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Subglacial drainage characterization from eight years of continuous borehole data on a small glacier… Rada Giacaman, Camilo A. 2019

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Subglacial drainage characterization from eight years ofcontinuous borehole data on a small glacier in the YukonTerritory, CanadabyCamilo A. Rada GiacamanMs.C. Geophysics, Universidad de Chile, 2009Bs. Astronomy, Universidad Cato´lica de Chile, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Geophysics)The University of British Columbia(Vancouver)September 2019© Camilo A. Rada Giacaman, 2019The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Subglacial drainage characterization from eight years of continuousborehole data on a small glacier in the Yukon Territory, Canadasubmitted by Camilo A. Rada Giacaman in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Geophysics.Examining Committee:Christian Schoof, Earth, Ocean and Atmospheric SciencesSupervisorGwenn Flowers, Earth Sciences, Simon Fraser UniversitySupervisory Committee MemberEldad Haber, Earth, Ocean and Atmospheric SciencesSupervisory Committee MemberRoger Beckie, Earth, Ocean and Atmospheric SciencesSupervisory Committee MemberUlrich Mayer, Earth, Ocean and Atmospheric SciencesUniversity ExaminerMichele Koppes, GeographyUniversity ExamineriiAbstractThe subglacial drainage system is one of the main controls on basal sliding butremains only partially understood, constituting one of the most significant sourcesof uncertainty in glacier dynamics models. Increasing the accuracy of such modelsis of great importance to correctly forecast the availability of water in glaciatedbasins and the global sea level rise.While current glacial hydrology models are successful in reproducing the gen-eral seasonal change in surface speed and the structure of the subglacial drainagesystem, they fail to reproduce significant features observed in boreholes. Here weuse an eight-year dataset of borehole observations on a small, alpine polythermalvalley glacier in the Yukon Territory, to assess which missing physical processesin current glacier hydrology models can explain borehole observations. Our pri-mary tool to analyze the borehole dataset and make inferences about the structureand evolution of the subglacial drainage system is a custom methodology to clusterwater pressure time series according to their similarities.We find that the standard picture of a distributed drainage system that pro-gressively channelizes throughout the melt season explains many features of thedataset. However, our observations underline the importance of hydraulically dis-connected parts of the bed. Different regions of the bed are generally either hy-draulically well-connected or disconnected, and the transition between the twostates is abrupt in time (minutes to a few hours) and space (<15 m), and the diffu-sivity at the bed has a significant fine structure at scales smaller than our minimumborehole spacing of 15 m.We found that some regions of the bed are more likely to become hydrauli-cally well-connected than others, and some areas can remain hydraulically discon-iiinected year-round, with a significant portion of these disconnected areas experienc-ing pressure variations due to normal stress transfers from hydraulically connectedareas. Using GPS measurements of surface speed, we found that the ratio betweenconnected and disconnected regions of the bed seems to have a greater influence onbasal sliding than the effective pressure within the connected drainage system, sug-gesting that a significant modification has to be made to the accepted ideas aboutbasal sliding.ivLay SummaryUnderstanding the physical processes that control the growth and decline of glaciersis vital to create models capable of forecasting how changing glaciers will affectthe geography, biosphere and climate.This PhD thesis investigates basal sliding, which corresponds to the increasein speed a glacier experience by sliding downhill over the underlying terrain. Wehave studied a glacier in the Yukon Territory, Canada, installing over 300 pres-sure sensors at the base of the glacier to understand how water can create there asubglacial drainage system and accelerate or slow-down the glacier.Contrary to the predictions of current subglacial hydrology models, we foundthat different regions of the bed are typically either hydraulically well-connectedor disconnected, and the transition between these two regimes is generally abrupt.Also, we found that the extent of areas hydraulically disconnected from the surfaceseems to have a very significant role in controlling basal sliding.vPrefaceThis PhD thesis is based on multiple datasets acquired over nine consecutive yearsstarting on 2008 by a group from the University of British Columbia led by Profes-sor Christian Schoof, and a group from Simon Fraser University led by professorGwen Flowers. I actively participated in fieldwork preparation, design, and exe-cution between the years 2012 and 2016. During this period, I collaborated in thedrilling of boreholes, and the installation and maintenance of pressure sensors, dataloggers, GPS towers, and repeat photography cameras.I designed and built the repeat photography cameras, the digital sensors, andtheir associated data loggers. I conducted all of the borehole data analysis andwrote the manuscript with the close collaboration of my supervisor Christian Schoof,who provided numerous corrections and suggestions. A version of Chapter 2has been published: Camilo Rada & Christian Schoof “Channelised, distributed,and disconnected: subglacial drainage under a valley glacier in the Yukon”, TheCryosphere 12(8):2609–2636, 10.5194/tc-2017-270.The kinematic solutions of our Global Position System (GPS) data used inChapter 4 were generated by Proffesor Matt King of the University of Tasmania.Other than that, I performed all the GPS data processing and analysis.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Fundaments of glacier dynamics . . . . . . . . . . . . . . . . . . 11.2 The subglacial drainage system and its influence on basal sliding . 51.3 Conceptual subglacial hydrology models . . . . . . . . . . . . . . 71.4 Evolution of the subglacial drainage system . . . . . . . . . . . . 121.5 Ground truth and challenges for models . . . . . . . . . . . . . . 131.6 Exploring the spatial structure of the subglacial drainage system . 151.7 Content overview . . . . . . . . . . . . . . . . . . . . . . . . . . 17vii2 South Glacier subglacial drainage characterization . . . . . . . . . . 192.1 Field site and methods . . . . . . . . . . . . . . . . . . . . . . . 212.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Modes of water flow: fast, slow and unconnected . . . . . 282.2.2 Spatial patterns in water pressure variations . . . . . . . . 362.2.3 Three-dimensional drainage structures . . . . . . . . . . . 392.2.4 Seasonal development of the subglacial drainage system . 412.2.5 Basal hydrology transitions and “switching events” . . . . 442.2.6 Inter-annual variability . . . . . . . . . . . . . . . . . . . 462.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3.1 Interannual variability . . . . . . . . . . . . . . . . . . . 552.3.2 Challenges to current subglacial drainage models . . . . . 552.3.3 Mechanically connected boreholes . . . . . . . . . . . . . 602.3.4 Data interpretation caveats . . . . . . . . . . . . . . . . . 612.4 Insights for models development . . . . . . . . . . . . . . . . . . 622.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643 Spatial structure and temporal evolution of the subglacial drainagesystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2.1 Clustering calibration, validation, and testing . . . . . . . 803.2.2 Cluster evolution in time . . . . . . . . . . . . . . . . . . 833.2.3 Hydraulic and mechanical cluster types . . . . . . . . . . 853.2.4 Spatial patterns in basal hydraulic connectivity . . . . . . 893.2.5 Pressure variation trends . . . . . . . . . . . . . . . . . . 923.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.3.1 Evolution of the subglacial drainage system . . . . . . . . 943.3.2 Diffusivity at the glacier bed and the two-dimensional na-ture of the drainage system . . . . . . . . . . . . . . . . . 993.3.3 Spatial patterns of connected and disconnected areas . . . 1023.3.4 Spatially averaged pressure trends . . . . . . . . . . . . . 1043.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106viii3.4.1 Diffusivity distribution at the glacier bed . . . . . . . . . 1093.4.2 Subglacial drainage evolution . . . . . . . . . . . . . . . 1113.4.3 Methodological caveats . . . . . . . . . . . . . . . . . . . 1133.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154 Dynamic effects of drainage system evolution . . . . . . . . . . . . . 1184.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.2.1 GPS array . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.2.2 Daily GPS solutions . . . . . . . . . . . . . . . . . . . . 1304.2.3 Sub-diurnal GPS solutions . . . . . . . . . . . . . . . . . 1334.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.3.1 Multi-day speed variations . . . . . . . . . . . . . . . . . 1444.3.2 Sidereal reconstruction: sub-diurnal speed variations . . . 1464.3.3 Glaciological results . . . . . . . . . . . . . . . . . . . . 1504.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614.4.1 South Glacier surface speed record . . . . . . . . . . . . . 1614.4.2 Glaciological controls on surface speed . . . . . . . . . . 1644.4.3 Sub-diurnal speed variations . . . . . . . . . . . . . . . . 1684.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1705 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Appendix A: Digital sensor pods design . . . . . . . . . . . . . . . . . . 197A.1 Main module . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199A.2 Instrument casing . . . . . . . . . . . . . . . . . . . . . . . . . . 200A.3 Power and data lines . . . . . . . . . . . . . . . . . . . . . . . . 200A.3.1 Analog communication . . . . . . . . . . . . . . . . . . . 200A.3.2 Digital communication . . . . . . . . . . . . . . . . . . . 202A.3.3 Input/Output board . . . . . . . . . . . . . . . . . . . . . 202A.4 Pressure sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 202A.5 Conductivity sensor . . . . . . . . . . . . . . . . . . . . . . . . . 204ixA.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . 204A.5.2 Computation of the conductivity . . . . . . . . . . . . . . 207A.5.3 Optimization of the design . . . . . . . . . . . . . . . . . 209A.6 Transmissivity sensor . . . . . . . . . . . . . . . . . . . . . . . . 210A.7 Temperature sensor . . . . . . . . . . . . . . . . . . . . . . . . . 210A.8 Orientation and motion sensor . . . . . . . . . . . . . . . . . . . 210A.9 Confinement sensor . . . . . . . . . . . . . . . . . . . . . . . . . 211A.10 Reflection spectrometer . . . . . . . . . . . . . . . . . . . . . . . 212Appendix B: Data quality . . . . . . . . . . . . . . . . . . . . . . . . . . 213Appendix C: Reliability of temperature as a melt proxy . . . . . . . . . 222Appendix D: Clustering evaluation and calibration . . . . . . . . . . . . 225D.1 Alternative clustering and data analysis techniques . . . . . . . . 225D.1.1 Empirical Orthogonal Functions (EOFs) . . . . . . . . . . 226D.1.2 Covariance analysis . . . . . . . . . . . . . . . . . . . . . 227D.1.3 Self Organizing Maps (SOMs) . . . . . . . . . . . . . . . 229D.1.4 K-means clustering . . . . . . . . . . . . . . . . . . . . . 230D.2 Distance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 230D.3 Entropy and Information Gain: Evaluating clustering performance 233xList of TablesTable 1.1 Summary of main drainage elements on glacier hydrology models 9Table 4.1 Symbols used . . . . . . . . . . . . . . . . . . . . . . . . . . 119xiList of FiguresFigure 1.1 Schematic representation of a glacier . . . . . . . . . . . . . 2Figure 1.2 Schematic representation basal sliding and internal deforma-tion in glaciers . . . . . . . . . . . . . . . . . . . . . . . . . 4Figure 1.3 Schematic representation of the main subglacial conduits . . . 7Figure 1.4 Modeled subglacial drainage system adapted from Werder et al.[2013] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.1 South Glacier general map . . . . . . . . . . . . . . . . . . . 22Figure 2.2 South Glacier study area map . . . . . . . . . . . . . . . . . 23Figure 2.3 Photographs of the study area on July 19th, 2012–2015 . . . . 25Figure 2.4 Analog and digital pressure sensors . . . . . . . . . . . . . . 27Figure 2.5 Long-term pressure record and positive degree day (PDD) . . . 29Figure 2.6 Locations and pressure time series for the boreholes associatedwith the fast-flow borehole during the summer of 2013 . . . . 31Figure 2.7 Locations and pressure time series for the boreholes associatedwith the slow-flow borehole on July and August 2014 . . . . . 33Figure 2.8 Switching event close-up in the borehole line of the slow-flowborehole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.9 Mean water pressure computed for 55 disconnected boreholesduring 2016 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 2.10 Locations and pressure time series for 102 boreholes on theplateau area and down-glacier during June–August 2015 . . . 38Figure 2.11 Relative pressure variations in a mechanical cluster . . . . . . 40xiiFigure 2.12 Locations and detailed views of the pressure time series for all42 boreholes shown in Fig. 2.10b . . . . . . . . . . . . . . . 42Figure 2.13 Relative amplitudes of pressure and temperature diurnal oscil-lations from May to September 2015 . . . . . . . . . . . . . . 45Figure 2.14 Extended pressure time series from the 2013 fast-flow borehole 47Figure 2.15 Overview of pressure variations on the lower portion of theplateau area from 2012 to 2015 . . . . . . . . . . . . . . . . . 49Figure 2.16 Confinement and pressure data for one of borehole from July2015 to September 2016 . . . . . . . . . . . . . . . . . . . . 50Figure 3.1 Horizontal normal stress transfers . . . . . . . . . . . . . . . 68Figure 3.2 Raw pressure and diurnal residual comparison . . . . . . . . . 75Figure 3.3 Dendrogram example . . . . . . . . . . . . . . . . . . . . . . 77Figure 3.4 Comparison of hierarchical clustering strategies . . . . . . . . 84Figure 3.5 Mechanical shapelet . . . . . . . . . . . . . . . . . . . . . . 86Figure 3.6 Correlated and anti-correlated subclusters . . . . . . . . . . . 87Figure 3.7 Scatter plots of mean effective pressure and pressure standarddeviation for mechanical and hydraulic clusters . . . . . . . . 88Figure 3.8 Relative positions of boreholes, connected boreholes and PDFfits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Figure 3.9 Probability density function for hydraulic connections . . . . 92Figure 3.10 Example of boreholes transitioning from correlated to anti-correlated . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Figure 3.11 Example of a cluster transitioning from hydraulic to mechanical 93Figure 3.12 Mean pressure in cluster H1 . . . . . . . . . . . . . . . . . . 95Figure 3.13 Cluster network during the melt season 2015 . . . . . . . . . 97Figure 3.14 Phase lag and amplitude distribution on window g of cluster H1 100Figure 3.15 Detailed spatial distribution of clustered boreholes . . . . . . 101Figure 3.16 Bed connectivity South Glacier . . . . . . . . . . . . . . . . 103Figure 3.17 Spatially averaged pressure trends . . . . . . . . . . . . . . . 104Figure 3.18 Hydraulic and Mechanical clusters in the frequency domain . 106Figure 4.1 Eight-year speed record at South Glacier . . . . . . . . . . . 121xiiiFigure 4.2 Multipath geometry for a GPS tower . . . . . . . . . . . . . . 125Figure 4.3 signal to noise ratio (SNR) example at tower R10C18 . . . . . 126Figure 4.4 Kinematic solutions at R14C18 for ten consecutive days . . . 126Figure 4.5 GPS tower set-up . . . . . . . . . . . . . . . . . . . . . . . . 129Figure 4.6 Approximate trajectories of GPS towers at South Glacier . . . 130Figure 4.7 Comparison of static solutions at R20C20 . . . . . . . . . . . 132Figure 4.8 Multipath pattern in kinematic solutions of consecutive days . 136Figure 4.9 Comparison of speeds computed from samples one day, oneSD and one ART apart . . . . . . . . . . . . . . . . . . . . . 137Figure 4.10 L-curve of the sidereal reconstruction inversion . . . . . . . . 142Figure 4.11 Synthetic multipath patterns . . . . . . . . . . . . . . . . . . 143Figure 4.12 Sidereal reconstruction of synthetic data . . . . . . . . . . . . 144Figure 4.13 Multi-base diurnal solutions for all GPS towers . . . . . . . . 145Figure 4.14 Static solution for tower R20C20 using different bases andprecise point positioning (PPP) . . . . . . . . . . . . . . . . . 146Figure 4.15 STD of the detrended trajectory as function of the mean speed 147Figure 4.16 Sidereal reconstruction for tower R22C18 using multiple λvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148Figure 4.17 Comparison of 8-year surface speed, PDD, and displacements . 149Figure 4.18 Reconstructed speed at four GPS towers . . . . . . . . . . . . 150Figure 4.19 Minimum and mean surface speed as function of PDD . . . . . 152Figure 4.20 Surface speed as a function of PDD . . . . . . . . . . . . . . . 153Figure 4.21 Comparison of mean pressure, surface speed, fraction of con-nected boreholes, and PDD . . . . . . . . . . . . . . . . . . . 154Figure 4.22 Mean daily speed as a function of mean daily pressure, pres-sure STD, PDD, and Jansson [1995] empirical model . . . . . 156Figure 4.23 Best fit of three different empirical models to surface speed . . 158Figure 4.24 40 days sidereal reconstruction for GPS tower R22C18 . . . . 160Figure A.1 Digital sensor pod PCB design . . . . . . . . . . . . . . . . . 199Figure A.2 I/O board PCB . . . . . . . . . . . . . . . . . . . . . . . . . 203Figure A.3 Example of a pulse with modulation (PWM) output . . . . . . 205Figure A.4 General schematics of the conductivity sensor . . . . . . . . . 205xivFigure A.5 States of the conductivity sensor . . . . . . . . . . . . . . . . 206Figure A.6 Conductivity signal amplitude . . . . . . . . . . . . . . . . . 206Figure A.7 Conductivity amplitude calculation . . . . . . . . . . . . . . 207Figure A.8 Conductivity resistor optimal value . . . . . . . . . . . . . . 209Figure B.1 Pressure records for the two sensors in borehole 13H16 . . . . 214Figure B.2 Pressure records for the two sensors in borehole 13H17 . . . . 214Figure B.3 Pressure records for the two sensors in borehole 13H58 . . . . 215Figure B.4 Pressure records for the two sensors in borehole 14H60 . . . . 215Figure B.5 Pressure records for the two sensors in borehole 14H62 . . . . 216Figure B.6 Pressure records for the two sensors in borehole 15HL07 . . . 216Figure B.7 Pressure records for the two sensors in borehole 15HU01 . . . 216Figure B.8 Pressure records for the two sensors in borehole 15HU04 . . . 216Figure B.9 Pressure records for the two sensors in borehole 15HU05 . . . 221Figure B.10 Pressure records for the two sensors in borehole 15HU17 . . . 221Figure B.11 Pressure records for the two sensors in borehole 15HU50 . . . 221Figure C.1 Comparison of Degree-day factors computed using the modelby Wheler and Flowers (2011), and PDD values . . . . . . . . 223Figure C.2 Alternative version of Fig. 2.13c using surface lowering asproxy for meltwater production . . . . . . . . . . . . . . . . 223Figure D.1 SOM clustering example . . . . . . . . . . . . . . . . . . . . 231xvGlossaryADC analog digital converterART aspect repetition timeAWS automatic weather stationDDF degree day factorDTW dynamic time wrappingELA equilibrium line altitudeEOF empirical orthogonal functionsGAMIT GNSS at MITGNSS global navigation satellite systemGPR ground-penetrating radarGPS Global Position SystemIGS international GNSS serviceKLRS Kluane lake research stationOBP overburden pressureOSP optimal split pointPDD positive degree dayxviPDF probability density functionPPP precise point positioningPWM pulse with modulationRIG relative information gainSD sidereal daySNR signal to noise ratioSOM self-organizing mapsSP split pointVDC volts direct currentxviiAcknowledgmentsI offer my enduring gratitude to those who offered their hard work and high spiritsto put together the extensive dataset that made this thesis possible. This datasetis an achievement that required thousands of hours of strenuous work and care-ful thinking by Manar Al Asad, Flavien Beaud, Ashley Bellas, Jeffrey Cromp-ton, Emilie Delaroche, Johan Gilchrist, Marianne Haseloff, Elisa Mantelli, NataliaMartinez, Arran Whiteford, Nat Wilson, Kevin Yeo, and of course by my advisorChristian Schoof and co-advisor Gwenn Flowers. Christian and Gwenn had thevision and perseverance to dedicate a decade to the careful research of a small for-saken glacier in the Yukon, humbly learning from the observations and generouslyteaching my fellow students and me along the way.I also need to thank my committee members Roger Beckie and Eldad Harber,and to professor Valentina Radic, whose questions and suggestions guided multiplebranches of my work. My fellow students Luz Caudillo, Noel Fitzpatrick andGabriela Racz for all their help and advice.I have to thanks my family and friends for their continuous support, affection,and appreciation for my work, that encouraged me to pursue and complete thisPhD. In particular to my Mother and above all to Natalia, my partner, companionand friend in the adventure of pursuing this PhD in Canada, and so many otheradventures, past and to come.Finally, I have to acknowledge CONICYT becas Chile for the scholarship thatallowed me to come here in the first place. Also, to UBC and Christian Schoof fortheir financial support during the last years of my program.xviiixixChapter 1IntroductionGlaciers are large ice masses that can be found from the polar areas to the tropics.At high latitudes, they can extend into massive ice sheets covering whole conti-nents, while in the tropics they are confined to the top of the highest mountains.However, regardless of their location, they play an essential role in shaping theirlocal ecosystems as well as the global environment. At a local scale, glaciers actas water reservoirs, storing snow and ice during the wet season and releasing waterduring the dry season. This modulation of the water supply results in the preven-tion or moderation of floods and droughts. At a global scale, glaciers constitutethe largest reservoir of water after the oceans, containing enough of it to raise thesea level by 66.5 m [Bamber, 2016]. The total mass of glaciers and ice sheets isshrinking at an accelerating rate, thus increasing their sea level contribution. Globalestimations suggest that the average contribution to sea level from glaciers and icesheets increased from 0.31± 0.35 mm yr−1 between 1992 and 1996 to 1.85± 0.13mm yr−1 between 2012 and 2016 [Bamber et al., 2018], and further increases areexpected.1.1 Fundaments of glacier dynamicsGlacier ice results from the accumulation and compaction of snow and the refreez-ing of meltwater. Therefore, any glacier relies on an area where snow accumulatesfaster than it melts. Over such an area, the glacier has an overall mass gain, and1the area is therefore termed as the “accumulation area” [Colbeck, 2002, Paterson,2002]. If the volume of ice in the accumulation area is large enough, the ice that weoften picture as a rigid and brittle material starts to display its viscoelastic proper-ties [Glen, 1952]. The stress generated by its own weight will set the ice in motion,flowing downhill and outside the accumulation area. The lower part of the glacierbelow the accumulation area constitutes the “ablation area”, where the surface isexposed to warmer temperatures and less snow accumulation, leading to a net lossof mass.Figure 1.1: Schematic representation of a glacier, including the accumula-tions and ablation areas, the equilibrium line altitude (ELA), and themain components of the surface drainage that routes meltwater into thesubglacial drainage system. Black arrows within the glacier illustratemagnitude and direction of ice flow through the glacier.Figure 1.1 illustrates the accumulation and ablation areas on the surface ofa glacier, as well as the equilibrium line altitude (ELA) separating the two. TheELA represents the elevation at which the annual gains of mass equal the losses.A glacier is in equilibrium when the total accumulation equals the total ablation,and the difference between the two quantities constitutes the mass balance of the2glacier. Glacier mass balance is a key indicator of the changes taking place in aglacier, and how those changes will affect the environment.Glacier mass balance is mainly controlled by precipitation and temperature.However, in any climatic scenario mass balance can be dramatically affected bychanges in the dynamics of the glacier. If the ice speed increases, the ice is trans-ported quicker to lower elevations, where higher temperatures accelerate the melt-ing. Also, higher speeds can change the shape of the glacier, enlarging the ablationarea at the expense of thinning the accumulation area. All these consequences of anincrease in speed have a negative effect on mass balance [Colbeck, 2002, Paterson,2002].The speed of a glacier is controlled by two processes: internal deformationand basal sliding. Internal deformation corresponds to a slow viscous creep ofthe ice driven by gravity and constrained by the mechanical properties of the ice[Lliboutry, 1958, Weertman, 1957, 1964]. This process is somewhat similar to theflow of a river, where the speed due to the internal deformation of the water peaksat centre line on the surface, while it decreases with depth, tending to zero at thebed. In contrast to internal deformation, basal sliding takes place exclusively at thesole of the glacier, referring to interfacial sliding of the glacier over the bedrock orthe deformation of a thin till layer.Figure 1.2 illustrates these two components of the glacier speed. Basal slidingcan only take place in glaciers whose base is at the melting point of water. Other-wise, the base of the glacier would be frozen to the bed and sliding would not bepossible, as it is the case for the so-called cold glaciers. Glaciers with bases at themelting point are instead referred to as temperate or polythermal glaciers, the latterreferring to glaciers that contains a mix of cold and temperate ice.While the physics controlling the internal deformation is relatively well under-stood, the one controlling basal sliding remains largely unknown and constitutesthe primary source of uncertainty in glacier flow models [Flowers, 2015]. Thisuncertainty arises partly from changes in basal conditions associated with the pres-ence of water. The base of temperate and polythermal glaciers generally containsliquid water, either in the form of a thin sheet between the ice and the bedrock, inwater pockets, inside conduits, or in the pore space between grains of till. All thosewater-filled volumes constitute the subglacial drainage system, and the water they3Figure 1.2: Schematic representation of the two processes that control thespeed of a glacier: basal sliding and internal deformation. The figureillustrate how each of these processes would affect the movement of anice column within the glacier.contain can come from two main sources: 1) basal melting as a consequence ofheat dissipated by basal friction, or geothermal heat, or 2) from the surface, wherewater can be routed to the bed through crevasses, cracks or other vertical pathways[Weertman, 1972].Figure 1.1 illustrates how surface meltwater in the ablation area forms sur-face streams that eventually find their way into the glacier through crevasses orsmall cracks that can be widened by the water flow to form vertical sinks known asmoulins. Routing of water from the surface can also happen through the snowpack,where water-saturated layers of snow can form and eventually drain into crevasses[Ro¨thlisberger, 1972, Fountain and Walder, 1998, Fountain et al., 2005]. Waterentering the subglacial drainage system can lead to changes in water pressure andin the geometry of the drainage system. In turn, these changes can affect the basalsliding speed.41.2 The subglacial drainage system and its influence onbasal slidingBasal sliding typically accounts for about half of the observed surface speed ofglaciers [Gerrard et al., 1952, McCall, 1952, Mathews, 1959, Shreve, 1961, Sav-age and Paterson, 1963, Vivian, 1980, Boulton and Hindmarsh, 1987, Blake et al.,1994, Harper et al., 1998]. Moreover, the contribution of basal sliding to overallice transport is especially important for large fast-flowing glaciers. For example,in the largest outlet glacier of the Greenland ice sheet (Jakobshavn Isbræ), basalsliding has been found to account for 44% to 90% of the measured surface speed[Lu¨thi et al., 2002, Ryser et al., 2014b]. Similarly, in Antarctic ice streams, basalsliding can account on average for about 69% of the observed surface speed [En-gelhardt and Kamb, 1998]. The sliding rate shows a marked seasonal variationin areas where seasonality is associated with significant changes in surface melt,with summer sliding speeds often two to three times faster than winter averages[Nienow et al., 1998a, Sole et al., 2011, Ryser et al., 2014b]. These variations area consequence of changes in the subglacial drainage system associated with theseasonal input of surface meltwater [Iken and Bindschadler, 1986, Gordon et al.,1998, Nienow et al., 1998b, Mair et al., 2001, Harper et al., 2005]. However, thosechanges in the subglacial drainage system are one of the least observed glaciologi-cal phenomena, and we have only a limited understanding of how they take place,which physical processes are involved, and how do they influence basal slidingrates.Traditionally, the main variable linking subglacial drainage processes to basalsliding is the effective pressure, defined as the difference between normal stress atthe bed and water pressure, where normal stress is usually taken to be equal to theoverburden pressure (OBP). In turn, the OBP corresponds to the weight of the icecolumn. Other variables that play a role in modulating basal sliding, include thesize and distribution of bedrock heterogeneities, the presence of basal till, and thesize and abundance of rock clasts embedded in basal ice [Weertman, 1957, Alleyet al., 1986, Alley, 1989]. Although these factors can change significantly from oneglacier to another, they are unlikely to control basal speed variations at seasonal orshorter timescales at a given glacier. Therefore, we will concentrate our attention5on the role of effective pressure.We can understand effective pressure as the strength of the mechanical cou-pling between the bed bedrock and the overlying ice. When effective pressure iszero, the glacier is basically afloat, and there is a negligible coupling. On the otherhand, high effective pressure corresponds to situations when most of the weight ofthe glacier is supported by the bedrock, leading to a strong mechanical coupling.When effective pressure is low, the corresponding high basal water pressureprovides partial support for the weight of the glacier, reducing the contact surfacewith the underlying bedrock, and therefore enhancing basal sliding [Lliboutry,1958, Hodge, 1979, Iken and Bindschadler, 1986, Fowler, 1987, Schoof, 2005,Gagliardini et al., 2007]. A similar effect is observed on glaciers resting on atill layer, where a lower effective pressure reduces the yield stress of the till, andtherefore also enhances basal sliding [Engelhardt et al., 1978, Iverson et al., 1999,Tulaczyk et al., 2000, Truffer et al., 2001]. Conversely, large effective pressuresenhance the mechanical coupling at the bed interface and therefore reduce sliding.The magnitude of the effective pressure is controlled by the combined effect ofthe rate of meltwater supply and the configuration of the englacial and subglacialconduits that drain the water out of the glacier [Iken et al., 1983, Kamb et al., 1985,Iken and Bindschadler, 1986]. For a given conduit configuration, an increase inwater supply is likely to decrease effective pressure. Specifically, water pressuregradients must increase in order to evacuate the additional water input from thesystem, requiring larger water pressures near locations of water supply to the bed.The extent to which increased water supply raises water pressure depends onthree factors: 1) the permeability of till underlying the glacier, 2) the configura-tion of conduits, both at the bed and within the ice, and 3) the storage capacity ofthe drainage system, which can act to buffer the effect of additional water supply.In turn, the conduits that make up the drainage system can change in response tochanges in water input: associated changes in discharge will affect the rate of con-duit enlargement by wall melting, and changes in effective pressure will affect therate at which conduits constrict by the viscous creep of the overlying ice. Changesin sliding (themselves perhaps due to changes in effective pressure) can also affectthe opening of some types of conduits [Hoffman and Price, 2014]. Therefore, overtime the drainage system can change, along with its response to a given water input6[Schoof, 2010a]. Those changes in the drainage system result from the action ofopening and closing mechanisms that control the size and extent of the differenttypes of conduits that compose the subglacial drainage system, where a conduit isany physical connection between two spatially separated parts of the glacier bed orinterior along which water can flow.1.3 Conceptual subglacial hydrology modelsFigure 1.3: Schematic representation of the main conduit types in currentconceptual glacier hydrology models.Many different subglacial conduits have been identified, some of them havebeen observed as water outlets at the edge of glaciers, inside natural caves andcrevasses [Carol, 1947], or from artificial tunnels excavated into the base of glaciers[Peterson, 1970, Vivian and Bocquet, 1973, McKenzie and Peterson, 1975, Vivian,1980], while others are hypothetical, and their existence has nor been fully con-firmed by direct observation. Although observed subglacial conduits vary widely7in shape, size, and location, they can be grouped according to the mechanismsthat allow them to form, grow, and close. Because ice behaves as a viscoelas-tic material, any opening is liable to close by viscous creep. How such conduitsform and stay open is one of the main questions in subglacial hydrology. Cur-rent drainage models typically consider a system composed of two main typesof conduits, R-(Ro¨thlisberger) channels [Ro¨thlisberger, 1972] and linked cavities[Lliboutry, 1968, Walder, 1986]. However, other types of conduits and modes ofwater transport have been proposed [Alley et al., 1986, Walder, 1982, Walder andFowler, 1994, Ng, 2000, Boulton et al., 2007, van der Wel et al., 2013]. Figure1.3 illustrates those most commonly referred to in the literature, while Table 1.1summarises their opening and closing mechanisms. Current understanding of thephysics of subglacial drainage is based on the interaction of these conceptuallydistinct conduits. However, for our research, we will work within the theoreti-cal framework used in current subglacial drainage models [Schoof, 2010a, Hewitt,2011, Schoof et al., 2012, Hewitt et al., 2012, Hewitt, 2013, Werder et al., 2013,Bueler and van Pelt, 2015]. This framework considers a subglacial drainage sys-tem composed only of R-channels and cavities. These two conduit types allow therepresentation of the two broad categories of channelized and distributed drainagesystems, all within a convenient and well developed mathematical framework. Achannelized drainage system would be composed of well developed R-channels,which are the end-member example of an efficient drainage system. In contrast, adistributed drainage can be composed of cavities, porous flow or films, and consti-tutes an end-member example of an inefficient drainage system.R-channels grow by turbulent dissipation of heat leading to wall melting andclose due to viscous creep in the surrounding ice. Creep closure is driven by thedifference between water pressure in the channel and far-field pressure in the ice[Ro¨thlisberger, 1972]. Multiple channels in close proximity are unstable. In sucha configuration, one channel that is slightly larger than its neighbours will alsocarry a larger discharge resulting in higher dissipation and a faster opening rate.The creep closure rate will also be faster in a larger channel than in a smaller one,but is less sensitive to size than the opening rate [Schoof, 2010a]. Therefore, thelarger channel will grow at the expense of the smaller ones. This process tends tofocus water flow into a few large channels, leading to the formation of an arterial8Table 1.1: Summary of main drainage elements on glacier hydrology modelsDescription Opening mechanism Closure mechanism When dischargeincreasesChannels Efficient, fast flow conduitsformed partially or entirelywithin the ice. They might varyin cross-sectional shape andthe association with bedrocktroughsViscous dissipation of heat Ice creep Effective pressureincreasesLinked cavities Inefficient, slow flow network ofwater-filled volumes formed inthe lee of bedrock obstaclesDynamic ice separationfrom the bedrockIce creep Effective pressuredecreasesSheets or films Inefficient, slow flow thin layerof water underlying the ice. Inits simplest form, this drainageelement is often considered un-stable [Walder, 1982]Water production by fric-tional or geothermal heatWater discharge Effective pressuredecreasesPorous medium Inefficient, slow flow throughpore space in underlying tillDilatation and deforma-tion, increasing pore spaceand hydraulic conductivityTill compression andcompactionEffective pressuredecreasesCanals Low-efficiency conduits incisedinto soft bed sediments, with iceroof and till floorViscous dissipation of heaton the ice and erosion ofsediments on the baseIce and till creep anddeposition of sedi-mentsEffective pressuredecreases9Figure 1.4: Simulation of the subglacial drainage in Gornergletscher on the2007 melt season at the time of peak input (July 19th) adapted fromWerder et al. [2013]. Blue lines represent channels and line widths areproportional to the channel discharge. Black dots represents moulinsand the background color represents the effective pressure.drainage system covering a small fraction of the glacier bed [Fountain and Walder,1998, Schoof, 2010a, Hewitt, 2011, 2013, Werder et al., 2013]. An example of anarterial drainage system is shown in Fig. 1.4.In an R-channel, steady state is reached at a higher effective pressure when thechannel discharge increases, as a faster melt rate has to be offset by a faster closurerate. By implication, when water drains through channels, an increase in watersupply should increase effective pressure around the channels, and slow the glacierdown [Nye, 1976, Spring and Hutter, 1982, Schoof, 2010a].On the other hand, linked cavity systems are thought to provide a less efficienttransport mechanism, where slow water flow provides negligible heat dissipation.Cavities are kept open by the sliding of ice over bed roughness elements (see Fig.1.3), which causes an ice-bed gap to open in their lee (see Fig. 1.3), while they alsoclose by viscous creep [Lliboutry, 1968, Kamb et al., 1985, Fowler, 1987].Unlike channels, multiple cavities can co-exist in close proximity, because a10larger cavity size facilitates faster creep closure rates, while the opening rate isgenerally assumed not to depend significantly on size. Therefore, larger cavitieswill tend to close faster and converge to equilibrium with small ones [Kamb et al.,1985, Fowler, 1987, Creyts and Schoof]. The opening of cavities is not assumedto depend significantly on effective pressure either, allowing them to coexist withchannels. This coexistence is possible even at the high effective pressures that canbe associated with channels, as long as the opening rate driven by basal sliding canoffset the closing rate.In contrast to channels, equilibrium in a linked cavity system is reached atlower effective pressure when discharge increases: cavities have to grow to accom-modate additional discharge, and this requires creep closure to be suppressed bya reduced effective pressure. Therefore, an increase in discharge should decreaseeffective pressure, and speed the glacier up [Kamb, 1987, Schoof, 2010a].If a cavity becomes disconnected, its fixed volume will result in a water pres-sure drop if sliding accelerates. Conversely, decelerating basal sliding will leadto relatively high water pressure in order to prevent creep closure, reducing basaldrag. In other words, isolated cavities could act either as sticky spots when basalsliding speeds up or as slippery spots when it slows down, working as a buffer forbasal sliding variations [Iken and Truffer, 1997, Bartholomaus et al., 2011].The formation of channels can be understood as an instability in the drainagethrough a distributed network of conduits and can be expected to occur when thewater supply is sufficiently large [Schoof, 2010a, Hewitt, 2011, 2013, Werder et al.,2013]. However, even under such conditions the formation of a well-developedarterial channel network requires time and may not be fully complete in a singlesummer melt season. Also, even where well-developed channels evolve, they canbe expected to be embedded in a remnant system of linked cavities.111.4 Evolution of the subglacial drainage systemThe seasonal evolution of the drainage system consists of the summer develop-ment of an arterial network of channels from the preexisting drainage pathways ofa linked cavity system. This is followed in winter by the closure of those channels,thus returning to a drainage system composed entirely by linked cavities [Foun-tain and Walder, 1998, Bingham et al., 2005, Irvine-Fynn et al., 2011]. Althoughthis cycle takes multiple months, the drainage system never really reaches equi-librium, because the seasonal increase in meltwater supply is composed of shorterpulses of meltwater production controlled by weather patterns and the diurnal cy-cle of incident solar radiation. In contrast, the time scale associated with channelevolution is longer than a day [Werder et al., 2013] and may be longer than the4–5 day timescale of typical weather systems. As a consequence, drainage sys-tems typically experience a pronounced diurnal pressure cycle, with low effectivepressures during the afternoon peak of daily meltwater production, and high effec-tive pressures during the minimum of meltwater production at night [Vivian andBocquet, 1973, Gordon et al., 1998, Harper et al., 1998, Fudge et al., 2005]. Inextreme cases, channels can eventually fail to fill up with water, at which pointinternal pressure drops to atmospheric pressure [Vivian and Bocquet, 1973, Radaand Schoof, 2018]. However, over multiple days the fast closure rates experiencedat night are compensated by intense wall melting during the afternoon.Although channels and cavities can be modelled as separate entities, the ap-proach used by the majority of current models use a unified conduit entity that canevolve both as a cavity or a channel, as in the framework put forward by Schoof[2010a]. Although such models have allowed significant progress in the under-standing of subglacial processes, they still fall short of reproducing the diversityof phenomena that take place in the subglacial drainage system [Flowers, 2015].These observations can consist of records acquired at the edge of glaciers, insidenatural caves, crevasses, artificial tunnels, and boreholes. Borehole observations,widely used since the late 1980’s, have become an important tool to study thesubglacial drainage system, providing a direct and reliable way to access the sub-glacial drainage system. These borehole observations will be a central part of thisPhD thesis.121.5 Ground truth and challenges for modelsDrainage models that study the interactions between cavities and channels [e.g.Werder et al., 2013], succeeded in reproducing the observed variations of glaciervelocities at a seasonal scale, and several other features of the drainage system.Such features include the seasonal development of a channelised drainage systemand its up-glacier propagation during the spring and summer [Gordon et al., 1998,Nienow et al., 1998b, Mair et al., 2001], and to some extent the relationship be-tween surface meltwater production, basal water pressure, and speed-up events.However, those models still fail to reproduce direct borehole observations ofsubglacial conditions [Flowers, 2015]. Some of these observations, such as thewater transport above the bed through englacial R-channels [Fountain and Walder,1998, Fountain et al., 2005], or the existence of unconnected areas of the bed, couldhave a direct effect on basal sliding.The most common observations that are not consistent with current modelsinclude:1. The existence of disconnected areas that show no signs of diurnal changesin water pressure [Hodge, 1979, Engelhardt et al., 1978, Murray and Clarke,1995, Gordon et al., 1998, Hoffman et al., 2016].2. The development of widespread areas of high water pressure during winter[Fudge et al., 2005, Harper et al., 2005, Ryser et al., 2014a, Wright et al.,2016].3. Boreholes exhibiting pressures exceeding the overburden pressure [Gordonet al., 1998, Kavanaugh and Clarke, 2000, Boulton et al., 2007].4. Large pressure gradients over short distances [Murray and Clarke, 1995, Ikenand Truffer, 1997, Fudge et al., 2008, Andrews et al., 2014].5. Sudden reorganisation of the drainage system [Leb. Hooke and Pohjola,1994, Gordon et al., 1998, Kavanaugh and Clarke, 2000].6. Boreholes exhibiting anti-correlated temporal pressure variations [Murrayand Clarke, 1995, Gordon et al., 1998, Andrews et al., 2014, Lefeuvre et al.,2015, Ryser et al., 2014a].137. Strong clustering of the pressure variations observed into distinct subsetsof boreholes [Gordon et al., 1998, Harper et al., 2002, Fudge et al., 2008,Huzurbazar and Humphrey, 2008, Schoof et al., 2014].8. Englacial conduits [Fountain and Walder, 1998, Nienow et al., 1998b, Gor-don et al., 1998, Fountain et al., 2005, Harper et al., 2010].These borehole observations offer hints of missing physics and flawed assump-tions in current models. These models assume the existence of a persistent andpervasive subglacial drainage system. In summer, this drainage system would becomposed of cavities and channels sustained by surface meltwater supply, and inwinter it would be composed exclusively of linked cavities. Cavities may indeedbe persistent features of the winter subglacial drainage system, both due to thelow effective pressure that characterises that period, and the year-round non-zerobasal sliding that is characteristic of temperate and polythermal glaciers. However,the pervasiveness of a linked cavity system relies on the assumption that cavitiesalways connect to each other. Field observations challenge the validity of this as-sumption: almost all the observations listed above suggest directly or indirectlythat some areas of the bed can become hydraulically isolated from others. In par-ticular, such isolation would impede the flow of water, allowing the developmentof high pressures in isolated areas as well as large pressure differences relative toother areas (observations #1,2, and 4). Additionally, isolated water pockets at thebase of the glacier can experience large pressure variations in response to changesin stress, potentially explaining the occurrence of anti-correlated pressure varia-tions (observation #6), in terms of horizontal transfers of normal stress [Murrayand Clarke, 1995, Lefeuvre et al., 2015]. Moreover, transitions between the dis-connected and connected states could explain the reorganization of the drainagesystem at time scales much shorter than those associated with the evolution of cav-ities and R-channels (observation #5). Finally, hydraulically isolated areas couldsplit the drainage system into multiple disjoined subsystems (observation #7).These consequences of hydraulic isolation show that most of the conflictingobservations can be reconciled with the traditional framework by disposing ofthe assumption that cavities always connect with each other forming a pervasivelinked cavity system. Alternatively, we can argue that the connections between14cavities form gradually and are controlled by multiple parameters, such as cav-ity size, effective pressure, bed roughness and porosity. However, acknowledgingthat portions of the bed can exist in hydraulic isolation seem to be necessary, butnot enough to explain all the observations. A complete explanation will requirethe consideration of physical phenomena such as mechanical transfers of normalstress. These stress transfers can either reduce or increase the effective pressure insome regions of the bed. The reduction of effective pressure can take place whenthe water pressure within a channel is higher than in their surroundings. Suchpressure excess would then offer partial support of the overlying ice, thus reduc-ing the effective pressure in neighbouring areas. This process was termed as “loadtransfer” by Murray and Clarke [1995], and subsequently observed and studied byothers [Gordon et al., 1998, Lappegard et al., 2006, Lefeuvre et al., 2015]. On theother hand, if the water pressure within the channel is lower than in its surround-ings, the unsupported weight of the ice above the channel would be transferred tothe neighbouring bed, thus increasing the effective pressure. This process was de-scribed by Weertman [1972] and later referred to as “bridging stress” [Lappegardet al., 2006]. Load transfer has been proposed as the explanation for boreholes ex-hibiting anti-correlated pressure variations, and bridging stresses might play a rolein the hydraulic isolation of some areas of the bed.1.6 Exploring the spatial structure of the subglacialdrainage systemThe explanation of single pressure records provides little insight into the overallevolution of the subglacial drainage system unless we can use them to study thechanges in the hydraulical connections that compose it. Such hydraulic connec-tions have been studied using salt and dye tracing [Hubbard and Nienow, 1997]and slug tests [Stone, 1993, Iken et al., 1996, Kulessa et al., 2005]. However, thesemethods are not suitable for the continuous study of the hydraulic connections be-tween boreholes. Alternatively, borehole pressure data has also been used to inferthe existence of hydraulic connections in the subglacial drainage system [Fudgeet al., 2008, Huzurbazar and Humphrey, 2008]. While this method is indirect, itallows the inference of hydraulic connections between boreholes based on their15pressure records alone. In particular, a hydraulic connection is implied betweenboreholes displaying similar pressure variations, and the degree of similarity be-tween the boreholes can be used to assess the relative efficiency of the identifiedconnection.The identification of hydraulic connections based on the similarities in pressurerecords is possible during the summer only, thanks to the quasi-diurnal meltwatersupply forcing that naturally probes the drainage system. Without such a forcing, itwould be impossible to use this approach to infer subglacial hydraulic connections.However, given that the whole drainage system is subject to a similar forcing, thisapproach also relies on the ability of each drainage pathway to modulate the forcingsignal in a significantly distinct way. This modulation can be the result of thespecific structure, permeability and distribution of storage capacity of the drainagesystem, or individual disjoined parts of it. Importantly, field observations supportthe existence of such modulation [Fountain, 1994, Gordon et al., 1998, Harperet al., 1998, Fudge et al., 2008, Schoof et al., 2014, and others].Previous studies have inferred subglacial hydraulic connections from the sim-ilarity of pressure variations patterns [Hubbard et al., 1995, Gordon et al., 1998,Harper et al., 2002, Fudge et al., 2008, Huzurbazar and Humphrey, 2008]. How-ever, similar pressure variations can also arise in hydraulically isolated boreholesdue to mechanical interactions, such as the already mentioned load transfer andbridging stresses, as well as changes in sliding speed [Murray and Clarke, 1995,Gordon et al., 1998, Lappegard et al., 2006, Lefeuvre et al., 2015]. In turn, thosestress and motion changes could be themselves driven by changes in the effectivepressure within a connected drainage system, potentially hindering the distinctionbetween connected and disconnected boreholes.The relative scarcity of borehole observations make it difficult to assess howcommon these phenomena are, and in some cases, the physical processes involved.Similarly, the relatively small number of boreholes used in most studies make it dif-ficult to reconstruct the general spatial pattern and the evolution of the subglacialdrainage system. These limitations highlighted the need for borehole observationswith denser spatial sampling and longer temporal coverages, as well as a method-ology to scale up the analysis techniques that have been used in smaller datasets.161.7 Content overviewThis PhD thesis takes a holistic view of an eight-year dataset of borehole waterpressure records and surface conditions obtained from a small polythermal valleyglacier in the Yukon Territory, Canada. This dataset includes 311 boreholes withup to 150 recorded simultaneously. We will present a comprehensive picture ofthe evolution and characteristics of the subglacial drainage system, describing andincorporating all the main features observed in the borehole record and ancillarydatasets such as atmospheric temperature, snow cover, and the surface speed of theice.We have organised the remainder of this thesis in three chapters: In Chapter 2,we describe the field site, the instrumentation used and developed, and the dataset.Then, we present the results of an exhaustive visual exploration of the pressurerecords. As part of those results, we characterize the main types of behaviour ob-served in the boreholes and advance a discussion of the physical processes thatmight underlay each of them. These types are fast-flow channels, slow-flow dis-tributed drainage and disconnected areas. We will show the evidence supportingthe existence of such disconnected areas of the bed and how they can explain mul-tiple features observed in our borehole dataset. Also, we will provide a first ap-proximation to the structure of the drainage system, showing that we can groupboreholes into subsets that share similar pressure variations. Additionally, we willshow that these subsets can evolve in space and time due to the opening and clos-ing of new hydraulic connections, a process that can happen gradually or abruptlytrough “switching-events”. Based on these observations we put forward a com-prehensive picture of the seasonal evolution of the subglacial drainage system andassess the extent to which the established understanding of drainage physics iscompatible with our observations. Finally, we will identify the main shortcomingsof current models and present improvements that could conciliate the observationswith a new generation of models.In Chapter 3 we build upon the bases presented in Chapter 2 and present, cali-brate, and validate a set of techniques that have enabled us to systematically estab-lish the relationship between the different boreholes and identify possible hydraulicconnections. Based on the characteristics and distribution of those hydraulic con-17nections in time and space, we study the evolution of the underlying drainage sys-tem. In particular, we will quantify the extent of connected and disconnected areasof the bed, showing that the latter are a significant component of the drainagesystem. We will also show the impact that horizontal stress transfers have in thedistribution of effective pressure along the glacier bed and from the study of dif-fusive pressure signals, we will show that our observations suggest that the beddiffusivity has a fine spatial structure that we cannot resolve with our 15 m sam-ple spacing. We will finally contrast our findings with the general picture of thesubglacial drainage evolution that we presented in the previous chapter, noting thatsome results are consistent with that picture while others suggest that some im-provements are required.Finally, in Chapter 4, we present the Global Position System (GPS) record ofthe surface speed of the glacier and study its relationship to the conditions in thesubglacial drainage system. We then discuss which aspects of this relationshipare consistent with the current conceptualizations of basal sliding and which onespoint to possible limitations. In particular, we will show that the relationship be-tween effective pressure and surface speed changes significantly throughout theyear, and it is, in general, a poor predictor of speed. In contrast, we will show howthe fraction of the bed hydraulically connected to the surface is a better predictor ofsurface speed. We also analyze the primary sources of error in velocity solutions atsub-diurnal timescales, and devise a method for removing, or at least minimizing,the dominant sources of such errors. Using this method we study the basic prop-erties of the sub-diurnal speed variations at South Glacier and their glaciologicalimplications.18Chapter 2South Glacier subglacial drainagecharacterizationThis chapter will be devoted to the description of the field site, the instrumentationused, and the dataset. Then, we present the results of an exhaustive visual explo-ration of the pressure records. Using those results, we will characterise the maintypes of behaviour observed in the boreholes and advance a discussion of the phys-ical processes that might underlay each of them. Additionally, based on these ob-servations we put forward a comprehensive picture of the seasonal evolution of thesubglacial drainage system. Finally, we assess the extent to which the establishedunderstanding of drainage physics is compatible with our observations, identifyingthe main shortcomings of current models and suggesting improvements. The mostsignificant discrepancy between observations and current models appears to be thedevelopment of hydraulically isolated patches of the bed [Hodge, 1979, Engelhardtet al., 1978, Murray and Clarke, 1995, Hoffman et al., 2016]. Incorporating thissingle feature to our conceptual model of the subglacial drainage system wouldallow explaining multiple features that seem to be at odds with current conceptualmodels, but are commonly observed in borehole data. Namely: anti-correlatedtemporal pressure variations [Murray and Clarke, 1995, Gordon et al., 1998, An-drews et al., 2014, Lefeuvre et al., 2015, Ryser et al., 2014a], the development ofwidespread areas of high water pressure during winter [Fudge et al., 2005, Harperet al., 2005, Ryser et al., 2014a, Wright et al., 2016], the existence of large pressure19gradients over short distances [Murray and Clarke, 1995, Iken and Truffer, 1997,Fudge et al., 2008, Andrews et al., 2014], and high spatial heterogeneity.The content is laid out as follows: in section 2.1, we describe the field site andobservational methodology. An overview of our observations is given in section2.2, with a physical interpretation presented in section 2.3. We will present anin-depth study of the evolution of the subglacial drainage system structure and itsrelationships with measured surface speeds in the following chapters.To help the reader to navigate through the numerous observations presentedhere, we provide below an extended overview of its contents, highlighting the mostimportant points to be considered:• The observed drainage system consists of three main components (sections2.2.1)1. Channelized: efficient, turbulent drainage at low water pressure2. Distributed: slow water velocities, damped response to diurnal melt-water input, high water pressure3. Disconnected: near-overburden mean water pressure with no diurnalvariations• The “disconnected” areas display a small but statistically significant and sus-tained drop in mean pressure during the melt season, suggesting weak con-nections potentially through porewater diffusion in the till (sections 2.2.1 &2.3.2).