Doctoral defence in Statistics - Giridhar R. Gopalan

Ph.D. student: Giridhar R. Gopalan

Dissertation title: Spatio-temporal statistical models for glaciology

Opponents:  Dr. Oksana Chkrebtii, Assistant Professor in Statistics, The Ohio State University.
Dr. Hilmar Gudmundsson, Professor in Geography and Environmental Sciences, Northumbria University

Advisor: Dr. Birgir Hrafnkelsson, Professor at the Faculty of Physical Sciences at the University of Iceland

Also in the doctoral committee: Dr. Christopher K. Wikle, Professor of Statistics, University of Missouri.
Dr. Guðfinna Th Aðalgeirsdóttir, Professor of Glaciology, University of Iceland.

Chair of Ceremony: Dr. Oddur Ingólfsson, Professor and the Head of the Faculty of Physical Sciences

Abstract

The purpose of this thesis is to develop spatio-temporal statistical models for glaciology, using the Bayesian hierarchical framework. Specifically, the process level is modeled as a time series of
computer simulator outputs (i.e., from a numerical partial differential equation solver or an emulator) added to an error-correcting statistical process, closely related to the concept of
model discrepancy. This error-correcting process accounts for spatial variability in simulator inaccuracies, as well as the accumulation of simulator inaccuracies forward in time.

For computational efficiency, linear algebra for bandwidth-limited matrices is used for evaluating the likelihood of the model, and first-order emulator inference allows for the fast approximation
of numerical solvers. Additionally, a computationally efficient approximation for the likelihood is derived. Analytical solutions to the shallow ice approximation (SIA) of the full Stokes equation 
system for stress balance of ice are used to examine the speed and accuracy of the computational methods used, in addition to the validity of modeling assumptions.

Moreover, the modeling and methodology within this thesis are tested on actual data sets collected by the University of Iceland Institute of Earth Science (UI-IES) glaciology team, including bi-yearly mass balance measurements at 22-25 fixed sites at Langjökull (a glacier) over 19 years, in addition to 100 meter resolution digital elevation maps. As a byproduct of the construction of the Bayesian hierarchical model, a novel finite difference method is derived for solving the SIA partial differential equation (PDE). Although the application domain of this work is glaciology, the model and methods developed in this thesis can be applied to other geophysical domains.

The thesis is structured around three papers. The first of these papers reviews dynamical modeling of glacial flow, introduces a second-order finite difference method for solving the SIA
PDE, presents a Bayesian hierarchical model involving this numerical solver, and validates the model with analytical solutions to the SIA PDE. The second of these papers generalizes the
statistical model of the first paper, probes higher-order random walks for representing model discrepancy, incorporates first-order emulators, and analyzes methods for efficient log-likelihood evaluation. The third of these papers applies the model framework of the first two papers to mass balance and surface elevation data at Langjökull. 

The major contributions of the thesis are the derivation of a new numerical method for solving the SIA PDE in two spatial dimensions and time, the use of a random walk to represent model
discrepancy (i.e., an error-correcting process), efficient methods for log-likelihood evaluation, and the application of spatio-temporal statistical modeling to Langjökull, one of Iceland’s main glaciers.

About the doctoral candidate

Before joining the University of Iceland, Giri obtained a B.S. in Applied and Computational Mathematics from Caltech and an A.M. in Statistics from Harvard. He has also had work experience in bioinformatics and data science. Giri would like to remain in academia and continue research in spatio-temporal statistics, Bayesian hierarchical modeling, and applications to geophysics.

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