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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Earth Sci.</journal-id>
<journal-title>Frontiers in Earth Science</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Earth Sci.</abbrev-journal-title>
<issn pub-type="epub">2296-6463</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.3389/feart.2021.610069</article-id>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Earth Science</subject>
<subj-group>
<subject>Brief Research Report</subject>
</subj-group>
</subj-group>
</article-categories>
<title-group>
<article-title>Bayesian Inference of Ice Softness and Basal Sliding Parameters at Langj&#x000F6;kull</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author" corresp="yes">
<name><surname>Gopalan</surname> <given-names>Giri</given-names></name>
<xref ref-type="aff" rid="aff1"><sup>1</sup></xref>
<xref ref-type="corresp" rid="c001"><sup>&#x0002A;</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/1097104/overview"/>
</contrib>
<contrib contrib-type="author">
<name><surname>Hrafnkelsson</surname> <given-names>Birgir</given-names></name>
<xref ref-type="aff" rid="aff2"><sup>2</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/1153920/overview"/>
</contrib>
<contrib contrib-type="author">
<name><surname>A&#x000F0;algeirsd&#x000F3;ttir</surname> <given-names>Gu&#x000F0;finna</given-names></name>
<xref ref-type="aff" rid="aff3"><sup>3</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/228492/overview"/>
</contrib>
<contrib contrib-type="author">
<name><surname>P&#x000E1;lsson</surname> <given-names>Finnur</given-names></name>
<xref ref-type="aff" rid="aff3"><sup>3</sup></xref>
<uri xlink:href="http://loop.frontiersin.org/people/901508/overview"/>
</contrib>
</contrib-group>
<aff id="aff1"><sup>1</sup><institution>Statistics Department, California Polytechnic State University</institution>, <addr-line>San Luis Obispo, CA</addr-line>, <country>United States</country></aff>
<aff id="aff2"><sup>2</sup><institution>Faculty of Physical Sciences, School of Engineering and Natural Sciences, University of Iceland</institution>, <addr-line>Reykjavik</addr-line>, <country>Iceland</country></aff>
<aff id="aff3"><sup>3</sup><institution>University of Iceland Institute of Earth Sciences, University of Iceland</institution>, <addr-line>Reykjavik</addr-line>, <country>Iceland</country></aff>
<author-notes>
<fn fn-type="edited-by"><p>Edited by: Alun Hubbard, Aberystwyth University, United Kingdom</p></fn>
<fn fn-type="edited-by"><p>Reviewed by: Douglas John Brinkerhoff, University of Montana, United States; Gwenn Flowers, Simon Fraser University, Canada</p></fn>
<corresp id="c001">&#x0002A;Correspondence: Giri Gopalan <email>ggopalan&#x00040;calpoly.edu</email></corresp>
<fn fn-type="other" id="fn001"><p>This article was submitted to Cryospheric Sciences, a section of the journal Frontiers in Earth Science</p></fn></author-notes>
<pub-date pub-type="epub">
<day>10</day>
<month>05</month>
<year>2021</year>
</pub-date>
<pub-date pub-type="collection">
<year>2021</year>
</pub-date>
<volume>9</volume>
<elocation-id>610069</elocation-id>
<history>
<date date-type="received">
<day>24</day>
<month>09</month>
<year>2020</year>
</date>
<date date-type="accepted">
<day>12</day>
<month>04</month>
<year>2021</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#x000A9; 2021 Gopalan, Hrafnkelsson, A&#x000F0;algeirsd&#x000F3;ttir and P&#x000E1;lsson.</copyright-statement>
<copyright-year>2021</copyright-year>
<copyright-holder>Gopalan, Hrafnkelsson, A&#x000F0;algeirsd&#x000F3;ttir and P&#x000E1;lsson</copyright-holder>
<license xlink:href="http://creativecommons.org/licenses/by/4.0/"><p>This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</p></license> </permissions>
<abstract><p>We develop Bayesian statistical models that are designed for the inference of ice softness and basal sliding parameters, important glaciological quantities. These models are applied to Langj&#x000F6;kull, the second largest temperate ice cap in Iceland at about 900 squared kilometers in area. The models make use of a relationship between physical parameters and ice velocity as stipulated by a shallow ice approximation that is generally applicable to Langj&#x000F6;kull. The posterior distribution for ice softness concentrates around 18.2 &#x000D7; 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>; moreover, spatially varying basal sliding parameters are inferred allowing for the decomposition of velocity into a deformation component and a sliding component, with spatial variation consistent with previous studies. Bayesian computation is conducted with a Gibbs sampling approach. The paper serves as an example of statistical inference for ice softness and basal sliding parameters at temperate, shallow glaciers using surface velocity data.</p></abstract>
<kwd-group>
<kwd>surface velocity</kwd>
<kwd>basal sliding</kwd>
<kwd>data analysis-methods</kwd>
<kwd>Bayesian inference</kwd>
<kwd>ice properties</kwd>
</kwd-group>
<contract-sponsor id="cn001">Icelandic Centre for Research<named-content content-type="fundref-id">10.13039/501100001840</named-content></contract-sponsor>
<counts>
<fig-count count="4"/>
<table-count count="0"/>
<equation-count count="6"/>
<ref-count count="36"/>
<page-count count="9"/>
<word-count count="6158"/>
</counts>
</article-meta>
</front>
<body>
<sec sec-type="intro" id="s1">
<title>1. Introduction</title>
<p>The dynamics of glaciers have become of greater scientific interest in recent times due to global climate change and its impact on the size and flow of glaciers; perhaps most crucially, melting glaciers have an effect on sea levels (Bj&#x000F6;rnsson et al., <xref ref-type="bibr" rid="B5">2006</xref>; Zammit-Mangion et al., <xref ref-type="bibr" rid="B36">2014</xref>; Hock et al., <xref ref-type="bibr" rid="B18">2019</xref>). Glacial dynamics are dependent on two main physical parameters: ice softness, related to the deformation of ice, and basal sliding, related to basal velocity (Cuffey and Paterson, <xref ref-type="bibr" rid="B11">2010</xref>). Both ice softness and basal sliding parameters cannot be measured directly and must be estimated in order to properly understand glacial dynamics and, consequently, globally relevant phenomena such as sea level rise. The purpose of this paper is to use surface velocity data in combination with Bayesian statistical inference in order to infer ice softness and basal sliding parameters, with particular application to Langj&#x000F6;kull, a prominent Icelandic glacier (Bj&#x000F6;rnsson, <xref ref-type="bibr" rid="B4">2017</xref>). Iceland, about 10 percent of whose area is covered by glaciers (Bj&#x000F6;rnsson and P&#x000E1;lsson, <xref ref-type="bibr" rid="B6">2008</xref>), is a natural laboratory for cryosphere science, particularly in an era of increasing temperatures and glacial melting.</p>
<p>A simplified definition of a glacier is a mass of ice that is situated upon bedrock or sediment, where ice is accreted from compacted snowfall and is lost due to melting. Ice slowly flows in the direction of the negative gradient of the glacier surface. The flow is thought to be a combination of deformation of the ice due to gravity and sliding of the glacier at the bed. For glaciers that are temperate such as Langj&#x000F6;kull, ice softness, which governs the rate of ice deformation, is also constant because softness is dependent on temperature. In contrast, the basal sliding parameter can vary both spatially and temporally, especially during glacial surge events (Bj&#x000F6;rnsson et al., <xref ref-type="bibr" rid="B8">2003</xref>).</p>
<p>We employ Bayesian statistical modeling of surface velocities in order to obtain posterior distributions for ice softness and basal sliding parameters; as such, the approach allows for uncertainty quantification of these pivotal glaciological quantities. Bayesian inference has appeared in the glaciology literature before, albeit for different aims. For instance, Brinkerhoff et al. (<xref ref-type="bibr" rid="B9">2016</xref>) use Bayesian inference to estimate subglacial topography, while Werder et al. (<xref ref-type="bibr" rid="B35">2020</xref>) and Guan et al. (<xref ref-type="bibr" rid="B16">2016</xref>) use Bayesian inference to estimate ice thickness, and Minchew et al. (<xref ref-type="bibr" rid="B24">2015</xref>) use Bayesian methods to infer velocity fields from multiple repeat-pass interferometric synthetic aperture radar (InSAR) measurements in Iceland. Petra et al. (<xref ref-type="bibr" rid="B28">2014</xref>) and Isaac et al. (<xref ref-type="bibr" rid="B21">2015</xref>) develop computationally efficient methods for inverse problems involving ice-sheet models, within a Bayesian framework. Perhaps the closest Bayesian inference papers in the literature to the work described here are in Pralong and Gudmundsson (<xref ref-type="bibr" rid="B31">2011</xref>) and Raymond and Gudmundsson (<xref ref-type="bibr" rid="B32">2009</xref>), which use Bayesian inference to estimate basal sliding and basal topography properties jointly. Building off of this work, we infer ice softness and basal sliding parameters jointly, and infer posterior probability distributions for ice softness and basal sliding parameters. In many instances subglacial topography is not known and must be inferred, but in the case of Langj&#x000F6;kull basal topography is known due to the radio-echo survey of the bed Bj&#x000F6;rnsson and P&#x000E1;lsson (<xref ref-type="bibr" rid="B7">2020</xref>) and surface elevation topography maps from various sources are also available (P&#x000E1;lsson et al., <xref ref-type="bibr" rid="B27">2012</xref>).</p>