• The connected drainage system (i.e. channelized and distributed compo-nents) consists of spatially distinct parts (subsystems) that appear to act in-dependently. Each is characterized by a common diurnal pressure variationpattern that differs markedly from other subsystems (sections 2.2.2 & 2.3).• Pressure variations in boreholes in disconnected areas can also occur due tobridging effects and potentially due to ice motion, the latter giving rise tolow-amplitude, high-frequency pressure variations shared by distant bore-holes (sections 2.2.2, 2.3.2 & 2.3.3).20• Observations suggest the existence of a dense network of englacial conduits,but it is unclear if these can transport water over extended distances horizon-tally (sections 2.2.3 & 2.3.2).• During a spring event, a large distributed drainage system quickly developsover a large fraction of the bed. This drainage system then splits into anincreasing number of subsystems over the summer season, each potentiallyfocusing around a channelised drainage axis. The extent of disconnectedareas of the bed grows as a result (sections 2.2.4 & 2.3).• The transition from connected to disconnected is abrupt, with the connectedparts of the bed having a high hydraulic diffusivity (sections 2.2.5 & 2.3.2).Disconnection and reconnection “events” typically occur as water pressure isfalling and rising, respectively. These observations motivate the modificationof existing drainage models presented in section 2.4.• The timing and degree of channelisation reached by the subglacial drainagesystem vary widely depending on weather and surface conditions duringsummer, and the spatial pattern of drainage can change from year to year(sections 2.2.6 & 2.3.1).• Abrupt growth of the distributed drainage system, analogous to that observedduring the spring event, can be observed during the summer in response toa sudden, abundant meltwater input following an extended hiatus, the latterusually caused by a mid-summer snowfall event (section 2.3).2.1 Field site and methodsAll observation presented were made on a small (4.28 km2), unnamed surge-typealpine glacier in the St. Elias Mountains, Yukon Territory, Canada, located at 60◦49’ N, 139◦ 8’ W (Fig. 2.1). We will refer to the site as “South Glacier” for con-sistency with prior work [Paoli and Flowers, 2009, Flowers et al., 2011, 2014,Schoof et al., 2014]. Surface elevation ranges from 1,960 to 2,930 m above sealevel (asl), with an average slope of 12.6◦. The ELA lies at about 2,550 m [Wheler,21Figure 2.1: WorldView-1 satellite image of South Glacier taken on Septem-ber 2nd, 2009. Borehole positions are marked according to the yearof drilling, showing the most recent year in repeatedly drilled loca-tions. Time-lapse camera positions (C1 & C2), automatic weather sta-tion (AWS), approximate equilibrium line altitude (ELA) are also indi-cated. The inset map shows the general location in the Yukon. The whitebox corresponds to the area shown in Fig. 2.2. Note that the differentsymbols indicating years of borehole drilling are used systematicallythrough the text.22Figure 2.2: Detailed map of the study area showing the location of selectedboreholes and instruments referred to in the text. The following symbolsindicate specific boreholes: those used for non spatially biased statistics(blue symbols), displaying behaviour similar to the fast-flow hole in Fig.2.6 (red symbols), re-drilled ones (light blue symbols), and those usedin Fig. 2.11 (orange symbols), the 2014 slow-flow borehole in Fig. 2.7(yellow triangle), the location of the Automatic Weather Station (yellowcircle), and the central GPS tower shown in Flowers et al. [2014] (yel-low square). The red outlines encompass all the boreholes displayed infigure 2.15, shown here using coloured and white markers. Black linesindicate major crevasses, blue lines indicate the surface streams. Con-tours show surface elevation, blue shading ice thickness. Grey shadingindicates the upstream area, calculated assuming a hydraulic gradientgiven by an effective pressure equal to half of the ice overburden pres-sure, and computed using the D∞method described by Tarboton [1997].232009]. Bedrock topography at the site was reconstructed from extensive ground-penetrating radar (GPR) surveys by Wilson et al. [2013], reporting an average andmaximum thickness of 76 m and 204 m respectively. Direct instrumentation andradar scattering [Wheler and Flowers, 2011, Wilson et al., 2013] reveal a polyther-mal structure with a basal layer of temperate ice. Exposed bedrock in the valleyconsists mainly of highly fractured Shield Pluton granodiorite [Dodds and Camp-bell, 1988, Crompton et al., 2015]. Borehole videos have also shown the presenceof granodiorite cobbles in the basal ice, and highly turbid water near the bottom offreshly drilled boreholes. Frozen-on sediments and a basal layer of till of unknowndepth are visible in some borehole imagery, and till thicknesses in excess of twometres are exposed near the snout.An AWS operated at 2,290 m next to the lower end of the study area betweenJuly 2006 and August 2015 [MacDougall and Flowers, 2011] as part of a simul-taneous energy balance study [Wheler and Flowers, 2011]. The average net massbalance over the whole glacier during the period 2008-2012 was estimated to bebetween -0.33 and -0.45 m/year water equivalent [Wheler et al., 2014], correspond-ing to 37-51 cm/year of average glacier thinning. Elevation changes in the studyarea derived from differential GPS measurements of borehole locations (taken afterdrilling) suggest a thinning of 59 cm/year over the same period, and 37 cm/year inthe period 2008-2015.We use air temperatures (specifically positive air temperatures, meaning themaximum of measured temperature and 0◦ C) and Positive Degree Days (positivedegree day (PDD), defined in the usual way as the integral with respect to time overpositive air temperatures) as the main proxy of the water input into the subglacialdrainage system. We estimated temperature values after the August 2015 removalof the on-glacier AWS by a calibrated linear regression of data from a second AWSoperated since 2006 by the Geological Survey of Canada and the University ofOttawa 8.8 km to the southwest, at an elevation of 1845 m.Surface velocities were measured with a GPS array [Flowers et al., 2014], anddisplay a strong seasonal contrast. The velocity at the GPS tower at the centre ofthe array (see Fig. 2.2) varied from 30.6 to 17.9 m/year between summer 2010 andearly spring 2011. Modelled basal motion in our study area accounts for 75–100%of the total surface motion (see Fig. 6b in Flowers et al. [2011], where our study24Figure 2.3: Photographs of the study area taken from camera C1 (see Fig.2.1) on July 19th, 2012–2015, as indicated in each panel. The interan-nual variability evident in the photo will be discussed in section 2.3.area is located between 1600 and 2500 metres).Between 2008 and 2015, 311 boreholes were drilled to the bed [Schoof et al.,2014] in the upper ablation area of the glacier between 2,270 and 2,430 m asl (Figs.2.1 and 2.3), covering an area of approximately 0.6 km2, with an average ice thick-ness of 63.4 m and a maximum of 100 m. During the first three years of the drillingcampaign, we deployed a sparse and evenly distributed grid of boreholes intend-ing to get a general idea of the subglacial drainage structure. In later years, wefocused drilling in the areas surrounding the main drainage axes suggested by pre-vious campaigns and by bed and surface topography. We organized the boreholesin dense lines (at 15–30 m spacing) perpendicular to the ice flow direction. Thesetransects aimed to intersect longitudinal drainage axes, which would be followeddownstream by other transects spaced 60–120 m along the direction of flow. Nomoulins are visible in or above this area. Instead, the surface meltwater is routedinto the glacier through abundant crevasses (Fig. 2.2). The basal layer of temperate25ice in the study area extends up to 30–60 m above the bed [Wilson et al., 2013].Boreholes were instrumented with pressure transducers providing continuoussubglacial water pressure records, with up to 150 boreholes being recorded simul-taneously. The inclination of the boreholes was not measured, but the drilling tech-nique used aims to ensure minimal deviations from vertical. A comparison withGPR data shows that borehole lengths were generally in agreement with ice thick-ness within a 6% margin [Wilson et al., 2013]. Contact with the bed was deemed tohave been established if water samples taken from the bottom of the holes showedsignificant turbidity. Otherwise, a borehole camera was used to assess bed contactvisually; a significant number of additional, unsuccessful drilling attempts termi-nated at englacial cobbles near the bed. With only a few exceptions, sensors wereinstalled only in holes that we were confident had reached the bed, and placed 10–20 cm from the bottom. Boreholes typically froze shut within one to two days,becoming isolated from the surface. The spatial distribution of new boreholes var-ied each year, not following a regular pattern. However, they were generally 15–60m apart along cross-glacier lines, with lines 60–120 m apart. A map of all bore-holes drilled is shown in Fig. 2.1. The region labelled as the “plateau” in Fig. 2.2was re-drilled every year between 2011 and 2015.Pressure data were acquired using Barksdale model 422-H2-06 and 422-H2-06-A and Honeywell model 19C200PG5K, MLH250PSB01A, and SPTMV1000PA5W02transducers. Each sensor was embedded in clear epoxy to provide mechanicalstrength and waterproofing (see Fig. 2.4 panel a ). Most transducers installed fromsummer 2013 onwards were equipped with a Ray 010B ¼” brass piston snubber asprotection against transient high-pressure spikes, without altering the signal at thesensor sampling frequencies as verified by doubly instrumented boreholes (see sup-plementary material section B). Data were recorded by Campbell Scientific CR10,CR10X and CR1000 data loggers, set to log at intervals of 2 minutes during sum-mer for CR10(X) loggers, switching to 20 minutes for the rest of the year, and atintervals of 1 minute for CR1000 loggers year-round. In the present thesis, we willreport water pressure values in metres of water (the height of the water column thatwould produce that pressure).During the summers of 2014 and 2015, a total of 10 custom-made digital sensorpods was installed in boreholes. These pods were built around an ATMega328P mi-26Figure 2.4: (a) Analog pressure sensors embedded in clear epoxy and readyfor deployment. The shown unit features a Honeywell pressure trans-ducer model 19C200PG5K. (b) Digital custom-made sensors pod em-bedded in clear epoxy and ready for deployment. The shown unitfeatures a Honeywell pressure transducer model MLH250PSB01A, anLSM303DLHCV accelerometer and magnetometer, an RS-485 commu-nication module based on an SP3485 transceiver IC, and a custom madegraphite conductivity probe. The remaining components are located onthe other side of the circuit board (visible in panel b). (b) Internal elec-tronics of one of the digital sensors pods. The main components shownare a 3V vibration motor, a DS18S20 temperature sensor, an ArduinoPro mini board and its ATMega328P microprocessor, four LED lightsources and a TSL2561 luminosity sensor. The remaining componentsare located on the other side of the circuit board and can be seen throughthe epoxy encapsulation in panel (b).27croprocessor and communicated via the RS-485 protocol with custom-made dataloggers constructed using the Arduino Mega open-hardware platform. The sensorpods recorded pressure, conductivity, turbidity, reflectivity in five spectral bands,tilt, orientation, movement, temperature, and confinement. The latter is a measureof the magnitude of the acceleration produced by an internal vibrating motor, usedto assess whether the sensor was hanging freely in water, or tightly confined withinsolid walls. Figure 2.4 panel (b) shows one of the digital sensor pods, embeddedin clear epoxy and ready for deployment. Panel (c) shows the bare electronic com-ponents of the pod, including the Arduino board hosting the microprocessor. Thecommunication, accelerometer and magnetometer modules are in the side of thecircuit board opposite to that shown in panel (c) and are visible through the epoxyencapsulation in panel (b). More details of the digital sensor pods design are pre-sented in Appendix A. Seven of these digital sensors were installed in the sameboreholes as the standard analogue transducers to assess data quality (see sectionB of the supplementary material).We have not used data from a stream gauge at the outlet of the glacier, main-tained for part of the observation period by the Simon Fraser University glaciologygroup, for two reasons: first, several surface melt streams and at least one majorlateral stream enter the glacier below the study site. Second, the instrumentation atthe stream site was destroyed on multiple occasions by flood waters, and a contin-uous record is not available.The limited available stream gauging data suggests typical summer flow around1-2 m3/s, with maximum values around 5 m3/s and minima below the measuringcapacity of the gauging station [Crompton et al., 2015]. However, the outlet streamwas never observed to run dry (Jeffrey Crompton, personal communication, 2018).2.2 Results2.2.1 Modes of water flow: fast, slow and unconnectedDespite a large diversity of borehole pressure records, a few general patterns areeasy to identify. The most common is the contrast between an inactive winterregime and an active summer period. During winter, most sensors show stable,28Figure 2.5: Pressure time series recorded in borehole D of Fig. 2.2 from 2011to 2015 (blue line). Daily positive degree day (PDD) values are shownas a red line, annual cumulative positive degree days as orange shad-ing, and fresh snow cover determined from time lapse imagery as lightblue shading. The fading blue at the end of the winter indicates theappearance of larger snow-free patches and the filling of a perennialsupraglacial pond in the study area, rather than the complete disappear-ance of the winter snowpack. Green bars indicate the count of boreholesdrilled each day on a scale from zero to 13.high (near overburden) water pressures, interrupted only during a 2-4 month periodof summer activity starting in June-July (Fig. 2.5). The onset of the active summerperiod (or “spring event”) occurs during rapid thinning of the snowpack underhigh summer temperatures. After the spring event, 20% of sensors show a drop indiurnal running mean pressure, and most start displaying diurnal oscillations.Pressure records alone do not allow us to determine the characteristics of waterflow at the bed, and visual observations at the bottom of boreholes often fail due tothe high turbidity of the water after drilling. However, in a few exceptional cases,we were able to observe water flow at the bed directly. We will describe the twomost clear-cut cases.On July 28th, 2013, while installing a sensor at the bottom of a borehole, strongperiodic pulls were felt through the sensor cable, revealing a conduit with turbulent,fast water flow in the bottom 50 cm of the borehole. This borehole was also the29only one in which there was an audible sound of flowing water. The location of thehole is marked as “Fast-Flow” in Fig. 2.2, it was drilled at the very end of the fieldoperations, and no further detailed on-site investigation was conducted.The fast-flow borehole was 93 m deep and drained at a depth of 87 m duringdrilling. On the first recorded diurnal pressure peak, the water reached a pressureof about 5.2 m (6% of ice overburden). A water sample retrieved from the bottomshowed moderate turbidity. Two pressure sensors were installed in this borehole,10 and 80 cm above the bed, the upper one with a snubber and the lower onewithout.Panel c of Fig. 2.6 shows the pressure recorded in the fast-flow borehole forthe first 33 days after installation, and panel 2.6d shows the pressure records inthree boreholes along the same line across the glacier at 15 m spacing. Note thelack of similarity between the fast-flow hole pressure record and those from othernearby boreholes. This lack of similarity contrasts with the typical behaviour ofboreholes exhibiting diurnal pressure oscillations. Such boreholes usually share asimilar pattern of pressure oscillations with one or more neighbouring boreholes,forming a cluster that extends some distance laterally across the glacier (see section2.2.2).However, in the case of the fast-flow borehole, somewhat similar temporalpressure patterns were observed down-glacier and at much larger distances thanthe 15 m lateral borehole spacing, as shown in panels 2.6e and 2.6g, and less so inpanel 2.6h. By contrast, a set of boreholes exhibiting very different variations closeto those in panel 2.6d is shown in panel 2.6f. For reference, panel 2.6i shows theremaining pressure time series recorded in the same area, highlighting the diversityof pressure patterns observed. No systematic time lags were found between peaksin the fast-flow borehole and pressure peaks of boreholes displayed in panels 2.6eand 2.6g.The grouping of boreholes into panels in Fig. 2.6 was done on the basis ofspatial proximity in panel c, and on the basis of a commonality of diurnal pressurevariations in the remaining panels. In particular, we have clustered the recordson the basis of commonality in how the amplitude of diurnal pressure variationschanges in time. For instance, the similarity between the records in panel 2.6gshould be obvious. However, note that there can be subtler similarities: panels30Figure 2.6: Locations and pressuretime series for the boreholes as-sociated with the fast-flow bore-hole during the summer of 2013.(a) The map uses the same schemeas Fig. 2.2, but omits the up-stream area shading. (b) Tem-peratures (grey) and fresh snowcover determined from time lapseimagery (light blue shading, fad-ing colour indicates partial cover).(c) Pressure in the fast-flow bore-hole (red) and its correlation withtemperature in grey, computed forany given time over a 3-day run-ning window. Note that two sen-sors were installed in the fast-flowborehole, offset vertically fromeach other by 70 cm, making thetwo lines indistinguishable mostof the time at the presented scale.Later in section 2.2.5, the completerecord will be displayed, wherethe two curves are more distin-guishable. (d-i) Pressure recordsfrom other boreholes marked onthe map. The colour of plots corre-sponds to borehole marker colourson the map; the same conventionis used in all subsequent figures.Symbol shapes represent drillingyears as in Fig. 2.2, uncoloured(grey) markers correspond to bore-holes with active sensors duringthe displayed period, the pressuretime series of which are not shownin the figure.312.6c, 2.6e and 2.6g at least partially share a period of larger diurnal amplitudesleading up to August 3rd, a hiatus lasting until August 10th punctuated by a diurnalpressure peak late on August 6th, and a period of renewed diurnal oscillationslasting until August 17th; this differs from the pattern of diurnal oscillations seenin panel 2.6h. Grouping boreholes in this way is partially a subjective measure, andwe will present a more systematic clustering method (which has helped to guide thegroupings here) in the Chapter 3 [Gordon et al., 1998, Huzurbazar and Humphrey,2008, see also]. All borehole groupings presented in the following figures weremanually selected using the same criteria as described for Fig. 2.6.Several features stand out in the pressure record from the fast-flow borehole:sharp diurnal pressure peaks and a small time lag between peak surface tempera-tures and diurnal water pressure maxima (1-3 hours), as well as the general simi-larity between the temporal variations in pressure and temperature. The correlationbetween the two, computed over a moving window, stayed above 0.8 for severaldays (Fig. 2.6c, grey shading). This high correlation was more pronounced late inthe season, also coinciding with the water pressure dropping to atmospheric valuesat night.A contrasting observation of water flow was made on July 23rd, 2014, whena clear water sample was retrieved from the bottom of a borehole (“Slow-flow”in Fig. 2.2) and the borehole camera was deployed. The resulting borehole video(see supplementary material) reveals a slowly flowing, thin layer of turbid water atthe borehole bottom overlain by clear water, an unusual condition that allowed theobservation, as the water in a bed-terminating hole is usually highly turbid due tothe basal sediments disturbed by the drill jet.The slow-flow borehole was 62 m deep, and the first recorded diurnal pressurepeak reached 48 m (85% of ice overburden). One pressure transducer with snubberwas installed 6 cm above the bed. Figure 2.7c shows the pressure recorded inthe slow-flow borehole (black line). Pressure records from three other boreholesin the same across-glacier line and one sensor downstream are shown in red inthe same panel, while the record of a fifth borehole in the same line is shown inpanel 2.7d. Note that there are four virtually indistinguishable records in panel 2.7cduring July 23rd-25th (see also figure 2.8). After a data gap caused by a corruptedcompact flash card, the records have become more dissimilar by August 2nd, but32Figure 2.7: Locations and pressure time series for the boreholes associatedwith the slow-flow borehole on July and August 2014, with the sameplotting scheme as Fig. 2.6 (see corresponding caption). Panel c showspressure in the slow-flow borehole (black) and three other boreholes inthe same line. The correlation with temperature has been calculatedusing the only borehole that remains connected over the whole interval.The remaining panels show pressure time series from other nearby holesas indicated by the line and borehole marker colours. The time seriesfrom boreholes S1–S4 are shown in more detail in Fig. 2.8.33Figure 2.8: Pressure records from the four sensors marked S1–S4 in 2.7b inJuly (a) and August (b) 2014. Colour coding is red (S1), yellow (S2),green (S3) and blue (S4). We have applied a constant value offset inpressure to each time series (meaning, added a constant to the directlymeasured pressure) to make the agreement between the records clearer.The offset values are, in order, 27, 26, 24, and 29 metres in panel (a), and27, 20, 22, and 27 in panel (b). Note that the S2–S4 time series in panel(a) agree so well with each other that they are barely distinguishable.continue to exhibit common pressure variations. The pressure time series from theborehole that is part of the line immediately below the slow-flow hole by contrasthas significantly higher mean water pressure and the diurnal pressure variationshave a much smaller amplitude. We have included it in panel 2.7c because it is theonly one in that lower line that matches one of the other pressure records in panel2.7c well, if we remove their means and scale them to have unit variance.Most boreholes showing diurnal pressure oscillations share the general featuresdisplayed by the slow-flow borehole, specifically 1) smooth pressure peaks andtroughs, 2) pressure patterns well differentiated from the atmospheric temperaturepattern, 3) mean pressures during periods with diurnal oscillations that lie between55-120% of the overburden ice pressure (much higher than in the fast-flow hole), 4)peak pressures that typically lag peak temperatures by 2-8 hours and 5) patterns oftemporal pressure variations that are often similar to neighbouring boreholes bothin the along- and across-glacier direction.On average, during summer, 71% of sensors showed the behaviour observed34in the slow-flow borehole at some point, as assessed visually from the presence ofsmooth diurnal pressure oscillations. Only eight boreholes (3% of the total, shownas red markers in Fig. 2.2) exhibited water pressures dropping to atmospheric pres-sure, one of the key characteristics of the fast-flow borehole. Six of them werefound during the three years with the highest cumulative positive degree day countin the dataset (2013: 437 ◦C days, 2009: 386 ◦C days, 2015: 297 ◦C days).However, these figures may not be representative as drilling was concentratedin some areas. For this reason, we have selected 70 boreholes in two across-glacierprofiles (blue markers in Fig. 2.2). Among those, 81% show a behaviour qualita-tively similar to the slow-flow borehole, and 4% that of the fast-flow one. However,note that even these statistics remain biased, as borehole spacing along these linesis concentrated in areas that were of interest due to likely drainage activity, andcrevassed areas are under-represented as sensor signal cables typically have a shortlife span there.We emphasize that the borehole in which fast flow was observed initially dis-played a relatively smooth diurnal cycle, and the statistics above are based onthe identification of diurnal pressure oscillations reaching atmospheric pressureat night: it is thus possible that more boreholes intersect conduits with fast-flowingwater, without the observed pressure records indicating as much.The remaining 26% (or 15% in the two cross-glacier lines in Fig. 2.2) of bore-holes do not show any significant diurnal pressure oscillations at any point duringthe year. These “disconnected” boreholes usually show year-round mean pressuresbetween 90-120% of ice overburden. Disconnected boreholes frequently show anear-constant pressure signal, but not always, with some exhibiting difficult-to-interpret temporal variability. In 2016 there were 55 disconnected boreholes, al-lowing us to treat their behaviour statistically. However, despite only slight dif-ferences in mean pressures between winter and summer, there is a slow but statis-tically significant decrease in water pressure during summer, starting around thespring event and amounting to about 6% of the overburden pressure in total (Fig.2.9).35Figure 2.9: Mean water pressure computed over a 1-day running window foreach of the 55 sensors that did not display diurnal oscillations during2016, shown in black. For legibility, we have subtracted the mean overall sensors and the time window shown. The blue line shows the meanover all the black lines at a given time (i.e. over all the sensors) andthe bootstrap confidence intervals [Efron and Tibshirani, 1993] of 90and 99% (dark and light pink shading, respectively). Gray shading rep-resents the period over which the initiation of diurnal oscillations wasobserved in connected boreholes, and the red vertical line is the mediantime at which diurnal oscillations first appeared in the 70 boreholes thatdid experience such oscillations during 2016.2.2.2 Spatial patterns in water pressure variationsWhen the whole dataset is viewed over a given time window during summer, it isoften possible to identify multiple clusters of boreholes, each exhibiting a specificpattern of temporal pressure variations. Often, these patterns are defined by the wayin which the amplitude of diurnal oscillations changes over time. While boreholesin a given clusters will share the pattern of temporal variability, this will differsignificantly from the pattern of temporal variability in the other clusters.One example of this phenomenon comes from the boreholes in Fig. 2.6f, wherewe can see a group of boreholes that display a very coherent signal but with adistinctive two-day period. However, those boreholes in figure 2.6f are directlyadjacent to those in 2.6e. The latter by contrast show a very different pattern of36diurnal pressure variations (that we have associated with the fast-flow borehole,along with panels 2.6c and 2.6g).Less clear-cut, though indicative of the same phenomenon is Fig. 2.7, where wesee boreholes in panels d-f that exhibit quite different diurnal pressure variationsfrom those observed in panel c (the group associated with the slow-flow borehole).Figure 3 of Schoof et al. [2014] also shows an example of the same phenomenonduring July and August 2011: borehole B in that figure is, in fact, one of a groupof 5 that exhibit almost identical diurnal water pressure oscillations that are quitedistinct from those in boreholes A1–A6 in the same figure.The spatial patterning of the drainage system into distinct clusters becomesmuch clearer when a dense borehole array with good spatial coverage is available.During the summer of 2015, there were 88 boreholes with active sensors on theplateau indicated in Fig. 2.2, and 66 further downstream. The corresponding pres-sure records are presented in Fig. 2.10. Between June 26th and August 27th, 42boreholes on the plateau (panel 2.10c) and 11 boreholes downstream (panel 2.10d)showed a highly coherent pressure signal that was qualitatively different from theatmospheric temperature signal (panel 2.10b) and the majority of other boreholesin the two areas (panel 2.10f). There was no consistent time lag between sensorsin the plateau and downstream. However, there was a clear drop in amplitude ofdiurnal oscillations (panels 2.10c and 2.10d), where the latter showed amplitudesaround 15-30% of those seen in the plateau.Five of the sensors on the plateau were capable of conductivity measurements(panel 2.10g). We emphasize that, in general, the spatial patterning was recog-nizable only in the pressure records, and pressure oscillations were not associatedwith conductivity changes. Although all five sensors showed very similar temporalvariations in pressure (panel 2.10c), the conductivity time series bear far less sim-ilarity to each other, with only a handful of abrupt conductivity changes commonto three of the sensors (S2 and S4 marked in panel 2.10g).The group of 14 boreholes in panel 2.10e also shares common diurnal pres-sure variation patterns, though this is not immediately clear as the mean pressuresand amplitude of pressure variations varies significantly. For that reason we havehighlighted one line in black that shows these variations clearly. Notably, thesevariations are “inverted” versions of the pressure variations seen in panel 2.10b,37Figure 2.10: Locations and pressuretime series for all 82 boreholeson the plateau area and 20boreholes down-glacier duringJune–August 2015, plotted usingthe same scheme as Fig. 2.6.(b) Temperatures (grey) andfresh snow cover (light blue),(c) pressure in 42 boreholes onthe plateau that shared similarwater pressure variations. Thehighlighted time series are fromboreholes D1 (black) and H1(yellow). Both boreholes areindicated on the map. S1–S7indicate ‘switching events’ in theD1 record (see section 2.2.4).(d) Boreholes downstream of theplateau showing similar pressurevariation to those in panel (b).(e) Pressure records that areanti-correlated to those on panels(c)–(d). (f) Pressure records fromthe remaining boreholes on themap, (g) conductivity recordsfrom the six digital sensorsincluded in panel c.with peaks becoming troughs and vice versa. These anti-correlated boreholes,in contrast to those in panels 2.10c and 2.10d, have smaller diurnal oscillationsamplitudes, and the oscillations are superimposed on a signal with near-constantrunning mean and high mean pressure, usually close to overburden. Therefore,if diurnal variations were filtered out, these pressure records would resemble the38winter regime. At 15 metres spacing between boreholes, we do not observe se-quences of boreholes smoothly transitioning from correlated to anti-correlated, inthe sense that there appears to be no continuous change in phase and amplitudefrom borehole to borehole: we observe a sharp boundary between correlated andanti-correlated boreholes, or correlated boreholes and boreholes exhibiting no di-urnal oscillations. Note that one of the records in panel f of Fig. 2.7 also anti-correlates strongly with the record in panel 2.7e during the later part of the timewindow shown, and the record in panel 2.7d anti-correlates strongly with the recordfrom the adjacent borehole S4 during August; anti-correlation of this kind is acommon feature of the dataset, often but not always involving boreholes in closeproximity.There is typically another set of boreholes that show very similar diurnal varia-tions in water pressure super-imposed on near-constant or slowly changing diurnalrunning mean values. The diurnal variations for this set have very small amplitude(typically 0.2–0.6 m, exceptionally up to 6 m), and resemble a square-wave withsuper-imposed high-frequency variations. Matching oscillations can be observed inmultiple boreholes spread over large distances, both along and across the glacier,and both diurnal and much higher frequency features in the pressure signal arepreserved between these boreholes. An example from 2011 involving boreholesacross the width of the study area, and recorded by different data loggers, is shownin Fig. 2.11. Clearly, the oscillations can be both correlated or anti-correlated witheach other. Not shown in Fig. 2.11 is the longer-term evolution of water pressurein the same boreholes. While they share short-time-scale temporal variability, theirlong-term pressure variations are generally not well correlated.2.2.3 Three-dimensional drainage structuresThe pressure observations primarily give us a two-dimensional picture of the drainagesystem. The drilling process itself as well as borehole camera investigation pro-vides additional information on englacial connections [Fountain et al., 2005, Harperet al., 2010]. 37% of all the boreholes drained completely or partially during thedrilling process, as did 39% of those in the cross-glacier lines marked as blue sym-bols in Fig. 2.2. For simplicity, we will give statistics for the entire dataset in run-39Figure 2.11: Relative pressure variations in 7 boreholes (orange symbols inFig. 2.2) displaying common small amplitude diurnal oscillations withhigh-frequency content during August 2011. To make these visible inthe same plot, we have applied offsets of 93, 71, 64, 61, 63, 66 and27 metres to the measured pressures. Three common high-frequencyfeatures are highlighted by grey vertical bands.ning text below, and the corresponding figure for the cross-glacier lines in paren-theses. Of the boreholes that drained during drilling, only 14% (0%) drained whenreaching the bed, and the remaining 86% (100%) drained at some point duringthe drilling process, suggesting connections to englacial conduits or voids. Suchconnections were also observed on multiple occasions using the borehole camera.Drainage events occurred at all depths during drilling, but with a slight preferencefor greater depths, with 60% (59%) happening in the lower half of the boreholes.This remains true for the 2012 drilling campaign, where the first sensors wereinstalled before the spring event and observations are likely to reflect winter condi-tions. Unfortunately, water level change and duration of drainage events were notrecorded.During the borehole re-freezing process, 29% (11%) of the boreholes showed apressure spike typically about 1.3 times overburden pressure, suggesting that freez-ing happened in a confined space. In total, 62% (73%) of these initially confinedboreholes showed diurnal oscillations during the first week, suggesting that somedegree of connection was developed with a drainage system experiencing diurnally40varying water input.In 2014 and 2015, three one-year-old boreholes were re-drilled, and the sensorswere recovered (boreholes A, B and C in Fig. 2.2). During this process, we foundthat holes A and C had sections about 8–12 m long near the bed that had remainedunfrozen for the entire year, suggesting that boreholes, as well as natural englacialconduits close to the bed, could remain open through the winter. In borehole A,contact with the bed had erroneously been assumed after the initial drilling basedon highly turbid water. However, borehole video footage taken after re-drillingshowed that the original borehole had terminated at an isolated rock. From thedepth of nearby boreholes, we estimate that the sensor was installed approximately4 m above the bed. Nonetheless, the diurnal water pressure oscillations recorded inborehole A continued to mimic other nearby bed-terminating boreholes that weredrilled in 2014 and 2015, indicating a persistent connection. Fig. 2.12i shows thepressure record in borehole A and the point at which the re-drilling took place.2.2.4 Seasonal development of the subglacial drainage systemWe have described the apparent spatial patterning of the drainage system above.However, this patterning is not fixed but evolves over time. In Fig. 2.10c, it isclear that all 42 boreholes show very coherent temporal pressure variations at thestart of the observation period. During late July and August, some the pressurevariations in some of the boreholes become more distinct until, by late August,there is no longer a common signal and all boreholes show dissimilar temporalpressure variations.In Fig. 2.10c, this emerging patterning is evident only in the more disorderedappearance of the plot for later times. In Fig. 2.12, we show the 42 boreholesof Fig. 2.10b separated into subgroups. Within these groupings, it is clear thatboreholes can switch from having closely correlated pressure records to behavingindependently and, less frequently, to being strongly correlated again (panels d–h inparticular); For simplicity, we refer to boreholes as being “connected” while theyexhibit the same temporal pattern of pressure variations, and as “disconnected”otherwise. For each grouping, we have computed a mean pressure displayed inblack, including only the boreholes that are connected at a given time; in some41Figure 2.12: Locations and pressuretime series for all 42 boreholesshown in Fig. 2.10b, groupedaccording to the similaritiesbetween their diurnal pressurevariations in July–August 2015,same plotting scheme as inFig. 2.6. (a) The map area isindicated as a box in figure 2.10a.Grey shading is the upstreamarea shown in Fig. 2.2 (b) Airtemperature (grey) and freshsnow cover (light blue). (d–k)Borehole pressure time series,the colour of plots correspondsto the colour of borehole markeron the map. Black lines are themean pressure in each panel,computed only over those bore-holes that are “connected” at agiven time (see main text); theblack lines frequently obscureone of the borehole time series.(c) The black “mean” curvesfrom panels (d–k), plotted inthe corresponding boreholemarker colour. The maximumcross-correlation coefficients,allowing for time lags of up to 6h, between all pressure recordsand air temperature computedover a moving three-day windowis shown in dark grey.42cases, no boreholes were connected to each other, and we still used the last boreholeto exhibit diurnal oscillations to define the set of connected boreholes. In thatcase, the black mean curve obscures the corresponding, coloured borehole pressuretime series. These mean curves for each panel are re-plotted in the correspondingborehole marker colour in panel 2.12c.The major dichotomy in Fig. 2.12 is between the groupings in panels d–g andi on one hand and panel k on the other. The distinctions between panels d–g inparticular are more subtle, and generally relate to the absence or subdued nature ofcertain diurnal peaks in them: for instance, panels d–f all show diurnal oscillationson August 1st and 2nd, while the larger group in g does not; there are other ex-amples close to the end of the summer season. For all groupings in panels d–g, itappears that the early season records resemble each other more closely than thoselate in the season, as was already evident in Fig. 2.10b, and that fewer boreholeshave disconnected early in the season.At a much smaller scale, a similar fragmentation of the drainage system isshown in Figs. 2.7b and 2.8, where we see four boreholes that are initially verywell connected during late July having become much less well connected in Au-gust, although with the diurnal pressure oscillations still showing some similari-ties between several of the boreholes. Interestingly, one borehole (S2, yellow) hasceased to exhibit oscillations by August, but is straddled by two that still do (S1 andS3, 15 m to either side), suggesting a relatively fine-scale structure to the drainagesystem locally.In addition to spatial patterning, Figs. 2.10 and 2.12 hint at an overall evolutiontowards lower mean water pressures and larger diurnal oscillations. The seasonalevolution of the drainage system may be evident not only in its spatial extent, butalso in the evolution of mean water pressure and its response to surface melt input.Perhaps the simplest measure of sensitivity to surface melt input is what we termthe relative amplitude of pressure to temperature oscillations: we compute standarddeviations of pressure time series from boreholes that exhibit diurnal oscillations atsome point of the season, and also standard deviations of positive air temperatures(the maximum of air temperature and 0 ◦C). We compute these standard deviations43over one-day running windows, and define the ratio of the two running standarddeviations to be the relative amplitude of pressure to temperature variations. Takingthe running standard deviation of air temperature as a marker of surface melt ratevariability (see section C in the supplementary material for a discussion about thisassumption), the relative amplitude defined in this way gives an indication of howsensitive the drainage system is to variations in water input.In Fig. 2.13, we see that the running standard deviation in pressure only vaguelytracks its temperature counterpart. However, the relative amplitude systematicallyincreases during much of the season (Fig. 2.13c), except during an interval ofcolder weather and surface snow around the beginning of August, while the meanwater pressure also decreases (Fig. 2.13d).2.2.5 Basal hydrology transitions and “switching events”Above, we have seen that boreholes can become disconnected from each other, go-ing from a state in which they undergo synchronous and virtually identical pressurevariations over time to a state in which borehole pressure appears to evolve inde-pendently. The reverse change also happens, though less frequently (except duringthe spring event). The change from connected to disconnected and its reverse cantake different forms. In a few cases, disconnection is gradual, with the boreholescontinuing to exhibit similar diurnal pressure oscillations that progressively be-come more dissimilar in amplitude, phase, and mean water pressure. The recordfrom H1 in Fig. 2.10c (yellow line in panel c) is one such example. However, inmost cases the transition is abrupt, and the same is true of boreholes connectingwith each other: a rapid change in water pressure can occur over the course of afew hours or less as a connection is established. We term such abrupt transitions“switching events”, following Kavanaugh and Clarke [2000].Figure 2.8 shows multiple examples in the boreholes labelled S1–S4 in Fig.2.7c, spaced 15 m apart. Perhaps unsurprisingly, the majority of switching eventsinvolving new connections seem to occur while water pressure is increasing or af-ter a recent increase, while disconnections tend to occur as water pressure is falling(see Fig. 2.12c,d,f for several obvious examples), though the two are rarely sym-metric, with disconnection usually occurring at a lower water pressure than the44Figure 2.13: Relative amplitudes of pressure and temperature diurnal oscilla-tions from May to September 2015. (a) Positive air temperature (grey),and its standard deviation over a 1-day running window (red). (b) Stan-dard deviation of pressure over a 1-day running window (thin blacklines) for each borehole in the plateau area, and the mean of thesestandard deviations (blue) with bootstrap confidence intervals of 90% (dark pink) and 99% (light pink). (c) Ratio between pressure andtemperature standard deviations shown in panels (a) and (b), computedonly where standard deviation of air temperature is non-zero. (d) Meanpressure computed over a 1-day running window. Light blue shadingrepresents fresh snow on the glacier surface.45original connection. The record from sensor D1 in Fig. 2.10c is one such exam-ple, where arrows labelled S1–S7 mark multiple switching events. The boreholeoriginally disconnected from the main group on July 11th, but reconnects on sev-eral occasions during periods of high water pressure in the active drainage system,disconnecting when water pressure subsequently drops. Note that for the first tworeconnections, S2 on July 21st and S4 on July 24th, the switching events are clearlyassociated with large drops in conductivity as seen in panel g, suggesting an inflowof meltwater that has spent less time in contact with the bed [Oldenborger et al.,2002].Towards the end of the melt season, most boreholes have become disconnectedfrom each other, and water pressure in them typically rises again towards over-burden, remaining nearly constant through the winter. However, in some cases,we observe quasi-periodic pressure variations in winter as previously reported inSchoof et al. [2014]. Figure 2.14 shows the winter pressure record for the twosensors installed in the fast-flow borehole, extending the summer record shownin Fig. 2.6c. As in other boreholes, we see water pressures rising at the end ofthe summer season. This is briefly interrupted during early September, when sur-face snow cover temporarily disappears, and a drop in water pressure occurs in theborehole, accompanied by the resumption of diurnal oscillations. This is followedonce again by the termination of diurnal oscillations and a sharp rise in water pres-sure towards overburden once the surface becomes snow-covered again. However,unlike in most other boreholes, that rise towards overburden is interrupted by os-cillations lasting from 2 to 12 days. During these oscillations, water pressure candrop rapidly to as little as a quarter of the overburden, followed by a slower rise inpressure back towards overburden, stabilization, and a renewed rapid drop.2.2.6 Inter-annual variabilityAs observed in Fig. 2.5, there is large inter-annual variability in positive air tem-peratures and hence, presumably, in surface melting, both in terms of onset andintensity. In addition, we expect that differences in the snow-pack can also affectwater delivery to the englacial system, because a thicker snow-pack can store orrefreeze surface meltwater, and leads to higher average surface albedo during the46Figure 2.14: Extended pressure time series from the 2013 fast-flow borehole(Fig. 2.6). (a) air temperature (grey) and fresh snow cover (light blue).