) is the deviatoric stress tensor, p is the pressure, \(\boldsymbol {I}\) is the identity matrix, and \(\eta \) is the viscosity. Boundary conditions for the mechanical model typically assume a stress-free ice surface and the specification of a friction or sliding law at the ice–bedrock interface.

A number of simplifications to the full Stokes equations exist, including the three-dimensional model from Blatter (1995) and Pattyn (2003), the two-dimensional shallow-shelf approximation (MacAyeal, 1989), and the two-dimensional shallow-ice approximation (Hutter, 1983).

Conservation of mass is described by the mass transport equation:

$$\displaystyle \begin{aligned} \frac{\partial H}{\partial t}+\nabla\cdot H\boldsymbol{v} = M_s + M_b, \end{aligned} $$
(14)

where \(\boldsymbol {v}\) is the horizontal velocity, H is the ice thickness, \(M_s\) is the surface mass balance, and \(M_b\) is the basal mass balance. For regional models of ice flow, the thickness is prescribed at the inflow boundaries, and a free-flux boundary condition is typically applied at the outflow boundary.

Finally, conservation of energy is described by the following equation:

$$\displaystyle \begin{aligned} \frac{\partial T}{\partial t} = (\boldsymbol{w}-\boldsymbol{v})\cdot\nabla T+\frac{k}{\rho c}\varDelta T+\frac{\varPhi}{\rho c}, \end{aligned} $$
(15)

where \(\boldsymbol {v}\) is the horizontal velocity, \(\boldsymbol {w}\) is the vertical velocity, T is the ice temperature, k is the constant thermal conductivity, c is the constant heat capacity, and \(\varPhi \) is the heat production term. The boundary conditions typically constitute a Dirichlet boundary condition at the ice surface, a relation for the geothermal and frictional heating at the base of the ice sheet, and a relation for the heat transfer at the ice–ocean interface.

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Gopalan, G., Zammit-Mangion, A., McCormack, F. (2023). A Review of Bayesian Modelling in Glaciology. In: Hrafnkelsson, B. (eds) Statistical Modeling Using Bayesian Latent Gaussian Models . Springer, Cham. https://doi.org/10.1007/978-3-031-39791-2_2

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