<p>An important result is that the estimated value of ice softness is comparable to the recommended value for temperate glaciers from Cuffey and Paterson (<xref ref-type="bibr" rid="B11">2010</xref>), despite that a nearly flat prior is used. In contrast to the previous works, the Bayesian estimate also comes with an uncertainty estimate, here represented with the posterior standard deviation. The posterior is substantially more precise (i.e., lower standard deviation) than the prior, suggesting that the learning of physical parameters is achieved. The inferred spatial variation in basal sliding appears to be consistent with prior work on characterizing horizontal velocities at Langj&#x000F6;kull (Minchew et al., <xref ref-type="bibr" rid="B24">2015</xref>).</p>
<p>The structure of the paper is as follows. In section 2, we begin with an overview of the equations that are used to relate ice softness and basal sliding parameters to glacier surface velocities computed with the shallow ice approximation of the stress balance, as derived in Marshall et al. (<xref ref-type="bibr" rid="B23">2005</xref>). A detailed description of Bayesian statistical models of surface velocity and an explanation of the Gibbs sampler (i.e., a Markov chain Monte Carlo approach for sampling from the posterior distribution) that is used for inference of physical parameters follows. Additionally, we describe the surface elevation, bed topography, and surface velocity data from the University of Iceland Institute of Earth Sciences (UI-IES). Section 3 discusses the results of applying the Bayesian statistical models to the Langj&#x000F6;kull data. Section 4 discusses these results in a more general glaciological context, points out limitations of the approach, and suggests future avenues for work. While the focus of this paper is Langj&#x000F6;kull, the delineated approach can be applied to other glaciers where data on surface velocity, bed, and surface topography are available, and the SIA is appropriate. The melting of such glaciers is expected to contribute to the rise of sea levels within the next century, supporting the scientific importance of the statistical modeling and inference within this paper.</p></sec>
<sec id="s2">
<title>2. Method</title>
<p>The Bayesian statistical models that are used in this analysis involve a shallow ice approximation (SIA) model for glacier surface velocity in 2 horizontal dimensions. The probability distributions for surface velocity data use these equations and take either a normal or t distribution. Prior distributions are used for the ice softness and basal sliding parameters, and a Markov chain Monte Carlo (MCMC) algorithm known as Gibbs sampling is used to conduct posterior inference. In the following subsections, we discuss these matters in more depth, and conclude with a description of the data sources that we apply the Bayesian statistical modeling and methodology to. We refer the reader to the Appendix, section 5, for further mathematical equations and details.</p>
<sec>
<title>2.1. Review of Exact Shallow Ice Approximation Surface Velocity Equations</title>
<p>We use an analytically exact SIA model for horizontal velocities without longitudinal coupling, as in Equation (12) of section 2.2 of Marshall et al. (<xref ref-type="bibr" rid="B23">2005</xref>), a section which presents the derivation of the SIA for a 2 dimensional model. The SIA is valid when the thickness of a glacier is much less than horizontal dimensions, as occurs at Langj&#x000F6;kull (Bj&#x000F6;rnsson and P&#x000E1;lsson, <xref ref-type="bibr" rid="B6">2008</xref>). A SIA (Hutter, <xref ref-type="bibr" rid="B20">1983</xref>) is a commonly used physical model for shallow glacial dynamics as for instance in Marshall et al. (<xref ref-type="bibr" rid="B23">2005</xref>), Bueler et al. (<xref ref-type="bibr" rid="B10">2005</xref>), Jarosch et al. (<xref ref-type="bibr" rid="B22">2013</xref>), and Gopalan et al. (<xref ref-type="bibr" rid="B15">2018</xref>), to give a few examples. A SIA has been applied specifically to Langj&#x000F6;kull in Flowers et al. (<xref ref-type="bibr" rid="B12">2007</xref>), for instance. By <italic>x</italic> and <italic>y</italic> directions, we are referring to their typical usage in a Cartesian coordinate system. In particular, Marshall et al. (<xref ref-type="bibr" rid="B23">2005</xref>) show that the exact surface velocities in the <italic>x</italic> and <italic>y</italic> directions, <italic>v</italic><sub><italic>x</italic></sub> and <italic>v</italic><sub><italic>y</italic></sub> respectively, are given by:</p>
<disp-formula id="E1"><label>(1)</label><mml:math id="M1"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003B3;</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>&#x003C1;</mml:mi><mml:mi>g</mml:mi><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E2"><label>(2)</label><mml:math id="M2"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>v</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>&#x003B3;</mml:mi><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>&#x0002B;</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x0002B;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>&#x003C1;</mml:mi><mml:mi>g</mml:mi><mml:mi>&#x003B1;</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>In the above equations, <italic>B</italic> is the constant ice-softness parameter and &#x003B3;(<italic>x, y, t</italic>) is the basal sliding field parameter, which varies spatially and temporally. Additionally, &#x003C1; is the density of ice, <italic>g</italic> is gravitational acceleration, <italic>n</italic> is a constant from the constitutive relation for temperate ice that is typically set to 3 (and is precisely 3 in this paper), and <italic>H</italic>(<italic>x, y, t</italic>) is glacial thickness.</p>
<p>Furthermore, if <italic>S</italic>(<italic>x, y, t</italic>) denotes the glacier surface elevation, basal shear stress is given by:</p>
<disp-formula id="E3"><label>(3)</label><mml:math id="M3"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>&#x003C1;</mml:mi><mml:mi>g</mml:mi><mml:mi>H</mml:mi><mml:mfrac><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<disp-formula id="E4"><label>(4)</label><mml:math id="M4"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:msub><mml:mrow><mml:mi>&#x003C4;</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi>&#x003C1;</mml:mi><mml:mi>g</mml:mi><mml:mi>H</mml:mi><mml:mfrac><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>and &#x003B1; is the surface slope, given by:</p>
<disp-formula id="E5"><label>(5)</label><mml:math id="M5"><mml:mtable class="eqnarray" columnalign="left"><mml:mtr><mml:mtd><mml:mi>&#x003B1;</mml:mi><mml:mo>=</mml:mo><mml:msqrt><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>&#x0002B;</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mo stretchy="true">(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo stretchy="true">)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msqrt><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>As such, glacial flow under this model is in the direction of the negative surface gradient, the <inline-formula><mml:math id="M6"><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula> and <inline-formula><mml:math id="M7"><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>&#x02202;</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:mfrac></mml:math></inline-formula> terms.</p></sec>
<sec>
<title>2.2. Bayesian Statistical Models for Surface Velocities</title>
<p>We construct and use Bayesian statistical models that allow for the inference of ice softness and basal sliding parameters. The models have two important pieces &#x02013; the first relates the analytical SIA surface velocities, <italic>v</italic><sub><italic>x</italic></sub> and <italic>v</italic><sub><italic>y</italic></sub>, which in turn depend on the physical parameters <italic>B</italic> and &#x003B3;, to the observed surface velocity data. The second important aspect of the Bayesian model is the prior distribution for parameters. In the next few subsections we describe both the data distributions and prior distributions, and follow up with how Bayesian inference is conducted computationally.</p>
