This is a description of the Ph.D. project Statistical Models for Glaciology. Giri Gopalan is the Ph.D. student that is working on the project. He started in August 2016.


The objective of this project is to develop statistical models that describe glaciological data in combination with theoretical ice flow dynamics. The value and originality of the project lies in applying state of the art Bayesian hierarchical dynamical spatio-temporal models to the field of glaciology. The student will work with a consortium of experts in Bayesian statistics and glaciology at the University of Iceland and experts in spatial statistics, spatio-temporal models and Bayesian computation from the University of Missouri and the Norwegian University of Science and Technology. The consortium is excited to combine their knowledge and expertise to create state of the art statistical methods for glaciological data and the PhD student will play a key role in developing these methods. The Glaciology group at UI possesses extensive data and knowledge about the Icelandic glaciers. The project is supported by the Icelandic Research Fund.


This project presents a unique opportunity to create new statistical methods for glaciological data. The Bayesian group and the Glaciology group at University of Iceland (UI), two international specialists in spatial statistics, statistical spatio-temporal models and Bayesian computation, one at University of Missouri and the other at Norwegian University of Science and Technology, are ready to combine their knowledge and expertise to create state of the art statistical methods for glaciological data. The Glaciology group at UI possesses extensive data and knowledge about the Icelandic glaciers.

Statistical models for ice flow based on glaciological data have not been studied to a great extent. Berliner et al. 2008a and Berliner et al. 2008b presented a Bayesian hierarchical model for ice velocities in one spatial dimension.

The objective of this project is to develop statistical models that describe glaciological data in combination with theoretical ice flow dynamics.

Knowledge from the field of statistics and the field of glaciology will be combined creating a common methodology through an active collaboration on this project. In particular, the plan is to model the surface elevation and its evolution in two spatial dimensions utilizing various degrees of approximations to the dynamics described by the full Stokes equations as well as to model the response to climatically forced changes in mass balance. To achieve this goal field data from Langjökull Ice Cap will be used, an ice cap relatively simple in terms of its topography and dynamics. However, the plan is to set forth methods that can be extended to other glaciers, such as Vatnajökull Ice Cap or the Greenland Ice Sheet.

The value and originality of this work lies in applying state of the art Bayesian hierarchical dynamical spatio-temporal models to the field of glaciology. Inclusion of uncertainty in glaciological models needs to be explored further. The Bayesian methodology has the potential to make best use of available observational data as it combines the theoretical ice flow dynamics and stochastic modeling.


Physical models of glacier dynamics are formulated by partial differential equations that are typically solved numerically on grids. The models describe force balance, constitutive relations (flow laws) and basal sliding coupled with mass continuity that is forced with evolving surface and basal mass balance. The time evolution of the glacier geometry is forced by the climatic mass balance, that is typically described by meteorological data such as temperature and precipitation. This is a free boundary problem. The inputs to these models are data on the geometry and mass balance of the glacier. The output is the spatial and temporal change in glacier geometry. Actual measurements of the surface elevation are not used in the modeling but rather applied to evaluate and validate the model performance by comparing the model output to them. This is in contrast to the methodology of statistics where the data always enter the model directly. By using state of the art statistics, information contained within the data could be extracted more efficiently than using surface elevation measurements only for comparison to physical model output.

Bayesian hierarchical models (BHMs) are well suited for learning about complex systems that can be described with dynamic models (e.g. Cressie and Wikle 2011; Berliner 2003). Information about the states within the dynamic models (e.g. surface elevation, mass balance) come through one or more data sources which are direct or indirect measurements of the states. BHMs often come in three levels, namely, data level, process level and parameter level. At the data level a description of the measurement error and a link to the states in the dynamic model are given. At the process level a probabilistic model, that takes into account both the dynamic model and a stochastic component, is specified. At the parameter level prior distributions are laid out. Berliner et al. 2008a and Berliner et al. 2008b presented a BHM for ice velocities in one spatial dimension. The physics were taken into account, however, the temporal component was not modeled.