(b) Pressure recorded by two sensors installed 10 cm and 80 cm abovethe bed (red and blue respectively).summer [Male and Gray, 1981, Fountain and Walder, 1998]. Figure 2.3 showsa view of the study area from an automated camera on July 19th in 2012–2015.These images illustrate very significant differences in surface snow cover at theheight of the summer melt season: the visible snow cover in each image is part ofthe remaining winter snowpack.Alongside the inter-annual variability in temperature and snow cover, there arealso significant season-to-season differences in the water pressure records. Differ-ences in drilling objectives from season to season make year-to-year comparisonsdifficult except in one part of the study area. Figure 2.15 shows a compilationof pressure records from a set of boreholes drilled in almost the same locationsevery year in the lower plateau area from 2012 to 2015, as indicated by two redpolygons in Fig. 2.2. There are four boreholes (surrounding the fast-flow hole inFig. 2.6) included here for 2013–2015 that were not drilled in 2012, and four thatwere drilled in 2012 but not in later years. Alongside air temperatures and snow47cover, we also indicate the total PDD count prior to June 15th, and at the end ofSeptember. Also shown are the median of the dates on which diurnal oscillationsappear and disappear over all boreholes with functioning sensors (red lines). Thelatter are clearly a crude measure of drainage system evolution as they are biasedby borehole locations and drilling dates. Despite these differences, the boreholepressure records clearly indicate some systematic differences, with a relative ab-sence of diurnal pressure oscillations in 2012 and 2013, though accompanied byvery low water pressures associated with the fast-flow borehole in 2013 (see alsoFig. 2.6), and a larger number of “connected” boreholes with large-amplitude diur-nal oscillations in 2015. We also note that we drilled new boreholes in 2014–2015in the location of the 2013 fast-flow hole without encountering more evidence ofturbulent water flow.2.3 DiscussionThe seasonal evolution of the drainage system we observe is broadly consistentwith existing ideas about drainage physics. A drainage system forms annually,triggered abruptly by the delivery of meltwater to the bed in a spring event [Ikenand Bindschadler, 1986]. The timing of the spring event varies significantly fromyear to year, taking place when most of the glacier surface is still snow covered,but always after the appearance of the first sizable snow-free patches (Fig. 2.3,see also Nienow et al. [1998b]). This suggests that the development of drainagepathways through the surface snow cover is a precursor to water delivery to thebed, with the timing most likely dictated by snow depth, temperatures, and earlyseason melt rates [see also Harper et al., 2005]. Additionally, the appearance ofbare ice will significantly lower albedo, and could lead to a significant increasein melt production. After the spring event, most boreholes show strongly corre-lated diurnal pressure variations, suggesting extensive hydraulic connections, andat least a slight drop in water pressure. However, when compared with late-seasondiurnal pressure fluctuations, these early season pressure oscillations have smalleramplitudes and lower correlation with the inferred surface melt rates, suggesting arelatively inefficient drainage system. We will refer to this initial state of the sub-glacial drainage system as stage 1. Note that the “stages” identified here are not48Figure 2.15: Overview of pressure variations on the lower portion of theplateau area from 2012 to 2015. Each panel includes air temperature(grey), coverage of fresh snow (light blue), and vertical red lines dis-playing the median date of initiation and termination of diurnal oscilla-tions on all active sensors each year. Cumulative positive degree daysare displayed for the beginning and end of the interval shown.49Figure 2.16: Confinement data (grey) and pressure (blue) for one of the digi-tal sensors in Fig. 2.10c from July 2015 to September 2016.the same as the “phases” discussed in Schoof et al. [2014], who focused only onthe later part of the melt season and the subsequent winter; for instance, phase 2 inSchoof et al. [2014] corresponds to the transition from stage 2 to 3 here.Later in the season, the drainage system becomes more focused, in what we willcall stage 2. During this stage, the mean water pressure in the system drops, andthe magnitude of diurnal pressure variations increases (see Fig. 2.13, also Harperet al. [2002]). Different parts of bed still exhibit diurnal oscillations but cease tobe mutually well-connected, as also observed by Fudge et al. [2008]. We willrefer to the parts of the bed that remain internally well-connected as hydraulicsubsystems (Fig. 2.6 panels f and g, Fig. 2.7 panels c and f, and Fig. 2.12d–h, j and k are examples of this behaviour, with panels d–g in the latter sharingmany features but appearing quite distinct from panel k). Subsystems progressivelyshrink, shutting down drainage over an increasing fraction of the bed. At mostboreholes, the drainage shut-down is marked first by the sudden disappearanceof diurnal cycles in a switching event, often followed by a sustained increase inpressure that takes approximately one to a few weeks to stabilize at a value closeto overburden.In high-melt years, the fragmentation of the drainage system can be extreme.Figure 2.6 shows only a handful of boreholes exhibiting diurnal oscillations to-wards the end of stage 2. Our data suggest these boreholes may align with down-glacier drainage axes. Had we sampled the glacier bed differently, we could havehad no boreholes showing diurnal oscillations during this period. This widespreaddrainage shut-down around highly focused drainage subsystems would explainwhy the end of diurnal oscillations in most boreholes precedes the decline in in-50ferred meltwater supply and pro-glacial river runoff as observed by Harper et al.[2002] and Fudge et al. [2005].The distinct response of different subsystems to the same surface conditionsmust be the result of peculiarities of each subsystem. The amplitude of the diurnalpressure cycle typically varies over periods of several days, but the temporal pat-tern of amplitude variations differs between subsystems, and generally, does notreproduce corresponding variations in diurnal melt amplitude (Fig. 2.12 here andFig. 3 of Schoof et al. [2014]).The systematic increase throughout stages 1 and 2 of the relative amplitude ofdiurnal pressure and inferred melt oscillations (Fig. 2.13), and the correlation withpositive temperatures (Figures 2.6 and 2.12) is consistent with an increase in thedrainage system efficiency.A widespread termination of diurnal oscillations in the remaining connectedholes is typically triggered by a marked drop in meltwater supply, usually coinci-dent with a snowfall event. We label this as the start of stage 3 in Figs. 2.6, 2.10, and2.12; in Fudge et al. [2008], this is referred to as the “fall event” (though their datamakes connections with snowfall less easy to establish). The termination of diurnaloscillations is often followed by a rise in borehole pressures towards overburden,marking the beginning of the winter pressure regime, where pressure variations areno longer closely correlated, suggesting an absence of hydraulic connections.However, the shrinking and fragmentation process during stage 2, and possi-bly the onset of stage 3, may, however, be partially reversed by brief episodes inwhich the reconnection of at least some boreholes is observed. These reconnectionepisodes are often associated with strong increases in meltwater supply, usually onhot days when temporary snow cover clears. During 2015, snow events during lateJuly and early August led to several episodes in which most boreholes shown inFig. 2.10 appeared to disconnect from each other, pressures in them not only ceas-ing to exhibit diurnal oscillations but also evolving independently. These episodesended with surface snow cover disappearing and melt supply resuming, leading towidespread and often abrupt reconnection at high basal water pressures.Similarly, we cannot exclude the possibility that highly focused drainage sub-systems remain open during the early parts of stage 3: the borehole array cannotsample all conduits directly, and we are only certain of having intersected a main51conduit in one instance. That conduit, the 2013 fast-flow borehole, remained closeto atmospheric pressure for 9 days at the start of stage 3 (Fig. 2.6). Subsequently,water pressure started to rise, but even then, the disappearance of snow cover andcontinued melting led to a pressure drop and renewed diurnal pressure oscillationscorrelated with surface temperatures from September 1st to 6th (Fig. 2.14).We have referred to snow cover on the glacier being a good indicator of a dropin water supply to the bed. Often this snow cover persists for a period of daysin positive temperatures. With the data we have, we cannot state unequivocallywhether the reduction in water supply is primarily due to the high albedo of snowsuppressing melt, or due to water retention in the snowpack.The spatial evolution of the drainage system is consistent with the drainagesystem becoming channelized during the melt season. By this, we mean the for-mation of individual Ro¨thlisberger-type (R) channels, incised into the base of theice by dissipation-driven melting [Ro¨thlisberger, 1972]. Formation of channelsshould cause the mean water pressure to drop, as the focusing of water dischargecauses larger channel wall melt rates that have to be offset by faster creep closure,driven by larger effective pressures [Nye, 1976, Spring and Hutter, 1982]. It canalso account for the increased sensitivity of the pressure response to the inferredmelt input, and the reduction of dye tracer transit time observed in other glaciers[Nienow, 1993].The clustering of boreholes into drainage subsystems indicates good hydraulicconnections between them. However, as channels cannot coexist stably in closeproximity [e.g. Schoof, 2010a, Hewitt, 2011], it is unlikely that all boreholes thatsample the same drainage subsystem are located in R-channels, or in an R-channelat all. A more obvious explanation is that in stage 2, each independent subsystemcontains a channel surrounded by a distributed drainage system consisting of linkedcavities or a similar conduit configuration [Kamb, 1987, Fowler, 1987, Hubbardet al., 1995, Schoof, 2010a, Hewitt, 2011]. Such a distributed system is consistentwith the observation of slow-moving water in the 2014 slow-flow borehole. Inaddition, the existence of narrow R-channels within those systems is also consistentwith the finding of the 2013 fast-flow borehole in stage 2.Pressure records alone are insufficient to determine if there is water flow andwhether a sensor is in a channel or a distributed system, even if the distributed sys-52tem is hydraulically well-connected. The pressure record shown in Fig. 2.6 is theone record of which we know that it almost certainly reflects pressure variationsin a channel. We know that highly turbulent flow occurred in the bottom 50 cm ofthe borehole, which we take to be the height of the channel, but its width is un-known. The first week of that time series resembles the smooth pressure variationsobserved in many other boreholes (albeit at fairly low water pressures), while itdevelops very distinct features later: water pressure drops to atmospheric at night,and there are unusually small time lags relative to and very high correlation withinferred surface melt rates.The 2013 fast-flow borehole does not connect hydraulically to other nearbyones that lie along an across-glacier line (Fig. 2.6d), but appears to connect toa narrow set of boreholes extending 500 m downslope (Fig. 2.6 e and g). Thepressure time series from those boreholes differ somewhat from that measured inthe channel, so there is probably a narrow distributed system close to the channel,the width of that system being less than the 15 m borehole spacing.These observations are consistent with a highly developed channel with higherwater discharge that has become hydraulically isolated from the neighbouring bed:the high effective pressures in the channel would favour the closure of cavitiesor other connections in the surrounding bed. This closure may also be enhanceddue to the effect of bridging stresses [Lappegard et al., 2006]. Bridging stressestransfer part of the weight of the ice overlying the channel to its surrounding bed,effectively increasing the ice overburden in those regions above its mean value[Weertman, 1972].The 2013 season was marked by high net inferred melt: the total PDD at the endof that season exceeded the PDD for 2014 and 2015 by 46% or more (Figs. 2.5 and2.15). The high inferred melt rates are consistent with channelization reaching anend-member state. The rapid flow of water in the borehole also made identificationeasier; it is unclear if a smaller channel would have been as easy to identify.Using the channel end-member feature of diurnal oscillations with pressuredropping to atmospheric at night, we have identified seven other boreholes wherethe drainage system is likely to have evolved into a well-developed channel (Fig.2.2, red symbols), in all cases during the second half of July or first days of Augustduring years with relatively high cumulative PDD, which ought to favour channel53formation. Their locations loosely match zones with high up-stream areas (Fig 2.2,dark shading), which correspond to portions of the bed likely to concentrate basalwater flow due to the expected hydraulic gradients.Late in the season, the shut-down of the now well-developed basal drainagesystem during a period of dwindling inferred melt supply is consistent with higheffective pressures causing the closure of subglacial connections, especially as dis-connection events often occur at low observed water pressures. Different bore-holes appear to become hydraulically isolated from each other during this process.We interpret the subsequent evolution of pressure records after disconnection asreflecting the response of an isolated water pocket in the borehole, presumablycontaining a fixed (or nearly fixed) volume of water exposed to the ambient stressfield. Initially, creep closure will reduce any volume still occupied by air in theborehole and pressure can rise gradually; once there is no air space left, changes inwater pressure must reflect the pressure required to maintain the borehole volumeconstant (assuming no further freezing) while the borehole may still deform un-der anisotropic stress conditions [Meierbachtol et al., 2016]. Intuitively, we wouldexpect the borehole to become flattened perpendicular to the direction of great-est compressive stress, requiring a larger borehole pressure to maintain a constantvolume, which could account for the slow rise observed in water pressure, andpossible for slightly above overburden values. Importantly, the pressure in an dis-connected borehole should depend on its shape and can, differ from borehole toborehole; abrupt creation of new storage volume for instance due to crevasse prop-agation could also lead to abrupt changes in pressure in disconnected boreholes.Therefore, we have to caution against interpreting the pressure in individual dis-connected boreholes as an indication of the conditions in the unconnected parts ofthe bed: instead the borehole pressure may be controlled predominantly by localstresses in the ice, and the orientation, volume and shape of the unfrozen portionof the borehole.During winter, a handful of boreholes exhibited large-scale quasi-periodic pres-sure oscillations as detailed in Schoof et al. [2014] and shown in Fig. 2.14. Wehave previously hypothesized that these multi-day winter oscillations indicate on-going drainage in a few locations, with the oscillations driven by the interactionbetween conduit growth and distributed water storage in smaller water pockets,54basal crevasses and moulins; such oscillations could be triggered when water sup-ply drops below a critical value in combination with a steady background watersupply [Schoof et al., 2014]. Winter oscillations are common in boreholes thatshowed end-member channel behaviour at the end of the summer, as is the casefor our 2013 fast-flow borehole shown in Fig. 2.14, and borehole D in Schoofet al. [2014]. However, similar winter oscillations can occur also in boreholes thatwere disconnected or belonged to a distributed drainage system during the previoussummer.2.3.1 Interannual variabilityThe timing of spring events and speed at which the evolution of the drainage systemoccurs appears to be linked systematically to the availability of meltwater. Cool,snowy summers are most obviously linked to a poorly developed drainage systemwith weak diurnal cycles (2012) and poor correlations between boreholes, as wellas the absence of a sharp spring event (Fig. 2.15).The spatial structure of the drainage system also varies from year to year. Theplateau area reliably has drainage activity, though upstream area pattern in Fig.2.2 does not directly agree with the observed drainage structure (Fig. 2.12), butis merely suggestive. Channel formation is influenced by pressure gradients con-trolled by surface and bed topography. However, changes in water supply geometryand the instability inherent in channel growth and competition between emergingchannels implies that channels need not form in the same location every year. Thisis consistent with our failure to find in 2014 and 2015 a channel at the 2013 fast-flow borehole location.2.3.2 Challenges to current subglacial drainage modelsBoreholes do not only disconnect from or reconnect to each other during the sum-mer, a significant number of boreholes never connect at all. Others disconnectfrom the drainage system as it becomes more focused and fragmented into subsys-tems during stage 2. Some boreholes even disconnect and reconnect multiple times(figures 2.8, 2.10 and 2.12).There is typically a very clear distinction between connected holes showing a55similar response to the diurnal input, and disconnected ones that do not. Within agiven drainage subsystem, there is typically no gradual phase shift or diminutionof oscillation amplitudes from borehole to borehole, as would be expected if thedrainage system were a diffusive system with a finite diffusivity [Hubbard et al.,1995]. Effectively, our data suggest that, if the distributed system is diffusive, itsdiffusivity is very high, or zero where the system has become disconnected.This observation contrasts with the interpretation of data in Hubbard et al.[1995], who identify a gradual phase shift and decay in amplitude of diurnal pres-sure oscillations away from an inferred subglacial channel location. However, inour view, the phase lag in their Fig. 5 can also be interpreted as showing diurnalswitching of their borehole 40 from being well-connected with their boreholes 29and 35 to being disconnected. The latter interpretation would be consistent withMurray and Clarke [1995], and with Fig. 2.8 here also showing an example ofswitching events with similar characteristics on South Glacier.Hubbard et al. [1995], suggest that the bed substrate at their study site is com-posed of glacial till of varying grain size distributions, acknowledging that “a net-work of small channels” on a hard bed could also account for their observations.However, in terms of hydrology, till and a distributed drainage system at the ice-bed interface share many characteristics: we expect both to give rise to a diffusivemodel for water pressure if water storage in the distributed system is an increasingfunction of water pressure. The primary difference is in how the permeability ofthat system evolves. In the ’hard-bed’ view, the permeability evolves over timein response to changes in effective pressure, whereas for a granular till, porosity,and thus permeability are simply functions of effective pressure and thus respondinstantly to changes in it [Flowers, 2015]. The main inconsistency of appealing todrainage through continuous till layer as the main pathway for water flow is thatwe would expect to see more standard diffusive behaviour, and certainly no sharpswitches between connected and disconnected portions of the bed. In addition, tillwith a sufficient coarse-grained fraction of cobbles and boulders would probablybe capable of supporting the formation of cavities in the lee of those larger grains.In short, if till is capable of creating cavities, or is interspersed with bedrock bumpsor somehow capable of supporting switching events by other means, then our in-terpretation would not be affected by assuming a hard or granular bed.56Pressure measurements at South Glacier thus suggest that the distributed partsof each drainage subsystem are hydraulically well-connected, with all connectedboreholes showing almost identical pressure variations. However, the limited elec-trical conductivity and turbidity measurements also indicate that relatively littlewater might actually flow in the distributed system [Oldenborger et al., 2002]. Un-like in the data in Hubbard et al. [1995], there are no diurnal variations in electricalconductivity. With a hydraulically well-connected system, this has to correspondto a low water storage capacity, where substantial variations in water pressure donot require similarly large changes in stored water. Alternatively, storage capac-ity could be relatively localized, so that water does not need to flow everywhere.Oldenborger et al. [2002] shows how water pressure variations with no correspond-ing change in conductivity can be observed over an impermeable bed. However,the proposed mechanism requires the boreholes to be disconnected, and would notoperate on hydraulically connected boreholes as in this case.While there are typically insignificant cross-glacier differences in diurnal pres-sure response within well-defined drainage subsystems, the same is not true inthe down-glacier direction, even where we believe a hydraulic connection can beidentified. The pressure time series along the inferred channel system in Fig. 2.6(panels c, e, and g) are merely suggestive of a hydraulic connection, but hardlyidentical. The amplitude of pressure variations decreases markedly downstreamfrom the fast-flow borehole, which would be consistent with a diffusive system,though it is unclear whether the change in amplitude occurs along the length of thechannel, or within a putative distributed system flanking the channel, as the holesfurther down-glacier most likely did not sample the channel directly. However,importantly, there is no systematic phase lag accompanying the decrease in ampli-tude, as would be predicted by a diffusion model [Hubbard et al., 1995]. However,it is conceivable that additional water input from surface sources along the flowpath can have a significant effect on the phase of the pressure signal.We have referred to boreholes that cease to exhibit diurnal pressure variationsas having disconnected. Connection and disconnection typically manifest them-selves very abruptly in time (Fig. 2.8, see also Fig. 5 of Murray and Clarke [1995]).This transition usually takes from few tens of minutes to a few hours. However, theinitiation of the transition, often identified as a clear change in the rate of change57of pressure with respect to time, can in many cases have the appearance of aninstantaneous phenomenon, even at our shortest sampling interval of one minute.Therefore, it is unclear if these time scales can be associated with the connection ordisconnection process, as they might only represent how fast the system respondsto a perhaps instantaneous switch between connected and disconnected states.Usually, disconnection occurs during a drop in water pressure in the subsys-tem, and reconnection during an increase (figures 2.8 and 2.10). This is consistentwith connection or disconnection resulting from viscous creep closing connectionsbetween individual cavities within the distributed system [Kamb, 1987], or presum-ably with elastic gap opening or closing if sufficiently rapid. Disconnection couldalso be the result of cavities shrinking while remaining connected, if the boreholesimply terminates on an ice-bed contact area between connected cavities and thosecontact areas are systematically larger than the ∼ 10 cm diameter of our bore-holes. This process has been observed previously by Meierbachtol et al. [2016].However, it seems unlikely that this effect, which should be random, would leadto a recognizable spatial structure of narrow drainage regions flanked by increas-ing large disconnected regions. Instead, we would expect a random distribution ofapparently connected and disconnected boreholes.The anti-correlated signals we observe in our data (Fig. 2.10e) have previouslybeen explained by a mechanical load transfer mechanism, where the ice arounda pressurized conduit redistributes normal load, reducing the normal stress overneighbouring areas of the bed. Unconnected water pockets in those areas wouldthus experience a drop in water pressure [Murray and Clarke, 1995, Gordon et al.,1998, Lefeuvre et al., 2015]. A three-dimensional Stokes flow model Lefeuvreet al. [2018] supports this interpretation, and suggest that the anti-correlation pat-tern depends on the bed slope, which can be one of the factors affecting the ob-served distribution of borehole displaying this behaviour. Boreholes exhibitinganti-correlated pressures must be effectively disconnected, so that a change in nor-mal stress mainly causes changes in the pressurization of the borehole rather thanwater exchange. The load transfer mechanism is consistent with our observations.An alternative explanation suggests that such signals are associated to en-hanced cavity opening due to basal sliding changes [Bartholomaus et al., 2011,Hoffman and Price, 2014, Iken and Truffer, 1997]. However, it is unlikely that a58variation in sliding would precisely mimic the local water pressure variations inthe adjacent drainage subsystem, as suggested by Fig. 2.10e: the force balance thatdetermines sliding velocities should be affected by changes in basal shear stressacross a larger portion of the bed.Note that we observe anti-correlated signals in boreholes that are not immedi-ately adjacent to boreholes showing a correlated signal (purple and blue markers inFig. 2.10). It would be difficult to explain the anti-correlated signal in these bore-holes by normal load transfer over larger distances, when other disconnected bore-holes nearby show no such behaviour. This suggests that the connected drainagesystem can contain fine structure (either as channels or narrow regions of dis-tributed drainage) with lateral extents smaller than the ∼ 15 m borehole spacing.The same is indicated by the formation of disconnected “islands” in lines of oth-erwise connected boreholes at the same spacing as seen in Fig. 2.8 for the Augustobservation period (see also Murray and Clarke [1995], for analogous observa-tions).We have referred to a borehole as disconnected when observations show thatpressure variations on a diurnal time scale are not communicated to a borehole.However, the evolution of the mean water pressure in disconnected boreholes isconsistent with a residual amount of water leakage into the connected drainagesystem: during the summer, that mean pressure gradually decreases. The end ofthe monotonic increase in water pressure of disconnected boreholes observed inFig. 2.9, coincides with the spring event, followed by a slow decrease. The largesample obtained in summer 2016 supports this trend up to a 99% confidence despitethe large variability of the observations.As in Hoffman et al. [2016], such a slow evolution could be accounted for byflow through a relatively impermeable till aquifer underlying a much more effectivebut less pervasive interfacial drainage system, and the magnitude of that leakagecould have a significant impact on basal sliding rates if disconnected areas act assticky spots.Widespread hydraulic isolation of the bed in winter is supported by high recordedwater pressures and the marked pressure drop at the spring event observed in 20%of boreholes. In contrast, theories based on a remnant “distributed” system wouldordinarily suggest relatively low water pressures in winter [Schoof, 2010a, Hewitt,592013]. Although it is possible that some boreholes do not connect because theywere not properly drilled to the bed, we believe that the existence of persistentlydisconnected areas is robust. Non-spatially biased samples suggest that up to 15%of the bed could remains unconnected year round. The existence of such uncon-nected holes, and the possibility of dynamic connection and disconnection, repre-sents a challenge to existing drainage models, which typically assume pervasiveconnections at the bed.In addition to conduits at the bed interface, englacial conduits are known toexist inside temperate glaciers [Fountain and Walder, 1998, Nienow et al., 1998b,Harper et al., 2010]. However, it is unclear whether they allow mostly verticalwater transport, or if horizontal water transport over significant distances is alsopossible through them. Frequent drainage events during drilling (also observed byIken and Bindschadler [1986]) suggest the existence of a large number of englacialconduits, and borehole re-drilling observations show that upward conduits can re-main open through the winter season in a layer extending several metres above thebed. However, we have no evidence of significant along-glacier drainage in win-ter, while we know that englacial connections can remain. This suggests that theenglacial connections remain isolated from each other during winter. It is unclearif they can connect in summer and establish an englacial drainage system capa-ble of supporting significant down-glacier drainage. The persistence of conduitsthrough winter is most likely related to the basal layer of temperate ice [Wilsonet al., 2013], and hydraulic isolation preventing creep closure. The apparent ubiq-uity of englacial conduits suggests a need to assess their role in downstream watertransport in future; if significant, this represents another area of improvement fordrainage models.2.3.3 Mechanically connected boreholesStrong correlations over long distances were observed in boreholes displaying allthe features of disconnected boreholes except a superimposed low-amplitude diur-nal pressure variations with high-frequency variations (Fig. 2.11). From their widespatial distribution, it appears impossible for them to be connected by hydraulicconduits. As such conduits would need an extremely high diffusivity to preserve60the observed high-frequency features over large distances (¿500 m as seen in Fig.2.2). Moreover, a high diffusivity is at odds with the diverging evolution of tempo-rally smoothed borehole water pressures in the same holes.These signals do not seem to be instrumental artifacts, and in many cases wererecorded by independent data loggers. We have also considered effects due to in-duction on non-twisted signal cable coils, temperature, or solar irradiation. How-ever, in those cases, such signals should also be superimposed on records fromdistributed drainage systems, contrary to our observations. Possible explanationsmust be related to periodic large-scale stress changes in the ice compressing dis-connected boreholes the volume of which must remain constant, thereby elicitingan instant water pressure response. The most likely cause of such large-scale stresschanges would appear to be the occurrence of periodic diurnal basal slip events assuggested by Andrews et al. [2014].2.3.4 Data interpretation caveatsWe generally assume that the sensors at the bottom of boreholes measure the waterpressure at the bed. However, this may not always be the case if the sensor becomesencased in ice, is connected to an englacial conduit, or if the borehole did not reachthe bed or has penetrated into the basal till.It is likely that with time, some sensors can become encased in ice, as sug-gested by the fact that older sensors are less likely to show diurnal oscillations (forthat reason, sensors in old disconnected boreholes were often decommissioned be-fore they ceased to produce a signal), and the observations in doubly instrumentedboreholes (see section B of the supplementary material). Digital confinement datasuggest that in some cases, as in Fig. 2.16, the termination and initiation of diur-nal oscillations is associated with an increase and decrease in confinement. Thisobservation would also be consistent with ice encapsulation of the sensor duringwinter.Although the upper end of the boreholes typically freezes shut within few days,the abundance of englacial conduits opens the possibility that the sensors couldconnect to an englacial conduit through the lower portion of the borehole while itis still open. In such a case, the pressure record could at least partially reflect the61evolution of englacial conduits instead of the basal drainage system.Alternatively, in the absence of englacial connections, a sensor in a boreholethat fell short of the bed would appear as disconnected, even if the underlying bed isnot. However, we believe this is not a common situation due to the strict proceduresfollowed to assess whether a borehole reached the bed or not (see section 2.1).Observations by Hart et al. [2015] using wireless pressure sensors installedacross the basal till layer in a glacier in Norway showed that, while a sensor at theice-till interface shows clear diurnal variations, another one placed a short distanceaway inside the till layer can show a signal very similar to our disconnected bore-holes. This could be a problem affecting some of our sensors, as borehole drillingcould eventually penetrate the till. Nevertheless, the lifespan of a sensor buried inthe till ought to be short if there is differential motion between ice and the sensorplacement in the till [e.g. Engelhardt and Kamb, 1998], causing the signal cable totear. Indeed, one sensor that was accidentally installed directly on the bed with lim-ited (1 m) cable slack, rather than suspended just above the bed, survived for onlyjust over a month, and showed uncharacteristic high-frequency noise superimposedon a smooth diurnal oscillation (see the lowest curve in Fig. 2.6g).Calibration drifts may affect in situ sensors over time, and differences in mea-sured water pressure may not be reflective of an actual pressure gradient betweentwo boreholes (supplementary material section B); consequently, we have takensimilarity in response to diurnal forcing as our indicator of connections, ratherthan looking directly at the evolution of pressure gradients.2.4 Insights for models developmentOur data show that the glacier bed not only contains regions that remain discon-nected from the subglacial drainage system during the melt season, but that thoseregions can evolve in time, and that disconnection from or reconnection to thedrainage system can be quite abrupt. By itself, that insight is not new. Previ-ous observational studies [Murray and Clarke, 1995, Gordon et al., 1998, Andrewset al., 2014, Meierbachtol et al., 2016] have pointed out the same set of phenom-ena. However, most models in their present form [Schoof et al., 2012, Hewitt et al.,2012, Hewitt, 2013, Werder et al., 2013, Bueler and van Pelt, 2015] do not capture62them: water can flow everywhere in the domain, although the permeability of thedistributed system varies with position and over time. The expected signature ofthe distributed system in borehole records is then a progressive decrease in ampli-tude of diurnal oscillations away from subglacial channels, with a correspondingphase lag [Hubbard et al., 1995, Werder et al., 2013]. This contrasts with the possi-bility of abrupt disconnection from the drainage system that appears to be the mainfeature of our field data, rather than a diffusive attenuation of pressure signals.The only exception is the model of Hoffman et al. [2016], which contains a‘weakly connected’ component that exchanges water with the active remainderof the drainage system through highly inefficient connections. Diurnal pressurevariations in that weakly connected system are primarily due to the effect of icemotion rather than through the exchange of water, as we have also inferred forthe groups of boreholes in our data that show common, mechanically transferredpressure variations (Fig. 2.11). However, the spatial extent of individual weaklyconnected parts of the bed is left unresolved in Hoffman et al. [2016], and waterexchange with the distributed system occurs locally, as is also the case in dual-porosity models [de Fleurian et al., 2014]. Instead of prescribing the physics bywhich the connection between distributed and weakly connected systems evolves,a simple linear increase in the exchange coefficient is assumed to occur during thesummer.Models of distributed drainage [Hewitt, 2011, Schoof et al., 2012, Werder et al.,2013, Bueler and van Pelt, 2015] typically describe a system of cavities, and modelthe mean cavity size at any given location. Crucially, these cavities are assumed toconnect whenever they have non-zero size. Using the results of the work presentedin this chapter, Rada and Schoof [2018] take a different approach and advance amodel capable to resolve connected and unconnected (or weakly connected) re-gions explicitly, and track their evolution. This is achieved by the implementationof a percolation limit that replaces the assumption that cavities always connect.This percolation limit allow cavities to form a connected system only once theyhave reached a critical size. Using a simple model that implements this approach,they were able to reproduce qualitatively some of the main features of our dataset:sharply-defined drainage subsystems with insignificant diffusive pressure signalattenuation and the existence of disconnected areas.632.5 ConclusionsWhile winter pressure record suggests that most boreholes remain disconnectedduring that period, a rapid springtime increase in melt overwhelms the water stor-age capacity of the snowpack, leading to the sudden supply of water to the bed andactivation of an extensive and well-connected distributed drainage system. Duringthis period, the majority of boreholes show similar diurnal pressure variations andexperience modest water transport (see section 2.2.1).Over time, water transport becomes concentrated in some areas, and probablybecomes channelized: water flow ends up focused in R-channels surrounded by adistributed drainage system that carries relatively low water fluxes. Borehole waterpressure data in most cases do not allow the direct identification of channels. Infact, in most cases, our borehole array probably fails to intersect the narrow R-channels. However, in one instance we were able to confirm the existence of achannel from direct observation in a borehole in which the lowermost 50 cm wereoccupied by turbulent water flow.The increase in effective pressure associated with channelization leads to theprogressive shut-down of drainage activity in the surrounding distributed drainagesystem, possibly due to basal cavities becoming isolated from each other as theyshrink under the effect of a larger effective pressure. During long and hot enoughsummers, most of the bed can become disconnected, concentrating drainage innarrow pathways.The eventual complete shut-down of the entire drainage system at the end ofthe summer season is presumably the result of low water supply: high effectivepressure and low dissipation rate in channels allow basal conduits to close. Thisappears to be strongly linked with the appearance of fresh snow cover, rather thanthe arrival of low temperatures alone (see section 2.3).Most of our observations are consistent with borehole data from other sites.However, the density of boreholes at South Glacier has allowed us to identify, inparticular, the prevalence of “switching events”, through which the drainage sys-tem focuses, and the disconnected areas enlarge. Such disconnected areas alwaysexist, even during the spring event. Disconnected parts of the bed are necessaryto account for many aspects of our data, including anti-correlation between bore-64hole pressure time series, above overburden water pressures, and the occurrenceof strongly correlated high-frequency pressure variations in sets of widely spacedboreholes (see section 2.3.3). As in Hoffman et al. [2016], our data suggest thatdisconnected areas need not be completely isolated, but can experience slow leak-age into the active drainage system (see section 2.3.2).In view of the above, perhaps the main shortcoming of most current drainagemodels is their inability to account for the evolution of an disconnected or weaklyconnected component [Hoffman et al., 2016]. Aware of this shortcoming, Rada andSchoof [2018] have shown that the ability to model the evolution of disconnectedareas can however be incorporated in the current modelling framework as a per-colation threshold, assuming that cavities only form a connected system once theyreach a critical size. This relatively minor modification of the existing frameworkprovides a starting point for the development of a new generation of models, ca-pable to reproduce borehole observations and ultimately provide a more completeand accurate description of the subglacial drainage system.However, the ability of the system to fully shut-down requires the incorpo-ration of other physical process that could allow the reactivation of the drainagesystem during the spring event, something that is probably accomplished by over-pressurization. Their model also requires a more careful treatment of normal stressredistribution, in particular in association with isolated and closely spaced cavitiesof very different water pressures.65Chapter 3Spatial structure and temporalevolution of the subglacialdrainage system3.1 IntroductionThe drainage system of a glacier consists of a network of conduits that exists un-der its surface. Here, we mean by a “conduit” any physical connection betweentwo spatially separated parts of the glacier along which water can flow. Someof these conduits are located within the ice and constitute the englacial drainagesystem, while the conduits located at the ice-bed interface constitute the subglacialdrainage system. The subglacial drainage system plays a significant role in regulat-ing the mechanical coupling between the ice and the underlying bed, influencingthe basal sliding rate of the glacier [Iken and Bindschadler, 1986, Gordon et al.,1998, Nienow et al., 1998b, Mair et al., 2001, Harper et al., 2005]. In particular,these conduits can change both the total contact area between the ice and the bed,and the effective pressure at the contact interface. Effective pressure is defined asthe difference between normal stress at the bed and water pressure, where normalstress is approximately equal to the weight of the ice column, also referred to asoverburden pressure. Effective pressure is usually regarded as the primary variable66controlling changes in basal sliding rates [Lliboutry, 1958, Hodge, 1979, Iken andBindschadler, 1986, Fowler, 1987, Schoof, 2005, Gagliardini et al., 2007].Current subglacial drainage models [Schoof, 2010a, Hewitt, 2011, Schoof et al.,2012, Hewitt et al., 2012, Hewitt, 2013, Werder et al., 2013, Bueler and van Pelt,2015, Downs et al., 2018, Sommers et al., 2018] consider a pervasive subglacialdrainage system that covers all of the ice-bed interface. Therefore, such a sys-tem can effectively transmit effective pressure variations across the entirety of theglacier bed. Drainage models of this type have succeeded in reproducing many ofthe observed variations of glacier velocities at a seasonal scale, and the seasonalup-glacier development of a channelized drainage system during the spring andsummer [Hewitt, 2013, Werder et al., 2013]. However, these models still fail toreproduce direct borehole observations [Flowers, 2015].In particular, borehole observations support the existence of hydraulically dis-connected areas at the bed. These areas are characterized by boreholes that showconstant or slowly varying water pressure, while other nearby areas display diurnalpressure variations in response to the surface meltwater supply [Hodge, 1979, En-gelhardt et al., 1978, Murray and Clarke, 1995, Gordon et al., 1998, Hoffman et al.,2016, Rada and Schoof, 2018]. As noted previously, other common observationsthat cannot be explained with a pervasive subglacial drainage system are:1. Large and sustained pressure gradients over short distances [Murray andClarke, 1995, Iken and Truffer, 1997, Fudge et al., 2008, Andrews et al.,2014].2. The development of widespread areas of high water pressure during winter[Fudge et al., 2005, Harper et al., 2005, Ryser et al., 2014a, Wright et al.,2016].3. Boreholes exhibiting persistent pressures that exceed the overburden pres-sure [Gordon et al., 1998, Kavanaugh and Clarke, 2000, Boulton et al., 2007].4. Boreholes exhibiting mutually anti-correlated diurnal pressure variations [Mur-ray and Clarke, 1995, Gordon et al., 1998, Andrews et al., 2014, Lefeuvreet al., 2015, Ryser et al., 2014a].67Figure 3.1: Effect of ice overburden and normal stress transfers on isolatedwater pockets. (a) In the absence of an active drainage system, the waterpressure in an isolated water pocket reaches equilibrium close to over-burden pressure. (b) A conduit with an internal water pressure aboveoverburden reduces the normal stress in the surrounding bed leading toa pressure below overburden on nearby isolated water pockets. (c-d)A conduit with an internal water pressure below overburden increasesthe normal stress in the surrounding bed leading to above overburdenpressures on isolated water pockets. Pressure variations in the channelwould produce anti-correlated variations in the water pocket.At South Glacier, more than 25% of the borehole observations fall under someof the above categories during summer and almost 100% during the winter, whenmost boreholes display pressures near or above overburden for several months[Rada and Schoof, 2018].In Chapter 2, we described how we could reconcile all these observations as aresult of hydraulic isolation. In particular, this can produce areas of the bed thatremain unaffected by variations in meltwater supply, allow the existence of largepressure gradients, and the persistence of near-overburden water pressures over lo-calized areas. All these effects are associated with the existence of isolated conduits68or “water pockets” existing within such disconnected regions of the bed. These iso-lated water pockets can develop equilibrium water pressures close to overburden(Fig. 3.1a).Isolated water pockets can also display significant pressure variations in re-sponse to changes in the stress field in the ice around them, providing an explana-tion for boreholes displaying above-overburden pressures. Mutually anti-correlatedpressure variations can also be understood as the response of isolated water pock-ets forced to keep a fixed water volume during changes in the normal stress inthe surrounding ice. Such changes in normal stress can be due to normal stresstransfers.These stress transfers can either reduce or increase the normal stress in some re-gions of the bed. Figure 3.1b illustrates how a decrease of the water pressure withinan isolated water pocket can take place when the water pressure of a nearby con-nected conduit is higher than the normal stress in its surroundings. Such pressureexcess would offer partial support of the overlying ice, thus reducing the normalstress around the water pocket and its internal water pressure. Murray and Clarke[1995] termed this process as “load transfer” [see also Weertman, 1972, Gordonet al., 1998, Lappegard et al., 2006, Lefeuvre et al., 2015]. Conversely, if the waterpressure within the connected conduit is lower than the normal stress in its sur-roundings, part of the unsupported weight of the ice above the connected conduitwill be transferred to the bed surrounding the water pocket, increasing the normalstress around it and thus the water pressure within it (see Fig. 3.1c-d). This processis referred to as “bridging stress” Lappegard et al. [2006].The existence of disconnected areas motivates the need for a better understand-ing of the spatial structure of the subglacial drainage system and its evolutionthrough time. In particular, how does the extent of connected and disconnectedareas evolve, and how does that evolution relate to the seasonal cycle of meltwatersupply?Generally, we cannot observe directly the geometry of the subglacial drainagesystem, and have to rely on inferences made from borehole observations. The mostcommon approach to this problem is to assess the efficiency of the connection be-tween pairs of boreholes based on their response to a common forcing signal, whichcan be natural or artificial. A connection is efficient if the two boreholes display69a similar response to the forcing, and inefficient otherwise. More specifically, aconnection is efficient when the hydraulic conductivity of the conduit system con-necting two boreholes is high and the water storage capacity of that system is low.During the spring and summer months, the subglacial drainage system is forcedby a quasi-diurnal cycle in surface meltwater supply. The distinct response of eachborehole to this forcing can be used to assess the efficiency of the connectionsbetween them. Boreholes showing a very similar pattern of pressure variationsare likely to be well connected, while boreholes showing a very different patternare poorly or nor at all connected. In Chapter 2, we described how we can usethis approach to arrange the South Glacier borehole dataset into distinct groups ofboreholes that show a similar response to meltwater forcing, while the responsesdisplayed by boreholes from different groups are very distinct.The traditional view of a homogeneous and fairly well connected subglacialdrainage system would suggest that we should see boreholes connecting to eachother with smoothly varying degrees of efficiency, and that efficiency should gen-erally decrease with distance between the boreholes. Instead, we find mostly abinary distinction: boreholes usually appear to be either well-connected or notconnected. Mutually well-connected boreholes then belong to a distinct drainage“subsystem”.It is important to note that this approach to the identification of subglacial con-nections relies on the ability of each drainage subsystem to modulate the forcingsignal in a distinct way. That modulation is the result of the specific geometricalstructure, permeability and storage capacity of each subsystem. However, differ-ences in water pressure variations between subsystems could also arise from differ-ences in the forcing, as melt water production can vary across the glacier surface.This variation can result from differences in albedo, slope or shadowing.Although we cannot distinguish between differences in observed pressure re-sponse that arise from internal properties or forcing changes, arguably, the two sub-systems would still represent areas that evolve with some degree of independence.Moreover, even if a highly spatially variable forcing could lead to the formationof apparent subsystems in the borehole record, this would still not explain the for-mation of disconnected areas with no diurnal pressure response, the behaviour ofmutually anti-correlated boreholes, or the absence of diffusive pressure waves in70the record.More problematic for the identification of subglacial hydraulic connectionsis the possibility that two distinct subsystems could display indistinguishable re-sponses to the same forcing. A method based on the similarity of diurnal pressureresponse can erroneously aggregate mutually-disconnected areas of the drainagesystem into a single subsystem. However, the extent of the differences observedin the responses of neighbouring subsystems to meltwater supply suggests that in-dependent subsystems generally modulate the forcing signal to a point where theybecome well differentiated from each other (see Figs. 2.6, 2.7, and 2.10). Thisobservation is also consistent with those reported from other glaciers [Fountain,1994, Gordon et al., 1998, Harper et al., 1998, Fudge et al., 