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The deforming bed beneath a surge-type glacier : measurement of mechanical and electrical properties Blake, Eric Weston 1992

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THE DEFORMING BED BENEATH A SURGE-TYPE GLACIER: MEASUREMENT OF MECHANICAL AND ELECTRICAL PROPERTIES by ERIK WESTON BLAKE B.A.Sc., The University of Toronto, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy  We accept this thesis as conforming to the required standard  Signature(s) removed to protect privacy  THE UNIVERSITY OF BRITISH COLUMBIA March 1992  ®  Erik Weston Blake, 1992  In presenting this thesis in  partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or  by  his  or  her  representatives.  It  is  understood  that  copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Signature(s) removed to protect privacy  (Signature)  Department of  ‘  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  MCH  Z7 ,19’2  kST,eO (‘toM (  ABSTRACT Glacier surging is a flow instability characterized by short periods of rapid glacier flow separating longer periods of normal flow. It is accepted that sustained high sub glacial water pressure causes glacier surging by decoupling the glacier from its bed, but how this high subglacial water pressure is developed and sustained is the subject of debate. The current focus of glaciological research is on the interaction of subglacial processes with the subglacial drainage system. We have developed new investigative techniques for exploring two subglacial pro cesses: basal deformation and electrical phenomena. These techniques have been ap plied in research undertaken on Trapridge Glacier, a small surge-type glacier in the St. Elias Mountains, Yukon, Canada; these are the first in situ measurements of deforma tion, electrical resistivity, and streaming potentials beneath a surge-type glacier. The development of a reliable rheological description of subglacial material re quired field observations of its stress—strain response; this was the motivation for our subglacial deformation experiments. We have demonstrated that no clear relationship exists between values of shear stress and effective pressure calculated using accepted methods and deformation rate; the absence of an expected relationship suggests that alternate methods for quantifying subglacial shear stress and effective pressure need to be found. Data from our subglacial electrical resistivity measurements and from the deforma tion measurements provide strong evidence that the distribution of normal and shear loading at the glacier bed is not even and that subglacial deformation rates can be affected by distant changes in subglacial pressure conditions. We have also observed temporal changes in electrical flow paths within subglacial sediments. We have shown that temporal variations in natural potential observed beneath Trapridge Glacier are caused by streaming potentials; streaming potentials result from cross-coupling between fluid flow and electric currents. Our data suggest that inexpen sive subglacial electrode arrays may be used to supplement pressure sensors. 11  TABLE OF CONTENTS Abstract  11  List of Figures  x  List of Tables  xl”  List of Symbols  xiv  Acknowledgements  xvi  Chapter 1. INTRODUCTION  1  1.1 What is glacier surging 7 1.2 How does one identify a surging glacier? 1.3 What happens during a glacier surge?  5  1.3.1 Hard bed versus soft bed  7  1.3.2 Seasonal timing  8  .  1.4 The study site  9  1.5 Thesis scope  •  •  .  .  13  1.5.1 Basal deformation  15  1.5.2 Electrical phenomena  15  1.5.3 Peripheral activities  16  1.5.4 How does it fit together?  •  .  .  .  Chapter 2. BED DEFORMATION THEORY AND METHOD  17 18  2.1 Introduction  18  2.2 Technique for sensor insertion  18  2.3 Qualitative measurements  21  .  2.3.1 Bed casting  •  21  2.3.2 Rubber rod  •  22  •  25  2.4 Quantitative measurements 2.4.1 Electrolytic tilt sensors  26  III  2.4.2 Leaf spring tilt sensors  34  2.5 Potential sources of error  38  2.5.1 Sensor scale effects  38  2.5.2 The ice—bed interface  39  2.5.3 Sensor attitude  41  2.5.4 Sediment intrusion into the borehole  42  2.5.5 Connecting wires  44  Chapter 3. BED DEFORMATION: DATA ANALYSIS  46  3.1 Introduction  46  3.2 Experiment design  47  3.2.1 Ancillary information  48 Internal deformation  49 Basal sliding  53  3.3 The 1988 experiment  53  3.3.1 Correlation with effective pressure 3.3.2 Effective viscosity  61  3.4 The 1989 experiment  66  3.4.1 1989 tilt results  70  3.4.2 1989 strain rates  70  3.4.3 Subglacial pressure  73  3.4.4 Effective viscosity  74  3.4.5 Net strain and mean strain rate  75  3.5 Negative strain rates  76  3.5.1 Fluid models  77 Sheet flow  77 Extrusion  78  3.5.2 Roller bearing models  82 iv  3.5.3 The shadow box computer  84  .  3.6 Discussion  85  3.6.1 Boulton and Hindmarsh flow models  86  3.6.2 Shear stress and normal stress  87  3.6.3 Effective viscosity  89  Chapter 4. ELECTRICAL PHENOMENA  THEORY  91  4.1 Introduction  91  4.2 The rock—electrolyte interface  91  4.2.1 Gouy—Chapman model  92  4.2.2 Stern model  94  4.3 Conduction mechanisms  96  4.4 Electrical resistivity  97  4.4.2 Measuring electrical resistivity  99  . Potential fields  99 Interpretation Current switching Transient effects 4.5 Natural Potentials 4.5.1 Irreversible thermodynamics 4.6 Electrokinetic phenomena 4.6.1 Observing streaming potentials  .  4.6.2 A theoretical development 4.6.3 Hydrodynamics and boundary layers 4.6.4 Zeta potentials 4.6.5 The reverse current The charge accumulation model The charge conservation model  V  4.6.6 Calculating the streaming potential  120  4.7 Electrical phenomena relevant to this study Chapter 5. ELECTRICAL PHENOMENA  —  121 METHOD  123  5.1 Introduction  123  5.2 The 1987 apparatus  123  5.2.1 Electrode configurations  124  5.2.2 Electrode design  124  5.2.3 Voltage measurement  125  5.2.4 Current source and current measuring  126  5.2.5 Additional control  128  5.2.6 Technical specifications  129  5.3 The 1988 apparatus  129  5.3.1 EPROM programs  129  5.3.2 The current multiplexer  130  5.3.3 The potential multiplexer  131  5.3.4 Electrode design  133  5.4 The 1989 apparatus  135  Chapter 6. ELECTRICAL PHENOMENA  DATA ANALYSIS  136  6.1 Introduction  136  6.2 Predicted forcing/response relationships  137  6.2.1 Diurnal phenomena  138  6.2.2 Episodic phenomena  139  6.3 Assumptions  139  6.4 The 1987 experiments  141  6.4.1 Forefield operational test  141  6.4.2 Experimental design  142  6.4.3 Diurnal cycling of d.c. resistivity  144  vi  6.4.4 Geometrical corrections  145  6.4.5 Polarity reversals  145  6.4.6 Streaming potentials  147  6.4.7 Recapping the 1987 field season  148  6.5 Dedicated electrode arrays  149  6.6 The 1988 Experiments  149  6.6.1 Experimental design  150  6.6.2 Telluric noise  154  6.6.3 Potential error  155  6.6.4 Potential gradients  157  6.6.5 Hole connections  158  6.6.6 Overwintering events  162  6.7 1989 Experimental design  164  6.7.1 Fall shutdown  166  6.8 Manipulation experiments  168  6.9 Conclusions  173  6.9.1 Apparent resistivity  173  6.9.2 Streaming potentials  174  Chapter 7. CONCLUSIONS  176  7.1 General comments  176  7.1.1 Shear stress and normal stress 7.2 Electrical phenomena  178  7.2.1 Streaming potentials 7.2.2 Electrical resistivity  176  178 .  .  .  7.3 Basal deformation  179 179  7.3.1 Rheology  179  7.3.2 Effective viscosity  180 vii  REFERENCES..  182  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 197  A.1 Introduction  197  A.2 Historical overview  198  A.2.1 Basic dip and azimuth measurements  198  A.3 Electronic incinometry  200  A.3.1 Measuring tilt  200  A.3.2 Measuring azimuth  200  A .3.2.1 External azimuth control  201  A.3.2.2 Internal azimuth control  202  A.4 The UBC inclinometer  203  A.5 Coordinate systems  205  A.6 Data analysis A.6.1 Calibration  •  .  .  .  •  •  .  .  207  208  A.6.2 Transformations A.6.2.1 Normalization A.6.2.2 Eulerian Angles  212 .  213  .  A.6.2.3 Eulerian Transformation A.6.2.4 Universal application of transformations A.6.3 Inverse problem  •  .  .  .  217  A.7 Interpolation Scheme  220  A.8 Sensitivity  223  A.9 Discussion  227  Appendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS 229 B.1 Niskin sampler  229  B.2 Subglacial vacuum sampler  230  B.3 Considerations  234 yin  Appendix C. SUBGLACIAL DRAG SPOOL  235  C.1 Introduction  235  C.2 The “Slide-O-Meter” or “drag spool”  235  ix  LIST OF FIGURES 1.1. Location of Trapridge Glacier  10  .  1.2. Topographic map of Trapridge Glacier 2.1. The borehole percussion hammer  12 •  .  20  2.2. Bed casts  23  2.3. Results from rubber rod experiment  24  2.4. Electrolytic tilt sensors  •  27  2.5. Data from electrolytic tilt cells  32  2.6. Leaf spring tilt sensor  35  2.7. Data from leaf spring tilt sensors  37  .  2.8. Drag spool data  43  3.1. Location of Trapridge Glacier  48  3.2. Velocity proffle through Trapridge Glacier  49  3.3. Internal deformation proffle  52  3.4. Location map for 1988 experiment  .  54  3.5. Data from 1988 experiment  56  3.6. Comparison with Boulton and Hindmarsh data  65  3.7. Location map of 1989 experiment  67  .  3.8. Data from 1989 experiment  69  3.9. Strain rate data from the 1989 experiment  71  3.10. Pressure data from the 1989 experiment  74  3.11. The vertical displacement record  81  .  3.12. Roller bearing model  83  3.13. The shadow box  85  4.1. The rock—electrolyte interface  93  4.2. Detail of the Stern model interface  .  x  95  4.3. Pseudo-depth calculation  .  4.4. The d.c. resistivity current waveform  104 105  4.5. Induced polarization 4.6. Streaming potential equivalent circuit 5.1. The 1987 d.c. resistivity apparatus configurations 5.2. High voltage current limiter 5.3. A Cu—CuSO 4 porous pot electrode 6.1. Forefield pseudo-section 6.2. Resistivity record P1, 1987 6.3. Resistivity record P2, 1987 6.4. Polarity reversing electrode pattern 6.5. Natural potential record from 1987 experiment 6.6. Relative locations of electrode arrays 6.7. Electrode array template 6.8. The 88DC01 electrode array 6.9. The 88DC02 electrode array 6.10. Effect of telluric noise on measurements  .  6.11. Electrode noise 6.12. Natural potential fluctuation 6.13. Borehole forcing response 6.14. Diurnal cycling  162  6.15. Overwintering apparent resistivity record 6.16. Overwintering natural potential record  .  .  163  .  164  6.17. The 89DC01 electrode array  165  6.18. Fall shutdown electrical phenomena 6.19. Map of 88DCO1B electrode array 6.20. Subglacial phenomena during manipulation xi  172  A.1. Block diagram of UBC inclinometer  204  A.2. The inclinometer coordinates systems  206  A.3. The inclinometer orientation vectors  208  A.4. Eulerian angles transformations  214  A.5. The circular arc interpolation method  .  A.6. Detail of circular arc interpolation method  .  .  220  .  221  A.7. Interpolation of coplanar vectors  222  A.8. Monte Carlo analysis of inclinometer sensitivity  226  B.1. Niskin sampler  231  B.2. Subglacial Hoover  233  C.1. The drag spool  237  XII  LIST OF TABLES 1.1. Geometrical characteristics of Trapricige Glacier  13  1.2. Characteristics of Trapridge Glacier study site  13  4.1. Established phenomenological relations  .  111  .  5.1. Control line function, 1987 apparatus 5.2. Control line function, 1988 current multiplexer  128 .  131  5.3. EPROM programs, 1988 current multiplexer  132  5.4. EPROM programs, 1988 potential multiplexer  133  5.5. Control line function, 1988 potential multiplexer  133  xlii  LIST OF SYMBOLS Meaning  Symbol a  glacier surface slope  /3  glacier basal slope  e  charge on an electron  E  electric field  e  dielectric constant strain rate mean strain rate dynamic viscosity  J  electrical current density  J  generalized flow density  g  gravitational acceleration  C  d.c. resistivity geometrical factor  7 h  sediment yield stress layer or glacier thickness  H  pore geometrical factor  I  electrical current  k  reflection coefficient  kB  Boltzmann’s constant  1 K  thermal conductivity  2 K  hydraulic conductivity phenomenological coefficients  u 1  ion chemical potential  n  number density of ions  fl  surface normal vector  P  water pressure potential  4  interface potential  o  interface potential at surface total electric potential  q  fluid flow density  xiv  Symbol  Meaning  p  scalar electrical resistivity  Pap  apparent resitivity  pi  density of ice  pv  volumetric charge density  pv.  density of water  o  electrical conductivity  T  temperature  r  shear stress  V  electric field potential  X  generalized flow forcing (-potential  z  valence of an ion  , 1 z  pseudo-depth  xv  ACKNOWLEDGEMENTS To my parents, Ir&grid and Wes, for giving me the motivation and skills to attempt a task such as this... .and to Sabine, for her support from afar. I wish to express my sincere gratitude to my graduate supervisor, Garry K. C. Clarke. Throughout my sojourn at U.B.C., Garry’s door has never been closed to me (although it has been locked). Because of his generosity, I have enjoyed five wonder-filled field seasons in the glorious Kluane mountains, a winter sabbatical at the Scott Polar Re search Institute, and numerous conferences. Perhaps Garry’s greatest attribute is his unobtrusive and gentle manner; I have enjoyed what must be an uncommon freedom to explore subjects that have not always served Garry’s or my best interests. I wish to thank my friends and peers in the Department, in particular Marc G&in, Francis Jones, Tavi Murray, B. Barry Narod, Jeffrey Schmok, and Dan Stone, for entertainment and thoughtful discussion. I am grateful for the accommodations my committee members made for a rather hasty completion schedule: Drs. Robert M. Ellis, Rosemary Knight, Douglas Oldenburg, and Tadeus Ulrych. I owe special thanks to K. Dieter Schreiber, William Siep, and Harry Verwoerd for their comments and help on instrumentation design and for their fine machining skills. I also want to thank the Department of Geophysics and Astronomy as a whole for providing a comfortable and friendly atmosphere eminently suitable for research and education; this is no small thing. Funding for the research presented in this thesis, and for food to feed its au thor, has been provided by the Natural Sciences and Engineering Research Council of Canada, the Canadian Northern Studies Trust of the Association of Canadian lini versities for Northern Studies, the University of British Columbia, and the Northern Science Training Program of Indian and Northern Affairs Canada. xvi  Chapter 1 INTRODUCTION  “I say, ‘Why don’t you write an anti-glacier book instead?’ -  Kurt Vonnegut, Jr., 1969, Slaughterhouse Five  “Make as many corrections as you consider needed.” -  Chinese fortune cookie, 17 February 1992  Glaciers have long held the interest of humans. Anyone who has travelled through a glaciated landscape has experienced the sense of awe and wonder that these rivers of ice inspire. But glaciers do not always play a benevolent role in the landscape; people who dwell near glacier fringes would be well advised to cultivate a sense of respect for glaciers and an appreciation for the dynamics of glacier flow. Encroachment of glaciers on structures is a clear threat, but it is likely that in direct consequences of glacier advance, such as ice-dammed lakes, ice avalanches, and outburst floods wreak greater destruction. In the Swiss Alps, alpine glaciers have sent lethal torrents of ice and water into the inhabited valleys below and sometimes the glaciers themselves have crept down into the trees (Hoinkes, 1969; Röthlisberger, 1969a, 1981). Similar events have been recorded in the Karakoram (Desio, 1954; Hewitt, 1969). Wonderful woodcuts, etchings, and paintings record the variations and consequences of glacier motion in the Alps since the late 16th century (e.g. Hoinkes, 1969; Zumbühl and others, 1981). On the west coast of North America, the flooding of land and the outburst floods associated with ice-dammed lakes (formed at the margins of glaciers and by the damming of rivers by advancing glaciers) have been recorded in Indian oral history (e.g. Cruikshank, 1981) and in geological records (Clarke and Mathews, 1981; Clarke, 1982, 1986; Clarke and others, 1984a; Waitt, 1984; Schmok, 1986). Catastrophic  1  2  Chapter 1. INTRODUCTION  sea-level rise resulting from melting of polar ice caps is perhaps the most stealthy of glaciological disasters because the fateful event would transpire in a remote and gener ally unfamiliar area. The subject of this thesis is glacier surging, an uncommon mode of glacier flow that has been associated with each of the glaciological mishaps listed above. 1.1 What is glacier surging?  Not all glaciers move with slow, steady persistence; some glaciers exhibit a periodic flow instability known as surging that is characterized by short periods of rapid glacier flow separating longer periods of normal flow. The quiescent phase of the cycle lasts from 10 to 100 years and the surge phase lasts for 1 to 6 years; for a given glacier, the periodicity of the surge cycle is thought to be regular (Post, 1960; Meier and Post, 1969). During a surge, the average flow speed of the glacier increases by one to three orders of magnitude and results in the transport of ice from a reservoir area to a receiving area (Meier and Post, 1969; Clarke and others, 1986; Raymond, 1987). This definition of glacier surging is practical rather than exhaustive because, as with most natural phenomena, variety abounds and exceptions exist. Surging glaciers are a subset of a larger group, broadly described as fast flow ing. Jacobshavns Isbr, a tidewater glacier in western Greenland, flows at more than 8000 myr’ and is considered the world’s fastest glacier (Lingle and others, 1981), but except for a tidal influence near the terminus, its flow is relatively constant (Bind schadler, 1984; Echelmeyer and Harrison, 1990). It is tempting to speculate that Ja cobshavns Isbr is surging since this sustained velocity is comparable to maximum observed flow rates of surging glaciers elsewhere  —  for instance, the upper parts of  Variegated Glacier, a surging glacier in the Alaska Panhandle, flowed at a similar speed during the 1982—1983 surge (Kamb and others, 1985)  —  but Jacobshavns Isbr  Chapter 1. INTRODUCTION  3  is not delivering more ice to the ocean than is accumulated in its drainage basin (Bind schadler, 1984) so we could reasonably expect the glacier to flow at this speed indef initely. Ice Stream B in Antarctica flows at a speed of about 800 m yr’ (Whillans and others, 1987) and has shown no signs of slackening its pace, yet this ice stream is operating at a deficit  —  it delivers more ice to the sea than is accumulated in its  cachment area (Clarke, 1987a). Perhaps Ice Stream B is destined to slow down and “shut off” as evidence indicates its neighbour, Ice Stream C, has done (Shabtale and others, 1987). This suggests that the ice streams controlling the mass balance (and perhaps the stability) of the Antarctic ice sheet are large analogues to surging glaciers, but operating at correspondingly large temporal scales. Surge-type glaciers tend to be clustered geographically, although why they are found in particular regions is not well understood. In Canada, concentrations of surging glaciers are found in the St. Elias mountains of the Yukon Territory, and on Ellesmere Island, N.W.T. Elsewhere, surging glaciers are found in the Pamirs, the Caucasus, the Tien-Shan, the Karakoram, the Himalayas, the Andes, Svalbard, Iceland, East Greenland, the Alaska Range, and Switzerland (Dolgoushin and Osipova, 1975; Pater son, 1981, p. 283; Clarke and others, 1986). Not all glaciers within these geographical concentrations surge, and glaciers in close association with surge-type glaciers (those that share accumulation zones, for instance) do not necessarily surge. The clustering of surge-type glaciers is probably related to subglacial conditions such as geology and hydrology rather than climatic ones (Post, 1969). For example, glaciers on the “wet” and “dry” sides of the St. Elias Range, Yukon, receive disparate amounts of annual snowfall, yet surging glaciers are found in both areas (Clarke and others, 1986).  4  Chapter 1. INTRODUCTION 1.2 How does one identify a surging glacier?  Surging glaciers are best identified by direct observation of one or more surges. Par ticularly in uninhabited regions, such information is not available and we must rely on other clues (Paterson, 1981; Clarke and others, 1986). The most striking features often found on large surging glaciers are wavey, looped, contorted, or sheared medial moraines.  These features are formed when tributary  glaciers surge or when the main trunk of the glacier surges and ice piles up against stagnant ice and debris near the terminus. Some caution must be used in attributing deformed medial moraines to surging since these features can also be formed by changes in mass balance, movement of ice divides, and other prosaic changes in glacier systems. During the quiescent phase, a surging glacier may develop a characteristic longitu dinal proffle that shows progressive thickening of the glacier in the reservoir zone and thinning in the receiving zone. This evolution of the longitudinal profile reflects the buildup of ice mass in preparation for the next surge phase. The proffles of Medvezhiy Glacier (Pamir Mountains) and Variegated Glacier (St. Elias Mountains), two wellstudied surge-type glaciers, have been shown to evolve in this manner (Dolgoushin and Osipova, 1975, 1978; Raymond, 1987; Raymond and Harrison, 1988) and Post (1960) has observed similar changes in Muldrow Glacier (Alaska Range). The longitudinal profile of Trapridge Glacier (St. Elias Range) has evolved in a different manner as the glacier approaches its next surge phase: a prominent bulge has developed where the glacier flows into the stagnant ice left over from the previous surge; the formation of the bulge is probably controlled by thermal conditions at the base of the ice (Clarke and others, 1984b; Clarke and Blake, 1991). Following a surge, the ice in the lower reaches of the glacier typically stagnates. Progressive melting during the quiescent phase can leave portions of this ice stranded below the main body of the glacier.  The stranded ice, often covered with debris,  may persist until the next surge cycle. It may also be possible to identify surge-type  Chapter 1. INTRODUCTION  5  glaciers by comparing the balance mass flux with the actual down-glacier transport of ice (Clarke, 1987a; Clarke, 1991). In accordance with the buildup of ice in the reservoir zone, the latter should consistently exceed the former for a quiescent surge-type glacier. 1.3 What happens during a glacier surge?  Glacier surging presents some fascinating questions: What causes the transition from a quiescent state to a surge state? How is a surge state maintained for a period of one or more years? How does a surge terminate? Most of the work on surging glaciers has focused on the first of these questions. Several hypotheses for a surge trigger have been proposed. Tarr and Martin, (1914, chapter 10) observed rapid advances of several glaciers in Alaska following the 1899 earthquake at Yakutat Bay and suggested that earthquakes may trigger surges. Sub sequent work by Post (1960, 1967) discredited this hypothesis, but Gardner and He witt (1990) argued that rockslides onto a glacier surface (which can be triggered by earthquakes) may cause a surge in a glacier that is predisposed to surge behaviour. Robin (1955) proposed thermal instability in a surge-type glacier could permit the rapid basal sliding rates which seem necessary to allow high surge speeds. A similar hypothesis developed by Clarke and others (Jarvis and Clarke, 1975; Clarke, 1976) is based on a model containing a transition from a warm bed at the pressure melting point (which allows sliding) to a cold frozen bed (which inhibits sliding), but this hy pothesis is suspect as a generalized surge trigger because temperate surge-type glaciers exist (Bindschadler and others, 1976) (the bulk of a temperate glacier is at the pres sure melting point). Later work (Clarke and others, 1977; Paterson and others, 1978; Cary and others, 1979; Yuen and Schubert, 1979) explored the effects of strain heating on the stability of glaciers and ice sheets (earlier suggested by Weertman, 1957), but the possibility that thermal instability could represent a surge trigger for glaciers was discounted in part because the predicted surge cycle periods were too long.  6  Chapter 1. INTRODUCTION  The only hypotheses that retain wide acceptance are those which focus on the interaction of subglacial water drainage, subglacial water pressure, and sliding processes as a means for explaining aspects of the surge cycle. The reason for this is that the high flow velocities encountered during a surge must be accompanied by rapid sliding at the glacier bed since internal deformation of the glacier cannot possibly contribute significantly to motion. It is now widely accepted that glacier surging is caused by high subglacial water pressure decoupling the glacier from its bed, and thus allowing the glacier to flow rapidly downhill. This high water pressure results from the destruction or disabling of the normal subglacial drainage system. The disabling of the drainage system also causes a rise in the quantity of water stored at the bed. What is not understood is how these alterations are initiated and sustained during a surge. High subglacial water pressure and water storage lead to fast flow rates by re ducing the normal force exerted by the glacier on its bed (thereby reducing the max imum available friction force available to retard ice flow), by lubricating the ice-bed interface, and perhaps by softening subglacial sediments. The positive correlation be tween water pressure and flow velocity has been observed experimentally (Iken, 1972; lodge, 1979; Iken and Bindschadler, 1986; Engelhardt and others, 1987; Kamb and En gelhardt, 1987, Meier, 1989) and explored theoretically (Bindschadler, 1983; Weertman and Birchfield, 1983). The forerunners of current sliding models were Weertman’s “tombstone” model (1957) and Lliboutry’s “washboard” model (1959)  —  the names give an idea of the rep  resentation chosen for basal roughness. Lliboutry (and later Weertman (1962, 1969)) introduced the idea that increases in the basal water pressure could raise the glacier clear of small obstacles in its path and thereby allow it to slide more easily. There fol lowed a period of heated exchange between Lliboutry and Weertman regarding whose model more accurately portrayed the processes of sliding over a hard bed and the disposition of subglacial water (Weertman, 1962, 1964a, 1966, 1967a, 1967b, 1969;  Chapter 1. INTRODUCTION  7  Lliboutry, 1964, 1966, 1967, [discussion following Weertman, 1967a}, 1968). During this period of debate, other researchers made interjections of a theoretical and ob servational nature (e.g. Nyc, 1958; Kamb and LaChapelle, 1964 [commented on by Weertman, 1964b]; Kamb, 1970). Reviews of this debate can be found in Lliboutry (1968, 1979) and Weertman (1979). The salient difference between the two models lies in how they deal with subglacial water: Lliboutry’s model proposes a system of liziked cavities in the lee of basal obstacles; the pressure in and the movement of water between cavities are controlled by narrow, self-adjusting channels, Walder (1986) and Kamb (1987) incorporated these ideas into their proposed glacier surge mechanisms. Weertman’s model proposes a largely continuous film of water under the glacier with obstacles protruding through the film serving to retard glacier flow. It is easy to see how Weertman argues that any thickening of the water film will increase glacier flow rate by drowning obstacles, but the principal flaw in this argument is that there is no feedback mechanism to prevent the excess water draining into subglacial channels  —  the surge would halt straightaway. 1.3.1 Hard bed versus soft bed Until quite recently, most of the theory concerning basal sliding and surge mechanisms has been based on a hard bed model where the glacier is assumed to move over a hard, non-deforming substrate (e.g. Fowler, 1987a, 1987b; Kamb, 1987). Some work began on soft (i.e. deformable) glacier bed models in the late 1970’s (Boulton, 1979a, Boulton and Jones, 1979), but it was not until the late 1980’s that the community at large shifted its thinking towards soft-bedded glaciers (Clarke and others, 1984b; Blankenship and others, 1986; Shoemaker, 1986; Alley and others, 1987a, 1987b; Boulton and Hind marsh, 1987; Clarke, 1987; Alley, 1989a, 1989b; Alley and others, 1989). A soft bed has interesting ramifications for glacier motion and subglacial hydrology. Movement of the ice can occui through deformation of the sediments as well as sliding. Subglacial  Chapter 1. INTRODUCTION  8  drainage can be through channels excised in the ice (Röthlisberger- or R-channels) or sediments (similar to Nye- or N-channels), through a linked cavity system, or through the sediments themselves, but the most important aspect of soft-bed glacier dynamics is that sediment deformation can cause rapid changes in the mode of drainage. Some glaciers and ice sheets are underlain by a deforming layer of water-saturated sediment (Boulton and Jones, 1979; Clarke and others, 1984b; Alley and others, 1986, 1987a; Blankenship and others, 1986; Blankenship and others, 1987; Brown and oth ers, 1987; Clarke, 1987b; Kamb and Engelhardt, 1989; Engelhardt and others, 1990; Alley, 1991; Blake and others, 1991; Clarke and Blake, 1991). High subglacial water pressure associated with glacier surges has been attributed to morphological change in the subglacial drainage system (Röthlisberger, 1969b; Clarke and others, 1984b; Kamb and others, 1985; Alley, 1989a), so we are investigating the possibility that subglacial deformation causes the rise in water pressure by disabling the normal drainage sys tem. Competition between processes that establish the drainage system and those that destroy it could easily degrade the efficiency of subglacial drainage. If the drainage system of the bed is adversely affected to the point where water input exceeds drainage capacity, then high water pressure and surge conditions might arise. 1.3.2 Seasonal timing In several instances, the initiation and termination of surges has been observed to occur in certain seasons. These observations may provide clues as to the nature of the surge mechanism. During the 1982—1983 surge of Variegated Glacier, both surge pulses began in winter and terminated in early summer (Kamb and others, 1985). Recent surges of Medvezhiy Glacier in 1963 and 1973 also conformed to this seasonal schedule (Dolgoushin and Osipova, 1975), but a recent surge of Peters Glacier initiated in summer and terminated in mid-winter (Echelmeyer and others, 1987). According to Raymond (1987), no contradictory evidence has been found for seasonal initiation  Chapter 1. INTRODUCTION  9  timing, but he notes that generalized statements concerning the timing of surge onset and termination should be avoided since observation of glacier surge events is haphazard and spotty. 1.4 The study site  The St. Elias Range, Yukon Territory, has one of the world’s highest concentrations of surge-type glaciers. Many of the larger ones, such as the Lowell, Tweedsmuir, Steele, and Donjek Glaciers, cause flooding and outburst floods when they surge across the rivers they abut. As recently as 1852, the land now occupied by the town of Haines Junction and a good portion of the Alaska Highway were flooded by an episode of Neoglacial Lake Alsek.  This lake is created when the Lowell Glacier surges across  the Alsek River (Schmok and Clarke, 1989) and pinches off the flow against Goatherd Mountain. Trapridge Glacier is a small subpolar surge-type glacier in the later stages of its quiescent phase (the temperature profile of a subpolar glacier begins with below-freezing temperatures in its upper layers and warms to the pressure melting point at the base). Trapridge Glacier is located in the St. Elias Range near the northern extent of Kluane National Park (Figure 1.1). The glacier flows eastward off a flank of Mt. Wood into a valley that contains two other glaciers that have been identified as surge-type: Rusty and Backe Glaciers (Clarke and Classen, 1970) (the names of the glaciers have changed since this paper was published; Trapridge, Rusty, and Backe Glaciers were earlier known as Hyena, Fox, and Jackal Glaciers respectively). Trapridge Glacier is the study site for the experiments discussed in this thesis.  Chapter 1. INTRODUCTION  10  Fig. 1.1: (a) The location of Trapridge Glacier in the St. Elias mountains, southwestern Yukon. (b) Trapridge Glacier is located near two large surgetype glaciers, the Steele and Hodgson Glaciers. (c) Trparidge Glacier flows roughly north by east off the flanks of Mt. Wood (ice is white, bare ground is shaded). Evidence for the surge behaviour of Trapridge Glacier is threefold: (1) Photo graphic evidence from two expeditions into the Steele Glacier/Mt. Wood area in the late 1930s through to 1941 indicate that Trapridge Glacier was then in a pre-surge state (Wood, 1940, cover plate; Sharp, 1947, Fig. 5; Sharp, 1951, Plate 5B). Sharp (1947) de scribes “Glacier 13” as advancing rapidly, but photographs of Trapridge Glacier taken in recent years look almost identical to those taken during this 1941 expedition, and no surge is currently in progress. The events of World War II disrupted exploration of the area and it was not until 1951, when the first vertical air photographs of Trapridge Glacier were taken (Clarke and others, 1984b, Fig. 3), that the observational record resumed (in 1943, a vertical air photography pass up the Steele Glacier valley stopped tantalizingly short of Trapridge Glacier). In the 1951 air photo, Trapridge Glacier has advanced far down the valley; sometime between 1941 and 1951, Trapridge Glacier surged. Subsequent vertical air photography through the 1970s shows the stagnation  Chapter 1. INTRODUCTION  11  and melting of the lower reaches of the glacier (Clarke and others, 1984b, Fig. 3). (2) Stagnant ice in the valley below the glacier is visible in both the 1941 photographs and those taken in recent years, although the amount of stranded ice present in the 1941 photographs is significantly greater. The vertical air photo sequence shows clearly that this ice is the remains of the ice delivered to the receiving zone during the pre vious surge. (3) Since 1969, surface measurements on Trapridge Glacier have followed the evolution of a wave-like bulge in the upper, active parts of the glacier (Clarke and others, 1984b; Clarke and Blake, 1991). The formation of this bulge reflects the accumulation of ice in preparation for the next surge. Trapridge Glacier is an ideal study site precisely because the glacier is small and thin. Its small size (roughly 1 km by 4km) allows us to roam the entire surface with ease (Figure 1.2). Because the glacier is only about 75m thick, we can also drill many holes to the glacier sole; our record for holes drilled was 83 in 1989. Researchers on larger (and thicker) glaciers and ice streams are limited to a fraction of this number of boreholes. Holes to the bed of the glacier are vital for studying basal phenomena and our ability to drill so many holes presents unique opportunities for the UBC glaciology group. Figures 1.1, 1.2 and 3.1 show the general location of the study area used for the bed deformation studies. Extensive hot-water drilling in this area indicates that the glacier has a very uniform thickness of about 72 m; during a given drilling season, we can often predict the depth of a new hole to within 10—20 cm. For several hundred metres to the north, west, and east, the glacier topography is gently undulating. To the south, there is a slight rise associated with heavy crevassing. The surface and basal slopes, as determined by surveying and drilling, are both 7° in the direction of glacier flow. The flow direction, as determined by stake surveys, is 11° north of east (bearing 79°). Data from vertical strain sensors indicates that the area is under moderate compression.  Chapter 1. INTRODUCTION  Fig. 1.2: Topographic map of Trapridge Glacier showing the entire catche— ment area. The contours and boundaries are based on 1981 air photography. The glacier has advanced about 300 m since that time in a north by east direction.  12  13  Chapter 1. INTRODUCTION  Tables 1.1 and 1.2 summarize the geometrical and thermal characteristics of Trapridge Glacier and the study site. Elevation range Elevation of equilibrium line Glacier length  2250—2900m —2400 m 3.5 km  Table 1.1: The geometrical characteristics of Trapridge Glacier (compare with Figures 1.1, 1.2 and 3.1).  Elevation Surface flow direction Width Thickness Surface slope Basal slope Temperature at lOm depth Basal temperature Net annual ablation (estimated from annual_survey_of flow marker_poles) Mean annual flow rate (1984—1991)  —‘2370m 110 north of east 0.95 km ‘-72 m 7° 7° —6°C to —3°C 0°C 0.5—1.Omyr’ 1 33 + 1 m yr  Table 1.2: Thermal and geometrical characteristics of the Trapridge Glacier study area (sources: Clarke and Blake, 1991; unpublished data)  1.5 Thesis scope It is our belief that a greater understanding of surging phenomena can be gained by studying the basal processes of surge-type glaciers. To date, theoretical developments in the study of surging have outpaced complementary and elucidating experimental work, principally because of the difficulties in making useful measurements in a subglacial environment. Experimental geophysicists are largely in the business of making remote sensing measurements of natural phenomena. When the object of investigation is the deform ing bed beneath a glacier, measurements are necessarily of a remote nature and can be  Chapter 1. INTRODUCTION  14  divided into two classes: (1) measurements made from the glacier surface and (2) mea surements made near the ice-bed interface. Surface measurements are remote in the classical sense; some energy field, either a natural one or one arising from the applica tion of an artificial primary field, is observed. Basal measurements may be similar to surface measurements or they may be direct measurements of basal properties in the vicinity of the sensor. Measurement techniques that have been used to investigate basal processes are listed together with representative references: Surface measurements include radar sounding (Jones, 1989), seismic refraction (Blankenship and others, 1987), electrical resistivity (Röthlisberger, 1967), and gravity (Bull and Hardy, 1956). Basal measure ments include temperature (Clarke and others, 1984b; Clarke and Blake, 1991), seismic tomography (Clarke and Blake, 1990), electrical resistivity (Haeberli and Fisch, 1984; Hooke and others,. 1988), water pressure (Iken and Bindschadler, 1986), and deforma tion (Boulton and Hindmarsh, 1987; Kohier and Proksch, 1991). Prior to the development of efficient hot-water drilling techniques, researchers were limited to making surface measurements and basal measurements requiring a small number of holes. In recent years, basal measurements, some involving many holes, have become in vogue because they provide an opportunity for detailed examination of the subglacial system. Nevertheless, surface measurements are still yielding astonishing results (for example, seismic refraction techniques were used to discover the deforming sediment layer beneath Ice Stream B, Antarctica). The glaciology group at UBC is currently using a suite of specialized instruments for making basal measurements. Many of the instruments developed in the course of this thesis work continue to be used; more instruments are being developed every year. In preparation for my first data collection season in 1987, we decided to pursue two little-used basal measurement techniques that we felt had great potential for clarifying basal processes associated with a surge-type glacier: basal deformation and electrical  Chapter 1. INTRODUCTION  15  resistivity. A serendipitous failure of the electrical resistivity equipment that summer led us to expand our investigations to include naturally occurring electrical potentials. 1.5.1 Basal deformation An understanding of subglacial deformation processes is crucial for developing a rheol ogy for subglacial sediments and for understanding the role that these sediments might play in surging. In 1987, we were aware of only one other research group who had made in situ measurements of subglacial deformation (Boulton, 1979a; Boulton and Hindmarsh, 1987). As will be discussed in Chapter 3, their work has various limitations that throw into question the value of the rheologies they derive; our experiments constitute an attempt to correct these faults and to make definitive measurements of subglacial deformation. Our measurements are also the first to be made beneath a surge-type glacier. 1.5.2 Electrical phenomena Subglacial electrical resistivity and streaming potential measurements have the ability to monitor the movement of water in the subglacial environment. Electrical resistivity provides a measure of the bulk water content of the sediments and streaming potentials provide a measure of the water pressure gradient through the sediments; measured over time, these electrical phenomena should be sensitive to the morphological changes in basal sediments that lead to elevated subglacial water pressure and surge initiation. In 1987, we were also only aware of one other researcher who had made basal mea surements of subglacial electrical resistivity (Haeberli and Fisch, 1984) although mea surements were not made over a period of time. Several researchers have made surface measurements of resistivity (e.g. Hochstein, 1967; Röthlisberger, 1967; Röthlisberger and Vögtli, 1967; Vögtli, 1967; Fisch and others, 1977), but there are obvious difficul ties in observing a conductive subglacial system through a thick and effective insulator  Chapter 1. INTRODUCTION  —  16  the glacier. Since the initiation of our work, Brand and others (1987) have published  measurements of subglacial resistivity beneath Storgiaciliren, Sweden, but their exper iment was on a much smaller scale, both temporally and spatially, and did not involve measuring the evolution of subglacial streaming potentials over time. Our observations of subglacial streaming potentials are a first for glaciology and our measurements of electrical phenomena in general are the only ones spanning a considerable period of time. Additionally, no other researchers have made electrical measurements of any sort on surge-type glaciers. 1.5.3 Peripheral activities The electrical phenomena experiments require not only the drilling of many holes, but also the calculation of borehole trajectories so that the position of subglacial sensors can be determined. This requirement became evident when the difficulty in interpret ing the data from 1987 was encountered. In 1988, we borrowed an inclinometer from the U. S. Geological Survey, and in 1988, we had our own prototype inclinometer built. The processing of inclinometer data is by no means a trivial procedure and consider able effort was expended in developing a robust processing algorithm. The analytical procedures were derived specifically for the UBC instrument but are applicable to in clinometers in general. This work is detailed in Appendix A. In the future, we hope that accurate measurements of internal deformation within Trapridge Glacier will be made with the UBC inclinometer. Interpretation of subglacial deformation and speculation concerning the surge mechanism are facilitated if samples of subglacial material are available. Samples from the forefield of the glacier provide a good hint as to what material is found under the ice, but it is desirable to have samples taken directly from the actively deforming bed. Two original designs for subglacial samplers have been developed at UBO: the  Chapter 1. INTRODUCTION  17  “Hoover” and the modified Niskin sampler, although I can only claim to have designed the former. These instruments are described in Appendix B. 1.54 How does it fit together? The relationship between subglacial electrical phenomena and subglacial sediment de formation might seem obscure, but both these investigative avenues provide informa tion concerning the disposition of water in the subglacial system and may supply clues relating to the glacier surge mechanism operating at Trapridge Glacier.  Chapter 2 BED DEFORMATION: THEORY AND METHOD  2.1 Introduction Deformation beneath soft-bedded glaciers may be a physical mechanism that con tributes to flow instabilities such as surging.  A rheological description is required  if the role of bed deformation is to be understood, but the development of a rheology is hampered by a paucity of in situ stress and strain measurements. In this chapter, we describe four new techniques for measuring subglacial strain. Three of these give continuous strain measurements, a capability that permits calculation of instantaneous strain rates and allows comparison of strain data with other time series.  We also  describe a technique for inserting sensors into subglacial sediments. We limit the scope of this chapter to a description of experimental techniques and include data only to illustrate strengths and weaknesses of each approach, and to draw obvious qualitative conclusions concerning sediment rheology. A more complete data set and analysis are presented in Chapter 3. The substance of this chapter has been published previously in the Journal of Glaciology (Blake and others, 1991). 2.2 Technique for sensor insertion A hollow percussion hammer allows a flexible instrument to be inserted into the stiff glacier bed (Figure 2.1). The hammer consists of a 2 m long tubular stainless steel body having an inside diameter of 10.87mm and an outside diameter of 17.3mm (0.375 in, schedule 80 seamless pipe). Brass blocks of 3.81 cm (1.500 in) diameter are fixed to both ends of the body. The lower block serves as an anvil for a tubular striker that  18  Chapter 2. BED DEFORMATION: THEORY AND METHOD  19  slides over the body. This striker, which is 60 cm long and weighs 2.1 kg in water, is suspended by a 1.59mm (0.0625 in) stainless steel wire yoke that threads through the upper stop block. The lower body thread extends below the anvil and allows various accessories to be attached. When installing a bed strain instrument, we attach an insertion sheath of sufficient internal diameter to allow a loose fit for the instrument. Each instrument is terminated with a conical brass tip that fits snugly into the sheath when the instrument and its cable are threaded through the percussion hammer. The tip has a larger diameter than the sheath, so that the lip of the cone forms a blunt annular barb. While the hammer, with the bed strain instrument loaded inside, is lowered down a hole, its weight is carried by the wire rope. The instrument cable is held just tight enough to keep the instrument from sliding out of the hammer. To avoid jostling the assembly, the rope and cable are laid out on the ice surface and the operator walks toward the hole from a distance greater than the thickness of the glacier, holding both lines. When bottom is reached, the taut instrument cable is marked level with the ice surface. As the wire rope is used to drop the striker repeatedly against the anvil, percussive forces are transferred to the brass tip; the instrument remains protected within the sheath. The cable is left slack during hammering so that impulsive advances of the hammer and instrument will not yank on the cable. Insertion depth is measured by observing the displacement of the mark with respect to the ice surface, although this displacement does not reflect the position of the instrument with respect to the ice—bed interface. Hydraulic excavation by our hot water drill creates a soft, decimetres-thick disturbed layer in the subglacial material through which the hammer penetrates by virtue of its static weight. The hammering procedure results in penetration of the underlying undisturbed basal material. Thus, even if the length of the instrument is less than the insertion depth into undisturbed material, the entire instrument may be below the ice—bed interface.  Chapter L BED DEFORMATION: THEORY AND METHOD  20  WIRE ROPE  STOP STRIKER  BODY  ANVIL INSERTION SHEATH INSTRUMENT TIP Fig. 2.1: Schematic diagram (not to scale) of the borehole percussion hammer used to insert flexible strain instruments into the soft bed of Trapridge Glacier. The tool is sufficiently narrow that a figure drawn to scale would hide details.  The hammer weight is light (2.1 kg in water), but repeated light blows have a cum mulative effect similar to a few heavy blows and are certainly gentler on the instrument. As many as one hundred blows may be required before the hammer ceases to advance. When the hammer is withdrawn, the lip of the instrument tip catches in the sediment and draws the instrument from the sheath. Once the harmner has been lifted to the  Chapter 2. BED DEFORMATION: THEORY AND METHOD  21  surface, the instrument cable is tied off at the surface with 1—1.5m of slack cable let down the hole. 2.3 Qualitative measurements  Bed strain measurements were first attempted during the 1987 field season. Our pri mary goal was to determine if measurements of bed deformation were possible using the percussion hammer insertion technique. With this in mind, we developed two simple methods for making qualitative strain measurements: the bed cast and the rubber rod. 2.3.1 Bed casting The bed cast is a simple method for measuring total strain. A length of vinyl tubing, containing freshly mixed casting resin and a heating wire, is hammered into the bed. At freezing temperatures the catalyzing reaction is effectively halted, so the tube remains flexible. After several days of deformation, the resin is heated and hardened by passing a current through the wire. The stiff tube, now a cast of the deformation to which it has been subjected, is then pulled out. In most cases, the cast retains enough elasticity to survive passage through the borehole undamaged; if the cast should crack, its shape is easily reconstructed. In our experiments, we use vinyl tubing having an outside diameter of 6.35mm (0.250 in) and an inside diameter of 4.76 mm (0.1875 in). The polyester casting resin is doped with the recommended amount of catalyst for hardening at room temperature. The nichrome heating wire has a resistance of 35 Il m . Hardening is accomplished 1 by circulating 0.9—1.3 A of current through the wire for —‘3 h. This represents a power dissipation of 28—59Wm’ into the resin. The nichrome wire is sheathed in teflon “spaghetti” tubing to prevent the vinyl tube from melting. Because the resin is attacked by water, care must be taken to ensure that the instrument is watertight.