The dynamic model mentioned above can be solved at various levels of complication, from the 0th order Shallow Ice Approximation (SIA) (Cuffey and Paterson, 2010; Hutter, 1982) that only takes into account vertical shear stresses to the full Stokes formulation (e.g. Le Meur et al., 2004 for comparison of SIA and Full Stokes models; Jarosch 2008) with various degrees of higher order models in between that include horizontal stress gradients (e.g. Hindmarsh, 1993; Blatter, 1995; Pattyn, 2000). The full Stokes model is computationally demanding as it requires solving five partial differential equations plus kinematic boundary condition for four variables at each grid point (isotropic pressure and 3D velocity). It is solved using the finite element method on an irregular 3D grid (e.g. Gillet-Chaulet et al., 2012). The SIA model on the other hand is often vertically integrated and solves one partial differential equation for one unknown variable at each grid point and is often solved with finite difference method on 2D regular grid (e.g. Mahaffy, 1976; Aðalgeirsdóttir et al., 2006). The climatic mass balance is usually calculated from gridded data of estimated weather (e.g. temperature and precipitation) using a temperature-index mass balance model (Reeh, 1991) and sometimes energy balance model, where radiation measurements are available (Hock, 2005). In some areas direct observations of climatic mass balance are available for some periods of time.


Figure 1. Surface topography of Langjökull Ice Cap: (A) Location of mass balance sites and the surface elevation tracks from field campaigns in 2013. (B) High resolution 2007 DEM, based on a LiDAR survey in 2007 (survey NERC: Gary M. Llewellyn, Ian Willis, Neil Arnold, DEM created by Finnur Pálsson).

Extensive knowledge of the glaciers in Iceland has been gathered, which makes Iceland an ideal place for developing BHM for glaciological data. It is proposed to conduct the model development for Langjökull Ice Cap in central west Iceland. The bedrock topography was mapped with radio echo sounding methods in 1997. Surface Digital Elevation Models (DEMs) with variable accuracy exist for the years 1937, 1945, 1986, 1997, 2004, 2007 and 2012. Figure 1 shows the 2007 high resolution DEM based on a LiDAR survey in 2007. These have already been used to estimate the long term mass balance of Langjökull Ice Cap (Pálsson et al. 2012). Since 1997 climatic mass balance has been measured every year. Annually there are two field campaigns to the glacier, in the spring to measure the amount of winter accumulation (winter mass balance) and again in the autumn to measure summer ablation. Winter, summer and net balance are deduced from the survey data acquired at the sites L01-L23 in Figure 1. In addition to collection of mass balance data, the surface elevation at all the survey sites is measured with Global Positioning System (GPS). Additionally the surface elevation on tracks between the surveys’ sites is also measured (Figure 1). The SIA model has been applied for Langjökull Ice Cap to model the evolution during different time periods in the past (Flowers et al. 2007, 2008) and possible future evolutions (Guðmundsson et al. 2009a, 2009b).


The SMG project is planned as a three year project with the work planned as three work packages. The three work packages are listed below along with their respective methodology and time schedules.

Work package 1

Construction of a BHM based on the shallow ice approximation by applying direct measurements of the glacier surface mass balance. Observations of the surface elevation will be modeled as a function of a surface height field and a measurement error term at the data level. The mass balance observations are modeled at the data level in the same way as the surface elevation.

At the latent level (process level) of the BHM the surface elevation field is modeled on a regular grid as a function of the mass balance field. Working on a regular grid the SIA dynamics can be described through a discretized finite difference form of the SIA equation. This creates spatio-temporal dependence in the surface elevation field through direct dependence between neighbor grid points of the field. In addition a spatio-temporally dependent stochastic component is needed for modeling possible deviation from the SIA dynamics and error resulting from its discretization. Prior distributions for unknownparameters at the two levels above are specified at the third level (parameter level). Such prior distributions can both reflect prior knowledge of parameter behavior or lack of knowledge.

