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Observations on Athabaska Glacier and their relation to the theory of glacier flow Paterson, William Stanley Bryce 1962

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OBSERVATIONS ON ATHABASKA GLACIER AND THEIR RELATION TO THE THEORY OF GLACIER FLOW by WILLIAM STANLEY BRYCE PATERSON M.A. (Hons.) The University of Edinburgh, 1949 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1962 In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of P h v s i c s  The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada. Date Mav ^ r d . 1962. GRADUATE STUDIES Field of Study: Glaciology Advanced Geophysics Isotope Geophysics Nuclear Physics Related Studies: J.A. Jacobs R.D. Russell J.B. Warren Differential Equations Computational Methods Geodesy C.A. Swanson C. Froese H.R. Bell The University of British Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL' ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of WILLIAM STANLEY BRYCE PATERSON M.A. Hons., University of Edinburgh, 1949, FRIDAY, MAY 4th, 1962, at 9:30 A.M. IN ROOM 302, PHYSICS BUILDING COMMITTEE IN CHARGE F.H. Soward, Chairman H.R. BELL W.H. MATHEWS R.W. BURLING J.V. ROSS J.A. JACOBS R.D. RUSSELL R.W. STEWART External Examiner: J.F. NYE University of Bristol, England OBSERVATIONS ON ATHABASKA GLACIER AND THEIR RELATION TO THE THEORY OF GLACIER FLOW ABSTRACT The objects of the present study were to collect adequate data concerning the distribution of velocity in a typical valley glacier, to relate these to theories of glacier flow, and i f necessary to suggest modifications to these theories. Surface movement, both horizontal and v e r t i c a l , was measured, and move-ment at depth was determined by measurements in bore-holes. Measurements of ice thickness were also available. It is shown that, on the Athabaska Glacier, the longitudinal strain rate is net constant with depth, and that; for about 100 metres below the surface, the horizontal velocity is slightly greater than its surface value. Present theory does not cover these cases. Possible modifications are suggested. The assumption, sometimes made in the past, that the width of a valley glacier can be regarded as inf i n i t e , is shown to be unjustified. The relation between the second invariants of the strain rate and stress deviator tensors is compared with the simple power law as determined by laboratory experiments with ice. Comparison is made both for borehole measurements and measurements of change of surface velocity across transverse lines= Agreement is satisfactory, within the limits of experimental error, for a l l the borehole results and some of the surface movement results. This is interpreted as evidence that the underlying theory is not seriously in error. In particular, the basic assumptions, made by Nye, seem to be reasonable approximations. Of three laboratory flow laws, that of Glen for quasi-viscous creep gives the most satisfactory f i t to the data. The f i t would be improved i f the mean temperature of the glacier were about -0,75°C rather than the pressure melting temperature. The results appear to show that the index in the power law is re duced at low stresses. Other interpretations of the data are possible, however, so that this result is not considered to be established. PUBLICATIONS' 1. W.S.B. Paterson. Altitudes on the inland ice in North Greenland. Meddelelser om Gronland. 137, 1, 1-12, 1955. 2. W.S.B. Paterson and C.G.M, Slesser. Trigonomet-r i c levelling across the inland ice in North Greenland. Empire Survey Review, 13, 3.00, 252-261, 1956. 3. W.S.B. Paterson. Atmospheric refraction above the inland ice in North Greenland. Bulletin Geo desique, 38, 42-54, 1956. 4. AoG, Bomford and W.S.B. Paterson. The survey of South Georgia. Empire Survey Review, 14, 107, 204-213 and 242-247, 1958. 5. W.S^o Paterson. Movement of the Sefstroms Glet cher, North East Greenland. Journal of Glaciolo 3, 29. 