2008].Subglacial hydraulic connections have also been studied using artificially in-duced signals. One approach is to use tracers such as salt or fluorescent dyes [Hub-bard and Nienow, 1997]. If the injected tracer is detected at a given location, theinjection site and that location must be hydraulically connected. However, hy-draulic connections that are not associated with significant water exchange cannotbe detected in this way, although they could be equally or more relevant to the con-trol of the overall effective pressure at the bed. An alternative approach is the useof slug tests [Stone, 1993, Waddington and Clarke, 1995, Stone and Clarke, 1996,Iken et al., 1996, Kulessa et al., 2005]. On glaciers, this method usually consists ofstudying the water level changes in an open borehole after an initial artificially in-duced level change. However, the logistical challenges associated with performingrepeated tracer injections or slug test year-round in hundreds of locations, have pre-vented them to be used in long-term studies of the subglacial drainage. In addition,slug tests may also actively alter the drainage system. In contrast, the relative sim-plicity and less invasive nature of continuous pressure measurements in boreholeshave made them common practice for the study of subglacial hydraulic connec-tions [Gordon et al., 1998, Harper et al., 2002, Fudge et al., 2008, Huzurbazar andHumphrey, 2008].In this chapter, we use a clustering method to define groups of “similarly-behaving” boreholes. The parameters used in the algorithm are chosen to optimallyreproduce sets of manually-picked borehole records that exhibit similar diurnal re-sponses to surface melt. While a process-based approach might be preferable, such71as the inversion of a forward model, doing so would require the forward model toaccount for the full phenomenology of the borehole records, and such a forwardmodel does not yet exist.To identify these similarly-behaving boreholes systematically, we will look forgroups of boreholes that display a similar pattern of diurnal pressure variations.We will refer to those groups as “clusters”. Then, for each cluster, we will tryto identify the physical process causing the similarity. As we have seen, we candistinguish two broad types of processes responsible for similarity: hydraulic con-nections, and mechanical interactions.When we have evidence that a group of boreholes shares a common patternof pressure variations as a consequence of mechanical interactions only, we willrefer to it as a “mechanical cluster”. Otherwise, we will refer to it as a “hydrauliccluster”. A disconnected borehole that displays the same pattern of pressure vari-ations as a hydraulic cluster but in inverted form (presumably due to a normalstress transfer) will also be included in the hydraulic cluster, although our clus-tering method will be able to distinguish connected and disconnected boreholeswithin the cluster. These disconnected boreholes together with their hydraulicallyconnected counterparts will define an area of influence that extends beyond thereach of the hydraulically connected part of the cluster.While the identification of hydraulic clusters provides useful information aboutthe structure of the subglacial drainage system, this is only a snapshot of the sub-glacial drainage system, and gives no insight into how it is evolving. As the sub-glacial drainage system changes continuously in response to the seasonal cycle, weneed a sequence of these snapshots to capture the evolution of the system through-out the year. For this reason, we repeat the identification of clusters over successivetime windows over the study period, and we also determine how clusters in onetime window have evolved into the clusters in the next window . We will discuss indetail how the length of this time window is chosen, as it has implications for theaccuracy of the clustering and the temporal resolution of the resulting sequence ofsnapshots.Hydraulic connections have previously been detected automatically throughthe identification of similar pressure records. This process has been done by group-ing similar boreholes into clusters using two different clustering techniques. The72first was K-means clustering [MacQueen, 1967] used by Fudge et al. [2008] onborehole pressure records from Bench Glacier, Alaska. Given a set of N boreholesand a distance function f that quantifies the similarity between pressure records.K-means clustering aims to partition the N boreholes into K (K ≤ N) sets to min-imize the within-cluster sum of distances. Although this is a simple and effectiveclustering technique, it is hard to automate due to the requirement that the numberof clusters within the dataset needs to be known a priori. This shortcoming waspointed out by Huzurbazar and Humphrey [2008], who instead used hierarchicalclustering. In contrast to K-means, hierarchical clustering groups together all thesensors that conform to a given degree of similarity. This method automatically de-termines the number of clusters by grouping first the most similar borehole recordsinto small clusters, and then iteratively adding similar borehole records or mergingsimilar clusters until there are no more borehole records with the required degreeof similarity.The dataset available at South Glacier is significantly larger than those used inthe studies mentioned above. To find the most suitable technique to identify bore-hole clusters in a large dataset, we have tested four different clustering methods:K-means [MacQueen, 1967], Hierarchical clustering [Rokach and Maimon, 2005],self-organizing maps (SOM) [Vesanto et al., 2000], and empirical orthogonal func-tions (EOF) [Jolliffe, 2002].We tested the capacity of each method to reproduce automatically a set of clus-ters picked by hand (see Section 3.2.1). This showed that hierarchical clusteringwas the best of the four clustering techniques for our application. We then devel-oped an automated algorithm to identify drainage subsystems based on hierarchicalclustering. We will describe this algorithm in Section 3.2.In Chapter 2, we found that disconnected regions of the bed exist, effectivelypartitioning the subglacial drainage system into an active area that carries the sur-face meltwater discharge, and disconnected areas. A consequence of such discon-nected regions is the reduction of the area of influence of the active subglacialdrainage system. Consequently, the extent of the disconnected regions of thebed could play an important role in controlling basal sliding and its sensitivityto changes in meltwater supply. Later, in Chapter 4, we will address how the extentof these disconnected areas relates to observed changes in basal sliding rates.73In this chapter, we study how the structure of the subglacial drainage evolves,how the extent of connected and disconnected areas changes, and how the effectivepressure varies within these two different regions of the bed. With this information,we will present a comprehensive picture of the seasonal evolution of the drainagesystem. This picture is broadly consistent with the standard description of an ex-tensive early-season distributed drainage system, that progressively evolves into achannelized system during the summer. However, it also provides further evidencefor the existence of extensive disconnected regions of the bed, and suggests thealmost complete shut-down of the subglacial drainage over winter. It suggests aswell that horizontal normal stress transfers play a more important role than pre-viously considered for the control of the effective pressure distribution at the bed,and it will present a novel window into the fine structure of subglacial diffusivitydistribution.The remainder of this chapter is divided into three sections: In section 3.2, wewill describe the methodology we have used to identify drainage subsystems andto create multiple snapshots of their spatial structure through the seasons. Then,in section 3.3, we will focus on the results obtained from the application of thismethodology on the extensive dataset available at South Glacier. In particular,we will study the evolution of the identified hydraulic connections, their areas ofinfluence, and their properties. Finally, in section 3.4 we will discuss the insightsand shortcomings of our results, and summarize what they can teach us about thesubglacial drainage system and its seasonal evolution on South Glacier.3.2 MethodsTo infer subglacial hydraulic connections, we will look for pressure time seriesthat display similar diurnal variations. However, these variations are, in general,time-limited. For this reason, we will look for this similarities over discrete timewindows. The exercise of identifying by eye which time series display similar pres-sure variations over a given time window becomes onerous as the number of timeseries and the differences between them increase. Figure 3.2, shows 60 time se-ries over a 6-day window. Among those time series, it is possible to identify somesimilarities. For example, the four green lines show very similar diurnal pressure74Figure 3.2: (a) Raw pressure records of 60 time series over a 6-day windowstarting on July 14th, 2014. Time series belonging to four manuallyidentified clusters are shown with thick lines. (b) to (e) Diurnal residualsof the sensors belonging to each of the four identified clusters (samecolour coding of panel a).variations, with similar amplitudes but a distinct offset. In contrast, the red lines donot appear to be similar to each other, and some of them seem to be flat lines. Inthis case, the similarity is difficult to identify because the pressure offset betweenthe boreholes is much larger than the amplitude of diurnal pressure variations. Forthat reason, the identification of similarities can be substantially facilitated by thesubtraction of the mean value from each time series. However, if the time series75consist of diurnal variations superimposed on a long-term trend, subtracting themean value might not be sufficient, because the pressure range covered by the trendcan also be large enough to render the diurnal variations imperceptible. Therefore,we subtract from each time series its running mean over a 1-day window. Mathe-matically, given a time series P with samples Pi at regular time intervals, we removethe running mean over a 1-day interval, defining a “diurnal residual” Ri throughRi =1σwindow(Pi− 1di+d/2∑j=i−d/2Pi)(3.1)where d is the number of samples contained in one day, and σwindow is the stan-dard deviation of the time series P within the window over which the similaritycomparison will be performed. The normalization by the factor σ−1window facilitatesthe identification of similar time series regardless of the amplitude of their diurnalvariations. This normalization is essential to reveal the similarities between hy-draulically connected boreholes and those affected by normal stress transfers con-trolled by the former. It also allow us to identify the similarities between boreholesaffected by other mechanical interactions. However, this normalization discardsamplitude information that could be relevant in distinguishing between differentdrainage subsystems if they display a similar pattern of diurnal pressure variations.We will term this diurnal residual transformation as “pre-processing”, referringto the fact that it is applied to the raw data before attempting to identify similarities.Panels b to e of Fig. 3.2 show the diurnal residuals of the four groups of similartime series that we found among the 60 shown in panel a. In panel b we can seehow the diurnal residual makes clear the similarity between the red lines and thesame happens for the other groups.The similarities between time series change in time as the structure of the sub-glacial drainage system evolve through the opening and closing of conduits. Tocapture this evolution, we first break the data set into discrete time windows overwhich we will search for similar time series. Even with the aid of the diurnalresidual pre-processing, the manual identification of similarities among hundredsof time series is time-consuming, difficult and prone to omissions, making it un-suitable for analyzing several hundreds of time windows. Therefore, to automate76Figure 3.3: Example of a dendrogram computed by agglomerative hierarchi-cal clustering over the 60 time series presented in panel a of Fig. 3.2.The coloured lines correspond to the four clusters shown in panels b–eof Fig. 3.2. The thick dotted line corresponds to a split point (SP) thatwould output the same four identified clusters.this process, we do the following. In each window, we find all the available timeseries, which are then interpolated to regular time stamps with 15 minute spacing.Data gaps up to 30 minutes were linearly interpolated. Longer data gaps resultedin the exclusion of the time series from the corresponding time window. Then, wecompute the diurnal residuals of each time series and then apply the agglomerativehierarchical clustering method [Rokach and Maimon, 2005]. This iterative clus-tering technique starts from a set of one-element clusters (single pressure residualtime series), and then, in each iteration, merges the pair of time series that displaythe higher degree of similarity into a larger cluster. This process effectively orga-nizes all the original time series into a tree-like structure termed a “dendrogram”.To identify the two most similar time series in each iteration, we need to quantifywhat we mean by similarity between two time series. This is done using a metricthat defines a generalized “distance” between two time series. The smaller the dis-tance the more similar the two time series are. We will define the metric we useshortly, but first we will illustrate the clustering process graphically.77Figure 3.3 shows a dendrogram computed for the 60 time series presented inFig. 3.2a. Lines in the dendrogram are termed “branches”, and the joints betweenbranches are “nodes”. The vertical position of a node represents the distance be-tween its lower branches. Therefore, similar clusters join lower in the dendrogramthan dissimilar ones. Once we have established the maximum distance that typ-ically exists between time series that belong to a single hydraulic subsystem, wecan then define a corresponding split point (SP) in the dendrogram. Once the SPis defined, we select the clusters forming below it as candidates for a hydraulic ora mechanical cluster. As an example, the coloured branches in Fig. 3.3 representclusters that would be selected using the SP defined by the dotted black line. Thoseclusters correspond to the time series shown on panels b–e of Fig. 3.2. It is impor-tant to note that we have also tested a scheme in which we define a separate SP forhydraulic and mechanical clusters. However, as the SP values found for each typeare very similar, we have preferred the use of a single SP for both types of clusters.The size of the time window over which the clusters are identified has a con-siderable impact in the resulting clustering. For our purposes, a useful windowsize must be longer than the main one-day period of the pressure variations, butshorter than the time required for significant changes in the subglacial drainage totake place. This criterion loosely constrains the window size from a few days to afew weeks, where the upper limit is fairly speculative. However, the observation ofchanges in the pressure records such as those of Fig. 2.12, suggest that the drainagesystem can undergo significant changes within two weeks, making that time scalea reasonable upper limit. Within that range, longer windows can better discrimi-nate between different subsystems, and shorter ones can resolve more stages in theevolution of the subglacial drainage. We use a time window of six days, that aimsto strike a balance between sensitivity and temporal resolution: it is long enoughto capture multiple diurnal cycles and the length of a typical weather system in thearea, but at the same time it is short enough to provide a detailed sequence of theevolution of the subglacial drainage.Our aim with the clustering is first to find all the boreholes showing similardiurnal residuals, and later discriminate which physical process is responsible fortheir similarity. In particular, we want to establish whether the similar time se-ries are consistent with the existence of a hydraulical connection or a mechani-78cal interaction. In the case of mechanical interactions such as normal pressuretransfer [Murray and Clarke, 1995, Gordon et al., 1998, Lappegard et al., 2006,Lefeuvre et al., 2015], basal slip events [Andrews et al., 2014], or bridging stresses[Weertman, 1972, Lappegard et al., 2006], the similarity between pressure recordsis limited to the relative pattern of pressure variations, while they can differ widelyin their absolute value, amplitude and long-term trend. It is important to notethat differences in absolute values and long-term trend are removed by the diur-nal residual pre-processing, allowing them to be clustered together. Mechanicalinteractions can also invert the direction of the variations, with peaks becomingtroughs and vice versa. For example, the pressure time series in Fig. 2.10c and2.10e show very different long-term trends, while they share a common pattern ofdiurnal variations. Notably, variations in 2.10e also represent inverted versions ofthose in 2.10c. This is consistent with the horizontal transfer of normal stress fromhydraulically connected conduits (sampled in 2.10c) to disconnected water pockets(sampled in 2.10e). Another example of time series with similar diurnal residualsand large variations in their mean value is the set of pressure records shown in Fig.2.11. These records display very similar pressure variations, yet some of them arein opposite directions, and their mean values differ up to 66 m. In Chapter 2, wespeculated that such records are the result of a mechanical interaction.Motivated by processes that can invert the pattern of pressure variations, wechoose a distance metric insensitive to that form of inversion. Therefore, we usean “absolute Euclidean distance”: given two time series A and B, with samples aiand bi respectively and with i = 1, ...N, we define the absolute Euclidean distancebetween A and B asD(A,B) = min(√1NN∑i=1(ai−bi)2,√1NN∑i=1(ai +bi)2))(3.2)This corresponds to the minimum of the Euclidean distance between A and B,and between A and −B. Therefore, the absolute Euclidean distance will assignsmall distances to pairs of similar time series, even if one of them is an invertedversion of the other. When operating over standardized time series (i.e. normalizedby the standard deviation) as in our case, the Euclidean distance is mathematically79equivalent to the correlation coefficient. Previous work in subglacial hydrology hasused Euclidean distance for clustering, either directly on the pressure time series[Fudge et al., 2008] or its first derivative [Huzurbazar and Humphrey, 2008].The absolute Euclidean distance as described above applies only to individ-ual time series. However, hierarchical clustering requires the calculation of thedistance between clusters of time series. The method used for such calculationsis known as the “linkage”. We use the average-link linkage [Rokach and Maimon,2005], where the distance between two clusters corresponds to the average distancebetween the time series in one cluster and the ones in the other.3.2.1 Clustering calibration, validation, and testingTo calibrate, validate and test the clustering approach, we manually identified clus-ters in 12-day time windows from August 2010 to August 2015. We choose herea window length longer than the 6-day interval used in our final analysis to reducethe number of windows to the point where we are able to process them manuallywhile still covering the whole dataset available after the 2015 field campaign. Over461 windows we identified a total of 613 clusters. In this process, we used all thedata available to assess if a group of sensors was likely to constitute a hydraulic ormechanical cluster. Therefore, if the similarity between the diurnal residuals wasnot clear but suggestive of a cluster, we consulted other data sources to inform thedecision. Those sources included the raw pressure records and the positions of thecorresponding boreholes. For example, if a group of boreholes seemed to form ahydraulic cluster but the similarities in their diurnal residuals were only marginal,we would also study the spatial distribution of the boreholes. If we found thatthe boreholes were very far apart across the width of the glacier, we would thenexclude the cluster, as such a configuration is unlikely for boreholes interactingthrough hydraulical connections. Analogously, ambiguous clusters were excludedor included based also on the similarities in their raw pressure records.It is important to note that similarity to the eye, as when performing manualcluster identification, is not the same as similarity in the strict sense defined by theAbsolute Euclidean distance. Both approaches are consistent, but to the eye thesimilarity seems higher when the shapes of the time series shows more numerous80and distinct features. As an example, imagine two pairs of time series A1, A2 andB1, B2, such that the distance between A1 and A2 is the same as between B1 andB2. A1 and A2 are mostly flat and show only one broad peak over the entire timewindow. In contrast, B1 and B2, show a very distinct pattern of multiple diurnalpeaks each one with a particular shape. In this example, B1 and B2 will appear tothe eye to have a higher degree of similarity than A1 and A2.After the manual clustering of this portion of the dataset, we then divided the613 identified clusters into three subsets of approximately 204 clusters each. Thesewere used for the calibration, validation, and testing of the clustering method.The manual identification of clusters on what constitutes almost the entirety of theSouth Glacier dataset was laborious, but necessary to obtain the three independentand statistically significant sets of clusters necessary for the calibration, validation,and testing. Then, after we automated the process, we applied it to the entire dataset, over shorter time windows, and with a larger overlap between windows.The calibration consists of finding the maximum distance between time seriesthat we believe to be associated with the same drainage subsystem or mechanicalcluster. Using the 205 clusters of the calibration subset, we found an optimal splitpoint (OSP) of 2.26 day−1.To find the OSP, we compared the performance of all possible SP, based onthe information gain achieved by each one. Information gain is a standard quan-tity used in decision tree analysis [Mitchell, 1997]. When elements belonging totwo classes are grouped according to their class, the information gain measuresthe quality of the grouping. The relative information gain (RIG) is the informationgain relative to the maximum possible gain, which is achieved when each elementis correctly assigned to the group it belongs. We can represent the RIG as a percent-age: a RIG of 100% means the information gain was maximum, and the clusteringstrategy reached its maximum possible performance (i.e. all elements were as-signed to the correct group). A more detailed explanation of these concepts andtheir mathematical implementation can be found in Appendix D.Using the calibration set, the OSP found (2.26 day−1) reached a RIG of 89%.The method was then applied to the 203 clusters of the validation set, achievinga RIG of 89% as well. The validation dataset serves the purpose of checking thatwe have not over-fitted our calibration data. However, as the RIG achieved by the81validation dataset is similar to the one by the calibration dataset, no over-fitting wasindicated, thus no modification of the clustering parameters was needed. On theremaining 205 clusters, which correspond to the testing set, the RIG achieved wasof 84%. For a more detailed description of the calibration process see AppendixD.Alternatively, we could use other clustering techniques. In addition to hierar-chical clustering, we tested EOFs [Jolliffe, 2002], SOMs [Vesanto et al., 2000], andK-means clustering [David and Vassilvitskii, 2007], concluding that hierarchicalclustering is the most suitable for this application (see Appendix D). To define thebest strategy to use within the hierarchical clustering framework besides the diurnalresidual, we also evaluated the effectiveness of clustering using the raw pressuretime series, and other pre-processing such as diurnal running mean, diurnal runningstandard deviation and the power spectrum. More details about the exact definitionof each alternative pre-processing method and their performance can be found inAppendix D. As an alternative distance metric, we also evaluated the performanceof dynamic time wrapping (DTW) [Mullin, 1983]. DTW is a technique often usedin voice recognition systems, allowing the user to measure the similarities betweenwaveforms ignoring changes in speech speed between the reference waveform andthe one being classified. In our context, DTW can help us to recognize similarrecords that have a small time shift (due to datalogger clock offsets), or a varyingtime lag due the propagation of the pressure signal in a diffusive medium. How-ever, DTW can also erroneously match two signals with different periods if theyhappen to have a similar shape over the analyzed time window.We also evaluated two alternative linkage methods in addition to average-link:complete-link, and single-link [Rokach and Maimon, 2005]. These methods definethe distance between two clusters respectively as the longest and shortest distancebetween members of each cluster. Finally, in addition to the distance SP criterion,we also evaluated the use of gap size and inconsistency [Zahn, 1971]. Gap sizeSP identifies clusters in the dendrogram by finding the ones with the largest dis-tance gap between its top node and the following node. In contrast, inconsistencyidentify clusters based on the “relative distance”. This relative distance is obtainedby normalizing the distance by the average distance of all other nodes at the samehierarchical level.82Figure 3.4 summarizes the performance achieved by each combination of thesehierarchical clustering alternatives. We see that diurnal residual pre-processingis by far the best at reproducing hand-picked clusters. Among the distance met-rics, absolute Euclidean performs better than DTW at a fraction of the computa-tional expense. For the two leading linkage methods, the performance differenceswere small. However, average-link performs significantly better than complete-linkwhen used in combination with the absolute Euclidean distance metric. Finally, thedistance SP criterion performed best at identifying clusters in a dendrogram. Moredetails about each evaluated option can be found in Appendix D.3.2.2 Cluster evolution in timeTo study the evolution of the drainage subsystems, we apply the calibrated hier-archical clustering method to the whole dataset over a moving window of 6 days,with neighbouring windows overlapping by 3 days. After independently cluster-ing successive time windows, we apply an algorithm to identify whether a clusteridentified in one window is newly formed or corresponds to a pre-existent clusteralready identified in previous windows. Without such a “tracking” algorithm, it be-comes challenging to follow the evolution of a particular area or set of boreholes.Also, continuity between successive windows is required to study the evolution ofparameters such as the mean diurnal pressure, amplitude of pressure oscillations orthe spatial extent of a given cluster.Determining whether a cluster is new or constitutes the continuation of an ex-isting one is somewhat ambiguous: if a cluster splits into two clusters of equal size,it is unclear which branch to follow when we want to describe the evolution of theproperties of the fragmented cluster.We have adopted an iterative approach: in the first iteration we consider thata given cluster continues in the following window as the cluster that shares themost sensors with it, and we resolve arbitrarily the ambiguities that arise when twosuccessor clusters share the same number of sensors with the original one. Thisfirst iteration step successfully links clusters but tends to create many short-livedclusters, instead of an equally consistent but more continuous sequence. For thisreason, in the subsequent iterations we again choose from all the possible suc-83Figure 3.4: Mean relative information gain (RIG) achieved over the 203 clusters of the validation dataset by eachevaluated hierarchical clustering strategy. Coloured triangles point to the reference RIG obtained on the calibrationdataset. Dashed black lines show the maximum RIG that a perfect dendrogram split method could reach on eachcase. The actual mean RIG achieved by each split method is shown for depth OSP (green), inconsistency OSP(Blue) and gap OSP (red).84cessors using the same criterion, but this time we look further into the followingwindows (that are now preliminarily linked), considering not only how many bore-holes a cluster shares with a potential successor, but also with the successor of thesuccessor and so on, through a total of four windows. When counting the numberof shared boreholes, we give different weights to each consecutive window: fromthe closest to the furthest, these weights are 0.5, 0.375, 0.25 and 0.125. After a fewiterations, the cluster structure converges to a more continuous sequence.3.2.3 Hydraulic and mechanical cluster typesClusters with similar pressure records may arise from different physical processes.In particular, they can be the result of hydraulic connections between boreholesor due to a common response of isolated boreholes to stress changes in the ice.For the automatic identification of these “mechanical clusters”, we use the TimeSeries Shapelets method [Ye and Keogh, 2009]. This method allows us to takeadvantage of the characteristic shape of the diurnal cycle observed in mechanicalclusters, especially during the melt season. The Time Series Shapelet method takesa dataset with time series belonging to multiple classes, in this case, Mechanical(M) and Hydraulic (H), and searches through all the sub-sections of a prescribedlength L within all time series. Each sub-section is termed a “shapelet”, and themethod tests the capacity of each shapelet to determine whether a given time seriesbelongs to the class M or H. This is based on the minimum absolute Euclideandistance found between the shapelet and all the sub-sections of length L within thegiven time series.Using a calibration dataset that contains 49 mechanical and 156 hydraulic man-ually identified clusters, we applied the Time Series Shapelets method with L = 1day to find the best shapelet to discriminate between the two classes. The bestshapelet found is shown in Fig. 3.5. We use this shapelet to classify time series au-tomatically as mechanical if their minimum absolute Euclidean distance (see Eq.3.2) to the shapelet of Fig. 3.5 is smaller than 12.9. This value corresponds to theOSP found for the discrimination between mechanical and hydraulic clusters withinthe calibration dataset. More details regarding the derivation of this threshold canbe found in Appendix D. Note that a shapelet is always a section of a single time85Figure 3.5: Best shapelet found for classification of mechanical connections(black line). This shaplet reached a 81% information gain (OSP = 12.9).For comparison, 23 time series of mechanical diurnal oscillations arealso shown (blue and red lines).series. Therefore, the shapelet shown in Fig. 3.5 corresponds to a 1-day long pieceof the pressure record observed at one of our boreholes.We label as hydraulic any cluster not classified as mechanical. Within most hy-draulic and mechanical clusters, we can identify two subclusters, where the peaksof one correspond to troughs of the other and vice versa. In the diurnal residu-als, the two subclusters show up clearly as inverted versions of each other. Figure3.6 shows a clear example of a cluster involving 50 sensors, with one subclustershown in red and the other in blue. Note that these two subclusters would have be-come independent clusters if the initial hierarchical clustering had been done usingordinary instead of absolute Euclidean distances.We separate the two subclusters by computing the matrix of correlation coef-ficients between all members of the clusters. Then, all positive values are set to 1and negative values to -1, effectively turning each row of the matrix into a sequenceof values that, for one particular borehole, indicate which boreholes are correlatedor anti-correlated with it. Then, these sequences are separated into two subclustersusing K-means clustering [David and Vassilvitskii, 2007], allowing us to achievethe separation shown in Fig. 3.6 without manual intervention.86Figure 3.6: Diurnal residuals during 8 days for 50 sensors belonging to a hy-draulic cluster. The two subclusters are presented in red an blue.Figure 3.7 shows in panel b the standard deviation of the diurnal residual andthe mean pressure of all the member time series of a hydraulic cluster that wastracked over 102 days. Panel a similarly shows a mechanical cluster that wastracked over 135 days. These clusters correspond respectively to the largest hy-draulic and mechanical cluster observed during the 2015 melt season. To facilitatefuture references to these clusters, we will refer to them as “H1” and “M1” re-spectively. The values for each borehole in Fig. 3.7 were computed using all thepressure records at that borehole during the periods of time where it was identi-fied as a member of the cluster, and the standard deviation of the diurnal residualis provided as a proxy of the amplitude of diurnal variations. As in Fig. 3.6, onesubcluster is shown in blue, and the other in red. We can see that there is a clearsegmentation between the two subclusters in panel b, one having large amplitudesand high mean effective pressures (in blue), and the second having small ampli-tudes and low mean effective pressure (in red).We interpret this as follows: large amplitudes and lower water pressures (highereffective pressure) are more likely to be associated with an active drainage systemthat drains surface meltwater, while low-amplitude pressure variations around over-burden are likely to be the result of horizontal normal stress transfers [Murray andClarke, 1995, Gordon et al., 1998, Lappegard et al., 2006, Lefeuvre et al., 2015].We will label the subclusters as correlated and anti-correlated, alluding to fact thatboreholes in the correlated subcluster display maximum water pressures late in theafternoon, when the peak in meltwater supply is expected. We have extended thislabelling also to mechanical clusters, where correlation or anti-correlation is de-87Figure 3.7: Scatter plots of mean effective pressure and pressure standard de-viation for (a) mechanical cluster M1, (b) hydraulic cluster H1, and (c)window f of hydraulic cluster H1. Each point represents a boreholewithin the cluster. Boreholes that display diurnal variations in-phasewith each other (i.e. belong to the same subcluster) are shown in thesame colour. Boreholes are plotted as circles if located north of thecentral GPS tower (see Fig. 2.2) or as triangles otherwise.termined based on which subcluster peaks at the time period when we expect themaximum meltwater supply.While the plots in panels a and b of Fig. 3.7 are useful to identify automat-ically the connected and unconnected subclusters, they do not offer an accuraterepresentation of the real variation in mean effective pressure and amplitude. Thismisrepresentation is due to the differences in data availability and the length oftime for which each sensor was part of the cluster. For example, if one sensor ispart of the cluster only in a period where all sensors show small amplitudes, it willshow up with an anomalously small amplitude. A representative example of thetypical distribution of mean effective pressure and pressure standard deviation inthe cluster H1 can be observed in Fig. 3.7c, where we display data only for onewindow of 6 days, from July 15th to July 21st, 2015. This corresponds to windowf in Figs. 3.12 and 3.13.Clusters were automatically identified in each window, tracked between win-88dows, classified as mechanical or hydraulic, and divided into correlated and anti-correlated members. Subsequently, we performed a manual check of the automatedoutput to correct apparent artifacts in the clustering process and handle exceptionslike boreholes switching from correlated to anti-correlated (see Fig. 3.10), or clus-ters that switch from hydraulic to mechanical (see Fig. 3.11) or vice versa.3.2.4 Spatial patterns in basal hydraulic connectivityOne of the questions we want to answer is whether the hydraulic properties ofthe ice-bed interface at South Glacier are homogeneous or if some areas are morelikely to develop hydraulic connections than others. To address this question, wecan study the spatial distribution of all the inferred hydraulic connections in ourclustering output. However, comparing the changes in connectivity between dif-ferent areas requires us to account for the spatial and temporal sampling biases inour dataset.The spatial sampling bias arises from the fact that short-distance connectionsare more likely than long-distance ones. Therefore, a borehole will be more likelyto make connections if it has many boreholes nearby than if it is relatively iso-lated. Similarly, the temporal sampling bias arises from the uneven data avail-ability. Therefore, a borehole with a long pressure records will capture better thetypical probability of connections than another with data limited to a few weeks,especially if the limited data covers a period of exceptionally high or low overallconnectivity.To overcome these sampling biases, we will first assume that the bed is homo-geneous and, under that assumption, we will estimate the probability of a hydraulicconnection between two arbitrary points of the bed based on their relative position(i.e. distance and direction between them). Later, we will be able to test howwell this probability can explain our observations, and assess the validity of thehomogeneity assumption.Note that we consider a hydraulic connection to have been identified betweentwo boreholes when both boreholes are correlated members of the same hydrauliccluster over a given time window.To estimate the probability of a connection across the bed, we consider how89Figure 3.8: (a1) Relative positions of all 718,341 possible pairs of boreholesin selected time windows. (a2) Gridded density map of the relative po-sitions in panel a1. (a3) Density map derived from the best bivariateGaussian distribution fit to the relative positions in panel a1. (b1) Rela-tive positions of all 9,514 pairs of boreholes identified as hydraulicallyconnected in selected time windows. (b2) Gridded density map of therelative positions on panel b1. (b3) Density map derived from the bestbivariate Gaussian distribution fit to the relative positions in panel a1.many hydraulic connections we have identified at a given relative position, andthen estimate the probability of those connections based on how many times wehave sampled for connections at such a relative position. In this calculation we useonly the part of the year where we observe activity within the drainage system. Weachieve this by only using time windows for which we have identified at least onehydraulic cluster. Therefore, we ignore the extended winter period where we at-tribute the lack of connections to the absence of meltwater supply. We also assumethat the connection probability can be represented as bivariate Gaussian probabilitydensity function (PDF).90We estimate this probability asP(r,θ) =Dconn(r,θ)Dboreholes(r,θ)(3.3)where r is distance and θ is the azimuth. Dconn is a bivariate Gaussian PDF fitto the relative positions of all 9,514 pairs of boreholes for which we identified ahydraulic connection in the selected time windows. Dconn represents how likely aborehole in our dataset was to establish a connection with other borehole at distancer and azimuth θ . Figure 3.8b1 show all these relative positions, Fig. 3.8b2 shows adensity map of the same positions gridded into 100 m by 100 m grid cells, and Fig.3.8b3 the density map expected from the bivariate Gaussian PDF fit for the samenumber of observations. Finally, Dboreholes is a bivariate Gaussian PDF fit to therelative positions of all the 718,341 connections that would have been possible inall selected time windows. Dboreholes represents how likely a borehole in our datasetwas to find another borehole at distance r and azimuth θ . Figure 3.8a1 shows allthe relative positions, Fig. 3.8a2 shows a density map of the same positions, andFig. 3.8a3 the density map expected for the same number of observations using thebivariate Gaussian PDF fit.Figure 3.9 shows the probability density function for hydraulic connections Pas defined in Eq. 3.3. This function will be used to estimate the number of con-nections we would have expected at a given borehole. That number of expectedconnections, corresponds to the sum of expected connections on each window inwhich that borehole contained a functioning pressure sensor. In turn, the number ofexpected connections for a given window corresponds to the sum of the probabilityof connection with each one of the other boreholes recorded during that window.For example, consider a borehole that had a functioning pressure sensor in two timewindows. In the first time window there were two other functioning boreholes andthe probability of connection with them was 0.3 and 0.2. In the second time win-dow, there were three other functioning boreholes with probabilities of connectionof 0.1, 0.6 and 0.2. In this case, the expected number of connections would be thesum of all these probabilities, this is: 1.4 connections.Differences between the expected and observed number of hydraulic connec-tions at each borehole will be used to characterize different regions of the bed and91Figure 3.9: Probability density function for hydraulic connections P as de-fined in eq. 3.3.assess the validity of the assumption of homogeneity implicit in our definition ofthe connection probability P.3.2.5 Pressure variation trendsThe study of the average pressure variation in multiple connected boreholes will bea useful tool to understand the evolution of the water pressure within the subglacialdrainage system. However, due to the many discontinuities in our pressure records,and the wide range of mean values observed, the study of a simple average of thepressure records would not be very informative. For example, if the data from aborehole with relatively high water pressure becomes unavailable at some point intime, the average pressure at that point would suffer a sudden drop. This pressuredrop would be unrelated to any physical pressure change within the subglacialdrainage system, and it would obscure the actual trend we are interested in.Therefore, we will calculate the mean pressure of a series of boreholes by aver-aging the instantaneous pressure differences between consecutive samples of eachborehole. Those averaged differences are then integrated in time to reconstruct arelative averaged pressure time series for the whole interval. This relative averaged92Figure 3.10: Diurnal residual data between July 9th and 17th 2015 for a hy-draulic cluster. Over this period the cluster consisted of 41 correlatedboreholes (blue), and 16 anti-correlated boreholes (red). Two addi-tional boreholes (green and yellow lines), transition from correlated toanti-correlated around July 12-13th.Figure 3.11: (a) Water pressure in a hydraulic cluster observed in August2015 that we then identified as a mechanical in mid-September 2015.(b) Diurnal residual for the period between September 14th and 18th2015, using the same colour coding of panel a. It can be seen howthe sensor in black transitioned from displaying a large amplitude hy-draulic signal to small amplitude one, characteristic of mechanicalclusters.pressure starts at zero but represents accurately the pressure variations within theboreholes. As a final step, we add constant value to the relative averaged pressureso that the mean value of it matches the mean of all the original pressure samples.To put this in mathematical terms, consider that each borehole is represented by apressure time series with samples at times ti, such that Pm,i is the pressure recordedin borehole m at time ti. At each time ti, the number of valid samples is Mi, such thatboreholes can be represented by the index m = 1...Mi. Then, the relative averagedrelative pressure Ri of all time series at time ti is93Ri =i∑k=21MiMi∑m=1Pm,k−Pm,k−1 (3.4)then, the final averaged pressure time series P′i is given byP′i = Ri−Ri +Pm,i (3.5)where Pm,i is the mean of all samples in all time series and Ri is the mean valueof the relative average time series defined by Ri. All pressure time series in thischapter that show the average pressure variation of more than one borehole, werecomputed with the averaging defined by Eq. Results3.3.1 Evolution of the subglacial drainage systemData from the 2015 melt season represents our best record of the onset and evo-lution of subglacial drainage of South Glacier. While we performed the clusteringon the whole dataset, we will concentrate here on the 2015 melt season, where wehad up to 157 working sensors, with about half of them (74) installed in previousyears. The latter group produced a detailed record of the spring event and drainagedevelopment during the early season. Also, the 2015 melt season was long andwarm enough to allow the formation of a well developed subglacial drainage sys-tem, something that does not occur every year (see Fig. 2.15) Nonetheless, resultsfrom the previous seasons are consistent with the observations of 2015.To illustrate the evolution of a cluster during 2015, Fig. 3.12 shows the changesin mean pressure (panel a) and spatial distribution (panels b–i) for the correlatedand anti-correlated boreholes of cluster H1. We can see how the mean pressurewithin the correlated portion of this long-lived cluster steadily drops during theseason, only punctuated by limited increases during periods of enhanced meltwa-ter supply observed after two snow events around June 30th and July 31st. Thisdecreasing trend in pressure through the season has also been observed by Gordonet al. [1998] at Haut Glacier d’Arolla.By contrast, the mean pressure of the anti-correlated boreholes does not show94Figure 3.12: (a) Water pressure as a fraction of overburden for all corre-lated (blue) and anti-correlated (red) sensors participating in clusterH1. Thick lines represent mean values. The black line highlights ahigh-pressure correlated sensor, and the light blue shading the fractionof the glacier covered by fresh snow. Panels b to i show snapshots ofthe spatial distribution of correlated (blue circles) and anti-correlated(red circles) boreholes in eight time windows. Empty circles representother boreholes that were recording pressure at the moment. The ex-tent of the time windows associated with each snapshot is shown bythe black bars at the top of panel a. The blue shading represents icethickness with the same scale as Fig. 2.2.95any significant trend. Also, we observe that, while individual correlated boreholesshare a common long-term pressure trend, anti-correlated boreholes display a widevariety of long-term trends, the most common consisting of a constant pressurevalue.The study of the evolution of individual clusters can only provide a limited pic-ture of the overall dynamics. This includes the split of larger clusters into smallerones, the merging of multiple clusters, or the appearance of numerous short-livedclusters. To visualize this processes, Fig. 3.13 organizes each cluster in a temporalnetwork, where each small coloured box represents one of the clusters identifiedin a given window throughout the 2015 melt season. Clusters identified in thesame time window are aligned vertically, and horizontally aligned series of boxescorrespond to the different stages of one individual cluster trough time. The timewindows used during the clustering process were 6 days long, and neighbouringwindows had a 50% overlap. However, for visualization purposes each box in Fig.3.13 only covers two days around the centre of the corresponding window. Theposition of each cluster along the vertical axis has no physical meaning and hasbeen chosen to improve visualization.The height of each box is proportional to the number of boreholes within acluster. However, changes in sampling through the season as new boreholes weredrilled and old sensors stopped working, can give a misleading idea of evolution.This effect can be seen in Fig. 3.12, where the growth of the cluster H1 betweenwindows e and f is mostly associated with the incorporation of a new line of re-cently drilled boreholes. To properly account for sampling effects on cluster sizes,in Fig. 3.13 we scaled the height of each box by two factors. The first is the num-ber of boreholes that are part of the cluster in a given window divided by the totalnumber of working sensors between May and November 2015. The second is theratio of the total number of boreholes that formed part of the cluster for any partof 2015, to the number of working sensors in that window. The first factor scalesclusters according to their relative size, and the second adjusts the scaling for thechanging number of working sensors through the season.We can see that the evolution of hydraulic clusters shows a quick onset, andrapid growth during periods of increasing meltwater supply. Panel c shows the PDDrecord, which is a good proxy for the rate of surface meltwater production (see Ap-96Figure 3.13: Cluster network for the melt season of 2015. In panel a, each sequence of aligned coloured boxes rep-resents snapshots of a cluster trough time. Boxes labelled b–i correspond to the maps in Fig. 3.12. Hydraulicclusters are presented by blue and red boxes, where blue and red represents the fraction of correlated and anti-correlated boreholes respectively. Mechanical clusters are presented in shades of yellow. Thin grey lines rep-resent the trajectories of individual boreholes. In the background, the light blue shading provides a qualitativerepresentation of the fraction of the glacier covered by fresh snow as derived from visual inspection of time-lapse imagery. Panel b shows the total fraction of correlated (blue) and anti-correlated (red) sensors per windowparticipating in hydraulic clusters. Panel c shows the daily PDD record.97pendix C), and also a good proxy for the rate of meltwater supply to the subglacialdrainage system over periods without fresh snow cover (white background in Fig.3.13). During the first week of July, a substantial increase in meltwater supplyled to the formation of an extended cluster (labelled H1) that incorporated all theconnected sections of the bed.We observe cluster growth mainly during the spring event, and to a lesser extentafter snow events followed by high temperatures later in the season, as illustratedby the cluster H1 after the snow event observed at the end of July 2015. Whendiurnally-averaged meltwater supply is steady or decreasing, hydraulic clusters ex-perience a progressive reduction and fragmentation. We can observe this processin the evolution of cluster H1 during July 2015, and again during the second halfof August 2015. The observed cluster size reduction happens by borehole discon-nection and cluster fragmentation.Boreholes that cease to be hydraulically connected to a cluster can connect toanother hydraulic cluster or become entirely disconnected. In some cases, discon-nected boreholes can turn into anti-correlated members in the same cluster, as isthe case for the two sensors shown in Fig. 3.10. In other cases, they can turn intomembers of a mechanical cluster, as illustrated in Fig. 3.11.In the snapshots of the spatial distribution of cluster H1 shown in Fig. 3.12b–i,we can see that the growth of cluster observed in Fig. 3.13 is not only explainedby the incorporation of new boreholes within the initial area of influence of thecluster, but also by the growth in the spatial extent of the area of influence across theglacier. We can see how anti-correlated boreholes tend to show up preferentiallyon the edges of the connected regions, but they can also occur as “islands” withinareas of the bed predominantly well connected to the subglacial drainage system(see Fig. 3.12e–h).The fraction of correlated and anti-correlated boreholes in each window ofcluster H1 is represented in Fig. 3.13 in blue and red respectively. Note that af-ter the cluster H1 reached its peak size during the first week of July, the fraction ofanti-correlated boreholes increases while the cluster reduces its size (Fig. 3.13b).The fraction of anti-correlated boreholes for windows b–i of cluster H1 are 5%,20%, 0%, 23%, 37%, 32%, 40% and 33% respectively.Recall that we identify correlated boreholes by the larger amplitude of their98diurnal variations, and by their mean pressures being generally lower than those ofanti-correlated boreholes. Figure 3.7 shows the mean pressure and the amplitudeof diurnal variations for all the boreholes in clusters M1 (panel a) and H1 (panelb).There are exceptions to this general pattern. Some anti-correlated pressurerecords can display large amplitude oscillations, and most notably, some correlatedboreholes can display high mean pressures, small amplitude diurnal variations, andmean pressure trends dissimilar to those shown by most of the correlated sensorswithin the cluster. Such correlated boreholes resemble anti-correlated ones in everyaspect but their phase. The black line in Fig. 3.12 shows an example of this unusualkind of correlated pressure record.3.3.2 Diffusivity at the glacier bed and the two-dimensional nature ofthe drainage systemPhase lags between hydraulically connected boreholes, as well as the changes inthe amplitude of diurnal variations, are the signature of the propagation of pressurewaves through a diffusive system [Hubbard et al., 1995, Werder et al., 2013]. Theirstudy allows us to assess to what extent diffusion processes (with finite diffusivity)control the propagation of pressure variations in the subglacial drainage system.Phase lags for each time series within a cluster were computed relative to the meandiurnal residual of the cluster, and the associated lag corresponds to the time offsetthat maximizes its correlation coefficient with the mean diurnal residual of thecluster.The phase lags in mechanical clusters are often very small and close to ourmeasurement error. However, in hydraulic clusters, we consistently observe phaselags up to six hours. Note that our clustering method suppresses the clusteringof time series with a time lags around 6 and 18 hours, as those would neither bewell correlated nor anti-correlated. However, manual inspection of the boreholesexcluded from cluster H1 and other large clusters suggests that lags larger than6 hours are extremely rare. Figure 3.14 shows the distribution of phase lags andamplitudes of diurnal variations for the correlated sensors in window g of clusterH1 (see Fig. 3.12). A diffusion model for pressure variations would predict thatobservations at increasing distances from an active drainage axis, such as a channel,99Figure 3.14: Spatial distribution of phase lag (a) and amplitude of diurnalvariations (b), for the correlated boreholes in window g (July 27th toAugust 2nd, 2015) of cluster H1 (see Fig. 3.12). Panel c shows theindividual diurnal residuals for each borehole as thin lines, and themean diurnal residual for the cluster as a thick black line. Panel dshows the relationship between phase lag and amplitude.would display increasing phase lags and decreasing amplitudes [Hubbard et al.,1995]. In general, we indeed observe that leading phases in correlated boreholestend to be associated with larger amplitudes. We present a typical example ofthis loose relationship in Fig. 3.14d. In this example, as well as in most cases,sequences of boreholes that clearly display a diffusive behaviour are the exception.One example of behaviour qualitatively consistent with diffusion is the line of fourboreholes pointed by a black arrow in the upper right corner of panels a and bof Fig. 3.14. By contrast, most groups of boreholes display a more complicatedpattern of phase lag and amplitude distribution. In other exceptional cases, thereare even groups of boreholes where we observe increasing phase lags accompaniedby increasing amplitudes, opposite to what we would expect in a diffusive system.For each of the many spatial patterns shown by the different clusters, we also100Figure 3.15: Detailed spatial distribution (a) and diurnal residuals (b-f) of fiveclusters observed in the plateau area between July 18th and 23rd, 2015.Only correlated boreholes are shown.evaluated whether they were compatible or not with a subglacial drainage systemon which horizontal conduits are confined to the bed interface only. We found thatin some cases, the clustered boreholes exhibit a structure seemingly incompatiblewith a two-dimensional drainage system. Figure 3.15 shows an example of fiveclusters in the plateau area, where the clusters in panels c, d, and maybe b, seem tostraddle the one on panel f.1013.3.3 Spatial patterns of connected and disconnected areasIf we accept momentarily that two boreholes that are correlated members of thesame hydraulic cluster are linked by a hydraulic connection, then it becomes ev-ident that some regions of the bed are more susceptible than others to forminghydraulic connections. Other regions seem to remain disconnected for multipleyears. However, quantifying these differences in connectivity requires us to ac-count for the spatial sampling bias of our dataset adequately. For this reason, usingthe whole dataset we have calculated the average probability of a hydraulic con-nection between two boreholes. This probability was calculated under the assump-tion that the bed is homogeneous, meaning that hydraulic connections are equallylikely anywhere along the bed (see Section 3.2.4). Here we contrast the predic-tions of the connection probability computed in Section 3.2.4 with the observednumber of identified hydraulic connections at each position, allowing us to testhow heterogeneous the drainage system is.Figure 3.16a shows the total number of hydraulic connections found in all win-dows of our clustered dataset for each borehole. However, this number is heavilybiased by our spatial sampling and data availability. Figure 3.16b shows the num-ber of connections that we expect for each borehole using the connection probabil-ity P (see Section 3.2.4) and considering the data available at each borehole. Wecan see that there are two areas where we would expect the highest number of con-nections: a large one in the plateau at the top of the study area, and a smaller one atthe bottom near the eastern surface stream. Nonetheless, only the one at the plateauactually shows a large number of connections (Fig. 3.16a). To better explore thisdifference, we have defined a connectivity index consisting of the ratio betweenactual and expected connections. Figure 3.16c shows the connectivity index for allboreholes. We observe that some regions display 2 to 6 times more connectionsthan expected. In particular, the region at the top of the study area, and the one atthe bottom between the two surface streams.102Figure 3.16: (a) Total number of hydraulic connections observed in eachborehole. (b) Number of hydraulic connections expected for eachborehole using the estimated connection probability P (see Eq. 3.3).