<sec>
<title>2.2.1. Normal Distribution for Surface Velocities</title>
<p>We first consider a normal distribution (i.e., a normal model) for surface velocity data. Specifically, the observed velocity in the <italic>x</italic> direction for a particular site <italic>i</italic> with spatio-temporal coordinates (<italic>x</italic><sub><italic>i</italic></sub>, <italic>y</italic><sub><italic>i</italic></sub>, <italic>t</italic><sub><italic>i</italic></sub>) is a normal distribution with mean <italic>v</italic><sub><italic>x</italic></sub>(<italic>B</italic>, &#x003B3;(<italic>x</italic><sub><italic>i</italic></sub>, <italic>y</italic><sub><italic>i</italic></sub>, <italic>t</italic><sub><italic>i</italic></sub>)) and standard deviation &#x003C3;<sub><italic>x</italic></sub>. Likewise, the observed surface velocity in the <italic>y</italic> direction for a particular site <italic>i</italic> at time <italic>t</italic> is also normally distributed with mean <italic>v</italic><sub><italic>y</italic></sub>(<italic>B</italic>, &#x003B3;(<italic>x</italic><sub><italic>i</italic></sub>, <italic>y</italic><sub><italic>i</italic></sub>, <italic>t</italic><sub><italic>i</italic></sub>)) and standard deviation &#x003C3;<sub><italic>y</italic></sub>. Expressions for <italic>v</italic><sub><italic>x</italic></sub> and <italic>v</italic><sub><italic>y</italic></sub> are given in Equations (2.1) and (2.2). Independence is assumed between <italic>x</italic> and <italic>y</italic> velocity components for a given site and between distinct sites. A Kendall correlation test between <italic>x</italic> and <italic>y</italic> velocity components fails to reject the assumption of no correlation with the data, which is consistent with the assumption of independence between <italic>x</italic> and <italic>y</italic> surface velocity components. The reader is referred to section 5, the Appendix, for a precise mathematical specification.</p></sec>
<sec>
<title>2.2.2. t Distribution for Surface Velocities</title>
<p>We next consider a t distribution (i.e., a t model) for surface velocity data. That is, the observed velocity in the <italic>x</italic> direction for a particular site <italic>i</italic> with spatio-temporal coordinates (<italic>x</italic><sub><italic>i</italic></sub>, <italic>y</italic><sub><italic>i</italic></sub>, <italic>t</italic><sub><italic>i</italic></sub>) is t distributed, with mean <italic>v</italic><sub><italic>x</italic></sub>(<italic>B</italic>, &#x003B3;(<italic>x</italic><sub><italic>i</italic></sub>, <italic>y</italic><sub><italic>i</italic></sub>, <italic>t</italic><sub><italic>i</italic></sub>)) and scale &#x003C3;<sub><italic>x</italic></sub>. To accommodate heavy-tailed deviations from the SIA mean, we use a small degrees of freedom parameter: 3.0001. This choice of degrees of freedom allows for heavy tails (heaviness of tails increases with decreasing degrees of freedom) and maintains a well-defined skewness (skewness is undefined for degrees of freedom less than or equal to 3). Likewise, the velocity in <italic>y</italic> direction for a particular site is also t distributed with mean <italic>v</italic><sub><italic>y</italic></sub>(<italic>B</italic>, &#x003B3;(<italic>x</italic><sub><italic>i</italic></sub>, <italic>y</italic><sub><italic>i</italic></sub>, <italic>t</italic><sub><italic>i</italic></sub>)) and scale &#x003C3;<sub><italic>y</italic></sub>. As with the normal distribution model, independence is assumed between <italic>x</italic> and <italic>y</italic> velocity components for a given site and between distinct sites. As discussed in the previous section, a Kendall correlation test between <italic>x</italic> and <italic>y</italic> surface velocity components fails to reject the assumption of no correlation with the data set, which is consistent with the assumption of independence between <italic>x</italic> and <italic>y</italic> velocity components. The reader is referred to the Appendix (Section 5) for a precise mathematical specification.</p></sec>
<sec>
<title>2.2.3. Prior Distributions for Physical Parameters</title>
<p>A normal distribution with mean 35 and standard deviation 30 that is truncated to be on the interval [1, 70] is used for the ice-softness parameter (units are 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>); this is a nearly flat prior on the interval [1,70] (see comparison of the prior and posterior in the next section) that encapsulates a range of temperate ice softness values presented in Cuffey and Paterson (<xref ref-type="bibr" rid="B11">2010</xref>) (24, 38, and 55 in units of 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>) based on ice-softness measurements. Additionally, the prior distribution for log&#x003B3; follows a Gaussian process distribution with mean 0 and Mat&#x000E9;rn kernel for the covariance, where correlation is a decreasing function of Euclidean spatial distance between measurements. Specific mathematical equations for all priors are given in the Appendix (section 5).</p>
<p>We use the parameterization log&#x003B3; so that a Gaussian process prior is sensible: a Gaussian process prior for &#x003B3; would still allow for spatial correlation but admit negative values for &#x003B3;, which is not physically plausible. The Mat&#x000E9;rn kernel is commonly used for geostatistical applications (Gelfand et al., <xref ref-type="bibr" rid="B13">2010</xref>; Bakka et al., <xref ref-type="bibr" rid="B3">2018</xref>) and subsumes both the exponential and Gaussian kernels, widely used for Gaussian processes. Specifically, a smoothness parameter of 0.5 corresponds to exponential and a smoothness parameter that approaches infinity corresponds to Gaussian. In this work, smoothness and range parameters of the Mat&#x000E9;rn kernel are selected by minimizing the posterior predictive mean absolute error over the set {100, 1000, 10000} for the range and {0.5, 2, 10} for the smoothness. Based on this criterion, a range of 100 is used for the normal and t distribution Bayesian models; the smoothness is 10 for the t distribution Bayesian model and 0.5 for the normal distribution Bayesian model. The marginal variance is set to 100, which is large enough such that sensible values of basal sliding (i.e., correct order of magnitude) are within one to two standard deviations of the mean. The prior sensitivity analysis in the Appendix (section 5) examines when priors are changed using an inverse-gamma prior (which is not truncated) for <italic>B</italic> as well as a smaller marginal variance for the Gaussian process prior for log&#x003B3;. The results indicate that the posterior is not substantially affected by changing the prior distributions.</p></sec>
<sec>
<title>2.2.4. Posterior Distribution and Inference via Gibbs Sampling</title>
<p>The overarching aim of this Bayesian analysis is to sample from the posterior distribution of <italic>B</italic>, log&#x003B3;, &#x003C3;<sub><italic>x</italic></sub>, and &#x003C3;<sub><italic>y</italic></sub>, which by Bayes&#x00027; theorem is:</p>
<disp-formula id="E6"><mml:math id="M8"><mml:mtable columnalign="left"><mml:mtr><mml:mtd><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mo class="qopname">log</mml:mo><mml:mi>&#x003B3;</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold-italic"><mml:mi>Y</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold-italic"><mml:mi>Y</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>&#x0221D;</mml:mo></mml:mtd></mml:mtr><mml:mtr><mml:mtd><mml:mtext>&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;&#x000A0;</mml:mtext><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mstyle mathvariant="bold-italic"><mml:mi>Y</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mstyle mathvariant="bold-italic"><mml:mi>Y</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:mi>B</mml:mi><mml:mo>,</mml:mo><mml:mo class="qopname">log</mml:mo><mml:mi>&#x003B3;</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mo class="qopname">log</mml:mo><mml:mi>&#x003B3;</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>&#x003C3;</mml:mi></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>.</mml:mo></mml:mtd></mml:mtr></mml:mtable></mml:math></disp-formula>