Time schedule:

1.1 Gathering, organizing and preparing the available data for Langjökull Ice Cap (January 2016 – May 2016)

1.2. Literature study and development (January 2016 – May 2016)

1.3. Propose a BHM for the SIA model which uses direct surface height and field mass balance measurements (May 2016 – July 2016)

1.4. Bayesian inference for the SIA and programming (August 2016 – October 2016)

1.5. Collect results. Write a paper on the BHM for SIA (November 2016 – March 2016)

Work package 2

The second task is to incorporate the mass balance component through meteorological observations as opposed to direct field measurements. The plan is to develop a new statistical model for the mass balance based on meteorological observations as covariates, i.e., temperature and precipitation observations from nearby meteorological stations. In this case the true mass balance component enters the BHM through a prior distribution that can be specified at the latent level. This prior distribution is based on the new statistical model. The motivation behind this task is the fact that direct field mass balance measurements are not always available. In case of Langjökull Ice Cap, the new statistical model can be validated by comparing its predictions to the in-situ data (1997-2013).

Time schedule:

2.1 Propose a new statistical model for mass balance based on meteorological data (April 2017 – July 2017)

2.2 Set forth a BHM for the SIA which incorporates the new mass balance model (August 2017 – September 2017)

2.3 Bayesian inference for the updated BHM for SIA and programming (October 2017 – November 2017)

2.4 Collect results. Write paper on the updated new mass balance model and the BHM for SIA (December 2017 – May 2018)

Work package 3

The third and last task involves progressing towards the full Stokes equations for the ice dynamics at the process level of the BHM instead of the SIA. There is an enormous difference in computational scope between the SIA and full Stokes model. Computational feasibility will be evaluated and possibly a search for a formulation of intermediate complexity between the SIA and Stokes models will be conducted. This involves investigating approximations of higher order than the zero-th order SIA.

Time schedule:

3.1 Computational feasibility of a BHM based on Stokes equation evaluated and an investigation of higher order approximations (June 2018 – September, 2018)

3.2 Write paper on a BHM for a higher order approximation (October 2018 – December 2018)

The PhD student will participate in field excursions to Langjökull Ice Cap in spring and autumn every year, in collaboration with the glaciology group at the IES (Finnur Pálsson). Three visits of PhD student to international collaborators are planned. The first visit is to Missouri, US (Christopher K. Wikle) in February-March 2015, and the second and the third visits are to Trondheim, Norway (Håvard Rue) in October-November 2015 and November-December 2016.

Planned are three conference presentations:

1. Workshop on Bayesian inference for latent Gaussian models (LGM) in UK August 2016.

2. European Geophysical Union (EGU) General Assembly in April 2017.

3. EGU General Assembly in April 2018 or alternatively LGM in August 2018.


The SMG project is organised in the work plan and the schedule shown in the previous section, which lists the major activities planned for the 36 months of the project. The result of each work package will be assessed in the last week of the planned period of the respective work package. The following list of milestones will be used to guide this assessment. Each milestone is shown with red color in the table above.

M1: Project start (Jan 2016).

M2: Ph.D. candidate visits Christopher Wikle in Missouri, USA (Aug-Sept 2016).

M3: Bayesian Hierarchical Model (BHM) for Shallow Ice Approximation (SIA) Model and Mass Balance measurements fully developed (Nov 2016).

M4: Ph.D. candidate visits Håvard Rue in Trondheim, Norway (April-May 2017).

M5: BHM for SIA with new mass balance model fully developed (Dec 2017).

M6: Ph.D. candidate visits Håvard Rue in Trondheim, Norway (May-Jun 2017).

M7: BHM for a higher order ice flow model fully developed (Oct 2018).

M8: Project end (Dec 2018)

The following deliverables are planned within the project. The deliverables are marked with

green colors in the table above.

D1: Conference presentation at LGM2016 (Aug 2016).

D2: Paper 1 submitted to an international journal (Mar 2017).

D3: Conference presentation at EGU general assembly 2017 (Apr 2017).

D4: Paper 2 submitted to an international journal (May 2018).

D5: Conference presentation at EGU general assembly 2018 (Apr 2018) or alternatively LGM 2018 (Aug 2018).

D6: Paper 3 submitted to an international journal (Dec 2018).