(c) Connectivity index for each borehole. (d) Number of hydraulicconnections expected for borehole where no connections were ob-served.103Figure 3.17: Spatially averaged mean pressure values for three types of bore-holes between May 15th and November 1st, 2015. In blue, the meanof 171 boreholes that at some point in the 2015 melt season werehydraulically connected. In green, the mean of 78 boreholes thatparticipated in mechanical clusters but were never hydraulically con-nected. In red, the mean of 33 boreholes that were at some point anti-correlated members of a hydraulic cluster but were never hydraulicallyconnected. All pressure values used to compute these means were nor-malized by the overburden pressure of each corresponding borehole.Finally, Fig. 3.16d shows the number of expected connections for boreholeswhere we found no connections at all. We can see that some regions include mul-tiple boreholes for which we would have expected to observe over a hundred con-nections but found none. The most notable example is the area next to the easternsurface stream.3.3.4 Spatially averaged pressure trendsAs we have pointed out, we cannot apply our clustering algorithm outside ofthe summer melt season due to the lack of diurnal forcing. However, a generaloverview of seasonal pressure changes through the year, can be obtained by av-eraging over the records of all sensors based on their behaviour during the meltseason (see Section 3.2.5 for the averaging method). We have selected three typesof boreholes whose means are displayed in Fig. 3.17:1. Boreholes that we identified at some point as correlated members of a hy-draulic cluster (in blue). We expect these boreholes to be representative of104the regions of the bed over which the summer drainage system develops dur-ing the melt season.2. Boreholes that were anti-correlated members of a hydraulic cluster, withoutever becoming hydraulically connected (in red). If these anti-correlated pres-sure variations are the result of horizontal normal stress transfers, the corre-sponding boreholes must be necessarily sampling disconnected portions ofthe bed. Therefore, they constitute our best proxy of the pressure variationsin such disconnected areas. We have excluded other disconnected boreholesdue to the concerns expressed in section 2.3.4, regarding the fact that someboreholes might be encased in ice and not directly sampling the bed.3. Boreholes that we identified at some point as members of mechanical clus-ters and were never hydraulically connected (in green). We include thiscategory to provide more information for the interpretation of mechanicalclusters.We can see how the three types of boreholes display mostly constant pres-sure before and after the melt season, differing in their mean value by up to 25%of overburden. Connected boreholes (in blue) show lower pressures, with signif-icant pressure variations during the melt season and an after-season value about15% lower than the pre-season mean value. In contrast, mechanical and anti-correlated boreholes show very similar pre and post-season mean values, and atseasonal timescales their mean pressure is typically anti-correlated to connectedboreholes. Note that this anti-correlation is true also at diurnal timescales but onlyfor the anti-correlated boreholes (red). The mean pressure in mechanical and anti-correlated boreholes remain close to the overburden pressure, however the formerare generally below it and the latter above.Pressure variations in “mechanical” and “hydraulic” cluster can also be stud-ied in the frequency domain. We have seen that mechanical clusters are charac-terized by more square-wave-shaped diurnal variations, as is clear in the shapeletof Fig, 3.5, and these variations have small amplitudes, typically below 2 meters.Spectrally, the time series produced by mechanical clusters also have a larger high-frequency content than hydraulic clusters. To quantify the difference between clus-105Figure 3.18: Power spectrum of clusters H1 (in blue) and M1 (in red).ter types, Fig. 3.18 presents the average power spectrum of the pressure time seriesin cluster M1 (in red) and H1 (in blue). For M1, the data comes from 30 boreholesproviding 492 time series on 44 different 6-day time windows, and for H1 it comesfrom 120 boreholes providing 865 time series on 33 different 6-day time windows.In each window, all time series belonging to H1 or M1 over that window weretapered using a turkey window [Bloomfield, 2004] with α = 1/3 and then fouriertransformed. The power spectra over all windows and available time series thereinfor each cluster were averaged to produce the H1 and M1 average spectra.We can see how pressure variations in the mechanical cluster M1 have a greaterpower that those of H1 for all periods shorter than one day. Conversely, cluster H1has a greater power than M1 for all periods longer than two days, reflecting agreater amplitude of low-frequency variability.3.4 DiscussionWe can robustly automate the picking of clusters based on the similarity of diurnalresiduals, which correspond to the normalized residuals of the raw pressure signalsrelative to a diurnal running mean (Eq. 3.1). The algorithm is based on hierarchicalclustering and an “absolute” version of the Euclidean distance metric (Eq. 3.2).106This distance metric defines how the similarity between time series is quantified.Different clusters differ in two respects: the details of the shape of diurnalpressure oscillations (in terms of properties such as how sharp the daily pressurepeak is), and the day-to-day variations in the amplitude of the the diurnal pressureoscillations (see Figs. 2.6 panels c and f, 2.12 panels e and f, and 3.15 panels dand f). We have chosen to normalize pressure variations (that is, not to take intoaccount the absolute amplitude of pressure oscillations) and to group boreholes thatare both well correlated and anti-correlated with each other.We can distinguish two cluster types, which we have suggestively named “me-chanical” and “hydraulic”. The former is characterized by a more square-waveshape of the diurnal oscillations (Fig. 3.5) with a significant high-frequency con-tent (Fig. 3.18) and small amplitude diurnal oscillations, typically 1–2 m (Fig.2.11). In contrast, the hydraulic cluster type is characterized by smoother pressurevariations that can reach large amplitudes (typically tens of meters). Both clus-ter types can also often be broken into two subclusters of mutually anti-correlatedpressure records. Hydraulic clusters differ from their mechanical counterpart inhaving one “correlated” subcluster for which the raw pressure records have signif-icantly higher effective pressure and larger diurnal amplitudes compared with theother, “anti-correlated” subcluster. For mechanical clusters, effective pressures anddiurnal amplitudes are relatively small and comparable on both subclusters, whichalso differ in their phase (see Fig. 3.7).We interpret hydraulic correlated subclusters as consisting mainly of boreholesthat are physically connected to an active subglacial drainage system. This con-nection is consistent with their almost always below-overburden internal waterpressure and the large amplitude of their diurnal pressure variations [Kamb, 1987,Hubbard and Nienow, 1997].The anti-correlated subcluster must then correspond to hydraulically isolatedboreholes that experience pressure oscillations due to normal load transfers froma nearby active drainage system. Within such an active drainage system, below-overburden water pressure increases the normal stress near these isolated bore-holes, and this increase then causes the water pressure in the boreholes to riseto keep their volume fixed. Pressure variations within the active drainage systemchange the strength of this load transfer, with higher water pressures in the ac-107tive system generating lower normal stress in the surrounding area and vice versa.This induces anti-correlated pressure variations in the isolated boreholes [Weert-man, 1972, Murray and Clarke, 1995, Lappegard et al., 2006]. Some pressurerecords suggest that the load-transfer mechanism can exceptionally generate cor-related pressure variations (see the black line on Fig. 3.12). We speculate that thisreflects a “second-order” load-transfer. In these cases, an active drainage systemwould induce anti-correlated pressure variations in an isolated water pocket, andthis water pocket would, in turn, induce pressure variations on a second isolatedwater pocket. These later variations would then be correlated with those of theactive drainage system. This kind of interaction would be possible only if the firstwater pocket extends far enough from the active drainage system, such that its in-fluence on the second isolated water pocket becomes stronger than that of the activedrainage system.Using a semicircular R-channel model, Weertman [1972] showed that the loadtransfer effect extends over a distance similar to the radius of the channel. In such acase, the probability of randomly drilling into a channel would be the same as thatof randomly drilling into a section of the bed under the influence of load transfersfrom the channel. However, in cluster H1 we observe that anti-correlated boreholesaccount typically for 20% to 40% of the cluster, numbers that are similar to whatwe observe in other large clusters. While this difference could arise from anti-correlated boreholes being detectable only at distances shorter than one channelradius, the distribution of correlated and anti-correlated boreholes suggests thatlarge clusters are not composed of a network of well developed R-channels. InFig. 3.12f–h, we can see how anti-correlated boreholes tend to appear in groupssurrounding the areas dominated by correlated boreholes, while for a network ofR-channels we would expect them to be finely interleaved in between the channels(assuming that the diameter of those channels would be much smaller than our 15m sample spacing). Therefore we interpret large clusters, such as H1 in windowse to h of Fig. 3.12, as consisting of a distributed drainage system where the gapsbetween conduits are generally small compared with the borehole bottom diameter,that we estimate to be between 25 and 50 cm approximately. These conduits couldcorrespond to a network of cavities, pore spaces, small channels, or a combinationof them.108We interpret “mechanical” boreholes as similarly isolated boreholes that expe-rience pressure oscillations due to changes in the stress field around them. Thisis consistent with their typically near-overburden water pressure (Fig. 3.17), theirwidespread spatial distribution (see orange triangles in Fig. 2.2 and time series inFig. 2.11), and their greater high-frequency content (Fig. 3.18). The diurnal ac-celeration and deceleration of the glacier could induce the changes in the stressfield that drive the pressure variations within mechanical clusters [Andrews et al.,2014]. If that is the case, their characteristic “square-wave” summer profile ofdiurnal variations (Fig. 3.5), would be suggestive of stick-slip basal motion.Basal conditions surrounding boreholes belonging to mechanical clusters orhydraulic anti-correlated subclusters would be essentially the same. However, thehigher mean pressure of the latter (see Fig. 3.17), is consistent with the increasednormal stress experienced in the proximity of an active drainage system (Fig. 3.1).3.4.1 Diffusivity distribution at the glacier bedOur observations suggest that hydraulic connections are more likely to be found insome areas of the bed than others (see Fig. 3.16). The areas with a high connectivityindex cannot be predicted by simple upstream area calculations that assume thatthe effective pressure is a constant fraction of the ice overburden pressure (seeFig. 2.2), and do not display a strong association with basal or surface topographyfeatures. We found the largest concentration of boreholes with high connectivityindex on the plateau area of Fig. 2.2. This area is characterized by a relatively flatsurface and low-angled bedrock topography. However, other similarly flat areasdo not show enhanced connectivity. A more detailed analysis to explain thesedifferences and the overall location of enhanced connectivity areas would requirefurther information about the points were meltwater supply enters the subglacialdrainage system.In contrast with the connected areas, a significant fraction of the glacier bed canremain disconnected year-round, even during the spring event. The concentrationof permanently disconnected boreholes in areas that we sampled densely and overlong periods (see Fig. 3.16d) suggests that their presence is robust and we cannotattribute it to sampling biases.109The location of permanently disconnected areas does not seem to be definedby the local geometry of the glacier either. We find the two most notable caseslocated beside the two areas with the highest connectivity. This association couldsuggest that areas with high connectivity might turn nearby sections of the bedinto disconnected areas: in the absence of local water input, the highly connectedparts of the bed might able to evacuate all incoming water from further up-glacierand would not require the adjacent disconnected areas to form part of the drainagesystem.Hoffman et al. [2016] also recognized a marked heterogeneity in the diffusivityof the glacier bed, proposing the existence of a disconnected or weakly connectedcomponent. In Chapter 2 we identified some features in the data that might beconsistent with a slow leakage from the disconnected parts of the bed (Fig. 2.9).The boreholes that most confidently represent the pressure variations in the dis-connected portions of the bed are the anti-correlated boreholes of hydraulic clustersbecause, if our interpretation is correct, such condition guarantee that they are sam-pling the bed-ice interface. Interestingly, the mean pressures in such boreholes (seered line in Fig. 3.17) does not show a significant leakage during the period in whichlow pressures dominate the connected drainage system. In a water volume that ishydraulically disconnected except for a slow leakage, the associated reduction inwater volume should show up as a gradual drop in water pressure. Alternatively, ifthe water pressure remains constant, the leakage should result in a reduction of thewater volume within these disconnected areas of the bed. However, we do not seethat the pressure in the red line of Fig. 3.17 drops in response to low water pres-sures within the connected drainage system. Similarly, Fig. 3.13b does not show areduction of the number of hydraulic anti-correlated boreholes in those periods oflow water pressure within connected boreholes.Additionally, the anti-correlated boreholes are in the proximity of the con-nected portions of the bed, arguably a factor that should increase the leakage rate,and the lack of a slow pressure response indicating leakage suggests that discon-nection is in effect complete. Alternatively, we can argue that the high effectivepressure in the connected areas would favour the closure of connections in the sur-rounding bed due to bridging stresses [Weertman, 1972, Lappegard et al., 2006],therefore, making disconnected areas less likely to leak if they are close to hy-110draulically connected ones.3.4.2 Subglacial drainage evolutionThe extent of the disconnected fraction of the bed changes through the melt sea-son. For the 2015 season, Fig. 3.13 suggests that the connected fraction of the bedincreases quickly at the start of the melt season in response to the initial rise inmeltwater supply in the last week of May. However, the lack of a variable melt-water supply before that period would have rendered any preexisting connectionundetectable by our method. Therefore, we do not know if the increase in ob-served connections is due to the establishment of new connections or an increasein the ability of our method to detect them.We can remove this uncertainty by observing the evolution of the subglacialdrainage during long periods of sustained diurnal meltwater supply. There weretwo of such periods in 2015: the second half of June, and most of July (see Fig.3.13 panel c). Notably, we observe a different behaviour in each one: during thesecond half of June, the connected areas of the bed undergo sustained growth. Incontrast, during July we observe a slow decline in the extent of connected areas.These two behaviours contrast in several other important aspects. (1) In the earlierperiod, the drainage system starts small and fragmented, while the later one startsas a single large connected subsystem. (2) In the earlier period, the drainage systemgrows and preserves or reduces its degree of fragmentation, while we observe theopposite trend in the later period. (3) The earlier period is characterized by highand relatively constant mean water pressure, similar to winter pressures (see Fig.3.17). In contrast, the later period starts with a significant drop in diurnal meanwater pressure, followed by a sustained downward trend (see Fig. 3.12). (4) Theearlier period is characterized by diurnal variations of much smaller amplitude thanthose in the second period. (5) In the earlier period, there is not a clear trend inthe fraction of anti-correlated boreholes in hydraulic clusters, while in the secondperiod there is a clear increase of anti-correlated boreholes relative to the correlatedones.We interpret the transition between these two behaviours, as a turning point inthe efficiency of the drainage system, possibly associated with the onset of viscous111heat dissipation as the dominant term for conduit growth [Ro¨thlisberger, 1972,Schoof, 2010a]. Therefore, this transition would mark the beginning of the “chan-nelization” of the drainage system, a process that is consistent with the sustaineddecrease in water pressure (see Fig. 3.12), and the increasing fragmentation.The increase in the fraction of anti-correlated boreholes is also consistent withthe perimeter enlargement associated with the development of an arborescent drainagesystem. It remains unclear whether the snowfall event that separates both periodsplayed a significant role in triggering this transition. In the following chapter, wewill explore possible links between this transition and changes in the surface speedof the glacier.The difference between these two periods suggests that sustained meltwatersupply might have a different effect on the development of the summer drainagesystem, enlarging the area occupied by the low-efficiency drainage system foundearly in the season, yet promoting fragmentation and focusing in the more efficientdrainage present later in the season.The lack of variable meltwater supply outside the melt season hinders the appli-cation of our method. Widespread near-overburden water pressures and insignif-icant correlation between pressure changes when they happen suggests that anydrainage system that persists over winter is highly fragmented and mostly discon-nected from the surface.While the spatial structure of the clusters identified in most time windows canbe described using a two-dimensional conduit network, some clusters seem to betopologically disconnected, such as clusters d and f of Fig. 3.15. Explaining thestructure of these clusters would require a subglacial drainage system that includeshorizontal englacial conduits at multiple levels. However, this topological discon-nection might be merely highlighting a shortcoming of our clustering technique.In particular, the method is unable to track changes in the drainage system at timescales shorter than the time window used for the clustering process. Therefore,cluster d could consist of mutually connected boreholes that connect and discon-nect from cluster f as a consequence of a switching event. In that case, cluster dwould not constitute an independent drainage subsystem, but a temporary exten-sion of cluster f. We consider that the evidence provided by Fig. 3.15 and otherexamples of clusters seemingly incompatible with a 2-D structure of the subglacial112drainage system is not strong enough to discard that interpretation. However, theyare suggestive of some degree of three-dimensional structure.3.4.3 Methodological caveatsOur clustering removes information about the mean water pressure and the ampli-tude of diurnal oscillation through the pre-processing step of forming normalizeddiurnal residuals. This step is necessary to identify mechanical clusters, and toincorporate all anti-correlated holes of a hydraulic cluster. Nonetheless, absolutepressure variations are relevant to the question of whether two boreholes have anactual hydraulic connection, which is one of the main objectives of our study.While differences in absolute pressure can arise from differences in the abso-lute elevation of the lower end of the boreholes or sensor calibration errors (seeAppendix B), two boreholes with well-matched diurnal residuals but with differentoscillation amplitudes necessarily experience variations in hydraulic head differ-ences that themselves resemble the diurnal residuals. For two hydraulically con-nected boreholes, such a hydraulic head difference implies that water will flow.If also there is water storage along the flow path with high water pressure corre-sponding to larger storage [Freeze and Cherry, 1979, Hubbard and Nienow, 1997,Werder et al., 2013] then we expect to see an attenuation in oscillation amplitudeand an increasing phase lag along the flow path.However, our clusters do not always conform to this expectation (see Fig. 3.14).While in general phase-leading boreholes display larger amplitude of diurnal vari-ations, suggesting that diffusion processes do play an important role in the prop-agation of pressure signals. There are numerous cases where phase lags and theamplitude of diurnal variations do not follow the pattern expected in a diffusivesystem. Those cases suggest that the spatial resolution our data is unable to distin-guish the heterogeneities of the distribution of basal diffusivity. This shortcomingsuggests that the diffusivity distribution has a fine structure at scales smaller thanthe minimum spacing between our boreholes ( 15m). Large tortuosities and abun-dant englacial connections could also contribute to the complex patterns of phaselags and amplitudes we observe.In a system dominated by diffusive pressure signals, our clustering technique113would also be a poor choice due to the significant phase lags that can be introducedby diffusion. However, we also tested other clustering variants that should performbetter in that scenario, yet they proved to do a worse job in reproducing our man-ually picked clusters (see Fig. 3.4). For example, the running standard deviationpre-processing quantifies variations in amplitude but is insensitive to phase lags.Similarly, the DTW distance metric assign small distances to signals with similarshapes, regardless of phase lags or stretching. The lack of diffusive signals is alsoconsistent with what we observed during the manual picking of clusters for thecalibration, validation and testing datasets.The stark contrast between the small number of apparently diffusive signals ob-served at South Glacier and those predicted by models is unlikely to arise from thelack of diffusion processes at the bed. Instead, it is most likely a result of the sim-ple conduit geometries assumed by models, such as sheets or straight lines betweengrid nodes. Therefore, this discrepancy also points to a diffusivity distribution thathas a fine structure that we cannot resolve with a 15 m sample spacing.Another shortcoming of our clustering technique is the disregard of diurnally-averaged pressures. While these are frequently uncorrelated for mechanical clus-ters and anti-correlated subclusters, our method can also spuriously identify bore-holes with poorly correlated diurnally averaged pressure as hydraulically connected,so long as their diurnal residuals resemble each other closely enough. Theoreti-cally, the diffusive picture of the drainage system would suggest that at medium-term timescales (on which the conduit configuration and diffusivity do not change)pressure variations should correlate. Our clustering can thus produce false posi-tives for hydraulic connections, as the black line on Fig. 3.12, that might insteadbe the result of a second-order load transfer.We should recall that our method relies on the ability of each drainage subsys-tems to modulate the forcing signal distinctly, as a result of their specific geometry,permeability, and storage distribution. The high fragmentation of the subglacialdrainage observed during some periods suggests that indeed a different subsystemoften produce a significant and distinct modulation of the forcing. However, suchapparent fragmentation could also arise from differences in the forcing itself.We have observed this phenomenon on rare occasions while manually identi-fying clusters. In such cases, we have found boreholes that display similar pres-114sure variations but are very far apart across the glacier, a geometry that makesa hydraulic connection improbable, especially if there are no other boreholes in-between showing similar pressure variations. These cases of similarity that is likelythe result of common forcing are expected to be more frequent between nearbyboreholes, a situation in which we would be unable to distinguish this phenomenonfrom true hydraulic connections. Therefore, we expect that some of the identifiedconnections are artifacts due to the similarity of the forcing signal in individualdrainage subsystems.3.5 ConclusionsWe were able to automatically pick clusters of boreholes based on the similari-ties between their pressure response to surface meltwater supply, and we classifiedthese clusters into two main types: hydraulic and mechanical. Both clusters typesare often composed of two subclusters of mutually anti-correlated boreholes. Formechanical clusters, the two subclusters differ only in their phase, while in hy-draulic clusters one subcluster shows higher mean water pressure and diurnal oscil-lations of smaller amplitude. We refer to this subcluster as anti-correlated becauseit displays pressure variations that are anti-correlated with the surface meltwatersupply.We interpret correlated boreholes of hydraulic clusters as being hydraulicallyconnected to the surface meltwater supply, while anti-correlated boreholes sam-ple disconnected areas of the bed. These disconnected areas can display smallwater pressure variations due to normal stress transfers associated with the pres-sure variations within correlated boreholes [Weertman, 1972, Murray and Clarke,1995, Lappegard et al., 2006]. In large hydraulic clusters, we generally find anti-correlated boreholes at the edge of groups of correlated boreholes, suggesting thatthe distributed drainage system associated to these clusters is composed of a net-work of small conduits with spacings smaller than the borehole bottom diameter(approximately 25–50 cm). Within these hydraulically connected areas of the bed,patterns of phase lag and amplitude attenuation suggest that the diffusivity dis-tribution at the bed presents a fine structure at scales smaller than our minimumborehole spacing of 15 m.115Boreholes in mechanical clusters are also disconnected from the surface melt-water supply, and their pressure variations are likely to be controlled by stresschanges associated with the glacier motion. In this case, the square-wave shapecould be suggestive of a stick-slip motion regime.The distribution of areas of the bed connected to or disconnected from thesurface meltwater supply changes throughout the year, and even during the meltseason. Some areas of the bed can show a large number of hydraulic connectionswhile others remain disconnected year-round. The distribution of these areas doesnot seem to be dictated by surface and bed topography alone. We hypothesize thatthe location of the meltwater supply input points plays an important role in deter-mining which parts of the bed are well-connected or disconnected. Disconnectedareas do not show a significant water leakage during the melt season, suggest-ing that the hydraulic disconnection is complete. However, if bridging stressesare a significant contributor to hydraulic disconnection around connected conduits[Weertman, 1972, Lappegard et al., 2006], it is possible that leakage can becomesignificant in disconnected areas unaffected by normal stress transfers.The evolution of cluster sizes and fragmentation of the drainage system dur-ing the melt season suggest that repeated diurnal pulses of meltwater supply pro-mote the growth of the low-efficiency drainage systems found early in the seasonwhile stimulating the shrinkage, fragmentation, and focusing of the more efficientdrainage systems that appear later in the season. Therefore, the increase in drainageefficiency would inhibit the growth of the connected areas of the bed. In 2015 atSouth Glacier, the transition between these two regimes took place during the firstdays of July, when the pressure within connected boreholes underwent a significantpressure drop (see Fig. 3.12). This turning point might be associated with the onsetof viscous heat dissipation as the dominant term for conduit growth.Our observations support some of the features shown by recent subglacialdrainage models [Schoof, 2010a, Hewitt, 2011, Schoof et al., 2012, Hewitt et al.,2012, Hewitt, 2013, Werder et al., 2013, Bueler and van Pelt, 2015], such as theexistence of a distributed drainage system early in the melt season that graduallyevolves into a progressively more channelized and focused system. However, themost notable difference with the models is the extremely heterogeneous distribu-tion of diffusivity that our results suggest, and the robust support for the existence116of disconnected areas. These disconnected areas invalidate the assumption of thesemodels that the distributed drainage system pervades the whole glacier bed. There-fore, in addition to the effective pressure within the connected parts of the drainagesystem, the extent of this system could also be an essential control on basal speedvariations. It is possible that even relatively small disconnected areas could havea disproportionate effect on basal speed. Also, while our observations cannot con-firm or refute the year-round persistence of a distributed drainage system, winterpressure variations suggest high fragmentation of the drainage system, imposing alimit to the extent of such a persistent distributed drainage.Although the analysis of pressure records we have presented can be improved,many outstanding questions will not find a definite answer unless we sample athigher spatial resolution and new measurements are incorporated to study in moredetail the structure of the subglacial drainage system. For this reason, we recom-mend future work on the development of methods and instrumentation to performthe following actions at the bottom of boreholes:• Tracer injection and detection: to allow the direct detection of hydraulicconnections between boreholes.• Slug tests: to assess hydraulic isolation of subglacial water pockets and esti-mate their storage capacity.• Repeat photography: to use image sequences to measure basal sliding di-rectly.These measurements were performed manually in open boreholes. However,developing the technical means to make such measurements continuously at thebottom of closed boreholes would result in a significant improvement of our ca-pacity to observe the subglacial drainage system and its evolution. Also, we rec-ommend the exploration of technical solutions to seal boreholes directly above thesensors, to avoid the problems posed by englacial connections.117Chapter 4Dynamic effects of drainagesystem evolution4.1 IntroductionImproving our knowledge of the physics that controls glacier dynamics is of greatrelevance for the development of physics-based models. In contrast to empiricalmodels, physics-based models are in principle capable of predicting glacier dy-namics even under climatic scenarios for which we have no data.However, some physical processes that are essential to the understanding ofglacier dynamics are confined to the ice-bed interface, where direct observation isdifficult. These basal processes, together with calving and ice-ocean interactions,are arguably the primary source of uncertainty in current models of glacier dynam-ics.The main influence of subglacial processes on glacier dynamics is through theirregulation of the basal sliding rate (see Fig. 1.2). Basal sliding typically accountsfor about half of the observed ice speeds at the surface of glaciers [Gerrard et al.,1952, Shreve, 1961, Vivian, 1980], and this contribution is even more significantin large glaciers and ice streams [Lu¨thi et al., 2002, Ryser et al., 2014b]. Althoughit is hard to measure basal sliding, changes in surface speed that take place overdays, weeks or a season, are generally associated with changes in basal slidingrates [e.g. Iken and Bindschadler, 1986, Anderson et al., 2004, Bartholomaus et al.,118Table 4.1: Symbols usedub basal speedτb basal shear stressA Glen’s flow law ice hardness parametern Glen’s flow law creep exponentC Constant that could depend on bedrock roughness and thermalconductivity, ice specific latent heat, ice density, and Aµ Coulomb friction coefficient2008]. This association relies on the assumption that changes in basal conditionscan generate a much larger speed increase as a result of increased basal sliding thanenhanced internal deformation.Current models assume that basal sliding variations are mainly controlled bychanges in the basal effective pressure, which is defined as the difference betweennormal stress and water pressure at the bed, where normal stress is usually taken tobe equal to the overburden pressure. Effective pressure is in turn controlled by thecombined effect of the meltwater supply rate and the configuration of the englacialand subglacial conduits that drain the water out of the glacier [Iken et al., 1983,Kamb et al., 1985, Iken and Bindschadler, 1986, Chapter 3 of this work].The relationship between basal sliding and effective pressure is traditionallyformulated as a friction law. This law is a functional relationship between basalshear stress (τb), basal velocity (ub), and effective pressure (N). The first basalfriction law was introduced by Weertman [1957] and took the form τb =C uab, witha > 0 (a = 1/n is commonly used). Subsequent empirical studies of ice sliding byBudd et al. [1979] motivated the introduction of a dependence on effective pres-sure, such that τb = C uab Nb. Later, Fowler [1987] provided a theoretical basissupporting this formulation. As part of efforts to incorporate this type of frictionlaw into numerical models, a bounded variant was introduced by Schoof [2005]:τb =Cuab N(ub +N1/a)a (4.1)119Alternatively, also a Coulomb friction law of the following form has been pro-posed:τb = µN if |ub|> 0τb ≤ µN if ub = 0(4.2)Coulomb friction is appropriate for failure in subglacial sediment, where theyield stress τc = µN is not attained by basal shear stress τb (the second case in Eq.4.2). The stress is then determined by solving the ice flow problem with a no-slipboundary condition rather than prescribed through the boundary condition. Thehard switch at ub = 0 comes about because sliding in this model is only possibledue to till failure. It can be also assumed [Iverson et al., 1998, Schoof, 2010b] thatWeertman-type sliding occurs before till failure, in which case:τb = µN if Cuab ≥ µNτb =Cuab if Cuab ≤ µN(4.3)This and Eq. 4.1 are examples of “regularized” Coulomb friction laws.However, there are many other variables besides effective pressure that mightplay a role in modulating basal sliding, including but not limited to: changes inthe amount and size of rock clasts embedded in basal ice, changes in the contactarea between ice and bedrock, and sub-grid inhomogeneities in the horizontal andvertical distribution of effective pressure.Current models assume that cavity and channel sizes are in steady state, whichimplies that the contact area between ice and bedrock is itself simply a function ofub and N [Fowler, 1986, Schoof, 2005, Gagliardini et al., 2007]. In reality, sub-glacial conduits are likely to be in a transient state most of the time, trying to catchup with the rapidly varying water discharge and effective pressure. Also, the resultspresented in Chapter 3 suggest the existence of significant small-scale (<15m) ef-fective pressure variations. Such variations could invalidate a key assumption ofmodels for basal sliding, namely that water pressure is a slowly varying function ofposition that can be treated as constant at the scale of the bed roughness elementsthat generate drag [Fowler, 1986, Schoof, 2005, Gagliardini et al., 2007]. It is thatassumption which allows a single effective pressure to appear as an argument inthe friction laws [Schoof, 2002, appendix A].120Figure 4.1: Eight-year speed record at South Glacier, showing the typicallong term speed behaviour (black line), and positive degree day (PDD)record (yellow shading). This speed time series was generated from acombination of the data from 16 receivers in our GPS array, and repre-sents the speed at the central GPS of the array (See Fig. 2.2). See section4.2.2 for processing details.Studying the relationship between changes in basal sliding and subglacial pa-rameters such as mean effective pressure or the extent of the hydraulically con-nected fraction of the bed can help us test our current understanding of basal slid-ing and guide improvements to basal sliding models. Ultimately, the goal would beto test basal friction laws directly using field data, although as we will show, thereare numerous issues to be overcome first.At a seasonal scale, the meltwater supply cycle is the main control of basalsliding variations. Figure 4.1 shows an 8-year record of surface speed at SouthGlacier. The main feature of the seasonal cycle is the spring event speed-up thathappens in late spring or early summer, associated with high positive degree day(PDD) values. PDD is a proxy of meltwater production at the surface. When theglacier surface is snow-free, or the snow is water-saturated, we can interpret PDDas a proxy for meltwater supply to the subglacial drainage system. However, ifthe snow-pack is not water saturated, it can store surface meltwater and delay thedelivery to the subglacial drainage. Also, the proportionality constant between PDDand meltwater production is different between snow and ice [MacDougall et al.,2011, Wheler and Flowers, 2011]. This proportionality constant is referred to asthe degree day factor (DDF) and in the temperature-index melt model for SouthGlacier in MacDougall et al. [2011], the DDF for ice was found to be 30–55%121larger than for snow.While the correlation between high PDD and speed is clear, there is a widerange of glacier speeds associated with similar PDD figures. These differences mostlikely arise from changes in the structure of the subglacial drainage system and theeffective pressure within it. The same water supply rate need not, for instance,correspond to the same effective pressure if the subglacial conduit configurationhas changed.For a given conduit configuration, an increase in the water supply is likely todecrease effective pressure. In turn, the conduits that make up the drainage systemcan change in response to changes in water input: associated changes in dischargewill affect the rate of conduit enlargement by wall melting, and changes in effectivepressure will affect the rate at which conduits constrict by the viscous creep ofthe overlying ice [Ro¨thlisberger, 1972, Schoof, 2010a]. These changes in conduitsize probably happen at time scales larger than the typically diurnal variations inmeltwater supply.Changes in sliding will also affect the opening of basal cavities [Iken, 1981,Schoof, 2010a, Hoffman and Price, 2014]. Therefore, the drainage system canchange over time, along with its response to a given water input. Those changesin the drainage system result from the action of opening and closing mechanismsthat control the size and extent of the different types of conduits that compose thesubglacial drainage system.In Fig. 4.1 we can see that the glacier has a background speed in winter of about4 cm/day, and higher speeds during summer that quickly peak at around 7 cm/dayduring the spring event to then fade out slowly. While we can associate the summerspeed excess to enhanced basal sliding [Rabus and Echelmeyer, 1997, Coplandet al., 2003], at South Glacier, it is very likely that basal sliding occurs year-roundand accounts for a significant fraction of the year-round motion of South Glacier.According to the model by Flowers et al. [2011, Fig. 6b between 1600 and 2500m], basal motion in the study area probably accounts for 75–100% of the totalsurface motion.Given the substantial contribution of basal sliding to the speed of South Glacierand the long timescales associated with changes in internal deformation, we canuse surface speed variations to estimate changes in basal sliding rates. We mea-122sure surface speed using global navigation satellite system (GNSS) receivers. Tounderstand how we calculate these speeds and how the different processing tech-niques impact our ability to study basal sliding variations, we have to consider theparticulars of the GNSS system, which is commonly referred to as GPS. GPS re-ceivers find their position by triangulation, using the travel time of signals emittedby a network of GPS satellites. The actual measurements performed by GPS re-ceivers are the travel time and phase of the received signals. These measurementsare termed “pseudorange” and “carrier-phase” respectively, and are collectivelytermed “observables”. Pseudorange observations measure the signal travel time,and carrier-phase observations allow the signal travel distance to be estimated toa fraction of its wavelength, enabling for high-accuracy solutions. These GPS ob-servables are recorded simultaneously at two frequencies for all visible satellitesat specific points in time referred to as epochs [Hinze and Seeber, 1988]. Thesetwo frequencies are known as L1 and L2 and have wavelengths of 19.05 and 24.45cm respectively. It is important to note that carrier-phase observations only pro-vide information about fractional part of the number of wavelengths travelled bythe signal, not about the total number of wavelengths. Therefore, a high-accuracysolution relies on “ambiguity fixing”, that is the process of fixing to integers thereal-value estimates of the total number of wavelengths travelled by the carrier sig-nal [Colombo, 1998]. In adverse conditions such as high noise levels, it is possibleto fix the ambiguity to an incorrect integer.There are two main ways to process GPS observables to compute the locationof the receiver: static and kinematic. Static solutions assume the receiver is notmoving and use all the observations to compute one single position. In contrast,kinematic solutions assume that the receiver is moving and compute one positionat every epoch.For very slow-moving objects such as tectonic plates, the static approach ispreferred. In these cases, the displacement over the observation period is minimalcompared with the error of a kinematic solution. Therefore, the static approachresults in improved accuracy. In contrast, when the displacements are significantcompared with the kinematic solution error, such displacements can be resolved,and the kinematic approach is preferred.Glacier speeds straddle these two regimes: the position of fast-moving glaciers123that advance one meter per day or more are generally computed with the kinematicapproach, while the static approach is often used for slow-moving glaciers advanc-ing from a few millimetres to a few centimetres a day. If GPS observations froma moving object are used to compute static solutions, time series of such solutionswill display biases and spurious periodic signals.These artifacts originate from the ambiguity parameter absorbing the effect ofthe velocity and fixing the ambiguity to an incorrect integer. The magnitude ofthese artifacts is proportional to the speed of the object [King, 2004]. Therefore,the static approach can be used only to compute the position of static objects or themean position of very slow moving objects, in which case the induced artifacts areoutweighed by the gain in accuracy on the mean position. Any gain in accuracyobtained by computing static solutions on a moving object will be at the expenseof temporal resolution.One of the main challenges to compute accurate positions comes from theeffect of changing ionospheric conditions on the propagation of GPS signals. Astandard method that solves this problem is precise point positioning (PPP), whichuses double-frequency GPS receivers and the frequency dependency of ionosphericdelays to estimate a “ionosphere-free” combination. PPP solutions reach high ac-curacy by using these ionosphere-free combinations together with carrier-phaseambiguity fixing. Alternatively, ionospheric delays can be cancelled out by com-puting a solution relative to a reference station, provided that this reference stationis located close enough to experience the same ionospheric delay. In such a differ-ential configuration, we refer to the receiver at the point of interest as the “rover”,and to the reference receiver as “base station”. When a base station and double-frequency observables are available, processing algorithms can combine these twoapproaches in order to maximize the accuracy of the solution [Colombo, 1998].One of the primary sources of error that remains after double-frequency dif-ferential GPS processing is multipath, and it is especially strong at glacier surfacesbecause for the wavelengths of GPS signals they can act as specular reflectors [Lar-son et al., 2015]. Multipath arises from the interference between direct signals andthose bouncing off the surfaces around the antenna, as depicted in Fig. 4.2. Oneof the manifestations of multipath interference in the GPS data is an unusual os-cillatory pattern in the signal to noise ratio (SNR) [Larson et al., 2015]. When the124Figure 4.2: Multipath geometry for a GPS tower. Grey arrows highlight theadditional path length of a reflected signal. Figure from Larson et al.