<p>In the above expression, the <italic>x</italic> surface velocity measurements at <italic>N</italic> sites are denoted by <inline-formula><mml:math id="M9"><mml:msub><mml:mrow><mml:mstyle mathvariant="bold-italic"><mml:mi>Y</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>&#x0211D;</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>, and the <italic>y</italic> surface velocity measurements are denoted by <inline-formula><mml:math id="M10"><mml:msub><mml:mrow><mml:mstyle mathvariant="bold-italic"><mml:mi>Y</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mi>y</mml:mi></mml:mrow></mml:msub><mml:mo>&#x02208;</mml:mo><mml:msup><mml:mrow><mml:mi>&#x0211D;</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:math></inline-formula>. The posterior samples of <italic>B</italic> and log&#x003B3; are substituted into Equations (2.1) and (2.2) for <italic>v</italic><sub><italic>x</italic></sub> and <italic>v</italic><sub><italic>y</italic></sub> to generate posterior samples of the <italic>x</italic> and <italic>y</italic> velocities.</p>
<p>The posterior distribution is not analytically tractable, and a Markov chain Monte Carlo algorithm (MCMC) is often used to sample from the posterior distribution in such a scenario (Tierney, <xref ref-type="bibr" rid="B33">1994</xref>). Specifically, we develop a Gibbs sampling approach to sample from the posterior distribution. The idea of the Gibbs sampling approach is to sequentially sample from the posterior distribution of each parameter conditional on the data and remaining parameters: <italic>p</italic>(<italic>B</italic>|&#x02212;), <italic>p</italic>(log&#x003B3;|&#x02212;), <italic>p</italic>(&#x003C3;<sub><italic>x</italic></sub>|&#x02212;), and <italic>p</italic>(&#x003C3;<sub><italic>y</italic></sub>|&#x02212;), where - indicates all other variables and data. If this is iterated for many steps, the samples will be drawn from the posterior distribution, <italic>p</italic>(<italic>B</italic>, log&#x003B3;, &#x003C3;<sub><italic>x</italic></sub>, &#x003C3;<sub><italic>y</italic></sub>|<italic><bold>Y</bold></italic><sub><italic>x</italic></sub>, <italic><bold>Y</bold></italic><sub><italic>y</italic></sub>). Section 5 contains the precise Gibbs sampler steps that are run iteratively.</p>
<p>The Gibbs sampler crucially relies on an elliptical slice sampling step for log&#x003B3;. Instead of sampling from each component of log&#x003B3; one-by-one for each of the <italic>N</italic> sites (which is usually computationally inefficient), we use elliptical slice sampling because of the Gaussian process prior on log&#x003B3;. Elliptical slice sampling (Murray et al., <xref ref-type="bibr" rid="B25">2010</xref>) is an extension of slice sampling (Neal, <xref ref-type="bibr" rid="B26">2003</xref>) that is designed to sample from a posterior distribution in which the prior distribution is a Gaussian process. This property holds since log&#x003B3; has a Gaussian process prior. The algorithm is shown to be a competitive alternative to Metropolis-Hastings (Tierney, <xref ref-type="bibr" rid="B33">1994</xref>) for Gaussian process priors, often times working better for real data in terms of a larger effective sample size of Monte Carlo output per unit compute time (Murray et al., <xref ref-type="bibr" rid="B25">2010</xref>).</p>
<p>400,000 posterior samples are drawn from the posterior of physical parameters using Gibbs sampling for both the normal and t distribution variations, after removing a burn in of 50,000 iterations from 450,000 total iterations. The effective sample size and <inline-formula><mml:math id="M11"><mml:mover accent="true"><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mo>^</mml:mo></mml:mover></mml:math></inline-formula> convergence diagnostic from Vats and Knudson (<xref ref-type="bibr" rid="B34">2018</xref>), a recent version of the convergence diagnostic from Gelman and Rubin (<xref ref-type="bibr" rid="B14">1992</xref>), provide evidence of convergence of Gibbs sampling; these results are detailed in section 5.</p></sec></sec>
<sec>
<title>2.3. Langj&#x000F6;kull Data</title>
<p>The models set forth in the previous section are applied to Langj&#x000F6;kull, which is Iceland&#x00027;s second largest temperate glacier by area (and third by volume) (Bj&#x000F6;rnsson and P&#x000E1;lsson, <xref ref-type="bibr" rid="B6">2008</xref>). Langj&#x000F6;kull is 900 km<sup>2</sup> in area, 190 km<sup>3</sup> in volume, 210 m mean thickness, and has a max thickness of about 650 m. In particular, we use:</p>
<list list-type="bullet">
<list-item><p>100 m resolution surface elevation topography maps of Langj&#x000F6;kull at 1997 (P&#x000E1;lsson et al., <xref ref-type="bibr" rid="B27">2012</xref>), 2007 (Pope et al., <xref ref-type="bibr" rid="B29">2016</xref>), and 2015 (Porter et al., <xref ref-type="bibr" rid="B30">2018</xref>).</p></list-item>
<list-item><p>100 m resolution topography map of the Langj&#x000F6;kull glacier bed, constructed from radio echo sounding profiles, nearly 1 km apart, surveyed in 1997 (Bj&#x000F6;rnsson and P&#x000E1;lsson, <xref ref-type="bibr" rid="B7">2020</xref>).</p></list-item>
<list-item><p>51 GPS surface velocity measurements taken across Langj&#x000F6;kull at 1997, 2007, and 2015, including (<italic>x, y, t</italic>) covariates. The UI-IES takes measurements twice a year: once in late spring (end of April to beginning of May) for winter mass balance measurements and once in the fall for summer balance measurements (end of September to mid-October). The displacement measured between the two yearly measurements divided by time yields an estimate of the horizontal surface velocity at a particular measurement site during the summer time, when basal sliding is expected to have a substantial role in velocity (Minchew et al., <xref ref-type="bibr" rid="B24">2015</xref>). A topographical map of Langj&#x000F6;kull with the surface velocity survey sites is included in <xref ref-type="fig" rid="F1">Figure 1</xref>.</p></list-item>
</list>
<fig id="F1" position="float">
<label>Figure 1</label>
<caption><p>Surface topographical map of Langj&#x000F6;kull along with surface velocity survey locations.</p></caption>
<graphic xlink:href="feart-09-610069-g0001.tif"/>
</fig>
<p>Notice that the <italic>x</italic> and <italic>y</italic> velocity equations given in Equations (2.1) and (2.2) require glacial thickness and the surface elevation derivatives with respect to <italic>x</italic> and <italic>y</italic>. The former is obtained with the difference between surface elevation and bed elevation, and the latter is obtained with central differences of the surface elevation topographical maps. The Langj&#x000F6;kull surface elevation ranges from 440 to 1,440 m above sea level. The grid that contains the surface elevation data and bed measurements is of dimensions 43,800 m by 46,400 m. Langj&#x000F6;kull is about 50 km long and 20 km wide, which explains the name &#x0201C;long glacier.&#x0201D;</p></sec>
</sec>
<sec sec-type="results" id="s3">
<title>3. Results</title>
<p>The posterior mean of ice softness using the t distribution is 18.2 with a posterior standard deviation of 0.847, and the posterior mean for ice softness using the normal distribution is 17.8 with a posterior standard deviation of 1.11 (units of 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>). For comparison, 23.7 is estimated by Gudmundsson (<xref ref-type="bibr" rid="B17">1999</xref>), 23.4 is found in A&#x000F0;algeirsd&#x000F3;ttir et al. (<xref ref-type="bibr" rid="B1">2000</xref>), 22.2 is found in Albrecht et al. (<xref ref-type="bibr" rid="B2">2000</xref>), and 20 is found in Hubbard et al. (<xref ref-type="bibr" rid="B19">1998</xref>), also in units of 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>. The recommended value from Cuffey and Paterson (<xref ref-type="bibr" rid="B11">2010</xref>) is 24 &#x000D7; 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>. The posterior distributions for ice softness appears to be in the vicinity of these values, though strictly less. Using the t distribution for surface velocities, the posterior distribution for ice softness is slightly closer to the recommended value for temperate glaciers from Cuffey and Paterson (<xref ref-type="bibr" rid="B11">2010</xref>) than when the normal distribution is used, and has a lower standard deviation (i.e., more precise) as is illustrated in <xref ref-type="fig" rid="F2">Figure 2</xref>. Moreover, plots of posterior means for <italic>x</italic> and <italic>y</italic> velocities vs. observed <italic>x</italic> and <italic>y</italic> velocities are displayed in <xref ref-type="fig" rid="F3">Figure 3</xref> and show a generally close agreement; the few outliers in the <italic>x</italic> velocity direction are at locations close to the outer edge of the glacier where the SIA is a poorer physical description. As velocities are dependent on both ice viscosity and basal sliding, the overall agreement between observed and predicted surface velocities indicates that inferences are physically realistic.</p>
<fig id="F2" position="float">