The SMG project will be a close collaboration between the Institute of Earth Sciences (IES) at the University of Iceland (UI) and the Bayesian statistics group at UI. Additionally two international colleagues will collaborate with the groups on the project. Information on the two groups, with participating personnel marked in bold font, is given below as well as information on the two international collaborators. The Bayesian Group at the Science Institute at UI is active in research and collaboration both domestically and internationally. In 2013 the group organized and hosted The third Workshop on Bayesian Inference for Latent Gaussian Models with Applications which was held in Reykjavik and attended by seventy scientists from around the world. The research focus of the group is on developing new Bayesian models for meteorological, geophysical and hydrological data as well as computational methods for these models. The current/ongoing projects involve models for precipitation, river discharge, floods and temperature. Birgir Hrafnkelsson, Associate Professor, is the head of Bayesian Group at UI. Other members are Egil Ferkingstad and Óli Páll Geirsson, both Research Scholars at the Science Institute at UI. The main collaborators of the group are Håvard Rue at Norwegian University of Science and Technology, Daniel Simpson at University of Bath, Christopher K. Wikle at University of Missouri, Einar Sveinbjörnsson at Vedurvaktin ehf., and a number of collaborators at the Icelandic Meteorological Office (IMO) and UI.

The Bayesian group at UI is currently working on a general Markov chain Monte Carlo sampling scheme for latent Gaussian models that takes advantage of sparse matrix structure (Geirsson et al. 2015a) and penalized complexity prior for latent Gaussian models. Recent projects include spatial modeling of annual 24 hour extreme precipitation (Geirsson et al. 2015b), spatial modeling of annual minimum and maximum temperatures (Hrafnkelsson et al. 2012a) and spatial predictions of monthly accumulated precipitation (Sigurdarson and Hrafnkelsson 2015). The precipitation projects utilized outputs from a meteorological model of orographic precipitation from IMO. A project on statistical rating curves that describe the relationship between discharge and water level at a given observation site in a river is presented in Hrafnkelsson et al. (2015). The study extends the traditional rating curve into a generalized rating curve based on a Bayesian hierarchical model (BHM) and looks at how these rating curves relate to the underlying hydraulics. This is based on the work by Hrafnkelsson et al., 2012b.

The Glaciology Group at the Institute of Earth Sciences is an active and growing research group with a glaciological research program that has taken advantage of the proximity and easy access to the largest glaciers in Europe. The group has actively participated in numerous collective research efforts on national and international levels. During the last decades valuable experience and data has been obtained in numerous field campaigns. The bedrock topography of the largest Icelandic ice caps has been surveyed with radio-echo sounding methods and the surface topography has been mapped a number of times with various methods, including airborne, satellite and ground based measurements. For the last two decades the surface mass balance (Vatnajökull Ice Cap since 1991-92 at about 60 survey sites and Langjökull Ice Cap since 1996-97 at 23 sites) and the surface energy balance in summer has been estimated from Automatic Weather Station data (~ 10 sites on Vatnajökull Ice Cap since 1996, and 2 sites on Langjökull Ice Cap since 2001). Thus detailed knowledge exists of the evolution of the glaciers during this period (e.g. Björnsson, 1986; Björnsson and Pálsson, 2008; Björnsson et al., 2002, 2013; Flowers et al., 2007, 2008; Guðmundsson et al., 2009a, 2009b, 2011; Hodgkins et al. 2012). Numerical models have been applied to simulate the surface mass balance and glacier dynamics to put the evolution into climate change context (Aðalgeirsdóttir et al., 2006, 2011; Flowers et al., 2005; Marshall et al., 2005).