[2007].direct and reflected signals arrive at the antenna, they interfere constructively ordestructively depending on their phase difference. That phase difference in turnsdepends on the difference in travel distance between both signals. Therefore, asthe satellite rises above the horizon and the travel distance between the two signalsvaries, the combined signal varies in power, transitioning from a maximum whenboth signals are in phase, to a minimum when they are in anti-phase, cycling everytime the travel distance difference changes by one wavelength. Figure 4.3 showsan example of this interference pattern on the SNR in data recorded at one of theGPS towers at South Glacier. This kind of interference pattern is a common featurein our GPS dataset, highlighting the relevance of multipath effects at South Glacier.The intensity of multipath depends on the characteristics of the surfaces sur-rounding the antenna, as well as the relative geometry of the terrain, antenna, andsatellites. Among these factors, the most predictable one is the geometry of theGPS satellite constellation, which repeats once every sidereal day. One sidereal125Figure 4.3: signal to noise ratio (SNR) for the signal of GPS satellite SV04 asseen at tower R10C18 on South Glacier on August 2nd , 2013. SmoothedSNR of the L1 frequency is shown in blue, and L2 in green (raw SNR ingray). The time of the maximum elevation (35.3o) is pointed by theyellow is the time it takes the Earth to complete a full rotation and lasts 23h 56m 4.1s.The difference with the 24h solar day arises from the contribution of the translationof Earth around the Sun. As seen by an observer on Earth, this makes the Sun take4 minutes longer to get to the same position on every rotation.Figure 4.4: Horizontal track of kinematic solutions at station R14C18 duringa 6 hour period for ten consecutive days in September 2008. Two outof three days have been greyed out to improve the visualization of thedisplacement pattern.In terms of instantaneous positions, changes in multipath associated with thegeometry of the GPS constellation induce a deviation from the true position thatvaries along a sidereal day but repeats each sidereal day. Figure 4.4 shows thekinematic solution of the position of one GPS tower during a 6-hour period on tenconsecutive days. We can see the repeating pattern of deviations from the trueposition, which lies somewhere in the middle of the point cloud. Notably, the126scatter of the cloud of instantaneous positions completely conceals the real trackof the GPS tower over each 6-hour period.These multipath-induced displacements are mostly random but periodic in time,and also repeatable as long as surface conditions remain unchanged. Therefore, ifwe compute a GPS solution over a period close to one sidereal day, the errorsmostly cancel out. This error cancellation significantly improves the accuracy ofstatic solutions that span one sidereal day. However, for kinematic and sub-diurnalstatic solutions, multipath remains as a significant error source.The objective of this chapter is to contrast the changes in the structure and prop-erties of the subglacial drainage system described in Chapter 3 with the observedvariations in surface speed. Due to the particularities of the GPS errors describedabove and the slow velocities observed at South Glacier, we will make this compar-ison following two different approaches. First, we will study the multi-day speedvariation of South Glacier using static solutions over time windows with a lengthof one sidereal day. While biases and spurious periodic signals can be expected toaffect these static solutions, we expect these artefacts to be negligible due to thelow speeds observed at South Glacier.In a sensitivity analysis performed by King [2004] to study the errors in 1-hour static solutions associated with displacement of 4 cm, he found that for a N-Smoving glacier there were no biases in the N-S speed and only a 5-10% bias inthe E-W speed. These solutions also presented spurious periodic oscillations thatwere estimated to have an amplitude of about 5% of the total speed. Over thestudy area, South Glacier roughly flows in a N-S direction, and displacements overone sidereal day are of 4–6 cm; about the same magnitude as those used by King[2004]. Therefore, we can expect negligible biases in the along-flow speed andthe existence of a relatively small spurious periodic signal with an amplitude up toabout 0.6 cm/day.Regardless of these small caveats, multi-day speed variations will allow us toreveal some of the limitations of current basal sliding models that rely only oneffective pressure to explain the observed variations in basal motion. Based on theresults of Chapter 3, we will discuss which variables, along with effective pressure,could play a significant role in controlling glacier speed variations.Our second approach will be to study the diurnal variations of surface speed127and its relationship to changes in effective pressure within the subglacial drainagesystem. However, if we use the static processing on time intervals short enough toresolve diurnal speed variations, multipath errors and spurious oscillations domi-nate the solution, drowning the real motion signal. Alternatively, if we compute akinematic solution of the glacier motion, multipath errors dominate the signal inthis case, also concealing the true motion of the glacier as in Fig. 4.4. To overcomethis problem, we will develop a new methodology that exploits the periodicity ofmultipath errors to resolve sub-diurnal speed variations using an inverse modellingapproach.4.2 MethodsIn previous chapters, we have studied the data recorded in a dense borehole networkdeployed at South Glacier. The amount of drilling required for such a network waspossible due to the moderate ice thicknesses found in the study area, generallybelow one hundred meters. The downside of a relatively thin glacier such as SouthGlacier is the very slow surface speed, in our case between 4–6 cm/day with evensmaller speed variations. These variations have a magnitude comparable to theerrors in the GPS data. Therefore, after describing the dataset, we will present twodifferent approaches to extracting dynamical information from our GPS record.The first approach aims to resolve speed variations at a multi-day scale, and thesecond and more ambitious looks to resolve these variations at a sub-diurnal scale.4.2.1 GPS arrayThe GNSS data at South Glacier comes from an array of 16 dual-frequency Trim-ble R7 GPS receivers equipped with a geodetic Zephyr antenna. Receivers wereinstalled permanently on the glacier surface mounted on a 2 m tall tower with 4 mlegs drilled into the ice as depicted in Fig. 4.5. Due to surface melt, we had to ex-tract each tower periodically and reinstall it in a new position usually within a fewmeters from the original location. Depending on the location of the tower withinthe glacier, this process was typically repeated as often as twice a year for sometowers, and as rarely as once in the whole study period for others. Towers wereinitially positioned in a regular grid approximately aligned with the ice flow. Each128Figure 4.5: GPS tower set-up.tower is identified by row and column numbers within this grid. Tower R18C18(Row 18, Column 18) is the central tower of the array (central GPS in Fig. 2.2).Figure 4.6 shows the trajectory of each tower during the study period. Each GPSreceiver recorded observables every 2 seconds. From summer 2008 to 2013, werecorded data only 6 hours a day, but after summer 2013 a new power managementsystem was installed, allowing continuous data recording during the periods whenenough solar power was available.A base station at Kluane lake research station (KLRS), located 45 km NE fromthe field site was available since the start of the measurements in 2008. In August2013, we installed a new base station (RIDG) at a distance of 800 meters fromthe center of the GPS array, on the bedrock ridge above it. In addition to thesetwo bases, three international GNSS service (IGS) base stations exist around SouthGlacier: AB35, AB42, and WHIT, at 198, 165 and 212 km respectively.129Figure 4.6: Approximate trajectories of GPS towers at South Glacier. Blackdots show locations on August 1st of each year. Empty circles repre-sent missing locations. Location sequences start in 2008 for all towersexcept R16C20 which was installed in 2012, and R16C16, which wasonly in place between 2010 and 2012. Yellow lines represent intervalswhere the GPS towers were relocated manually. Grey markers representboreholes, and marker shapes identify the year they were drilled.4.2.2 Daily GPS solutionsWe obtained daily GPS solutions following standard procedures with the opensource GNSS package RTKLIB. The positions were computed as post-process dif-ferential static solutions over one sidereal day period (23h 56m 4.1s), using finalephemeris and clock files [Dow et al., 2009].We rejected solutions for which less than 4 hours of data was available within130the corresponding one-sidereal-day period. Solutions derived from only 4 hoursof data would have position uncertainties much larger than those derived from afull sidereal day of data. Nevertheless, we chose this low threshold to maximizethe coverage of the data for long term analysis. For detailed analysis, where highaccuracy is required, we applied further filtering of the solutions only to keep po-sitions computed from at least 12 or 23 hours of data. Such additional constraintswill be specified case by case. We also computed redundant differential solutionsusing each one of the available base stations, as well as PPP static solutions. Figure4.7 shows speed time series derived from the differential solutions and PPP. We cansee that, while all solutions follow a similar trend, at specific points in time eachsolution shows one or more significant deviations from the trend. We observe thisbehaviour also in the time series relative to the RIDG base station (in yellow). Thisstation is very close to the glacier and should produce the most accurate solutions.These speed deviations observed in Fig. 4.7 are most likely the cause of a localperturbation around the base station. For example, such perturbations can take theform of snow accumulation near the antenna or the installation and removal of ob-jects in the vicinity of the antenna. When such perturbations displace the positionof a single solution, it will show up in the velocity time series as two anomalouspoints: one too high and the other too low. To remove these outliers without erasingreal diurnal speed-up events, we have combined all solutions through a weightedmean, where the weight of a point is inversely proportional to the standard devia-tion of its corresponding time series within a 5-day window around it. If speed s iscomputed at fixed one-day time steps denoted by subindex i using one of M basesdenoted by the subindex b such that sbi is the speed at time step i using base b. Theweight of sbi is given bywbi =(14i+2∑j=i−2(sb j− sbi))− 12(4.4)where sbi is the mean speed over the time window given bysbi =15i+2∑j=i−2sb j (4.5)131Figure 4.7: Comparison of differential GPS solutions and PPP solution forGPS tower R20C20. Bases are RIDG (yellow), KLRS (light green),WHIT (dark green), AB42 (red), AB35 (blue), and PPP (magenta).Also, the speed weighted mean is shown as a thick gray line.Then, the weighed speed Si is given bySi =∑Mb=1 sbiwbi∑Mb=1 wbi(4.6)Figure 4.7 shows this weighted mean S as a thick grey line.Individual GPS towers have multiple data gaps over the 8-years span of the data.These gaps were due to multiple reasons, for example, power loss, tower resetting,receiver failure, mechanical tower failure, or memory shortage. For long termseasonal comparisons, we have combined time series from all GPS towers. Thiscombined time series is shown in Fig. 4.1. The reference tower was the centralGPS (see Fig. 2.2), and for all remaining towers, a linear regression was calculatedto account for the typical differences in mean value and amplitude. After all timeseries were adjusted to match the central GPS tower, they were combined using thesame weighted mean approach described above (see Eq. 4.6).1324.2.3 Sub-diurnal GPS solutionsHere we will describe a novel methodology that attempts to remove multipath ef-fects from kinematic GPS solutions to recover the sub-diurnal motion of the glacier.This methodology exploits the periodicity of multipath to obtain displacementsover periods close to one sidereal day, on which multipath effects cancel out. Thesedisplacements are the only multipath-free proxy we have of glacier speed and areequivalent to mean speed of the glacier over a moving window of one siderealday (SD). However, given that one SD has a very similar length to one day, thismean speed provides very little information about the sub-diurnal speed variationsand is effectively a smoothed version of the high- temporal-resolution speed recordwe are seeking. Therefore, we need to “de-smooth” this mean speed record, butthe solution to this de-smoothing problem is not unique. We will tackle this prob-lem using an inverse model approach and selecting one solution among the infinitepossibilities by using a regularization term chosen to be consistent with simplephysical considerations about the expected behaviour of the glacier.Kinematic GPS solutionsTo obtain sub-diurnal velocity time series, we need first kinematic solutions of ourGPS data. These solutions were computed by Matt King (University of Tasma-nia) using GNSS at MIT (GAMIT) and Track v1.29 software [Chen, 1998, Herringet al., 2010], and the nearest base station (RIDG) when available, or KLRS baseotherwise. He produced time series of coordinates at each glacier site every 30 s,or 2 s over some periods. This kinematic processing strategy avoids the genera-tion of the small amplitude spurious signals that can appear in speed time seriesderived from static solutions [King, 2004]. As the distance between RIDG andthe glacier is less than 2 km, we assumed that ionospheric delays were equal inboth stations. Therefore we did not need to use the two frequencies L1 and L2 tocompute a ionosphere-free combination when RIDG data was available, instead,we used both carrier frequencies L1 and L2 directly in the analysis. We processedeach GPS tower in a separate solution. For the satellite positions, we used the IGSprecise orbits [Dow et al., 2009].Track software uses a Kalman Filter for the estimation of unknowns, consid-133ering their temporal evolution a random walk process. We loosely constrainedthe temporal evolution of the GPS via Kalman Filter process noise at a level of0.1mm/√30s. In addition to GPS tower positions, we estimated troposphericzenith delays every epoch with process noise of 0.1mm/√30s, mapping the line-of-sight delays to the zenith with the Global Mapping Function [Boehm et al.,2006]. We modeled the antenna phase center offsets and variations using stan-dard calibrations [Schmid et al., 2004]. The output was customized to include theidentification numbers of the GPS satellites used in each solution.Multipath effect on kinematic solutionsAt a sampling rate of 2 seconds, the displacement of the glacier between samples(about 0.001 mm) is well below the estimated positioning accuracy. Therefore,the displacements between consecutive samples are completely dominated by er-ror. The standard approach to this problem would be to average positions overa period long enough to get the error below the actual displacement in the sameperiod. However, at South Glacier this approach does not reach an acceptable ac-curacy unless the averaging is performed over a whole day, thus smoothing outthe speed variations that might have existed in response to diurnal water pressureoscillations. Nevertheless, a closer look at the irregular trajectories defined by thesolutions reveals that most of the dispersion follows a pattern that repeats everyday, suggesting that the primary source of error is multipath (see Fig. 4.4).GPS satellite orbits repeat every SD relative to the Earth. However, due toperturbations such as solar radiation pressure or anomalies in the terrestrial grav-itational field, the orbits need to be adjusted regularly. Also, due to these pertur-bations and adjustments, the satellites do not necessarily return to the exact sameposition every day, and the time that a particular satellite takes to return to the clos-est position to the previous day can differ by several minutes from one SD. Theexact repetition time is termed the aspect repetition time (ART), and for a particularsatellite, we can compute it from precise ephemeris data. In general, the averageART is 8 seconds shorter than one SD [Agnew and Larson, 2007]. We will refer tothis period as the “typical ART” (23 h 55 m 56.1 s). But each satellite has its ownART that determines the repetition period of the associated multipath contribution.134The actual ART from individual satellites can differ up to a few minutes from thetypical ART.To compute the ART of each satellite, we follow the method described by Ag-new and Larson [2007]. For this, we read in sp3 format files [Remondi, 2000],which contain the satellite positions in Earth-fixed coordinates tabulated every 15minutes. To compute positions at any arbitrary time, we use Neville’s algorithmfor polynomial interpolation [Press et al.] as suggested by Schenewerk [2003]. Fi-nally, we use the Matlab minimization routine fminbnd to find the time on the nextday that minimizes the distance to the line in that passes by the receiver and theposition of the satellite at the reference time (in the previous day). The fminbndalgorithm is based on golden section search and parabolic interpolation [Brent,Forsythe et al.]. Finally, the computed ART is the time elapsed between the ref-erence time and the moment of the following day that minimizes the mismatchbetween the two apparent positions as seen from the receiver.The periodicity of the geometry of the satellite constellation results in multipatheffects repeating every one ART, provided that the terrain surrounding the GPSantenna does not change over that period. The idea of using this periodicity inmultipath corrections was first suggested by Genrich and Bock [1992], and led tothe standard technique of sidereal filtering. This technique can be used in areaswhen a reference day with no displacements can be recorded, and the terrain doesnot change. In this scenario, the multipath pattern from the reference day can besubstracted from the position time series of any of the following days [Larson et al.,2007], allowing an easy way to remove multipath effects. Unfortunately, glaciersettings do not meet any of these two requirements: the terrain does change, andall days present displacements.However, in a glacier setting, we can assume that the terrain does not changesignificantly between any two consecutive days, except in the eventuality of asnowfall event. To illustrate this fact, we present in Fig. 4.8 the position (East-ing coordinate) of one GPS tower between 8 PM and 10 PM during two consec-utive days that were subject of strong surface melting (September 5th (blue) and6th (red) 2008). We can see how the drift of the solution shows a very similar pat-tern on both days. Moreover, these patterns aligned perfectly only after we appliedan offset of one typical ART. The repeatability of these variations, together with135Figure 4.8: Relative easting solutions for September 5th (blue) and 6th (red)2008 every 2 seconds at tower R14C18 on South Glacier. The timeoffset applied to the second-day time series was one typical ART (23h55m 56.1s).their period precisely matching a typical ART, confirms that the pattern is indeeda multipath artifact, and does not represent real glacier motion. The true motion,in this case, could be approximated by the vertical displacement between the twolines. However, this vertical displacement also varies significantly depending onthe chosen time window.On the other hand, the minimal differences observed between the two patternssupport the assumption that surface melting over one day has only minor influ-ence on the shape of the multipath pattern, even on days with high melting rates.Note that September 5th and 6th 2008 were exceptionally hot days, with maximumtemperatures of 7oC and 11oC respectively. The meltwater production on these twodays was 4.5 and 14.3 mm of water equivalent respectively, as estimated by Whelerand Flowers [2011] using a temperature-index model. As a reference, accordingto the same model between May 2007 and September 2012, 95% of the days thatpresented non-zero melt registered a total melt below 5 mm of water equivalent.Extracting “multipath-free” displacementsGiven that most of the position bias generated by multipath repeats every ART, wewill compute displacements over that period in order to cancel out the multipathinfluence and obtain multipath-free displacements. However, this cancellation will136Figure 4.9: (a) Comparison of speeds computed from samples one day(black) and one sidereal day (red) apart. (b) Comparison of speeds com-puted from samples one sidereal day (black) and one typical aspect rep-etition time (ART) (red) apart. (c) ART deviations from the typical ARTfor each satellite used in the solution (coloured thin lines), and meanART of all satellites used in the solution (thick red). (d) Mean differencebetween positions of consecutive days as a function of the offset appliedto the time series. The offset is relative to one day. Vertical lines rep-resent the offset that would correspond to one typical aspect repetitiontime (ART) (green) and the offset that minimizes the mean difference(red). (e) Same as d but computed only during the shaded period.137not be perfect, because the observed multipath is a combination of the effect frommultiple satellites, each one with a slightly different ART. Here we will present amethodology to maximize the multipath cancellation and extract the most accuratedisplacements the data can provide.Figure 4.9 shows how the cancellation of multipath errors leads to smootherspeed time series when we compute those speed over periods closer to one ART. Inpanel a, the mean speed is computed using samples on day apart (black), and onesidereal day apart (red). The latter is significantly smoother, due to the cancella-tion of multipath errors. It is interesting to note that the main features observed inthe one-day mean speed time series (black) were, in fact, a consequence of mul-tipath. We can obtain further improvements by just reducing the period betweensamples by 8 seconds, equivalent to one typical ART. These improvements areshown in Fig. 4.9b. This high sensitivity to the period used is a consequence of thehigh-frequency variations in the multipath pattern, which lead to significant dis-placements between samples only a few seconds apart. Analysis of South Glacierdata shows that the displacement between samples 8 seconds apart is greater than6 mm in 5% of cases. Notably, these short-time-scale displacements can producespurious speed spikes of 0.6 to 1.2 cm/day, which are of the same order of mag-nitude than the observed speed variations between consecutive days. Therefore, itbecomes essential to use accurate ART values, as they can deviate up to a few tensof seconds from the mean value.To identify the source of the remaining high-frequency noise in the mean speedscomputed over one typical ART (Fig. 4.9b), it is useful to note that the reductionin the variability of the speed time series observed in Fig. 4.9b is mostly confinedto the second half of the interval, while there was no improvement over the greyshaded period. We can explain this contrast based on the differences in ART be-tween each of the satellites used to compute these solutions. Figure 4.9c, showsthe ART of each satellite used in the solution and the mean (thick red line). Notethat the section with large high-frequency variations of panel b (mean speed overone typical ART), coincides with the period where there is a significant differencebetween the individual and the typical ART. These differences highlight two issuesthat we need to address: the first is the need to use accurate ART values to findpairs of positions that are affected by the same multipath bias. The second issue138is related to the fact that the multipath effect on the signal has contributions fromeach satellite used in the kinematic solution, and they all have a slightly differentART. Therefore, even if we can accurately find the ART of each satellite, we can-not separate the contribution of each one, and no single ART value can be used toremove the multipath contribution of all satellites.To disentangle the contribution of each individual satellite to the multipathpattern, these contributions should be accounted for prior to the computation ofthe position. However, this would require changes in the processing softwareGAMIT, something that is beyond the scope of this work. Therefore, we propose aworkaround that can satisfactorily solve these two issues using only the availablefinal positions. First we identify sections of the GPS record that are separated byone typical ART and also use the same set of satellites. Then we compute the best“empirical ART” in the range between the shortest and longer ART of the satellitesused in the solution. To find this empirical ART, we perform a cross-correlation tofind the offset that minimizes the mismatch between the two position time series.To quantify this mismatch we use the mean displacement between correspondingsamples of each day.Figure 4.9d presents this mismatch as a function of the offset applied to thesecond time series using the whole time window presented in panels a–c. In thiscase, we see a well-defined minimum (red line) that closely matches the typicalART. In contrast, over the shaded interval, the minimum is not as well defined (Fig.4.9e), due to the superposition of different ART values. For each interval wheresolutions use the same satellites in both days, we pick an empirical ART that mini-mizes the mismatch, but the search for the minimum is constrained to the range ofthe computed ART values of the satellites used in the solution. Intervals that do notshare the same satellites are generally very short, and we do not use displacementsover those periods for the inversion. This rejection is due to our reduced capabilityto remove multipath effects in those cases. Our GPS data allows the calculation ofone displacement every 2 seconds, which provides more than enough data points ifwe want to resolve speed variations at roughly one-hour resolution. This selectionof intervals that share the same satellites effectively filters the data, leaving onlythe displacements we can compute with higher accuracy.139Recovering high-resolution trajectories: Sidereal reconstructionAs discussed above, in a glacier setting it becomes difficult to separate the multi-path pattern from the true signal. The only multipath-free measurements we canharvest from the data are the displacements over one ART, because the multipatheffects will mostly cancel out if the surface around the antenna has not changed sig-nificantly over that period. These displacements do not provide enough informa-tion to recover the underlying displacement signal, but they do provide a constraintthat can be used to reconstruct the signal provided that we make some assumptionsabout the properties of the signal.Here we propose a new processing technique we have termed “sidereal recon-struction”. It consists of recovering the original signal from the displacementsduring one ART using an inverse problem approach. From the infinite number ofsolutions that would satisfy the constraints imposed by these displacements, weselect the one that minimizes the acceleration experienced by the glacier along itstrajectory. This constraint is our regularization in inverse problem terminology.With this regularization, there is now a unique solution that satisfies the dis-placement constraints, and it can be computed using the least squares approachby the computation of the Moore–Penrose pseudoinverse matrix. However, thisanalytic approach cannot account for the white noise in the signal and does not al-low for other constraints. Therefore, we will focus on solutions computed throughconvex optimization.If Xobs is a vector with the observed positions that define the trajectory of theglacier, and X is the modelled multipath-free trajectory, the optimization problemcan be written as:minX‖DART X−DART Xobs‖22STD(‖DART Xobs‖2)2︸ ︷︷ ︸1st data mismatch term+‖X−Xobs‖22STD(‖Xˆobs‖2)2︸ ︷︷ ︸2nd data mismatch term+λ‖Dacc X‖2︸ ︷︷ ︸Regularization (4.7)where DART is the linear operator that produces the differences between selectedsamples one empirical ART apart, STD is the standard deviation function, Xˆobsis the detrended trajectory (i.e. Xobs minus a linear fit of Xobs), Dacc is a linear140operator that computes the acceleration along the trajectory, and λ is the weight ofthe regularization term.The first data mismatch term requires that the modelled trajectory fits the ob-served displacements over one ART. The second data mismatch term requires thatthe trajectory must be similar to the measured trajectory. The need for this secondterm is not obvious. In its absence, oscillating trajectories with large departuresfrom the observed trajectory can also minimize the first term. The denominatorof each data mismatch term is a measure of the error associated with each dataconstraint.In the expression above, the only free parameter is the regularization factor λthat sets the weight of the regularization term, which effectively controls the degreeof smoothness of the solution. We executed the optimization using CVX, a Matlabsoftware package for specifying and solving convex problems [CVX Research,2012, Grant and Boyd, 2008].Choosing the right value for the regularization factor λ would require calibra-tion against independent data, which we do not have. In these cases, a reasonablechoice can be made exploring multiple λ values and studying how the magnitude ofthe regularization term varies for different magnitudes of the data mismatch terms.This relationship is known as the “L-curve”, and the recommended λ lies in thepoint of maximum curvature of the curve.Figure 4.10 shows the L-curve for the inversions of the data of tower R22C18based on 54 different λ values. Unfortunately, it does not present any sharp changesin curvature, which would be required to derive an optimal λ value. Therefore, tofind a reasonable λ , we will follow a different approach: first we find one λ largeenough to produce the same output that would be obtained with a one day run-ning mean of the solution. Then we look for the closer smaller value that showssignificant sub-diurnal structure. Although this does not provide a robust way to re-construct the amplitude of sub-diurnal speed variations, it does allow us to recoverthe timing and direction of such variations, as such properties remain unchangedthrough a wide range of λ values (see also Fig. 4.16).141Figure 4.10: L-curve for GPS tower R22C18 sidereal reconstruction betweenAugust 3rd and 17th 2014.Synthetic dataTo test the capabilities of the sidereal reconstruction method, we have created syn-thetic data generated by superimposing a motion signal with a multipath patternand incorporating the different elements of the real GPS solutions that might limitthe performance of the method, such as white noise, and the slow change of themultipath pattern expected when the glacier surface changes due to ice/snow meltor snow accumulation.The speed of the synthetic time series resembles the shape of a typical pres-sure time series. We created this synthetic speed by the superposition of Gaussianfunctions over a base background speed of 5 cm/day. We then calculate the posi-tions by numerical integration of the speed. The multipath patterns were modelledas a random walk and scaled to have a standard deviation of 0.84 cm/day, whichcorresponds to the standard deviation of the real multipath pattern shown in Fig.4.8. For each experiment we computed two multipath patterns, as the ones shownin Fig. 4.11, and used a linear combination of them in the synthetic experiments,142Figure 4.11: Example of two different synthetic multipath patterns. The mul-tipath inserted into our synthetic data consists in a linear combinationof both patterns, transitioning from (a) to (b) in an interval we variedbetween one and 20 days.where the transition time between the two patterns was varied from one to 20 days,to account for different rates of surface geometry change.We performed multiple synthetic experiments to test the sidereal reconstruc-tion method. In these experiments, the method successfully recovered the truedisplacements in most cases, and in all cases, it performed better than any runningmean/median approach. The method failed only in two scenarios: (1) When thetransition time between the two multipath patterns was one day, and (2) When thetrue signal was perfectly periodic over most of the analyzed interval.Figure 4.12 shows a representative example of a sidereal reconstruction of syn-thetic data, showing that the actual velocity signal can be recovered much moreaccurately than with any running mean choice.143Figure 4.12: Sidereal reconstruction (thick red) of synthetic raw measure-ments (yellow) and moving averages (12 h cyan and 6 h magenta).There is a good agreement between the true synthetic speed signal(blue) and its sidereal reconstruction (thick red). The speed computedfrom samples one ART apart is shown in for the raw synthetic data(black) and the reconstructed data (thin red) showing that the solutionsatisfies the constrains.4.3 Results4.3.1 Multi-day speed variationsMultiple differential static solutions were computed for each GPS tower: one rela-tive to each of the available bases, plus a PPP solution. We computed one solutionper day, each one using data over the slightly shorter period of one typical ART.Multi-base solutions were then calculated using the weighted-mean approach de-scribed in section 4.2.2. Figure 4.13 shows these solutions for the 2015 season,as well as the mean motion (in black) using all towers adjusted to match R18C18,following the procedure described in section 4.2.2. For this computation of themean motion, we excluded data from tower R18C26 due to the small amount ofdata available. We also discarded solutions derived from fewer than 12 hours ofraw GPS data. Figure 4.1 also shows the mean motion of the glacier but over an ex-tended period between 2008 and 2006. This extended time series was also adjustedto match tower R18C18.These static solutions are probably affected by biases and spurious periodic144Figure 4.13: Multi-base diurnal solutions for all GPS towers (colour lines)during the 2015 melt season. The black line represents the weightedmean of all towers but R18C26, using R18C18 as reference.variations as a result of the GPS receiver not being truly static, and the amplitude ofthese variations is expected to be proportional to the speed of the receiver [King,2004]. To study these effects, over two weeks we computed 24 static solutions perday, each one using raw data collected over a window of one typical ART, whereeach window was offset by one hour relative to the previous one. Figure 4.14shows these static solutions for tower R20C20 using all base stations. It is indeedpossible to identify some periodic variations, however, they seem to be of smalleramplitude than non-periodic variations.As it is difficult to quantify the amplitude of such variations from position timeseries like those in Fig. 4.14, we have used the standard deviation of the detrendedposition (as in Fig. 4.14a) as a proxy for the upper limit of the amplitude of spuriousvariations. This proxy constitutes an upper limit because any divergence froma straight line trajectory would increase the standard deviation, including othererrors and real motion variations. Figure 4.15 shows the standard deviation ofthe detrended along-flow position as a function of the mean speed for all towersavailable. The best linear fit to the data (dotted line) suggests that the amplitude145Figure 4.14: Static solution for tower R20C20 between August 3rd and 17th2014, using RIDG base (magenta), WHIT (yellow), AB35 (blue),AB42 (red), KLRS (green), and PPP solution (black). (a) Along-flowdeviations from a straight line trajectory. (b) Across-flow deviationsfrom a straight line trajectory. (c) Vertical deviations from a straightline trajectory.of the variations roughly corresponds to a 10% of the mean speed. Note that thelinear fit was required to pass by the origin.4.3.2 Sidereal reconstruction: sub-diurnal speed variationsWe have applied the sideral reconstruction method to a period of time where we haduninterrupted GPS measurements for four towers over two consecutive weeks. Fig-ure 4.16 shows the reconstruction of ice speed for tower R22C18 over two weeks inAugust 2014. The four coloured lines represent solutions with different values ofthe regularization parameter λ . The yellow line (λ = 1.0×109), has no significantsub-diurnal variations and its shape does not change substantially if we perform a24-hour running mean smoothing. The following three lower values for λ show di-urnal variations with increasing amplitudes. Figure 4.16b shows the displacementsover one ART for each solution (coloured lines) and the measurements (grey). The146Figure 4.15: Standard deviation of the detrended along-flow trajectories asfunction of the mean speed for all the towers available between August3rd and 17th 2014. The dotted line has a slope of 0.10 and representsthe best linear fit passing by the dots represent displacements computed over periods of time where the em-pirical ART computation was possible (i.e. solutions in consecutive days used thesame set of satellites). The difference between solution lines and raw data in Fig.4.16b corresponds to the first data mismatch term of Eq. 4.7. Figure 4.16c showsthe detrended along-flow positions for the solution and measurements. To detrendthe trajectory we computed the best linear fit to the data and removed the associ-ated trend, which was equivalent to a constant speed of 8.6 cm/day. The detrendingwas necessary to visualize the subtle deviations from a straight line trajectory. Thedifference between solution lines and the raw data in Fig. 4.16c corresponds to thesecond data mismatch term of Eq. 4.7.147Figure 4.16: Sidereal reconstruction of the motion of GPS tower R22C18 be-tween August 3rd and 17th 2014 with four different regularization pa-rameters λ : 6.0× 108 (blue), 1.0× 109 (green), 2.5× 109 (magenta),and 6.0× 109 (yellow). (a) Reconstructed speeds (colour lines) andmeasured speed derived from position averages over 12 hours (grey).(b) Reconstructed displacements over one ART (colour lines), displace-ments over one typical one ART using all samples (grey line), and dis-placements over one empirical ART (back dots). (c) Reconstructed(colour lines), and measured (grey line) along flow detrended relativeposition.Figure 4.18 presents the reconstructed speeds over two weeks for four differentGPS towers using λ = 2.5×109 (as for the magenta line in Fig. 4.16), showing thatsolutions are consistent between different GPS towers. These reconstructed speedsare also consistent if we vary the size or the limits of the inverted period (see Fig.4.24a).148Figure 4.17: (a) Accumulated positive degree day (PDD) for each calendar year (red) and accumulated surface dis-placement for each period between spring speed-up events (blue). Only the displacements additional to thosecorresponding to a base speed of 4.85 cm/day are displayed. Values on the top indicate the final accumulatedvalues and the time elapsed between consecutive speed-up events. Black dots on accumulated displacementlines mark the displacement after 329 days, that corresponds to the shortest interval registered between speedup events. (b) Daily PDD. (c) Surface speed record (gray) and smoothed record (black). Red dots representmanually picked minimum speed values before spring speed-up events. The horizontal black line represents theminimum registered speed of 4.85 cm/day. Yellow background represent the same PDD values shown in panel(b).149Figure 4.18: Reconstructed speed between August 3rd and 17th 2014 for tow-ers R22C18 (blue), R26C18 (red), R20C20 (green), and R18C22 (ma-genta) using λ = 2.5× Glaciological resultsTo study how the basal conditions of the glacier affect the surface speed, we presenthere the surface speed dataset in a broader glaciological context, comparing its vari-ations with PDD, effective pressure and other properties of the subglacial drainagesystem derived in the previous chapters. We will start from an annual timescale tothen study processes at diurnal and finally sub-diurnal timescales. In this section,we will refer to the correlated boreholes of hydraulic clusters (see Section 3.3.1) as“connected boreholes” (alluding to the inferred hydraulic connection between themand the surface). Analogously, we will refer to the fraction of connected boreholesrelative to all available boreholes as the “connected fraction of the bed”. The nu-merical values used here for the connected fraction of the bed were corrected forsampling biases as described in section 3.3.1.Figure 4.17 shows an overview of the whole speed record available from SouthGlacier, presented alongside with the PDD. The main feature of this speed recordis the abrupt speed-up that takes place each year between late May and early July,followed by a period of high summer speeds and a gradual slow down that contin-ues until the next speed-up. We have manually identified the point that correspondsto the minimum speed and the initiation of the speed up, represented by red dotsin Fig. 4.17c. These inflexion points naturally divide the study period into distinct150intervals that will refer to as “glaciological years”. Therefore, a glaciological yearstarts with the initiation of the speed-up of the corresponding calendar year andextends until the start of the next year. For example, the glaciological year 2010starts on May 28th 2010 (that corresponds to the minimum speed of the year andthe initiation of the speed-up), and ends in June 17th 2011, when the next speed-upstarts.At the top of Fig. 4.17 we can see that glaciological years during the studyperiod span between 329 and 385 days, and the minimum speed recorded at theend of any of the eight glaciological years studies was 4.85 cm/day, observed at theend of the glaciological year 2014. If we consider this value as the upper bound forthe surface motion associated with internal deformation of the glacier, any speedin excess of that value will be associated with basal sliding. At South glacier, 18%of the total annual motion is the product of these excess speeds.Figure 4.17b shows the daily PDD, and Fig. 4.17a, shows the accumulated dis-placement over each glaciological year and the accumulated PDD. Note that wehave computed the accumulated PDD over the corresponding calendar year, as themeltwater produced during the weeks before the speed-up event is presumablystored in the snowpack and delivered to the subglacial drainage system at the startof the following glaciological year. Figure 4.17 suggests that glaciological yearswith high accumulated PDD values are associated with lower minimum speeds andsmaller total displacements.To explore in more detail this relationship between accumulated PDD and speedat annual timescales, Fig. 4.19 shows the relationship for the minimum surfacespeed (corresponding to the red dots on 4.17c), mean surface speed, and meansummer surface speed, where summer for this purpose consists of the set of dayswith non-zero PDD during the corresponding calendar year. We see a negativecorrelation between the minimum speed and the total cumulative PDD (blue dots),meaning that hot summers lead to slower winter speeds. This correlation has a p-value of 0.02, meaning that if there were no correlation at all, there would be onlya 2% probability that the observed correlation arose from pure chance. In contrast,there does not seem to be a significant correlation between accumulated PDD andmean annual speed (orange dots and p-value of 0.54) or the mean summer speed(red dots and p-value of 0.93). However, the mean summer speed data show hints151Figure 4.19: Minimum surface speed (blue), mean surface speed (orange),and mean surface speed on days with non-zero PDD (red), as a functionof accumulated PDD. Dots aligned vertically corresponds to the sameyear. Speed-derived values correspond to the period after the springevent speed-up of the corresponding year and before the following yearspeed-up. We calculated PDD values over the corresponding calendaryear. Black lines show the best linear fit to each set of points.of being low for both low and high annual PDD (cold and hot summers) and highin summers with intermediate PDD values.At diurnal timescales, Fig. 4.20 shows the surface speed as a function of dailyPDD, with data points color-coded according to the time elapsed since the springevent speed-up of the corresponding year. The point cloud corresponding to daysbefore the speed up does not shows an obvious correlation with surface speed (p-value of 0.08), while the days after the speed-up event show a strong correlation(p-value of 2× 10−35) but with a large scatter. This correlation shows that highPDD days are associated with fast surface speeds.The large scatter is not a surprise, as basal sliding is controlled by the basalconditions that depend both on the meltwater supply and the configuration of the152Figure 4.20: Surface speed as a function of positive degree day (PDD) foreach day between June 2008 and July 2016. The colour of each dotrepresents the time elapsed since the spring event speed-up. Negativenumbers represent days before the speed-up in a given calendar year.subglacial drainage system. While PDD constitutes a proxy of water supply, it pro-vides no information about the configuration of the subglacial drainage. Therefore,we will focus now on the relationship between surface speed and variables directlyrelated to the state and configuration of the subglacial drainage system, such aswater pressure and the connected fraction of the bed. Figure 4.21 shows thesevariables in context with the speed record presented in Fig. 4.13 and with the dailyPDD. Fig. 4.21c also shows the fraction of hydraulic anti-correlated boreholes (red)along with the fraction of connected boreholes (blue). We have corrected all thesevalues for sampling biases as explained in Section 3.3.1.Figure 4.21a shows the speed record with arrows to the four main peaks. Thefirst and smallest, took place during the last days of May and the first days ofJune. The second and largest, was observed in the second half of June. The thirdpeak was observed during the first half of July, and the last one in the first weekof August. Interestingly, the first two speed peaks do not seem to be associated153Figure 4.21: (a) Weighted-mean speed from Fig. 4.13. (b) Mean pressurewithin connected boreholes (see Fig. 3.17). (c) Fraction of connectedboreholes (blue) and hydraulic anti-correlated boreholes (red). Frac-tional magnitudes corrected as in Fig. 3.13. (d) PDD record. In allpanels light blue shading represents periods of fresh snow coverage,and arrows point to the time of the four main speed peaks. Verticalgrid lines are spaced by one week.with any significant increase in water pressure, but a water pressure drop occurs attheir termination. In contrast, the third and fourth speed peaks are associated witha significant water pressure peak. Between the second and third speed peaks, asharp drop in water pressure marks the transition between relatively constant high-pressure values with small diurnal oscillations (similar to the period before the startof the melt season), to lower mean values with large diurnal variations. In Chapter2, we associated this transition with the shift between a low efficiency distributeddrainage system and a more efficient one, as a result of developing channelization.The fraction of connected boreholes presented in Fig. 4.21c (blue bars) shows154a rapid increase associated with the first speed peak, followed by drop coincidentwith a long period where the glacier surface was covered in fresh snow. Subse-quently, the fraction of connected boreholes rose again to reach a maximum at thetime of the third speed peak. This peak was followed by a month of steady declineleading to a dramatic drop at the beginning of a long period of fresh snow coverand low temperatures that marked the start of winter conditions, starting in lateAugust.Finally, the PDD record shows that significant surface melt happened betweenearly May and late August. During this time, at timescales of one or two weeks, wecan associate all the main peaks in PDD to a corresponding speed peak. However,increased PDD over periods shorter than a week do not show a response in surfacespeed.To quantify the correlation between surface speed and the other variables pre-sented in Fig. 4.21, we present in Fig. 4.22 how the surface speed during the 2015melt season relates to: (a) the mean daily pressure within connected boreholes, (b)the daily pressure standard deviation within connected boreholes, (c) the PDD, and(d) the empirical function of effective pressure proposed by Jansson [1995]:< us >day= KN < N >−bday (4.8)where KN and b are empirical constants, us is the surface speed, and <>day rep-resent the mean value over one day. He found that the best fit to data acquiredat Storglacia¨ren and Findelengletscher was achieved in both cases with b = 0.4,which is the value we have used in Fig. 4.22d. In contrast, KN was specific toeach site. Jansson [1995] quantified the quality of the fit using the coefficient ofdetermination R2, finding a value of 0.61 for Storglacia¨ren and 0.77 for Findelen-gletscher. However, given the limitations of the coefficient of determination forassessing the quality of an empirical fit, after R2 values in the text, we also reportthe root-sum-squared misfit in brackets.It is important to note that we are calculating effective pressure as N = hiρig−Pw where hi is ice thickness, ρi ice density (910 kg/m3), g is the acceleration ofgravity (9.8 m/s2), and Pi is the water pressure measured at the bottom of the bore-hole. However, in the case of the effective pressure used in Fig. 4.22d, the corre-155Figure 4.22: Mean daily speed as a function of: (a) Mean daily pressurewithin connected boreholes. (b) Daily pressure standard deviationwithin connected boreholes. (c) positive degree day (PDD). (d) Em-pirical function of effective pressure proposed by Jansson [1995]. Dif-ferent colours represent different periods of time along the 2015 meltseason: April 1st to July 1st (blue), July 1st to August 18th (red), andAugust 18th to October 31st (green). Each panel contains the best lin-ear fit (black line) for which the coefficient of determination R2 is alsopresented.sponding Pw is a mean value over multiple boreholes, and we do not know whichhi should be used in the computation of N for a spatially averaged Pw. Note that thehi to be used in these cases is not necessarily the average hi over all the boreholes.Therefore, we have treated hi as a free parameter that has been minimized togetherwith the linear fit parameters. We used the same approach for all the linear fits tofunctions of N shown in this section.Each point in Fig. 4.22 represents a day, and the colors distinguish betweendays before the main pressure drop in July 1st (blue), days between that point andthe end of diurnal pressure variations in August 18th (red), and later days (green).Note that days from each one of these periods form distinct clusters in Fig. 4.22a156and d. In particular, if we consider the whole season, Fig. 4.22a suggests the thathigher water pressures are associated with lower surface speeds, contrary to thecommon understanding of basal friction and all the friction laws presented in Sec-tion 4.1.This inconsistency highlights the fact that the relationship between speed andwater pressure changes dramatically over the season and we cannot treat early,mid, and late season in the way. Notably, if we look only at days in the middle ofthe season (red dots), we observe the expected relationship of high water pressureleading to high surface speed, but this is not true elsewhere, explaining why themean pressure turned out to be the worst predictor of surface speed in Fig. 4.22,with R2 = 0.23 (17.01 cm/day). As we had anticipated, PDD is also a poor predictorof speed, with R2 = 0.27 (16.57 cm/day). The standard deviation of the waterpressure and the model by Jansson [1995] result in a slightly higher R2, but the fitis still very poor, with R2 = 0.39 (15.13 cm/day) and R2 = 0.32 (15.98 cm/day)respectively.In Fig. 4.21 we can see that the fraction of connected boreholes is also relatedto surface speed. Therefore, we adapted the empirical model by Jansson [1995] toquantify how well we can predict the surface speed from measurements of the ef-fective pressure and the connected fraction of the bed. We define the first modifiedversion