<label>Figure 2</label>
<caption><p>Illustration of posterior densities for ice softness using a normal distribution and t distribution for surface velocities. The prior, nearly flat, is also shown for comparison. Units are 10<sup>&#x02212;25</sup><italic>s</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;3</sup>.</p></caption>
<graphic xlink:href="feart-09-610069-g0002.tif"/>
</fig>
<fig id="F3" position="float">
<label>Figure 3</label>
<caption><p>An illustration of the posterior mean for <italic>x</italic> and <italic>y</italic> glacier surface velocities compared to the observed <italic>x</italic> and <italic>y</italic> glacier surface velocities (using a t distribution for surface velocities). Units for velocities are in meters per year. Overall, the observed and inferred values agree, particularly for the <italic>y</italic>-velocity values.</p></caption>
<graphic xlink:href="feart-09-610069-g0003.tif"/>
</fig>
<p>An important consequence of the model output is the ability to decompose posterior ice velocity into a ice deformation component (referred to as viscous flow in Minchew et al., <xref ref-type="bibr" rid="B24">2015</xref>) and a sliding component. The panels of <xref ref-type="fig" rid="F4">Figure 4</xref> map the posterior mean of deformation velocities and sliding velocities across Langj&#x000F6;kull during the melt seasons of 2007 and 2015, and are overlaid on surface elevation topography. Several important observations are apparent based on these maps. First is that the direction of velocity in the negative of the surface gradient, evident based on the directions pointing toward areas of rapid elevation descent. The second main observation is that there are two important regions of significant basal sliding in the southwest and north of the glacier. In the southwest (i.e., W-Hagafellsj&#x000F6;kull) the bed consists of gentle slopes of layers of lava flows of a shield volcano (now even visible at the glacier edge), and in the north region hard bed is also likely, whereas much of Langj&#x000F6;kull bed is probably of more permeable bedrock or sediments. Less permeability makes it more likely for meltwater to heighten the water pressure at the bed, at the start of melting periods (which may occur more than once during the melt season), thus creating conditions for basal sliding.</p>
<fig id="F4" position="float">
<label>Figure 4</label>
<caption><p>Illustration of posterior mean horizontal surface velocities across Langj&#x000F6;kull during the 2007 and 2015 melt seasons. <bold>(A,B)</bold> illustrate 2015 velocities due to deformation (i.e., viscous flow) and sliding, respectively. <bold>(C,D)</bold> illustrate 2007 velocities due to deformation (i.e., viscous flow) and sliding, respectively. The velocities are overlaid on surface topographical maps (elevation in meters), and the arrow directions are that of the steepest descent. The lengths of the arrows are proportional to speed: the <bold>black arrow</bold> on panel <bold>(A)</bold> corresponds to <bold>12.4 m/a</bold>. Substantial sliding appears to occur in the southwest and north; the sliding magnitude appears to be slightly less in 2015 compared to 2007, which corresponds to about 1/3 less melt during 2015 based on mass balance measurements.</p></caption>
<graphic xlink:href="feart-09-610069-g0004.tif"/>
</fig>
<p>In Minchew et al. (<xref ref-type="bibr" rid="B24">2015</xref>) Figures 4C,D, maps of speed assuming only viscous (i.e., deformational flow) are compared to maps of measured speeds with InSAR at Langj&#x000F6;kull; the difference between the two is indicative of the amount that basal sliding contributes to surface velocity. These maps also show evidence of significant basal sliding velocity in the north and southwest of Langj&#x000F6;kull, consistent with the spatial variation we have observed in posterior basal sliding speeds. Additionally, we see a significant posterior deformation velocity component in the southeastern part of Langj&#x000F6;kull (see the arrow that runs off of the body of Langj&#x000F6;kull in the southeast), where basal sliding is quite low. This result is also consistent with Minchew et al. (<xref ref-type="bibr" rid="B24">2015</xref>) because deformation speed is quite close to total speed in this region (Figures 4C,D of Minchew et al., <xref ref-type="bibr" rid="B24">2015</xref>). We do not notice significant temporal variation in posterior velocities except for what appears to be a slight reduction in basal sliding velocities in the north and southwest regions. This finding is likely to be related to less summer melt (about a third) in summer 2015 than in 2007 observed via mass balance measurements.</p>
<p>In addition to maps of posterior basal sliding and deformation, Section 5 <xref ref-type="supplementary-material" rid="SM3">Figures A7&#x02013;A10</xref> provide illustrations of the posterior uncertainty in the basal sliding parameter for 2007 and 2015. The numbers on the maps indicate the measurement sites, and the boxplots that follow have numbering on the horizontal axis corresponds to the numbering on the map. The boxplots plot the posterior distribution for the basal sliding parameter at the corresponding site, illustrating the uncertainty in the posterior distribution. The regions that show large amounts of basal sliding, both in 2007 and 2015, are in areas with large basal sliding speeds, as illustrated in <xref ref-type="fig" rid="F4">Figure 4</xref>; these regions are in the southwest and north. Moreover, uncertainty in basal sliding is much larger in the north than in the southwest, both in 2007 and 2015. In the southwest, the basal sliding parameters appear to be larger in 2007 than in 2015 (just under 4 &#x000D7; 10<sup>&#x02212;4</sup> in 2007 vs. 3 &#x000D7; 10<sup>&#x02212;4</sup> in 2015 in units of <italic>ma</italic><sup>&#x02212;1</sup><italic>Pa</italic><sup>&#x02212;1</sup>). The reduced amount of basal sliding corresponds to about a 1/3 reduction in melt as measured by mass balance measurements in 2015 and 2007.</p></sec>
<sec sec-type="discussion" id="s4">
<title>4. Discussion</title>
<p>The objective of this paper is to develop Bayesian models for glacier surface velocity, based on a shallow ice approximation, that allow for the statistical inference of important glacial parameters: ice softness and basal sliding. Specifically, one model stipulates a normal distribution for surface velocities, whereas the other specifies a heavy-tailed t distribution.</p>
<p>Overall, a notable result is that the posterior distribution for ice softness is close to values derived in the literature (for instance, the recommended value from Cuffey and Paterson, <xref ref-type="bibr" rid="B11">2010</xref>), albeit by different methods. In contrast to these works, the estimate of ice softness is accompanied with an uncertainty estimate as well via the posterior standard deviation. An additional differentiating aspect of our work in comparison to other Bayesian approaches involving glacial surface velocity is the use of a t distribution in addition to a normal distribution (an assumption used in Pralong and Gudmundsson, <xref ref-type="bibr" rid="B31">2011</xref>; Brinkerhoff et al., <xref ref-type="bibr" rid="B9">2016</xref> to give a few examples). The pattern of spatial variation in basal sliding is also consistent with prior studies of surface velocity at Langj&#x000F6;kull, in particular Minchew et al. (<xref ref-type="bibr" rid="B24">2015</xref>). However, it is important to note that these patterns of spatial variation in basal sliding are specifically during the melt season, so it is not possible to verify how the approach performs for inferring sliding velocities during the winter-spring season when the total velocity is expected to be almost due to deformation only.</p>
<p>The residual analysis detailed in the Appendix (section 5) provides a means of assessing the normal distribution and t distribution used for surface velocities. Residuals are defined as centered and scaled versions of the <italic>x</italic> and <italic>y</italic> observed surface velocities (i.e., subtracting off the posterior mean of surface velocity and dividing by the posterior mean of the scale parameter for both directions). A Shapiro-Wilk test is conducted for the residuals, and the assumption of normality is rejected for both the <italic>x</italic> and <italic>y</italic> surface velocity residuals, providing evidence against the use of the normal distribution for surface velocities. Additionally, <xref ref-type="supplementary-material" rid="SM3">Figure A11</xref> of section 5 provides quantile-quantile (QQ) plots comparing the sample quantiles with theoretical quantiles for the residuals assuming that a normal distribution holds. A good fit occurs when there is close agreement between sample and theoretical quantiles but is not observed for the normal distribution residuals, particularly for the <italic>x</italic>-velocity components. We also include a QQ-plot for the t distribution residuals in the second row of <xref ref-type="supplementary-material" rid="SM3">Figure A11</xref>. There is some improvement in the QQ-plot for the t distribution residuals of <italic>x</italic>-velocities, but not as much for the <italic>y</italic>-velocities, it appears. Overall the residual analysis supports the claim that a normal distribution is not a good fit for surface velocities and a heavy-tailed t distribution improves goodness-of-fit.</p>