Guðfinna Aðalgeirsdóttir, Associate Professor, joined the group in summer 2012. She works with numerical models of glacier dynamics, climate models and the coupling of ice sheet and climate models to simulate evolution of glaciers in response to climate change and the resulting sea level rise. Alexander Jarosch, Research Associate Professor, joined the group in September 2013. He specializes in geophysical fluid dynamics and numerical models of glaciological systems. Eyjólfur Magnússon, post-doctoral researcher focuses on ice dynamics, glacier changes and application of remote sensing data in glaciology. Finnur Pálsson is an electronic engineer who has been in charge of IES field work (including radio echo sounding, GPS and mass balance surveys, AWS) in excess of three decades. He has produced surface and bedrock maps for all of Iceland´s larger ice caps and the mass balance records from the field data. Þorsteinn Jónsson and Sveinbjörn Steinþórsson are research technicians with a long experience in glaciological field work. Helgi Björnsson, a world renowned glaciologist, who initiated the glaciological work at IES in the 1970’s, recently retired and is now Professor Emeritus but is still actively working with the group. His continued efforts throughout the years form the foundation of many future projects within the Glaciology Group.

Christopher K. Wikle is a Professor of Statistics at University of Missouri. He is the author of more than 100 articles on the topics of spatio-temporal methodology, spatial statistics, and Bayesian hierarchical models and the co-author of the book Statistics for Spatio-Temporal Data. In his work these methods are applied to problems in the fields of climatology, meteorology and oceanography among other. Ph.D. candidate will visit Prof. Wikle at University of Missouri while he/she is working on the project.

Håvard Rue is a Professor of Statistics at Norwegian University of Science and Technology (NTNU) within Department of Mathematical Sciences under the statistics group. His research interests include Gaussian Markov Random Fields and Bayesian computation. He is the author of the book Gaussian Markov Random Fields: Theory and Applications (Rue and Held, 2005) and numerous articles. The statistics group at NTNU has offered the Ph.D. candidate to visit NTNU over a few periods while he is working on the project.


The central part of the SMG project is a PhD project, set to start in Januar 2016. The main research in the project is to be carried out by the Ph.D. candidate under the supervision of the principal investigator, co-proposers and other participants. We recognize the importance of professional networking and the exposure gained at international conferences and therefore a part of the project’s budget is dedicated to travel expenses due to conference attendance and visits to international colleagues.


Glaciers in Iceland cover currently about 11% of the land area, but the rapid mass losses that are being observed can lead to dramatic changes in the landscape and appearance of the country. Improved understanding and capability of projecting future changes are therefore of high importance. An important aspect of this is to assess the uncertainty in models and predictions and this is the aspect we aim to improve.

The scientific impact we aim to achieve is to open a new dimension of research to the field of glaciology by developing state of the art statistical methods for glaciology data and create a new methodology that is based on methodology from both fields of statistics and glaciology. In order to achieve this impact the plan is to work with leading researchers in both fields and to present the research in international journals and at international conferences representing both fields.

Secondly, the project will strengthen collaboration between the glaciology group and the statistics group at UI and encourage further collaboration between the two groups. Furthermore the project has the potential to encourage collaboration between the fields of statistics and glaciology on an international level. Finally, this project involves the training of a young scientist, i.e. the PhD candidate, in an international collaborative research.


The primary results of the research are to be submitted to peer-reviewed scientific journals with an emphasis on open-source journals. The target journals in glaciology will be the Cryosphere ( and the International Glaciological Society Journal of Glaciology ( The target journals in statistics are Environmetrics, The Annals of Applied Statistics (, Bayesian Analysis ( and Journal of the American Statistical Association.

The PhD thesis is planned as three peer-reviewed publications. The planned publications are:

1. Paper on the BHM for SIA based on direct field mass balance measurements (Mar 2017)

2. Paper on the updated new mass balance model and the BHM for SIA (May 2018)

3. Paper on a BHM for a higher order approximation (Dec 2018)

As detailed above, the plan is to present the research at least 3 international conferences. Additionally the research will be presented by the PhD student at visits to international colleagues. Domestically the research will be presented in lectures for researchers and students in both geophysics and statistics.


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Aðalgeirsdóttir G., T. Jóhannesson, H. Björnsson, F. Pálsson, O. Sigurðsson, 2006, The response of Hofsjökull and southern Vatnajökull, Iceland, to climate change. J. Geophys. Res., 111, F03001, doi:10.1029/2005JF000388

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