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  22  Figure 2.2a shows a cast which was hammered 12 cm into the bed and then cat alyzed immediately. As expected, no deformation is evident. The slight curling of the cast was caused by shrinkage of the resin as it set. Figure 2.2b shows a cast which was inserted 7 cm into the undisturbed bed, and left 92 h before heating began. The dashed line in the figure represents the surface of the undisturbed bed material, and the dotted line indicates the estimated location of the ice—bed interface (20 cm above the undisturbed material). The distinct double bend offset near the bottom of the cast suggests that the zone containing the bend was deforming, but because the cast may have extended up into the ice, the bending could be attributed to glacier sliding. During the experiment, the glacier surface above the instrument moved approximately 40cm. Two other bed casts were successfully installed in 1987 (a fifth cast was not properly anchored in the bed and detached when the hammer was withdrawn). One, installed to a depth of 12 cm, had a break in its heating circuit and could not be catalyzed. The other, installed to a depth of 35 cm, was held so tightly by the glacier bed that it snapped in two and only the upper 10cm of the cast was recovered. 2.3.2 Rubber rod To ascertain whether strain rate varies with time, we built a rudimentary strain gauge capable of making continuous qualitative strain measurements. The rubber rod consists of a 20 cm length of 6.35 mm (0.250 in) square-section rubber onto which two strain gauge networks are bonded. The gauge networks are bonded onto the faces of the rod near its midpoint and can record fiexure along two axes. The rod is sheathed in a glycerin-filled protective vinyl tube. Results from two days of observation using this instrument confirmed that tempo ral changes in the strain rate occur (Figure 2.3). The experiment ended when excessive  Chapter 2. BED DEFORMATION: THEORY AND METHOD  a  bñ -4---  ••e•....  .— — — — —  23  .  -_  ....  .....  I  I  .... ......  I  I  I  — •  4 1 Ocm  Fig. 2.2: Tracings, from photographs, of Trapridge Glacier bed casts on a 10 cm by 10 cm grid. (a) A bed cast catalyzed and withdrawn immediately after insertion. (b) A bed cast left in the deforrning layer for 92 h before re moval (the tip was lost upon withdrawal). The dashed line indicates the upper boundary of the undisturbed bed into which the bed casts were hammered; the dotted line indicates the estimated location of the ice—bed interface. Glacier motion is to the left. strain caused breaks in the strain bridge wiring. The glacier surface velocity was ap proximately 12 cm day 1 during the course of the experiment. The axes of strain are  Chapter 2. BED DEFORMATION: THEORY AND METHOD  24  labelled down-flow and cross-flow assuming that the upward trending record (solid line) represents strain in the direction of ice flow, and the second record (dotted line) repre sents strain perpendicular to this direction. We believe these assumptions are correct because the physical characteristics of the rod cause it to twist within the vinyl tube so that the bending strain is shifted to one pair of faces. A striking feature of the downflow strain record is that it is not monotonically increasing. The slope of the curve, which is proportional to strain rate, is occasionally zero or negative. Because defor mation counter to the direction of ice flow is unlikely, the negative strain rates suggest sporadic extrusive flow within the deforming layer. Extrusive flow within the layer is not excluded because, unlike the glacier itself, the layer is confined from above and be low by rigid boundaries. The glacier forms a rigid upper boundary, and non-deforming sediment or bedrock a lower boundary.  Cl) I  Zz  —D  10  fri —I ‘dl  9  10  11  12  13  AUG 1987 Fig. 2.3: Results from the 1987 rubber rod experiment. The vertical scale is in arbitrary linear strain units, as the device was uncalibrated. The solid line represents strain in the direction of ice flow. The dotted line represents cross-flow strain.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  25  2.4 Quantitative measurements  Our success with qualitative techniques led us to design two types of tilt sensors capable of making continuous quantitative measurements of strain rate at several levels within the bed. Data from subglacial tilt sensors can be used to compute instantaneous strain rates (averaged over the length of the cell) by numerical differentiation of the tilt time series. For the purposes of this derivation, we assume that the tilt cells are experiencing simple laminar shear in a fluid. A fluid model is reasonable for a fine-grained material where the tilt cell is much larger than the largest clast, but as the subglacial material beneath Trapridge Glacier is inhomogeneous and contains clasts of a size comparable to that of the tilt cells, this model is not particularly suitable. It is likely that at the scale of observation of the tilt cell, the deformation cannot be treated as simple shear and that the tilt cells are influenced by the rotation or rocking of neighbouring clasts. It is also likely that the strain experienced by the tilt cell is not uniform over its length. Nevertheless, the fluid model does provide a framework for the interpretation of basal deformation data. The question of scale of observation is addressed in a subsequent section. Consider a coordinate system having its x axis positive in the direction of ice flow, and the z axis normal to the bed and positive upward. In a macroscopically homogeneous material, the strain rates  =  and lfOu  are defined as t9w’\  (2.1)  + and =  lfOv  thv’\ +  (2.2)  Chapter 2. BED DEFORMATION: THEORY AND METHOD  26  where u, v, and w, are the down-flow, cross-flow, and upward components of velocity respectively. If Ow/Ox and Ow/Oy are negligible, these equations become 1 Ou  1 OtanOd Ut  1 Ov  1 Otan8  (2.3)  and (2.4)  where the velocity gradients are expressed as the rate of change of down-flow tilt angle 9, and cross-flow tilt angle 9. Tilt angle is measured with respect to the z axis; positive tilt corresponds to tilt in the positive x and y directions. 24.1 Electrolytic tilt sensors Electrolytic tilt cells operate on the principle that the conductance between two elec trodes immersed in an electrolyte is proportional to the total wetted surface area of the electrodes. A single aids electrolytic tilt cell consists of three electrodes partially immersed in an electrolyte (Figure 2.4). Electrodes may descend from the top of the cell or ascend from the bottom. As the cell is rotated about the plane of the electrodes, the central electrode remains immersed to the same level and the lateral electrodes are dipped in and out of the electrolyte. To make a tilt measurement, the tilt cell is connected to an alternating cycle (a.c) bridge containing two reference resistors (a.c. excitation prevents electrode polarization). The magnitude of the output voltage of the cell is roughly proportional to the tangent of the tilt angle, and the sign determines the tilt direction. A dual axis tilt cell has five electrodes arranged in a with the central electrode common to both circuits.  “+“  pattern  Chapter 2. BED DEFORMATION: THEORY AND METHOD  27  BRIDGE CIRCUIT  ELECTRODES ELECTROLYTE Fig. 2.4: Schematic diagram of the electrolytic tilt cell and electronics used in the 1988 experiment. The cell is 57mm long and 16 mm in diameter. An a.c. excited bridge is formed between the cell and two reference resistors. As the cell is tilted, the lateral electrodes dip in and out of the electrolyte and their resistance with respect to the central electrode changes. The bridge imbalance is measured to give the degree and polarity of tilt. For dual axis tilt sensitivity, a second pair of lateral electrodes is mounted at right angles to the first pair and the central electrode is shared by both circuits. Our tilt cells have copper electrodes suspended from the top of an acrylic cell. The electrolyte is a weak solution of Alconox, a common laboratory detergent. The space above the electrolyte is filled with naphtha to prevent water inifitration under pressure. When microfractures are etched in the surface of the acrylic by rinsing it for 5 s in methanol, no visible meniscus forms between these two fluids; the fluid interface remains horizontal as the cell is tilted. The presence of a meniscus would interfere with  Chapter 2. BED DEFORMATION: THEORY AND METHOD  28  accurate measurements of tilt because the electrodes are in close proximity to the wall of the cell. Our choice of immiscible fluids represents the best formula picked during a series of desperate in-field tests. During the 1988 field season, we discovered that the original electrolyte choice  —  saturated copper sulphate solution  —  did not behave properly in  the presence of naphtha or any other insulating fluid available to us. We are certain that better choices exist (such as those in commercial devices) and do not recommend that this electrolyte formulation be used. Since the tilt cells are not recovered after the experiment is completed, the use of commercial tilt cells for these experiments was too expensive. Three cells, each 57 mm long and 16 mm in diameter, are assembled into a string at a centre-to-centre spacing of 10 cm. The string is cased in a protective sheath of heat shrink tubing, and the lowermost cell is fitted into the insertion tip. Prior to insertion, the cells are calibrated on a special jig. The jig allows a cell to be rotated about its long axis while the tilt of the axis is fixed at various angles. Our cells are calibrated at tilt angles ranging between  00  and 60° from vertical. For each calibration tilt angle Oj,  the output voltages from each tilt circuit are fitted, using least squares, to functions of the form V(8, q ) 5  =  A sin(4i + B) + C  (2.5)  where the subscript i enumerates the discrete tilt angles, A are the fitted signal ampli tudes at those tilt angles, B are the azimuth offsets, C are voltage offsets, q’ represents the azimuth of the tilt, and V is the predicted voltage output. It is important to note that the azimuth q is measured in a local coordinate system where the z-axis aligned with the long axis of the tilt cell; the azimuth is in no way equivalent to orientation in a geographical coordinate system.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  29  For each tilt angle 8, the least squares fit is found usin g the equations (m  sin qf  sin ci) n  7%  Os  n  a  (2.6a)  -  7%  7%  n  /3  (2.6b)  A  =  i/(a2 +132)  B = 2 tan  (2.6d)  —  a  —  where =  (  (2.6c)  a  sin  (2.6e) j=1)  /7%  7%  :1=1  —  ( >:  (2.6f)  \j1  /7%  sin  =  (nsinc  (2 .6g)  -.  In siIt.cos  injcosqj  =  \  —  , 5 sinq  (2.6h)  j=1  and where Vj are the measured output voltages at azimuths cEj. The symmetry evident in the expressions for a and /3 suggests, as is inde ed the case, that the least squares fit  Chapter 2. BED DEFORMATION: THEORY AND METHOD  30  is derived for the equation V(9, 4)  =  a sin(q5) + /3 cos(4) + C  (2.7)  Equations (2.6c) through (2.6e) are used to determine A, B, and C. A tilt cell consists of two tilt circuits, so each cell has two sets of functions of this form associated with it. For a given tilt angle, the circuits in a perfect cell would have  values of B differing by exactly 90°. In practice, the electrode groups are imperfect and the values of B differ by roughly 90°. When the field data are analyzed, a natural cubic spline (Press and others, 1986, p. 86) interpolates values of A, B, and C at tilt angles intermediate to those for which calibrations were performed. This transforms the two sets of discrete calibration functions for a specific cell into the continuous system of equations V(,4) =A(8)sin[4+B(6)] +C(O) V,(8, q.’)  =  A(O) sin{ + B(9)] + C(O)  (2.8)  where the subscripts x and y distinguish the orthogonal tilt circuits. The Newton Raphson method for nonlinear systems of equations (Press and others, 1986, p. 269) is used to calculate the values of 9 and q that would give the observed output voltages V and Vi,. The error on tilt and azimuth for these sensors was estimated by using the calibration data as input to the inversion scheme. The tilt angle error is +1° at 0° of tilt, increasing to +2° at 30° of tilt, and to ±3° at 60° of tilt. The azimuth error is ±5°. Though these errors are large, they are systematic. We believe that the relative error between successive measurements is negligible because the tilt and azimuth records appear smooth. Based on zero relative error, the error in strain rate is 1% at 0° of tilt, increasing to 4% at 30° of tilt, and to 20% at 60° of tilt. On 10 August 1988, an electrolytic tilt sensor string was hammered 8 cm into the undisturbed bed of Trapridge Glacier. Although this insertion depth is shallow, this is insertion depth into undisturbed sediments; because the insertion hammer penetrates  Chapter 2. BED DEFORMATION: THEORYAND METHOD  31  through disturbed sediments before coming to rest at the bottom of the borehole, the tilt cells are actually inserted more than 8 cm below the ice—bed interface (section 2.5.2). Figure 2.5 shows the results for 23 days of observation. The data are presented in the raw form generated by applying the calibration functions: the solid lines indicate tilt from vertical, and the dotted lines indicate the azimuth of the tilt with respect to the internal coordinates of the cell. During the course of the experiment, the glacier surface in the study area moved about 10 cm day’. The data for the lowest two tilt cells indicate an initial net tilt angle of about 45°; this is a surprisingly large value. The deformation instrument is inserted co-axially with the bottom of the borehole, but this borehole had a 3° up-glacier tilt. Some other disturbance caused the tilt cells to “fall over” shortly after the percussion hanuner was removed. We believe that the heat shrink sheath stiffened the instrument sufficiently so that when slack cable was lowered down the borehole, the instrument was pushed over. Even the bottom cell, which was inserted into undisturbed material, was affected. Evidently, the disturbed layer of the bed was not firm enough to prevent this motion. The rapid onset (within one day) of independent cell motion suggests that the disturbed layer recovered quickly, allowing further shear deformation to be recorded. If we assume that the principal direction of tilt is in the down-flow direction, we can further decompose the net tilt and azimuth values into down-flow and cross-flow components of tilt. Variations in the azimuth of a tilt cell result from three motions: (1) rotation of the tilt cell about its long axis, (2) shear-like movement of the tilt cell in a direction perpendicular to the vertical plane passing through its axis (this will also change the net tilt angle), and (3) pivoting of the tilt cell through a near-vertical orientation. This last movement results in a distinctive 180° shift in azimuth and a sharp dip in the net tilt angle. In the absence of information on the orientation of the tilt cell in a geographical coordinate system, the relative contribution of each of the first two motions to changes in azimuth are unknown. In order to interpret the data,  Chapter 2. BED DEFORMATION: THEORY AND METHOD  180  90  0.  w w 90  111111111  0  a  •  (I)  32  I  11111111111  Cl)  I  —180w w 80c. w  w 180  0 I-  = 1—  90  180  0  80 10  20 AUG 1988  30  glacier flow  —..  Fig. 2.5: The 23 day record from the 1988 electrolytic tilt sensor string. The upper, middle, and lower cells are labelled (a), (b), and (c), respectively. Solid lines indicate tilt from vertical (left scale); dotted lines indicate the azimuth of the tilt with respect to the internal coordinates of the cell (right scale). The records from the two lower cells indicate that they were within a deformation zone that sometimes experienced zero and negative strain rates. The arrows indicate these times. The cartoon below the graphs shows the position of the individual cells within the deforming layer at specific times. Ice movement is to the right. we make the assumption that azimuth changes due to rotation are generally long-term (over several days) and that changes due to shearing are generally short-term (less than  Chapter 2. BED DEFORMATION: THEORY AND METHOD  33  one day). The basis for this assumption lies in the observation that the azimuth time series tends to be a slowly-varying function upon which fluctuations are superimposed. By attributing the azimuth drift to cell rotation (which has no effect on strain rates), we can isolate the azimuth fluctuations, which in turn are attributed to both rotation and shear. This assumption is of little consequence because altering the partitioning of azimuth change between these two motions has a surprisingly small effect on the computed strain rates. In our analysis, the azimuth drift is removed by subtracting a linear drift function. The decomposition process must also account for the initial orientation of the tilt cell relative to the glacier flow direction; the initial cross-flow tilt may be non-zero. The azimuth records shown in Figure 2.5 show little variation from piecewise straight lines and, for this sensor string, the dip direction of the hole is aligned with the glacier flow direction. We conclude that almost all the tilt for this particular data set is in the down-flow direction so the data separate naturally into down-flow and cross-flow tilts. The component decomposition for these data are not shown because we wish to show the smooth character of the raw data, but the decomposition would show a near-zero cross-flow tilt and a down-flow tilt mimicking the net tilt. Early in the experiment, each of the lower two cells experienced a  G 180  shift in azimuth associated  with a sharp decrease in tilt angle. This suggests that these two cells originally tilted up-glacier; the azimuth shift indicates when each cell pivoted to tilt down-glacier. It is evident that all the cells quickly reached an orientation outside their calibrated range. Indeed, the uppermost cell seems to have passed the whole experiment resting on its side. Late in the experiment, this cell was tipped over, probably by the glacier pulling on the cable. As with the strain record from the rubber rod, brief intervals of zero and negative strain rate are observed. These are indicated by arrows on the figure.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  34  24.2 Leaf spring tilt sensors The disadvantages of our electrolytic tilt cell are that it has a limited range over which tilt measurements can be made and that the measurement error is large.  Special  attention must be paid to sealing the cell and care must be taken in choosing a suitable stable electrolyte. These problems are overcome with the leaf spring sensor illustrated in Figure 2.6. Two leaf spring pendula measure the tilt along perpendicular axes. Each leaf spring consists of a small clamped strip of 50.8 um (0.002 in) spring brass having a mass attached to the free end. A pair of small strain gauges is bonded to the faces of the spring. As the cell tilts, the weight of the mass bends the spring, and strains the gauges. The output voltage is roughly proportional to the sine of the tilt angle, so that the tilt angle itself is ambiguous when the cell is tilted more than 90° from vertical. These cells are 47mm in length and 16 mm in diameter. Each cell is made of acrylic, and is filled with Dow Corning 200 fluid. This inert silicone fluid is non-conductive and non-corrosive, serves to dampen vibration of the masses, and thwarts water inifitration (all these characteristics prove necessary). The damping fluid has a kinematic viscosity u of 50 cSt (1 cSt  =  1 mm 2 s’; the centi-Stoke is the cgs unit often used to describe  fluids in chemical supply house catalogues). Since the density p of the silicone fluid is close to that of water, the dynamic viscosity  =  p is about 0.05 Pa s. The pendulum  weights are made of split lead shot. The cells are assembled into strings of three having a centre-to-centre spacing of 7.5 cm. The cells are connected using 36 AWG solid copper wires. The strain bridge circuit is such that eight wires lead up to the surface from the upper cell, six wires connect the upper and middle cells, and four wires connect the middle and lower cells. Because the protective heat shrink sheath created problems in 1988, no sheath was placed over these sensor strings. Some permanent straining of the brass springs was observed to occur during the hammering process, but this shift in the calibration is easily corrected by using tilt recordings made before, during, and  Chapter 2. BED DEFORMATION: THEORY AND METHOD  35  LEAF SPRING DAMPING FLUID BULKHEAD STRAIN GAUGES PENDULUM MASS  Fig. 2.6: Schematic diagram of the leaf spring tilt cell used in the 1989 experiment. The cell is 47 mm long and 16 mm in diameter. As the cell is tilted, the pendulum masses bend the two leaf springs mounted at right angles to each other. The leaf springs are sensitive only to bending in one direction, so the two pendula respond to tilt along different axes. The bending is measured by strain gauges bonded to the surfaces of the leaf springs. after the insertion. In future, this problem could be avoided by using a higher viscosity damping fluid and plastic leaf springs. Certain plastics have a much higher elastic limit than does brass and would accommodate greater strain. The calibration process for leaf spring tilt cells is identical to that for electrolytic tilt cells with the exception that the calibration range is extended to  900  of tilt. The pro-  cessing of in situ deformation data is somewhat different because the Newton-Raphson  Chapter  .  BED DEFORMATION: THEORY AND METHOD  36  method does not yield a stable inversion. When the field data are analyzed, values 9 and ç for the cell are computed from Equation (2.8) using a two-dimensional simplex algorithm (Press and others, 1986, p. 289) that minimizes the difference between pre dicted and observed output voltages. As with the electrolytic tilt cells, the error on tilt and azimuth for these sensors was estimated by using the calibration data as input to the inversion scheme. The sine of the tilt has a maximum error of +0.005, which translates into a +0.3° error at  00  of tilt, increasing to ±0.6° error at 60° of tilt, and  to ±5° error at 90° of tilt. The azimuth error is ±4° at 0° of tilt, decreasing to +1° at 9Q0  of tilt. The error in strain rate is 0.003% at 0° of tilt, increasing to 2% at 600 of tilt. Two sensor strings were hammered into the glacier bed during the summer of  1989. Unlike our experience in 1988, the initial tilt of these sensors was close to that of their respective insertion angles. Omitting the stiff protective sheath appears to have resulted in instruments that are flexible enough so as not to be forced from their insertion position. Figure 2.7 shows the down-flow strain rate data sensed by one of these leaf spring tilt cell strings. This string was inserted 20 cm into the undisturbed basal sediments and had an overall length of 27 cm. After the insertion, the instrument cable was tied off with 1.5 m of slack. The signals from all three tilt cells indicate that they lay within actively deforming basal material. Because the insertion depth was not sufficient to place the uppermost cell within solid basal material, and because the cell did not have support from the cable, it must have been supported by the disturbed sediment layer. This sets the minimum thickness of the disturbed layer for this hole at about 10 cm. We estimate that, in general, the thickness of the disturbed layer is 15—25 cm.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  37  25O 1%  0 1  v’1sJv  —250  w cr  z 250 I Cl) w  V  —250  I  6  I  I  I  I  10  8  I  12  AUG 1989 Fig. 2.7: Sample results from the 1989 leaf spring tilt sensor experiment. Tilt records for each of the three tilt cells have been differentiated to give strain rate parallel to ice flow. Strain rates normal to ice flow have been omitted for clarity, but are of comparable magnitude. These records were produced by performing a decomposition of the raw tilt and azimuth data, as was discussed above. Because the tilt measurements are made with respect to vertical, we modify Equation (2.3) to include the basal slope of the glacier /3 so that  Ez =  18 j— tan(Od  —  /3)  (2.9)  Chapter 2. BED DEFORMATION: THEORY AND METHOD The basal slope in the region surrounding the study area is  70,  38  as determined by drilling  depths. Cross-flow strain rates are not shown (see Figure 3.10 for complete data set), but e are of comparable magnitude. This is a surprising result, but one we accept becaus n is the alternative seems to be unacceptable; if the down-flow/cross-flow decompositio s constrained to prohibit cross-flow strain, then the implied motion of the cells require 1000. back-and-forth rotation about their long axes by as much as  There may be  is partitioning between cell rotation and cross-flow tilt, but the effect on down-flow tilt slight. Any fraction of cross-flow tilt that is assigned to cell rotation reduces cross-flow tilt by a similar fraction and down-flow tilt increases slightly to maintain the correct net tilt. Strain rates were computed by applying a five point running first derivative ifiter an to the strain record (Abramowitz and Stegun, 1965, p. 914), followed by a Gaussi smoothing filter having a standard deviation of 50 mm. 2.5 Potential sources of error Here we discuss sources of error that may arise from the insertion procedure and from sensor characteristics. 2.5.1 Sensor scale effects When making detailed measurements of deformation within an inhomogeneous mate rial, a fundamental concern is choosing an appropriate scale of observation. If the scale of observation is too small (on the scale of individual clasts in the material), the con cept of bulk viscosity becomes meaningless because there is no macroscopic, smooth velocity gradient along which shear stress may be transferred and viscosity defined. Undersized sensors would measure the random wanderings of individual clasts rather than the macroscopic properties of the deforming material. Conversely, if we observe  Chapter 2. BED DEFORMATION: THEORY AND METHOD  39  the bed on too large a scale, the deforming layer appears as a single unit with no internal structure. We expect the deforming layer beneath Trapridge Glacier to be decimetres thick (Boulton and Hindmarsh, 1987; Clarke, 1987b), so we want to observe deformation on a scale of centimetres. Were the basal material beneath Trapridge Glacier homogeneous, our 5 cm long tilt sensors would make clear measurements of subglacial strain, but this is not the case. A dry volume fraction analysis of Trapridge Glacier basal material shows that 50% of the clasts are less than 0.25 mm in diameter and that 80% of the clasts are less than 2 cm in diameter (Clarke, 1987b); the size of our tilt sensors is such that they behave as large clasts. There is no alternative but to accept this limitation because any technique for making local deformation measurements will be subject to the same conditions. Tilt cells placed near a clast of similar or larger size will be influenced by the movement of the clast, and Equations (2.3) and (2.4), which rely on the homogenous nature of the basal material, may not be directly applicable. 2.5.2 The ice—bed interface The base of Trapridge Glacier in the study area is at the pressure melting point (Clarke and others, 1984b, Fig. 4, above site 11). For the bed to remain at this temperature, the net heat input to the basal ice must be positive and the basal ice must be melting. Basal melting produces a clean contact between the ice and the bed unless ice creep into subglacial sediments is faster. Nevertheless, we believe that the ice—bed interface beneath our study site is sharp; the basal sliding and deformation discussed elsewhere in this thesis will inhibit ice intrusion. When using the percussion hammer to insert instruments into the glacier bed, the final position of the instrument relative to the ice—bed interface is uncertain. We cannot observe the insertion procedure directly, and must rely on physical constraints and tactile information to infer what is happening at the bottom of the borehole.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  40  Our hot water drill uses a 1.14mm (0.045in) diameter jet with a pressure drop of 7—14 MPa (1000—2000 psi) across the nozzle. We believe that hydraulic excavation by the drill loosens material to a depth of several decimetres below the ice—bed interface. Evidence for this excavation includes the sediment laden basal water samples that we obtain shortly after completing a borehole. When the percussion hammer is lowered to the bottom of a borehole, its dead weight exerts a pressure of about 1 MPa on the tip (for comparison, the ice overburden pressure beneath our study site is 0.64 MPa). This pressure pushes the hammer through the loosened material; the insertion process begins, and is measured from, the upper surface of the undisturbed bed material. Thus, the upper parts of the instrument may be below the ice—bed interface, even if the insertion depth into the undisturbed material is less than the length of the instrument. Typically, an insertion proceeds quickly for the first 5 cm, and then slows down, suggesting that there is a gradation between the disturbed and undisturbed material. Evidence for the existence of the disturbed layer is found in the data and was discussed above. Unfortunately, we have not yet developed a reliable technique for measuring the distance between the top of undisturbed sediment and the ice—bed interface. In 1990, a crude device for measuring this distance was constructed. The device consists of a 15 cm long steel rod with two holes drilled through it at positions 5 cm and 10 cm along its length. By suspending this device from two thin wire ropes, the bar can be oriented in a vertical or horizontal position. The bottom of the borehole is measured with the bar in the vertical position and the level of the ice—bed interface with the bar in the horizontal position (when in the horizontal position, the effective diameter of the bar is 15 cm and the device is too large to fit back up the borehole). The elevation difference between these two positions is a measure of the separation we seek, but the results must be interpreted with two limitations in mind: (1) We cannot guarantee that the rod is binding at the ice—bed interface. (2) As the weight of the rod is much less  Chapter 2. BED DEFORMATION: THEORY AND METHOD  41  than that of the percussion hammer, the penetration depth of the rod into disturbed sediment is apt to be less than that of the hammer. Therefore, the measurements probably underestimate the separation between the ice—bed interface and the top of the undisturbed sediment. In the two holes where we managed to make this measurement (in the third hole, the wire ropes became irretrievably tangled), separations of 13±5cm and 63 ± 5 cm were measured. Additional information about the ice—bed interface is revealed during the drilling process. When completing the drilling of a “connected” hole, the meitwater in the hole drains rapidly into the bed; the equilibrium level of the water column represents the subglacial water pressure. On a number of occasions we have encountered artesian outflow conditions where silty subglacial water flowed up through the borehole at a rate of 2—3Lmin’; in these holes, the subglacial water pressure was well above ice flotation pressure and an equilibrium was not possible (Clarke and Germ, 1989). Con versely, the water in an “unconnected” hole does not drain into the bed, or does so very slowly. For some holes, the characterization of a given borehole as connected or unconnected is subjective, but for most holes, it is quite clear. The holes that drain quickly are connected to a well developed drainage system, but we have evidence that the spatial extent of connected regions of the glacier bed changes with time (Smart and Clarke, 1988; Clarke and Blake, 1991). lodge (1979) has observed similar behaviour at South Cascade Glacier. 2.5.3 Sensor attitude The diameter of the percussion hammer is 3.81 cm and, based on our experience with larger subglacial instruments, the diameter of the bottom of a borehole is just over 5 cm. These radial dimensions, in combination with the 2 m length of the percussion hammer, ensure that the hammer fits snugly in the borehole and that the instrument is inserted into the subglacial sediment co-axially with the bottom of the hole. Each  Chapter 2. BED DEFORMATION: THEORY AND METHOD  42  of the holes used for bed deformation experiments was profiled using an inclinometer equipped with dual-axis tilt sensors and a compass. By determining the azimuth of the borehole as it intersects the ice—bed interface, we determine the initial azimuth of the bed deformation instrument relative to the glacier flow direction. The inclinometer is capable of determining the location of the bottom of a 70 m borehole to within 60 cm. As they reach the bed, our boreholes commonly deviate from vertical by as much as 50  2.54 Sediment intrusion into the borehole After the borehole is completed, several hours typically pass before the insertion proce dure begins. This raises the concern that we may not insert our sensors into subglacial material, but rather into sediment that has squeezed into the borehole. If we accede that the sensors are located within the borehole, then the deformation observed by the sensors would be controlled by ice deformation; since the observed shear deformation rates exceed reasonable ice deformation rates by several orders of magnitude, the sen sors cannot be located above the ice—bed interface. Moreover, using similar insertion techniques, we install other types of sensors into subglacial sediment and have unequiv ocal evidence that these sensors are implanted in bed material rather than extruded sediment. We discuss these two observations in turn. The mean basal shear stress beneath the study site is 77 kPa (based on geometrical calculations) and the basal temperature of Trapridge Glacier in the study area is close to the pressure melting point (Clarke and others, 1984b, Fig. 4, above site 11). Evaluating Glen’s flow law for simple shear (2.10)  Chapter  .  BED DEFORMATION: THEORY AND METHOD  at 000 gives a strain rate of , and n 3 5.3 x 1O’ skPa  0.076yr when the shear stress r =  43  =  77kPa, A  =  3 (Paterson, 1981, p. 39). Assuming Newtonian simple  viscous shear deformation of the form T  (2.11)  this strain rate represents a dynamic viscosity of 2 x 1013 Pa s. This ice viscosity value is two orders of magnitude greater than published values of measured or predicted till viscosity (Boulton and Hindmarsh, 1987, fig. 7; Clarke, 1987b). Given the low strain rate of the ice, we could not expect to measure strain rates exceeding 100 yr 1 within a borehole (see Figure 2.7) if the straining of extruded material within a borehole were controlled by ice deformation. In 1990, we ran a series of experiments using “drag spooi” instruments. Our intent was to obtain some measure of the sliding rate beneath Trapridge Glacier. A drag spool consists of a multi-turn potentiometer connected to a spooled string (Fig. 0.1). The drag spool is suspended within the ice close to the bed, and measures continuously the length of string payed out to an anchor in the bed. Figure 2.8 shows an example of data obtained from a drag spool. 70  E 60 C.) •— 50 C  0 40 20  10 208  212  216 Time (days)  220  Fig. 2.8: Relative displacement between a drag spool and its anchor recorded in the summer of 1990. The data is from 901133, located within the study area.  Chapter 2. BED DEFORMATION: THEORY AND METHOD  44  The percussion hammer was used to insert three drag spooi anchors in differ ent boreholes to a depth within the undisturbed basal material similar to that of the 1989 deformation instruments. The data indicate that, on average, the anchor points moved away from their respective boreholes at a rate of about 4 cm day’. This large relative velocity could not persist for many days if the anchor were placed in intruded material. Although these measurements were made at different subsurface locations and during a different field season than the 1987—1989 bed deformation experiments, surface velocity in each season was about 10 cm day’ and we have not made any changes to our drilling and insertion techniques during this time; it seems unlikely that basal conditions have undergone dramatic changes between the 1987—1990 field seasons. Thus we conclude that the anchors for our bed deformation instruments are also below the ice—bed interface. If the upper parts of a deformation instrument were to extend above the ice—bed interface, the data should show this as a rapid increase in tilt to 90° as the tilt cells are drawn out of the borehole and in under the ice; we do not see this behaviour in any of our instruments. 2.5.5 Connecting wires The two quantitative instrument designs require that thin wires connect the individual tilt sensors together and that a multi-conductor cable transmit the strain information to the surface. The main instrument cable freezes to the side of the borehole, and could conceiv ably pull the entire sensor string out of the bed as the glacier slides forward. By letting slack cable down the hole after the instrument insertion is completed, we alleviate this problem. There is evidence from the 1988 results that this slack cable technique is sufficient to prevent interference from the cable for several weeks. The wires connecting the individual sensors are thin enough (36 AWG) so that they cannot prevent the sensor string from bending. Additional cladding, such as the  Chapter 2. BED DEFORMATION: THEORY AND METHOD  45  heat shrink sheath used in 1988, may stiffen the sensor string so that problems arise during the insertion procedure, but if the cladding is omitted, no problems arise. The connecting wires may also act under tension to align the sensors with each other. With the sensor string transfixing a shearing layer, any tension effect will become more acute as time passes. There is not much that can be done about this problem, other than using telemetry techniques that do not require wires. We have considered coiling the wires between the sensors in order to provide stretch, but in order to maintain the spacing between sensors during insertion, a strong non-extensible connection between the cells is required. Despite precautions, the insertion process places considerable tensional force on the instrument.  Chapter 3 BED DEFORMATION: DATA ANALYSIS  3.1 Introduction  In order to develop models for subglacial deformation and drainage, we must have a rheological model for the subglacial material. Quantifying the sediment rheology requires measurements of shear stress, subglacial water pressure (an indirect measure of normal stress), and strain rate. Boulton (1979b) has, to our knowledge, published the only measurements of sub glacial shear stress. The fixing of shear stress sensor plates onto a glacier substrate calls for some particular circumstances: (1) physical access to the bed and (2) a solid substrate into which the sensors can be set. Boulton installed his sensors on bedrock underlying short-lived cavities beneath the Glacier d’Argentière, France. He succeeded in measuring time-varying normal and shear stresses, although these variations were attributed to clasts scraping over the sensor plates rather than gross changes in basal shear stress. The demanding logistical requirements for this type of experiment make it highly improbably that measurements of shear stress beneath a soft-bedded glacier will ever be made. Those in situ measurements of strain rate that have been made (Boulton and Hindmarsh, 1987; Fahnestock and Humphrey, 1988) are of total strain over a number of days. An average strain rate can be computed from this “before and after” look at the bed, but any time-varying behaviour is obscured. Temporal changes in strain rate are expected because subglacial water pressure, and hence effective pressure on the bed, can fluctuate dramatically (Mathews, 1964; Iken, 1972, 1978; lodge, 1976, 1979; Iken and Bindschadler, 1986; Kamb and Engelhardt, 1987; Engelhardt and others, 1990).  46  Chapter 3. BED DEFORMATION: DATA ANALYSIS  47  Boulton and Hindmarsh have fitted total strain measurements made near the ter minus of Breidamerkurjökull to two non-linear viscous fluid models. A problem with their analysis is that they combine data from different years and different sites to pro duce a single rheology (Boulton and Hindmarsh, 1987, Figure 7). The necessary but unstated assumption is that the sediment is both spatially and temporally homoge neous. A proper rheological description may require continuous and contemporaneous strain measurements at a number of spatially distributed sites. In our experiments, we have succeeded in making continuous measurements of subglacial deformation, but lo gistical constraints have limited the spatial coverage to one or two sites per field season (Chapter 2; Blake and others, 1991). Nevertheless, our data reveal some interesting aspects of the basal processes beneath Trapridge Glacier. This chapter presents data collected from in situ strain instruments placed in the actively deforming bed of Trapridge Glacier during the 1988 and 1989 field seasons. Additional deformation measurements were made in the surnnier of 1990, but the data analysis has not been completed. Several models are presented that seek to explain the observed deformation patterns. The deformation data were collected using electrolytic and leaf-spring tilt cells inserted into the glacier bed; the function, design, calibration, and data processing of the tilt cells and the design of the percussion hammer used to insert them are described in Chapter 2. 3.2 Experiment design  Figures 1.1 and 3.1 show the general location of the study area used for the bed defor mation studies. Extensive hot-water drilling in this area indicates that the glacier has a very uniform thickness of about 72 m; during a given drilling season, we can often predict the depth of a new hole to within 10—20 cm. For several hundred metres to the north, west, and east, the glacier topography is gently undulating. To the south,  Chapter 3. BED DEFORMATION: DATA ANALYSIS  48  there is a slight rise associated with heavy crevassing. The surface and basal slopes, as determined by surveying and drilling, are both  70  flow direction, as determined by stake surveys, is  in the direction of glacier flow. The  110  north of east (bearing 79°). Data  from vertical strain sensors indicates that the area is under moderate compression.  Fig. 3.1: Trapridge Glacier is located in the St. Elias range of south western Yukon, Canada. The rectangle indicates the area where most holes, including those used in the subglacial deformation experiments, were drilled during the 1988, 1989, and 1990 field seasons. The dotted line indicates the approximate location of the fall snowline. The purpose of the magnetometer and telluric array will be clarified in Chapter 6.  3.2.1 Ancillary information Hard-bedded glaciers can move only by processes of basal sliding and internal deforma tion of the ice. Figure 3.2 shows that three processes contribute to the surface velocity of a soft-bedded glacier: internal deformation, basal sliding, and basal deformation  Chapter 3. BED DEFORMATION: DATA ANALYSIS  49  (Alley and others, 1986; Alley, 1989a). The experiments described in this paper at tempt to elucidate the process of basal deformation, but estimates of basal sliding and internal deformation are necessary for data interpretation.  z I  internal deformation  /  ICE BED  )\\N  sliding  veIoty  basal deformation Fig. 3.2: The surface velocity of Trapridge Glacier (about 10 cm day’ during the summer) is made up of contributions from internal deformation (0—1 cmday’), basal sliding (‘—‘ 4cmday’), and subglacial deformation 5 cmday). (‘—i Internal deformation Information on internal deformation of Trapridge Glacier is largely empirical. Much of the glacier is cold (Clarke and others, 1984b; Clarke and Blake, 1991), which discourages plastic deformation.  Although we do have problems with instrument cables being  50  ChapterS. BED DEFORMATION: DATA ANALYSIS  cut, this is an annoyance primarily in crevassed areas. In the region where the bed deformation experiments were conducted, several cables connecting pressure sensors placed at the glacier bed to the surface have survived intact for more than a year. Following the development in Chapter 6 of Paterson (1981.), we can compute a theoretical estimate of the internal deformation character of Trapridge Glacier based on Glen’s flow law for ice (Glen, 1952). Computing internal deformation in this way is notoriously error-prone because the parameters describing glacier ice rheology are known to vary between different ice samples and because the behaviour of ice near the pressure melting point is not well described or understood. The derivation of a theoretical deformation rate proceeds as follows: consider a coordinate system with the x axis pointing down-flow and parallel to the ice—bed  interface; the z axis is positive upward. We assume deformation in the x-z plane only, with no variation in strain, stress, or temperature with x or y. Under these conditions, Glen’s flow law may be written as  =  A(T)r’r  (3.1)  =  r 1 A(T)r’  (3.2)  with r2=rI+r Tz =  A(T)  =  pgsina(h  —  (3.3) (3.4)  z)  5.2 x 1016 exp  {(  —  263.15)  }  3 s_i kPa  (3.5)  2 is the normal deviatoric stress along the x axis and r, is the only non and where r zero shear stress component. The activation energy constant R  =  acceleration g  Q  8.314 J mol’ K—’, the density of ice p =  9.81 ms , the glacier thickness h 2  =  =  =  139 kJ mol’, the ideal gas  , the gravitational 3 900 kg m  72m, the surface slope a  =  70,  ChapterS. BED DEFORMATION: DATA ANALYSIS and the flow law exponent n A(T  =  51  3. Temperature T is in degrees Kelvin; the value of  —10°C) is 5.2 x 10_16 kPa 3  (Paterson, 1981).  We estimate the value of è by measuring the vertical strain rate c . Verti 2 cal strain rate is measured by freezing a 1.2m long, 0.076mm (0.003 inch) diameter constantan wire into a shallow 15 m borehole. The wire forms one arm an electrical bridge whose output is monitored over time; if the glacier is under compression, the wire will be stretched and its resistance will increase. This vertical strain instrument was designed by W. D. Harrison (personal communication). Equations (3.1) through (3.5) were used to calculate the internal deformation of the glacier assuming a linear temperature distribution between 0°C at the bottom and —6°C at the surface (Clarke and Blake, 1991). The dotted line in Figure 3.3 shows the computed velocity profile with no longitudinal strain; the solid line shows the effect of adding a longitudinal strain rate of  =  —0.019 yr’ (equal in magnitude to the  observed vertical strain rate of +0.019 yr ); this implies a compressive longitudinal 1 stress.  The computed velocity contribution from internal deformation accounts for  2.8myr’ (0.76 cmday’) of the average annual surface flow rate of about 3Omyr’. This rate of internal deformation seems consistent with the lifespan of our subglacial sensors. We have analyzed only one long-term field record of internal deformation. Hole 891165 was redrilled a year later as 901140 by following along a conductivity sensor cable; over a full year, the total lateral deformation of the hole was less than the error on the inclinometer which is about 30 cm (Appendix A) (the hole designations 881156  —  —  e.g.  indicate the year of drilling and the ordinal position of the hole in the glacier  drilling program for that year). Theory predicts that the largest deformation occurs near the bed where the ice is warmest. Should there be any debris in this basal ice (note that we do not belive that there is debris near the ice—bed interface beneath our study site), evidence suggests that  52  Chapter 3. BED DEFORMATION: DATA ANALYSIS  basal deformation should be even higher (Hubbard and Sharp, 1989; Echelmeyer and Wang, 1990; Huang and Wang, 1990; van der Veen and Whillans, 1990). It is therefore surprising that the theoretical prediction overestimates the observed deformation. Over a year, the theoretical calculation predicts lateral borehole deformation of at least 1.5 m even if longitudinal compression is neglected  —  deformation of this magnitude should be  clearly evident in the inclinometry data. This incongruity means that Trapridge Glacier ice is stiffer than is predicted by Glen’s flow law (using parameters from Paterson, 1981) and that almost the entire surface motion of the glacier is accommodated by sliding and deformation processes at the bed.  70  ci) -Q  I  ci) > 0  I  -o o 30  /  ci)  /  20 cm 10  -  0 0.0  0.2  0.4  0.6  0.8  Velocity (cm/day) Fig. 3.3: The computed internal velocity distribution for Trapridge Gla cier. The solid line indicates the solution with the observed compressive lon gitudinal strain included and the dotted line indicates the solution with no longitudinal strain.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  53 Basal sliding Experiments with basal drags spools in 1990 (Chapter 2; Appendix C; Blake and Clarke, 1991a) placed an upper limit on glacier sliding of about 4 cm day’ (because the spool anchors were placed within the deforming sediments, some of the observed relative motion between the anchors and the ice could have been caused by deforma tion of the interposed sediment). During the summer, Trapridge Glacier moves at a speed of about 10cm day’ (this speed was consistent during the 1988, 1989, and 1990 field seasons), so the above estimates for glacier sliding and internal deformation leave 1 of motion to be accounted for by basal deformation. 