<p>We must also lay forth limitations of our approach. Perhaps the greatest limitation is the use of the shallow ice approximation, which will not always hold. Another issue is that of scalability; the Gibbs sampler for posterior inference takes on the order of a few hours on 2020 iMac with 3.8 GHz 8-Core Intel Core i7 processor with a data set of 51 surface velocity observations. The approach may be more time consuming with a larger data set, especially for the Gaussian process prior for basal sliding parameters, whose probability density involves finding the inverse and determinant of a covariance matrix, generally computationally costly operations.</p>
<p>There are a number of future steps that can be taken to extend this analysis. First, the posterior distributions for the physical parameters can be used in Monte Carlo simulations involving a time dynamical model of glacier evolution in order to derive projections for how the shape of Langj&#x000F6;kull will evolve in the future. Second, alternate continuous distributions may be tested besides the normal or t distribution used within this work (perhaps even a non-parametric distribution). Third, additional correlation structures besides the Mat&#x000E9;rn kernel can be investigated for the prior distribution on the basal sliding field.</p>
<p>While the models have been exclusively applied to Langj&#x000F6;kull, we believe this work serves as a guide for selecting physical parameters and quantifying their uncertainty at other temperate, shallow glaciers around the globe, the melting of which is expected to contribute significantly to sea level rise in the next century. Additonally, the <monospace>R</monospace> code written for performing Bayesian inference with a Gibbs sampler may be reapplied for other scenarios where surface topography, bed topography, and surface velocity data are available. Despite that the models we have used rely on a SIA, different equations for <italic>v</italic><sub><italic>x</italic></sub> and <italic>v</italic><sub><italic>y</italic></sub>, for instance based on a numerical solver, may be used.</p></sec>
<sec sec-type="data-availability-statement" id="s5">
<title>Data Availability Statement</title>
<p>The original contributions presented in the study are included in the article/<xref ref-type="supplementary-material" rid="SM1">Supplementary Material</xref>, further inquiries can be directed to the corresponding author/s.</p></sec>
<sec id="s6">
<title>Author Contributions</title>
<p>GG developed the model, completed programming, and wrote the manuscript. Additional authors provided valuable feedback, guidance, and manuscript help. FP provided glacier data for conducting analysis. All authors contributed to the article and approved the submitted version.</p></sec>
<sec sec-type="COI-statement" id="conf1">
<title>Conflict of Interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. The reviewer GF declared a past co-authorship with several of the authors GA and FP to the handling editor.</p></sec>
</body>
<back>
<sec sec-type="supplementary-material" id="s7">
<title>Supplementary Material</title>
<p>The Supplementary Material for this article can be found online at: <ext-link ext-link-type="uri" xlink:href="https://www.frontiersin.org/articles/10.3389/feart.2021.610069/full#supplementary-material">https://www.frontiersin.org/articles/10.3389/feart.2021.610069/full#supplementary-material</ext-link></p>
<supplementary-material xlink:href="Data_Sheet_1.ZIP" id="SM1" mimetype="application/zip" xmlns:xlink="http://www.w3.org/1999/xlink"/>
<supplementary-material xlink:href="Data_Sheet_2.zip" id="SM2" mimetype="application/zip" xmlns:xlink="http://www.w3.org/1999/xlink"/>
<supplementary-material xlink:href="Data_Sheet_3.pdf" id="SM3" mimetype="application/pdf" xmlns:xlink="http://www.w3.org/1999/xlink"/>
</sec>
<ref-list>
<title>References</title>
<ref id="B1">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>A&#x000F0;algeirsd&#x000F3;ttir</surname> <given-names>G.</given-names></name> <name><surname>Gudmundsson</surname> <given-names>G. H.</given-names></name> <name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name></person-group> (<year>2000</year>). <article-title>The response of a glacier to a surface disturbance: a case study on Vatnaj&#x000F6;kull ice cap, Iceland</article-title>. <source>Ann. Glaciol.</source> <volume>31</volume>, <fpage>104</fpage>&#x02013;<lpage>110</lpage>. <pub-id pub-id-type="doi">10.3189/172756400781819914</pub-id></citation></ref>
<ref id="B2">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Albrecht</surname> <given-names>O.</given-names></name> <name><surname>Jansson</surname> <given-names>P.</given-names></name> <name><surname>Blatter</surname> <given-names>H.</given-names></name></person-group> (<year>2000</year>). <article-title>Modelling glacier response to measured mass-balance forcing</article-title>. <source>Ann. Glaciol.</source> <volume>31</volume>, <fpage>91</fpage>&#x02013;<lpage>96</lpage>. <pub-id pub-id-type="doi">10.3189/172756400781819996</pub-id></citation></ref>
<ref id="B3">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bakka</surname> <given-names>H.</given-names></name> <name><surname>Rue</surname> <given-names>H.</given-names></name> <name><surname>Fuglstad</surname> <given-names>G.-A.</given-names></name> <name><surname>Riebler</surname> <given-names>A.</given-names></name> <name><surname>Bolin</surname> <given-names>D.</given-names></name> <name><surname>Illian</surname> <given-names>J.</given-names></name> <etal/></person-group>. (<year>2018</year>). <article-title>Spatial modeling with R-INLA: a review</article-title>. <source>Wiley Interdiscipl. Rev. Comput. Stat.</source> <volume>10</volume>:<fpage>e1443</fpage>. <pub-id pub-id-type="doi">10.1002/wics.1443</pub-id></citation></ref>
<ref id="B4">
<citation citation-type="web"><person-group person-group-type="author"><name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name></person-group> (<year>2017</year>). <source>The Glaciers of Iceland: A Historical, Cultural and Scientific Overview. Springer</source>. Available online at: <ext-link ext-link-type="uri" xlink:href="https://www.springer.com/gp/book/9789462392069">https://www.springer.com/gp/book/9789462392069</ext-link></citation></ref>
<ref id="B5">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>A&#x000F0;algeirsd&#x000F3;ttir</surname> <given-names>G.</given-names></name> <name><surname>Gu&#x000F0;mundsson</surname> <given-names>S.</given-names></name> <name><surname>J&#x000F3;hannesson</surname> <given-names>T.</given-names></name> <name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name> <name><surname>Sigur&#x000F0;sson</surname> <given-names>O.</given-names></name></person-group> (<year>2006</year>). <article-title>Climate change response of vatnaj&#x000F6;kull, hofsj&#x000F6;kull and Langj&#x000F6;kull ice caps, iceland</article-title>, in <source>European Conference on Impacts of Climate Change on Renewable Energy Sources</source> (<publisher-loc>Reykjavik</publisher-loc>).</citation></ref>
<ref id="B6">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name></person-group> (<year>2008</year>). <article-title>Icelandic glaciers</article-title>. <source>J&#x000F6;kull</source> <volume>58</volume>, <fpage>365</fpage>&#x02013;<lpage>386</lpage>.</citation></ref>
<ref id="B7">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name></person-group> (<year>2020</year>). <article-title>Radio-echo soundings on Icelandic temperate glaciers: history of techniques and findings</article-title>. <source>Ann. Glaciol.</source> <volume>61</volume>, <fpage>25</fpage>&#x02013;<lpage>34</lpage>. <pub-id pub-id-type="doi">10.1017/aog.2020.10</pub-id></citation></ref>
<ref id="B8">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name> <name><surname>Sigur&#x00111;sson</surname> <given-names>O.</given-names></name> <name><surname>Flowers</surname> <given-names>G. E.</given-names></name></person-group> (<year>2003</year>). <article-title>Surges of glaciers in iceland</article-title>. <source>Ann. Glaciol.</source> <volume>36</volume>, <fpage>82</fpage>&#x02013;<lpage>90</lpage>. <pub-id pub-id-type="doi">10.3189/172756403781816365</pub-id></citation></ref>