as< us >day= KN < N−b >day (4.9)where, in contrast to the original model, the exponentiation is applied on N beforecomputing the mean of the day, as in general < X >n 6=< Xn >. This differencebecomes especially relevant when the exponent is significantly larger than one. Inthat case, high effective pressures can have a disproportionate effect on the meanspeed over a day, even if such pressures happen only over a relatively small period.This model is shown in Fig. 4.23a, were the best fit was found for b = 0.79,resulting in R2 = 0.32 (15.96 cm/day). Fig. 4.23b show the residual plot for this fit,where the substantial structure in the distribution of the residuals suggests that themisfit is not due to random error but to the unsuitability of this model to our data.Figure 4.23c–d use an analogous model that describes 〈us〉day as a function of157Figure 4.23: Best linear fit (left panels) and residual plots (right pan-els) of surface speed and three different empirical models: (a–b)〈us〉day ∝ 〈N−b〉day, where the best fit was found for b = 0.79 andhi = 77.1 m. (c–d) 〈us〉day ∝ 〈Φa〉day, were the best fit was found fora = 0.22 and hi = 89.0 m. (e–f) 〈us〉day ∝ 〈Φa〉day 〈N−b〉day, were thebest fit was found for a = 1.24, b = 0.10, and hi = 77.1 m. Differentcolours represent different periods of time as described in Fig. 4.22.Each panel on the left column contains the best linear fit (black line)for which the coefficient of determination R2 is also presented.158the connected fraction of the bed instead of effective pressure:〈us〉day = KΦ〈Φa〉day (4.10)where KΦ is an empirical constant, a is an empirical exponent, andΦ is the fractionof connected boreholes. The best fit was found for a = 1.05 , reaching R2 = 0.62(11.93 cm/day). The connected fraction of the bed performed significantly betterthan effective pressure as a predictor of surface speed. Also, the residual plotshows a more random distribution of residuals, suggesting that the misfit could bein large part due to measurements errors. Note that our clustering technique givesus a measure of Φ over a time window of six days, with this time window movingforward in steps of three days. Therefore, taking the mean over a day in our caseis irrelevant, as we compute only one value of Φ per day by linear interpolation ofour coarser resolution samples.Finally, Fig. 4.23 uses an extended model that combines contributions by Φand N, that we have defined as:〈us〉day = K〈Φa〉day 〈N−b〉day (4.11)where K is an empirical constant, and a and b are empirical exponents. The bestfit to this model was found for a = 0.95 and b =−0.08. Interestingly, the additionof the dependence in N did not improve the quality of the fit, which as before, hasR2 = 0.62 (11.94 cm/day). Note that contrary to what we would have expectedfrom the friction laws described in Section 4.1, the parameter b is in this case neg-ative, resulting in a positive exponent for N. However, we refrain from interpretingthe sign of b, given that its absolute value is very small and the data misfit is large.Next, we look into sub-diurnal time scales using the speed recovered by thesidereal reconstruction method. Fig. 4.24a shows the sub-diurnal surface speedvariations reconstructed over 40 days during the summer of 2014 (blue). For com-parison, it also presents the multi-day speed record derived from static solutions(dotted black line) and an analogous speed record computed from the reconstructedspeed using position averages over one sidereal day (dotted green line). Addition-ally, to study the relationship between these short timescale speed changes and ourborehole observations, Fig. 4.24b shows the normalized mean pressure within con-159Figure 4.24: (a) Sidereal reconstruction of the speed of GPS tower R22C18 over 40 days between July 20th and August29th 2014 (blue) using λ = 2.5× 109. The red line shows the reconstruction for the same GPS tower shownin Fig. 4.18 for the period between August 3rd and 17th 2014 using the same λ value. The black dotted lineshows the multi-base static solution shown in Fig. 4.13, and the green doted line shows an analogous speedrecord derived from the reconstructed data. (b) Normalized mean pressure of all sensors hydraulically connectedduring the period (blue), and for the main mechachanical cluster identified during the period (red). The lightblue background represent periods with fresh snow cover.160nected boreholes during the period (blue), and the normalized mean pressure of themain mechanical cluster identified during that period (red).The reconstructed speed is consistent with the multi-day speed record derivedfrom static GPS solutions (see dotted lines in Fig. 4.24a). However, the recon-structed sub-diurnal speed variation seems to be unrelated to other parameters, suchas the pressure in connected or mechanical boreholes, and surprisingly, it lacks aclear diurnal cycle.4.4 Discussion4.4.1 South Glacier surface speed recordIn general, static GPS solutions achieve higher accuracy than kinematic solutionsbecause they solve for a single position using a large number of observations ac-quired over multiple hours. With this purpose, the processing assumes that the GPSreceiver was stationary during the data acquisition. Therefore, if a moving receiveracquires the observations, that assumption is violated, and this gives rise to pro-cessing artifacts that worsen as the speed of the receiver increases. At high enoughreceiver speeds, these artifacts can outweigh the accuracy gain of the static pro-cessing, as would be the case in typical GPS applications such as people or vehicletracking.However, these artifacts are minimal and often neglected when computing theposition of extremely-slow-moving objects such as tectonic plates (<0.05 cm/day),for which static GPS processing is the preferred approach. In general, this approachis not recommended for glaciers [King, 2004]. However, to measure the velocityat diurnal timescales of slow-moving glaciers such as South glacier, we have foundthat a static approach yield betters results than using kinematic solutions.This advantage arises from the large error that multipath induces on the kine-matic solutions in glacier settings, and from the small displacements of the glacierover one sidereal day. Such small displacements allow us to compute static GPSsolutions over a whole sidereal day without the generation of significant process-ing artifacts. At the same time, these solutions are favoured by the partial can-cellation of multipath-induced errors, thanks to the repeatability of multipath over161one sidereal day. Also, position biases that result from the static processing of amoving object cancel out in the velocity computation. Therefore, when we calcu-late the velocity from static GPS solutions computed over one sidereal day we canachieve better accuracy than using kinematic solutions, provided that the speed ofthe glacier is small enough, which turns out to be the case for South Glacier, wherespeeds are typically below 10 cm/day in the study area.As described above, the primary concern of using static processing solutions tocalculate the speed of a glacier is the generation of spurious periodic variations inthe computed positions, as these spurious variations could create significant speedartifacts. These spurious variations arise from the effect of the glacier motion onthe in the ambiguity resolution performed during the static GPS processing. Forthe case of South Glacier, we estimated that the upper limit of the relative error inspeed is of a 10%.King [2004] performed a sensitivity analysis on synthetic data to isolate andstudy the artifacts that appear in static solutions of moving objects. In his analysis,he computed 1-hour static solutions of a receiver moving at a speed of 1 m/day,which results in a displacement of 4.2 cm over the solution period. He found thatthe solutions contained biases and spurious oscillations that were proportional tothe speed of the object. The biases were also different between East and North co-ordinates and were influenced by the direction of the motion. For a glacier movingin the north-south direction, there were no biases in the N coordinate and 5-10%bias in the E coordinate. Spurious periodic oscillations were estimated to have anamplitude of about 5% of the total speed.At South Glacier, the displacement over one ART is typically between 3 and10 cm. These displacements have the same order of magnitude as those used inthe sensitivity analysis by King [2004]. Thus, our upper limit estimation for theamplitude of spurious periodic oscillations of 10% is consistent with the 5% foundin that sensitivity analysis. Such spurious variations with a maximum amplitudeof a 10% of the magnitude of the speed would produce in our speed time series amaximum speed error of around 1 cm/day in a worst-case scenario where the periodof the spurious variations is a semi-integer multiple of the interval over which wecomputed the speed. Such errors would not affect signicantly the main features ofthe speed records presented, and the actual errors are likely to be smaller than this162upper limit estimate. Note that the period of these spurious variations is unknownto us and unrelated to the multipath repetition period.Even at slow-moving glaciers such as South Glacier, static solutions cannotbe used to compute sub-diurnal speed variations, as this would require solutionsover shorter time periods that suffer a significant increase in multipath errors aswell as a reduced accuracy due to the smaller amount of raw data used. Also,when positions are differentiated over small periods of time, any errors will besignificantly amplified.Kinematic solutions are also affected by strong multipath errors that dominatethe signal. These errors again make it impossible to detect small diurnal speed vari-ations, even if smoothing filters are applied. The preeminence of multipath errorsin kinematic solutions at South Glacier arises from both the strength of reflectedsignals (which is common in glacier settings), and the small amplitude of the speedvariations. At South Glacier, these speed variations are expected to be of just a fewcentimeters per day, relative to a base speed of only 3–6 cm/day.We observed that the error induced by multipath consists mostly of a peri-odic high-frequency position bias with a repetition period close to one siderealday. Therefore, using solutions at the high-frequency rates that can be achieved bykinematic processing, we can get the multipath bias to cancel out when we computedisplacements over an interval matching the repetition period of the multipath.It is worth noting that due to the periodicity of the multipath, when usingkinematic solutions of slow-moving glaciers, particular care has to be taken whenchoosing the interval over which the speeds are computed. The periodicity, re-peatability, and strength of multipath can generate significant spurious speed peaksif speeds are computed over a day (rather than a sidereal day, only 4 minutesshorter) or any other arbitrary period of time. Such artifacts can be removed orsubstantially reduced if the speeds are computed over one sidereal day or one typ-ical ART (see Fig. 4.9). As the multipath-induced biases can change significantlyover short periods of time, the difference of just 4 minutes between one day andone sidereal day is enough to produce a much more effective multipath cancella-tion, thus significantly reducing the multipath-induced arctifacts in the speed timeseries. Similarly, even the 8 seconds difference between one sidereal day and onetypical ART can generate additional improvements in the speed record.163Synthetic experiments suggest that the sidereal reconstruction method we haveproposed to exploit the repeatability and periodicity of multipath can effectivelyrecover sub-diurnal motion signals from GPS data strongly influenced by multi-path. When applied over real data, the reconstructed speeds at South Glacier areconsistent with each other when computed over different time intervals or differentGPS towers. The reconstructions are also consistent with the speed derived fromstatic GPS solutions differenced over a single day (See Figs. 4.16 and 4.24).However, the sidereal reconstruction of the speed is sensitive to the choice ofthe regularization parameter λ . While the lack of calibration data and the char-acteristics of the L-plot hinders the recovery of the absolute amplitude of speedvariations, their timing and sign can be recovered reliably as these properties donot change over broad ranges of λ .4.4.2 Glaciological controls on surface speedBasal sliding seems to be significant at South Glacier. The numerical model byFlowers et al. [2011] suggests that basal motion in the study area accounts for75–100% of the total surface motion. Although we cannot measure basal slidingdirectly, we can estimate a lower bound to its contribution to the overall glaciermotion if we assume that the minimum winter speed over the study period of 4.85cm/day corresponds to the upper bound of the contribution of internal deformationto surface speed. In such case, our long-term speed record shows that the speeds inexcess of that value can explain about 18% of the total annual motion, a value thatwould constitute an empirical lower bound for the contribution of basal sliding atSouth Glacier.On annual timescales, we see a negative correlation between the minimum win-ter speed value and the total cumulative PDD of the previous summer (Fig. 4.19,blue dots), meaning that hot summers lead to slower winter speeds. This nega-tive correlation have been observed as well in Greenland [Sole et al., 2013], andis consistent with the idea that more abundant meltwater supply produces a higherdegree of channelization, which in turn is associated with higher effective pres-sure, increased basal friction, and slower sliding speed. Note that while a highlychannelized drainage system is associated with efficient connections, these are re-164stricted to very narrow pathways. Therefore, such channelized drainage systemsare associated with a small connected fraction of the bed (Φ).In contrast, there no significant correlation between summer PDD and meanannual or mean summer speeds (Fig. 4.19, orange and red dots respectively). Thesummer data suggest relatively low speed both for cold and hot summers (lowand high annual PDD), and higher speeds in summers with moderate temperatures.This pattern would be consistent with cold summers not developing an extensivedistributed drainage system, associated with high water pressure and a large con-nected fraction of the bed. On the other hand, low mean speeds in hot summersare consistent with a fast transitioning from a distributed drainage system to a low-pressure channelized one, where the connected areas are small and restricted to theproximity of a few well-developed channels.At a daily timescale, the relationship between PDD and surface speed has aquick transition around the time of the spring speed-up event (see Fig. 4.20), show-ing a very weak correlation with speed before that event and a significant correla-tion afterwards. This transition is probably associated with the water saturationof the snowpack and the initiation of surface meltwater delivery to the subglacialdrainage system. However, even after the spring speed-up, PDD remains a poorpredictor of surface speed (see also green and red dots in Fig. 4.22c). This weakcorrelation between PDD highlights the role of the subglacial drainage in modulat-ing how surface meltwater supply affects basal sliding. PDD probably constitutes agood proxy of meltwater supply after the water saturation of the snowpack, but itprovides no information about the configuration of the subglacial drainage.The importance of effective pressure in controlling basal sliding is well estab-lished [Lliboutry, 1958, Hodge, 1979, Iken and Bindschadler, 1986, Fowler, 1987,Schoof, 2005, Gagliardini et al., 2007]. However, we observe that the relationshipbetween effective pressure and speed significantly changes throughout the meltseason. In particular, the correlation between effective pressure and surface speedthat we expected based on the friction laws described in section 4.1, is only ob-served after the noticeable pressure drop that took place at the beginning of July2015 (see Fig. 4.21b). This pressure drop is likely to correspond to the start thechannelization of the subglacial drainage system (see Chapter 3). However, beforethat pressure drop, the observed acceleration of the glacier does not seem to re-165spond to changes in the effective pressure. In particular, the main speed-up eventduring the second half of June is not associated with a corresponding increase inobserved water pressure (Fig. 4.21).This lack of correlation between effective pressure and speed during the earlymelt season suggests that other factors are responsible for the acceleration of theglacier. In particular, the extent of connected areas of the bed shows a strong cor-relation with surface speed (Fig. 4.21c). Note that an increase in the extent ofconnected regions of the bed and the corresponding reduction of disconnected re-gions could affect basal sliding by changing contact area between the ice and thebedrock or by connecting formerly isolated cavities. In turn, isolated cavities candampen speed variations as a consequence of their opening rate being proportionalto basal speed, while they keep a fixed volume [Iken and Truffer, 1997, Bartholo-maus et al., 2011]. Consequently, the effective pressure within isolated cavitiesincreases when the glacier accelerates and decreases when it decelerates.While this speed dampening mechanism could explain the observed correlationbetween bed connectivity and surface speed, we have not been able to identify thisbehaviour directly in any of our boreholes. Nevertheless, several factors couldmake this detection difficult, such as limited cavity size, small pressure variations,or pressure variations hard to distinguish from the seemingly random variationsobserved in the many boreholes that are not members of any identified cluster. Anaccurate measure of sub-diurnal speed variations would be necessary to know howthe water pressure record in such cavities would look.Alternatively, it might be precisely this isolated cavity behaviour what we ob-serve in mechanical clusters. In this case, one subcluster would be sampling dis-connected cavities, and the other would be sampling isolated water pockets thatexperience normal stress transfer from such cavities. However, we have no proofof this hypothesis, and an independent measure of basal sliding would be requiredto test it.Considering the well-established relationship between effective pressure andbasal sliding, Jansson [1995] proposed a power-law empirical relationship betweeneffective pressure and surface speed as described in Eq. 4.8. Using this empiricalmodel on the 2015 data at South Glacier, we obtain a poor fit with R2 = 0.31(16.2 cm/day) if we use the exponent b = 0.4 that was found to fit the data at166Storglacia¨ren and Findelengletscher (see Fig. 4.22d). Allowing b to vary and usingthe modified model of Eq. 4.9, we found that b = 0.79 provides the best fit to ourdata. However, this fit results in a similarly poor fit of R2 = 0.32 (16.0 cm/day), asshown in Fig. 4.23a.Interestingly, there is a significant amount of structure in the residual plot as-sociated to this model, suggesting that the poor fit is not due to random errors, butinstead due to the unsuitability of the model to our dataset. The distinct segmen-tation of values observed in early, middle, and late season also explains the rela-tively good performance of this model found by Jansson [1995] at Storglacia¨renand Findelengletscher, where the pressure data was collected in a more limitedperiod during the melt season. For Storglacia¨ren the data was collected betweenJuly 14th and August 18th 1987 and between May 15th and June 20th 1982 forFindelengletscher.We have tested the capacity of a similar empirical power-law model (see Eq.4.10) to predict surface speed based on our measure of the connected fraction ofthe bed. This model fits the data significantly better than Eq. 4.9, resulting inR2 = 0.62 (11.9 cm/day), as shown in Fig. 4.23c. This improved fit, combined withthe absence of significant structure in the residual plot, suggests that connectivity isa better predictor of surface speed than effective pressure. This result is remarkable,especially if we take into consideration that the uncertainties in our measurementsof connectivity are likely to be much larger than those of pressure.We can attribute this difference in accuracy to the sparsity of our boreholenetwork and the limitations of our clustering methodology. First, the temporalcoverage of our connectivity measure is limited, because our method requires a di-urnally variable meltwater supply to identify hydraulic connections. Therefore, wecannot measure connectivity very early or very late in the melt season. Similarly,the temporal resolution of our connectivity measure is limited as well, because itis constrained by the length of the time window used in our clustering algorithm.In contrast, we were able to measure pressure year round and at very high tem-poral resolution (2–20 minutes). Additionally, our connectivity measurement isvery sensitive to the number and distribution of our boreholes, while the pressurewithin hydraulically connected regions of the bed does not change as much acrossthe study area, especially in periods like the 2015 melt season when most of the167connected boreholes form only one large cluster.The two power-law empirical models tested above suggest that both connec-tivity and effective pressure can be used to predict surface speeds. Connectivityis a much better predictor than effective pressure, and its performance is sustainedthroughout the melt season. Nevertheless, we expected that the empirical rela-tionship described in Eq. 4.11 with dependence on both connectivity and effectivepressure would have allowed a better fit to the data. This relationship also containstwo exponents (a and b) instead of one, giving us one additional free parameter tominimize the data misfit. However, there was no improvement at all and the expo-nent associated with the effective pressure was an order of magnitude smaller thanthe one associated with the connected fraction of the bed (-0.08 and 0.95 respec-tively, see Fig. 4.23e). The small value of the exponent suggests that the effectivepressure within the connected regions of the bed does not play a significant role inmodulating basal sliding, which seems to respond more to the extent of the con-nected drainage system than its effective pressure.Alternatively, a power-law might not be the appropriate functional relationshipbetween effective pressure within the connected drainage system and basal sliding,as is generally assumed by proposed fiction laws.4.4.3 Sub-diurnal speed variationsSub-diurnal reconstructed speed variations do not display a diurnal cycle as we ex-pected based on the friction laws presented in section 4.1. The absence of a diurnalperiodicity immediately raises concerns about the accuracy of the reconstructedspeed. Unfortunately, we cannot dispel these concerns due to the lack of validationdata. Nevertheless, there are valuable insights to learn from this reconstructionregardless of whether it was successful or not in recovering the sub-diurnal speedvariations.If the reconstructed speed is a good representation of the motion of the glacier,the lack of a diurnal cycle would suggests that instead of smooth variations con-trolled by effective pressure, basal sliding is controlled at this time scale by anunidentified stochastic process, as could be the case if basal sliding follows a stick-slip motion regime. However, if basal sliding follows such a stick-slip regime,168the regularization we used in the sidereal reconstruction would not be adequate,because it strongly penalizes sudden accelerations. In such a scenario, our recon-structed speed would not be valid.Alternatively, if some other process controls sub-diurnal speed variations in away smooth enough that our regularization is suitable for the problem, then ourreconstructed speed would be reliable, and it would have an interesting glaciologi-cal implication: there would not be a clear association between speed and effectivepressure. In this scenario, one hypothesis that could explain such sub-diurnal speedvariations is that they are associated with short timescale changes in connectivity,such as those observed in switching events. These switching events could be inturn affected by changes in effective pressure and basal speed, but not in a purelydeterministic way. Interestingly, this hypothesis is not limited to the scenario inwhich basal sliding varies smoothly. Similarly, changes in connectivity could alsoplay an essential role in controlling the dynamics of a stick-slip regime.Another possibility to explain the lack of a diurnal cycle in the reconstructedsub-diurnal speed record is that the real motion presents a very regular diurnalcycle with constant amplitude. In such cases, our synthetic experiments failed torecover the true motion signal. If the regular pressure variations observed withinmechanical clusters are somehow associated with basal speed, they would suggestthat speed variations are very regular but abrupt and the sidereal reconstructionmethod would not be well suited to recover them.Finally, we have to consider the possibility that the sidereal reconstructionmethod failed due to other reason, such as the multipath not repeating as assumed,the surface changing faster than we expected, snow events having a large influencein the solution, or some other unforeseen reason.In summary, there are four main scenarios regarding the sidereal reconstructionoutput: it succeeded, it failed due to a stick-slip basal motion regime, it failed due tothe regularity of speed variations, it failed due to rapidly changing multipath. Notethat the first three of these scenarios would suggest that sub-diurnal basal slidingvariations are not a smooth function of effective pressure within the connectedfraction of the bed.Finally, we need to acknowledge that with or without the existence of a stick-slip regime, our lack of validation data leaves open the possibility that the sidereal169reconstruction does not accurately represent the motion of the glacier. To addressthis possibility, we plan to do further work to validate the reconstructed motionwith an independent measure of the position such as those derived from total sta-tions. Alternatively, we are planning to test the method in a controlled experimentalsetup, such as platform surrounded by ice that we can move at will. Ultimately, theideal approach to remove multipath errors would be to consider this effect withinthe GNSS processing software (GAMIT or RTKLIB in this case), and correct for thecontribution of each satellite individually, in which case we can calculate the ARTeasily. Also, the SNR could be used to discard the data from satellites sufferingfrom heavy multipath effects. In the static approach, we use the data from approx-imately 43,000 epochs to solve for one position: Easting, Northing, and Elevation(in addition to processing unknowns such as ambiguities, ionospheric delays, andclock corrections), and in the kinematic approach we solve for 43,000 positions.Slow-moving objects such as South Glacier could benefit from an intermediate ap-proach that assumes a constant velocity during the observation period and solvesfor a linear trajectory: this means the method has to solve for initial position andmean velocity. However, standard GNSS processing software does not offer suchintermediate approach.4.5 ConclusionsWe have found that the static GNSS processing approach yields better results thanthe kinematic approach to measuring the velocity at diurnal timescales of a slow-moving glacier such as South glacier. In this case, the advantage of the staticprocessing is a result of the small displacement observed at the glacier surface overtime intervals similar or longer than the repetition period of multipath effects, thatis roughly equal to one sidereal day (23h 56m 4.1s).Nevertheless, the static processing of moving objects results in processing ar-tifacts that are proportional to the speed [King, 2004]. At South Glacier, we ex-pect the errors in the speed record associated with these processing artefacts tobe smaller than 1 cm/day. This error estimation is valid for the mean speed com-puted over one sidereal day, and the error would grow rapidly for shorter intervals.Therefore, this static processing approach does not allow the computation of accu-170rate sub-diurnal speed variations.To reconstruct sub-diurnal speed variations, we have proposed an inverse modelapproach termed “sidereal reconstruction” that exploits the repeatability and peri-odicity of the multipath bias observed in kinematic GPS solutions. We tested thismethod using synthetic data, and the results suggest that we can successfully re-cover the timing and direction of sub-diurnal speed variations. However, recover-ing the absolute amplitude of such variations requires the calibration of the methodagainst validation data to establish the magnitude of the regularization parameterλ .Using the static processing approach we computed an 8-year-long surface speedrecord at South Glacier. This speed record suggests that basal sliding is responsiblefor more than 18% of the year-round glacier motion, and numerical models indi-cate that the actual contribution is significantly higher, most likely between 75%and 100% [Flowers et al., 2011].On annual timescales, we found that hot summers (i.e. high accumulated PDD)are associated with low minimum speeds during the following winter. This corre-lation is consistent with the idea that a more abundant meltwater supply producesa higher degree of channelization, which in turn is associated with enhanced basalfriction. In contrast, there is no significant correlation between annual accumulatedPDD and mean annual or mean summer speeds (see Fig. 4.19). However, the re-lationship between accumulated PDD over a year and mean summer speeds showshints of low speeds being associated with relatively cold and hot summers, whilehigh summer speeds would be associated with summers with intermediate temper-atures. If confirmed, this pattern would be consistent with cold summers failingto develop an extensive connected distributed drainage system, and hot summerstransitioning fast from such a drainage system to a channelized one.If we look at diurnal time scales, the correlation between PDD and speed be-comes significant only after the spring event speed-up (see Fig. 4.20). This tran-sition is probably associated with the water saturation of the snowpack and theinitiation of surface meltwater delivery to the subglacial drainage system. In spiteof this significant correlation, we observe a wide range of surface speeds associ-ated with the same diurnal PDD values. This variability is likely to be the result ofa gradual modification of the subglacial drainage system, that in turn can changes171how the surface meltwater supply affects basal sliding.In general, our observations suggest that basal sliding is controlled by the ex-tent and effective pressure within two distinct regions of the bed: disconnectedareas, where water conduits and reservoirs are hydraulically disconnected from thesurface, and the “active drainage”, where conduits and reservoirs are hydraulicallyconnected to the surface. This active drainage includes all the conduits responsiblefor the meltwater transport through the glacier.At a glacier-wide scale, the relationship between effective pressure within ac-tive drainage and the basal sliding is more complex than suggested by friction laws.In particular, because this relationship seems to change throughout the year, withthe main change taking place at a point that we have associated with the initiationof the channelization of the drainage system (the beginning of July for the 2015melt season). Before this point, the effective pressure seems to have no significantinfluence on basal sliding, but afterwards, the changes in observed speed are con-sistent with a power-law functional relationship with the effective pressure withinthe active drainage.In contrast, we observe a significant correlation between surface speed and theextent of the active drainage system throughout the melt season, with the connectedfraction of the bed providing a predictor of surface speed significantly better thanthe effective pressure within the active drainage. Using a power-law approachsimilar to the one proposed by Jansson [1995] (Eq. 4.9), the effective pressurewithin the active drainage can only explain 32% of the variance observed in thesurface speed (R2 = 0.32). Meanwhile, a similar model based on the connectedfraction of the bed (Eq. 4.10), can explain 62% of the surface speed variance (R2 =0.62). Interestingly, adding a dependence on effective pressure to this empiricalmodel (Eq. 4.11) produces no increase in the performance of the model.The influence of the connected fraction of the bed over basal speed might arisefrom the effect of disconnected cavities, which can restrain glacier forward move-ment. This restriction would be a consequence of the opening rate of the cavitiesbeen proportional to basal speed, while they keep a fixed volume [Iken and Truf-fer, 1997, Bartholomaus et al., 2011]. Consequently, the effective pressure withinisolated cavities would increase when the glacier accelerates, leading to strongerbasal friction. However, among observed boreholes, the only ones that could be172capturing the water pressure within disconnected cavities are the members of me-chanical clusters. Nevertheless, we lack the independent measure of basal speedthat would be required to test this possibility.The reconstructed sub-diurnal speed variations do not display a diurnal cycleas we expected, neither a significant correlation with effective pressure within con-nected or mechanical boreholes. This unexpected behaviour raises concerns aboutthe validity of our regularization choice and other assumptions of the sidereal re-construction method. There are four main scenarios regarding the sidereal recon-struction output: it succeeded, it failed due to a stick-slip basal motion regime, itfailed due to the regularity of speed variations, or it failed due to rapidly changingmultipath. While we cannot distinguish between these options without an indepen-dent measure of surface or basal speed, it is important to note that the first threescenarios would suggest that sub-diurnal basal sliding variations are not a smoothfunction of effective pressure within the connected fraction of the bed.Further work is required to test, calibrate and validate the sidereal reconstruc-tion output. Ideally, a new GNSS processing method should be developed for theparticular case of slow-moving objects, incorporating the capability to solve for themean speed during the observation period and to deal with multipath effects.173Chapter 5ConclusionUsing the borehole water pressure dataset of South Glacier, we were able to iden-tify the most common phenomena within these boreholes and make informed infer-ences about the underlying physical processes and the evolution of the subglacialdrainage system. While none of the observed phenomena was new to the glaciolog-ical literature, we have been able to place them in the context of a comprehensivedescription of the subglacial drainage system and its evolution throughout the year.Our most significant contribution to the field was to provide strong support tothe existence of hydraulically disconnected regions of the glacier bed, establishingtheir spatial extent, endurance in time, and most importantly, the significant influ-ence they have on the basal speed. Notably, adding these disconnected areas to ourconceptual understanding of subglacial hydrology allows us to explain most of theborehole observations within a consistent framework. Previously, several of theseobservations were at odds with the available conceptual and numerical models ofglacier hydrology.In particular, these conflicting observations were the development of widespreadareas of high water pressure during winter [Fudge et al., 2005, Harper et al., 2005,Ryser et al., 2014a, Wright et al., 2016], boreholes exhibiting pressures exceedingthe overburden [Gordon et al., 1998, Kavanaugh and Clarke, 2000, Boulton et al.,2007], large pressure gradients over short distances [Murray and Clarke, 1995, Ikenand Truffer, 1997, Fudge et al., 2008, Andrews et al., 2014], sudden reorganisation174of the drainage system [Leb. Hooke and Pohjola, 1994, Gordon et al., 1998, Ka-vanaugh and Clarke, 2000], boreholes exhibiting anti-correlated temporal pressurevariations [Murray and Clarke, 1995, Gordon et al., 1998, Andrews et al., 2014,Lefeuvre et al., 2015, Ryser et al., 2014a], and strong clustering of the pressurevariations observed in particular subsets of boreholes.We found that distinct regions along the bed seem to be either hydraulicallywell-connected or disconnected, with the transition between these to stages oftenhappening abruptly through a “switching-event”. Spatially, the transition betweenwell-connected and disconnected areas also appears abrupt at a sampling resolutionof 15 m, which corresponds to the smallest spacing between our boreholes. Ingeneral, the hydraulic diffusivity distribution at the bed seems to have significantfine structure at spatial scales shorter than 15 m, explaining the notable lack withinour dataset of pressures signals that show a clear diffusive behaviour.The basal water pressure in winter is generally high and close to overburden.These high pressures and the lack of correlation between the pressure variations ofdifferent boreholes suggest that in winter the subglacial drainage system is frag-mented into many small subsystems, most of them hydraulically isolated from thesurface. These high water pressures in winter (low effective pressure) are generallyassociated with low surface speeds, seemingly contradicting the accepted idea thatlow effective pressure leads to reduced basal friction and increased basal sliding.However, there would be no such contradiction if we consider that disconnectedregions of the bed can restrain glacier movement. This restriction would be a con-sequence of the opening rate of cavities being proportional to basal speed, whilethey must keep a fixed volume. Therefore, any acceleration of the glacier wouldlead to increased effective pressure within disconnected cavities [Iken and Truffer,1997, Bartholomaus et al., 2011].This winter regime changes rapidly in spring or early summer, as a result ofa dramatic increase in meltwater supply to the subglacial drainage system. Thisincrease is a consequence of the water saturation of the snowpack under conditionsthat support continued surface meltwater production. This abundant meltwater sup-ply results in the activation of an extensive and well-connected distributed drainagesystem, where the majority of boreholes show similar diurnal pressure variationsand experience modest water transport (see section 2.2.1). This activation is prob-175ably accomplished by overpressurization and hydraulic jacking. Further increasesin meltwater supply to the distributed drainage system, lead then to low effectivepressures and a rapid reduction of the disconnected areas of the bed. These twoprocesses contribute to the high surface speeds observed during this period andproduce the spring event speed-up that characterizes the annual speed cycle. Whilesome areas of the bed become activated by the meltwater supply, others remaindisconnected year-round. The distribution of these areas could be controlled by thelocation of the input points of meltwater supply.During this period, high water pressures and cavity growth associated withhigh basal speed, lead to the onset of viscous heat dissipation as the dominantterm for conduit growth. Over time, water transport becomes concentrated in someareas, and the drainage system becomes increasingly channelized. The increasein effective pressure associated with channelization leads to the progressive shut-down of the activity in the surrounding drainage system. This shut-down might beaccelerated by exceptionally large effective pressures produced in the vicinity ofchannels as a consequence of bridging stresses [Lappegard et al., 2006].Eventually, if the meltwater supply continues, the water flow ends up focusedin R-channels surrounded by a narrow distributed drainage system that carries rel-atively low water fluxes, and beyond this active part of the drainage, the bed be-comes hydraulically disconnected from the surface. Consequently, during long andhot enough summers, most of the bed can become disconnected. Our observationssuggest that even during periods where the drainage system was strongly channel-ized, all our boreholes failed to intersect the narrow R-channels. However, in oneinstance we were able to confirm the existence of a channel from direct observationin a borehole.The eventual complete shut-down of the connected drainage system at the endof the summer season is presumably the result of low water supply, often associatedwith the appearance of fresh snow on the glacier surface. The lack of meltwatersupply results in high effective pressures and low dissipation rates in channels,leading to the closure of all basal conduits unless they become hydraulically dis-connected.During the period of the melt season characterized by diurnal meltwater pro-duction, we were able to automatically pick clusters of boreholes based on the sim-176ilarities of their pressure response to surface meltwater supply. We classified theseclusters into two main types: hydraulic and mechanical. Both clusters types areoften composed of two subclusters of mutually anti-correlated boreholes. For me-chanical clusters, the two subclusters differ only in their phase, while in hydraulicclusters one subcluster shows higher mean water pressure and diurnal oscillationsof smaller amplitude. We refer to this subcluster as anti-correlated.We interpret correlated boreholes of hydraulic clusters as being hydraulicallyconnected to the surface meltwater supply, constituting the “active drainage”. Incontrast, anti-correlated boreholes correspond to disconnected areas of the bed thatdisplay small water pressure variations due to normal stress transfers associatedwith the pressure variations within the active drainage [Weertman, 1972, Murrayand Clarke, 1995, Lappegard et al., 2006].Boreholes in mechanical clusters are also disconnected from the surface melt-water supply, and their pressure variations are likely to be controlled by stresschanges associated with the glacier motion. We hypothesize that these boreholesmight be sampling disconnected cavities at the bed. In this case, their square-waveshaped pressure variations would be suggestive of a stick-slip motion regime. How-ever, we lack an independent measure of basal speed to test this hypothesis.Combining the conceptual model of the subglacial drainage evolution pre-sented above with standard friction laws (see section 4.1), and the dynamic effectof disconnected cavities, it is clear that basal sliding should be controlled by theextent of the active drainage and the effective pressure within it. However, theinfluence of the extent of the active drainage on the basal speed seems to be signif-icantly larger than the influence of its effective pressure.Using a power-law approach similar to the one proposed by Jansson [1995](Eq. 4.9), the effective pressure within the active drainage can only explain 32% ofthe variance observed in the surface speed. Meanwhile, a similar model based onthe connected fraction of the bed (Eq. 4.10), can explain 62% of the surface speedvariance. Interestingly, adding a dependence on effective pressure to this empiricalmodel (Eq. 4.11) produces no increase in the performance of the model.The relationship between effective pressure within active drainage and thebasal sliding is, at a glacier-wide scale, more complex than suggested by frictionlaws. In particular, because this relationship seems to change throughout the year,177with the main change taking place at a point that we have associated with the initi-ation of the channelization of the drainage system.These relationships between basal speed and the state of the subglacial drainagesystem rely on accurate measurements of surface speeds. In a very slow movingglacier such as South Glacier, the computation of speed variation at diurnal andsub-diurnal timescales stretches the limits of standard GNSS processing methods.To study the speed variations at daily timescales, we found that the static GNSSprocessing approach yields better results than the kinematic approach. In this case,the advantage of the static processing is a result of the small displacement observedat the glacier surface over time intervals similar or longer than the repetition periodof multipath effects, which is roughly equal to one sidereal day (23h 56m 4.1s).However, this static approach is not suitable to compute sub-diurnal speed vari-ations. With this particular purpose, we developed a method termed “sidereal re-construction” that exploits the repeatability and periodicity of the multipath biasobserved in kinematic GPS solutions. We tested this method using synthetic data,and the results suggest that we can successfully recover the timing and direction ofsub-diurnal speed variations. However, an independent speed measurement is re-quired to establish the magnitude of the regularization parameter λ , to test differentregularization terms, and to validate the output of the method.Nevertheless, the lack of a clear diurnal period in the reconstructed sub-diurnalspeed record suggests that basal sliding variations are not a smooth function of ef-fective pressure within the active drainage. This result, combined with the pressurevariations observed within mechanical clusters, suggest that basal sliding at SouthGlacier might consist of discrete “stick-slip” motion events.Our observations support some of the features shown by recent subglacialdrainage models [Schoof, 2010a, Hewitt, 2011, Schoof et al., 2012, Hewitt et al.,2012, Hewitt, 2013, Werder et al., 2013, Bueler and van Pelt, 2015, Downs et al.,2018, Sommers et al., 2018], such as the existence of a distributed drainage systemthat exists year-round and during the melt season it evolves into a progressivelymore channelized and focused system. However, the most notable differences arethe extremely heterogeneous distribution of diffusivity that our results suggest, theimportance of normal stress transfers, and the existence of disconnected areas. Wealso showed that the extent of the active drainage is – at South Glacier – the pa-178rameter that better explains the observed basal speed variations. In contrast, theeffective pressure within the active drainage seems to have a more limited effectthan previously thought.Our observations are also consistent with borehole data from other sites. How-ever, the density of boreholes at South Glacier has allowed us to identify the preva-lence of “switching events”, through which the drainage system focuses, and thedisconnected areas enlarge.We propose that the main shortcoming of current drainage models is their in-ability to account for the evolution of a disconnected component of the subglacialdrainage system. Aware of this shortcoming, Rada and Schoof [2018] have shownthat the ability to model disconnected areas can be incorporated in the current mod-elling framework as a percolation threshold, assuming that cavities only form aconnected system once they reach a critical size. This relatively minor modifica-tion of the existing framework provides a starting point for the development of anew generation of models, capable of reproducing borehole observations and ulti-mately provide a more complete and accurate description of the subglacial drainagesystem.Although the analysis presented here is can be improved, many outstandingquestions will not find a definite answer unless we carry on further field studies,sampling at higher spatial resolution over some areas, and incorporating new mea-surements. For this reason, we plan future work on the development of new meth-ods and instrumentation to achieve the following objectives:• Perform repeated tracer injection and detection at the bottom of boreholes.• Perform repeated slug tests at the bottom of boreholes.• Perform repeat photography at the bottom of boreholes.• Develop a GNSS processing method that can accurately solve for the initialposition and mean velocity of a slow-moving object and correct for the mul-tipath effect on the measured observables of individual satellites.• Develop a method to extend the life of sensor pods at the bottom of bore-holes.179• Develop a cost-effective way to deploy boreholes networks at higher spatialresolution and with measurements at multiple vertical levels.These improvements would allow us to study the three-dimensional structureof the glacial drainage system at higher spatial resolution, detect hydraulic con-nections directly, assess hydraulic isolation, and estimate the storage capacity ofsubglacial conduits. 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Surfacemelt-induced acceleration of greenland ice-sheet flow. Science, 297:218–222,2002.196Appendix ADigital sensor pods designMonitoring the subglacial environment is always a very difficult task. It involvesplacing instrumentation near the bed, which is almost always inaccessible, dark,unstable and subject to high pressures and strain rates. The glacier bed have beenaccessed in several places by natural tunnels or crevasses, but that is possible invery limited cases and even in such cases, they only allow access to the bed undershallow ice. In other instances, tunnels were excavated in the bedrock, but highcosts limit the number of areas that can be accessed in such a way. Also, none ofthese methods allow the access to permanently flooded areas of the bed, which arethe ones concentrating most of our interest.In order to gain access to the bed, boreholes need to be drilled, either withmechanical, steam, or hot water drilling methods. When boreholes stay open (nat-urally or artificially), the water level in them have been used as a measurement ofthe water pressure at the bed (Kamb et al. [1985]). However, the borehole wa-ter level may instead reflect the pressure within englacial conduits (Fountain andWalder [1998]) that could have little or no relation to the basal water pressure.Also, continuous measurements over long periods are difficult due to the boreholefreezing shout or closing due to ice creep.There is also several indirect measurement techniques that we will not coverin detail, based on the properties of the reflection of electromagnetic or acousticwaves on the bed. We will only mention that such methods have big footprints thatlead to low spatial resolution, and cannot recover some of the variables that influ-ence basal sliding, such as effective pressure, stress, strain rates, or basal speed.197Therefore, to have direct measurements at unbiased and representative pointsof the glacier bed, and over extended periods of time, it is necessary to drill bore-holes and place permanent instrumentation at or near the bed, which impose severaltechnical challenges.• High pressure: Areas of the glacier that are important in the overall dynam-ics are usually thick, typically between hundreds to thousands of meters,leading to very high pressures of 100–1000 PSI or more.• Communication: Data and power must be transferred trough the overlyingice, due to the long time and high costs associated with hardware retrievalfrom the bed.• High strain rates: Ice surface moves faster than the basal ice due to internaldeformation. Therefore, any cables connected to the instruments will stretchand eventually snap. Also, most of the internal deformation happen near thebed, exposing the instrumentation to very strong mechanical stress• Turbid environment: The aqueous media under the glacier is usually heav-ily laden with fine sediments (also know as “glacier flour”), rendering im-possible any optical observation.• Autonomy and reliability: With no access to the instrumentation after place-ment, it is impossible to service or replace the hardware. Therefore, all in-struments must be autonomous and reliable for all its expected operationallifetime.• Cost: Instruments are basically disposable with a short lifetime ranging fromweeks to few years (due mostly to snap cables and dead batteries if a wire-less approach is used). Therefore, a dense continuous monitoring networkrequires a large number of instruments, which need to be available at lowcost for the operation to be economically viable.