5—6 cm day 3.3 The 1988 experiment  On 10 August 1988, a deformation instrument consisting of three vertically-spaced electrolytic tilt sensors was hammered 8 cm into the bed of Trapridge Glacier. The tilt cells were spaced at 10 cm intervals and the overall length of the sensor string was 27cm. A signal cable led from the instrument to the surface; following insertion, 1.5 m of slack cable was fed into the borehole. In the weeks prior to inserting the tilt sensor string, two subglacial water pressure sensors and a vertical strain sensor were installed in neighbouring holes. Figure 3.4 shows a location map of the experimental site. The indicated positions of subglacial sensors are those determined by borehole inclinometry. The indicated position of the vertical strain sensor, placed 15 m below the surface, is that of the borehole collar (the top of the borehole). Hole 881104, which contains one of the pressure sensors, was the only connected hole; the other holes reaching the bed (881135 and 881156) were unconnected.  54  Chapter 3. BED DEFORMATION: DATA ANALYSIS  I  88H04  .  PRESSURE  88H37 88H35  .  VERTICAL  PRESSURE  88H56 0 TILT  0  10  Fig. 3.4: The location map for the 1988 experiment. The indicated po sitions of the subglacial sensors is that of the hole bottom as determined by borehole inclinometry. Solid dots indicate connected holes and holes not reaching the bed; circles indicate unconnected holes. Data from the 23 day experiment are shown in Figure 3.5. Figure 3.5a shows the tilt angle from vertical decomposed into down-flow (heavy line) and cross-flow tilt (light line). All of the observed azimuth changes were attributed to cross-flow tilt (see sec tion 2.4.1). Inclinometry results indicate that the hole through which the deformation instrument was lowered intersected the glacier bed in a near-vertical orientation. The design of the percussion hammer ensures that the instrument is inserted coaxially with the bottom of the borehole (Chapter 2; Blake and others, 1991), which in this case tilted  30  up-glacier. The large initial tilt angle reported by the tilt cells suggests that  the disturbed sediment at the bottom of the borehole was not able to support the sensor string once the hammer was withdrawn, perhaps because the stiff heat shrink sheath on the instrument allowed the slack instrument cable to press down on the instrument.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  55  Nevertheless, the rapid onset of independent motion in the lower two cells suggests that the bed quickly reconsolidated following the drilling disturbance. The uppermost tilt cell appears to have passed most of the experiment resting on its side, perhaps lying at the ice—bed interface. Late in the experiment this cell was tipped over to point down-glacier, probably by the glacier pulling on the cable. The two lower tilt cells quickly reached their calibration limit of  600  tilt from vertical.  Figure 3.5b shows strain rate curves generated by applying Equations (2.4) and (2.9) to the tilt data from the lower two sensors (the basal slope /3 is 7°). The time series were truncated at the time when the net tilt of each cell exceeded the calibration limit of 60° and then smoothed using a Gaussian smoothing filter having a standard deviation of 50 mm. The two striking characteristics of the strain rate curves are that the strain rates are large and sometimes negative. Assuming a 1 m thick deforming subglacial layer supporting 5 cm day 1 of uniform shear deformation, the expected down-flow strain rate is +9yr’, but in Figure 3.5b, we observe strain rates ranging between —200 yr 1 and +400 yr . The large positive down-flow strain rates observed on both 1 traces for the first day are probably associated with consolidation of the disturbed subglacial sediments in which the sensors were located, but large excursions are also found at later times. For instance, early on 15 August, the middle tilt cell experienced rapid changes in down-flow tilt angle that translated into large strain rate fluctuations. Negative strain rates are expected in the cross-flow record  —  we can picture the  tilt cell wobbling from side to side as down-flow deformation progresses  —  but the  negative excursions in the down-flow strain rate record indicate that, at times, the down-glacier velocity gradient through the deforming layer is negative. Both the strain rate extremes and negative strain rates are probably caused by interaction between the tilt cells and large neighbouring clasts; we shall explore this and other options below.  56  Chapter 3. BED DEFORMATION: DATA ANALYSIS  -90-J +90 C,) Ui Ui  C, Ui  MIDDLE .1  -90 +90  a  -  BOTFOM  0-  -90.. 1111111111  II  1111111111  >‘  Ui  b  z FCl)  1111111111111111111111  E Ui  C  D U) U) Ui 0.  AUG 1988 Fig. 3.5: Data records from the 1988 experiment. (a) The tilt records for the three tilt sensors indicating tilt from vertical. The uppermost record is for the uppermost tilt sensor. The heavy line indicates down-flow tilt and the light line cross-flow tilt. (b) The strain rate records corresponding to the lower two tilt cells. Strain rate is not calculated for the top cell, so the uppermost record is for the central tilt cell. Down-flow strain rates are drawn with heavy lines and cross-flow strain rates with light lines. (c) The pressure records from holes 881104 (light line) and 881135 (heavy line).  Chapter 3. BED DEFORMATION: DATA ANALYSIS  57  3.3.1 Correlation with effective pressure Effective pressure is a poorly-defined term used to describe the distribution of ice overburden pressure on the two load-bearing components of the subglacial system: the water and the sediment. The effective pressure Fe is a measure of the loading experienced by the sediment. In a definition analogous to that describing the partial pressures of constituents making up a gas, the conventional expression for Fe is F=Fe+Fw  (3.6)  where F is the overburden pressure of the ice and F is the pore water pressure. Sub glacial measurement of pore pressure poses obvious logistical difficulties, so researchers usually substitute measurements of water pressure made at the bottom of boreholes. Unfortunately, the low hydraulic conductivity usually associated with glacial sedi ments means that changes in subglacial water pressure take some time to diffuse down into the sediments. In addition, sediment deformation can drastically alter the pore pressure without affecting subglacial water pressure. Pore water pressures at depth within the subglacial material may be quite different from the borehole water pressure. We can compute an estimate of the rate of intrusion of water at the ice—bed interface into subglacial sediments by using Darcy’s Law (Equation (4.24c)). Darcy’s Law is usually written as q  =  —KVh  (3.7)  where h = F/pg is the pressure head, P is the pressure potential (fluid pressure less the hydrostatic pressure), pm is the density of water, g is gravitational acceleration, q is fluid flux, and K is the hydraulic conductivity. Typical values of K for glacial till and clay range between  10—12  and  106  ms (Freeze and Cherry, 1979, table 2.2),  although there is evidence from borehole response tests that a thin (5 cm) layer of  Chapter 3. BED DEFORMATION: DATA ANALYSIS  58  material with K = 10—2 to 10_i ms (Daniel Stone, personal communication) may exist next to the ice—bed interface. If we choose a large head gradient of 50 (this represents the entire overburden pressure of Trapridge Glacier expressed across a 80 cm 1 (suggested by Tavi thick sediment layer) with a hydraulic conductivity of 1O m s Murray, personal communication), then the intrusion rate is about 0.05 IIms. This is a low flow rate; if this pressure head gradient is sustained for an entire day, water at the ice—bed interface will penetrate only 4 mm into the bed. Translating fluid flux into temporal changes in pore pressure requires knowledge of the hydraulic diffusivity of the bed and an understanding of the effect of deformation on pore pressure. Since these are both unknown quantities, we carl oniy make qualitative statements about pore pressure: we know that deformation can cause dilation which lowers the pore pressure (Murray, 1990), and, based on the above calculations, we expect that diffusion of pressure from the ice-bed interface will be slow (on the order of hours or days). Equation (3.6) underestimates the loading actually experienced by the contact surfaces of the sediment grains; it is the grain-to-grain contact pressures that control the rheological behaviour of the sediments. A more accurate definition for effective pressure is obtained by scaling the contributions of effective pressure and water pressure to overburden support. It can be argued that some fraction of the planar bottom surface of the glacier (if indeed the bottom is planar) is supported by water and the remainder by sediment. A trivial proof demonstrates that if a planar cut is made through a representative volume of material, the ratio of the area of the cut intersecting water to the area of the cut is equal to the porosity. The weighted effective pressure is thus defined as P = (1  fl)Pe +flPu,  (3.8)  Chapter 3. BED DEFORMATION: DATA ANALYSIS  59  where n is the porosity of the sediment. Typical values of porosity for glacier till are 20—30% for consolidated sediment and 40% for dilated sediment (Kamb, 1991), so Equation (3.8) produces larger values of Fe than does Equation (3.6) for the same values of P and P,. Nonetheless, as Scott (1963) points out, Equation (3.8) also underestimates the grain-to-grain contact pressure. In a water-saturated sediment, neglecting surface ten sion effects (which may not be insignificant), the proper expression for effective pressure is P  =  aPe + (1  —  a)F  (3.9)  where a is the area ratio of the grain contacts. Since a << 1, Equation (3.9) is often written as P=aPe+Pw  (3.10)  Clearly, if a is small, Fe can get very large indeed. Practically, we cannot make in situ measurements of n or a, so we make do with Equation (3.6) using water pressure measured with subglacial pressure sensors rather than with pore pressure sensors. We must keep in mind that the effective pressure calculated in this way is wrong. Figure 3.5c shows the contemporaneous pressure records collected from the two neighbouring subglacial pressure sensors. The heavy line represents the pressure in hole 881135 and the light line the pressure in 881104. The similarity of the two pres sure records, except for a brief four-day period, suggests that hole 881135 has become connected to the subglacial drainage system since it was drilled. Migration of con nected zones has been observed at Trapridge Glacier and is not unusual (Smart and Clarke, 1988). The two pressure sensors are separated by 23 m. It is noteworthy that late on August 15th, the pressure recorded by these two sensors diverged sharply for  Chapter 3. BED DEFORMATION: DATA ANALYSIS  60  a period of four days after which they resumed strong mutual agreement; the pres sure pulses (or dips) produce a strong diurnal cycling of horizontal pressure gradient greater than 1 m(H . It is unlikely that such a large pressure gradient could exist 1 o) m 2 within a single connected zone; the hydraulic properties of connected zones are such that a pressure disturbance in one part is felt elsewhere in the zone almost immediately (Smart and Clarke, 1988). To explain this pressure event, we imagine the following scenario: On August 15, the drainage system in the neighbourhood of 881104 becomes temporarily unconnected; at the same time, some morphological change in the subglacial drainage system or some change in diurnal meltwater input causes the pressure in 881135 to begin large fluctuations, raising the local water pressure near to the flotation pressure of 65 m(H o) 2 (flotation pressure is equal to the ice overburden pressure). The separation between holes 881104 and 881135 is roughly half the thickness of the ice; over this distance and on a diurnal time scale, the glacier is stiff enough to support a lateral transfer of normal stress on the bed. In response to an upwards force on the glacier at 881135, the overburden pressure in the neighbourhood of 881104 is reduced; the water pressure in 881104 drops accordingly. Unfortunately, we have no pressure record from 881156, the location of the bed de formation instrument, but because 881156 is roughly equidistant from the two pressure sensors, we expect that some changes in water pressure —  —  and hence effective pressure  occurred at 881156 during this four-day period. Given a change in effective pressure  and no change in driving stress, we also expect that the basal strain rate should change. This response is not evident in Figure 3.5b. The only discernable correlation between the onset of the diurnal pressure cycles and strain rate is a transition from near-zero values of strain rate to positive values. We are not convinced that this correlation is anything but coincidental, particularly since the strain rate of the middle tilt cell does not revert to near-zero values once the pressure event terminates.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  61  3.3.2 Effective viscosity The classical definition for dynamic viscosity  Tb =  of a linear viscous fluid is  ?7—  (3.11)  2ij  (3.12)  or, substituting Equation (2.3),  Tb =  The velocity gradient ôtt/Oz is measured along a line perpendicular to the interface across which the shear stress  Tb 15  measured. This relationship can be derived by  considering the transfer of momentum across the interface by the motion of particles making up the fluid (e.g. Joos, 1988, p. 568). The statistical analysis fails when the number of particle interactions considered is too low; this situation arises when the dynamic system is observed at too large a magnification (so that only interactions with a few particles are visible) or when the system is observed for too short a period of time. In situ measurements of deformation within granular subglacial material suffer from a problematic scale of observation. The basal material contains particles having a size comparable to that of the tilt sensors. Since it is these macroscopic particles that are the vehicles of momentum transfer, determining sediment viscosity by observing the motion of a tilt cell is analogous to determining the viscosity of a gas by watching one molecule bouncing about. In addition, subglacial particle interactions are not elastic collisions as envisaged in the momentum transfer model, but are bumping and grind ing interactions. These aspects of the deformation process suggest that the concept of viscosity may be entirely inappropriate for describing the proportionality between stress and strain in granular material. Nevertheless, we can attempt to mitigate these weaknesses by averaging the observed instantaneous strain rates over time, thereby  62  Chapter 3. BED DEFORMATION: DATA ANALYSIS  increasing the number of particle interactions. We must keep in mind this limitation of the experimental design. We can compute an estimate of  which we call the effective viscosity. The mean  down-flow strain rate ê for the middle tilt cell over the four day period beginning on . This is a period during which both subglacial water pressure 1 20 August is 36±16 yr and strain rate appear relatively stable. The average basal shear stress 0)  =  Tb  77kPa is computed using Equation (3.4) assuming a glacier thickness h of 72m,  a glacier slope a of gives a value for  j  70,  and no longitudinal stress gradient. Applying Equation (3.12)  of 1.7 ± 0.8 x 1010 Pa s. The failure of the viscosity concept becomes  clear when one considers individual points on the strain rate curve. We have no reason to believe that the shear stress r changes, yet the strain rate is sometimes zero (resulting in infinite viscosity) or negative (persisting negative viscosity would suggest that the glacier is being dragged down-slope by fast flow of subglacial sediments  —  this is a  preposterous idea). To estimate the thickness of the deforming layer by assuming that the strain rate is uniform throughout the layer (although the two strain rate records in Figure 3.5b appear to contradict this), we can rewrite Equation (3.11) as  =  (3.13)  If we assume that shear deformation accounts for 5 cm day’ of glacier motion (Lu) then .z, the deforming layer thickness, is 12 ± 5 cm at this time and place. If this is indeed the thickness of the deforming layer, it might explain why we were able to drive the tilt sensor string only 8 cm into undisturbed subglacial sediments. Boulton and Hindmarsh (1987) surmnarize a series of subglacial deformation exper iments undertaken between 1977 and 1983 from within ice tunnels excavated beneath the terminus of Breidamerkurj6kull. They fit the observed average strain rate  over  Chapter 3. BED DEFORMATION: DATA ANALYSIS  63  the course of each experiment to computed values of effective pressure Fe and basal shear stress  Tb.  The seven data points are fitted to equations of the form = A(m  —  (3.14)  Th ) 7 P  and (3.15)  = AP 1  which represent a Bingham fluid model and a nonlinear viscous model respectively (-y is a yield stress). Since there are a small number of data points and a relatively large number of free parameters, it is perhaps not surprising that Boulton and Hindmarsh achieved a rather good fit of their data to Equations (3.14) and (3.15).  With the  parameters m, n, and A substituted, Equations (3.14) and (3.15) become = 7.62  (T7)  0.625  where 7 = 0.625Pe +  (-)  —1.25  (3.16a)  (3.16b)  3750  and =  The strain rate  (rb)1.33  is in yr’, and Fe and  Tb  ()  —1.80  (3.17)  are in Pa. According to these equations,  strain rate is proportional to stress, as we expect, and inversely proportional to effective pressure. The average effective pressure Fe, as calculated using Equation (3.6), for holes 881104 and 881135 over the four day period beginning on August 20 is 292+6 kPa. If we substitute this value of Fe into Equations (3.16) and (3.17), together with  Tb  = 77 kPa,  we get a predicted strain rate of 0.41 yr for the viscous fluid model and no deformation for the Bingham fluid model (the shear stress is below the predicted yield stress 7). Since we have ample evidence that the strain rate was about 36 yr during this time,  Chapter 3. BED DEFORMATION: DATA ANALYSIS  64  it is clear that Equations (3.16) and (3.17) are not appropriate rheological descriptions for subglacial material under Trapridge Glacier. Boulton and Hindmarsh based their values of effective pressure on measurements taken from pressure sensors buried in the subglacial sediments; they were attempting to measure pore pressure rather than water pressure at the ice—bed interface, but we suspect that their pressure measurements are no better at estimating pore pressure than ours are. Boulton and Hindmarsh placed their pressure sensors less than 10 cm into the sediments. If Breidamerkurjökull has a thin layer of more hydraulically conductive sediment near the ice bed interface, as we believe Trapridge Glacier does, then we would expect shallow pressure sensors to measure the same pressure as at the ice—bed interface. Indeed, Boulton and Hindmarsh observed diurnal pressure cycling with peaks in the afternoon; this pattern of pressure change is consistent with daily meltwater input and shows no time lag resulting from diffusion of pressure into the sediments. Figure 3.6 reproduces the results for the Bingham fluid model (Boulton and Hindmarsh, 1987, Fig. 7) with the our data point for the 1988 experiment superimposed. Assuming that the fluctuations in basal water pressure observed in 881135 were present in 881156, the effective pressure near the deformation instrument varied over the range indicated by the dotted line. As noted above, we observed strain rate variations between —200 and +200 yr’ with no correlated change in effective pressure. For the effective pressure and shear stress at our study site, the rheology functions proposed by Boulton and Hindmarsh greatly underestimate the observed strain rate. The different experimental locale is surely responsible for part of this discrepancy. For instance, a comparison of the particle size distributions for the two glaciers (Boulton and Hindmarsh, 1987, Fig. 3; Clarke, 1987b, Fig. 4; Murray, 1990; Tavi Murray, personal communication) reveals that Trapridge Glacier has a greater fraction of silt and clay particles; at high effective pressures, clay-rich sediments are more easily deformed than coarser sediments because the clay has a lubricating effect (more precisely, the internal  65  Chapter 3. BED DEFORMATION: DATA ANALYSIS  C  C  C C  (C  Effective pressure  —  kPa  Fig. 3.6: A facsimile of Boulton and Hindmarsh (1987), Figure 7, with the data from the 1988 experiment superimposed. Note the discontinuity in the effective pressure axis. The dotted line indicates the range of effective pressure encountered in hole 881135 during the 1988 experiment, and the solid bar indicates the effective pressure range for the 1989 experiments. friction angle of clay is low (Murray, 1990)). In addition, it is possible that Boulton and Hindmarsh have overestimated the shear stress present beneath the Breidamerkurjökull ice tunnels. Driving stress calculations involve using longitudinal stress gradients and the difference between surface and basal slope angles to make corrections to the simple gravitationally-driven basal shear stress. Boulton and Hindmarsh may have improperly used the slope of the over-steepened ablating glacier snout (Boulton, 1979a, Fig. 6) and neglected the longitudinal compressive stresses usually present at a glacier terminus (Robin, 1968; Collins, 1968; Paterson, 1981, Chapter 6, page 98). The mean basal shear stress for Trapridge Glacier is computed from knowledge of the glacier thickness, surface slope, and basal slope. We have recorded a near-surface longitudinal compressive strain of 0.019 yr’, but determining the effect of this longi tudinal stress on basal shear stress requires a knowledge of how  and  vary along  the glacier flow line (x axis); this is information we do not have. Because our experi mental site is located under a uniform part of the glacier and only 10—20 m in elevation  Chapter 3. BED DEFORMATION: DATA ANALYSIS  66  below the equilibrium line, we suspect that there are no significant longitudinal stress gradients that we need to consider; we can make due with a gravitationally-driven basal shear stress (Equation (3.4) evaluated at z  =  0).  The unfortunate corollary to having a uniform glacier is that our study site provides only one mean basal shear stress value; this makes the development of a basal rheology for Trapridge Glacier difficult. Nevertheless, we have evidence from both electrical experiments (Chapter 6) and deformation experiments (discussed below) that shear stress beneath Trapridge Glacier has a far from uniform distribution. Had we a method for measuring in situ local shear stress, we might be able to gather the data necessary for deriving a rheology; unfortunately, we have no such method. 3.4 The 1989 experiment On 5 August 1989, a deformation instrument consisting of three vertically-spaced leaf spring tilt sensors was hammered 20 cm into the glacier bed at the bottom of hole 891178 (Figure 3.7). A subglacial water pressure sensor was placed in the same hole. The next day, a second deformation instrument was hammered 17 cm into the glacier bed at the bottom of hole 891180, located 17.5m northeast of 891178. No pressure sensor was placed in this second hole. For both deformation instruments, 1.5 m of slack cable was fed into the borehole before securing the cable at the surface. Hole 89H79 was drilled on 5 August, but the attempt to install a deformation instrument in this hole failed; the instrument did not penetrate into the bed far enough for the subglacial sediments to provide a sufficient anchor.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  67  89H80  /  N  TILT  89H50 0  PRESSURE  89H78 PRESSURE TILT  89H14 • PRESSURE  I  0  I  m  10  Fig. 3.7: The location map for the 1989 experiment. The indicated posi tions of the sensors are those of the hole bottoms as determined by borehole inclinometry. Solid dots indicate connected holes; circles indicate unconnected holes. A second pressure sensor had been installed on 20 July in hole 891150, located midway between the two deformation instruments. As a demonstration of the large lateral variability in the subglacial drainage system, holes 891178 and 891180 were con nected holes and 891150 was unconnected; there is no evidence in the 891150 pressure record that the drainage system at 891150 had evolved into a connected system by the time holes 891178 and 891180 were drilled. A third pressure sensor, installed on  Chapter 3. BED DEFORMATION: DATA ANALYSIS  68  12 July in connected hole 891114, was located about 40 m southwest of the deformation experiment. Tilt data from the six-day experiment (terminated upon our departure from the glacier) is shown in Figure 3.8. The sampling interval is 5 mm. Figure 3.8a shows the tilt data from 891178 (due to data logger failure, the first day of data was lost) and Figure 3.8b shows the tilt data from 891180. Note that the vertical scale in Figure 3.8b is twice that in Figure 3.8a. As in Figure 3.5, the heavy line indicates down-flow deformation and the light line indicates cross-flow deformation. All the tilt cells registered initial tilt angles close to the terminal tilt angles of their respective boreholes (7° for 891178; 5.5° for 891180) which suggests that the in strument cable did not interfere with the deformation instruments. The improvement in maintaining initial orientation may have resulted from a design change: the two 1989 deformation instruments did not have a stiff heat-shrink protective sheath as did the 1988 instrument and this may have prevented the slack cable from pushing on the instruments. The 1989 deformation instruments were also haimnered further into the undisturbed sediments. Hole 891178 intersected the glacier bed at a 7° tilt angle pointing roughly up-glacier, but 12° away from the flow direction. This initial orientation produces an offset in the cross-flow tilt as is evident in Figure 3.8a. Hole 891180 pointed more directly up-flow, so the initial cross-flow tilt angle in Figure 3.8b is almost zero. The tilt records for the two upper cells are truncated because as the tilt cells approached a net tilt of 90°, the inversion scheme that recovers tilt and azimuth (Chapter 2; Blake and others, 1991) became unstable. Because of the significant cross-flow tilt excursions present in these records, we chose to attribute only 50% of the observed tilt cell azimuth fluctuations to cross-flow tilt; the remaining azimuth changes were attributed to rotation of the tilt cells on their  Chapter 3. BED DEFORMATION: DATA ANALYSIS  69  40  TOP C  0  Ui  40n  C!,  0  I— C,)  w  MIDDLE  a  Ui  40  BOflOM  0 I  I  I  I  C,) Ui Ui Ui  I  I  MIDDLE -  8O  0 I  I  I  6  7  8  I  10 9 AUG 1989  I  I  11  12  13  Fig. 3.8: Tilt data from the 1989 experiment. (a) The tilt records from 891178. The uppermost record is for the uppermost tilt cell, and the lowest record is for the lowest tilt cell. The heavy line indicates down-flow tilt and the light line cross-flow tilt. The labels on the top record show how different mean strain rates can be computed for different time intervals (section 3.4.5). (b) The tilt records from 891180. The data are arranged as in part(a). axes (Chapter 2; Blake and others, 1991). The consequence of this decision is to reduce the amplitude of cross-flow tilt fluctuations by a factor of 2.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  70  34.1 1989 tilt results The tilt records in Figure 3.8 suggest that the three tilt cells within each deformation instrument experienced the same local deformation events but that basal deformation at the two sites was of quite different character. The three tilt sensors in 891178 were moving in almost perfect concert; even the finer details are present in all three records. Unlike the tilt data from 1988, there is a diurnal signal evident throughout the 891178 traces, although it is strongest in the last three days; in general, there appears to be an increase in tilt angle in the morning and then a decrease in tilt angle in the afternoon. The three tilt records from 891180 also exhibit some similarity; for instance, all tilt sensors experienced a sharp increase in tilt angle early on 8 August followed by a sharp decrease in tilt angle at noon on the same day. Unlike the 891178 data, fine details are not consistent across the three records and there appears to be more high-frequency tilt activity. In the final three days, the lowest tilt sensor shows the same diurnal behaviour as the records from 891178, but the tilt oscillations are delayed by several hours. 34.2 1989 strain rates The strain rates computed from the tilt records are shown in Figure 3.9. As in Fig ure 3.5b, the strain rate curves have been smoothed. Figure 3.9a shows the strain rates from 891178 and Figure 3.9b shows the strain rates from 891180. Note that the vertical scale in Figure 3.9b is ten times that in Figure 3.9a. The synchronized motion of the tilt cells in 891178 is again evident in Figure 3.9a.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  71  >  w  a  z I U)  I  I  I  I  I  I  +1500  TOP  0-150Q  w +1500  MIDDLE  b  0-  z  -1500 +1500  BOTTOM  U)  0-  -  -1500  ib  I  11  I  12  13  AUG 1989 Fig. 3.9: Strain rate data from the 1989 experiment. (a) The strain rate records from 891178. The uppermost record is for the uppermost tilt cell, and the lowest record is for the lowest tile cell. The heavy line indicates down-flow strain rate and the light line cross-flow strain rate. (b) The strain rate records from 891180. The data are arranged as in part (a). The data for both 891178 and 891180 share several interesting characteristics: (1) Negative and zero strain rates are observed. Although the mean down-flow strain  Chapter 3. BED DEFORMATION: DATA ANALYSIS  72  rate for each cell is positive, indicating net down-flow strain within the deforming layer, the periods of negative strain rate are frequent. (2) In the second half of the exper iment, there is evidence of diurnal synchronization. (3) The cross-flow strain rate is comparable in magnitude to the down-flow strain rate. For the 891178 sensor string, all three records are similar; this indicates that each sensor was moving in concert with the others and suggests uniform deformation over the 20 cm length of the sensor string. When contemplating the large strain rates, it is important to keep in mind that these peaks result from the differentiation and smoothing of step-like changes in tilt angle  —  these high strain rates are not sustained over long periods of time. Because  these step changes in tilt angle are often observed to occur simultaneously for all the tilt sensors in one deformation instrument, we are confident that the events are not artifacts of unstable data inversion. In subsequent sections, we will discuss mechanisms that might generate these tilt events. Net cross-flow strain over the period of the experiment is small for all the tilt sensors, but the raw data indicate unequivocally that the tilt sensors were experiencing large azimuth fluctuations.  As mentioned above, we have attributed 50% of these  azimuth changes to rotation of the tilt cells about their long axes and feel that it would be unreasonable to increase this fraction. As the only other source of azimuth changes is cross-flow tilt, large cross-flow tilt changes must have occurred; large cross-flow tilt may not be an unreasonable consequence of deformation of granular material. In the absence of overwhelming, persistent extrusive flow within the subglacial sediments, we expect that the average down-flow strain rate over some reasonable period of time will be positive. Although their appearance may not be convincing, all the strain rate curves in Figure 3.9 have positive integrals over the course of the experiment. The surprising aspect of these curves is that the deformation must sometimes be integrated over many days to ensure the expected positive net down-flow strain.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  73  34.3 Subglacial pressure Figure 3.10 shows the data from the three subglacial pressure sensors. The stepped character of the 891114 and 891178 records is caused by the limited resolution of the digital data logger. The pressure records from 891114 and 891178 are well matched, although the observed pressure fluctuations are very much smaller in amplitude than those observed in 1988. Measurements of borehole depth and surveying results indicate that the pressure sensor in 891114 is roughly 3 m above the sensor in 891178.  For  ) 1120 a horizontal piezometric surface, the pressure sensor in 891178 should read 3 m(  greater than that in 891114. Since the measured pressure difference is only 1 m(H o), 2 we infer that a steady pressure gradient exists between these two holes  —  the polarity  of this gradient is appropriate for driving water down-glacier. The pressure record from 891150 indicates a falling pressure that is consistently well above the flotation pressure of 65 m(H o). Recalling that 891150 was an uncon 2 nected hole and that 891178 was a connected hole, it is astonishing to see such a large lateral pressure gradient. We are certain that 891150 was drilled to the bed and sus pect that the high pressure in the hole was caused by freezing of the water in the upper reaches of the hole. Because both 891178 and 891180 were connected holes, and because previous experience with local connected systems at Trapridge Glacier sug gests that neighbouring connected holes can be remarkably well interconnected (Smart and Clarke, 1988), we will assume that the pressure record from 891178 reflects the subglacial water pressure in the vicinity of 891180.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  !68  74  89H50  58-  AUG 1989 Fig. 3.10: The pressure records from 891114, 891150, and 891178 are shown. Note that the 891150 has a different vertical scale. The correlation between the 891178 pressure record and the 891178 and 891180 strain rate curves is not impressive, but they share a clear diurnal cycling towards the end of the experiment. Periods of higher subglacial water pressure in the early morning hours of 10 August and 11 August correlate well with positive strain rates in 891178 and less well with those in 891180. Because the pressure fluctuations are small in amplitude (about 10 kPa), we are surprised to see such a large change in strain rate. Compared with the effects of the pressure event in 1988, the strain rate variations seem completely out of proportion. We suspect that the correlation between local subglacial water pressure and strain rate is not direct, but that there is some other mechanism at work; we will discuss this in a later section.  34.4 Effective viscosity Using the same procedure as for the 1988 experiment, we can derive an effective vis cosity value for the bed. The mean strain rates for the three sensors in 891178 are +8.5yr’, +10.8yr’, and +4.7yr’ for the top, middle, and bottom sensors respec tively. The mean of the down-flow strain rates measurements from all three sensors is 1 (the error estimate is based on the deviation of the three sensor means, +8 + 2yr  Chapter 3. BED DEFORMATION: DATA ANALYSIS  75  not on the standard deviation of all the data points). Using Equation (3.11) we find the effective viscosity is 8 + 2 x 1010 Pas. This estimate of effective viscosity is almost five times that found in 1988. If we make the same calculations for the 891180 sensor string, we get a “global” mean strain rate of +30 + 15yr and an effective viscosity of 2 + 1 x  1010  Pas.  34.5 Net strain and mean strain rate Rather than integrating strain rate curves over an interval of interest, it is simpler to compute the mean strain rate by recasting Equations (2.4) and (2.9) in the form 1 (tan 82 —tanSi) 2  (3.18)  and applying this equation to the tilt values at the endpoints of an interval. The subscripts in Equation (3.18) indicate the values of 8 and time t at the endpoints of the interval [1,2];  is mean strain rate over that interval. Note that for down-flow strain,  the tilt angle 8 must be corrected for basal tilt as in Equation (2.9). When we apply Equation (3.18) to the down-flow tilt angle records for the top tilt cell in 891178 (Figure 3.8a), we can arrive at a number of estimates of mean subglacial strain rate. The mean strain rate using interval [A, B] is 32 yr’, but the mean value using interval [A,C] is 4.1yr. This latter value is close to the strain rate predicted by the Boulton and Hindmarsh models for an shear stress of 77kPa and an effective pressure of 78 kPa (the mean pressure in 891178 over the course of the experiment gives an effective pressure of 78kPa, as calculated using Equation (3.6)). The eight-fold difference in mean strain rate that arises simply by considering net strain over different time intervals demonstrates the danger in relying on measurements of net strain for the development of basal rheologies, but the tale is even worse: it is also clear that intervals can be chosen so that the mean strain rate is zero (an infinite effective viscosity) or negative (an irrational effective viscosity).  Chapter 3. BED DEFORMATION: DATA ANALYSIS  76  3.5 Negative strain rates  Strong negative strain rates are striking characteristics of Figures 3.5b and 3.9. A negative strain rate results when the down-flow velocity of the top of the cell is negative with respect to the cell bottom; the bottom of the cell is displaced, in a positive direction, relative to the top of the cell. Note that we have not specified any constraints on the net velocity of the tilt cell with respect to the geographical coordinate system; the tilt cell can be moving down-glacier, up-glacier, or rotating in place. In this section, we explore several mechanisms that could produce the observed data. In the process, we discover that negative strain rates and extreme values of strain are not difficult to produce. Before describing basal deformation mechanisms, the effects of instrument design on tilt data should be discussed. Problems related to the relative size of the tilt sensors and basal clasts have been addressed above. We also believe that the stiff protective sheath used on the 1988 deformation instrument may have caused some problems at the start of the experiment before the disturbed sediments reconsolidated, but once this had occurred, the sheath would have had negligible effect on bending of the instrument because the effective viscosity of the sediment was so large. It is clear that tension in the wires became important late in the 1988 experiment when all the tilt sensors had high tilt angles (shortly after the tilt records in Figure 3.5a end, the instrument was pulled apart). No protective sheath was used in 1989 and the tilt sensors were connected only by flexible 36 AWG wires. These deformation instruments were very delicate; several of the connecting wires were broken during careful handling on the glacier surface. Since none of these wires broke during the course of the experiment, we believe that almost no tension existed between tilt cells in the 1989 experiment. For this reason, we have ignored any possible interference from connecting wires and/or sheaths.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  77  3.5.1 Fluid models Imagine, for the moment, that the particles making up the bed are all much smaller than the tilt sensors; this allows us to model the deformation as a fluid flow. Equations (2.4) and (2.9) require that the lateral velocity gradient normal to the ice—bed interface be negative in order that negative strain rates be produced. There are at least two mechanisms that can generate these velocity proffles: extrusion and sheet flow. Sheet flow Sheet flow is the flow of fluid between two parallel surfaces under the influence of a lateral pressure gradient. In this case, the upper surface is the ice—bed interface and the lower surface is the underlying bedrock or non-deforming sediment. We introduce a coordinate system with the origin at the bottom of the deforming layer and the x axis pointing down-glacier parallel to the ice-bed interface. The z axis is positive upward normal to the bed. We assume that the subglacial material can be considered a fluid and that the deformation zone has infinite extent in the x  —  y plane.  For an incompressible linear viscous isotropic fluid, the relationship between the deviatoric shear stress tensor -r’ and the deviatoric strain rate tensor  ‘  is (3.19)  =  where  is the viscosity and where i and  j  denote the terms of interest. The deviatoric  quantities may be expanded to give  —  where  Sjj  =  —  (3.20)  kk6)  is the Kronecker delta and a double subscript indicates summation over  the diagonal terms of the tensor. In the steady state and in the absence of body forces, momentum balance considerations give us the relation Or/Ox  =  0 For an  Chapter 3. BED DEFORMATION: DATA ANALYSIS incompressible fluid,  kk  =  78  0. In addition, hydrostatic water pressure P  =  . 3 —Tkk/  If  we restrict velocity and pressure fields to functions of the form u(z) and P(x), we can differentiate Equation (3.20) to give OP  82u  (3.21)  =— --  For boundary conditions u(hb) deforming layer, and u(0)  =  =  U, the velocity of the material at the top of the  0, the solution to Equation (3.21) is u(z)  =  /OP 1 -h-— —z(z  =  /OP1 —(2z  —  hi,) +  Uz  ---  (3.22)  and 2e  Ou =  —)  — hb) + —U  (3.23)  where hb is the thickness of the deforming layer. The first term of Equation (3.22) repre sents the parabolic laminar flow solution (the planar analogue to Poiseuille’s equation) and the second term represents the linear simple shear solution. If we use values ofj and z  =  =  2.0 x  1010  Pas, hb  =  lm,  =  +40yr, U  =  5cmdaf’,  0 (the location of maximum strain rate), then the required value of OP/Ox is  —180 kPam 1 or about  m(H 1 — o 2 8 )  m. This is an unreasonably large lateral pres  sure gradient. We have observed lateral pressure gradients as large as 100 kPam’ sustained over a day, but we believe that maintaining a hydrostatic pressure gradient of —180 kPam’ over large distances and for long periods of time is unrealistic. Nev ertheless, pressure gradient driven deformation could contribute to basal deformation. Extrusion If the ice—bed interface moves vertically then subglacial material tends to move later ally. This is much the same mechanism that forces toothpaste out of a tube. Rapid surface elevation changes of several centimetres (which presumably reflects vertical  Chapter 3. BED DEFORMATION: DATA ANALYSIS  79  motion of the ice—bed interface) have been observed on Trapridge Glacier. Iken and others (1983) have observed similar changes on Unteraargletscher as have Iken and Bindschadler (1986) on Findelengletscher and Kamb and Engelhardt (1987) on Varie gated Glacier. If the entire glacier is permitted to rise and fall, we expect to see quantities of basal material sluicing in and out at the margins of the glacier. As this large-scale displacement of basal material is not observed, we assume that local changes in ice— bed interface elevation are compensated by elevation changes of opposite polarity in neighbouring areas, by compression and/or expansion of subglacial sediments, or by the growth and decay of subglacial cavities and/or channels. In deriving a mathematical expression for this extrusive flow, we will make certain simplifying assumptions: (1) Sediment flows only in the x direction. (2) The change in elevation is small compared to the thickness of the deforming layer. (3) Material flow resulting from extrusion has a parabolic velocity profile between the upper and lower boundaries, just like the solution for sheet flow. (4) Subglacial sediments are incompressible. At first, we assume that the velocity U, as defined above, is zero. The velocity distribution through the layer is then u(x,z,t)  =  A(x,t)z(z  —  (3.24)  hi,)  where A is some function of x and time t. We can position the coordinate system at the spreading centre of the extrusive flow such that u(O, z, t)  =  0. The rate of volume  change, per unit width, of the deforming layer between the origin and a point x is then defined as V(x,t)  =  w(t)x  (3.25)  80  Chapter 3. BED DEFORMATION: DATA ANALYSIS  where w(t) is the velocity of the ice—bed interface along the z axis. Mass conservation for an incompressible fluid requires that the change in volume be equal to the quantity of material entering or leaving the region. Thus, ,.  (3.26)  u(x,z,t)dz  V(ct)=_J 0  Combining Equations (3.24), (3,25), and (3.26) gives the solution u(x,z,t)  6w(t)xz =  (z  —  (3.27)  hb)  1  We now superimpose the linear simple shear solution to give u(x,z,t)=  6w(t)xz  U (z—hb)+---z  (3.28)  b 11  and 2e  6w(t)a  ãu =  LiZ  =  (2z  —  he,) +  U  (3.29)  -—  Figure 3.11 shows the displacement in the z direction required to reproduce the strain rate curve of the top sensor in 891178; the solution is for x U  =  , and z 1 5 cm day  layer). The value of  =  =  10 m,  Jib  =  1 m,  hb (i.e. the top tilt sensor is at the top of the deforming  was chosen somewhat arbitrarily, but is of the same order of size  as we believe connected zones to be. We note that the displacements in Figure 3.11 are small compared to displacements that have been observed at Trapridge, which suggests that extrusion could easily account for all aspects of the observed strain rates. Unfortunately, we have no survey data with sufficient temporal or spatial resolution to determine whether Figure 3.11 is representative of glacier surface motion.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  81  0 —1  d (mm)  -2 —3  -4 6  13  Aug 1989  Fig. 3.11: The vertical displacement record required to reproduce the strain rate record for the upper tilt cell in 891178 at a point 10 m from the spreading centre of the extrusion. See text for a complete explanation. Although the extrusion model seems to offer an explanation for large, and some times negative, strain rates, it does predict coherent deformation along the glacier bed, even if deformation instruments are located in different horizons within the bed. An examination of the records in Figures 3.8 and 3.9 reveals that there is little correlation between the movement of the tilt sensors in the two holes; the diurnal fluctuations are not synchronized. Since the basal sediments cannot be considered homogeneous on our scale of observation, we can reasonably argue that the smooth velocity profile defined by Equation (3.25) is not preserved over appreciable distances. Unfortunately, the extrusion model has one major drawback: Equation (3.28) does not take into account the viscosity of the subglacial material. For  =  2.0 x 1010 Pas,  the glacier must exert a local downward pressure far greater than its weight in order to achieve the vertical motion described in Figure 3.11. Extrusion is therefore unable to explain the observed deformation.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  82  3.5.2 Roller bearing models In an attempt to derive a deformation model which more accurately describes the inhomogeneous nature of the subglacial material, we turn to models of granular de formation. There is evidence that deformation of subglacial material may result in the formation of weak, dilated shear planes along which most of the deformation is acconunodated (Murray, 1990). If the layer of deforming material beneath Trapridge Glacier exhibits this tendency, then we can imagine that the layer splits up into long blocks that extend across the direction of glacier flow. In this model, the blocks have polygonal cross-sections and churn against each other in accordance with the frictional, translational, and rotational forces to which they are subjected. The blocks may break up and coalesce, again in accordance with the rheological properties of the block and the forces acting upon it. Herrmann and others (1990) examined the problem of packing cylinders together in such a way that all cylinders roll against their neighbours. They solved this problem using a fractal distribution of cylinder sizes.  Since subglacial sediments contain a  variety of particle sizes, this fractal packing arrangement seems to offer a reasonable deformation model. Unfortunately, rolling friction will exist between the cylinders; with an infinite number of cylinder-to-cylinder contacts, the fractal packing will seize. A simplified version of this model is illustrated in Figure 3.12. The prismatic blocks are represented by infinitely long cylinders of equal radius that are packed in a hexagonal fashion between two bounding plates (representing the ice above and the non-deforming material below). Some variable friction coefficient function is assigned to the surface of each cylinder and the two boundaries. The upper plate is loaded vertically and moved horizontally at some velocity U. This will cause the cylinders to begin rotating in response to the normal and tangential forces at their contacts with neighbouring objects. If the separation between the top and bottom plates is fixed and the cylinders forming the lateral bounds of the packing are forced to maintain their  Chapter 3. BED DEFORMATION: DATA ANALYSIS  83  relative positions, then no rearrangement of the hexagonal packing is possible. This has two consequences: (1) The geometry of the packing will force some cylinders to slip against neighbouring objects. (2) The centres of all the cylinders will move horizontally at velocity U/2.  ICE  BASEMENT Fig. 3.12: The deforming layer modelled as a series of hexagonally-packed cylinders. Some cylinders will be forced to rotate in a reverse sense. A tilt sensor next to a cylinder rotating in reverse will experience a negative strain rate. Solving for the rotation direction of each cylinder and how this direction might change over time is a difficult problem. An equilibrium analysis of the packing quickly shows that the system of dynamical equations describing the packing is under-deter mined. Nevertheless, it should be clear is that some cylinders will rotate in a normal sense (clockwise in Figure 3.12) and others will rotate in a reverse sense (anti-clockwise). In addition, the direction of rotation of a given cylinder will change with time as the cylinder rotates and areas of different frictional coefficient are presented to its neighbours. Clearly, a tilt sensor placed next to one of these cylinders can experience both positive and negative strain rates and any transition is strain rate sign will be rapid.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  84  If some method of solution were found, then this representation of the deforming glacier bed could perhaps be used to model basal shear stress and the space- and timeaveraged properties of basal deformation. Unfortunately, the model has limitations which would render suspect any simulations: (1) Rearrangement of particles is not possible. (2) All particles translate at the same speed. (3) All particle rotate at the same speed. (4) Even if the cylinders are conglomerates, there is effectively only one particle size. (5) The interstices between cylinders are not dealt with by the model. 