<ref id="B9">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Brinkerhoff</surname> <given-names>D. J.</given-names></name> <name><surname>Aschwanden</surname> <given-names>A.</given-names></name> <name><surname>Truffer</surname> <given-names>M.</given-names></name></person-group> (<year>2016</year>). <article-title>Bayesian inference of subglacial topography using mass conservation</article-title>. <source>Front. Earth Sci.</source> <volume>4</volume>:<fpage>8</fpage>. <pub-id pub-id-type="doi">10.3389/feart.2016.00008</pub-id></citation></ref>
<ref id="B10">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Bueler</surname> <given-names>E.</given-names></name> <name><surname>Lingle</surname> <given-names>C. S.</given-names></name> <name><surname>Kallen-Brown</surname> <given-names>J. A.</given-names></name> <name><surname>Covey</surname> <given-names>D. N.</given-names></name> <name><surname>Bowman</surname> <given-names>L. N.</given-names></name></person-group> (<year>2005</year>). <article-title>Exact solutions and verification of numerical models for isothermal ice sheets</article-title>. <source>J. Glaciol.</source> <volume>51</volume>, <fpage>291</fpage>&#x02013;<lpage>306</lpage>. <pub-id pub-id-type="doi">10.3189/172756505781829449</pub-id></citation></ref>
<ref id="B11">
<citation citation-type="web"><person-group person-group-type="author"><name><surname>Cuffey</surname> <given-names>K. M.</given-names></name> <name><surname>Paterson</surname> <given-names>W.</given-names></name></person-group> (<year>2010</year>). <source>The Physics of Glaciers, 4 Edn</source>. <publisher-loc>Oxford, UK</publisher-loc>: <publisher-name>Academic Press</publisher-name>. Available online at: <ext-link ext-link-type="uri" xlink:href="https://www.elsevier.com/books/the-physics-of-glaciers/cuffey/978-0-12-369461-4">https://www.elsevier.com/books/the-physics-of-glaciers/cuffey/978-0-12-369461-4</ext-link></citation></ref>
<ref id="B12">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Flowers</surname> <given-names>G. E.</given-names></name> <name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>Geirsd&#x000F3;ttir</surname> <given-names>&#x000C1;.</given-names></name> <name><surname>Miller</surname> <given-names>G. H.</given-names></name> <name><surname>Clarke</surname> <given-names>G. K.</given-names></name></person-group> (<year>2007</year>). <article-title>Glacier fluctuation and inferred climatology of Langj&#x000F6;kull ice cap through the little ice age</article-title>. <source>Q. Sci. Rev.</source> <volume>26</volume>, <fpage>2337</fpage>&#x02013;<lpage>2353</lpage>. <pub-id pub-id-type="doi">10.1016/j.quascirev.2007.07.016</pub-id></citation></ref>
<ref id="B13">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Gelfand</surname> <given-names>A. E.</given-names></name> <name><surname>Diggle</surname> <given-names>P.</given-names></name> <name><surname>Guttorp</surname> <given-names>P.</given-names></name> <name><surname>Fuentes</surname> <given-names>M.</given-names></name></person-group> (<year>2010</year>). <source>Handbook of Spatial Statistics</source>. <publisher-loc>Boca Raton, FL</publisher-loc>: <publisher-name>CRC Press</publisher-name>.</citation></ref>
<ref id="B14">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Gelman</surname> <given-names>A.</given-names></name> <name><surname>Rubin</surname> <given-names>D. B.</given-names></name></person-group> (<year>1992</year>). <article-title>Inference from iterative simulation using multiple sequences</article-title>. <source>Statist. Sci.</source> <volume>7</volume>, <fpage>457</fpage>&#x02013;<lpage>472</lpage>.</citation></ref>
<ref id="B15">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Gopalan</surname> <given-names>G.</given-names></name> <name><surname>Hrafnkelsson</surname> <given-names>B.</given-names></name> <name><surname>A&#x000F0;algeirsd&#x000F3;ttir</surname> <given-names>G.</given-names></name> <name><surname>Jarosch</surname> <given-names>A. H.</given-names></name> <name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name></person-group> (<year>2018</year>). <article-title>A Bayesian hierarchical model for glacial dynamics based on the shallow ice approximation and its evaluation using analytical solutions</article-title>. <source>Cryosphere</source> <volume>12</volume>, <fpage>2229</fpage>&#x02013;<lpage>2248</lpage>. <pub-id pub-id-type="doi">10.5194/tc-12-2229-2018</pub-id></citation></ref>
<ref id="B16">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Guan</surname> <given-names>Y.</given-names></name> <name><surname>Haran</surname> <given-names>M.</given-names></name> <name><surname>Pollard</surname> <given-names>D.</given-names></name></person-group> (<year>2016</year>). <article-title>Inferring ice thickness from a glacier dynamics model and multiple surface datasets</article-title>. <source>ArXiv pre-prints</source>. <pub-id pub-id-type="doi">10.1002/env.2460</pub-id></citation></ref>
<ref id="B17">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Gudmundsson</surname> <given-names>G. H.</given-names></name></person-group> (<year>1999</year>). <article-title>A three-dimensional numerical model of the confluence area of unteraargletscher, bernese alps, Switzerland</article-title>. <source>J. Glaciol.</source> <volume>45</volume>, <fpage>219</fpage>&#x02013;<lpage>230</lpage>.</citation></ref>
<ref id="B18">
<citation citation-type="web"><person-group person-group-type="author"><name><surname>Hock</surname> <given-names>R.</given-names></name> <name><surname>Rasul</surname> <given-names>G.</given-names></name> <name><surname>Adler</surname> <given-names>C.</given-names></name> <name><surname>C&#x000E1;ceres</surname> <given-names>B.</given-names></name> <name><surname>Gruber</surname> <given-names>S.</given-names></name> <name><surname>Hirabayashi</surname> <given-names>Y.</given-names></name> <etal/></person-group>. (<year>2019</year>). <article-title>High mountain areas</article-title>, in <source>Ipcc Special Report on the Ocean and Cryosphere in a Changing Climate</source>. Available online at: <ext-link ext-link-type="uri" xlink:href="https://www.ipcc.ch/srocc/chapter/chapter-2/">https://www.ipcc.ch/srocc/chapter/chapter-2/</ext-link>.</citation></ref>
<ref id="B19">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Hubbard</surname> <given-names>A.</given-names></name> <name><surname>Blatter</surname> <given-names>H.</given-names></name> <name><surname>Nienow</surname> <given-names>P.</given-names></name> <name><surname>Mair</surname> <given-names>D.</given-names></name> <name><surname>Hubbard</surname> <given-names>B.</given-names></name></person-group> (<year>1998</year>). <article-title>Comparison of a three-dimensional model for glacier flow with field data from haut glacier d&#x00027;arolla, Switzerland</article-title>. <source>J. Glaciol.</source> <volume>44</volume>, <fpage>368</fpage>&#x02013;<lpage>378</lpage>. <pub-id pub-id-type="doi">10.1017/S0022143000002690</pub-id></citation></ref>
<ref id="B20">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Hutter</surname> <given-names>K.</given-names></name></person-group> (<year>1983</year>). <source>Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets</source>. <publisher-loc>Mathematical Approaches to Geophysics. Heidelberg</publisher-loc>: <publisher-name>Springer</publisher-name>.</citation></ref>
<ref id="B21">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Isaac</surname> <given-names>T.</given-names></name> <name><surname>Petra</surname> <given-names>N.</given-names></name> <name><surname>Stadler</surname> <given-names>G.</given-names></name> <name><surname>Ghattas</surname> <given-names>O.</given-names></name></person-group> (<year>2015</year>). <article-title>Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the antarctic ice sheet</article-title>. <source>J. Comput. Phys.</source> <volume>296</volume>, <fpage>348</fpage>&#x02013;<lpage>368</lpage>. <pub-id pub-id-type="doi">10.1016/j.jcp.2015.04.047</pub-id></citation></ref>
<ref id="B22">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Jarosch</surname> <given-names>A. H.</given-names></name> <name><surname>Schoof</surname> <given-names>C. G.</given-names></name> <name><surname>Anslow</surname> <given-names>F. S.</given-names></name></person-group> (<year>2013</year>). <article-title>Restoring mass conservation to shallow ice flow models over complex terrain</article-title>. <source>Cryosphere</source> <volume>7</volume>, <fpage>229</fpage>&#x02013;<lpage>240</lpage>. <pub-id pub-id-type="doi">10.5194/tc-7-229-2013</pub-id></citation></ref>