• Environmental impact: Ultimately, instrumentation components will beincorporated into the glacier and the drainage system. Therefore, no toxicmaterials can be used neither in the hardware or its batteries if any.198The remainder of this section will describe the design of the digital sensor pods,presenting also some aspects of our analogous sensors that motivate our designdecisions.A.1 Main moduleThe main module consists on a circuit board (see Fig. A.1) that host all the elec-tronic components of the sensor pod. Including the microcontroller, sensors, andcommunications module. Figure 2.4c shows this module, and 2.4b shows the mod-ule in the final casing ready for deployment. The PCB was designed using theOpen Software EAGLE, and printed on a local facility.Figure A.1: PCB design of the top (upper panel) and bottom (lower panel),sides of the PCB board that host all the electronic components and sen-sors.199A.2 Instrument casingTo provide mechanical strength and waterproofing, all instrument were embeddedin clear encapsulating Epoxy compound (figure 2.4a–b). The encapsulation processis key for the reliability of the instrumentation, as bubbles or imperfections canquickly produce instrument failure due to water leakage.The inevitable damage in the cable jacket (due to internal ice deformation) canallow pressurized water to flow inside the cable jacket while the instrument is stillin operation. Water have even been observed to come out from the datalogger endof the cables in a few opportunities. To reduce the chances of water reaching theinstruments, an ”S” shape bend in the cable is added inside the epoxy and beforethe instrument, with the cable jacket stripped the whole length of the ”S” bend.Most transducers were equipped with a 010B Ray piston snubber, with the aimof protecting them from transient high-pressure spikes.A.3 Power and data linesInternal deformation of the glacier impose a limit on the life of any cable connect-ing the surface and the sole of the ice. In the other hand, wireless communicationsand its power requirements have a life limited by internal batteries capacity, whichcould easily be shorter that the life of a cable, and impose great environmentalconcerns depending on the battery chemistry used.On South Glacier, we have opted for the use cables for data and power, inparticular, a four AWG24 solid conductors with cross section of 0.205 mm2 andresistivity of 84.22 Ω/km, and with PVC jacket, in lengths varying between 60 and300 meters.Most of our sensors are based on unamplified ratiometric pressure transducers;that will produce an output voltage linearly proportional to pressure and input volt-age. With a standard input of 10 volts, they will produce 100 mV at full scale (200PSI for most of our sensors).A.3.1 Analog communicationWhen analog communication is used, the digital to analog converter (DAC) ofthe datalogger can directly read the unamplified millivolt output of the pressure200transducer (or conductivity or any other prove) and store it. The advantage ofthis approach is that only the pressure transducers are left under the ice, reducingthe overall cost of the instrumentation. The downside is that the resistivity of thecable becomes part of the calibration parameters, and it can be affected by cablestretching, cable corrosion and finally due to water bridging between cable threads.Stretching effects are small, as the resistivity increase inversely proportional to itsdecreasing cross section, the increase in voltage drop would be given byVDrop = R(IinVIORPPmax+ IOUT)(A.1)where R is cable resistivity, Iin the current in the input lines, VIOR the ratiometricfactor of the sensor, P is the pressure, Pmax the full range maximum pressure and Ioutis the current on the output lines. The left component inside the parenthesis is thecontribution of voltage drop in the input lines and the right one is the contributionof the output lines.In our case R = 26.8 Ω for a 300 meters cable, Iin = 338 µA at 5 volts supply,VIOR = 100, Pmax = 200 psi, Iout = 0.1 µA (it can vary depending on the internalresistance of the datalogger). That leads to a maximum voltage drop of 0.1 mVwhich would be included in the sensor calibration. Now if we consider that thecable can stretch to reduce its cross-section, assuming a maximum reduction of90% before it snaps, this would lead to and increase to 1 mV, which can still beconsidered small, producing a maximum error of a 2% at full scale. Corrosion ef-fects are much more difficult to quantify, but in extreme cases, you could expect apotentially large increase in the resistance. However, the effect of large increasesin resistance due to cable thinning or corrosion would produce a reduced ampli-tude pressure time series, and we do not commonly observe that kind of artifact,suggesting that if it happens it is a rare case.The biggest problem with analog communication comes when two or morecable jackets fail, allowing current transfer between input and output lines, whatwill very likely happen soon or later as the cable stretches. This can produce ahigh amplitude noise that completely overrides any pressure signal. Laboratoryexperiments have shown that resilient digital communication protocols are capableof transferring information reliably trough this kind of faulty lines.201A.3.2 Digital communicationTo eliminate the influence of the cable on the readings, and to be able to transferaccurate data trough faulty transmission lines, a microcontroller with analog dig-ital converter (ADC) and digital communication capabilities have to be includedin the down borehole instrument. The microcontroller will read the output of thetransducer and transmit it digitally to the datalogger on the surface. Therefore, thedata will be received unaffected, as long as the cables can support data and powertransfer.For this propose, for in the digital sensor pods we have implemented the espe-cially well suited RS-485 protocol. This is the standard communication protocolmost resilient to faulty transmission lines, being able to support data transfers overcable lengths of 1,200 metres and more. Our digital pods used the SP3485 IC forconverting the RS-232 serial stream output by the microcontroller into an RS-485data stream and them back into RS-232 on the datalogger side.A.3.3 Input/Output boardDigital data transmission is much more reliable than analogous transmission, also,multiple sensors can be communicated using the same transmission lines. How-ever, is due to ice deformation a cable snaps and the two lines short-circuit, datatransmission would become impossible for other sensors using the same lines.Such short-circuit in the power lines can also lead to the draining of the datalog-ger battery. Therefore, we designed and built an input/output board that allows toisolate the communication and power to each sensor pod and limit the maximumpower consumption. With this board, the datalogger can automatically turn ONand open the communication channel with each sensor pod when a measurementis required. Figure A.2 shows the circuit board design of the input/output board.A.4 Pressure sensorThe working principle of the pressure transducers is based on a stainless steel di-aphragm being deflected by pressure variations, which in turn deflects a diffusedsilicon sensing element. The piezoresistive properties of diffused silicon then allowtransforming pressure variations into a voltage signal.202Figure A.2: PCB circuit board design of the Input/Output boardOur analog pressure sensors use standard industrial Barksdale transducers (se-rie 422, Absolute or gauge, 200 psi) or Honeywell transducers (serie 19C, gauge,200 or 1000 psi). These units have standard excitation of 10 volts direct cur-rent (VDC) and 100 mV output (ratiometric) at full range. The actual excitationused is 5 VDC due to datalogger output constraints (Campbell CR1000, CR10Xand CR10). In contrast, our digital sensors use a Honeywell pressure transducermodel MLH250PSB01A, that produce an amplified output between 0.5 and 4.5VDC, that can be directly read using the 10 bit AD converter of the ATMega328Pmicrocontroller and transformed intro pressure values using calibration parame-ters. To reduce noise and spurious values, we perform 9 readings separated by 5ms, then, the median of these readings is reported as the pressure value.203A.5 Conductivity sensorThe conductivity of a liquid describe how well it conducts electricity. It is a prop-erty of the medium independent of the amount of liquid. When a conductivityprobe is used to measure the liquid conductivity it will in fact only measure theconductance between the probe terminals, which will of course depend on the con-ductivity of the liquid but also on the distance between both terminals and theactual geometry of the probe. Therefore for any given probe, a calibration processis needed in order to transform measured conductance in specific conductivity.The simplest way to measure the conductivity of a medium is to pass a electriccurrent trough it and measure the voltage drop, but this approach is problematicon liquid mediums and useless on aqueous solutions, because a direct current willchange the chemical composition and conductivity of the medium by water hydrol-ysis and other chemical reactions. Due to the nature of the ions produced by waterhydrolysis, this might also produce corrosion on the probe. The solution to thisis use a rapidly alternating current (AC current), were the direction of the currentflow reverses periodically leading to a zero net current flow.We have designed and built a custom conductivity sensor and probe, specifi-cally designed to measure the resistivity of the subglacial waters across a pair ofelectrodes. The sensor is implemented over the ATmega328 micro-controller run-ning at 5 volts and its inbuilt 10bit ADC.A.5.1 ImplementationMost micro-controllers provide pulse with modulation (PWM) output pins, that al-lows to control the power of peripherals (as dimming LED’s or controlling thespeed of a motor) by producing a square signal with a variable duty cycle. Thisis to achieve half of the power the square signal will spend 50% of the time in theOFF state and 50% in the ON state. This is therefore a perfect source of an ACpower source for conductivity measurements, with the only problem that the volt-age will vary between 0 volts and VCC volts (where VCC is typically 5.0 or 3.3volt for most Arduino compatible micro-controllers), leading to a current flowingalways in the same direction. We have solved this by the use of a “virtual ground”a higher voltage reference that can be easily created using a voltage divider. If for204example in a 5 volt microprocessor we set the reference for our square signal at+2.5 v, then the signal will be perceived by the medium at the probe as jumpingbetween -2.5 and +2.5 volts.Figure A.3: Example of a PWM output with 50% duty cycle, with a VCC/2virtual ground referenceFigure A.3 shows an example of such square signal, which will allow us to ac-curately measure the conductivity without altering the chemical nature of the liquidmedium being measured. Figure A.4 shows the schematics of this implementationusing three micro-processors pins. One for reading the voltage trough the probe(analog input), other to create the virtual ground trough the voltage divider createdby the resistors R1 and R2 (digital output) and a third one to create the square signal(PWM output).Figure A.4: General schematics of the conductivity meter set-upIn order to have a AC signal symmetric around zero, from now on we will as-sume that R1 = R2. During a measurement, the medium between the two terminalsof the probe will act as a third resistor in the system, that will effectively be in205parallel to R1 during the OFF part of the PWM cycle, and in parallel to R2 duringthe ON part of the cycle as shown on figure A.5.Figure A.5: Schematic configuration of the two states of the system: PWMON cycle (left) and PWM OFF cycle (right). Rw is the resistance ofthe probe once submerged on the medium to be measured. RE f f is theeffective resistance in the side of the voltage divider affected by theprobe.The resistance of the probe once submerged on the medium to be measured(Rw) will change both the maximum and minimum value read at the analog inputpin. Better than using any of those values we can use the amplitude of the signalas shown on figure A.6.Figure A.6: Amplitude of the signal oscillating between VHigh and VLow.If we solve for the amplitude A we getA =RR+2RwVCC (A.2)206were VCC is the operating voltage of the micro-processor (i.e. the maximum valueof the square signal).A.5.2 Computation of the conductivityTo compute the amplitude A, we have to perform tens to hundreds of measurementof the voltage at the analog input pin and find the amplitude of the square signalthe readings describe. In principle you can assume the that VHigh is the maximumread value and VLow the minimum, but in our experiments, the presence of outliersmake this approach useless. Therefore, we use an algorithm analogous to the com-putation of a median to find the the most common high and low values as describedin figure A.7.Figure A.7: To find the most common value corresponding to VHigh and VLowwe perform N readings and sort them. Then the one in the position 1/3N and 2/3 N will represent typical high and low values (1/4 N and 3/4N could be also reasonable values).Using Eq. A.2, the resistance of the probe can be computed from A asRw =R(VCC−A)2A(A.3)207where its inverse is the conductance (Cw):Cw =2AR(VCC−A) (A.4)Assuming there is a linear relationship between conductance of the probe (theinverse of the resistance) and the conductivity of the medium (which should be thecase and also with a zero linear constant) we can say that the conductivity of thewater cw is given bycw =Cw ∗F +K = 2AR(VCC−A) ∗F +K (A.5)where F and K are the constants of the linear relationship, that are dependent ofthe probe.Given that this is not a linear relationship for our measured A, in order to getmore accurate results using just linear interpolation between calibration values, wecan define the relative amplitude AR asAR =AVCC−A (A.6)Thereforecw =2RARF +K (A.7)At this point we can in fact take any arbitrary value for R (in our design we usedR = 100kΩ to measure conductivities between 10 and 100 µS), as any differencewith the real value will be absorbed in F during the calibration. With R = 2 we getcw = ARF +K (A.8)Which is a linear relation, and should lead to accurate interpolation/extrapola-tion results using a simple linear interpolation/extrapolation algorithm. F and Kmust be computed during the calibration of the probe.To improve measure stability, we perform 100 measurements separated by only1 ms. From these measurements the median high and low values are computed.Figure 2.4b shows the conductivity probe of our sensor pod. The ring config-uration of the graphite electrodes is recommended in situation were other electric208currents might be present in the measured medium.A.5.3 Optimization of the designUsing Eq. A.2 we can find the best value of R in order to maximize the range ofamplitudes in a given range of Rw, thus maximizing the resolution of the sensor inthe range of interest. The optimal R will be given byRoptimal = 2∣∣∣∣∣√Rw1Rw2(Rw2−Rw1)2Rw1−Rw2∣∣∣∣∣ (A.9)Figure A.8 shows the optimal value of R for a range of probe resistances Rw.Figure A.8: Optimal value of the voltage divider resistor R as function of therange of probe resistance for the least and most conductive expectedmedium. This value of R maximize the amplitude range and thereforethe resolution in the desired range.209A.6 Transmissivity sensorTransmissivity or turbidity sensor measure the amount of light transfer by themedium. The light source used is a superbright white led, that goes trough themedium by a 1 cm gap to a broad band light sensor TSL2561. This sensor reportsthe total radiant energy received during an specified time interval, referred to as“integration time”, ranging between 1 ms and 2 seconds.The sensor is first calibrated in clear water, and the integration time required forsensor saturation is established are stored. Then, to take a transmissivity reading,the luminosity is measured using the saturation time as integration time (typicallyaround 105 ms). If the luminosity reading is less than 10% of the saturation valuewe repeat the reading using the maximum integration time (2 seconds). The finaltransmissivity value is computed as the percentage that the received light representout of the reference received light on clear water.A.7 Temperature sensorAs long as the glacier bed is water filled, the temperature should be stable at zerodegrees Celsius. Temperature sensors are included to detect dry bed events onwhich cold air can reach the glacier sole or eventually transitions to cold base;even we do not expect such events on South Glacier. The temperature sensors weuse are silicon bandgap temperature sensors, which are part of the one-wire digitalsensor DS18B20 with ±0.5 oC accuracy and 12 bits resolution (in the -55oC to+125oC range).A.8 Orientation and motion sensorA 3-axis accelerometer and a 3-axis magnetometer integrated into the LSM303sensor are used to measure the absolute orientation of the instrument relative tothe local earth magnetic field. Accelerometer readings provide a second source ofinclination data and allow to partition the orientation changes in inclination andazimuth (without assuming that the reference magnetic field recorded on surfacedoes not change and is also valid at the bed). Median values are reported as orien-tation and standard deviations as a measure of motion caused by water flow around210the instrument.Particular care has to be taken to calibrate the 3-axis magnetic readings, whichare subject to multiple errors and biases, leading to completely wrong headings ifno calibration is applied. The calibration procedure used is based on a completeerror model, including magnetometer errors (scale factors, nonorthogonality, andoffsets) and magnetic deviations (soft and hard iron) induced by the rest of theinstrument [Renaudin et al., 2010].A.9 Confinement sensorFor proper interpretation of pressure, turbidity, and conductivity measurements, itis of great interest to know if the instrument is surrounded by an aqueous mediaor confined in ice or sediments. To distinguish between this two cases, a 3 voltvibration motor have been included in the instrument, and acceleration readingsare taken while the motor is vibrating. When the sensor is free and surrounded bywater or air, mean acceleration magnitudes rise to about 1.3 g, and when confinedthey remain at 1 g. We base the confinement measure in the standard deviationof the accelerations values recorded while the vibration motor is operating. Largestandard deviations are associated to a sensor pod freely hanging in water, andsmall values to one tighly confined in ice or sediments. Partial and total confine-ment cannot be clearly differentiated, but this measurement still provides valuableinformation for the correct interpretation of the data.We scale confinement in order to be 0 when free and 100 when the standarddeviation of the acceleration is zero, thenConfiment = 100(1− RR0)(A.10)where R is the raw reading of acceleration standard deviation, and R0 is the ref-erence reading taken during calibration with the sensor pod free hanging in water.Figure 2.16 shows data from one of this sensors.211A.10 Reflection spectrometerThe broadband luminosity sensor TSL2561 is sensitive to light between 200 and1100 nm. We estimate the reflectivity of the aqueous media at different wavelengthby measuring the reflected intensity using different narrow band light sources, asthe one provided by inexpensive LEDs. The spectrometer is equipped with 5 LED:infrared (940 nm), Red (623 nm), Green (517 nm), Blue (466 nm) and Ultraviolet(400 nm).212Appendix BData qualityThe analysis of the pressure records from South Glacier presented in this paperdoes not rely on the accuracy of absolute pressure values, in the sense that we havenot attempted to use the differences in water pressures between different boreholesas an indication of hydraulic potential gradients. If a sensor is incorrectly calibratedand, for instance, the amplitude of recorded diurnal pressure oscillations is miscal-culated, our interpretation of connection between boreholes would be unaltered insections 2.2.2 and 2.2.4 of the main paper. Unless the calibration coefficients arewildly incorrect, variations in the amplitude of diurnal oscillation over longer peri-ods of time will still resemble each other even if the computed pressure differencebetween the boreholes varies spuriously. However, we do rely on the calibrationparameters being stable over time. Otherwise, inferred variations in the amplitudeof diurnal oscillations would be affected by calibration drift, and would not neces-sarily represent the changes in the drainage response to surface forcing. Similarly,we rely on calibration coefficients being stable in order to infer whether isolatedboreholes experience a seasonal drift in water pressure during summer in section2.2.1.There are numerous mechanisms by which a sensor can become corrupted. Anobvious cause is signal cable damage, which is sometimes visible at the glacier sur-face due to crevasse opening. We have visually identified records that show signsof such corruption, such as large, random jumps in pressure between successivemeasurements, and large negative water pressures.Transient high-pressure spikes [Kavanaugh and Clarke, 2000] are likely to have213Figure B.1: Pressure records P1 and P2 for the two sensors in borehole 13H16,installed 20 cm apart. (a) A phase plot with points colour-coded bytime. The expected relationship P1 = P2 is shown as a black dottedline. The linear regression model constructed over the first month afterinstallation is shown as a red dotted line, with parameter values givenin inset box. (b) Pressure time series for P1 (blue) and P2 (orange). Theresidual between P2 and the linear regression model is shown in yellow,scaled by a factor of 10 for visibility. Unexplained variance over a one-month moving window is shown as a purple line.Figure B.2: Pressure records for the two sensors in borehole 13H17, installed34 cm apart, plotted using the same scheme as in Fig. B.1caused abrupt calibration changes in four of the recorded pressure time series.From 2013 onwards, most sensors were equipped with snubbers, and only onesensor displayed this issue afterwards. In all four cases, instantaneous offsets weremanually identified and corrected.Eleven boreholes were equipped with two sensors, logged independently. Thesesensors recorded in total more than 13 years of duplicated data. Seven of the bore-holes included a digital transducer, in which the measurement is made in the sen-214Figure B.3: Pressure records for the two sensors in borehole 13H58 (the“fast-flow” borehole of the main paper), installed 70 cm apart, plottedusing the same scheme as in Fig. B.1.Figure B.4: Pressure records for the two sensors in borehole 14H60, installed20 cm apart. P2 was recorded by a digital sensor. The plotting schemeused is the same as in Fig. B.1.sor, while the remaining sensors were analogue sensors, which rely on voltagemeasurements at the surface and can, therefore, be corrupted by damage to thesignal cable introducing partial short circuits. Assuming that the two sensors inthese boreholes remain hydraulically connected, their records allow us to assessdata quality and calibration drift.All presented pressure values were computed from differential voltage readingsusing a linear transformation of the formP = MV −P0, (B.1)where P is the pressure at the sensor and V the ratio of a measured differentialvoltage to an applied excitation voltage on a Wheatstone bridge circuit, one of215Figure B.5: Pressure records for the two sensors in borehole 14H62, installedat the same elevation. P2 was recorded by a digital sensor. The plottingscheme used is the same as in Fig. B.1.Figure B.6: Pressure records for the two sensors in borehole 15HL07, in-stalled at the same elevation. The plotting scheme used is the same asin Fig. B.1.Figure B.7: Pressure records for the two sensors in borehole 15HU01, in-stalled at the same elevation. P2 was recorded by a digital sensor. Theplotting scheme used is the same as in Fig. B.1.Figure B.8: Pressure records for the two sensors in borehole 15HU04, in-stalled at the same elevation. P2 was recorded by a digital sensor. Theplotting scheme used is the same as in Fig. B.1.216whose limbs is a strain gauge bonded to the pressure sensor diaphragm. The cal-ibration constants are the offset P0 (corresponding to the voltage measured at at-mospheric pressure), and the multiplier M. Our reported measurements rely onpre-installation calibrations and assume no change in calibration constants.Panel a of each of Figs. B.1–B.11 shows the pressure at one sensor in a givenborehole, computed using pre-deployment calibration values of M and P0, plottedagainst the corresponding value for the other sensor in the same borehole, for alldoubly-instrumented boreholes. If both sensors record the same pressure, thenthese phase plots should lie along the black dotted line P2 = P1, as the expectedhydrostatic pressure difference between the two sensors is minimal (except perhapsin figure B.3). Given that there could be small errors in the calibration parameters,we estimate a transformation between P1 and P2 with a linear regression over thedata recorded during the first month after installation (red dotted line). Deviationsfrom that line indicate that one or both of the sensors are not behaving linearly as inequation B.1, the calibration constants have changed, or that the two sensors havebecome isolated from each other due to freezing or ice creep.In panel b of each figure, the pressure recorded at each sensor as computedfrom pre-installation calibration is presented as a blue or orange line. Also plottedis the difference between the recorded pressures. This difference is calculated afterapplying the linear regression model correction computed over the first month ofsensor operation (indicated in the inset in panel a). This residual has been multi-plied by a factor of 10 for easier visibility and plotted as a yellow line.To quantify changes in the similarity of both records, we have also computedlinear regressions over moving one-month windows, and subsequently calculatedthe fraction of unexplained variance (FUV) over that period. The FUV is the frac-tion of the variance in P2 that cannot be explained by variations in P1. As such, itis a measure of the quality of a linear fit: if samples are taken at times t1, t2, ..., tNthe FUV would beFUV =∑Ni=1(P1(i)−P′2(i))2∑Ni=1 (P1(i)−µ1)2(B.2)where P1(i) is the value of P1 at time ti, P′2(i) is the value of P2 at time ti, transformedlinearly using regression model over the moving one-month window. N is thenumber of samples in the window and µ1 is the mean of P1.217We see that except for Fig. B.4, all differences in pressure are smaller than 5meters, and deviations in the multiplier (the difference between the coefficient ofP1 in the formula in panel a and the expected value of one) are in most cases smallerthan 2%. In figures B.4 and B.7, where anomalously large slopes are observed, wecan see that there is a clear change in the slope below a certain pressure, suggestingthat the sensors used most likely did not behave linearly over their nominal cali-bration pressure range. A similar effect probably affects the sensors in Fig. B.5,where the two sensors may have also become disconnected from each other. In allcases, the non-linearity seems to occur only in the digital sensors, and these makeup a small fraction of the whole dateset.In the first month of each time series, the residual remains below 5 metersand the FUV below 1%. Such consistent records can extend for more than oneyear (Fig. B.7), but the agreement between the sensors can also degrade after afew months. The increase in the residual usually starts at a very specific point intime. This is suggestive of a loss of the connection between the two sensors. Someexamples of possible disconnection are evident in Fig. B.2 in July 2014, in Fig.B.3 in December 2013, Fig. B.6 in late July 2015, and Fig. B.9 in October 2015. Itis unlikely that the difference observed in these cases arise from a single, transienthigh-pressure spike changing the calibration of the sensor instantly: the pressurerecords should then still be linearly related to each other, but with different offsetsand multipliers, and the FUV should remain small.A few records (for instance those in Figs. B.1 and B.5) are consistent with agradual change in the offset and/or multiplier. However, this effect seems to besmall in comparison with the differences arising from possible disconnection.Disconnection can be explained by sensors becoming encased in ice due toice creep or due to freezing, the latter being less likely if we consider that radarmeasurements indicate the presence of temperate ice at the base. The fact that withmost disconnections happen after the end of the summer season, and happen soonerin sensor pairs that are consistently recording low water pressures (for instanceFig. B.6), is consistent with disconnection by creep closure; the ice surroundingthe corresponding boreholes should be subject to higher creep rates. It is importantto note that residuals and FUV are typically very small over periods where diurnalvariations are present, and that significant inconsistencies between sensor pairs in218the same borehole appear in the absence of such variations, indicating a loss ofconnection to the drainage system.Doubly instrumented boreholes also allow us to assess the effect of pressuresnubbers on pressure records. In figures B.3, B.4, and B.7 to B.11, sensor P1(blue) was equipped with a snubber and P2 (orange) was not. In figures B.1 andB.2, both sensors had snubbers, and in figures B.5 and B.6 neither of them did. Inthe cases where only one sensor had a snubber, it can be seen that no smoothingof the pressure signal is observed. Close examination of the pressure time seriesshows that even spikes lasting a few minutes are well-reproduced by both records.Therefore, at the sampling frequencies of our sensors (1 to 20 minutes), the effectsof the snubber are negligible. By contrast, among the sensors not equipped with asnubber, one out of 48 suffered large, “instantaneous” pressure offsets, contrastingwith only one in 174 experiencing the same among sensors equipped with a snub-ber. Therefore, pressure snubbers seem to be effective at filtering out the transienthigh-pressure spikes (“fluid hammer”) that are thought to be responsible for thoseoffsets through damaging the sensor diaphragm, but without affecting the accuracyof the instruments for measuring slower pressure variations.Tests of the sensors extracted by re-drilling after one year also found that smalloffsets had developed, in all cases smaller than 3 m. Multiplier changes were alsoobserved, but they account for an even smaller error within the measured pressurerange. The observation of a few records in the dataset gradually drifting up tonearly 200% of overburden is therefore difficult to explain and we were not ableto correct for it. The effect may be due to errors in initial calibration or hard-to-diagnose sensor damage, or due to calibration drift that is sufficiently rare or ex-treme not to have been captured by the relative small sample of sensors in doubly-instrumented or re-drilled boreholes. One possible cause of large calibration errorsdeveloping could be the permanent deformation of the sensing diaphragm by iceformation against it.Instrumental accuracy and precision aside, our interpretation relies on waterpressures having been measured at the bed, except in cases where the sensor isknown to have been installed englacially (such as hole A in Fig. 2.2). Recall thatsensors were typically installed 10-20 cm above the bed; we are relying on theconnection to the bed not becoming closed off. We observe that sensors in doubly-219instrumented boreholes can start to exhibit independent pressure variations duringwinter, sometimes reverting to a common signal. One example can be seen in thetwo sensors displayed in Fig. B.3 and Fig. 2.14 of the main text, both installed70 cm apart in the same hole (the fast-flow borehole). One possible explanationfor the mismatched data could be the sensors becoming encased in ice during thewinter and thus separated from each other.There is direct evidence for processes that could lead to sensors becoming en-closed in ice, for example, the digital confinement data shown in Fig. 2.16 of themain text (see section 2.3.4 of the main text). Note that the fraction of boreholesshowing diurnal variation in their second summer season decreases compared withthe first, dropping from 71% in the first season to 58% in the second. This drop inactivity could also be a consequence of sensors becoming encased in ice or other-wise isolated from the bed.220Figure B.9: Pressure records for the two sensors in borehole 15HU05, in-stalled at the same elevation. P2 was recorded by a digital sensor. Theplotting scheme used is the same as in Fig. B.1.Figure B.10: Pressure records for the two sensors in borehole 15HU17, in-stalled at the same elevation. P2 was recorded by a digital sensor. Theplotting scheme used is the same as in Fig. B.1.Figure B.11: Pressure records for the two sensors in borehole 15HU50, in-stalled at the same elevation. P2 was recorded by a digital sensor. Theplotting scheme used is the same as in Fig. B.1.221Appendix CReliability of temperature as amelt proxyIn section 2.2.4 of the main text, we use the standard deviation of the temperatureover a 1-day running window as a measure of diurnal variations in meltwater input.That variation is then used to compute the curve in Fig. 2.13c, showing the ratio ofthe amplitude of pressure variations in the connected part of the subglacial drainageto variations in meltwater input. Later, in section 2.3 of the main text, we attributethe increase in the relative amplitude to an increase in the efficiency of the drainagesystem. However, it could be argued that the same signal can arise from changes inthe degree-day factor through the season. To address this question, we present anassessment of the variability of degree-day factors and a calculation of the relativeamplitudes using an independent proxy of melt variability.To assess the variability of the degree-day factor, we used melt output of theenergy balance model described by Wheler and Flowers [2011], covering the years2007-2009 and 2011-2012. For each day, we computed the degree-day factor thatwould explain the modelled melt from that day’s PDD value. Fig. C.1 shows the re-sults during four different summer seasons (2009 was excluded due to the existenceof many outliers in the melt record). It can be observed that after an initial increase,the degree-day factors are fairly stable in some years (2007 and 2012), while in oth-ers (2008 and 2011) they can have significant variations, with the monthly runningmean changing by a factor up to 2.5. Such variations would however be insufficientto explain the signal of Fig. 2.13c, in where the relative amplitude increases by a222Figure C.1: Degree-day factors (blue dots) computed using melt outputs ofthe model by Wheler and Flowers (2011), and PDD values (orange line)derived from the temperature record. A monthly running median of thedegree-day factors is also shown (green line).Figure C.2: Alternative version of panel c of the Fig. 2.13c of the main text,using total daily surface lowering as the proxy for the diurnal amplitudeof surface meltwater production.factor of 4.2.As we have no melt model results for 2015, we cannot however directly ruleout larger variations in the degree-day factors for that year. Nevertheless, largevariations in the degree-day factor are less likely in the period between July 4thand 30th where there were no snowfall events, or significant changes in the relativeextent of old snow and bare ice evident in the time-lapse images. During the sameperiod, the relative amplitude shown in Fig. 2.13c increased by a factor of 1.6, avariation unlikely to be explained by a degree-day factor change.223We also re-computed the relative amplitude of Fig. 2.13c of the main text overa shorter period (until August 27th) for which surface elevation changes were mea-sured by a sonic ranger SR50 at the AWS location (see Fig. 2.2 in the main text). Inthis alternative approach, we used surface lowering as a proxy for melt amplitudeinstead of the standard deviation of the positive part of temperature. The caveatof this approach is that again we cannot measure melt rate in mass of water perunit time directly, but now have to account for density variations between betweenfresh snow, old snow and ice. In the model by Wheler and Flowers [2011], thesecan vary up to a factor of 3.6 (from 250 kg/m3 for the lightest snow to 900 kg/m3 forice). However, in the July period considered above, we are confident that surfaceconditions did not change, and surface lowering is a reliable proxy for meltwaterproduction at the AWS location.Figure C.2 shows the relative amplitude computed using the surface elevationrecord as melt proxy, showing that the increase we have attributed to drainage ef-ficiency is also robust under this alternative approach. This includes the periodbetween July 4th and 30th, where no significant changes in surface conditions oc-curred in the main study area. Although both approaches have uncertainties andflaws, the increase in the relative amplitude we associate to increased efficiency ofthe drainage system seem to be robust. Due to its overall magnitude and persis-tence in periods were the density or degree-day factor are unlikely to have changedconsiderably.224Appendix DClustering evaluation andcalibrationD.1 Alternative clustering and data analysis techniquesThe hierarchical clustering technique presented in Chapter 3 was not adopted arbi-trarily. On the contrary, it was the result of the careful testing and comparison ofseveral alternative approaches. Here we will present each one of those approaches,explain why we choose hierarchical clustering and then describe in detail the dif-ferent hierarchical clustering variations we tested and how we compared their per-formance.The data exploration approaches presented in this appendix are not restrictedonly to clustering techniques, but to all the statistical methodologies we have testedin the search for tools that can help to reduce the complexity of the dataset andexpose the spatial structure and phenomenon underlying the complex response ob-served in boreholes water pressures.Each method was tested using one or more versions of the pressure data werefer to as “pre-processings”. These pre-processings are the following:• Raw The pressure is directly fed to the clustering algorithm.• Diurnal running mean The pressure signal is smoothed using a runningmean with a 24 hours window.225• Diurnal residual The residual is computed by subtracting the diurnal run-ning mean from the raw pressure data. A normalized version of the diurnalresidual is presented in Eq. 3.1.• Residual standard deviation A 24 hours running standard deviation (STD)computed over the diurnal residual.• Power spectrum Power spectrum of the raw pressure in the frequency space.Each of these pre-processing can additionally be “standardized”, that corre-sponds to the normalization to the standard deviation.D.1.1 Empirical Orthogonal Functions (EOFs)EOFs, also known as Principal Component Analysis (PCA) was our first attempt toreduce the complexity of the dataset and find the main modes that drive the pressuresignals. After attempting the EOF analysis using the raw pressure and multiple pre-processing options such as standardized raw pressure, daily running mean, diurnalresidual, and residual STD. We found that the similar sets of boreholes that wewere aiming to capture by a single principal vector, usually corresponded to smallprincipal values, because most of the variance in the dataset was explained bydisconnected sensors showing large pressure variations.After a thoughtful analysis of the results, EOFs proved of little help to under-stand the ongoing processes due to three main reasons:1. The difficulty to link a specific process or signal (sometimes very well rep-resented by one or more boreholes) with one eigenvector.2. The tendency of single boreholes with high variability to take over the biggesteigenvalues.3. The significant change in the eigenvectors as a result of changes in the do-main (when one sensor was added or removed to the set). Making it difficultto follow one particular group of sensors in time.EOF analysis was able to group similar pressure signals together reasonablywell. However, it did not help in reducing the complexity of the data as expected.226Also, EOFs did not provide a simple way to identify which principal vectors wereassociated to groups of very similar boreholes or groups of very distinct ones.Given that our objective is to track the evolution of these groups in time, we needa technique capable to deal with a changing number of sensors in the data set. Thechange of the domain size is a known problem in EOF analysis, as the principalvectors are likely to change dramatically in such circumstances, making it difficultto follow a group of similar boreholes trough time.Despite the countless improvements that can be done to our algorithms to an-alyze EOF data, we decided to try more advanced techniques that overcome someof the problems that EOF present for out particular application.D.1.2 Covariance analysisOne of the simplest approaches to identify efficient hydraulic connections (i.e. find-ing boreholes showing identical or very similar pressure pattern profiles), is troughtheir covariance. We calculated covariances matrices (equations D.1 and D.4) overa moving time window, including all the boreholes with valid data in the interval.High covariance cases we examined to assess their actual similarity. The exercisewas repeated with normalized covariances (or correlation coefficients, equationsD.2 and D.6), which produced better results.The covariance Cov between two time series P(m) and P(n) with N samples isgiven byCov(P(m),P(n)) =1N−1N∑i=0(P(m)i −P(m))(P(n)i −P(n))(D.1)where P(m) is the mean value of the time series m. If m = n the above equationleads to the variance of P(m), or Var(P(m)).Also, we can obtain the normalized covariance (correlation coefficient) byCorr(P(m),P(n)) =Cov(P(m),P(n))Var(P(m))Var(P(n))(D.2)We defined also the covariance matrix P. For a set of M time series each withN samples, each column j of P corresponds to one time series and each row i227corresponds to the samples at one time step for all time series. For simplicity wewill define also the matrix P were the mean of each column have being removed,this isPi j = Pi j−P( j) (D.3)now the covariance matrix C is given byC =PP′N−1 (D.4)where Ci j = Cov(P(i),P( j)). Let’s now define the column vector D composed ofthe square root of the inverse of the diagonal elements of PP′, this isDi =√1(PP′)ii(D.5)Then, the normalized covariance matrix C would be given byC= PP′DD′ (D.6)where  represent the Hadamard element-wise product of matrices. Then we haveCi j =Corr(P(i),P( j)).We developed a special visualization tool of the spatial distribution of the mag-nitude of the covariance, which allowed us to identify multiple sets of boreholesshowing very similar pressure time series, suggesting they are connected or re-sponding to the same forcing. Simple covariances and correlation coefficient haveproved to be effective to find boreholes apparently connected, but several limita-tions have been identified:• Common forcing. All boreholes somehow connected to surface water inputswill show a high correlation as long as their pressure variations are in phase.Although, the similarity of pressure variations due to a shared forcing is anissue affecting all the methodologies we tried, the covariances performanceis particularly poor at resolving distinct drainage systems.• Amplitude data loss. Normalized covariances (correlation coefficients, equa-228tion D.2) are useful to detect connections between boreholes with very dif-ferent oscillation amplitudes, but they discard useful amplitude informationas efficient connections should lead to similar amplitudes between boreholes.• Time shifts. Small time shifts as expected for diffusive hydraulic connec-tions or datalogger clocks drift can hide strong correlations. Cross-covariancescould be an option to find maximum covariances at different offsets within areasonable range, but it would be computationally intensive.In conclusion, an approach based only in the covariance is not sensitive enoughto the different properties that make two time series similar or not. Neverthelessimprovement to overcome those limitations can be envisioned. For example, bydefining a “connectivity index” that combines the covariances between differentpre-processed versions of the pressure signal, such as the envelope and the diurnalresiduals.D.1.3 Self Organizing Maps (SOMs)Self-Organizing Maps is a form of neural network that uses competitive learningto produce a topologically accurate representation of a dataset reducing its dimen-sionality, becoming a very powerful unsupervised clustering algorithm. SOM isan iterative algorithm that for given numbers N and M, creates a “map” consistingof N×M nodes organized in a two-dimensional array. Each node or “unit” con-tain a time series that is optimized to minimize the Euclidean distance (minimumsquares), to a subset of time series in the data set. In this way, the final map, con-tain the N×M set of time series that better represent the data, and each data series(boreholes in our case) can be assigned to one map unit, and similar time serieswould likely be grouped in the same unit. An advantage of this method is that itcan easily be applied independently or simultaneously to the raw data, and any setof pre-processed versions of it.SOM clustering was very successful finding similar data series. However, thesize of the map has to be pre-determined, which turned out to be very problematicfor our application. For example, if a reasonable map size is adopted based onthe number of sensors for a given time window, if there are abundant connectionsas is common in early spring, the map would be oversized, and it could force the229separation of one coherent cluster into multiple groups, therefore, requiring a re-clustering. On the other hand, in a typical winter situation with many isolatedsensors, there will be not enough map units to capture the diversity of the system,and if there is a coherent cluster, it will likely be contaminated with other unrelatedsensors.We envisioned and attempted few approaches to an adaptive map size, findingiteratively the size that minimizes quantization and topological errors, but this,together with all the tunable parameters of the SOM algorithm made the approachcumbersome and computationally expensive. For this reason, before committingto a highly specialized SOM clustering approach, we decided first to explore othersimpler techniques that might produce equivalent or better results.Figure D.1 show a map with the 53 sensors operating over ten days, weredistinct clusters were captured in units #4, #7 and #10. An example of a groupcatching unrelated sensors is #6, suggesting that the map is too small. However, ifrecomputed on a larger size, group #10 would be split into two map units, requiringre-clustering.D.1.4 K-means clusteringK-means clustering is a very popular and perhaps the simplest clustering algo-rithm. In combination with a standard Euclidian metrics, it was used by Fudgeet al. [2008] on a small array of 16 boreholes on Bench Glacier, Alaska. However,as with SOM’s it requires the number of clusters to be defined beforehand, runningin the same kind of problems described above. For this reason, it was not tested,given that hierarchical clustering was a more promising technique.D.2 Distance metricsHere we presented a more detailed account of the precise formulation of each ofthe tested distance metrics, beside the absolute Euclidean distance that has alreadybeen described in Eq. 3.2.• Euclidian Differences between time series a and b, each one with N samples230Figure D.1: 3 by 4 units SOM computed from the pressure residuals of 53sensors operating between July 28th and August 8th 2011. The time se-ries representative of each unit (thick gray line) is shown together withthe sensor data matching the unit. For each unit an average quantiza-tion error is shown (average Euclidean distance between each line andthe one representative of the unit) and the number of hits (number ofsensors matching that unit). The overall quantization error of the mapis 15.78, and the topological error is 0.13.are measured as Euclidian distance in an N-dimensional space.DE(a,b) =√N∑i(a1−bi)2 (D.7)• Correlation Time series differences are quantified by the covariance com-plement. Therefore, distance is small for highly correlated signals. It isworth mentioning that for clustering proposes of standardized time series,the Correlation and Euclidian distances are mathematically equivalent.DC(a,b) = 1− cov(a,b)σaσb = 1−∑Ni (a1− a¯)(b1− b¯)√∑Ni (a1− a¯)2√∑Ni(b1− b¯)2 (D.8)231where cov(a,b) is the covariance between series a and b, σa and σb are theirrespective standard deviations and a¯, b¯ their mean values.• Absolute correlation Time series differences are quantified by the absolutecovariance complement. This produce small distance between well corre-lated and also anti-correlated signals.DAC(a,b) = 1− |cov(a,b)|σaσb = 1−∣∣∑Ni (a1− a¯)(b1− b¯)∣∣√∑Ni (a1− a¯)2√∑Ni(b1− b¯)2 (D.9)where |x| stands for the absolute value of x.• Dynamic Time Wrapping DTW is a standard technique used in voice recog-nition systems, allowing to measure the similarities between waveforms butinsensitive to differences and changes in speed between the reference wave-form and the one being classified [Mullin, 1983]. In our context, DTW canhelp recognizing similar signals that have a small time shift (due to datalog-ger clock offsets or physical processes).The following pseudo-code describe the used DTW algorithm232a,b Vectors of size N containing the two time seriesw Maximum offset allowed as number of samplesD Matrix of size (N +1)× (N +1) fill up with infinitesD(1,1) = 0for i = 1 to Nfor j = max(i−w,1) to min(i+w,N)distance = (a(i)−b( j))2D(i+1, j+1) = distance+min(D(i, j+1),D(i+1, j),D(i, j))end j loopend i loopDDTW = D(N +1,N +1)• Absolute Dynamic Time Wrapping The minimum between the standardDTW distance (as described before), and the DTW distance to the reversedtime series (i.e. up/down flipped). This allows detection of anti-correlatedsensors.DADTW (a,b) = min(DDTW (a,b),DDTW (a,−b)) (D.10)D.3 Entropy and Information Gain: Evaluatingclustering performanceInformation gain is a standard quantity used in decision tree analysis [Mitchell,1997]. When elements belonging to two classes are grouped according to theirclass, the information gain measures the quality of the grouping. The RIG is theinformation gain relative to the maximum possible gain, which is achieved wheneach element is correctly assigned to the group it belongs. We can represent theRIG as a percentage: a RIG of 100% means the information gain was maximum,233and the clustering strategy reached its maximum possible performance (i.e. allelements were assigned to the correct group).If we have a group G made out of N objects that belong to M classes C1,C2, ...CM,the entropy E of a group of them, is a measure of how ”pure” is the group and isgiven byE(G) =−M∑i=1NCiNlog(NCiN)(D.11)where NCi is the number of objects belonging to the ith class. If we have only twoclasses A and B, and we call fA and fB to the fraction of the objects that belong toeach class respectively, then the entropy will beE(G) =− fA log( fA)− fB log( fB) (D.12)where two relevant extremes are fA = fB in which case we have the maximumpossible entropy E = 1, or when either fA or fB is zero, then we have only oneclass and minimum entropy E = 0.This two-classes case is the one we have if we classify all time series in thepressure dataset for a given time window as either belonging to a cluster or not.Lets call this two classes C for “connected” or U for “unconnected”. If a givenclustering strategy splits the dataset G into two sub-groups: IN for the ones iden-tified as connected to a single system and OUT for the one unconnected to thatparticular system, the entropy of the dataset after the splitting G′ will beE(G′) = fINE(IN)+ fOUTE(OUT) (D.13)Now the effectiveness of the clustering strategy for separating the dataset intosub-groups purely composed by connected or unconnected sensors can be mea-sured by the entropy loss or information gain (Gain) given byGain = E(G)−E(G′) = E(G)−(fINE(IN)+ fOUTE(OUT))(D.14)Similarly, the relative information gain (RIG) can be defined as234RIG =E(G)−E(G′)E(G)= 1−(fINE(IN)+ fOUTE(OUT))E(G)(D.15)which can be represented as a percentage: a RIG of 100% means the informationgain was maximum, and the clustering strategy reached its maximum possible per-formance, putting all sensors connected to a particular system in the IN sub-group.235


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