3.5.3 The shadow box computer Rather than attempting to solve the roller bearing problem numerically, Tavi Murray (personal communication) suggested that an analogue computer could be built to solve the problem. An assemblage of two-dimensional particles cut out of 3 mm thick particle board is placed in a gap between two verticle plexiglas sheets. A ruler is passed between the plates and is used to apply normal and shear forces on the cut-outs. By projecting a bright light from behind the sheets, the silhouettes of the cut-outs can be recorded on video tape for viewing and digitizing at a later time. This apparatus is dubbed the “shadow box computer”. In addition to solving the roller bearing problem with alacrity, the shadow box has two great advantages over the numerical approach: (1) gravitational forces are included and (2) a variety of particle shapes and sizes can be used. Figure 3.13 shows a pair of images recorded from the shadow box computer. The silhouettes clearly show particles rotating in the normal and reverse senses as well as particles not rotating at all. Dilation of the particle packing is also evident.  Chapter 3. BED DEFORMATION: DATA ANALYSIS  85  Fig. 3.13: A pair of frames from a video record of the shadow box. The upper panel shows the initial configuration of the blocks between the plates of the box. As the upper boundary, with the particle marked “A” attached, is moved to the right, the particle configuration in the lower panel results. Notice that particle “B” has experienced normal rotation, whereas particle “C” has experienced reverse rotation. Some dilation of the sediments is also evident. 3.6 Discussion  Basal strain rate is observed to change polarity, and to fluctuate wildly. If a defor mation rate of 6 cm day’ (surface velocity less sliding velocity as determined by the  Chapter 3. BED DEFORMATION: DATA ANALYSIS  86  slidometers; internal ice deformation neglected) is associated with uniform shear defor mation in a 1 m thick subglacial layer, then Equation (2.4) gives a basal strain rate of 1 in Figure 3.9 are consistent with about 11 yr. The mean strain rates of 8—30 yr this estimate, but the instantaneous strain rate often exceeds the mean value by an order of magnitude and can be of opposite polarity. The coherence of the three traces in Figure 3.9a suggests that this is not a consequence of local heterogeneity in the sediments. The erratic behaviour might be caused by rapid changes in the coupling between the glacier and the bed, or changes in the thickness of the actively deforming layer. 3.6.1 Boulton and Hindmarsh flow models Our in situ measurements of subglacial deformation demonstrate that the rheological relations derived by Boulton and Hindmarsh (Equations (3.16) and (3.17)) are inap propriate for the sediments beneath Trapridge Glacier. Because the sediments beneath Breidamerkurjökull and Trapridge Glacier are without doubt different, this inapplica bility is perhaps not surprising. Nevertheless, we have demonstrated that measurements of net strain, such as those made by Boulton and Hindmarsh, are fundamentally flawed because they ignore the time-varying nature of subglacial deformation. In section 3.4.5, we showed that quite different values for mean strain rate can be obtained simply by considering deformation over different time intervals. We make no statements regarding the utility of Equations (3.14) and (3.15) for describing subglacial deformation other than these: (1) the definition of effective pressure must be rigorous (i.e. Equation (3.6) is probably inadequate), (2) the effective pressure cannot be reliably inferred from mea surements of either subglacial water pressure or pore pressure (see discussion below), and (3) a good understanding of the shear stress exerted on the sediments is required.  ChapterS. BED DEFORMATION: DATA ANALYSIS  87  Although we have demonstrated that the Boulton and Hindmarsh relations as not applicable to sediments underlying Trapridge Glacier, we have not derived an alterna tive rheology. The reasons for this are two-fold: (1) our data demonstrate no correlation between effective pressure (as defined by Equation (3.6)) and strain rate and (2) we have no measurements of local shear stress. 3.6.2 Shear stress and normal stress The use of shear stress in these experiments warrants some discussion. We have been using a mean shear stress calculated using Equation (3.4), but there is strong evidence that the basal shear stress is far from uniform  —  this has obvious implications for  values of effective viscosity computed with Equation (3d1) and is a problem faced by all glaciologists. Consider hole 891150. As we noted above, the pressure in this hole exceeds the nominal flotation pressure by  ). Sm(H 0 2  It is clear that the glacier does  not accelerate upwards as a result of this apparent force imbalance, so the overburden pressure at 891150 must be greater than the thickness of the ice suggests. Since the glacier, on a diurnal time scale and over distances less than the glacier thickness, is a somewhat rigid body, lateral transfer of loading on the bed is possible; the load can be greater at some points and less at others, so long as the mean loading is equal to the theoretical overburden pressure pgh. We must also bear in mind that the shear stress on the bed is not necessarily proportional to the normal loading on the bed, as Equation (3.4) implies. A sticky patch (with a high shear stress) can have a low normal load and a slippery patch (with a low shear stress) can have a high normal load; again, the stability requirement is only that the mean shear stress averaged over the glacier sole be equal to that computed by Equation (3.4). The folly of relying on subglacial water pressure to determine effective pressure and of relying on geometry to determine subglacial shear stress is revealed! If the overburden pressure cannot be determined, then it is impossible to compute the effective  Chapter 3. BED DEFORMATION: DATA ANALYSIS  88  pressure by any of Equations (3.6), (3.8), or (3.10). Similarly, a non-uniform shear stress distribution makes theoretical shear stress calculations of dubious value. We expect that slippery patches are associated with connected zones and sticky patches are associated with unconnected zones. The characteristic overpressure con ditions observed in unconnected boreholes (boreholes drilled into unconnected zones) demonstrate that the normal loading over unconnected zones is generally larger-thanmean (this is why over-pressurized water in unconnected holes does not simply force its way out). The larger-than-mean normal loading implies a capacity to support largerthan-normal shear loading. We have, however, observed overpressure situations in con nected zones (on occasion, we encounter artesian outflow conditions when completing a connected borehole). Furthermore, the significant pressure gradient observed between 891114 and 891178 indicates that a connected zone is not just a puddle of water under the glacier; some resistance to hydraulic flow is present. This suggests that connected zones consist of a basal layer that has been washed clean of fine particles, leaving an open hydraulic aquifer capable of supporting both normal and shear loading. This sticky/slippery—connected/unconnected model of the bed provides an expla nation for the diurnal strain rate fluctuations observed in 1989 when there were no corresponding fluctuations in water pressure. Because hole 891150 is unconnected, the normal loading at this point on the bed must be high. Holes 891178 and 891180, the locations of the two deformation instruments, may also be experiencing a larger-thanmean normal loading. Although we have no measurements of local shear stress, we can infer that this area is also capable of supporting larger-than-mean shear stress; this could result in greater deformation. The question arises: how does lateral transfer of shear stress occur? In the weeks prior to the bed deformation experiments, pressure sensors installed at various locations near the deformation experiment site recorded dramatic diurnal fluctuations  —  each day during the afternoon and evening, the subglacial water pressure  Chapter 3. BED DEFORMATION: DATA ANALYSIS rose to a peak at about 21:OOh  —  89  but on 1 August, four days prior to the beginning  of the deformation experiment, all these pressure sensors began reporting quiescent pressure levels. As we will show in Chapter 6, subglacial electrical phenomena inch cated that the subglacial environment was anything but quiescent; large fluctuations in apparent resistivity and natural potential ‘occurred until 10 August. Since meitwater is the only significant diurnal force acting on the subglacial environment (tidal effects and thermal stress, both certainly negligible, are the only other candidates we can think of), we strongly suspect that subglacial water pressures continued to rise during the afternoon between 1 August and 10 August, but that none of our pressure sensors happened to be located where these pressure cycles could be observed. Increases in water pressure elsewhere in the subglacial environment would, through processes dis cussed in Chapter 1, reduce the shear stress in those areas; the principle of constant mean shear stress then requires that the shear stress increase elsewhere. If the region of the deformation experiment experienced a rise in shear stress, we could expect to see diurnal cycling in deformation without any change in subglacial water pressure at the strain measurement site. in the latter part of the 1989 deformation experiment, the strain rate makes a transition from negative to positive late in the evening, a few hours after the presumed maximum in subglacial water pressure; this correlation is consistent with the idea of positive strain rate indicating down-flow deformation of the sediments. As the water pressure elsewhere in the system fails, the inferred enhanced shear stress on our sensor array vanishes and we see a negative-going trend in strain rate. This trend is interpreted as being some form of sediment relaxation phenomena. 3.6.3 Effective viscosity In closing this chapter, we wish to reiterate our distrust of using effective viscosity in discussions of sediment deformation. Although we have derived these values of effective  Chapter 3. BED DEFORMATION: DATA ANALYSIS  90  viscosity for Trapridge Glacier sediment, we believe that effective viscosity is only useful as a aid in visuaiizing the ability of subglacial material to resist deformation and glacier flow. There is considerable evidence that the rheology of subglacial sediments is non linear (e.g. Boulton and Hindmarsh, 1987; Murray, 1990; Kamb, 1991) and we have discussed the difficulties in applying the very concept of viscosity to deformation of granular material.  Chapter 4 ELECTRICAL PHENOMENA  —  THEORY  4.1 Introduction Subglacial measurements of electrical phenomena are sensitive to the movement and distribution of water within the glacier bed. This is because the water is in large measure the electrically active component of the subglacial environment. By making measurements at intervals over a period of time, we can also monitor temporal changes in the subglacial hydraulic system. This chapter presents a theoretical foundation for direct current (d.c.) electrical resistivity and natural potentials; we will give particu lar attention to electrokinetic phenomena. The objectives and justifications for, and data from, our subglacial electrical measurements are discussed in Chapter 6. We be gin we discussions of the microscopic structure of the rock—electrolyte interface and mechanisms for electrical conduction through rocks and sediments. 4.2 The rock—electrolyte interface Fluid filling pores within rocks and sediments can be divided into two components with distinctly different properties: bulk fluid and fluid associated with the interface between the two phases. Since we will be using the term electrolyte in association with the pore fluid, we call this interface the rock—electrolyte interface. Whenever two phases of dissimilar electrical properties are in contact, a separation of charge arises at the interface  —  this separation is called an electrical double layer be  cause it consists, in crude terms, of two planar regions of opposite charge. The simplest conceptual model for the electrical double layer is usually attributed to Helmholtz: two  91  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  92  layers of opposite charges are arranged in parallel to form a sort of molecular capac itor. Such a double layer can easily form between two solid phases, but it is difficult to imagine how a plane of charges could persist in a fluid electrolyte where thermal diffusive forces compete with electrostatic forces for the control of ions. Indeed, the ions will spread out into a diffuse layer. 4.2.1 Gouy—Chapman model  The theory for a diffuse double layer was developed independently by Gouy (1910) and Chapman (1913). Figure shows the ion and potential distributions associated with the Gouy—Chapman model. The significance of the  potential will be discussed in  section 4.6. The Gouy—Chapman model represents the solid surface as a uniform planar charge distribution with potential o relative to the bulk electrolyte. The charges in the pore fluid are modelled as point charges. The surface attracts ions of opposite charge (counter-ions) and repels ions of like charge. To solve for the potential distribution, we begin with Poisson’s equation for the potential  ‘  arising from a volumetric charge distribution Pv =  _  (4.1)  where e 0 is the permittivity of free space and D is the relative permittivity of the fluid (though D  ‘s-’  80 for bulk water, it may be significantly different close to an interface as  a result of poorly understood interactions with the solid surface). At equilibrium, the electrical and diffusional forces on each ion species i must balance, i.e. = —zeV  (4.2)  where z is the valence of the ion, e is the charge on an electron, and the chemical potential  is defined as gui =  gu + kBT1nn  (4.3)  Chapter  4. ELECTRICAL PHENOMENA  e  —  THEORY  93  e  e :1  e e e  e  C  e  b  Fig. 4.1: Two models for the rock—electrolyte interface. The diagrams on the left indicate the ion distribution and those on the right indicate the potential distribution. In this example, the surface potential o is negative. (a) The Gouy—Chapman or diffuse model. The potential at the hydrodynaniic shear plane (dashed line) is the potential. (b) Stern’s model. The outer limit of the Helmholtz layer is indicated by dotted line (after Morgan and others, 1989). where T is the temperature, kB is Boltzmann’s constant, of the ion species, and  4  is the chemical potential for n  =  =  is the number density  1 (Keizer, 1987, P. 107).  All ions, not just the counter-ions, are included in Equations (4.2) and (4.3) because they all contribute to the electric field structure.  Chapter  4. ELECTRICAL PHENOMENA  — THEORY  Taking the gradient of Equation (4.3) and solving for rz (where 4  94  =  0 and n  =  in the bulk electrolyte) gives Boltzmann’s equation  =  exp  (—::)  (4.4)  We notice that 0 p  =  (4.5)  >Zmjzje  and so the distribution obeys the Poisson—Boltzmann distribution  =  where the permittivity is e  =  -Jnzjeexp  (_-,)  (4.6)  D. 0 e  The analytic solution to Equation (4.6) involves hyperbolic trigonometric functions (these arise because ion species of like valency and opposite charge can usually be paired off), but the solution is essentially exponential in form (see Figure 4.2. Stern model The Gouy—Chapman model is sufficient for explaining the origin of electrokinetic phe nomena, but it is not the currently accepted model of the rock—electrolyte interface. The Gouy—Chapman model does not make allowance for adsorption of ions onto the interface, the non-point nature of the ions, or for secondary effects such as the align ment of polar molecules in the electric field and the resulting dampening of the field (the most common polar molecule in a groundwater system is water). Stern (1924) was the first to propose that the ion distribution has two major components: (1) an inner or compact layer of ions (termed the Helmholtz layer) where the charge arrangement and potential distribution are controlled by short-range or adsorptive forces and by geometrical restrictions; (2) an outer Gouy—Chapman or diffuse zone that follows a  Chapter  .  ELECTRICAL PHENOMENA  —  THEORY  95  Poisson—Boltzmann distribution. The ion and charge distributions found in the Stern model are shown in Figure Figure 4.2 shows a detail of the Stern model interface and the associated potential distribution. The inner Helmholtz plane (IHP) marks the limit of closest approach of any adsorbed ions or molecules and the outer Helmholtz plane (OHP) marks the limit of influence of adsorptive forces. The diffuse layer extends outward from the OHP.  C •  —  ©  —  c0  POSITIVE ION NEGATIVE ION WATER MOLECULE  Fig. 4.2: Detail of the Stern model rock—electrolyte interface. The IHP demarcates the distance of closest approach of ions and molecules. The OHP is the limit of influence for adsorptive or short-range interaction between ions and molecules. This example has negative surface charge 4 and negative adsorbed ions. Notice the preferential orientation of water molecules near the interface. The diffuse layer ion distribution has been omitted (after Bockris and others, 1963; Hunter, 1981).  Chapter  .4. ELECTRICAL PHENOMENA  —  THEORY  96  The preferential orientation or polarization of water molecules near the interface is evident in Figure 4.2. Bockris and others (1963) note that the permittivity e of the water drops from about 80 beyond the OHP to 32 between the IHP and OHP, and to 6 within the IHP; since the dielectric properties of water depend in part on the freedom of the molecules to move about, this decrease in permittivity reflects the increasing orientation restrictions placed on the water molecules as the surface is approached. The advantage of the Stern model is that it permits changes in the magnitude and polarity of the  potential with changes in concentration and pH; it is hard to  accomplish this with the diffuse model alone since the surface charge o is relatively constant. In general, as the pH is lowered (more acidic conditions), the increases; the relationship between  potential  potential and ion concentration and species is  more complex (Hunter, 1981, Chapter 6). Under certain conditions, unusually high ion concentration and low pH, the  potential can be made equal to zero, and streaming  potentials vanish; this is known as the point of zero charge. The potential distribution in the Stern model is troublesome to describe mathe matically because the double layer incorporates so many interacting processes; we will not delve into the mathematics here (readers can refer to Hunter, 1981, Chapter 2), but this omission is irrelevant since the conceptual model we have developed is sufficient to understand electrokinetic phenomena. 4.3 Conduction mechanisms There are three mechanisms by which electrical current can propagate in natural set tings: electronic, dielectric, and electrolytic conduction. Electronic conduction requires the presence of free electrons, as found in metals and suiphide minerals. These materials have very low resistivities (less than 10 1 m). Although suiphide minerals are found in the moraines surrounding Trapridge, the pre dominant rock types are carbonaceous and siliceous; the minerals composing these rocks  Chapter  4. ELECTRICAL PHENOMENA  have resistivities above 1010  —  THEORY  97  m (Telford and others, 1990, p. 285—289). We do not  expect electronic conduction to be a significant contributor to subglacial conductivity. Dielectric conduction results from the slight displacement of bound electron rel ative to their nuclei when in the presence of an electric field. The displacement is necessarily small, and the effect is not observable as a current except when the external electric field varies. The displacement current is analogous to current flowing through a capacitor. Since our measurements are made with static or near-static electric fields, we do not observe displacement currents and we do not measure dielectric properties of the subglacial material. The subglacial material beneath a glacier is a mixture of water and minerals. The pore water is an electrolyte containing dissolved mineral salts that may or may not be in equilibrium with the surrounding minerals. Under the influence of an external electric field, the ions in the water will move. Electrolytic conduction has two components: bulk conduction and surface conduction. Bulk conduction arises from the movement of ions beyond the hydrodynamic shear plane. Surface conduction is the enhanced conductivity within the hydrodynamic shear plane that arises because of the high concentration of charge carriers. Although the ions themselves do not move, current is carried by protons (Hj hopping from place to place, and perhaps also by H3O (Parks and others, 1966; Anderson and Parks, 1968). Since most of the subglacial minerals are poor conductors, subglacial conduction is dominated by electrolytic conduction; the resistivity of the subglacial material will vary with the mobility, concentration, and degree of dissociation of the ions, as well as with the properties of the rock—electrolyte interface. 4.4 Electrical resistivity Electrical resistivity is a material property that quantifies the resistance of a material to the flow of electrical current. This general empirical relationship, known as Ohm’s  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  98  Law, is expressed as (4.7)  —VV = pJ  where p is the electrical resistivity tensor, V is the electrical potential field (a macro scopic potential field rather than a microscopic surface potential field), and J is the electrical current density. Anisotropy in the resistivity tensor indicates current flow in a direction other than that of the applied electrical field B = —VV. For most substances, and for the analyses in this thesis, this effect is neglected and resistivity is treated as a scalar quantity p. Equation (4.7) is sometimes written as (4.8)  J = —oVV where o- = i/p is the electrical conductivity.  As discussed in section 4.3, current flow through water-saturated sediment is car ried by bulk conduction and surface conduction.  If we neglect surface conduction  (Morgan and others (1989) note that neglecting surface conduction in clay-rich sedi ments is risky) and remember that the minerals in the sediment are poor conductors, we can imagine that the electrical resistivity of the sediment will depend on the amount of water in the sediments and on how well pockets of water are connected together. Archie (1942) proposed the formula an_mS_dpw  (4.9)  to describe the resistivity of porous rock; n is the fractional pore volume (porosity), S is the fraction of the pores containing water (saturation), m is a cementation factor, d is the saturation exponent, and Pw is the resistivity of the pore water. The range of values for the constants d, a, and m are: ci water-saturated subglacial sediments, S  =  2, 0.5  a < 2.5, and 1.3  m <2.5. For  1. The value of m is roughly proportional  Chapter  4.  ELECTRICAL PHENOMENA  —  99  THEORY  to the age of the rock and is about 1.3 for Tertiary sediments (Parasnis, 1986), but is probably even smaller for deforming sediments with high porosity.  4  ..  2 Measuring electrical resistivity  Two- or four-electrode methods may be used to measure electrical resistivity in the field. Although two-electrode systems are logistically simple, they suffer from two im perfections: (1) It is difficult to ensure a good consistent electrical connection between the electrodes and the ground; a significant contact potential, which is indistinguishable from the potential arising from current flow through the medium, often arises. (2) As current passes through the electrodes, charge carriers become depleted in the vicinity of the electrodes and the resistance of the circuit increases over time; this effect is known as electrode polarization. In consequence, two-electrode systems are used rarely and only for specialized purposes (e.g. Neftel and others, 1985). Both of the problems mentioned above are associated with the flow of electrical current through electrodes. By separating the electrodes measuring potential from those delivering current, four-electrode systems avoid these problems entirely. One pair of electrodes is used to inject a metered current into the ground using as much voltage as is required; a second pair is used to measure the resulting electrical field. 4.2.1 Potential fields The equation describing the potential distribution V resulting from a current source I on the surface of a half-space is V  2irr  (4.10)  where r is the distance from the current source (Telford and others, 1990). The singu larity at r  =  0 is of no importance since real current sources are not point sources.  Chapter  .4. ELECTRICAL PHENOMENA  —  THEORY  100  Since Equation (4.7) is linear, the output of a four-electrode array with all four electrodes on the surface of a half-space can be computed by superimposing the voltage contributions from multiple current sources. The two current electrodes are numbered 1 and 2 and the two potential electrodes are numbered 3 and 4; current enters at elec trode 1 and electrode 4 is at zero potential. The voltage V at electrode 3 is given by  14 r  2ir \r  23 r  24 r j  (4.11)  where rmn represents the scalar distance between electrodes m and m. Equation (4.11) is often written as  v  (4.12)  =  where  13 \r  14 r  23 r  r24J  (4.13)  is a geometrical factor. We will use the term d.c. potential to describe the induced potential V; this term reflects the anthropogenic nature of the potential and its asso ciation with constant (direct) current levels. Interpretation In practice, the Earth is not an isotropic homogeneous medium. Although this an isotropy introduces complications, a homogeneous medium would not be particularly interesting. Establishing the relationship between Pap and the true resistivity structure of a half-space is the burden placed on geophysicists. The resistivity computed from a pair of current and voltage observations is called apparent resistivity; the apparent resistivity is equal to the homogeneous half-space  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  101  resistivity that would yield the same voltage and current readings and from Equa tion (4.12) is given by Pap  =  2irV IG  (4.14)  The simplest form of anisotropy is layering; the resistance encountered by current flowing along the layers (so the layers act as resistors wired in parallel) may be quite different from the resistance encountered by current flowing across the layers (so the layers act as resistors wired in series). Analytic solutions for voltage response can be de rived for simple Earth models (e.g. buried spheres, simple layered models), but for more complex models, other approaches are required. A basic characteristic of four-electrode d.c. resistivity arrays is that the farther apart the current electrodes are, the deeper the current penetrates into the ground. The apparent resistivity reflects some spatially averaged resistivity near this “pseudo-depth”. For many years, geophysicists relied on extensive sets of type curves compiled for various simple layered Earth structures (e.g. one, two, and three layers over a half-space) and standard electrode configurations (e.g. Wenner, Schiumberger, pole-dipole, etc.). With the advent of powerful comput ers has come the ability to infer more complex two- and three-dimensional subsurface structures using inverse techniques and forward modelling. In our subglacial experiments, the electrodes are not placed in standard, or even rectilinear, configurations. This is not because we wanted disorderly arrays, but be cause the holes drilled to the bed of the glacier are rarely plumb. For this reason, we have developed a simplistic measure for the depth of observation, or pseudo-depth, for a generalized d.c. resistivity array. The method involves computing the apparent resistivity over a simple layered-Earth model. At an interface between two homogeneous regions of differing electrical resistivity, the tangential electric field must be continuous. This condition gives =ñxE 1 ñxE 2  (4.15)  Chapter  .4. ELECTRICAL PHENOMENA  —  102  THEORY  The normal current flow must be continuous across the interface giving 1 ñ•E P1  =  2 ñE  (4.16)  P2  Analytic solutions to Equation (4.7) subject to Equations (4.15) and (4.16) are possible, but the solutions are unwieldy. Current density J falls of with an inverse-square relation as we move away from a current source. This relation is the same as for light and it happens that the equations describing reflections between semi-transparent mirrors can be applied to currents in the Earth. Telford and others (1990, Chapter 8) show the development of this method-of-images technique for solving d.c. resistivity problems. We chose a single layer over a half-space for our pseudo-depth calculations. This model was chosen because the ice above the array can be considered infinitely resistive (section 6.1). For very thin layers, the array in question will respond primarily to the half-space, or basement, resistivity, and for a thick layer, the array will respond to the layer resistivity. By calculating the apparent resistivity for a varying layer thickness, we can determine what layer thickness corresponds to transition from layer response to basement response. The method-of-images equation for the potential distribution on the top of the layer resulting from a current source I (also on the top of the layer) is I V=— +2 ./(2mb)2 + 2ir r mO  (4.17) r2  where kP2I)1 P2 + P1  and where p’ is the layer resistivity,  P2  (4.18)  is the half-space resistivity, b is the layer thick  ness, and r is the separation of the current source and voltage measurement point. The value of k is analogous to a reflection coefficient for the layer-basement interface:  Chapter k  =  4. ELECTRICAL PHENOMENA  —1 for a conductive basement and ic  =  —  103  THEORY  +1 for a resistive basement. Equation (4.17)  is analogous to Equation (4.10) for a homogeneous half-space. Following the develop ment of Equation (4.14), but using Equation (4.17) in place of Equation (4.10), we derive an expression for the apparent resistivity of a generalized d.c. resistivity array over the layered Earth model:  Pap  [1+  (2b)2 00 -  +r3  k’ (2mb)2 +  T3  +  (2mb)2 + r  }]  (4.19)  where G is defined as in Equation (4.13). 2 Figure 4.1 shows the apparent resistivity curves for a Wenner array (ri 34 r  =  =  23 r  =  1 m, electrodes all in a line). The solid line is the curve for a resistive layer and  the dotted line is the curve for a conductive layer. Since the resistive layer provides a sharper transition, we will use a resistive layer in our calculations of pseudo-depth. The pseudo-depth z, is defined as the point where, on a logarithmic scale, the apparent resistivity is midway between p’ and and  =  P2.  In Figure 4.3, that point is at Pap  =  0.480 m. Notice that z depends only on the electrode geometry  3.16 1 m —  it is  independent of the resistivity values chosen. Current switching As will be discussed in section 4.5, natural electric potentials exist in the Earth; these natural potentials are added to the d.c. potentials. The standard method for removing this natural potential offset is to measure the d.c. potential with the current being driven in both directions. Because Equation (4.10) is linear, changing the polarity of the current will change the polarity of the resulting potential. If the natural potential  Chapter  4.  ELECTRICAL PHENOMENA  101  -  —  104  THEORY  —-------  -  \ \  E  \ \ \ 100  I I  io—  102  101  100  101  I  I  102  LAYER THICKNESS (m) Fig. 4.3: The apparent resistivity for a layer over a half-space model as observed by a Wenner array configuration with a 1 m electrode spacing. The horizontal axis shows the variation in the layer thickness. The solid line is the curve for a resistive layer (p = 10 fm, p2 = 1 1 m), and the dotted line is the curve for a conductive layer (p1 = 111 m, P2 = 10 Il m). remain unchanged, subtracting the potentials observed at each current polarity removes the natural potential component. In order to make these measurements, a typical electrical resistivity apparatus uses the current waveform illustrated in Figure 4.4. The forward biased current is applied for a period of time followed by a period of no current flow. Then the current supply is reversed and the pattern is repeated. Transient effects The Earth’s voltage response to changes in applied electric current is not immediate. There are at least two processes that create a delay in establishing a stable voltage; electrode polarization and membrane polarization (see Marshall and Madden (1959);  Chapter  .  ELECTRICAL PHENOMENA  —  105  THEORY  I  z w  D C.)  TIME  Fig. 4.4: The alternating, pseudo-static current waveform traditionally used for making d.c. resistivity measurements. Keevil and Ward (1962) for concise overviews). These processes are known collec tively as induced polarization (IP) effects and are often used in mineral exploration, principally in the search for disseminated suiphide minerals. Electrode polarization was mentioned in section 4.4.2 in connection with chargecarrier depletion at the current injection electrode; the same phenomenon occurs when current passes between electrolytic conductors (e.g. pore water) and electronic conduc tors (e.g. graphite and sulphide minerals). We distinguish these two situations where electrode polarization arises by speaking of ore-body polarization when referring to the source of IP effects. Ions drawn to a metallic interface under the influence of an external electric field exchange electrons with the metallic body. Since the metallic body does not interrupt the electric or current fields, but does present a physical barrier to ion movement, ions accumulate on the boundaries of the metallic body and cause a gradual increase in the observed surface potential. After the electric current is turned off, a period of time elapses before the ions return to their initial positions; the resulting potential decay can be observed (Figure 4.5). Most time-domain IP exploration equipment measures  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  106  the IP signal on its decaying limb after the current is shut off, either by integration or sampling over time. The dependence of ore-body polarization on ion concentration, mineralogy, mineral dissemination, porosity, pore size, etc. is not fully understood.  LU  0  >  TIME  Fig. 4.5: The effect of induced polarization on the observed voltage re sponse of the Earth. The deviation from the square wave in Figure 4.4 is the IP effect. Physical barriers to ion movement can also be generated by a pre-existing charge dishibutions in the ground. The electrical double layers discussed in section 4.2 can block pores if the pore size is comparable to that of the double layer thickness; this effect is known as membrane polarization. Ore-body and membrane polarization are indistinguishable, but ore-body polarization effects are generally the larger of the two. In 1987, the first year we took d.c. resistivity measurements under Trapridge Glacier, we made oscilloscope observations of potential waveforms measured at the glacier bed and found no evidence for IP potentials. Despite an apparent absence of IP effects, we chose to measure d.c. potentials at least 1 s after turning on the current; this delay is large enough to eliminate all but the most persistent IP effects. Our cau tion was motivated by a concern that IP effects might develop during the course of our experiments; owing to a lack of metallic minerals, we did not expect to observe  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  107  ore-body polarization potentials, but the presence of streaming potentials (discussed in section 4.6) implies the presence of electrical double layers and suggests that the membrane polarization mechanism might be active. Because of instrument limitations, we made no attempt to measure IP effects. 4.5 Natural Potentials  Naturally occurring potentials are often referred to as self potentials or spontaneous potentials, but to avoid confusion of these terms, both having the acronym “SP”, with streaming potentials, we will use the term natural potentials to describe them. Sources of natural potentials include: streaming potentials, thermoelectric potentials, electrochemical potentials, and telluric potentials. Thermoelectric and diffusion potentials are phenomena related to the activity of ions in the pore fluid electrolyte. Under isothermal conditions, the differing mobility of the ions creates a liquid-junction or diffusion potential; as the ions diffuse in response to the concentration gradient, the more mobile ions (usually the cations, as they are physically smaller) will move faster; a charge separation develops. There are several mechanisms producing thermoelectric potentials. Since the mo bility of ions is temperature dependent, temperature gradients under conditions of constant concentration will produce a thermoelectric potential. Thermoelectric po tentials also appear when two dissimilar metals or metallic minerals come in contact; this effect, named after Seebeck, is the operational principle of thermocouples, but the Seebeck effect is not of concern here. In the ground, thermoelectric potentials cre ated by differing ion mobility are typically 0.2—0.3 mV °C’ (Nourbehecht, 1963) and have been used in geothermal exploration (e.g. Corwin and Hoover, 1979; Corwin and others, 1981).  Chapter (. ELECTRICAL PHENOMENA  —  THEORY  108  Corrosion or mineralization potentials are produced by electrochemical cells formed between the pore fluid electrolyte and certain conductive minerals (principally sul phides, graphite, and some oxides). These potentials can be very large millivolts  —  —  hundreds of  and are analogous to the reactions occurring in primary and secondary  batteries. Typically, corrosion potentials arise over large ore bodies where the condi tions of burial cause one part of the body to undergo oxidation and another to undergo reduction (Telford and others, 1990). Corrosion potentials are the principal justifica tion for using metal/metal salt electrodes since corrosion potentials on an electrode can be large compared to natural potentials (see section 5.3.4). Telluric potentials are caused by the electric currents induced in the Earth by fluctuations in the Earth’s magnetic field. The frequency of these fluctuations ranges from iO Hz up into the audio range and the potentials resulting from the telluric currents are of the order lOmVkm’ (Telford and others, 1990). There are other sources of higher frequency electric signals  —  for instance tn  boelectric and seismoelectnic phenomena associated with rock fracture and transient stress (Butler, 1991)  —  but because we are making measurements at intervals over a  period of many days, it is unlikely that these transient potentials could contaminate our data. Irreversible thermodynamics Streaming potentials result from a cross-coupling between pressure gradients and elec tric potentials; several of the other sources of natural potential mentioned above also derive from similar cross-coupling mechanisms. To fully understand these coupled phe nomena, we must delve into the thermodynamics of irreversible processes. In a pair of seminal papers, Onsager (1931a, 1931b) considered the interaction of two or more irreversible transport processes (e.g. heat conduction, diffusion, elec trical conduction, and fluid flow) and laid out the framework for thermodynamics of  Chapter  4.  ELECTRICAL PHENOMENA  —  THEORY  109  irreversible processes. Onsager’s first postulate of the thermodynamics of irreversible processes is that, in a system where many flows and forces mingle, it is possible to express each flow J as a linear combination of all the extant forces X,. The resulting system of equations =  L,X,  (4.20)  constitutes the thermodynamic equations of motion or the scalar phenomenological relations. L 3 are phenomenological coefficients. The phenomenological coefficients L 3 can be divided into two classes: where i where i  0 j,  j,  L, are generalized conductivities and  are coupling coefficients between the different flows and forces. The  forces X 2 are usually written as negative gradients of potential fields. Equation (4.20) is a low-order Taylor expansion of the general equation J (X X 2 f , 1 ,X 2 ,. 3  .  .  ,  =  X,,j when we stipulate that no flow occurs in the absence of forcing.  The linear relationship between forcing and flow is not preordained and must be tested for each physical system, but there are many empirical relations of this form in common use. For instance, Jcurrent  =  —uVV  Ohm’s Law  (4.21a)  Jheat  =  VT 1 —K  Fourier’s Law  (4.21b)  Jflujd  =  VP 2 —K  Darcy’s Law  (4.21c)  --DVC  Fick’s Law  (4.21d)  soiute =  where o, K , and D are appropriate conductivities, and VV, VT, VP, VC are 2 , K 1 gradients of electric potential, temperature, pressure potential (fluid pressure less hy drostatic pressure), and concentration. In general, linear relationships should hold for small displacements from equilibrium, but for larger excursions, higher order terms must be introduced. For example, at high fluid flux non-linearities in Darcy’s Law.  Jfiuid,  turbulence can introduce  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  110  Onsager’s reciprocal relation states that the phenomenological coefficient matrix L is symmetrical:  =  (4.22)  This relation was first proposed by Thomson (Lord Kelvin) (1853) in connection with thermoelectric phenomena and later by Helmholtz (1882) in connection with diffusion potentials (diffusion potentials are sometimes referred to as Nernst potentials in hon our of Nernst (1888) whose kinetic model derivation of Equation 4.22 corroborated experimental results). Onsager’s second postulate of irreversible thermodynamics is the principle of mi croscopic reversibility. Microscopic reversibility states that for conservative force fields (a conservative force field is one where the work required to move a particle from one location to another is independent of the path taken), the dynamical laws are always reversible. That is to say that if the velocities of all the particles are reversed simulta neously, the particles will retrace their former trajectories, reversing the entire sequence of prior states. Using the example of a chemical equilibrium, Onsager (1931a) showed how Equation (4.22) can be derived from the principle of microscopic reversibility (see Haase, 1969 and de Groot and Mazur, 1984, for discussions of microscopic reversibility). Nourbehecht (1963) presents a clear development of how irreversible thermody namics can be applied to study a host of seemingly unrelated phenomena occurring in the ground. Equation (4.20) does not determine which forces are relevant for a par ticular flow; these choices must be made experimentally. Table 4.1 lists some of the interdependences between forces and flows that have been shown to exist. As has been discussed above, the linearity of the phenomenological relations must be verified.  Chapter  .  ELECTRICAL PHENOMENA  FLOW fluid  pressure Darcy’s Law  current  streaming potential Isothermal heat transfer streaming current  heat flow solutes  —  THEORY  FORCING electric temperature electroosmotic thermoeffect osmosis Ohm’s Law Seebeck potential Peltier Fourier’s effect Law electroSoret phoresis effect  111  concentration normal osmosis diffusion potential Dufour effect Fick’s Law  Table 4.1: Some established phenomenological relationships between the indicated forcing fields and flows. It is appropriate to note that Onsager’s linear reciprocal relations are not uni versally accepted. Truesdell, a respected authority on thermodynamics, has coined the terms “Onsagerism” and “Onsagerist” to describe concepts and individuals making use of these principles; the tone of these words conveys Truesdell’s opinion! Truesdell (1984, p. 404) quotes Jaynes (1980) as follows: “This approach (Onsager’s work), therefore, reached a dead end. The logic of using equilibrium relations in nonequilibrium situations was hardly an advance over that used by Thomson in 1854; indeed, we are unable to see wherein they differ at all. To make further progress beyond this point, it was necessary to go back to first principals and reason things out all over again, much more carefully. The coup de grace and final benedictions were administered by Wei and Truesdell.” We must admit that we too are disturbed by the use of equilibrium relations in nonequilibrium situations, but we cannot dispute the appearance of streaming poten tials in our data (Chapter 6)  —  as many others do, we will apply Onsager’s reciprocal  relations to our observations. As we will show in Chapter 6, we do observe a linear relationship between pressure gradient and streaming potential response.  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  112  4.6 Electrokinetic phenomena There are two electrokinetic phenomena that arise out of the phenomenological equa tions for electrical current flow and water flow. If we write Equation (4.20) for these flows we get J  L1  £12  —VV  £21  L 2 2  —VP  (4.23)  =  q  where J is the current density, q is the fluid flow, V is the electric potential, and P is the pressure potential (the fluid pressure less hydrostatic pressure). Comparing Equation (4.23) with Ohm’s Law and Darcy’s Law, we see that conductivity o-, and that  £22  is the electrical  is the hydraulic conductivity.  Caution is required in isolating a pair of flows as we have done in Equation (4.23) since the presence of any one gradient will generate, through coupling, other gradients and flows. For example, Equation (4.23) neglects thermoelectric effects and the Nernst effect. This separation is necessary, however, in order to simplify the analysis. By Onsager’s reciprocal relation, 1 L 2  =  . 2 L 1 If these two phenomenological coeffi  cients are non-zero, then we will observe two coupled phenomena: (1) The electric field will contribute to water flow. This effect is known as electro-osmosis. (2) The pressure gradients will contribute to the electric current. This contribution can be modelled as an additional electric potential field known as the streaming potential. We can write J  where C  =  =  -(VV + GVP)  (4.24)  /L 1 L 11 2 is the streaming potential coefficient and V is the electric field.  We will note here that some early work on streaming potentials suggested that the phenomenon is not linear (Wyllie, 1951), but this nonlinearity is probably associated with streaming potentials in turbulent flow (Boumans, 1957a, 1957b, 1957c, 1957d; Kurtz and others, 1976).  Chapter  .  ELECTRICAL PHENOMENA  —  113  THEORY  J,.6.1 Observing streaming potentials The dependence of streaming potentials on pressure gradients makes for some inter esting consequences. If a pressure source is located within a homogeneous half-space, then streaming potentials will exist throughout the medium, but no streaming poten tial anomaly will be observed on the surface; this is because no tangential pressure gradients can exist on a free surface. Since most geophysicists are concerned with mea surement of natural potentials on a free surface, the theory developed in the literature concentrates on how subterranean inhomogeneities can result in surface potentials. We recognize that the subglacial environment does not have any extensive free surfaces (the closest approximation to a free surface might be a large, low pressure-gradient subglacial conduit), yet it is instructive to follow the development of streaming poten tial theory as it applies to free-surface measurements. Indeed, the ice—bed interface may represent the ideal locale for making measurements of streaming potentials. .6.2 A theoretical development From Equation (4.24), we can define a total electric potential =V+CP  (4.25)  such that the current flow is given by J  =  —crV&  (4.26)  Equations (4.25) and (4.26) disregard all sources of electric fields except for the stream ing potential V. Nourbehecht (1963) and Fitterman (1978, 1979a, 1979b, 1984) argue that in the absence of electrical current sources, the current field is divergence-free (V J .  =  0) and the governing equation for any homogeneous region becomes (4.27)  Chapter  .  ELECTRICAL PHENOMENA  —  THEORY  114  Note that pressure sources are not considered electrical current sources because we assume that the water injected is electrically neutral. It is important to realize that the potential b cannot be directly measured; elec trodes placed in the medium will measure V. For instance, in a homogeneous medium with no current sources, we will observe streaming potentials, but J and & are every where identically zero since V  =  —CF.  Equation (4.27) ensures that sources of  can only be found at discontinuities in  the medium. At any interface, continuity of normal current flow requires that =0  (4.28)  where 1 and 2 denote the two media on opposite sides of the interface. Continuity of F and V at the interface requires that  L=cLF=s  (4.29)  where S is a generalized source function. Because the source function is a step change in potential, it can be modelled as a dipole charge distribution along the surface. This is indeed what Nourbehecht and Fitterman have done. The upshot is that for a surface anomaly to exist on a free surface, the necessary and sufficient conditions are that (1) there exist a boundary separating regions of different streaming potentials coefficient, and (2) there exist a gradient of pressure parallel to this boundary (Nourbehecht, 1963; Fitterman, 1978). Because the ice—bed interface is not a free surface, streaming potentials can be generated and observed at the interface, but this “free-surface” result reveals that contacts below the ice—bed interface can give rise to additional streaming potentials. These additional streaming potentials may also be observed at the ice—bed interface.  Chapter  4. ELECTRICAL PHENOMENA  —  115  THEORY  4.6.3 Hydrodynamics and boundary layers Next to the rock—electrolyte interface, there is a boundary layer within which no fluid motion occurs. The limit of this boundary layer is indicated by the dashed line in Figure 4.1 and is known as the hydrodynamic slipping or shear plane.  