<ref id="B23">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Marshall</surname> <given-names>S. J.</given-names></name> <name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>Flowers</surname> <given-names>G. E.</given-names></name> <name><surname>Clarke</surname> <given-names>G. K. C.</given-names></name></person-group> (<year>2005</year>). <article-title>Simulation of Vatnaj&#x000F6;kull ice cap dynamics</article-title>. <source>J. Geophys. Res. Earth Surf.</source> <fpage>110</fpage>. <pub-id pub-id-type="doi">10.1029/2004JF000262</pub-id></citation></ref>
<ref id="B24">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Minchew</surname> <given-names>B.</given-names></name> <name><surname>Simons</surname> <given-names>M.</given-names></name> <name><surname>Hensley</surname> <given-names>S.</given-names></name> <name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name></person-group> (<year>2015</year>). <article-title>Early melt season velocity fields of Langj&#x000F6;kull and Hofsj&#x000F6;kull, central Iceland</article-title>. <source>J. Glaciol.</source> <volume>61</volume>, <fpage>253</fpage>&#x02013;<lpage>266</lpage>. <pub-id pub-id-type="doi">10.3189/2015JoG14J023</pub-id></citation></ref>
<ref id="B25">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Murray</surname> <given-names>I.</given-names></name> <name><surname>Adams</surname> <given-names>R. P.</given-names></name> <name><surname>MacKay</surname> <given-names>D. J.</given-names></name></person-group> (<year>2010</year>). <article-title>Elliptical slice sampling</article-title>. <source>J. Mach. Learn. Res.</source> <volume>9</volume>, <fpage>541</fpage>&#x02013;<lpage>548</lpage>.</citation></ref>
<ref id="B26">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Neal</surname> <given-names>R. M.</given-names></name></person-group> (<year>2003</year>). <article-title>Slice sampling</article-title>. <source>Ann. Statist.</source> <volume>31</volume>, <fpage>705</fpage>&#x02013;<lpage>741</lpage>. <pub-id pub-id-type="doi">10.1214/aos/1056562461</pub-id></citation></ref>
<ref id="B27">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>P&#x000E1;lsson</surname> <given-names>F.</given-names></name> <name><surname>Gu&#x000F0;mundsson</surname> <given-names>S.</given-names></name> <name><surname>Bj&#x000F6;rnsson</surname> <given-names>H.</given-names></name> <name><surname>Berthier</surname> <given-names>E.</given-names></name> <name><surname>Magn&#x000FA;sson</surname> <given-names>E.</given-names></name> <name><surname>Gu&#x000F0;mundsson</surname> <given-names>S.</given-names></name> <etal/></person-group>. (<year>2012</year>). <article-title>Mass and volume changes of Langj&#x000F6;kull ice cap, iceland, 1890 to 2009, deduced from old maps, satellite images and <italic>in situ</italic> mass balance measurements</article-title>. <source>J&#x000F6;kull</source> <volume>62</volume>, <fpage>81</fpage>&#x02013;<lpage>96</lpage>.</citation></ref>
<ref id="B28">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Petra</surname> <given-names>N.</given-names></name> <name><surname>Martin</surname> <given-names>J.</given-names></name> <name><surname>Stadler</surname> <given-names>G.</given-names></name> <name><surname>Ghattas</surname> <given-names>O.</given-names></name></person-group> (<year>2014</year>). <article-title>A computational framework for infinite-dimensional bayesian inverse problems, part II: stochastic newton mcmc with application to ice sheet flow inverse problems</article-title>. <source>SIAM J. Sci. Comput.</source> <volume>36</volume>, <fpage>A1525</fpage>&#x02013;<lpage>A1555</lpage>. <pub-id pub-id-type="doi">10.1137/130934805</pub-id></citation></ref>
<ref id="B29">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Pope</surname> <given-names>E. L.</given-names></name> <name><surname>Willis</surname> <given-names>I. C.</given-names></name> <name><surname>Pope</surname> <given-names>A.</given-names></name> <name><surname>Miles</surname> <given-names>E. S.</given-names></name> <name><surname>Arnold</surname> <given-names>N. S.</given-names></name> <name><surname>Rees</surname> <given-names>W. G.</given-names></name></person-group> (<year>2016</year>). <article-title>Contrasting snow and ice albedos derived from modis, landsat etm&#x0002B; and airborne data from Langj&#x000F6;kull, iceland</article-title>. <source>Remote Sens. Environ.</source> <volume>175</volume>, <fpage>183</fpage>&#x02013;<lpage>195</lpage>. <pub-id pub-id-type="doi">10.1016/j.rse.2015.12.051</pub-id></citation></ref>
<ref id="B30">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Porter</surname> <given-names>C.</given-names></name> <name><surname>Morin</surname> <given-names>P.</given-names></name> <name><surname>Howat</surname> <given-names>I.</given-names></name> <name><surname>Noh</surname> <given-names>M.-J.</given-names></name> <name><surname>Bates</surname> <given-names>B.</given-names></name> <name><surname>Peterman</surname> <given-names>K.</given-names></name> <etal/></person-group>. (<year>2018</year>). ArcticDEM. <pub-id pub-id-type="doi">10.7910/DVN/OHHUKH</pub-id>. Harvard Dataverse, V1.</citation>
</ref>
<ref id="B31">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Pralong</surname> <given-names>M. R.</given-names></name> <name><surname>Gudmundsson</surname> <given-names>G. H.</given-names></name></person-group> (<year>2011</year>). <article-title>Bayesian estimation of basal conditions on Rutford Ice Stream, West Antarctica, from surface data</article-title>. <source>J. Glaciol.</source> <volume>57</volume>, <fpage>315</fpage>&#x02013;<lpage>324</lpage>. <pub-id pub-id-type="doi">10.3189/002214311796406004</pub-id></citation></ref>
<ref id="B32">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Raymond</surname> <given-names>M. J.</given-names></name> <name><surname>Gudmundsson</surname> <given-names>G. H.</given-names></name></person-group> (<year>2009</year>). <article-title>Estimating basal properties of ice streams from surface measurements: a non-linear bayesian inverse approach applied to synthetic data</article-title>. <source>Cryosphere</source> <volume>3</volume>, <fpage>265</fpage>&#x02013;<lpage>278</lpage>. <pub-id pub-id-type="doi">10.5194/tc-3-265-2009</pub-id></citation></ref>
<ref id="B33">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Tierney</surname> <given-names>L.</given-names></name></person-group> (<year>1994</year>). <article-title>Markov chains for exploring posterior distributions</article-title>. <source>Ann. Statist.</source> <volume>22</volume>, <fpage>1701</fpage>&#x02013;<lpage>1728</lpage>.</citation></ref>
<ref id="B34">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Vats</surname> <given-names>D.</given-names></name> <name><surname>Knudson</surname> <given-names>C.</given-names></name></person-group> (<year>2018</year>). <article-title>Revisiting the Gelman-Rubin diagnostic</article-title>. <source>arXiv preprint</source> arXiv:1812.09384.</citation></ref>
<ref id="B35">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Werder</surname> <given-names>M. A.</given-names></name> <name><surname>Huss</surname> <given-names>M.</given-names></name> <name><surname>Paul</surname> <given-names>F.</given-names></name> <name><surname>Dehecq</surname> <given-names>A.</given-names></name> <name><surname>Farinotti</surname> <given-names>D.</given-names></name></person-group> (<year>2020</year>). <article-title>A Bayesian ice thickness estimation model for large-scale applications</article-title>. <source>J. Glaciol.</source> <volume>66</volume>, <fpage>137</fpage>&#x02013;<lpage>152</lpage>. <pub-id pub-id-type="doi">10.1017/jog.2019.93</pub-id></citation></ref>
<ref id="B36">
<citation citation-type="journal"><person-group person-group-type="author"><name><surname>Zammit-Mangion</surname> <given-names>A.</given-names></name> <name><surname>Rougier</surname> <given-names>J.</given-names></name> <name><surname>Bamber</surname> <given-names>J.</given-names></name> <name><surname>Sch&#x000F6;n</surname> <given-names>N.</given-names></name></person-group> (<year>2014</year>). <article-title>Resolving the Antarctic contribution to sea-level rise: a hierarchical modelling framework</article-title>. <source>Environmetrics</source> <volume>25</volume>, <fpage>245</fpage>&#x02013;<lpage>264</lpage>. <pub-id pub-id-type="doi">10.1002/env.2247</pub-id><pub-id pub-id-type="pmid">25505370</pub-id></citation></ref>
</ref-list>
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<fn fn-type="financial-disclosure"><p><bold>Funding.</bold> This work is from the lead author&#x00027;s Ph.D. work with the University of Iceland. The work was funded by the Icelandic Research Fund (RANNIS) by a grant entitled: Statistical Models for Glaciology. Grant number was 152457-051.</p>
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