The shear  plane does not necessarily coincide with the OHP discussed in section 4.2.2. Under the influence of pressure gradients, only the fluid beyond the shear plane will move, carrying with it an excess of either positive or negative charges. This movement constitutes a convective flow of electric current, and as we will discuss below, gives rise to streaming potentials. By Onsager’s reciprocal relation, we know that if a system can produce streaming potentials, then it can also produce electro-osmotic effects. If we apply an external electric field, the mobile ions outside the shear plane will begin to drift. Viscous drag will cause the pore fluid to be entrained; the electric field causes fluid flow. Since the the late nineteenth century, electro-osmosis has been used in geotechnical applications for stabilising water-saturated embankments and other earthen structures (Adamson and others, 1966; Lewis and others, 1975). Note that the electric field established by streaming potentials will create an electro-osmotic effect that will tend to accelerate or retard fluid flow (depending on the predominant charge polarity found beyond the shear plane). Fortunately, this effect is very small and may be neglected. 4.6.4 Zeta potentials The potential at the hydrodynamic shear plane is the  potential. As we will show,  it is this potential that controls the magnitude and polarity of streaming potentials. We note that the  ( potential does not have to be equal to 4’ or even have the same  sign; Stern (1924) attributes the first discussion of this relationship to Freundlich, although no reference is given. For a given solid surface, the value of  will depend  on temperature, ion concentration, ion species, and pH. For ion concentrations and pH  Chapter  4. ELECTRICAL PHENOMENA  — THEORY  116  is negative and ranges between —100 and  values generally found in soils and rocks,  —2OmV (Hunter, 1981; Ishido and Mizutani, 1981; Morgan and others, 1989). Consider the laminar flow through a single cylindrical capillary tube (similar de velopments may be found in: Levine and others (1975); Hunter (1981); Morgan and others (1989)  — this is Helmholtz-Smoluchowski theory). The axial velocity at a dis  tance r from the axis of the tube is given by Poiseuille’s equation  ()  —r 2 (R )  (4.30) is the  where R is the radius of the capillary at the hydrodynamic shear plane and  dynamic viscosity of the fluid. The streaming current I is found by integrating the product of charge density and velocity over the cross-sectional area of the tube: I,  =  2rJ  (4.31)  rvz(r)pv(r)dr  0  where pv(r) is the volumetric charge density distribution. We are only interested in the charge distribution near the shear plane since the bulk fluid in the tube is electrically 2 neutral, so if we substitute the approximation (R  —  ) 2 r  2R(R  — r) for R  Equation (4.30) and substitute the result into Equation (4.31) while letting r’  r into =  R  —r  (r is distance away from the shear plane) we get  ()  is =  Now, since  pv(r’)  j’(R  —  r’)r’p(r’)dr’  (4.32)  is small for moderate and large values of r’, we can approximate  Equation (4.32) with I  2 /0P\ irR =  ----) j  R  r’p(r’)dr’  (4.33)  —  Chapter (. ELECTRICAL PHENOMENA  117  THEORY  Substituting Equation (4.1) into Equation (4.32) and integrating by parts gives = —  = =  4 2 e /p\ R ,d 2 irR r —dv (%\) 77 Oz 2 dr’ o  (4.34)  j ‘  e (OP 2 7rR 77 \Oz)  (ylR  (\  dr’Jr,_o  —  R 1  --dr’l  0 dv’ J  J  e 2 irR 77  (4.35) (4.36)  ZJ 9 \  The definite term vanishes because d4’/dz  =  0 at r’  =  R. Normalizing Equation (4.36)  by the cross-sectional area of the pore gives =  . (r 77  \OzJ  Surprisingly, this expression has no dependence on the surface area of the pore  (437)  —  the  implication is that grain size has no effect on the magnitude of the streaming potential. The Kozeny—Carman relation (e.g. Bear, 1972, section 5.10; Berryman and Blair, 1987) is a microphysical model of permeability that is consistent with Darcy’s Law. As with the Helmholtz-Smoluchowski theory, the derivation of the Kozeny-Carman relation assumes a Poiseuille distribution of flow qp within a cylindrical pore; to arrive at the bulk fluid flow q, an empirical tortuosity correction Co is applied to the flow within the pore giving q = conqp  (4.38)  where n is the porosity and c 0 is taken as 8/5. Making a similar adjustment for porosity and tortuosity in Equation (4.37) and assuming that preferred electrical current flow paths are parallel to the pressure gradient (since current flows by surface conduction and bulk conduction, this is equivalent to assuming that water flow paths are parallel to the pressure gradient), we get Js  =  coE(VP  (4.39)  Chapter  4.  ELECTRICAL PHENOMENA  —  THEORY  118  If the geometry and orientation of the fluid conduction paths force fluid flow oblique to the pressure gradient, then anisotropic effects must be considered (see Section 4.6.6). Equation (4.39) indicates that it is difficult to measure the value of  (,  even in a  laboratory setting, because measurements of J 3 (or rather E) do not isolate  from  geometrical, dielectric, and viscosity effects. We have already discussed variability in e —  it is also likely that the viscosity of a fluid in a narrow pore is different from that of  a bulk sample of the same fluid. It is interesting to note that the fundamental streaming potential proportionality constant L 12  =  21 derived for e</ in Equation (4.37) is the same as the value of L  electro-osmosis, also using Helmholtz-Smoluchowski theory (Mitchell, 1976, Chapter 15 —  note that Mitchell uses c.g.s. electrostatic units). This agreement probably has no  value in confirming or refuting Onsager’s reciprocal relation, since both results are based on the same initial equations and models, but it is comforting that the result 12 L  =  21 is consistent with our electrostatic distribution model for the pore (see L  Section 4.4.5). 4.6.5 The reverse current The streaming current J 3 does not generate an electric field; J 3 is a simple transport of charged particles and has an effect no different from that of a stream of charged particles drifting through a vacuum in the absence of any force fields. The streaming potential appears because the transport of charge gives rise to a separation of charge. We know that charge conservation principles must be satisfied over some spatial scale, yet streaming potentials seem to involve the movement of electrically charged fluid; we need some mental image that reconciles these seemingly conflicting obser vations. There are at least two conceptual models for understanding how streaming potentials are generated, but neither is particularly gratifying.  Chapter  .  ELECTRICAL PHENOMENA  —  THEORY  119 The charge accumulation model Consider a system in equilibrium with no fluid flow and no electric field. When a pressure field is impressed on the system, the fluid flow will begin carrying off the excess charges found outside the hydrodynamic shear zone. The destination of these charges is not clear, but imagine that these charges eventually begin to accumulate somewhere; as the flow progresses, an electric field resulting from the charge separation is established. This electric field produces an electric counter-current that eventually grows to balance the streaming current. If we neglect electronic and surface conduction mechanisms and if we assume a homogeneous medium (which ensures that the return current flows parallel to the fluid flow), we can imagine that the convective streaming current and the ionic counter current cancel; the excess ions in the pore fluid will remain stationary and the water will flow past them; in other words, once the charge distribution is established, the charge carriers stop moving. The charge conservation model Consider a volume of material as a system through which pore water flows. Figure 4.6 illustrates an equivalent electric circuit for the streaming potential phenomenon. The streaming current I acts as a current source, continuously moving charge carriers from one end of the system to the other. We can reasonably impose charge conservation on the system: the net charge on the fluid entering the system must equal the net charge leaving it. Charge conservation demands a counter-current I to balance I and electrical resistance in the conduction paths available to I necessitates the existence of an electric field.  Chapter  .  ELECTRICAL PHENOMENA  —  THEORY  120  Is >  0—  RBULK  RSURFACE  Wr  <  STREAMING POTENTIAL  Fig. 4.6: An equivalent circuit for streaming potentials. The convective streaming current I, is balanced by a reverse current I which flows via bulk and surface conduction. A potential difference arises across the circuit. J.6.6 Calculating the streaming potential We can compute the streaming potential V resulting from the streaming current defined in Equation (4.37) by appealing to Equation (4.8), Ohm’s Law. The resulting expression is  =  PEHvp 77  where p  (4.38)  1/u is the electrical resistivity of the material. Notice that Equation (4.30)  implies that streaming potential gradients are parallel to pressure gradients. In gen eral, anisotropy in the phenomenological coefficients o and K 2 (Equations (4.21a) and (4.21c)) and the quantities derived from them (p and H respectively) produces streaming potential gradients and pressure gradients that are not parallel.  Chapter  4. ELECTRICAL PHENOMENA  —  121  THEORY  4.7 Electrical phenomena relevant to this study Although streaming potentials can develop in any porous system (e.g. Ahmad, 1964; Bogoslovsky and Ogilvy, 1972, 1973; Korpi and Bruyn, 1972; Hunter, 1981; Ishido and Mizutani, 1981), streaming potentials are greatest where clay is present (Hunter, 1981, p. 17—21). There are two reasons for this: (1) substitution of lower-valence metal atoms (typically magnesium for aluminium) in the crystal structure of clay minerals creates a large negative surface charge o, which results in a large negative  potential;  (2) the specific surface area of clay-rich sediments is large so that the volume fraction of pore water under the influence of double layers is greater than for other sediments. The sediments beneath Trapridge Glacier are not particularly clay rich, but do con tain quantities of calcite, kaolinite, illite, and montmorillonite (Tavi Murray, personal communication). Pressure gradients parallel to the ice—bed interface are a natural consequence of the down-slope orientation of the glacier, so we can expect streaming potentials beneath this glacier. We also expect negative  potentials.  Natural potentials have been used for many years for mapping groundwater flow and transients events that cause changes in subterranean pressure fields such as earth quakes (Mizutani and others, 1976) and thermonuclear explosions (Nourbehecht, 1963). Discussions of natural potential field studies concentrate on streaming potentials (e.g. Bogoslovsky and Ogilvy, 1973; Ishido and others, 1983; Schiavone and Quarto, 1984) and thermoelectric potentials (e.g. Corwin and Hoover, 1979; Corwin and others, 1981; Ernstson and Scherer, 1986). The same holds true for theoretical treatments (Nourbe hecht, 1963; Fitterman, 1978, 1979a, 1979b, 1984; Morgan and others, 1989). We have not located any references to field studies of diffusion potentials, but conditions beneath Trapridge Glacier suggest that such potentials may be of concern. In the summertime, fresh solute-free surface melt water can enter the subglacial system beneath Trapridge Glacier, often on a diurnal cycle. Until dissolution processes mineralize this fresh wa ter, concentration gradients will exist and diffusion potentials along with them. Since  Chapter  4. ELECTRICAL PHENOMENA  —  THEORY  122  we have a very poor understanding of how surface water reaches the bed (i.e. is it a distributed source, line source, or point source), we cannot gauge how these diffusion potentials will influence natural potential measurements. Data presented in Chapter 6 will show that diffusion potentials are a small component of the total natural potential. The paucity of ore-bearing rock in the moraines surrounding Trapridge Glacier suggest that there are no significant ore body outcrops under the glacier and that corrosion potentials are unlikely to contribute significantly to the natural potential signal. Even if corrosion potentials exist, they are not likely to be time-varying and  will therefore not disrupt our measurements. We can expect telluric currents since the cold glacier ice presents no significant obstacle to electromagnetic radiation at frequencies up to several hundred kllz. Moni toring changes in the Earth’s magnetic field allows us to attempt compensation for tel luric currents, although as stated above, telluric potentials are of the order lOmVkm’ (Telford and others, 1990) and are probably insignificant. Thermoelectric potentials are unlikely to exists beneath the glacier since the ice—bed interface in the experimental site is at a constant temperature: the pressure melting point (Clarke and Blake, 1991). In addition, thermoelectric potentials would not be time-varying.  Chapter 5 ELECTRICAL PHENOMENA  METHOD  5.1 Introduction The electrical phenomena experiments carried out under Trapridge Glacier measured d.c. resistivity and natural potentials at the glacier bed. These measurements were performed using four-electrode apparatus at a number of locations and over a period of several years. For these experiments, we used two different measurement systems. The fundamental difference between the two systems concerns the nature of the elec trodes: the 1987 apparatus employed the same electrodes for both current injection and potential measurement whereas the apparatus used in later years employed dedicated electrodes for the two tasks. For all our experiments, data were collected on Campbell Scientific CR10 data loggers. These data loggers have eight differential analogue input channels (each dif ferential channel can also be configured as two single-ended channels), eight digital input/output channels, and some other interface channels not relevant here. Periph eral equipment, such as d.c. resistivity equipment, can be controlled using the digital output channels and the data recorded through the analogue input channels. The data loggers are programmed to execute a series of instructions at periodic intervals; these instructions may initiate measurements, change the status of the digital ports, insert delays, or process data. 5.2 The 1987 apparatus The 1987 experiment was our initial foray into making long-term measurements of temporal variations in subglacial d.c. resistivity. We also wanted to investigate the  123  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  124  spatial variation in d.c. resistivity, both laterally and with depth. Because the electrodes we place at the glacier bed are fixed in position and are not recoverable, this last objective requires that we have many electrodes at our disposal, of which we can choose four as appropriate. Thus, our d.c. resistivity apparatus requires three major components: (1) a switchable, current-limited, and polarity-reversible high voltage supply for injecting current into the subglacial material, (2) a way to measure the induced potential field, and (3) a mechanism for selecting which electrodes are “active” at any given time. Since the equipment is battery powered, we must also have a method for turning off the equipment. 5.2.1 Electrode configurations To fulfil these requirements, we designed and built an instrument that uses eight four pole/double throw relays to select four of eight identical electrodes for use as a d.c. resistivity array. Figure 5.1 lists the eight electrode configurations available. The desired configuration is selected by setting three digital output lines on the data logger so that they represent, in binary code, the configuration number. A digital logic circuit decodes the three binary inputs into signals to energize the various relays. If the electrodes are spaced evenly and sequentially in a line, then configurations P1 through P5 represent Wenner arrays stepping along a profile and configurations Si through S3 represent a depth sounding centred at the middle of the line. Notice that configuration Si duplicates configuration P3. 5.2.2 Electrode design The eight electrodes were simply 15 cm sections of standard half-inch copper pipe sol dered to a length of single-conductor stranded wire leading to the surface. The elec trodes were placed 0.5 m above the bottom of the borehole.  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  125  SWITCHING CIRCUIT  7//I  I V V I IVVI P3 IVV P4 I VV I P5 IVVI Si IVVL S2 I VV I S31 VV I P1  z P2 0 D  C, LI..  0  C)  1  2 3456  78  ELECTRODE  Fig. 5.1: The electrode configurations available with the 1987 d.c. resis tivity apparatus. The eight electrodes are numbered 1 through 8 across the top, and the eight configurations are numbered P1 through P5 and Si through S3. An “I” indicates that the electrode is used for current injection and a indicates that the electrode is used to measure voltage. Copper electrodes corrode and are therefore not eminently suitable for making voltage measurements, but we did not deem this a problem since the alternating d.c. resistivity measurement cycle removes pseudo-static natural potentials, of which elec trode corrosion potentials are one component. 5.2.3 Voltage measurement The two electrodes chosen as voltage electrodes are simply routed to a differential input channel on the data logger. During the 1987 field season, the only difficulty encountered was the large electrode polarization voltage that develops when an electrode is used to inject current. This potential decays steeply after the current is interrupted; if this  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  126  same electrode is used to make voltage measurements a short time later, the polarization potential decays significantly during the d.c. resistivity measurement cycle and renders this alternating cycle ineffective in cancelling natural potentials. We eliminated these problems by judiciously selecting the sequence in which the eight electrode configurations were used. Measurements were made in three groups separated by 10 mm  “rest” intervals to allow electrode polarization potentials to de  cay to near-constant levels. The configurations used within each group avoided using electrodes as both voltage and current elements. The groups were arranged as follows: (1) configurations P1 and P4; (2) configurations P2 and P5; (3) configurations Si, S2, and S3. 5.24 Current source and current measuring Recalling the theoretical development in Chapter 4, we realize that the high voltage supply (HV circuit) and potential measurement sections (LV circuit) must be electri cally isolated  —  in other words, the potential of the current electrodes must be able  to float relative to that of the voltage electrodes. This isolation is also necessary to protect the data logger from the high voltages present in the current supply. In our apparatus, the high voltage is generated using a d.c. to d.c. voltage converter. Voltage converters operate by chopping the incoming voltage, thereby creating a squarewave voltage signal. This pseudo-sinusoidal voltage is fed into the primary coil of a step-up or step-down transformer. The resulting secondary coil voltage is then rectified and filtered to produce the output voltage of the converter. A high chopping frequency (typically 10—20 kllz) permits the use of small transformers and reduces the physical size of the converter. The transformer also serves to electrically isolate the input and output voltages. The converter used in our apparatus produces a 250 V output from a 12 V input. This high voltage is fed through a coarse two-terminal current limiter. The  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  127  circuit for the limiter is shown in Figure 5.2. In typical operation, voltag es of 70-100 V are required to inject 10 mA of current into subglacial sediments.  CURRENT IN  TIP5O  Rb 250k 1/2W  Re 180R  CURRENT OUT  2.5V ZENER LM285-2.5  Fig. 5.2: The two-terminal current limiter placed in series with the high voltage current supply. Current flowing through Rb and the 2.5 V Zener diode ensures that the base of the NPN transistor is held at 2.5 V relative to point X. Current also flows into the base of the transistor causing collector-emitter cur rent, but the voltage drop across Re will limit this current to about 1.9/Re mA (1.9 V equals 2.5 V less the approximate base-emitter voltage of the transis tor). One technical hurdle remains and that is the measurement of the curren t being delivered to the current electrodes. in the HV circuit, the current is easily measured by monitoring the voltage drop across a small resistor, but this voltage must be conveyed to the data logger while maintaining the electrical isolation mentioned above. We accomplish this by feeding the voltage drop representing the current into a voltage to frequency converter. The resulting frequency is used to drive an opto-is olator circuit whose detector is in the LV circuit; the voltage drop is recovered using a phase-locked loop (PLL). Whenever a set of d.c. resistivity readings is made, a measurement of the output of the PLL is measured before the high voltage current supply is turned on. The tem perature stability of the isolation circuit can be determined by monito ring the changes  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  128  in this zero-current PLL output as the seasons change; the temperature stability is estimated at 100 ppm °C’. This temperature drift is negligible compared to the un certainty introduced by electrode position errors. The HV current sense circuitry is powered by a separate 12 V battery, whereas the LV circuit and the high voltage supply are powered from the same 12 V battery as the data logger. 5.2.5 Additional control Three additional digital outputs from the data logger (for a total of six) are used to complete control of the d.c. resistivity apparatus . One control line is used to turn the entire apparatus on and off. A second control line is used to turn on the high voltage supply. A third control line is used to change the polarity of the high voltage output (this function is implemented using a single double pole/double throw (DPDT) relay). The control line usage is summarized in Table 5.1.  Control Line 1 2 3 4 5 6  Function Apparatus on/off High voltage on/off High voltage polarity Configuration select line 1 Configuration select line 2 Configuration select line 3  Table 5.1: A summary of the control line function for the 1987 d.c. resis tivity apparatus. Light-emitting diodes (LED) on the front panel of the box indicate when the high voltage supply is active, what current polarity is selec ted, and which electrodes are being used as voltage and current elements. An analo gue voltage meter connected to the output of the current regulator is also mounted on the panel. These indicators facilitate monitoring the operation of the apparatus .  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  129  5.. 6 Technical specifications The entire d.c. resistivity apparatus fits into a plastic box having dimensions 6 cm by 12 cm by 18 cm. The precision of voltage measurements is determined by the data logger (better than 1 part in 7500). The precision of current measurements is lim ited by the opto-isolator circuit calibration; we estimate the error at ±2 %. 5.3 The 1988 apparatus  For the 1988 experiments, we wished to mak e accurate measurements of both d.c. re sistivity and natural potentials. Because the metal/metal salt electrodes suitable for voltage measurement are unsuitable for current injection, this task led to a natu ral separation of the current injection and volta ge measurement functions. To replace the 1987 apparatus, we designed and built seve ral pairs of current and potential multip lex ers capable of controlling 8 and 32 electrode s respectively. The primary functional difference between these multiplexers and the 1987 appa ratus is that the electrode configurations cannot be set directly by the data logger. Rather, the data logger commands the multiplexer to step through one of a num ber of programmed electrode configuration sequences; these configuration sequences are stored in erasable programmable read-only memory (EPROM) chips within the mu l tiplexers. The multiplexers were con structed using wire-wrapping techniques , with soldered discrete components in the HV circuit. 5.3.1 EPROM programs Both types of multiplexer use 27C64 EPROMs to store the configuration progra ms. Each 27C64 EPROM has 13 address lines and stores 8192 (213) bytes (a byte contain s 8 bits of digital on/off codes). The upper 4 address lines are tied to a set of switches inside the multiplexer and the lower 9 addr ess lines are connected to a binary cou nter. The counter is reset to zero as the multip lexer is turned on and advances when dig ital  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  130  clock pulses are received from a digital output on the data logger. When the multiplexer is on, the EPROM chip is continuously enabled; the data appear ing at the output of the EPROM chip always reflects the data stored at the location indexe d by the composite address (made up the switch settings and counter value). Thus, the switches can be used to manually choose one of sixteen multiplexer “progr ams” (each program is 512 steps long) and the data logger is used to step through the progra m; at each step, the eight digital outputs of the EPROM are used to select the active electrode(s). EPROM programs are a compromise between flexibility and simpli city of control. Although the data logger cannot select specific electrodes directl y, inteffigent program ming of the EPROM allows most, if not all, desired functio ns to be programmed into the multiplexer. The characteristics of the multiplexer can be changed easily by sim ply reprogramming the EPROM. Control of this type of multiplexer is very simple : for simple electrode selection operations, only two control signals are required: the first signal both controls power to the multiplexer and resets the program counter; the second signal is used to step through the program. 5.3.2 The current multiplexer The 1988 high voltage apparatus uses the same circuits for generating the high voltage and for limiting, measuring, and reversing the current, but instead of using a digital number provided by the data logger to select the electrode config uration (see Table 5.1), the output from an EPROM is used to designate which two electrodes are to be used as current electrodes. Table 5.2 lists the control signals used to operate the current multiplexer.  Chapter 5. ELECTRICAL PHENOMENA Control Line 1 2 3 4  —  131  METHOD  Function Apparatus on/off and clock reset Clock input High voltage on/off High voltage polarity  Table 5.2: The control line functions for the 1988 current multiplexer. Three of the output lines from the EPROM select the electrode connected to the positive terminal of the high voltage supply and another three output lines select the electrode connected to the negative terminal; the two remaining output lines are not used. The multiplexer uses 16 single pole/single throw relays (8 for each terminal of the high voltage supply) to link the electrodes. This arrangement allows each of the 8 electrodes to be connected to either terminal of the high voltage supply. The multiplexer can even connect both terminals of the high voltage supply to the same electrode, effectively short circuiting the supply; of course, the EPROM should never contain such a program step! Notice that the current multiplexer does not need a full complement of eight current electrodes; it can operate with as few as two electrodes. For the 1988 experiments, the EPROM programs were designed to connect the attached electrodes to the high voltage supply in all possible combinations. Table 5.3 lists the 16 EPROM programs used in the current multiplexer. 5.3.3 The potential multiplexer The operation of the potential multiplexer is very similar to the current multiplexer, but because its capacity has been increased to 32 voltage electrodes, two EPROMs are needed to designate the electrode pair. One EPROM is used to select which voltage electrode is connected to the positive side of the data logger’s differential input channel and the other selects which electrode is connected to the negative side of the channel. Only seven bits of each EPROM are used to select one of the 32 electrodes (32  =  2).  Chapter 5. ELECTRICAL PHENOMENA Program number 0 8 1 9 2 10 3 11 4 12 5 13 6 14 7 15  —  METHOD  132  Function 3 electrodes 4 electrodes 5 electrodes 6 electrodes 7 electrodes 8 electrodes source sweep sink sweep  Table 5.3: The EPROM programs for the 1988 current multiplexer. The two sweep programs connect electrode 0 (as current source or sink) with each of the other electrodes in sequence. Voltage electrodes are connected to the data logger, not with relays, but with solid-state analogue switching circuits; the analogue switches used in our potential multiplexer are the same as those used to perform multiplexing tasks inside the data logger. Table 5.4 lists the 16 EPROM programs used in the potential multiplexer. As with the current multiplexer, each program selects all combinations of electrode pairs from a set voltage electrodes, but unlike the current multiplexer, internal short-circuit settings are included (the two sweep programs connect the first electrode to each of the others in sequence). The short-circuit settings should give a zero voltage reading; they are used to check the proper operation of the multiplexer and to estimate the noise level introduced by the multiplexer. Instrument noise levels observed under field conditions were less than 30 V. Note that proper operation of the data logger requires that it be grounded; this is to ensure that the potentials reaching the apparatus do not exceed the common mode ranges of the multiplexer (+7.5 V) and data logger (+5 V). We accomplish this by grounding the apparatus to one of the electrodes, picked at random; this electrode is called the grounding electrode. Table 5.5 lists the control functions required for operation of the potential multi plexer.  Chapter 5. ELECTRICAL PHENOMENA Program number 0 1 2 3 4 5 7 8 9 10 11 12 13 14 15  —  METHOD  133  Function 4 electrodes 5 electrodes 6 electrodes 7 electrodes 8 electrodes 9 electrodes 10 electrodes 12 electrodes 14 electrodes 16 electrodes 20 electrodes 24 electrodes 28 electrodes 32 electrodes positive sweep negative sweep  Table 5.4: The EPROM programs for the 1988 potential multiplexer. The two sweep programs connect electrode 0 (as positive or negative) with each of the other electrodes in sequence. Control Line 1 2  Function Apparatus on/off and clock reset Clock input  Table 5.5: The control line functions for the 1988 potential multiplexer. 5.3.4 Electrode design The current electrodes used in 1988 were the same copper pipe sections used in 1987. The potential electrodes were of a different design. As discussed in section 4.2, there are a number of reasons relating to electrical noise for choosing metal/metal salt electrodes over metallic electrodes for voltage mea surements. In a laboratory setting, inert metals such as gold and platinum may be used as electrodes, but fiscal restraint is required in a field setting where the electrodes are not recovered. The standard-issue electrode in geophysical exploration is the cop per/copper sulphate porous pot electrode; a shallow porous ceramic pot is ifiled with  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  134  copper sulphate crystals, some water, and a copper electrode. The pot is allowed to age for a period of days. During this time, the dissolved copper sulphate approaches equilibrium with the excess copper sulphate crystals and the copper electrode. In use, the outer surface of the pot is placed in contact with the ground and the potential measuring instrument is connected to the copper electrode. Petiau and Dupis (1980) investigated the long-term stability of various metal/ metal salt electrodes for making telluric measurements. In measuring the noise levels produced by various electrodes over a frequency range of 0.001—100 Hz, they found 4 electrodes. noise levels of 3 V at 0.01 Hz for Cu—CuSO  Ag—AgC1 and Pb—PbC1 2  , steel, brass, and 2 electrodes exhibited slightly lower noise levels whereas Cd—CdC1 graphite electrodes showed higher noise levels. 4 porous pot design that was Figure 5.3 shows the variation on the classic Cu—CuSO used for our experiments. Since the electrode is to be immersed in the water present at the bottom of every borehole, the porous pot must be sealed. We contracted with a Vancouver potter to furnish a number of small porous vase-like pots (the porosity of the vessel is controlled by the kiln temperature). After coating the neck of each pot with a resin epoxy, we fed in a small piece of copper tubing soldered to the electrode lead. The pot was then completely filled with copper sulphate crystals and water, followed by an annular paper wad and a final plug of epoxy. Since the pot gets saturated with water when being ifiled, the initial hydrophobic epoxy coating allows the epoxy plug to bond well to the pot.  Chapter 5. ELECTRICAL PHENOMENA  —  METHOD  135  ELECTRODE LEAD  COPPER ELECTRODE -COPPER SULPHATE POROUS CERAMIC POT II  1 cm  Fig. 5.3: A cutaway view of the Cu—CuSO 4 porous pot electrode used as the potential electrode in the 1988 and later experiments. Data will be presented in Chapter 6 showing that our Cu—CuSO 4 electrodes intro duce a voltage offset level of about 1 mV between the fluid and the electrode lead. We have no measure of the frequency-dependent noise potential of these electrodes. 5.4 The 1989 apparatus In 1989, the 1988 multiplexer designs were fabricated as printed circuit boards. This was done to increase the reliability of the circuits, especially in cold winter operation. Slight modifications were made to the 1988 designs; these included: (1) rational ization of the logic circuits that translate the EPROM output into electrode selection signals, (2) using a second d.c. to d.c. converter to provide an isolated 12 V power supply for the HV circuit, and (3) modification of the EPROM addressing scheme so that larger-capacity EPROMs could be used in the multiplexers (27C128 and 27C256 chips). Larger EPROMs allow for a greater number of programs or longer programs. The 1989 apparatus was employed during the 1989 and 1990 field seasons using the 1988 EPROM programs.  Chapter 6 ELECTRICAL PHENOMENA  DATA ANALYSIS  6.1 Introduction Some of the first glaciological applications of electrical resistivity techniques included locating ice-cored moraines (østrem, 1959, 1964) and inferring temperature and den sity distributions (Hochstein, 1967), but most researchers have used electrical resis tivity sounding to determine glacier thickness (e.g. Röthlisberger and Vögtli, 1967; Vögtli, 1967; Fisch and others, 1977). Röthlisberger (1967) gives a good overview of such work. The resistivity of glacier ice, as determined by the aforementioned workers, is in the M1 m range, whereas the resistivity of water-saturated sediments and high porosity rocks is in the k m range (e.g. Telford and others, 1990). Assuming that the glacier sole is made of similar water-saturated material, we can expect a rather large resistivity contrast between the ice and the bed. Electrical resistivity techniques are well suited for determining the thickness of a layer having a resistivity much larger than the material below, but little information about the properties of the underlying material can be obtained. To study the material underlying the glacier, it is desirable to skirt the resistive ice overburden and to place the electrodes near the ice—bed interface. As far as we know, Haeberli and Fisch (1984) and Brand and others (1987) have published the only accounts of such experiments. Both reports discuss the spatial variation of subglacial electrical resistivity; Brand and others also made two sets of measurements separated by a period of two days and noted a temporal variation in electrical resistivity.  136  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  137  In Chapter 4 we discussed the importance of pore water in controlling the elec trical properties of the glacier bed. Both streaming potential and electrical resistivity measurements made over time should provide clues concerning the development and evolution of the subglacial drainage system. The importance of this drainage system in relation to glacier surging was discussed in Chapter 1. Glaciologists investigating subglacial processes are faced with the problem of maxi mizing the amount of information they obtain from each borehole within the constraints of available resources. The Trapridge Glacier research group is lucky in that large num bers of boreholes are drilled every summer (the glacier is relatively thin and our hot water drill is efficient), but fiscal constraints prevent us from instrumenting each bore hole as we would like. For example, a network of 50 or more pressure sensors would be unprecedented in glaciological research and would certainly provide invaluable data relating to the nature and evolution of subglacial drainage systems, but the sensor cost is too great. One of our aspirations for the subglacial streaming potential experiments is that they will prove an inexpensive method for expanding the coverage of a modest pressure sensor network: the cost to instrument a borehole with a pressure sensor is more than twenty times the cost for a potential electrode (‘.$15). The data presented in this chapter represent the first electrical resistivity measure ments made beneath a surge-type glacier and the first streaming potential measure ments made beneath any glacier. This work also represents the first attempt to make detailed long-term measurements of subglacial electrical phenomena. 6.2 Predicted forcing/response relationships Based on arguments presented in Chapter 4, we can speculate on a number of electrical responses to hydraulic forcing. These forcing phenomena can be divided into diurnal and episodic events.  Chapter 6. ELECTRICAL PHENOMENA  —  138  DATA ANALYSIS  6.2.1 Diurnal phenomena During the summer, given appropriate meteorological conditions, there is a diurnal melting of surface snow, firn, and ice. Solar noon, the time of maximum solar energy input, occurs at 21:2OhUCT (Universal Coordinated Time) because the longitude of Trapridge Glacier is 140.3°W, but summer daylight savings time and unruly time zones conspire to place chronological noon at 19:OOhUCT. This means that solar noon occurs at 14:20 h local time. Peak melt, then, occurs sometime in the afternoon. Surface meitwater can percolate down to the bed of the glacier, probably through deep crevasses and englacial channels (e.g. Seaberg and others, 1988; Hooke and oth ers, 1988); this water should reach the bed in the afternoon or evening. Surface meltwa ter from glaciers is quite resistive; since it originates as precipitation, it has almost no dissolved minerals. In contrast, water that has spent time in contact with minerals at the glacier bed has a considerable dissolved ion content and will be relatively conduc tive. Therefore, we expect the resistivity of the fluid at the ice—bed interface to increase in the afternoon as the resistive meitwater reaches the bed, and then to decrease again during the night as this meltwater mineralizes or mixes with conductive water deeper in the subglacial sediments. These changes in fluid resistivity may affect measurements of apparent resistivity. The input of meitwater will also cause the concentration of charge carriers to decrease and we might expect a corresponding increase in  (  potential. This  could result in larger streaming potentials during the afternoon and evening. Evidence presented in Chapter 3 suggests that deformation rates can have diurnal fluctuations. In section 4.4, we discussed the dependence of electrical resistivity on porosity. Since porosity is affected by deformation, we might expect diurnal fluctuations in electrical resistivity. The diurnal input of water can also cause diurnal changes in the subglacial wa ter pressure, which might change the lateral pressure gradients. The largest pressure  Chapter 6. ELECTRICAL PHENOMENA  —  139  DATA ANALYSIS  gradient changes will occur at the margins of unconnected zones in the bed (see sec tion 2.5.2); the water pressure in the connected zone will change quickly whereas the pressure in the unconnected zone will not. A large pressure gradient can result, with a corresponding increase in streaming potential. 6..2 Episodic phenomena In addition to diurnal cycling, pressure records from beneath Trapridge Glacier some times exhibit large sudden increases  —  and less often, sudden decreases  —  in subglacial  water pressure. The pressure change resulting from these events typically decays expo nentially over a period of a day or so. We will show examples of such pressure pulses below. When we are drilling a borehole, we sometimes record a pressure pulse in neigh boring holes when the drill reaches the glacier bed. During drilling, the borehole is usually completely filled with water. If the borehole reaches a connected zone at the bed, the water usually drains rapidly into the subglacial hydraulic system, causing a rise in local water pressure. If the borehole reaches an unconnected zone, the water column in the borehole drains much more slowly into the glacier bed and pressure dis turbances are small or non-existent. In either case, rapid changes in lateral pressure gradient can develop with corresponding sudden changes in streaming potential. We have also observed naturally-occuring pressure pulses that, depending on the distribu tion of subglacial pressure gradients, could also cause streaming potential variations. 6.3 Assumptions  In analyzing the results from all our electrical experiments, we have made two primary assumptions: (1) the resistivity of the glacier ice is very high compared to that of the sediments; (2) although each electrode is placed about 0.5 m above the glacier bed in  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  140  order to protect it from being torn apart, the electrode is effectively at the ice—bed interface. The resistivity of the glacier ice can be tested by making d.c. resistivity measure ments in truncated boreholes that terminate within the glacier. This was done in 1987. Four holes were drilled to a depth of 34 m and electrodes were installed. After waiting for the boreholes to freeze shut (so as to eliminate waterborne conduction paths), the resistivity of the glacier ice was measured as 2.0 + 0.5 x 1010 f m. This value compares favourably with laboratory measurements on clean, cold glacier ice by Cagniard (1959) and østrem (1967) and is an order of magnitude lower than the values obtained by Queille-Lefèvre and others (1959). The resistivity of ice varies over several orders of magnitude depending on its temperature and purity, so our value of resistivity is in the expected range. Close to the ice-bed interface, the resistivity of the ice might decrease because of englacial debris, increasing temperature, and/or increasing water content; we have not made measurements of ice resistivity variations with depth, but since the bed beneath our study site is at the pressure melting point (Clarke and Blake, 1991), the ice there is melting continuously and it is unlikely that a high concentration of englacial debris exists. We will assume that the electrical resistivity contrast between the glacier and its bed is large enough that the glacier can be treated as an infinitely resistive half-space. Our assumption concerning effective position of the electrode also requires that the ice be resistive compared to the basal water. If this is the case, then the water in the borehole forms an extension to the electrode that reaches the glacier bed at a point directly beneath the electrode. The borehole itself freezes shut overnight, sealing the electrode wire into a chamber having only one good electrical connnection to its surroundings: the bottom of the borehole. Judging from our experiences with redrilling boreholes after they have frozen shut, the first parts of the borehole to freeze shut are 10—20 m from the surface  —  this is where the glacier is coldest. Because the bottom  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  141  of the glacier is at the pressure melting point, we expect that the borehole around the electrode itself does not freeze shut during the summer field season. Creep closure of the borehole will take several months to close the ice in around the electrode. 6.4 The 198T experiments  In the summer of 1987, we made our first attempts to measure subglacial electrical resistivity using the apparatus described in section 5.2. We had several objectives for this first field season: (1) to test the apparatus, (2) to test the practicability of making subglacial measurements of electrical resistivity, and (3) to determine if our boreholes are drilled vertically. This was also the field season during which we discovered timevarying natural potentials. 64.1 Forefield operational test On 25 July 1987, we placed a rectilinear eight-electrode array across the terminus of Trapridge Glacier, about 10 m in front of the ice. The exposed sediments in this area were completely water saturated by meitwater seeping off the glacier and by ongoing precipitation; there was standing water on the surface. Since 1987, the sediments at the test site have been overridden by the advancing glacier. The electrodes, 7 cm long zinc-plated wood screws, were spaced at 4 m intervals. Figure 6.1 shows the d.c. resistivity values obtained for the seven distinct electrode configurations (see Fig. 5.1) in “pseudo-section” form. The horizontal axis indicates the position of the midpoint of the array and the vertical axis represents the pseudo depth calculated using the method described in section  Chapter 6. ELECTRICAL PHENOMENA  -4  -8 I  1940  1660  —  142  DATA ANALYSIS  ORDNATE (m) Q  +4  I  I  1730  1660  +8  1600  L93  2020  3.47  2260  5.00  i I-  Fig. 6.1: The d.c. resistivity pseudo-section through forefield sediments. The values of resistivity are in IZ m. A cursory interpretation of this pseudo-section reveals a fairly uniform, moderately conductive surface underlain by progressively more resistive material. The reported resistivity values are comparable to reported values for subglacial sediment (Haeberli and Fisch, 1984). The apparatus was functioning as expected. 64.2 Experimental design Eight boreholes, numbered 1 through 8 were drilled through the glacier at a spacing of 4m in a line parallel to the glacier flow direction. A copper electrode was placed in each of these holes. Two duplicate holes, numbered 2B and 7B were drilled next to the 2nd and 7th boreholes respectively; electrodes were also placed in these holes. We had no borehole inclinometer with which to measure the position of the electrodes. The experiment ran from 20 July 1987 to 14 August 1987. During the course of the experiment, only eight of the ten available electrodes were  iii  use at any moment,  but substitutions of electrode 2 for 2B and electrode 7 for 7B were made occasionally. The duplicate electrodes were drilled to help determine whether the boreholes were plumb  —  our rationale was that if the apparent resistivity reported by an electrode  Chapter 6. ELECTRICAL PHENOMENA  —  143  DATA ANALYSIS  configuration involving either the 2nd or 7th electrode positions was independent of which duplicate electrode was used, then the electrodes must be in the same position. Unfortunately, it became clear during the data analysis that the boreholes were far from plumb and that we had little idea of where the electrodes were located at the glacier bed  —  the duplicate electrodes served no purpose other than to confuse the  data records. Figure 6.2 shows the apparent resistivity data record for electrode configuration P1 (see Fig. 5.1) calculated according to Equation (4.14) assuming that the relative elec trode positions at the ice—bed interface are the same as those at the surface. The resistivity values are lower than those found in the forefield test, but they are not unreasonable for wet sediment. The reader will notice large sudden jumps in electri cal resistivity from time to time; these jumps result from switching between duplicate electrodes as indicated on the figure. Recall that configuration P1 selects electrode positions 1, 2, 3 and 4; the jumps occur whenever electrode 2B is substituted for elec trode 2. It is clear that electrode 2 is nowhere near electrode 2B; the boreholes are not plumb.  300  -  2,7  2,7B  E  2B,7 %_.._  .,  —F,-—  2B,7B  0  11111 I I IllillIllill iii IE  20  JULY  10 5 30 AUG 1987  Fig. 6.2: The apparent resistivity record for electrode configuration P1. The gaps and sudden jumps are explained in the text.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  144  The error introduced by an imperfect knowledge of electrode location is entirely one of scaling; the error is in computing the geometric factor G. For this reason, we cannot determine absolute values of apparent resistivity (or attempt to recover true resistivity), but we can discuss temporal variations. 64.3 Diurnal cycling of d.c. resistivity Figure 6.2 shows a clear diurnal fluctuation. The resistivity readings increase in the afternoon and decrease around midnight. These oscillations correlate well with the predicted meitwater-driven cycling discussed in section 6.2.1, but diurnal fluctuations Pap resulting from meitwater input are unlikely because the d.c. resistivity array  probably has a pseudo-depth on the order of 1 m (for an evenly-spaced linear electrode array with 4 m spacing, the pseudo-depth for the P1 configuration is 1.93 m) and is therefore relatively insensitive to the fluid conductivity at the ice—bed interface. In section 3.3.1, we showed that the penetration of water down into the basal material is a very slow process; it is possible that we are observing the diurnal migration of resistive water deep into subglacial sediments, but such rapid water transport runs counter to our understanding of the hydraulic permeability of the bed. Another mechanism which could produce diurnal fluctuations in resistivity is de formation. At the conclusion of Chapter 3, we showed that diurnal cycling of deforma tion occurs beneath Trapridge Glacier (the 1989 data showed this best). Recall that the deformation rates tended to be positive in the early morning and negative in the early afternoon. Although no significant pressure fluctuations were observed during the deformation experiments, we postulated that pressure fluctuations must have been occurring nearby; increases in subglacial pressure elsewhere can cause translocation of basal shear stress to other areas, such as our experimental site. We speculated that the observed down-flow deformation (indicated by positive strain rate) occurs in re sponse to such an increase in shear stress. We also postulated that negative strain  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  145  rates represent a relaxation or consolidation phenomenon that occurs when the shear stress drops. Murray (1990) discusses the increase in porosity and permeability associ ated with dilation  —  an increase in porosity should also cause an decrease in resistivity  (Equation 4.9). If we assume that positive deformation is driven by glacier flow and negative deformation is a relaxation process, then we would expect lower resistivity when deformation occurs and higher resistivity when relaxation occurs. 644 Geometrical corrections We attempted to determine the location of the 1987 electrodes using iterative forwardmodelling techniques. Using a model of a conductive layer between two half-spaces (the ice above, with a known high resistivity, and the basement below), we tried to fit the apparent resistivity observed by each electrode configuration to data from one set of observations. For a given set of eight electrodes, the model had 27 degrees of freedom (resistivity of the layer and basement; thickness of the layer; and the three coordinates of each electrode). We tried to solve the problem using non-linear minimization techniques (Press and others, 1986, p. 289) and Monte-Carlo simulations, but all solutions obtained were unable to obtain satisfactory predictions of observed apparent resistivity values. We conclude that the resistivity structure of the subglacial material is more compli cated than a three-layer model can realistically depict. Morphological changes resulting in alteration of conduction paths and inhomogeneity of the sediments are two possible explanations for our failure to divine the electrode positions. 64.5 Polarity reversals We have evidence for time-varying changes in electrical conduction paths. Figure 6.3 shows the apparent resistivity record for configuration P2. As in Figure 6.2, the vertical scale is incorrect because we go not know the correct value of G. The substitution of duplicate electrodes is indicated on the figure.  Chapter 6. ELECTRICAL PHENOMENA  —  146  DATA ANALYSIS  +150 -  2B,7B  -AJ  c 2,7  —150— 20  ,7B i i  JULY  i i i  10 30 5 AUG 1987  Fig. 6.3: The apparent resistivity record for electrode configuration P2. Notice the diurnal changes in resistivity polarity in early August. In early August, polarity reversals are evident. We are confident that these rever sals are not caused by equipment malfunction because (1) the transitions from positive to negative resistivity are smooth and (2) no modifications to the apparatus or elec trode array were made during this time. For a homogeneous, isotropic medium, polarity reversals are impossible. Figure 6.4 shows the plan view of a cross-shaped electrode configuration that will produce polarity reversals for slight changes in the path taken by the electric current. If the current flows as indicated by the solid line, a positive resistivity will be recorded since, as measured along the current flow line, the positive voltage electrode is closer to the positive current electrode; the reverse holds true if the current flows along the dashed line. Unfortunately, we do not have any information on absolute electrode position, so we cannot describe precisely how the water flow paths are changing. In all three years of data collection, this apparent resistivity record, as well as a similar record from configuration P4, are the only observed instances of polarity reversal.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  147  .  .-I  +1  .-v Fig. 6.4: An electrode configuration that can produce polarity reversals in apparent resistivity. If the current flows along the solid line, a positive potential difference will be recorded; if the current flows along the dashed line, the potential difference will be negative.  6.4.6 Streaming potentials Between 25 July 1987 and 27 July 1987, the d.c. resistivity apparatus was removed from the glacier for testing in the forefield; this hiatus is evident in Figures 6.2 and 6.3. The longer data gap beginning on 3 August 1987 was caused by a malfunction of the apparatus  —  a loose wire prevented the high voltage current supply from activating.  Five days passed before the fault was noticed, but during this time, the apparatus continued making voltage measurements. When these data were plotted, we noticed that rather large temporal variations in voltage were occurring. Figure 6.5 shows the potential record for configuration P2 during this period. Note that variations on several time scales (our sampling interval is one hour) and polarity changes are evident.  Chapter 6. ELECTRICAL PHENOMENA  —  148  DATA ANALYSIS  25 >0 vV///1v/w 25 E —50  I  4  I  I  5  I  I  I  I  .7 6 AUG 1987  l-t  8  9  Fig. 6.5: The natural potential record from configuration P2 during the time the high voltage supply failed to operate. Because the electrodes are pieces of copper metal, it is possible that these timevarying potentials were produced by electrode corrosion, but because the electrodes had already been in place for more than 14 days, we expect that corrosion potentials would have stabilized somewhat by the time these measurements were made. We decided to study these natural potential variations in more detail the following year. 64.7 Recapping the 1987 field season The results of the 1987 field season demonstrated that measurements of subglacial elec trical phenomena are possible. We realized that the subglacial sediments are electrically inhomogeneous and that our boreholes are not plumb. These realizations, together with our observations of time-varying natural potentials, suggested several modifications to our experimental design: (1) the use of dedicated current and potential electrodes, (2) the use of non-rectilinear electrode configurations, and (3) the use of a borehole inclinometer to determine electrode location.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  149  6.5 Dedicated electrode arrays In 1988 and 1989, we installed electrode arrays having separate electrodes for making potential measurements and for injecting current. Different apparatus were also used (see Chapter 5). Figure 6.6 shows the relative locations of the four electrode arrays installed during 1988 and 1989. Note that the group of electrodes marked 88DCO1B are 1989 additions to the 88DC01 array; between 1988 (when the positions indicated for the 88DC01 electrodes were determined) and 1989, the 88DC01 electrode array had moved downglacier by 33m so in 1989, the 88DC01 and 88DCO1B arrays were coincident. The flow direction of the glacier is marked, as are the surface locations of all the electrodes in the arrays. Notice that each array has a characteristic “L” shape. This b-shaped pattern is designed to allow examination of anisotropic character istics of the glacier bed. Figure 6.7 shows a detail of the electrode pattern. Current electrodes (marked with a circle) are interspersed between potential electrodes (marked with a cross) in such a way that electrical resistivity and natural potentials can be mea sured along the perpendicular arms of the array. Electrode arrays may contain a greater or lesser number of electrodes depending on logistical considerations at the time the boreholes were drilled. 6.6 The 1988 Experiments  The apparatus used in 1988 employed separate potential and current electrodes (see section 5.3). By separating the electrode functions, we could make measurements of natural potentials without worrying about interference from contemporaneous electrical resistivity measurements. In 1988, we had an inclinometer on loan from the U.S. Geological Survey for mea suring the trajectory of boreholes. Unfortunately, this instrument was not as accurate as the UBC fluxgate inclinometer (discussed in Appendix A) used in 1989. Based  Chapter 6. ELECTRICAL PHENOMENA  —  150  DATA ANALYSIS  89DC02 N o’N  89DC01  ;  .  88DC02  ‘BBDCOlB 88DC01  m  50  Fig. 6.6: The relative locations of the four electrode arrays installed in 1988 and 1989. The dots indicate the surface locations of the boreholes con taining electrodes. The designations 88DC01, etc. indicate the year and ordi nal number of the given electrode array. on comparisons of the multiple inclinometer measurements performed for each bore hole, we estimate that the error on 1988 electrode locations is 1—1.5m; some electrode positions have even larger error. 6.6.1 Experimental design Two separate arrays of electrodes were installed in 1988. Figures 6.8 and 6.9 show the electrode distributions for the 88DC01 and 88DC02 arrays. Both the surface posi tions of the borehole collars and the subsurface location of the electrodes are indicated.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  151  +  +  +  +  —  . +  I  I  0  —  potential electrode current electrode  +  m  Fig. 6.7: Electrode array template for the 1988 and 1989 electrode ar rays. Current electrodes are marked with a circle, and potential electrodes with a cross. The “L” shape is designed to allow observation of anisotropic characteristics of the glacier bed. The locations of other subglacial sensors (e.g. pressure sensors and deformation instru ments) are also marked. Notice that borehole deflections have completely obliterated the intended spatial arrangement of the electrodes. Streaming potential and d.c. resistivity records were collected during the summer and through the 1988—89 winter; records were collected for all possible electrode com binations. Although there are periods where the apparatus malfunctioned, the data set is large. We will not show all the data collected from these two electrode arrays, but will instead show selected records from the full data set.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  152  88H04  /  N  0 88H05  88H56  88H1 I 88H09  r  0 88H12  I  m  10  88H1 4 88H1 8 88H1 88H1 7  Fig. 6.8: The 88DC01 electrode array. Current electrodes are marked with a circle and potential electrodes with a cross. The locations of other subglacial sensors are marked with triangles. The heavy markings indicate the subglacial locations of the electrodes and the light markings the surface locations.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  153  —o C) CD  E —o 0) CD C) N 2: CD CD  N 2: CD CD  CD N  =  N CT)  2:  tO N CD CD  0 N 2:  CD CD  N N  z  CD CD  Fig. 6.9: The 88DC02 electrode array. Current electrodes are marked with a circle and potential electrodes with a cross. The heavy markings in dicate the subglacial locations of the electrodes and the light markings the surface locations. Note that no inclinometry data is available for 881134.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  154  6.6.2 Telluric noise In Chapter 4, we indicated that telluric currents were a source of natural potentials and that these telluric potentials might be indistinguishable from streaming potentials. In order to gauge the effects of telluric current noise on our measurements, and possibly to remove the noise should it prove severe, we installed a telluric monitoring station at our field camp in the summer of 1988. The telluric monitoring station was located to the north of the glacier on a south facing slope and consisted of a three-component fluxgate magnetometer on loan from Dr. T. Watanabe and a ground potential array (see Figure 3.1). The ground potential array consisted of three Cu—CuSO 4 electrodes arranged at the vertices of a right triangle; the electrodes were of the same type as those used under the ice. It is likely that the ground potential array recorded streaming potentials resulting from down-slope groundwater flow on the slope, but the telluric noise we were most concerned with was that at higher frequencies: those above our sampling Nyquist frequency (the streaming potential sampling interval ranged from 1—10 mm  during the summer). Recordings of the three magnetic field components and  two ground potential components were made at 2s intervals throughout the summer, yielding a voluminous data set (the telluric station was also installed in the summer of 1989 and another data set was collected for that summer). Figure 6.10 shows three panels of data. Panel (a) shows the variations of the three components of the Earth’s magnetic field during a 2.5 d period. The x axis of the magnetometer points towards magnetic north (a compass bearing of about 27°) and the z axis points downward. The vertical scale indicates only the deviations of the magnetic field components; it is not an absolute scale. Panel (b) shows the potentials recorded along the two perpendicular axes of the ground potential array over the same period of time; as in panel (a), the vertical scale shows oniy relative fluctuations in potential. The a axis points roughly towards magnetic north. Panel (c) shows the natural potential recorded between three pairs of electrodes in the 88DC01 array. For  Chapter 6. ELECTRICAL PHENOMENA  —  155  DATA ANALYSIS  all three natural potential traces, electrode 881108 is used as a reference potential; the top trace is the potential at 881106 (an electrode 93 m north of 881108  —  this electrode  is not shown in Figure 6.8), the middle trace is the potential at 881103 (an electrode 3.5 m to the north of electrode 881108), and the bottom trace is the potential at 881113 (an electrode 6.3 m upstream of electrode 881108). Recall that our estimated error on electrode position is 1—1.5 m. A minor geomagnetic storm is indicated by the arrow. We find significant noise correlated with magnetic field fluctuations only on electrode 881106 (i.e. along the 93m baseline) and even then the noise is only a few millivolts. Electrodes 881103 and 881113 (electrodes much closer to the reference electrode 881108) show insignificant disturbance. During the storm, the telluric potential between 881106 and 881108 amounts to about 7OmVkm’; we conclude that, for natural potential measurements within the spread of a single electrode array, telluric potentials are not a concern. The ground potential records, particularly for the a axis, show bursts of high frequency noise  —  we strongly suspect that these bursts are produced by the elec  tric generator used at the base camp. Fortunately, the generator does not appear to introduce noise at the subglacial electrode arrays. 6.6.3 Potential error Figure 6.10 gives us the first indications that streaming potentials vary with time. Be fore discussing these fluctuations, a brief discussion of measurement error is timely. In Chapter 5, we noted that the sum of multiplexer noise (section 5.3.3) and Cu— 4 electrode noise (Petiau and Dupis, 1980) should be about 30 pV. One method CuSO for estimating the error introduced by the entire electrical apparatus (i.e. data logger, multiplexer, wires, and potential electrodes) is to measure the potential difference be tween two seasoned electrodes located in the same borehole. Figure 6.11 shows such a record from hole 891115 (recorded in 1989). The potential difference is about 0± 1 mV;  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  156  1000  a  I  10>  b  E 0+15  C  E —15 26  27 JULY 1988  28  Fig. 6.10: The effect of telluric noise on natural potential measurements. The arrow indicates the location of a magnetic storm. (a) The three com ponents of the Earth’s magnetic field measured at 2 s intervals. (b) Voltages measured along the two axes of the ground potential array. (c) Natural po tentials relative to 881108 at the indicated boreholes. Only hole 881106, 93 m removed from 881108 do we find significant telluric noise. Measurements were made on the 88DC01 array. the error on our natural potential measurements is, for whatever reason, much larger than expected. Nevertheless, a potential error of ±1 mV is tolerable.  Chapter 6. ELECTRICAL PHENOMENA  —  157  DATA ANALYSIS  By seasoned electrodes, we mean electrodes that have been placed in a borehole for several days. Since the electrodes are assembled on the glacier surface just before being 4 solution are not initially at lowered down the hole, the copper electrode and Cu50 equilibrium. New potential electrodes typically show an initial potential of 50—200 mV relative to neighboring electrodes; this potential decays over 1—3 days towards a stable, near-zero potential with time-varying fluctuations superimposed.  +2.5 E° —2.5  OCT  AUG’ SEPT  NOV  1989 Fig. 6.11: The potential difference recorded between two seasoned poten tial electrodes located in the same borehole.  6.64 Potential gradients Equation (4.38) shows that for a negative  potential, the pressure potential and stream  ing potential gradients are of opposite sign; the other coefficients (p, e, H, and all positive. Since we expect a negative  ) are  potential under Trapridge Glacier (see sec  tion 4.6.4), the streaming potential component of the natural potential should increase with decreasing pressure (i.e. as one moves down-flow). The lowest trace in Figure 6.lOc begins with a reasonably steady negative potential of about —3 mV; electrode 881108, the down-glacier electrode, is at a higher potential than electrode 881113. Since it is reasonable to assume that subglacial water is flow ing down-glacier, this natural potential polarity is consistent with that arising from  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  158  a streaming current. Late on 26 July, the potential drops quickly by about 6 mV to establish a new level. If this sudden change in natural potential is caused by a change in streaming poten tial, we would expect a corresponding increase in the down-glacier pressure gradient. In this case, neither of the two nearby pressure sensors noted any sudden changes in pres sure, but the pressure sensor in unconnected hole 881135, 1.1 m upstream from 881113, changed from a decreasing pressure regime to an increasing one, so some change in the basal pressure distribution may have occurred. 6.6.5 Hole connections Because the completion of a borehole introduces a sudden increase in subglacial water pressure, whether or not the hole connects, we expect that the completion event should cause changes in pressure gradients and hence changes in streaming potential. Figure 6.12 shows an assemblage of pressure, natural potential, and apparent re sistivity data for an 18 day period in late summer, 1988; the data were collected on electrode array 88DC01. The “L” on the plan map at the right shows the surface lo cation of the electrode array; the  “+“  (881106) and  “—“  (881105) indicate the locations  of the natural potential electrodes. The trapezoid indicates the d.c. resistivity array (the dots are the current electrodes, 881109 and 881117, and the ticks are the potential electrodes, 881112 and 881116). The hatched line indicates the approximate boundary between an unconnected basal patch and a connected patch; at the time the boreholes were drilled, all holes north of (above) the line connected to the subglacial drainage system whereas all the holes south of (below) the line did not connect.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  159  50—  E  FJPRESSURURE 40  —  1111111  I  I  I  liii  III  25  >  E  10  lull  I  11111  Il  I  I  I  I  1.9  E 1.6 11111111111111  23  JULY  311  AUG  10  Fig. 6.12: Pressure, natural potential, and apparent resistivity data from electrode array 88DC01. The plan map at the right shows the surface location of the electrode array. The “+“ and “—“indicate the locations of the natural potential electrodes. The location of the pressure sensor is indicated by a “P”. The trapezoid indicates the d.c. resistivity array (the dots are the cur rent electrodes and the ticks are the potential electrodes). The hatched line indicates the boundary between an unconnected bed and a connected bed. We can see a number of features in these records: (1) Beginning on 25 July, there are a series of pressure pulses that are correlated with pulses in the natural potential record. (2) Beginning on 4 August, a series of diurnal fluctuations in natural potential and pressure are evident.  (3) The apparent resistivity is slowly dropping over the  period. The decrease in resistivity may indicate a slow migration of water down into subglacial sediments. Figure 6.13 shows the earlier part of the record in more detail. Note that data from a different d.c. resistivity array is shown in this figure (the dots are the current elec trodes, 881109 and 881117, and the ticks are the potential electrodes, 881113 and 881114). Two boreholes were drilled through the glacier during this four day period. Hole 881135 was completed at 18:12 h on 24 July; this hole was unconnected and reached the bed  Chapter 6. ELECTRICAL PHENOMENA  —  160  DATA ANALYSIS  1.1 m from potential electrode 881113, which formed part of the apparent resistivity array. The completion time is indicated by the first arrow on the time scale.  55  -  PRESSURE  E 40  E  c .  I  I  I  I  I  20  >  E  I  15  RESISTIVITY  1.00 0.95 I  24  I  I  JULY  28  Fig. 6.13: Pressure, natural potential, and apparent resistivity data from a drilling episode. The “C” indicates the location of the connected hole 88H36 and the “U” indicates the location of the unconnected hole 881135. See Fig ure 6.12 for an explanation of other features on the plan map. The apparent resistivity of the bed was being measured at 5 mm  intervals; be  tween 18:05h and 18:10h, a 4% drop in apparent resistivity was noted. The apparent resistivity record does not recover after this drop; some permanent change in the bed, perhaps only in the vicinity of 881113 occurred. Since an unconnected borehole does not drain quickly, the bed experiences a sig nificant over-pressure; in other words, the water pressure at the bottom of the borehole is significantly greater than the overburden pressure of the ice. Many unconnected holes begin to drain slowly shortly after completion; we conclude that a borehole overpressure condition can establish a temporary basal drainage system near the borehole. In this instance, at 20:04h, the pressure sensor notes a modest increase in pressure and  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  161  the natural potential exhibits a small pulse; these events may be associated with the slow outward progression of a water pressure wave from the unconnected borehole. On 25 July, at the time indicated by the second arrow, hole 881136 connected to the basal drainage system as it was being drilled. The hole connected at 13:58 h (according to the driller’s wristwatch); sharp increases in water pressure and natural potential were recorded at 14:OOh (according to the data logger clock). Within the resolution of our clock synchronization, estimated at 2 mm, the response to the hole completion was immediate. Notice that for more than a day after the hole was completed, the natural potential response was almost identical to the pressure. Even small details at the peak of the pressure pulse are reproduced in the natural potential record. The quick response of the pressure sensor to the completion of 881136 indicates that the connected patch is very efficient at distributing pressure at the bed; we suspect that the increase in water pressure also appears at the reference (negative) electrode for the natural potential measurement since this electrode was also placed in a connected hole. Because the positive electrode is in the unconnected patch, the pressure rise will create a large pressure gradient between the electrodes; the polarity and magnitude of the natural potential fluctuations are consistent with a streaming current. Figure 6.14 shows the diurnal cycling period in more detail. We notice that ap parent resistivity peaks in the early afternoon, just as in 1987. Natural potential peaks a few hours later at about midnight. The pressure sensor shows no clear diurnal sig nals, but records from several other pressure sensors suggest that when diurnal cycling in subglacial pressure occurs, the peak pressures usually happen between 21:00 h and midnight. Since the natural potential variations in Figure 6.14 are consistent with streaming potentials arising from diurnal cycling of pressure in the connected zone, we speculate that morphological changes in the bed may have disabled the hydraulic con nection between the pressure sensor and the 88DC01 array. Note that in Figure 6.12, the pressure record changes from a sawtooth pattern to a jittery, bumpy pattern on  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  162  1 August; perhaps this change indicates the hydraulic separation of the pressure sensor and electrode array.  45-  E 40 I  I  I  I  I  25-  >  E  POTENTIAL  -  10-  E  I  I  I  I  I  I  I  RESISTIVITY 1.7-  c .  —i 6 . 1  i  3  i  AUG  i  10  Fig. 6.14: Pressure, natural potential, and apparent resistivity data from a diurnal cycling episode. See Figure 6.12 for an explanation of the plan map.  6.6.6 Overwintering events Overwintering of electrical resistivity arrays is a difficult proposition, primarily because the electrodes do not last through the winter. Failure of an apparent resistivity con figuration is characterized by a month-long period where the current delivered by the high voltage supply slowly drops to zero and where asymmetry in the d.c. potential curve (see Fig. 4.5) increases dramatically; these characteristics probably result from breakage of the wires leading to the electrodes. Figure 6.15 shows an overwintering record from the 88DC01 array; on 11 August, the sampling interval was changed from once per hour to one daily reading at noon (the reduced sampling rate is necessitated by limited data storage capacity for the 11 month-long overwintering period). Notice  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  163  that prior to 11 August, the diurnal cycling of apparent resistivity, with a minimum around midnight and a maximum in the afternoon, is evident. The choice of noon as the sampling time during the winter was unfortunate, since before 1 August, the noon readings appear erratic. In retrospect, midnight readings, when the resistivity is at a minimum, would have been a better choice.  2000 1500 E  c 1000 500  SEPT  AUG 1988  Fig. 6.15: A record showing the overwintering character of an apparent resistivity record and the eventual failure of the array. The arrow indicates the time at which the d.c. potential record begins to become asymmetrical. The failure of natural potential electrode configurations is characterized by the onset of rapid oscillations in the observed potential. Figure 6.16 shows the continuation of the record shown in the middle trace of Figure 6.lOc (electrodes 881103 and 881108) through the winter. The failure of the electrodes on 17 June, 1989 is evident. Other electrode pairs failed on other days, so this is not a failure of the apparatus. The flat, near-zero natural potential shown in Figure 6.16 is typical of the winter period; if the primary source of natural potentials is streaming potentials, then this indicates low pressure gradients in the winter.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  164  >  E  J’A’S’O’N’DIJ ‘F’M’A’M  1988  1989  Fig. 6.16: A record showing the overwintering character of the natural potential and the eventual failure of the array. We do not understand why the current and potential electrodes appear to break so easily. As noted in Chapter 3, pressure sensors regularly survive through the winter without having their cables cut. Pressure sensors use a four-conductor cable; perhaps the single-conductor cable connecting an electrode to the surface is not strong enough or stretchy enough to endure the internal stresses of the glacier. Overwintering data from the 88DC02 array is similar in character, but the batteries supplying the apparatus expired in late October, 1988, so the data record is not as long as for array 88DC01. 6.7 1989 Experimental design  Two new electrode arrays were installed in 1989.  Some additional electrodes were  also added to the 88DC01 array (Fig. 6.6) for the purposes of some manipulation experiments. Unfortunately, persistent equipment malfunctions rendered worthless all data from the 89DC02 array and much of the data from the 89DC01 array. Figure 6.17 shows a plan map of the experimental site.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  165  E  1  WI  -o  a,  U, 0)  0)  a,  Fig. 6.17: The 89DC01 electrode array. Current electrodes are marked with a circle and potential electrodes with a cross. The locations of other subglacial sensors are marked with triangles. The heavy markings indicate the subglacial locations of the electrodes and the light markings the surface locations. A pressure sensor is located in hole 891114 and a conductivity sensor in hole 891127.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  166  6.7.1 Fall shutdown For the 1988 through 1991 field seasons, we have noticed a dramatic pressure-related event occurring near the end of July; within a few days of each other, pressure sensors that recorded magnificent diurnal fluctuations during the summer months suddenly in dicate that subglacial pressures are quiescent. We interpret this cessation of activity as a shutdown of the subglacial hydraulic system. Diurnal pressure cycling often resumes temporarily in late August and early September, probably driven by water input from melting early-winter snowfall, but we have not seen any evidence of these fluctuations in our natural potential records. Figure 6.18a shows the pressure records for holes 891106 and 891114 for a 30 day pe riod beginning on 29 July 1989. Hole 891106 is located 66 m upstream from hole 891114. The shutdown phenomenon is evident on 2 August, especially in 891106. Unfortunately, difficulties with the new 1989 apparatus, constructed on printed circuit boards, were not properly resolved until 1 August, so we were unable to monitor electrical phenom ena through this transition. Figure 6.18b shows the natural potential recorded between electrodes 891115 and 891117. Between 1 August and 10 August, the natural potential shows a weak diurnal signal; this period is followed by the characteristic winter regime discussed in section 6.6.7. The data gap between 10 August and 11 August results from rewiring and reprogramming activities relating to preparations for overwintering. It is frustrating not having electrical data prior to 1 August so that changes in electrical phenomena as the pressure fluctuations ceased could have been monitored, but we note with surprise that large diurnal natural potential fluctuations are present when the subglacial pressure is so calm. In the absence of pressure variations, we must seek another source for streaming potential variations. Perhaps subglacial deforma tion creates pressure gradients and zones of differing streaming potential coefficient C within the subglacial sediments and it is these effects we are observing. Deformation, if accompanied by sediment dilation, generates strong internal pressure gradients as  Chapter 6. ELECTRICAL PHENOMENA  DATA ANAL YSIS  —  167  70  a  E 0 >50  b 3200 E  C  1200 I  I  I -I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  60 Cl)  27  d  4IlIlIllIllIllIll III 111111111 I  1  20 10 AUG 1989  Fig. 6.18: Subglacial conditions at the end of the summer. (a) Subglacial pressures recorded by pressure sensors 891106 and 891114. (b) The natural po tential record at 891168 relative to 891114. (c) An apparent resistivity record. 891125 and 891128 are the current electrodes; 891121 and 891168 are the po tential electrodes. (d) A conductivity record from hole 891127. pore fluid is moved to fill the void spaces being created; these pressure gradients can  Chapter 6. ELECTRICAL PHENOMENA  —  168  DATA ANALYSIS  generate streaming potentials. Figure 6.18c shows that the apparent resistivity is also experiencing dramatic di urnal variations. The apparent resistivity fluctuations are peaking several hours earlier than we have seen in other records; sometimes the peak is before noon. The natu ral potentials, which we have previously observed peaking at about midnight are now peaking in early morning, about half a day out of phase with the apparent resistivity. Although no pressure fluctuations are evident, water supply is the only significant di urnal forcing acting on the glacier bed  —  if we do not observe pressure fluctuations in  the area of our experiment, it must mean that our study site has become disconnected from the still-active subglacial drainage system. Figure 6.18d shows the record from a conductivity cell located in 891127. Note that no diurnal signal in fluid conductivity is discernable either before or after 1 Au gust. This is our first evidence that changes in subglacial fluid conductivity (which can generate diffusion potentials) are probably not responsible for the natural poten tial fluctuations we observe. The calibration of the conductivity cell probably has an incorrect scaling factor, so the rather low conductivity values reported should not be trusted.  6.8 Manipulation experiments In Chapter 4, we discussed the dependence of streaming potentials on  value. The  (  potentials  potential can be manipulated in at least three ways: (1) changing pH;  (2) changing ion concentration; (3) adding surfactant. The first two possibilities were discussed in Chapter 4, but the last method begs attention; our discussion of surfactants is based on Hunter (1981, Chapters 6 and 8). In the mineral processing industry, one of the common methods for separating pure minerals from their ores is flotation. The surfactants used in flotation processes are highly active hydrocarbon chains having a charged terminal group. These chemicals  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  169  are added to the ore/mineral slurry and displace the adsorbed ions on the minerals. When enough hydrophobic surfactant molecules are attached, the mineral grains floats to the surface of the slurry and can be skimmed off. Common mineral processing surfactants include dodecylammonium chloride, sodium dodecyl sulphate, and sodium dodecyl suiphonate; the first is used in basic solutions with negative last two are used in acidic solutions with positive the  ( potentials and the  potentials. For acid surfactants,  potential becomes more negative as the concentration of surfactant increases  (Hunter, 1981, Fig. 8.3). The surfactants used in household cleansing operate on a similar principle, except the surfactant molecule has one end that displaces adsorbed molecules and another that is hydrophillic. When washing clothing or kitchenware, adsorbed particles of dirt are surrounded by surfactant molecules and carried off into the wash water. Although we do not know the effect of household surfactants on  potentials, we were certain  that some change in streaming potential would result. In early August, 1989, we executed two manipulation experiments at the old 88DC01 site; a salt injection and a surfactant injection. We had also planned an alkali injection experiment using quicklime (CaO), but this experiment was not performed. CaO turns to CaOH, a powerful base; had the experiment been carried through, we would have expected a more strongly negative streaming potential (see section 4.2.2). Since most, if not all, of the electrodes in the 88DC01 array were cut off, we installed a series of new electrodes in the array. Figure 6.19 shows the locations of the injection electrode array electrodes.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  170  89H65  89H67  I 0  m  10  Fig. 6.19: Location map of 88DCO1B electrode array. Electrode and sen sor designations are the same as in Figures 6.8 and 6.9. Hole 891165 contains a conductivity sensor. Between 18:38 h and 19:00 h on 4 August, approximately 3.5 kg of table salt was flushed down into the bottom of hole 891176. The quantity of salt is uncertain since some spillage occurred on the surface. The salt solution was delivered to the bottom of the borehole using a special apparatus constructed for that purpose. One end of a sealed section of 6 inch plastic plumbing pipe was attached to the pressurized water line from the hot water drill and the other end to a section of hose reaching to the bottom. By filling the pipe section with the salt solution and starting the pump, the solution was pumped to the bottom.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  171  Five days later, between 21:26h and 22:17h on 9 August, approximately 4kg of dishwasher detergent, also dissolved in hot water, was injected into the bottom of hole 891182. Again, the quantity of detergent injected is uncertain because of spillage. Dish washer detergent was chosen because it contains a high concentration of surfactants; no foaming agents, such as those used in hand dish-washing detergents, are used. The detergent used in our injection contained metasilicates, tripolyphosphate, and sodium carbonate as active ingredients. Both injections were made into poorly connected holes. Ideally, the injections would have been made into connected holes so that the salt and detergent could travel freely through the subglacial drainage system. In retrospect, we should have made the injections at one of the other two main electrode arrays, but at the time the injections were made, we wanted to avoid contamination of these two arrays. Figure 6.20a shows one of the natural potential records from the 88DC01 array (the potential of 891172 relative to 891161). The injection times for the salt and factant are indicated. Unfortunately, an error in collecting information from the data logger resulted in the ioss of data between 4 August at 19:29 h and 5 August at 17:40 h. Nevertheless, no response to either the salt or surfactant injections is evident. The sud den rise in potential on 9 August occurs at 17:39h, 4h before the surfactant injection, but the rise may have been associated with the driffing of 891182 between 16:03 h and 18:05 h. We note that electrode 891172, the down-glacier electrode, is at a potential higher than that of electrode 891161. Once more, this is consistent with a negative down-glacier pressure potential gradient and a negative  <  potential.  Chapter 6.  ELECTRICAL PHENOMENA  —  172  DATA ANALYSIS  20  a >10 E 0  —  I  I  I  I  I  I  I  I  I  I  300 0  LI)  250  b  SALT I  —  1  I  I  I  I  I  I  5 AUG 1989  I —  I  10  Fig. 6.20: (a) A sample natural potential record from the manipulation experiments (891172 relative to 891161). The increase in apparent noise level late on 3 August is caused by setting the data logger to a wider input range; the digital resolution has dropped to 0.33 mV from 33 V. (b) The subglacial conductivity cell record from array 88DCO1B. Figure 6.20b shows the conductivity record from 891165 (as in Figure 6.18d, the scaling of the fluid conductivity is uncertain). Some ambiguous diurnal fluctuations are visible, with maxima around noon, but the vertical scale has been expanded so much so that the resolution limit of the data logger is visible; these changes are small and do not correlate with either injection. Although electrode array 88DCO1B contains current electrodes, the HV supply (1988 vintage) did not function properly, so no apparent resistivity data are available for the injection experiments.  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  173  We belive that the absence of any observable response to our manipulations reflects poor hydraulic conductivity in the bed rather than a failure of the electrical phenomena to respond  —  we believe that the added chemicals did not reach the electrodes during  the course of our measurements. 6.9 Conclusions  We have demonstrated the practicability of long-term monitoring of subglacial electrical measurements. Our measurements have revealed diurnal and episodic fluctuations in electrical properties of the glacier bed; we have been able to correlate these events with changes in other subglacial processes. 6.9.1 Apparent resistivity  Apparent resistivity measurements indicate that the glacier bed is electrically inhomo geneous; this inhomogeneity is not on the scale of particles in the bed, but on a scale of at least several metres. We have also observed time-varying changes in electrical flow paths within the bed. We are not confident about the values of apparent resistivity calculated from our electrode arrays. Even in 1989, when we used a relatively accurate inclinometer to lo cate the electrodes, we still computed unreasonably large or small (sometimes even neg ative) apparent resistivities. Apparent resistivity records seem to fall into three classes: (1) 1000 <Pap <2000gm, (2) Pap  >  10000 tim, and (3) —200 <Pap  <  200am. Two  or more electrode configurations encompassing the same subglacial sediments often in dicated very different values of Pap  —  we suspect that electrode positioning error and  basal inhomogeneity are both responsible for these variations. Because the terms of Equation (4.11) are linear, any fractional error in electrode separation will generate a corresponding fractional error in Pap. In 1988, the positioning error was of the same order of magnitude as the electrode spacing, so errors in Pap were  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  174  extreme. In 1989, errors of 10—30% are expected. We estimate, with much reservation, that the resistivity of the sediments beneath Trapridge Glacier lies in the range of 1000—2000 f m.  This range is consistent with some of our subglacial records, with  the forefield measurements in 1987, and with measurements by others (Haeberli and Fisch, 1984; Brand and others, 1987). Brand and others (1987) installed a rectilinear electrical resistivity array of eight electrodes on Storglaciären, a non-surge-type glacier in northern Sweden. The surface spacings they used for their linear electrode array were similar to those used for our experiments. They used a half-Schlumberger electrode arrangement where the two potential electrodes were placed close together in holes separated by 1 m at the surface. One current electrode was placed in the lateral moraine of the glacier, 300 m from the other electrodes; the other current electrode was moved between the remaining six holes. They report values of pp ranging between 3  m for small pseudo-depth and 1O f m  for large pseudo-depth; the lowest value of Pap is unreasonably low. We suspect that the two holes containing their potential electrodes were not separated by 1 m at the ice—bed interface; if our experience with borehole deviation is any indication, their holes were not plumb (Storglacliiren is 120 m thick, about 50% thicker than Trapridge Glacier). Nevertheless, there can be no doubt that the two data sets they collected two days apart demonstrate that time-varying electrical resistivity occurs beneath Storglaciären. We believe that the most interesting aspect of our apparent resistivity measure ments is the correlation between sediment deformation and electrical resistivity, and the inferred correlation with subglacial water pressure. 6.9.2 Streaming potentials  We have shown that the natural potential variations observed beneath Trapridge Gla cier are caused by streaming potentials and not telluric or diffusion potentials. The  Chapter 6. ELECTRICAL PHENOMENA  —  DATA ANALYSIS  polarity of the observed streaming potentials indicate that the glacial sediments is negative, as expected. information to estimate the value of the  175 potential of the sub  Unfortunately, we do not have enough  potential for the sediments or how it may  vary with time. The hole connection data from 1988 indicate that streaming potentials can be used to supplement pressure sensors, but it is clear that there are other subglacial processes that cause changes in streaming potentials; sediment deformation may be one of these.  Chapter 7 CONCLUSIONS  7.1 General comments This thesis documents the development of two novel methods for exploring the sub glacial environment: measurements of subglacial deformation and electrical phenom ena. Using these techniques, we have recorded several heretofore unobserved subglacial processes. Foremost among these is evidence for time-varying lateral transfer of nor mal and shear stress beneath the glacier; these transfers of stress are controlled by the mechanical properties of the glacier bed and the behaviour of the subglacial hydraulic system. Such lateral variations in basal stress have not been properly addressed by current subglacial rheology hypotheses; if the study of subglacial rheology is to provide a greater understanding of glacier surge mechanisms, rather than localized stress—strain responses, then these lateral variations must be considered. At first glance, subglacial deformation and subglacial electrical phenomena appear unrelated, but we have shown that both these phenomena may be interpreted inde pendently, or in complementary fashion, to investigate the deformation and hydraulic properties of the glacier substrate. 7.1.1 Shear stress and normal stress Glaciologists have traditionally used mean values of stress calculated from the geometry and flow patterns of glaciers to describe the forces acting on the glacier bed, but we have demonstrated that such calculations risk being inaccurate, at least for Trapridge Glacier. This is of great concern since both of the variables fundamental to sediment rheology  —  effective pressure and shear stress  176  —  are obtained from these calculations.  Chapter 7. CONCLUSIONS  177  We caution those wishing to derive sediment rheologies based on in situ deformation experiments to ensure that, in addition to measurements of subglacial water pressure or pore pressure, they have some method for monitoring the normal and shear stresses at the bed. In particular, we wish to emphasize the uncertainty in computing any mea sure of effective pressure (Equations (3.6), (3.8), or (3.10)) based solely on subglacial pressure measurements. As discussed in Chapter 3, we have noted subglacial water pressures in excess of the nominal flotation pressure in several unconnected holes. It is clear that the glacier does not accelerate upwards as a result of this apparent force imbalance, so the local overburden pressure must exceed that suggested by the ice thickness (if it did not, the pressurized water would quickly leak out into the lower-pressure subglacial drainage system). Since the glacier is somewhat rigid on a diurnal time scale over spatial scales smaller than its thickness, lateral transfer of normal loading on the bed is possible; the load can be greater at some points and less at others, so long as the mean loading is equal to the theoretical overburden pressure. In addition, the shear stress on the bed is not necessarily proportional to the normal loading on the bed, as implied by Equation (3.4). A sticky patch (with a high shear stress) can have a low normal load and a slippery patch (with a low shear stress) can have a high normal load; again, the stability requirement is only that the mean shear stress averaged over the glacier sole be equal to that computed by Equation (3.4). We expect that slippery patches are associated with connected zones and sticky patches are associated with unconnected zones. We have, however, observed overpressure situations in connected zones (on occasion, we encounter artesian outflow conditions when completing a connected borehole). Evidence for uneven time-varying distribution of subglacial shear stress is found in our data for 1989. In the weeks prior to our experiments, pressure sensors installed at various locations in the study site recorded dramatic diurnal fluctuations, but on 1 August, four days prior to the beginning of the deformation experiment, all these  Chapter 7. CONCLUSIONS  178  pressure sensors began reporting quiescent pressure levels. As shown in Figures 3.9 and 6.18, other subglacial sensors indicated that the subglacial environment was any thing but quiescent; large fluctuations in deformation rate, apparent resistivity, and natural potential were observed. Since meitwater is the only significant diurnal force acting on the subglacial environment, we strongly suspect that diurnal cycling of sub glacial water pressure continued after 1 August, but that none of our pressure sensors was located where these pressure cycles could be observed. Increases in water pressure elsewhere in the subglacial environment would reduce the shear stress in those areas; the principle of constant mean shear stress then requires that the shear stress increase elsewhere. If the region containing our subglacial experiments experienced such a rise in shear stress, we could expect to see diurnal fluctuations in strain rate, streaming potentials and resistivity; corresponding fluctuations in local subglacial water pressure (or computed effective pressure) do not necessarily occur. The need for direct local measurements of shear stress and overburden pressure becomes clear; subglacial water pressure and geometry constitute insufficient data to compute local shear stress and effective pressure. T.2 Electrical phenomena  7.2.1 Streaming potentials We have shown that streaming potentials are the primary source of natural poten tial variations in the subglacial environment. The streaming potential measurements have proved useful in measuring pressure gradients at the glacier bed, although more work needs to be done to discover how other subglacial processes, such as sediment deformation, affect these potentials. The dramatic correlation between subglacial water pressure gradients and stream ing potentials suggests a number of experiments to further investigate this cross-coupled  Chapter 7. CONCLUSIONS  179  phenomenon. We envisage an array with the electrodes arranged on a rectangular grid. At locations along the periphery of the grid, pressure sensors would be installed togeth er with the potential electrodes. This arrangement of electrodes would allow a detaile d analysis of how streaming potentials can be used to interpolate pressure gradien ts and might also permit an analysis of time-varying hydraulic and electric conduction paths. It would also be interesting to try to place a string of electrodes vertically though the deforming sediments; such an array could, for several days, monitor changes in the vertical pore pressure distribution. By measuring the diffusion rates of pressu re down into the sediments, we could infer the hydraulic diffusivity of the sediments. 7.2.2 Electrical resistivity Measurement of subglacial electrical resistivity presented the most difficulty. We have observed exciting changes in polarity and magnitude in subglacial resistivity; these re sults suggest that not only is the subglacial environment electrically inhom ogeneous, but that this inhomogeneity changes with time. Absolute measurements of electri cal resistivity suffered from insufficiently accurate electrode position information; with better position control, we are certain that much could be learned about subgla cial deforming sediments using this technique. 7.3 Basal deformation  7.3.1 Rheology We have demonstrated that the rheological relations derived by Boulton and Hindm arsh (Equations (3.16) and (3.17)) are inappropriate for the sediments beneat h Trapridge Glacier, but we have not developed an alternative rheology for the substra te of this glacier because we do not have the necessary local shear stress and effective pressure  Chapter 7. CONCLUSIONS data  —  180  this is a problem faced by all glaciologists investigating subglacial deformation,  Boulton and Hindmarsh included. Were we to synthesize mean strain rate values from our continuous records of shear deformation, in order to mimic the type of data collected by Boulton and Hindmarsh, we still would be unable to develop a rheology. As discussed in section 3.1, we have only one theoretical shear stress value and a small range of effective pressure values inferred from subglacial water pressure; we cannot fit mean strain rates to conceivable rheological functions. Again, we stress that local measurements of subglacial shear stress and overburden pressure must be available. 7.3.2 Effective viscosity We have calculated effective viscosity values for the sediments beneath Trapridge Gla cier, but we are not convinced that effective viscosity values are meaningful since they are derived assuming a linear rheology and, in our case, by averaging strain rate over an arbitrary length of time (the length of our data set). Having shown that (1) mean strain rate of almost any value desired, both positive and negative, can be computed from a continuous strain record and (2) basal shear stress cannot be calculated reliably, we are loath to suggest that the effective viscosity values given in Chapter 3 are significant. Our deformation data do indicate that experiments measuring net strain over a period of time are vulnerable to large errors because of the time-varying nature of basal deformation. We have shown how negative net strain rates can be synthesized from our data. There is a considerable amount of work to be invested in developing bed deforma tion techniques further. It is possible, with today’s miniature radio telemetry technol ogy, to build tilt sensors that do not require a cable leading to the surface; such devices would remove all worries about inter-cell wires interfering with deformation measure ments. Gregory and Stubbs (1983) built an electromagnetic positioning system for use  Chapter 7. CONCLUSIONS  181  in permafrost studies; Kohier and Protsch (1991) have used a similar system beneath Storglaciären, Sweden. Although the electromagnetic devices used by these researchers employed connecting wires, we believe that this is an appropriate technology for the future development of wireless deformation experiments. Evidence presented in Chapters 3 and 6 indicates that measurements of pore pres sure, rather than subglacial water pressure, would be tremendously useful in under standing the rheology of deforming sediments. Future tilt sensor designs might in corporate a pressure sensor that could record pore pressure in the vicinity of the tilt cell.  REFERENCES Abramowitz, M. and I. A. Stegun, eds. 1965. Handbook of mathematical functions with formulas, graphs, and mathematical tables. Cambridge University Press, Cam bridge. Adamsom, L. G., G. V. Chilingar, C. M. Beeson, and R. A. Armstrong 1966. Electroki netic dewatering, consolidation and stabilization of soils. Eng. Geol., 1, 291—304. Ahmad M. U. 1964. A laboratory study of streaming potentials. Geophys. Prospect., 12, 49—64. Alley, R. B. 1989a. Water—pressure coupling of sliding and bed deformation: I. Water system. J. GlacioL, 35(119), 108—118. Alley, R. B. 1989b. Water—pressure coupling of sliding and bed deformation: II. Velo city—depth profiles. J. Glaciol., 35(119), 119—129. Alley, R. B. 1991. Deforming-bed origin for southern Laurentide till sheets?. J. Glaciol., 37(125), 67—76. Alley, R. B., D. D. Blankenship, C. R. Bentley, and S. T. Rooney. 1986. Deformation of till beneath ice stream B, West Antarctica. Nature, 322(6074), 57—59. Alley, R. B., D. D. Blankenship, C. R. Bentley, and S. T. Rooney. 1987a. Till beneath ice stream B: 3. Till deformation: evidence and implications. J. Geophys. Res., 92(B9), 8921—8929. Alley, R. B., D. D. Blankenship, C. R. Bentley, and S. T. Rooney. 1987b. Till beneath ice stream B: 4. A coupled ice—till flow model. J. Geophys. Res., 92(B9), 8931—8940. Alley, R. B., D. D. Blankenship, S. T. Rooney, and C. R. Bentley. 1989. Water—pressure coupling of sliding and bed deformation: III. Application to ice stream B, Antarc tica. J. Glaciol., 35(119), 130—139. Anderson, J. H. and 0. A. Parks. 1968. The electrical conductivity of silica gel in the presence of adsorbed water. J. Phys. Chem., 72, 3662—3668. Archie, 0. E. 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. A.I.M.E. Transactions, 146, 54—64. Bear, J. 1972. Dynamics of fluids in porous media. American Elsevier Publishing Com pany, Inc., New York. Berryman, J. 0. and S. C. Blair. 1987. Kozeny—Carman relations and image processing methods for estimating Darcy’s constant. J. Appi. Phys., 62(6), 2221—2228. Bindschadler, R. A. 1983. The importance of pressurized subglacial water in separation and sliding at the glacier bed. J. Glaciol., 29(101), 3—19.  182  REFERENCES..  183  Bindschadler, R. A. 1984. Jakobshavns Glacier drainage basin: A balance assessment. J. Geophys. Res., 89(C2), 2066—2072. Bindschadler, R. A., W. D. Harrison, C. F. Raymond, and C. Gantet. 1976. Thermal regime of a surge-type glacier. J. GlacioL, 16(74), 251—259. Blake, E. W. and G. K. C. Clarke. 1991a. Subglacial water and sediment samplers. J. Glaciol., 37(125), 188—190. Blake, E. W. and G. K. C. Clarke. 1991b. Theory of calibration and interpretation of borehole inclinometers. J. GlacioL, (in press). Blake, E. W., G. K. C. Clarke, and M. C. Germ. 1991. Tools for examining subglacial bed deformation. J. Glaciol., (in press). Blankenship, D. D., C. R. Bentley, S. T. Rooney, and R. B. Alley. 1986. Seismic mea surements reveal a saturated porous layer beneath an active Antarctic ice stream. Nature, 322(6074), 54—57. Blankenship, D. D., C. R. Bentley, S. T. Rooney, and R. B. Alley. 1987. Till beneath ice stream B: 1. Properties derived from seismic travel times. J. Geophys. Res., 92(B9), 8903—8911. Bockris, J. O’M., M. A. V. Devanathan, and K. Muller. 1963. On the structure of charged interfaces. Proc. Royal. Soc., Ser. A, 274, 55—79. Bogoslovsky, V. A. and A. A. Ogilvy. 1972. The study of streaming potentials on fissured media models. Geophys. Prospect., 20, 109—117. Bogoslovsky, V. A. and A. A. Ogilvy. 1973. Deformations of natural electric fields near drainage structures. Geophys. Prospect., 21, 716—723. Boulton, G. S. 1979a. Processes of glacier erosion on different substrata. J. Glaciol., 23(89), 15—38. Boulton, G. S. 1979b. Direct measurement of stress at the base of a glacier. J. Glaciol., 22(86), 3—24. Boulton, G. S. and R. C. A. Hindmarsh. 1987. Sediment deformation beneath glaciers: rheology and geological consequences. J. Geophys. Res., 92(B9), 9059—9082. Boulton, G. S. and A. S. Jones. 1979. Stability of temperate ice caps and ice sheets resting on beds of deformable sediment. J. Glaciol., 24(90), 29—43. Boumans, A.A. 1957a. Streaming currents in turbulent flows and metal capillaries. I. Theory (1). Distribution of charge in the liquid. Physica, XXIII, 1007—1026. Boumans, A.A. 1957b. Streaming currents in turbulent flows and metal capillaries. II. Theory (2). Charge transported by the flow of liquid. Physica, XXIII, 1027—1037. Boumans, A.A. 1957c. Streaming currents in turbulent flows and metal capillaries. III. Experiment (1). Aim and procedure. Physica, XXIII, 1038—1046.  REFERENCES..  184  Boumans, A.A. 1957d. Streaming currents in turbulent flows and metal capillaries. IV. Experiment (2). Results and conclusions. Physica, XXIII, 1047—1055. Brand, G., V. Pohjola, and R. LeB. Hooke. 1987. Evidence for a till layer beneath Storglaciären, Sweden, based on electrical resistivity measurements. J. GlacioL, 33(115), 311—314. Brown, N. E., B. Hallet, and D. B. Booth. 1987. Rapid soft bed sliding of the Puget Sound glacial lobe. J. Geophys. Res., 92(B9), 8985—8997. Bull, C. and J. R. Hardy. 1956. The determination of the thickness of a glacier from measurements of the value of gravity. J. GlacioL, 2(20), 755—763. Butler, K. E. 1991. Enhancement and modelling of signals from natural piezoelectric targets. M.Sc. thesis, University of British Columbia. Cagniard, L. 1959. Abaque pour sondages électriques sur glace. Annales do Ge’ophy sique, 15(4), 561—563. Cary, P. W., G. K. C. Clarke, and W. R. Peltier. 1979. A creep instability analysis of the Antarctic and Greenland ice sheets. Can. J. Earth Sci., 16(1), 182—188. Chapman, D. L. 1913. A contribution to the theory of electrocapillarity. Phil. Mag., 25, 475—481. Clarke, G. K. C. 1976. Thermal regulation of glacier surging. J. Glaciol., 16(74), 231— 250. Clarke, G. K. C. 1982. Glacier outburst floods from “Hazard Lake”, Yukon Territory and the problem of flood magnitude prediction. J. Glaciol., 28, 3—21. Clarke, G. K. C. 1986. Professor Mathews, outburst floods, and other glaciological disasters. Can. J. Earth Sci., 23(6), 859—868. Clarke, G. K. C. 1987a. Fast glacier flow: ice streams, surging, and tidewater glaciers. J. Geophys. Res., 92(B9), 8835—8841. Clarke, G. K. C. 1987b. Subglacial till: a physical framework for its properties and processes. J. Geophys. Res., 92(B9), 9023—9036. Clarke, T. S. 1991. Glacier dynamics in the Susitna River basin, Alaska, U.S.A. J. Glaciol., (in press). Clarke, G. K. C. and E. W. Blake. 1990. Temporal changes in subglacial acoustic properties. EOS, 71(43), 1314—1315. Clarke, G. K. C. and E. W. Blake. 1991. Geometric and thermal evolution of a surgeJ. Glaciol., type glacier in its quiescent state: Trapridge Glacier 1969—1989. 37(125), 158—169. Clarke, G. K. C. and D. F. Classen 1970. The Fox Glacier project. Can. Geog. J., 81(1), 26—29.  REFERENCES..  185  Clarke, G. K. C., S. G. Collins, and D. E. Thompson. 1984b. Flow, thermal structure, and subglacial conditions of a surge-type glacier. Can. J. Earth Sci., 21, 232—240. Clarke, G. K. C. and M. C. Germ. 1989. Presurge fluctuations in water pressure and flow velocity, Trapridge Glacier, Yukon Territory. EOS, 70(43), 1084. Clarke, G. K. C., W. H. Mathews. 1981. Estimates of the magnitude of glacier outburst floods from Lake Donjek, Yukon Territory, Canada. Can. J. Earth Sd., 18, 1452— 1463. Clarke, G. K. C., W. H. Mathews, and R. T. Pack. 1984a. Outburst floods from glacial Lake Missoula. Quaternary Research, 22(3), 289—299. Clarke, G. K. C., U. Nitsan, and W. S. B. Paterson. 1977. Strain heating and creep instability in glaciers and ice sheets. Rev. Geophys. Space Phys., 15(2), 235—247. Clarke, G. K. C., J. P. Schmok, C. S. L. Ommanney, and S. G. Collins. 1986. Charac teristics of surge-type glaciers. J. Geophys. Res., 91(B7), 7165—7180. Collins, I. F. 1968. On the use of the equilibrium equations and flow law in relating the surface and bed topography of glaciers and ice sheets. J. Glaciol., 7(50), 199-204. Corwin, R. F., G. T. DeMoully, R. S. Harding, Jr., and H. F. Morrison. 1981. In terpretation of self-potential survey results from the East Mesa geothermal field, California. J. Geoplzys. Res., 86(B3), 1841—1848. Corwin, R. F. and D. C. Hoover 1979. The self-potential method in geothermal explo ration. Geophysics, 44(2), 226—245. Cruikshank, J. 1981. Legend and landscape: convergence of oral and scientific traditions in the Yukon Territory. Arctic Anthropology, XVIII(2), 67—93. de Groot, S. R. and P. Mazur. 1984. Non-equilibrium thermodynamics. Dover, New York. Desio, A. 1954. An exceptional glacier advance in the Karakoram—Ladakh region. J. Glaciol., 2(16), 383—385. Dolgoushin, L. D. and G. B. Osipova. 1975. Glacier surges and the problem of their forecasting. Snow and Ice (Proceedings of the Moscow Symposium, August 1971), IAHS Publ. no. 104, 292—304. Dolgoushin, L. D. and G. B. Osipova. 1978. Balance of a surging glacier as the basis for forecasting its periodic advances. Mater. Glyatsiologicheskikh Issled. Khronica Obsuzhdeniya, 32, 260—265. Echelmeyer, K., R. Butterfield, and D. Cuillard. 1987. Some observations on a recent surge of Peters Glacier, Alaska, U.S.A. J. Glaciol., 33(115), 341—345.  186  REFERENCES..  Echelmeyer, K. and Wang Zhongxiang. 1987. Direct observation of basal sliding and deformation of basal drift at sub-freezing temperatures. J. Glaciol., 33(113), 83— 98. Echelmeyer, K. and W. D. Harrison. 1990. Jacobshavns Isbre, West Greenland: sea sonal variations in velocity or lack thereof. J. Glaciol., 36(122), 82—88. Engelhardt, H. F., M. Fahnestock, N. Humphrey, B. Kamb, and M. F. Meier. 1987. Rapid basal sliding in a large tidewater glacier caused by high subglacial water pressure. EOS, 68(44), 1272. Engelhardt, H. F., N. Humphrey, B. Kamb, and M. Fahnestock. 1990. Physical condi tions at the base of a fast moving Antarctic ice stream. Science, 248, 57—59. Ernstson, K. and H. TJ. Scherer. 1986. Self-potential variations with time and their re lation to hydrogeologic and meteorological parameters. Geophysics, 51(10), 1967— —  1977. Fahnestock, M. and N. Humphrey. 1988. Borehole water level measurements, Columbia Glacier, Alaska. Ice(86), 25—26. Fisch, W. Sen., W. Fisch Jun., and W. Haeberli 1977. Electrical d.c. resistivity sound ings with long profiles on rock glaciers and moraines in the Alps of Switzerland. Z. Gletscherkd. Glazialgeol., Bd. 13, Ht. 1/2, 239—260. Fitterman, D. V. 1978. Electrokinetic and magnetic anomalies associated with dilatant regions in a layered earth. J. Geophys. Res., 83(B12), 5923—5928. Fitterman, D. V. 1979a. Calculations of self-potential anomalies near vertical contacts. Geophysics, 44(2), 195—205. Fitterman, D. V. 1979b. Relationship of the self-potential Green’s function to solutions of controlled-source direct-current potential problems. Geophysics, 44(11), 1879— 1881. Fitterman, D. V. 1984. Thermoelectric self-potential anomalies and their relationship to the solid angle subtended by the source region. Geophysics, 49(2), 165—170. Fowler, A. C. 1987a. A theory of glacier surges. J. Geophys. Res., 92(B9), 9111—9120. Fowler, A. C. 1987b. Sliding with cavity formation. J. Glaciol., 33(115), 255—267. Freeze, R. A. and J. A. Cherry 1965. Groundwater. Prentice Hall, New Jersey. Gardner, J. S. and K. Hewitt 1990. A surge of Bualtar Glacier, Karakoram Range, Pakistan: A possible landslide trigger. J. Glaciol., 36(123), 159—162. Garfield, D. E. and H. T. Ueda. 1976. Resurvey of the “Byrd” Station, Antarctica, drill hole. J. Glaciol., 17(75), 29—34.  187  REFERENCES..  Gerrard, J. A. F., M. F. Perutz, and A. Roch. 1952. Measurement of the velocity distribution along a vertical line through a glacier. Proc. Royal. Soc., Ser. A, 213, 546—558. Glen, J.W. 1952. Experiments on the deformation of ice. J. Glaciol., 2(12), 111—114. Gouy, G. 1910. Sur la constitution de la charge électrique Ann. Phys. (Paris), Se’rie 4, 9, 457—468.  a la surface d’un electrolyte.  Gregory, E. C. and C. W. Stubbs. 1983. A high-resolution technique for measuring motion within the active layer. Permafrost: Proceedings of the Fourth International Conference, July 17—22, Fairbanks, Alaska, 389—394. Haase, R. 1969. Thermodynamics of irreversible processes. Dover, New York. Haeberli, W. and W. Fisch. 1984. Electrical resistivity soundings of glacier beds: a test study on Grubengletscher, Wallis, Swiss Alps. J. Glaciol., 30(106), 373—376. Helmholtz, H. L. F. v. 1882. Uber galvanische Str6me, verursacht durch Concentra tionsunterschiede; Folgerungen aus der mechanischen Wârmetheorie (aus: Wiede mann’s Annalen, Monatsberichte der Berliner Akademie, 1877, Bd. III, 201—216). in: Wissenschaftliche Abhandlungen von Hermann Helmholtz, Vol. 1, J. A. Barth, Leipzig, 840—854. Herrmann, H. J., G. Mantica, and D. Bessis. 1990. Space-ifiling bearings. Phys. Rev. Lett., 65(26), 3223—3226. Hewitt, K. 1969. Glacier surges in the Karakoram Himalaya (Central Asia). Can. J. Earth Sci., 6, 1009—1018. Hochstein, M. 1967. Electrical resistivity measurements on ice sheets. J. Glaciol., 6(47), 623—633. lodge, S. M. 1976. Direct measurement of basal water pressures: a pilot study. J. Glaciol., 16(74), 205—218. lodge, S. M. 1979. Direct measurement of basal water pressures: progress and problems. J. Glaciol., 23(89), 309—319. Hoinkes, H. C. 1969. Surges of the Vernagtferner in the Otztal Alps since 1599. Can. J. Earth Sci., 6(4), 853—861. Hooke, R. LeB. 1973. Structure and flow in the margin of the Barnes Ice Cap, Baffin Island, N.W.T., Canada. J. Glaciol., 12(66), 423—438. Hooke, R. LeB. 1981. Flow law for polycrystalline ice in glaciers: Comparison of theo retical predictions, laboratory data, and field measurements. Rev. Geophys. Space Phys., 19(4), 664—672. Hooke, R. LeB. and B. Hanson. 1986. Borehole deformation experiments, Barnes Ice Cap, Canada. Cold Regions Sci. Tech., 12, 261—276.  188  REFERENCES..  Hooke, R. LeB., P. Holmiund, and N. R. Iverson. 1987. Extrusion flow demonstrated by bore-hole deformation measurements over a riegel, Storglaciären, Sweden. J. Glaciol., 33(113), 72—78. Hooke, R. LeB., S. B. Miller, and J. Kohier. 1988. Character of englacial and subglacial drainage system in the upper part of the ablation area of Storglaciären, Sweden. J. Glaciol., 34(117), 228—231. Huang Maohuan and Wang Zhongxiang. 1987. Research on the tunnel excavated in Urumqi Glacier No. 1, Tianshan Glaciological Station, China. J. Glaciol., 33(113), 99—104. Hubbard, B. and M. Sharp. 1989. Basal ice formation and deformation: a review. Progress in Pliys. Geog., 13(4), 529—558. Hunter, R. J. 1981. Zeta potential in colloid science. Principles and applications. Aca demic Press, London. Iken, A. 1972. Measurements of water pressure in moulins as part of a movement study of the White Glacier, Axel Heiberg Island, Northwest Territories, Canada. J. Glaciol., 11(61), 53—58. Iken, A. 1978. Variations of surface velocities of some alpine glaciers measured at inter vals of a few hours. Comparison with arctic glaciers. Z. Gletscherkd. Glazialgeol., Bd. 13, Ht. 1—2, 1977, 23—25. Iken, A. and R. A. Bindschadler. 1986. Combined measurements of subglacial water pressure and surface velocity of Findelengletscher, Switzerland: conclusions about drainage system and sliding mechanism. J. Glaciol., 32(110), 101—119. Iken, A., H. Röthlisberger, A. Flotron, and W. Haeberli. 1983. The uplift of Unteraar gletscher at the beginning of the melt season a consequence of water storage at the bed?. J. Glaciol., 32(110), 101—119. Ishido, T. and H. Mizutani. 1981. Experimental and theoretical basis of electrokinetic phenomena in rock—water systems and its applications to geophysics. J. Geophys. Res., 86(B3), 1763—1775. Ishido, T., H. Mizutani, and K. Baba. 1983. Streaming potential observations, using geothermal wells and in situ electrokinetic coupling coefficients under high tem perature. Tectonophysics, 91, 89—104. —  Jarvis, G. T. and G. K. C. Clarke 1975. The thermal regime of Trapridge Glacier and its relevance to glacier surging. J. Glaciol., 14(71), 235—250. Jaynes, E. T. 1980. The minimum entropy principle. Ann. Rev. Plzys. Chem., 31, 579— 601. Jones, F. M. H. 1989. Digital impulse radar for glaciololgy: instrumentation, modelling, and field studies. M.Sc. thesis, University of British Columbia.  REFERENCES..  189  Joos, G. 1988. Theoretical Physics, 3rd Edition. Dover, New York. Kamb, B. 1970. Sliding motion of glaciers: theory and observation. Rev. Geophys. Space Phys., 8(4), 673—728. Kamb, B. 1987. Glacier surge mechanism based on linked cavity configuration of the basal water conduit system. J. Geophys. lies., 92(B9), 9083—9100. Kamb, B. 1991. Rheological nonlinearity and flow instability in the deforming bed mechanism of ice stream motion. J. Geophys. lies., 96(B10), 16585—16595. Kamb, B. and H. Engelhardt. 1987. Waves of accelerated motion in a glacier ap proaching surge: the mini-surges of Variegated Glacier, Alaska, U.S.A. J. Glaciol., 33(113), 27—46. Kamb, B. and H. Engelhardt. 1989. Flow mechanics of West Antarctic ice streams: observations by borehole geophysics. EOS, 70(43), 1081. Kamb, B. and E. LaChapelle. 1964. Direct observation of the mechanism of glacier sliding over bedrock. J. Glaciol., 5(38), 159—172. Kamb, B., C. F. Raymond, W. D. Harrison, H. Engelhardt, K. A. Echelmeyer, N. F. Humphrey, M. M. Brugman, and T. Pfeffer. 1985. Glacier surge mechanisms: 1982—83 surge of Variegated Glacier, Alaska. Science, 227(4686), 469—479. Keevil, N. B. Jr. and S. H. Ward 1962. Electrolyte activity: its effects on induced polarization. Geophysics, 27(5), 677—690. Keizer, J. 1987. Statistical thermodynamics of nonequilibrium processes. Springer—Ver lag, New York. Kohler, J. and R. Proksch. 1991. In situ measurement of subglacial till deformation beneath, N. Sweden. EOS, 72(44), supplement to October 29 issue, 158. Korpi, G. K. and P. L. de Bruyn. 1972. Measurement of streaming potentials. J. Colloid Interface Sci., 40(2), 263—266. Kurtz, R. J., E. Findl, A. B. Kurtz, and L. C. Stormo. 1976. Turbulent flow streaming potentials in large bore tubing. J. of Colloid and Interface Sci., 57(1), 28—39. Levanto, A. E. 1959. A three-component magnetometer for small drill-holes and its use in ore prospecting. Geophys. Prospect., 7(2), 183—195. Levine, S., J. R. Marriott, G. Neale, and N. Epstein. 1975. Theory of electrokinetic flow in fine cylindrical capillaries at high zeta-potentials. J. Colloid Interface Sci., 52(1), 136—149. Lewis, R. W., C. Humpheson, and J. C. Bruch, Jr. 1975. Applications of electro-osmosis to ground-water flow problems. Groundwater, 13(6), 484—491.  190  REFERENCES..  Lingle, C. S., T. J. Hughes, and R. C. Kollmeyer 1981. Tidal flexure of Jacobhavns Glacier, West Greenland. J. Geophys. Res., 86(B5), 3960—3968. Lliboutry, L. 1959. Une théorie du frottement du glacier sur son lit. Annales de Ge’o physique, 15(2), 250—265. Lliboutry, L. 1964. Sub-glacial ‘supercavitation’ as a cause of the rapid advances of glaciers. Nature, 202(4927), 77. Lliboutry, L. 1966. Bottom temperatures and basal low-velocity layer in an ice sheet. J. Geophys. Res., 71, 2535—2543. Lliboutry, L. 1967. Discussion of paper by J. Weertman, ‘Sliding of nontemperate glaciers’. J. Geophys. Res., 72(2), 525—526. Lliboutry, L. 1968. General theory of subglacial cavitation and sliding of temperate glaciers. J. Glaciol., 7(49), 21—58. Lliboutry, L. 1969. Contribution 6(4), 943—953.  a la thorie des  ondes glaciaires. Can. J. Earth Sci.,  Lliboutry, L. 1979. Local friction laws for glaciers: a critical review and new openings. J. Claciol., 23(89), 67—95. Madden, T. R. 1976. Random networks and mixing laws. Geophysics, 41(6A), 1104— 1125. Marshal, D. J. and T. R. Madden. 1959. Induced polarization, a study of its causes. Geophysics, 24(4), 790—816. Mathews, W. H. 1964. Water pressure under a glacier. J. Glaciol., 5(38), 235—240. McConnell, A. 1982. No sea too deep; The history of oceanographic instruments. Adam Huger Ltd., Bristol. Meier, M. F. 1989. Relation between water input, basal water pressure, and sliding of Columbia Glacier, Alaska, U.S.A. (abstract). Ann. Glaciol., 12, 214—215. Meier, M. F. and A. Post. 1969. What are glacier surges?. Can. J. Earth Sci., 6(4), 807—817. Miller, M. M. 1957. Phenomena Associated with the deformation of a glacier bore-hole. Intern. Assoc. Sci. Hydrol., Publ. 46, 437—452. Mitchell, J. K. 1976. Fundamentals of soil behaviour. John Wiley and Sons, Inc., New York. Mizutani, H., T. Ishido, T. Yokokura, and S. Ohnishi. 1976. Electrokinetic phenomena associated with earthquakes. Geophys. Res. Lett., 3(7), 365—368. Morgan, F. D., B. R. Williams, and T. R. Madden. 1989. Streaming potential properties of westerly granite with applications. J. Geophys. Res., 94(B9), 12449—12461.  REFERENCES..  191  Murray, T. 1990. Deformable glacier beds: measurement and modelling. Ph.D. thesis, Univ. of Wales, Aberystwyth. Narod, B. B. and R. D. Russell. 1984. Steady-state characteristics of the capacitively loaded flux gate sensor. IEEE Trans. Magn., MAG—20(4), 592—597. Narod, B. B., J. R. Bennest, J. 0. Strom—Olsen, F. Nezil, and R. A. Dunlap. 1985. An evaluation of the noise performance of Fe, Co, Si, and B amorphous alloys in ring-core fluxgate magnetometers. Can. J. Phys., 63(11), 1468—1472. Neftel, A., M. Andrée, J. Schwander, and B. Stauffer. 1985. Measurements of a kind of DC-conductivity on cores from Dye 3. in: Langway, C. C. Jr., H. Oeschger, and W. Dansgaard (eds.) Greenland ice core: geophysics, geochemistry, and the environment, AGU, Washington, DC (Geophysical monograph 33), 32—38. Nernst, W. 1888. Zur theorie der in Lösung befindlichen Körper. Zeits. Physik. Chem., 2, 613—637. Nourbehecht, B. 1963. Irreversible thermodynamic effects in inhomogeneous media and their applications in certain geoelectric problems. Ph. D. thesis, MIT, Cambridge. Nye, J. F. 1952. The mechanics of glacier flow. J. Glaciol., 2(12), 82—93. Nye, J. F. 1953. The flow law of ice from measurements in glacier tunnels, laboratory experiments and the Jungfraufirn borehole experiment. Proc. Royal. Soc., Ser. A, 219, 477—489.  Nye, J. F. 1957. The distribution of stress and velocity in glaciers and ice-sheets. Proc. Royal. Soc., Ser. A, 239, 113—133. Nye, J. F. 1958. Surges in glaciers. Nature, 181(4621), 1450—1451. Onsager, L. 1931a. Reciprocal relations in irreversible processes. I.. Phys. Rev., 37, 405—426. Onsager, L. 1931b. Reciprocal relations in irreversible processes. II.. Phys. Rev., 38, 2265—2279. østrem, G. 1959. Ice melting under a thin layer of moraine, and the existance of ice cores in moraine ridges. Geografiska Annaler, 41(4), 228—230. østrem, G. 1959. Ice-cored moraines in Scandinavia. Geografiska Annaler, 46(3), 282— 337. østrem, G. 1967. Laboratory measurements of the resistivity of ice. J. Glaciol., 6(47), 643—650. Parasnis, D. 5. 1986. Principles of applied geophysics, th edition. Chapman and Hall, London. Parks, G.A., B. K. Jindal, J. H. Anderson, Jr. 1966. Temperature and humidity in electrical separation of oxide minerals. A.I.M.E. Transactions, 235, 451—457.  192  REFERENCES..  Paterson, W. S. B. 1981. The physics of glaciers, 2nd edition. Pergamon Press, Toronto. Paterson, W. S. B. 1983. Deformation within polar ice sheets: An analysis of the Byrd Station and Camp Century borehole-tilting measurements. Cold Regions Sci. Tech., 8, 165—179. Paterson, W. S. B., U. Nitsan, G. K. C. Clarke. 1978. An investigation of creep instabil ity as a mechanism for glacier surges. Mater. Glyatsiologicheskikh Issled. Khronica Obsuzhdeniya, 32, 201—209. Paterson, W. S. B. and J. C. Savage. 1963a. Geometry and movement of the Athabasca Glacier. J. Geophys. Res., 68(15), 4513—4520. Paterson, W. S. B. and J. C. Savage. 1963b. Measurements on Athabasca Glacier relating to the flow law of ice. J. Geophys. Res., 68(15), 4537—4543. Perutz, M. F. 1947. Report on problems relating to the flow of glaciers. J. Glaciol., 1(2), 47—51. Perutz, M. F. 1949. Direct measurement of the velocity distribution in a vertical profile through a glacier. J. Glaciol., 1(5), 249. Perutz, M. F. 1950. Direct measurement of the velocity distribution in a vertical profile through a glacier. J. Glaciol., 1(7), 382—383. Petiau, G. and A. Dupis. 1980. Noise, temperature coefficient, and long time stability of electrodes for telluric observations. Geophys. Prospect., 28, 792—804. Post, A. 1960. The exceptional advances of the Muidrow, Black Rapids, and Susitna Glaciers. J. Geophys. Res., 65(11), 3703—3712. Post, A. 1965. Alaskan glaciers: recent observations in respect to the earthquakeadvance theory. Science, 148(3668), 366—368. Post, A. 1967. Effects of the March 1964 Alaska earthquake on glaciers. U. S. Geological Survey, Professional Paper 54—D. Post, A. 1969. Distribution of surging glaciers in western North America. J. Glaciol., 8(53), 229—240. Press, W. H., B. P. Flannery, S. A. Teukoisky, and W. T. Vetterling. 1986. Numerical recipes: the art of scientific computing. Cambridge University Press, Cambridge. Girard. 1959. Premier essai de mesure électrique A. Bauer, and d’paisseur d’un glacier. Annales de Ge’ophysique, 15(4), 564—567. Raymond, C. F. 1971a. Determination of the three-dimensional velocity field in a glacier. J. Glaciol., 10(58), 39—53.  Q ueille—Lefèvre,  —.  Raymond, C. F. 1971b. Flow in a transverse section of Athabasca Glacier, Alberta, Canada. J. Glaciol., 10(58), 55—84.  REFERENCES..  193  Raymond, C. F. 1971c. A new bore-hole inclinometer. J. Glaciol., 10(58), 127—132. Raymond, C. F. 1987. How do glacier surge? A review. J. Geophys. Res., 92(B9), 9121—9134. Raymond, C. F. and W. D. Harrison. 1988. Evolution of Variegated Glacier, Alaska, U.S.A., prior to its surge. J. Glaciol., 34(117), 154—169. Rivero, R. T. 1971. Use of the curvature method to determine true vertical reservoir thickness. J. Petrol. Tech., 23, 491—496. Robin, G. de Q. 1955. Ice movement and temperature distribution in glaciers and ice sheets. J. Glaciol., 2(18), 523—532.  Q. 1968. Surface topography of ice sheets. Nature, 215(5105), 1029—1032. de Q. 1986. A soft bed is not the whole answer. Nature, 323, 490—491.  Robin, G. de Robin, G.  Röthlisberger, H. 1967. Electrical resistivity measurements and soundings on glaciers: introductory remarks. J. Glaciol., 6(47), 599—606. R6thlisberger, H. and K. Vögtli. 1967. Recent d.c. resistivity soundings on Swiss gla ciers. J. Glaciol., 6(47), 607—621. Röthlisberger, H. 1969a. Evidence for an ancient glacier surge in the Swiss Alps. Can. J. Earth Sci., 6(4), 941—942. Röthlisberger, H. 1969b. In discussion of: J. Weertman, Water lubrication mechanism of glacier surges. Can. J. Earth Sci., 6(4), 941—942. Röthlisberger, H. 1981. Destructive power of glaciers. Switzerland and her glaciers, Kümmerly and Frey, Berne. Russell, R. D., B. B. Narod, and F. Kollar. 1983. Characteristics of the capacitively loaded flux gate sensor. IEEE Trans. Magn., MAG—19(2), 126—130. Savage, J. C. and W. S. B. Paterson. 1963. Borehole measurements in the Athabasca Glacier. J. Geophys. Res., 68(15), 4521—4536. Schiavone, D. and R. Quarto. 1984. Self-potential prospecting in the study of water movements. Geoexploration, 22, 47—59. Schmok, J. P. 1986. Sedimentology and chronology of Neoglacial Lake Alsek, Yukon Territory. M.Sc. thesis, University of British Columbia. Schmok, J. P. and G. K. C. Clarke. 1989. Lacustrine sedimentary record of ice-dammed Neoglacial Lake Alsek. Can. J. Earth Sci., 26(10), 2092—2105. Scott, R. F. 1963. Principles of soil mechanics. Addison-Wesley publishing Company, Inc., Reading, Massachusetts. Seaberg, S. Z., J. Z. Seaberg, R. LeB. Hooke, and D. W. Wiberg. 1988. Character of englacial and subglacial drainage system in the lower part of the ablation area  REFERENCES..  194  of Storglaciären, Sweden, as revealed by dye-trace studies. J. Glaciol., 34(117), 217—227. Shabtale, S., I. M. Whillans, and C. R. Bentley. 1987. The morphology of ice streams A, B, and C, West Antarctica, and their environs. J. Geophys. Res., 92(B9), 8865— 8883. Sharp, R. P. 1947. The Wolf Creek glaciers, St. Elias Range, Yukon Territory. Geog. Rev., 37(1), 26—52. Sharp, R. P. 1951. Glacial history of Wolf Creek, St. Elias Range, Canada. J. Geol., 59(2), 97—117. Sharp, R. P. 1953a. Deformation of bore hole in Malaspina Glacier, Alaska. Bull. Geol. Soc. Am., 64, 97—100. Sharp, R. P. 1953b. Deformation of vertical bore hole in a piedmont glacier. J. Glaciol., 2(13), 182—184. Shoemaker, E. M. 1986. Subglacial hydrology for an ice sheet resting on a deformable aquifer. J. Glaciol., 32(110), 20—30. Shreve, R. L. 1958. The borehole experiment on Blue Glacier, Washington. Intern. Assoc. Sci. Hydrol., Publ. 47, 530—531. Shreve, R. L. and R. P. Sharp. 1970. Internal deformation and thermal anomalies in Lower Blue Glacier, Mount Olympus, Washington, U.S.A. J. Glaciol., 9(55), 65— 86. Smart, C. C. and G. K. C. Clarke. 1988. Basal hydrology of a surge-type glacier. EOS, 69(44), 1210—1211. Stern, 0. 1924. Zur theorie der elektrolytischen doppelschicht. Zeit. Elektrochem., Bd. 30(21/22), 508—516.. Tarr, R. S. and L. Martin 1914. Alaska Glacier Studies in the Yakutat Bay, Prince William Sound and lower Copper River Regions. National Geographic Society, Washington, D. C. Telford, W. M., L. P. Geldart, and R. E. Sheriff. 1990. Applied Geophysics, 2nd edition. Cambridge University Press, Cambridge. Thomson, W. (Lord Kelvin). 1853. Dynamical theory of heat, part VI continued. A mechanical theory of thermo-electric currents in crystalline solids. Proc. Roy. Soc. Edinburgh, 3(44), 255—260. Throckmorton, P. 1969. Shipwrecks and archaeology; The unharvested sea. Little, Brown and Company, Boston. Truesdell, C. 1984. Rational thermodynamics, 2nd edition. Springer—Verlag, New York.  REFERENCES..  195  Van der Veen, C. J. and I. M. Whillans. 1990. Flow laws for glacier ice: comparison of numerical predictions and field measurements. J. Glaciol., 36(124), 324—339.  Vonnegut, K., Jr. 1969. Slaughterhouse Five. Dell, New York. V6gtli, K. 1967. D.c. resistivity soundings on Devon Island, N.W.T., Canada. J. Gla ciol., 6(47), 635—642. Waitt, R. B., Jr. 1984. Periodic jökulhlaups from Pleistocene glacial Lake Missoula new evidence from varved sediments in northern Idaho and Washington. Quater nary Research, 22, 46—58.  —  Walder, J. S. 1986. Hydraulics of subglacial cavities. J. Glaciol., 32(112), 439—445. Walker, B. II. 1986. Factors controlling hole angle and direction. J. Petrol. Tech., 38(12), 1171—1173. Walstrom, J. E., R. P. Harvey, and H. D. Eddy. 1972. A comparison of various direc tional survey models and an approach to model error analysis. J. Petrol. Tech., 24, 935—943. Weertman, J. 1957. On the sliding of glaciers. J. Glaciol., 3(21), 33—38. Weertman, J. 1962. Catastrophic glacier advances. Variations of the Regime of Existing Glaciers (Proceedings of the Obergurgi Symposium, September 1962), IAHS Pubi. no. 58, 31—39. Weertman, J. 1964a. The theory of glacier sliding. J. Glaciol., 5(39), 287—303. Weertman, J. 1964b. Discussion on Kamb and LaChapelle’s paper “Direct observation of the mechanism of glacier sliding over bedrock”. J. Glaciol., 5(39), 374—375. Weertman, J. 1966. Effect of basal water layer on the dimensions of ice sheets. J. Glaciol., 6(44), 191—207. Weertman, J. 1967a. An examination of the Lliboutry theory of glacier sliding. J. GlacioL, 6(46), 489—494. Weertman, J. 1967b. Sliding of nontemperate glaciers. J. Geophys. Res., 72(2), 521— 523. Weertman, J. 1969. Water lubrication mechanism of glacier surges. Can. J. Earth Sd., 6(4), 929—942. Weertman, J. 1979. The unsolved general glacier sliding problem. J. Glaciol., 23(89), 97—115. Weertman, J. and G. E. Birchfield. 1983. Basal water film, basal water pressure, and velocity of travelling waves on glaciers. J. Glaciol., 29(101), 20—27. Whillans, I. M., J. Boizan, and S. Shabtaie. 1987. Velocity of ice streams B and C, Antarctica. J. Geophys. Res., 92(B9), 8895—8902.  REFERENCES..  196  Wilson, G. J. 1968. An improved method for computing directional surveys. J. Petrol. Tech., 20, 871—876. Wood, F. J. 1940. An attempt on Mt. Wood, St. Elias Range. Am. Alpine J., 4(1), 1—8. Wyckoff, R. D. 1948. The Gulf airborne magnetometer. Geophysics, 13(2), 182—208. Wyffie, R. J. 1951. An investigation of the electrokinetic component of the self potential curve. Petroleum Transactions, AIME, 192, 1—18. Wyffie, R. J., and M. B. Spangler. 1952. Application of electrical resistivity measure ments to problem of fluid flow in porous media. Bull. Amer. Assoc. Petrol. Geol., 36(2), 359—403. Yuen, D. A. and G. Schubert. 1979. The role of shear heating in the dynamics of large ice masses. J. Glaciol., 24, 195—212. Zaremba, W. A. 1973. Directional survey by the circular arc method. J. Soc. Petrol. Eng., 13(1), 5—11.  Appendix A INCLINOMETER DESIGN AND DATA PROCESSING  Borehole inclinometers are standard equipment for field glaciologists and are com monly used for investigating the flow law of ice and for measuring the spatial position of englacial and subsurface sensors. The recent development, at the University of British Columbia (UBC), of a prototype inclinometer that employs a three-component fluxgate magnetometer to obtain a compass bearing has stimulated our interest in borehole in dinometry. Following a review of various approaches to glacier incinometry, we present a unified theory of data interpretation that can be applied to all inclinometers, discuss the application of the theory to the UBC inclinometer, and discuss the sensitivity of the theory to error in the data. The substance of this appendix has been published previously in the Journal of Glaciology (Blake and Clarke, 1991b). A.1 Introduction Whenever a deep hole is drilled into the surface of the Earth, there is uncertainty about its trajectory. Asymmetries in the drill, layering in the material being drilled, and shortcomings in driffing technique may cause the borehole to stray from its intended path (e.g. Walker, 1986). Incinometry tools remove spatial uncertainty by mapping the deviation of the borehole. Typically, two pieces of information are collected at a series of stations, or locations, along the length of the hole: the tilt of the inclinometer from vertical and the azimuth of that tilt in a geographical coordinate system. Of the many techniques available for making these measurements, we present a brief overview of those that have been used for glaciological work. Sources for some of this inclinometry equipment are given.  197  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  198  In this appendix, we attempt to formalize some aspects of inclinometry data analy sis and, at the same time, introduce a new indinometry tool that was recently assembled at the University of British Columbia (UBC). We describe the calibration procedure for the UBC inclinometer, formulate a general theory for interpretation of inclinometer data, and examine the sensitivity of this theory to error in the data. A.2 Historical overview Inclinometry of glacier boreholes has long been associated with field investigations of the flow law for ice (Perutz, 1947, 1949, 1950; Gerrard and others, 1952; Sharp, 1953a, 1953b; Nye, 1952, 1953, 1957; Miller, 1957; Savage and Paterson, 1963; Paterson and Savage, 1963a, 1963b; Shreve and Sharp, 1970; Raymond, 1971a, 1971b; Hooke, 1973, 1981; Garfield and Ueda, 1976; Paterson, 1983; Hooke and Hanson, 1986; Hooke and others 1987; Van der Veen and Whillans, 1990). Many of these researchers used in cinometry data to infer the internal velocity field of the glacier and thereby deduce parameters of the flow law. Increasing interest in basal processes has created a new ap plication for incinometry, that of accurately determining the position of englacial and subglacial sensors. As an example, the measurement of subglacial resistivity (Brand and others, 1987) requires knowledge of the spatial position of current and voltage electrodes within adjacent boreholes. Glaciologists are fortunate in that drilling through ice is relatively inexpensive and easy when compared with drilling through other crustal materials. Hydro-thermal drilling is likely the most common method in use today, but our experience on Trapridge Glacier has shown that boreholes drilled with hot water are often far from plumb. A.2.1 Basic dip and azimuth measurements A rudimentary fluid-level inclinometer can be built by partially ifihing a sealed glass vial with a solution of hydrofluoric acid. If the vial is kept at a fixed tilt angle for  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  199  an appropriate time, the acid will etch the glass, leaving a record of the fluid surface. If several vials are positioned along the length of a borehole while the etching occurs, they provide a discrete record of the borehole tilt (e.g. Savage and Paterson, 1963, p. 4522). An exotic variation of the above approach substitutes a warm solution of gelatin (“Jello” is a suitable commercial product) for the acid; the vials are left in place until the gelatin hardens and tilt is preserved in the surface of the gelatin (Shreve and Sharp, 1970, p. 71). Such fluid levels have three major drawbacks: (1) The vials may freeze to the walls of the borehole. (2) The precision of the tilt measurements is poor. (3) The vials do not measure tilt azimuth, although in the case of the gelatin level, a magnetic compass needle suspended within the gelatin can provide a magnetic bearing. “Floating-compass” inclinometers improve on fluid-level instruments by providing azimuth information, but they are laborious to use. Sperry-Sun Corporation (Houston, Texas) and Parsons Survey Company (South Gate, California) manufacture floatingcompass inclinometers that employ a miniature camera. The camera in the Sperry-Sun instrument is focused on a weighted spherical magnetic compass ball that has azimuth and tilt markings, much like lines of latitude and longitude. At predetermined time intervals, the camera takes a photograph of the illuminated compass ball; it is left to the operator to ensure that the instrument is at the proper position along the borehole at the time each picture is taken. The camera in the Parsons instrument photographs an illuminated plumb bob against a compass graticule. The single-frame version of this instrument must be brought to the surface after each picture has been taken (e.g. Shreve, 1958; Savage and Paterson, 1963), but an improved multi-frame version uses electrical signals from the surface to advance the film (Garfield and Ueda, 1976). With both the Sperry-Sun and Parsons instruments, azimuth and tilt readings are transcribed from the developed film images, an obvious impediment to in-field data acquisition.  ESSING Appendix A. INCLINOMETER DESIGN AND DATA PROC  200  inclinometer is brought Clamping the compass so that it can be examined once the position. The inclinometer to the surface is an alternative to ifim recording of compass l-mounted horizontal built by Pajari Instruments Ltd. (Orillia, Ontario) uses a gimba , a locking mechanism compass needle. After a predetermined interval of time has passed sets the timer, lowers the arrests the gimbal rings and the compass. The operator to lock the compass. The instrument to the desired location, and waits for the timer is removed from the inclinometer is then brought to the surface, the gimbal mount scales in the unit. This pressure casing, and azimuth and tilt are read from graduated le. R. M. Koerner laborious procedure is repeated for each station within the boreho the Agassiz Icecap. (personal communication, 1987) has used this instrument on A.3 Electronic inclinometry enabled the development of in recent years, advances in electronic technology have y to the surface. Com inclinometry systems that transmit position information directl nic inclinometry are pared with earlier techniques, the time and labour savings of electro le site, the trajectory appreciable; with appropriate equipment in place at the boreho is collected. of the borehole can even be computed and displayed as the data A.3.1 Measuring tilt s of tilt to be made Electronic tilt transducers allow precise and repeated measurement chanics inc. (Santa from a glacier surface. The Fredericks Company and Applied Geome are suitable for bore Cruz, California) manufacture electrolytic tilt transducers that Indicator Company hole inclinometry. General Oceanics (Miami, Florida) and Slope (Seattle, Washington) manufacture force-balance tilt transducers. A.3.2 Measuring azimuth s must be combined To reconstruct the borehole trajectory, each set of tilt measurement meters, tracked with a measurement of instrument azimuth. Rigidly-coupled inclino  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  201  inclinometers, magnetically-oriented inclinometers, and gyroscopically-oriented inch nometers are the four design categories known to us.  A .3.2.1 External azimuth control The orientation of a rigidly-coupled inclinometer is controlled by a rod leading from the surface. This method is suitable for shallow holes, but as the distance between the instrument and the operator increases, azimuth control deteriorates and logistical problems grow (Gerrard and others, 1952). Tracked incinometers require that a special grooved casing be installed in the borehole before incinometry begins. Spring-loaded wheels or pins keep the inclinometer aligned with the grooves in the casing as the instrument moves through the hole. For shallow holes, twisting of the casing is assumed to be negligible; it follows that the azimuth of the inclinometer is constant and equal to the azimuth of the casing at the top of the hole. For deeper holes, a torsion tool can be used to measure the twist in the casing. This tool consists of two sections, each of which has a set of tracking wheels, that are connected by a sensitive torsion transducer. Any twisting of the casing is recorded by the transducer as the tool is moved through the hole. If a torsion tool is not available, the presence or absence of casing twist must be inferred from the data. Although the effort and expense of installing an incinometry casing is an inconvenience, the casing provides a uniform borehole geometry for the inclinometer, serves to smooth out small perturbations in the borehole trajectory, and allows the operator to cancel any offset error in the instrument by executing precisely reversed runs through the hole; great accuracy can be achieved. Cased holes are also advantageous in long-term deformation studies because they are not prone to closure from creep and freezing. The biaxial inclinometer manufactured by Slope Indicator Co. (Seattle, Washing ton) is an excellent example of a tracked inclinometer. Hooke and Hanson (1986) and  Appendiz A. INCLINOMETER DESIGN AND DATA PROCESSING  202  Hooke and others (1987) have made precise measurements of glacier ice deformation using this instrument, but we are certain that they have cursed the need for a casing.  A.3.2.2 Internal azimuth control Gyroscopically and magnetically-oriented inclinometers do not require a casing, hence their appeal for glacier work. A gyroscopically-oriented inclinometer was built by the research team that made the historic measurements of deformation within Jungfraufirn (Gerrard and others, 1952). Their paper contains a brief description of this instrument, but we have been unable to locate the detailed instrumentation paper to which they refer (Broad, Jason, and Perutz, 1952, J. Sci. Instrum., designated “in press”) and suspect it remains unpublished. Indinometers that use a magnetic compass to determine azimuth and that report the azimuth electrically are a vast improvement over the Sperry-Sun, Parsons, and Pa Jan designs; the output from the tilt sensors and the compass can be recorded while the inclinometer is on station. The primary disadvantage of magnetic orientation is that the compass is sensitive to geomagnetic disturbances, as well as magnetic field varia tions caused by remnant magnetization and magnetic susceptibility in nearby materials. For ice, which has no magnetic signature, only geomagnetic disturbances are of conse quence. Glaciologists have used several designs for magnetically-oriented incinometers. Raymond (1971c) describes an instrument that was used to measure the internal ye locity structure of Athabasca Glacier (Raymond 1971b). Philip Taylor (Hydro-Tech, Seattle, Washington) has built several magnetically-oriented inclinometers specifically for use in glaciers. These instruments employ two orthogonal tilt sensors and a gimbal mounted compass that produces a voltage signal proportional to the magnetic bearing.  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  203  Further improvements in reliability can be gained by using a non-mechanical com pass, such as a magnetometer. Levanto (1959) used a downhole three-component fluxgate magnetometer to measure the in situ magnetic susceptibility of rock. The in strument did contain a tilt sensor, but the sensor was used to maintain the attitude of the magnetometer rather than measure inclination. With minor modification, this instrument would have served well as an inclinometer. A.4 The UBC inclinometer In the spring of 1989, the glaciology group at the University of British Columbia, in collaboration with Slope Indicator Canada Limited (Richmond, British Columbia) and Narod Geophysics (Vancouver, British Columbia), developed a prototype inclinometer. This instrument is similar to the Hydro-Tech instrument in that it uses the Earth’s magnetic field as an azimuth reference, but the field sensing device is a fluxgate mag netometer rather than a mechanical compass. Two force-balance tilt transducers and the magnetometer are enclosed within a non-magnetic stainless steel pressure tube (Fig. A.1). Brass centring springs help keep the inclinometer centred in the borehole and prevent the instrument from spinning when on station. The tube is 1.5 m long and has an outside diameter of 2.54cm (1.00 in). The tilt transducers are Slope Indica tor devices, identical to those used in their tracked incinometers. The magnetometer is a Narod Geophysics miniature ring-core fiuxgate magnetometer. (Wyckoff (1948) gives an overview of the principals of operation for a fluxgate magnetometer; more  detailed treatments related to this magnetometer are found in: United States Patent 3800213 (26 March 1974), UK Patent 2044460 (16 Oct 1979), Russell and others (1983), Narod and Russell (1984), and Narod and others (1985)). The sensing element of this three-component magnetometer is a 12.7 mm (0.50 in) cube. As the tilt sensors contain permanent magnets, the magnetometer is mounted as far from the tilt sensors  Appendia A. INCLINOMETER DESIGN AND DATA PROCESSING  204  as possible. Ideally, the magnetometer would be placed at the bottom of the instru ment, far from any secondary magnetic fields, but the tilt sensors completely ifil the interior of the pressure tube so that no wires can pass by them; thus the tilt sensors must be positioned at the bottom and the magnetometer at the top. The space between these sensors is occupied by the electronics package. The instrument cable attaches to the top of the instrument. This cable carries power to the instrument and analogue data to the surface, and has a steel strength member. The magnetic bias created by the steel cable and the electric currents flowing in the cable is compensated by the magnetometer calibration.  Fig. A.l: A block diagram of the peripheral devices attached to the TJBC inclinometer (inclinometer not drawn to scale). The power module provides power to the instrument from a small 6 Ah sealed lead-acid battery. Analogue signals are fed to the data logger which, on command from the handheld control box, records data from the tilt sensors and magnetometer into the storage module. The operator keys in depth information on the control box.  Appendiz A. INCLINOMETER DESIGN AND DATA PROCESSING  205  Power for the inclinometer is provided by a 6 Ah sealed lead-acid battery that can be connected to a solar panel for charging. The five analogue data signals (two from the tilt sensors and three from the magnetometer) are wired to data logger input channels. The inclinometer operator holds a small weather-tight control box which is connected to the data logger by a 5 m cable. The control box contains two SPST switches with indicator lights, a momentary switch, a 100—position push button digital potentiometer, an acknowledgement lamp, and a piezoelectric buzzer. For each hole, the inclinometer is stepped down and up at one metre intervals, resulting in a two-fold redundancy in the measurements. The depth of the inclinometer at each station, as marked on the cable, is coded manually on a digital potentiometer that is mounted in a handheld control box. After the operator closes a momentary switch, indicating that the inclinometer is on station, the data logger records the time, the digital values of the tilt and magnetometer signals, the voltage on the potentiometer, the state of the two SPST switches, and the voltage of the inclinometer battery. This data is copied immediately to a solid state storage module connected to the data logger. The acknowledgement lamp and buzzer are then activated briefly to signal the operator that the instrument can be moved. The two SPST switches are used to indicate special conditions, such as the bottom of the hole (which often does not fail on an integral number of metres) or a calibration reading. At the end of the day, the storage module is carried to our field camp where the data are transferred to a computer for analysis. This inclinometer system is very efficient. A single operator can make two inch nometry passes through a 70 m hole at 1 m depth intervals in 25 minutes. A.5 Coordinate systems Three right-handed Cartesian coordinate systems are required to process data from a tracked or gyroscopically-oriented inclinometer. Two additional coordinate systems as sociated with the Earth’s magnetic field and the magnetometer are required to process  Appemc1i A. INCLINOMETER DESIGN AND DATA PROCESSING  206  data from a magnetically-oriented inclinometer (Figure A.2). The coordinate system introduced by Nye (1957), and used in several subsequent papers by Nyc and others, is aligned with the flow direction of the glacier. This is natural for studies of inter nal deformation, but because our primary concern is locating sensors placed within boreholes, we choose a system related to map coordinates.  H  H  Fig. A.2: The five right-handed inclinometer coordinate systems dis played as a stereo pair. Top: the geographical coordinate system U and the geomagnetic coordinate system 7-1. The angle “d” between the grid north axis Yu and the magnetic north axis X is the magnetic declination. Bottom: the tilt sensor (T), case (C), and magnetometer (B) coordinate systems share a common z axis pointing downward along the body of the inclinometer. The geographical coordinate system U is tied to the Universal Transverse Merca tor (UTM) coordinate system used on Canadian topographic maps. The u, Yu, and z axes point east, north, and upward, respectively. The geomagnetic coordinate sys tem (labelled 7-1 after the standard designation for magnetic field) is tied to the local declination of the Earth’s magnetic field. The z axis is positive downward and the XH  axis points along the magnetic declination. This definition follows the international  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING convention for geomagnetic coordinate systems and results in positive  ZH  207 readings in the  northern magnetic hemisphere. The remaining three coordinate systems have z axes that point downward along the axis of the inclinometer. The tilt sensor coordinate system T has its  XT  and  YT  axes aligned with the sensitive axes of the tilt sensors. The  0 axis aligned with some feature on the inclinometer. case coordinate system C has the x In the case of a tracked inclinometer, this would be the alignment mechanism. The 0 axis aligned with a machined facet on the inclinometer UBC instrument has its x case (the use of this facet is discussed below). The magnetometer coordinate system (labelled B after the standard designation for magnetic flux density) represents the three axes of sensitivity of the fluxgate magnetometer. For the UBC inclinometer, the instrument-based coordinate systems are distinct, and the rotational angles between the  XT,  , and x axes must be known in order to process data. For other instruments 0 x  these three systems may or may not coincide. A.6 Data analysis The analysis of inclinometer data proceeds in three distinct steps: (1) The instrument calibrations are used to compute the two tilt angles and the azimuth at each station. (2) The tilt and azimuth values are transformed into vectors representing the orientation of the inclinometer at each station. (3) An inversion scheme computes the continuous trajectory of the borehole based on the discrete set of orientations. In the following two sections, we discuss in detail the calibration and transforma tion procedures for the UBC inclinometer. Although these procedures vary from instru ment to instrument, all share the common goal of determining the vertical unit vector ii and the orientation unit vector ih in an inclinometer coordinate system (Fig. A.3). For a magnetically-oriented inclinometer, ili represents the Earth’s magnetic field vector and for a tracked inclinometer, th represents the orientation of the track grooves. The UBC inclinometer yields the true magnetic vector, whereas an instrument that uses a  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  208  gimbal-mounted compass yields the magnetic vector projected onto the gravitationally horizontal plane.  n  n YT XT  m  ZT  Fig. A.3: A stereogram showing the two unit vectors measured by an inclinometer and the tilt coordinate system T in which they are resolved. The tilt sensors yield ii and the orientation apparatus yields ih.  A.6.1 Calibration The tilt sensors are calibrated using a calibration frame that allows positioning of the inclinometer at precise tilt angles (to within 1 mm  of arc) throughout its operating  range of +30° of tilt from vertical. Repeated calibration of the instrument between field seasons indicates negligible drift in the calibration. The output voltage from the tilt sensors is proportional to the sine of the tilt angle from vertical. In the T system, the components of fl are given by =  sinI  =  sinI,  I n=1(n2+n2)  }  (A.1)  where I and I are the two tilt angles. The calculation of the orientation vector  ii  would be trivial if the three axes of  the magnetometer were to have zero offset and equal sensitivity, but in practice, the  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  209  design of the magnetometer sensor makes it very difficult to realize these characteristics, particularly with regard to voltage offsets. A calibration is required. If a series of measurements of the Earth’s magnetic field are taken with the mag netometer positioned at many orientations within the field, then the measured points will map out a surface in the 13 system. If we assume that the axes of sensitivity of the magnetometer (with their non-zero offsets and unequal sensitivities) are mutu ally orthogonal, then the surface is a translated ellipsoid. The calibration function we seek transforms this ellipsoid into a unit sphere centred on the origin; applying the calibration to any given point will produce the corresponding orientation vector. The calibration data are best collected on the glacier surface where the magnetic environment can be controlled. The operator, stripped of magnetic clothing, holds the inclinometer at a succession of orientations. At each position, the inclinometer is held steady while the three magnetometer outputs are recorded. The orientations are chosen such that the Earth’s magnetic field intersects the magnetometer sensor from as many directions as the patience of the operator allows. The specific calibration orientations are not important, so a hand-held calibration can be used. In addition to instrument error, reading errors result from fluctuations in the Earth’s magnetic field during the calibration and movement of the inclinometer as data is recorded (the bandwidth of the magnetometer is 5 Hz and the three components are measured sequentially within a span of ll4ms). Typically, about 200—500 calibration triplets are collected. A calibration transfor mation that maps a data point  M(B)  on this ellipsoid onto a unit sphere centred on the  origin is given by m(B)  ) 3 m( mZ(B)  =  S  0  0  M,(B)+TZ  0  S,  0  M(B) + T  0  0  MZ(B)  +T  (A.2)  210  Appendia A. INCLINOMETER DESIGN AND DATA PROCESSING where T is an offset vector, S is a scaling matrix, and  the transformed point.  m(B) is  S, S,, T, T, T) are determined by minimizing the  The six free parameters  least squares objective function [m + m(B) + m(B)  —  1j2  = S, and S are non-negative). An alge  for the n transformed calibration triplets  braic approach to solving the minimization is algebraically horrific, if not intractable, so we use an iterative six-dimensional simplex algorithm (Press and others, 1986, p.289). The expanded objective function is given by =n+S  +  —  (E 2S  (  + mT + 4T + nT)  —  z +4T 3 2S  (  +  y + nT)  —  2S  (  z  + ThTz2)  +xy +Ty +Tx +2Txy  + 2T  yjx + 2TT +T  +2SSL(TT +  + 2T  zjy + 2TT  +2SS(T:T: +yz + T  +2Tzx  x + 2TT +T  yj + 2TT +T  y + 4TT  XiYi)  + 2T  z  + 4TT  + 2T  YiZi)  211  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  —  4ST  >  x  —  4S,T  —  (A.4)  4ST  Note that the summation terms are constants for a given data set. For the UBC inclinometer, the transformed calibration data fit a unit sphere to within 2%. Removing the constraint on magnetometer axis orthogonality by allowing non-zero off-diagonal terms in S results in a negligible improvement in fit, at the expense of greatly increasing the geometrical complexity of data analysis. A.6. 2 Transformations In this section, we discuss the transformations that are applied to data from the UBC inclinometer. Some of these transformations are generic in nature and can be used with all inclinometers. In order to simplify the analysis, the scaled magnetic vector  m(B) must  be rotated  about the inclinometer axis from the B system into the T system (Fig. A.2). The angle 1, measured from the  XB  axis to the  XT  intermediate angles. The angle between the  axis, is calculated as the sum of two  XT  and the x axes can be measured  accurately on the tilt calibration frame. The angle between the x and  XB  axes is not as  well constrained. Repeated sightings on known geographical reference points are made through a telescope clamped to the machined facet on the body of the inclinometer. By reconciling the magnetic declination published on the 1:50 000 topographic map (Energy, Mines and Resources, Canada map sheet 115 F/i, edition 1, 1987) with the coordinates of the reference points and the coordinates of the observation site, the value of this angle can be computed to within ±0.5°. Weather permitting, these reference sightings are made before and after each hole is logged. This allows correction for temporal variations in magnetic declination caused by geomagnetic disturbances; these disturbances can cause several degree of variation in declination, especially in polar regions where the Earth’s magnetic field is near-vertical.  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  212  A.6.2.1 Normalization The calibrated magnetic vector  m(B), from  Equation (A.2), is nominally a unit vector,  but errors in the data will result in a slightly erroneous vector magnitude. If the magnitude is not corrected, then the components of m(B), and vectors derived from it, cannot be used as direction cosines. Therefore, the unit magnetic vector th is defined by rotating and normalizing m(B) using the equation m m  =  —  cos  sin1  0  m(B)  sin 1  cos  0  m(B)  0  1  mZ(B)  0  m  (A.5)  where Im(B)I denotes the scalar length of vector m(B). Examination of Figure A.3 reveals that, in the case where  iii  is the full magnetic  field vector, there is redundancy in the data. The three degrees of freedom representing the orientation of the inclinometer body are constrained by the four degrees of free dom embodied in the components of th and fl, which in turn are derived from the five instrument readings (I,,, Is,,  MZ(B), M(B),  and MZ(B)). The two additional degrees of  freedom governed by the instrument readings are the magnitude of the magnetic field vector (lost in normalizing  iii)  and the spin orientation of the inclinometer on its axis  (in our analysis, this information is discarded). For the UBC inclinometer, the value of 11 is far better determined than th, so we use ii to define the inclination angle of the  ZT  axis. Vector th is used to constrain the azimuth of the  The functions of  ih  ZT  axis about vector ñ.  and ñ can be exchanged, but were th used as the inclination refer  ence, accuracy would be lost. Note that data from magnetically-oriented inclinometers become indeterminate when th and ñ are parallel or anti-parallel.  213  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING A.6.2.2 Eulerian Angles  The relationship between two arbitrarily-rotated coordinate systems that share the same origin can be uniquely described using Eulerian angles. Figure A.4 shows how a triplet of Eulerian angles (4’, 0, 4,) defines the transformation between any two righthanded Cartesian coordinate systems. The transformation is achieved by three succes sive right-handed rotations about specified axes. Defined as a matrix operation, the transformation from the unprimed coordinate system to the primed is (A.6)  x’=Ax  and A is the transformation matrix  where x and x’ are coordinates of a point in cos  ( A  =  —  cos 4’  sin b cos 4,  —  —  cos 8 sin 4’ sin & cos 8 sin 4, cos  —  cos b sin 4’ + cos 8 cos 4, sin &  sin b sin 0  sin & sin 4’ + cos 8 cos 4, cos  cos  sin 8 sin 4,  —  sin8 cos 4,  sin 8  cos 0 (A.7)  1 Because A is an orthogonal matrix, A  =  AT,  so the inverse transformation x  =  A’x’ is simply x The inverse matrix A—’ satisfies AA’  =  ATxI  =  (A.8)  I, where I is the identity matrix, and the  transposed matrix AT is obtained by writing the rows of A, in order, as columns.  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  214  z I  ::::::::::::::::::::::::  Fig. A.4: A set of Eulerian angles describes the transformation between two right-handed coordinate systems sharing a common origin. The transfor mation from an unprimed to a primed coordinate system is defined as a set of three right-handed rotations about specified axes. The three rotations are: (1) A rotation by angle g about the z axis. (2) A rotation by angle 0 about about the the intermediate x axis (dotted line). (3) A rotation by angle z’ axis. This level of complexity is only necessary when moving from an instrument-based coordinate system to a map coordinate system. Transformations between coordinate systems within these two groups is accomplished by simple rotation about the common z axis (9  =  =  0).  Appencli A. INCLINOMETER DESIGN AND DATA PROCESSING  215  A.6.2.3 Eulerian Transformation Our objective now is find a set of Eulerian transformation angles q, 8, and b which will map  Iii,  as expressed in the T system, onto the U system such that the horizontal  projection of th has the proper magnetic declination d. By casting the transformation in this way, we avoid directly implicating the 7( system. The transformation must also map ñ, as expressed in the T system, onto the vertical in the U system. We will treat the U system as unprimed and the T system as primed (Equation (A.7)). The angle 9 defines the net tilt of the  ZT  axis with respect to the vertical z,., axis.  This net tilt is simply 9 The angle  ‘  =  (A.9)  1n cos  controls the relative contribution of the two tilt angles to the net tilt,  defined as  =  The angle  tan’’  (A.1O) \\ny  I  rotates the T system about the z axis; in other words, q is the Eulerian  angle that controls the declination of the transformed magnetic vector. Given 9 and ‘ç&, the value of ç must satisfy Fsind Fcosd /[ZF2  where 0 <F  <  1 is the magnitude of  iii  m =  AT  m,  (A.11)  m when projected onto the horizontal (Usystem)  plane and d is the magnetic declination east of UTM north. This system of equations has an explicit solution for  4  and F, but the algebra can be simplified considerably by  216  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING first solving for the case where d  =  0 and subsequently adjusting the solution. This trick  results in an indirect association with the 71 system. The equation to solve becomes 0 F  =  AT  /1_F2  m  (A.12)  m  and the solution is q=tan  F  =  1 .  (m sin b + m cos  [(mi cos&  —  cos 9  (13)  .  —  m sin 9  m sinb) 2  2 9 (mi, cos + cos  —  + m sin 29 (m cos  m sin &)2 —  m sin  ‘v’)  1/2  2 9] + m sin  (A.14)  The non-zero declination ci is reinstated by modifying Equation (A.13) to give ç=tan —1  m,cos&—msin& 1i—d (m sin b + m cos Ø) cos 0 m sin Oj .  .  A.15 1  —  Equations (A.13) and (A.15) are curiously insensitive to the value of m in the sense that a value of  iui derived from a gimbal-mounted compass  give the same value for  (i.e. th is horizontal) will  as a value of th representing the full Earth field, but this in  sensitivity is expected. Both gimbal-mounted and full-field magnetometers are equally adept at determining magnetic bearing. As the final step in the data transformation procedure, the orientation vector the inclinometer (which corresponds to the  ZT  of  axis), as expressed in the U system, is  computed by the transformation t  sin9sing  0 =  AT  0  1  =  —sin9cosq cosO  (A.16)  ESSING Appendiz A. INCLINOMETER DESIGN AND DATA PROC  217  (A.7), (A.9), (A.1O), The Eulerian transformation matrix A is defined by Equations and (A.15). For this analysis the value of F is irrelevant.  A .6.2.4 Universal application of transformations to any incinometry Note that Equations (A.7), (A.9), (A.1O), (A.13), and (A.16) apply orientation vector ill. data that can be expressed in terms of a tilt vector ii and an meters. This includes data from tracked and gyroscopically-oriented inclino A.6.3 Inverse problem all inclinometers, for com In this section, we develop a general theory, applicable to rements of borehole puting a continuous borehole trajectory based on discrete measu of this step, it is depth and inclinometer orientation. Before considering the details orientation and the appropriate to consider the relationship between the inclinometer borehole axis. always parallel to The simplest assumed relationship is that the inclinometer is without additional data the borehole axis. Although this presumption is erroneous, is difficult to improve describing the borehole geometry and centring device geometry, it axis lies along a straight upon. At any given station along the borehole, the inclinometer s operate identically, line drawn between the centring devices. If the two centring device curvature of the borehole the borehole has smooth walls of constant diameter, and the the borehole axis at is a circular arc, then the inclinometer axis will lie parallel to three conditions are a point midway between the centring devices. In practice, these ed performance over rarely satisfied. It is here that tracked inclinometers exhibit improv accurately position instruments such as the UBC tool; the groove-tracking wheels can es both smooth walls the inclinometer in the centre of the casing and the casing provid described below and regular borehole geometry. The trajectory computation method  PROCESSING Appendix A. INCLINOMETER DESIGN AND DATA  218  coincide; the sensitivity of the will assume that the borehole and inclinometer axes of the appendix. method to positioning error will be discussed at the end ctory from a set of discrete The task of computing a continuous borehole traje problem. An inffnite number of tilt and azimuth measurements constitutes an inverse data set exactly. An infinite possible trajectory solutions exist that will satisfy a given fy the data exactly, but which number of additional solutions exists that do not satis solution of the first type can be do fall within the error bounds of the data. A single on the trajectory; the solution isolated by placing appropriate geometrical constraints me. All treatises of which we to the inverse problem becomes an interpolation sche al constraints to choose an exact are aware, this development included, use geometric . This method for solving an solution that will hopefully reflect the true trajectory a priori structural model is inverse problem differs from traditional methods in that an . Traditional methods can solely responsible for isolating one of the possible solutions function, such as minimizing find solutions of the second type by setting an objective assessment of error in the data. the rate of change of borehole tilt, and by allowing for inexact trajectory solutions. Data smoothing is another possible approach to finding polation of orientation Unlike interpolation between fixed points in space, inter cting the trajectory of the vectors can lead to cumulative error in position; in reconstru ng the relative position between hole from the surface downward, the error in determini two measurement stations accumulates. ssed in the petroleum in Many methods for interpolating slope angles are discu is often mentioned as the dustry literature. The “terminal angle tangential method” val, the orientation of the best known interpolation method: over any station inter lower station. In effect, no inclinometer axis is assumed to be equal to that of the d trajectory is discontinuous at interpolation is performed and the slope of the compute nometer motion, this method every station. Because of its faulty representation of incli ss five different methods results in appreciable error. Waistrom and others (1972) discu  Appendia A. INCLINOMETER DESIGN AND DATA PROCESSING  219  for interpolating the tilt angle and azimuth between stations. The underlying premise of many of their mathematical models is that tilt angle and azimuth can be treated as independent quantities. This premise does not always hold true, as is revealed by considering the radius-of-curvature method developed by Wilson (1968) and later expanded by Rivero (1971). Their method maps the borehole trajectory onto the sur face of a vertical cylinder such that the desired azimuth and tilt are preserved at the endpoints of the spiral segment. The resulting trajectory is not independent of the coordinate system in which the projection is made, indicating that in this case the tilt angle and azimuth cannot be treated separately. Angle interpolation methods are also cluttered with special treatment for the multivalued nature of the inverse trigonometric functions. We assume that the path of the inclinometer between stations can be described by a series of circular arc sections (Fig. A.5). Figure A.6 shows one of these arcs in ) are tangential to the normalized orientation 2 1,P detail. The endpoints of the arc (P , 2), and the length of the arc L equals the measured depth increment on 1 vectors (t the cable. Successive arcs join in a smooth manner with no discontinuities in slope, although discontinuities in the direction of slope change are present. If the position of 3 along the arc can be 2 or any intermediate point P 1 is known, then the position of P P computed. This method has been discussed by Zaremba (1973), but Zaremba derives an unnecessarily complicated solution.  220  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  x  Fig. A.5: A perspective view of the circular-arc interpolation model. Be tween pairs of measurement stations, the borehole is assumed to follow a circular trajectory. The arc length is constrained by the measured distance between stations, and the plane of the arc is defined by the instrument ori entation vectors at the endpoints of the arc. These orientation vectors are tangent to the arc A.7 Interpolation Scheme 1 and the unit tangent 3 given the initial position P 2 and P We derive solutions for P vectors to the circular arc  and  2  (Fig. A.6). Consider the general problem of solving  for the intermediate coplanar vector between two non-parallel unit vectors (Fig. A.7). The partitioning factor e ranges between 0 and 1. The unit vector ia is defined by the linear combination =  i 2 1 +k k 1  (A.17)  221  Appendiz A. INCLINOMETER DESIGN AND DATA PROCESSING  P  e  3 P 2 P ti  Fig. A.6: A detailed view of one circular arc in the borehole trajectory. , the tangent vectors I and 2, 1 Knowing the position of the starting point P the endpoint P 2 can be computed. the of of position length arc L, the and the 3 on the arc can also be computed. The position of any intermediate point P and t . 2 The angle 7 is defined by 2 must satisfy 1 and k where k 2 are appropriate scalar values. The solutions for k 1 and k (A.18)  = COS(E7)  =  cos [(1  —  (A.19)  yj  where (A.20)  2 cos-y=ti.t Equations (A.18) and (A.19) ensure that  is a unit vector coplanar with  i  and  2•  The solutions are =  2 k  cos(E-y) sin(E7)  —  Sin  7  —  cos-ycos [(1 sin 7  —  (A.21) (A.22)  222  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING In the special case of bisection (e  =  1 k  0.5),  =  2 k  =  (A.23)  2 cos(7/2)  ti C  3 t  Y  2 t 3 between two arbitrary Fig. A.7: The intermediate coplanar unit vector t and 2 is shown. The value of 0 < e < 1 defines non-parallel unit vectors the position of the intermediate vector. 2 bisects the two vectors In Figure A.6, the chord drawn between Pi and P  and  . The length of the chord is given by 2 t (A.24)  C=2()sin() and Equations B.1 and B.7 yield the unit vector along the chord 2 tl+t  C —  (A.25)  2cos(7/2)  so that we can write the solution for P 2 as /7’\ £ Pi+—tan(—) P = 2  [1+2]  (A.26)  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  223  , located at some fractional distance 0 3 The position at any intermediate position P e  1 is given by a similar construction as 1. along the arc from P 3 P  =  tan  1 + P  ()  22 k 1 + 1) + ] [(k (A.27)  , and k 1 2 are defined by Equations (A.20), (A.21), and (A.22). where -y, k 2 In the special case where t  —*  £ (the anto-parallel case is unlikely), the circular  3 are 2 and P arc will have infinite radius, and the solutions for P P =L 2 P + 1  (A.28)  =Pi+eLi 3 P  (A.29)  The trajectory computed by this circular-arc method is independent of the coordi nate system. The trajectory of the borehole is reconstructed by successively applying Equation (A.26) or (A.28), as appropriate, beginning at the glacier surface, and working downward. A.8 Sensitivity The trajectory of the borehole as determined by the circular-arc method, or by any other interpolation method, is inherently incorrect  —  even if the borehole orientation data  are error-free. This is because the continuous borehole trajectory is sampled at a finite number of points. In the case of the UBC inclinometer, we do not believe that sampling density is a major source of error because our 1 m sampling interval is short compared to the length of the inclinometer and to the length of the drill stem used to drill the holes; we do not expect perturbations in borehole trajectory on a scale smaller than 1 m (this could be checked by excluding every second data point from the analysis). For this dense sampling interval, and given error-free borehole orientation data, the divergence between the true trajectory and the computed trajectory is expected to be  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  224  at most a few centimetres. Problems arise because the borehole orientation data is not error-free. The UBC inclinometer has logged 125 boreholes during the 1989 and 1990 field seasons at Trapridge Glacier, Yukon. Each hole was logged at least twice. By comparing the inclinometry results, we estimate that the instrument locates the bottom of a 70 m hole to within 20—30 cm. The error tends to be largest in the azimuthal sense; the radial deviation of a borehole is determined to within 15 cm. Based on this evidence, we identify two major sources of error: random positioning error of the inclinometer in the borehole and systematic orientation error. Superimposed on these error terms are the measurement errors of the tilt sensors and magnetometer. Accurate positioning of a non-tracked inclinometer within a borehole is dependent on the texture of the borehole wall and the performance of the centring devices. Glacier borehole walls are not necessary smooth and the centring devices allow the inclinometer to cant relative to the borehole axis. The discrepancy between the axis of the borehole and the axis of the inclinometer is estimated at +0.5°.  This estimate is based on  the diameter of the borehole and that of the inclinometer with the centring devices fully compressed. It is also likely that the inclinometer tends to underestimate the borehole tilt since tension in the cable and the instrument weight work to force the instrument towards a vertical orientation. As this effect depends on the lay of the cable and the local hole geometry, it is impractical to quantify; we satisfy ourselves with the +0.5° positioning error. The tilt sensor error (less than one part in 10000) is neglected. The magnetometer error is appreciable and amounts to (±2%) on the and  YB  axes, and (+3%) on the  ZB  XB  axis. These estimates are based on the calibration  fit and the specifications for the magnetometer. Neglecting stretch in the cable, the error in positioning along the borehole is estimated at 1 cm for each 1 m interval. The effects of the random errors can be modelled using Monte Carlo techniques. Synthetic data are generated for a sequence of stations with error terms superimposed  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  225  on the “correct” data. The standard deviation of the Gaussian-distributed error for each reading is given above. Relative contributions to the net deviation can be examined by selectively removing the error terms. Figure A.8 shows the results from a series of Monte Carlo tests on a 70 station hole (69 m long). The figure shows horizontal maps of the computed bottom locations for each Monte Carlo test. The underlying model is a linear hole dipping  100  to the south with the top at the origin. Each test contains  1000 runs through the hole. Figure A.8a shows the effects of applying all error terms simultaneously. The cross hair indicates the location of the error-free hole bottom at 11.98m south. Note the slight stretching of the “data cloud” in the azimuthal (east west) sense and the offset towards the north (origin). Both of these effects are primarily due to error introduced by the magnetometer, as can be demonstrated by removing the magnetometer and depth error (Fig. A.8b). This results in a more symmetric cloud, with less noticeable offset. Figure A.8c shows how the magnetometer introduces these two distortions. The east-west lineation is actually a short section of a circle with its centre at the origin. This is easily seen in Figure A.8d where gross errors (±50%) in the magnetometer over one station interval result in a circular scattering envelope. Again, the cross hair indicates the error-free solution. The foreshortening (drawing in towards the origin) observed in Figures A.8a, A.8c, and A.8d results from the fact that error in azimuth will always cause the radius of the interpolating circular arc to decrease; this results in a horizontal projection that is always smaller than, or equal to, the true projection. On average, error in tilt measurements also produces foreshortening (Fig. A.8b). Note that for holes of this depth, the foreshortening amounts to no more than a few centimetres. In Figure A.8a, the net depth error amounts to no more than ±2cm.  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  226  —11.8  U,  a) a) E  0)  a) a) E  . .:i ..&.  —“C,  —12. —6  —.4  —.2  ‘T.!lI_L  .0  .4  .2  metres .2C  .10  0  a, a,  .00  E —.10  =20  —.10  .00  .10  .20  metres  Fig. A.8: Monte Carlo modelling of the sensitivity of the inclinometer to error in its sensors is shown as a series of scatter plots. Panels (a), (b), and (c) show the horizontal deviation of the computed hole bottom from its true position for 1000 Monte Carlo runs down a 69 m hole dipping southward at 10 (station interval of 1 m). The hole top is at the origin and the true bottom position is indicated by the cross hair. (a) The combined effect of a +0.5 1 error error in tilt, a 2% error in m and m, a 3% error in m, and a 1 cmm r error magnetomete in depth control. (b) The effect of tilt error alone. (c) The alone. (d) Scatter produced by allowing gross (50%) error in the components of ih over one station interval. Systematic error is caused by the non-random orientation of the inclinometer as  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  227  it moves through the borehole. The orientation of the inclinometer does not change greatly as it moves from station to station, but in separate passes through the hole, the general orientation of the inclinometer may be quite different. It follows that any offset error in determining the angle 1 between the  XT  and  XB  axes will accumulate  differently in different inclinometer passes, and this will result in additional azimuthal error. By adjusting the value of Il such that the azimuthal error is minimized for all holes, we can remove much of the systematic error. Our estimated positioning error of 20—30 cm, with slightly better radial control, is consistent with the results of the Monte Carlo test in Figure A.8a. This leads us to conclude that we have a good understanding of the factors influencing the performance of our inclinometer. A.9 Discussion Our experience with the UB C inclinometer has shown that it is an efficient and accurate instrument for glaciological work. Surveys are performed quickly and do not require any special attention to the instrument or preparation of the borehole. For these reasons, we expect that magnetically-oriented inclinometers using fluxgate magnetometers will gain popularity with glaciologists. For rigorous applications, some researchers may wish to combine the better tilt accuracy of a tracked inclinometer with the additional check on orientation provided by a magnetometer. The principal shortcoming of these instruments is that they become confused at very high magnetic latitudes. We estimate that the UBC inclinometer will not operate reliably where the dip of the magnetic field exceeds 85°. This constraint probably ex cludes its use throughout most of the Canadian arctic archipelago, and the Wilkes Coast and George V Land in Antarctica. We know that the accuracy of the magnetometer is being degraded by its close proximity to electrical currents inside the pressure tube. Redesigning the internal configuration of the instrument so as to avoid this magnetic  Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING  228  noise would reduce the error observed in Figure A.8c, and might allow the instrument to be used closer to the Earth’s magnetic poles. These restrictions aside, the error in determining the position of subglacial and englacial sensors with the UB C inclinometer is small enough for all but the most exacting experiments. We are seeking to improve the performance of the instrument, primarily by redesigning the centring springs. If this results in reduced error in the tilt readings, then the instrument might approach the sensitivity represented by Figure A.8c. The circular-arc interpolation scheme is simple to apply and easy to visualize. On a 12 Mhz IBM-PC compatible computer equipped with a numerical co-processor, the time required to calibrate, process, and plot the data from a 70 m borehole takes a fraction of a second. The foreshortening effect of the circular-arc method is the only drawback of which we are aware, but this effect is not unique to this interpolation method. We expect that improvements in the accuracy of inclinometry data inversion can be obtained in at least three ways: (1) Incorporate additional borehole geometry and centring device geometry. (2) Study the effects of centring device spacing relative to station spacing. (3) Introduce full inversion techniques that produce trajectories having an imperfect fit, within error, to the data. Data smoothing is one strategy for finding these imperfectly-fitting trajectories, but it is important that the smoothing process does not result in violation of the error bounds on the data. We invite other interested parties to consider these approaches.  Appendix B SUBGLACIAL WATER AND SEDIMENT SAMPLERS  The current focus of glaciological research on basal processes and hydrology makes the acquisition of samples from the subglacial environment a vital enterprise. In this letter, we describe two devices for obtaining samples of basal water and sediment within the confines of a narrow borehole. The samplers have been operated at depths ranging from 70—300 m. They are lightweight and require only a single operator. The substance of this appendix has been published previously in the Journal of Glaciology (Blake and Clarke, 1991a). B.1 Niskin sampler  Collecting water at depth has been a concern of oceanographers for centuries (see McConnell, 1982); designs for sampling bottles abound and are slowly modified by generations of researchers. The modern Niskin sample bottle consists of an open-ended tube which can be closed on command by a pair of stoppers. A Niskin bottle, attached to a wire rope, is lowered into the water and when the bottle reaches the desired depth, a messenger block is dropped along the rope. The block strikes the Niskin bottle and trips the sampling mechanism. We have designed a modified Niskin sampler having a trigger mechanism that operates axially. This action allows the device to operate in a narrow borehole. Fig ure B.1 shows the sampler in its open, cocked position. The sampler consists of four major units that move relative to one another: (1) The lower stopper is fixed to a hollow central rod. The central wire rope upon which the sampler is suspended passes through the rod and is held by a crimp at the bottom. (2) Two perforated brass disks are fixed within a Plexiglas sampling tube. The disks slide on the central rod and the  229  Appendiz B. SUB GLA CIAL WATER AND SEDIMENT SAMPLERS  230  perforations allow water to move through the tube when the stoppers are open. (3) The head block, with the upper stopper attached, is free to slide on the central rod, but two spring-loaded catches hold the block in a cocked position at the top of the rod. Two lengths of fine wire rope suspend the Plexiglas tube below the head block. The wire ropes are attached to small eye-hooks on the block and upper disk (for clarity, these fixtures are not shown in Figure B.1). (4) A brass messenger block slides along the central wire rope. The sampler, in a cocked position, is lowered into position at the borehole bottom and the messenger block is dropped along the wire rope. When the messenger strikes the catches, they spread apart and release their grip on the central rod; the head block falls against the sampling tube, the tube falls against the lower stopper, and a 220 mL water sample is trapped inside the device. The weight of the upper block ensures a watertight seal between the two stoppers and the tube. The sampler is opened, and the sample collected, by pushing on the top of the rod. Experience has demonstrated that wire rope  (‘-‘-i  0.5 mm diameter) must be used to connect the block and disk because  weaker materials (such as fine chain or string) will break. The modified Niskin sampler is simple to operate and performs reliably; accidental triggering of the mechanism is unusual. Since the introduction of the Niskin sampler to our field program in 1986, we have obtained basal water samples from more than 300 boreholes of 70 m depth and one sample from a 300 m borehole. Samples from the glacier bed often contain significant quantities of fine particulate matter, and sometimes subglacial material is found adhering to the lower stopper. The axial design allows the device to take samples in a borehole as narrow as 31.8 mm (1.25 in) in diameter. B.2 Subglacial vacuum sampler The modified Niskin sampler excludes the sand, pebble, and gravel fractions of the bed. In order to collect these larger-sized particles, we built an active vacuum sampler that  Appendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS  231  *E RE ES5ENGO OG(  cArd-Es  1-EAD BLOOC STCPF  IE RE  F9FORATED XK  W4TRAL ROD SAfl1 REE  TOFF  cRtp  Fig. B.1: The modified Niskin sampler in a cocked position. As the messenger block strikes the catches, the head block is released from the top of the central rod and slides down to seal the top of the sample tube. The sample tube falls against the lower stopper; this seals the bottom of the sample tube and traps a water sample inside the device. is powered by the high pressure pump on our hot-water drill. Because it works like a vacuum cleaner, we dubbed this device the “Hoover”. The Hoover design was inspired by the airlifts used to clear sediment from shallow submarine archaeological sites. According to Throckmorton (1969, p. 175), airlifts were invented at the turn of the century for clearing mine sumps and for mud-pumping in  Appendiz B. SUB GLA CIAL WATER AND SEDIMENT SAMPLERS  232  harbours. The first archaeological application was by Jacques Cousteau in the early 1950s at the Mediterranean site of Grand Congloué near Marseilles, France. An airlift consists of a large diameter flexible pipe (perhaps 15—20 cm in diameter) leading from the archaeological site at the sea floor to the deck of a support vessel. Injection nozzles mounted inside the pipe force air into the pipe. The presence of air into the pipe has two consequences of interest: (1) Viscous drag between the rising air bubbles and the water in the pipe causes upward flow. (2) The bulk density of the mixture in the pipe is lowered to such a degree that buoyancy forces lift the mixture up the pipe. Buoyancy forcing is the stronger of the two effects. Given sufficient air content, water in the pipe can be lifted well above the free water surface and onto the deck of the vessel. Artifacts small enough to be entrained by the water flow can then be collected in a sieve. An advantage of this system is that no impellers are required to drive water through the pipe; artifacts are not damaged and the pipe is unlikely to clog. The subglacial Hoover uses viscous drag from water jets (rather than air bubbles) to move water up through a Plexiglas tube that has a one-way valve at the bottom. Figure B.2 shows the design of the Hoover with an exploded view of the valve assem bly. Cold water from our hot-water drilling system is fed into the 37°JIC swivel hose fitting. The water travels down the central feed pipe to the nozzle assembly where the direction of flow is reversed as the water emerges through two 1.59 mm (0.0625 in) nozzles. Vibration of the nozzle assembly is prevented by three pins that reach out to meet the inner walls of the Plexiglas sample tube. The pressure drop across the nozzles is about 2 MPa at a flow rate of 18 L miri’. The valve consists of a cylindrical rip-stop nylon fabric sock. The lower edge of the sock is sewn to a sock cage; the cage consists of a brass ring that is brazed to a tetrahedral wire frame. The three wires in the cage prevent eversion of the sock, but also reduce the maximum particle size that can be admitted by the valve. A threaded retainer ring holds the valve assembly onto the valve seat. The bulkhead at the top of the Hoover has eight 11.1 mm (0.4375 in)  Appendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS  233  holes drilled around its periphery; these holes provide an outlet for the injected and entrained water. HOSE RTTtIG  BLLKI-EAD C  FEED PFE  0  0  C” 1 U’  NOZZLE ASSEtLY  SAMPLE  VALVE SEAT  VALVE SOOC  Oil  RETMER  Fig. B.2: An exploded view of the subglacial Hoover. Upward- directed water jets are created by forcing water through the feed pipe and nozzle assem bly. Viscous drag between the jets and the water in the sample tube creates a vigorous upward flow of water through the one-way sock valve. Sediment entrained by the flow is trapped inside the tube when the jets are turned off. When the Hoover is operated in air, a noticeable suction develops at the inlet. This suction is much greater when the device is operated in a water-ifiled borehole; we have found clasts wedged into the bulkhead outlet holes, indicating that clasts of  Appendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS  234  considerable size are being driven forcibly through the sample tube. The largest clasts that we have collected have a typical grain diameter of —20—25 mm. Samples are taken using the following procedure. With the Hoover at the bottom of the borehole, the water supply is turned on and the pressure hose is gently moved up and down causing the Hoover to move against the glacier bed. After 20—40 s, the water supply is turned off, and the sampler is hauled to the surface. Examination of the closed valve assembly suggests that the valve does not close suddeiily as the water jets stop, but rather that the valve is slowly pinched shut by sediment falling between the valve sock and the sample tube. The sediment is released from the Hoover by removing the retaining ring and valve assembly. Samples as large as 500 g have been obtained (for a representative sample, see Clarke and Blake, 1991, Fig. 11). B.3 Considerations The hot-water drilling system used at our field site on Trapridge Glacier, Yukon Ter ritory, Canada, alters the glacier bed as each borehole is completed; surface water is introduced to the subglacial environment and some fine material may be flushed away. In addition, the geometry of the both sampler inlets further biases the sampling of solid matter from the bed. These effects must be considered during sample analysis. The Hoover tends to eject finer particles through the water outlet, although our samples do contain a significant amount of coarse and fine sand. A screen placed over the outlet might improve the retention of finer material.  Appendix C SUBGLACIAL DRAG SPOOL  C.1 Introduction One of the problems discussed in Chapters 2 and 3 is the partitioning of ice motion at the base of the glacier between basal sliding and deformation. The instruments described in Chapter 2 make measurement of basal deformation, but the analysis of the data from these sensors cannot be complete without some measure of basal sliding (Figure 3.2). To our knowledge, only one prior attempt to measure basal sliding has been re ported in the literature: Boulton and Hindmarsh (1987) screwed an auger-like anchor into the sediments beneath the terminus of Breidamerkurjökull. The anchor was in serted through a sealed hole in the floor of an artificial ice tunnel and connected by string to a chart recorder; as the anchor string payed out, the recorder took note. The author is also aware of measurements made in a natural ice cave beneath the terminus of Isfallsglacinren, near the well-studied Storglaciren in Swedish Lapland. At Isfalls glaciären, an ice screw was fixed in the moving ceiling of the cave; a string connected the ice screw to a chart recorder on the floor of the cave. In this case, the glacier was sliding on a bedrock base; there was no subglacial deformation. C.2 The “Slide-O-Meter” or “drag spool” The instruments we designed for use under Trapridge Glacier is similar to the Boulton and Hindmarsh design only in that an anchor is placed in the bed and the amount of string payed out is measured. Our instrument can be installed in situ at the bottom of  235  Appendix C. SUBGLACIAL DRAG SPOOL  236  a deep borehole and has been honoured with two names: the “Slide-O-Meter” and the “drag spool”. This last term is the more dignified of the two and will be used here. Figure C.1 shows a schematic diagram of the drag spool instrument. The percus sion hammer (described in Chapter 2), fitting with a 0.635 cm (0.25 in) dowel attach ment, is used to hammer an anchor into the glacier bed. A 0.635 cm socket is drilled into the back of the anchor tip; the dowel fits into this socket and the drag spool casing is loosely attached to the dowel. After inserting the anchor, the hammer and dowel are withdrawn, leaving the anchor and the drag spool at the bottom of the borehole. The drag spool cable is then pulled tight. As the cable freezes into the borehole, the drag spool becomes fixed in the ice; as sliding occurs, the anchor will distance itself from the glacier borehole and the string spool will turn as the string pays out. A 5 k1 potentiometer is connected to the spool so that turns of the spool can be regis tered electrically from the glacier surface. Approximately 2 m of thin nylon string are stored on the spool  —  enough to last about four weeks at anticipated sliding rates of  0—l0cmday’. It should be noted that the drag spool will place only an upper limit on glacier sliding because the spool anchors are placed within the deforming sediments and some of the observed relative motion between the anchor and the ice can be caused by deformation of the intervening sediment.  Appendix C. SUBGLACIAL DRAG SPOOL  CABLE CASE SPOOL POTENTIOMETER  STRING ANCHOR II  1 cm Fig. C.1: A schematic diagram of the drag spool. As the string attached to the anchor is payed out, the potentiometer screw is turned and the resis tance change can be measured.  237  


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