845-849, 1960. i i ABSTRACT The objects of the present study were to c o l l e c t adequate data concerning the d i s t r i b u t i o n of v e l o c i t y i n a t y p i c a l v a l l e y g l a c i e r , to r e l a t e these to current theories of g l a c i e r flow, and i f necessary to suggest modifications to these theories. Conventional f i e l d methods were used. Surface movement, both horizontal and v e r t i c a l , was measured by tr i a n g u l a t i o n of markers i n the ice from f i x e d points on bedrock around the perimeter of the g l a c i e r . Movement at depth was determined by measurements i n boreholes of the change of i n c l i n a t i o n with time. Seismic and gravity measurements of ice thickness were also a v a i l a b l e . The methods of measurement and computation are described and t h e i r accuracy i s assessed. It was observed that the v e r t i c a l v e l o c i t y of the top of the pipe i n each borehole i s equal to that of the i c e i n i t s v i c i n i t y . Methods of analysing borehole data are c r i t i c a l l y reviewed i n the l i g h t of t h i s f a c t . A correction term f o r the curvature of the pipe i s also used i n the analysis. It i s shown that, on the Athabaska Glacier, the longitudinal s t r a i n rate i s not constant with depth, and that, for about 100 metres below the surface, the horizontal v e l o c i t y i s s l i g h t l y greater than i t s surface value. Present i i i theory does not cover these cases. Possible modifications are suggested. The assumption, sometimes made in the past, that the width of a valley glacier can be regarded as i n f i n i t e , i s shown to be unjustified. In the absence of a complete stress and velocity solution for the case of f i n i t e width, the stress solution i s modified by the introduction of the "shape factor" in the stress solution. The relation between the second invariants of the strain rate and stress deviator tensors is compared with the simple power law as determined by laboratory experiments with ice. Comparison i s made both for borehole measurements and measure-ments of change of surface velocity across transverse lines. Agreement is satisfactory, within the limits of experimental error, for a l l the borehole results and some of the surface movement results. This i s interpreted as evidence that the underlying theory i s not seriously in error. In particular, the basic assumptions, made by Nye, that the components of strain rate and stress deviator tensors are proportional, that the constant depends only on the second invariant of the stress deviator, and that the shear stress i s only a slowly varying function of distance down the glacier, seem to be reasonable approximations. / Of three laboratory flow laws, that of Glen for quasi-viscous creep gives the most satisfactory f i t to the data. i v The f i t would be improved i f the mean temperature of the g l a c i e r were about -0.75°C rather than the pressure melting temperature. This point has not been checked because of technical d i f f i -c u l t i e s . The r e s u l t s appear to show that the index i n the power law i s reduced at low stresses ( i . e . less than about 0.5 bar). Other interpretations of the data are possible, however, so the r e s u l t i s not considered to be established. V TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES ix ACKNOWLEDGEMENTS / x i 1. INTRODUCTION 1.1. The Flow of Ice in Glaciers 1 1.2. The Athabaska Glacier 4 2. FIELD METHODS 2.1. General 9 2.2. Surface Movement 9 2.3. Ablation and Accumulation 17 2.4. Ice Thickness 18 2.5. Borehole Measurements 22 2.6. Other Data 27 3. COMPUTATIONS 3.1. Position 28 3.2. Velocity 34 3.3. Slope 37 3.4. Strain Rates 39 3.5. Ice Thickness 42 3.6. Borehole Measurements 43 3.7. Miscellaneous Quantities 49 4. ACCURACY 4.1. General 52 4.2. Position 52 4.3. Velocity 60 4.4. Slope 63 4.5. Strain Rates 63 4.6. Ice Thickness 65 4.7. Borehole Measurements 68 4.8. Ablation and Accumulation 72 v i Page 5. THEORY 5.1. Distribution of Stress and Velocity 74 5.2. Effect of Valley Sides 85 5.3. Reduction of Borehole Data 87 6. RESULTS 6.1. General 91 6.2. Configuration of Surface 91 6.3. Configuration of Bed. 93 6.4. Surface Velocity 93 6.5. Strain Rate 99 6.6. Borehole Results 101 6.7. Ablation and Accumulation 103 7. DISCUSSION 7.1. Analysis of Borehole Data 105 7.2. Validity of Assumptions 110 7.3. Variation of Longitudinal Strain Rate with Depth 114 7.4. Variation of Velocity with Depth 120 7.5. Numerical Check of Equation 18 122 7.6. The Flow Law of Ice 123 7.7. Effect of Valley Sides 132 7.8. Stress and Strain Rate at Boreholes 136 7.9. Comparison of Data with Different Flow Laws 139 APPENDIX 1 Alternative Derivation of Equation 16 145 APPENDIX 2 Generalizations of the Theory 148 BIBLIOGRAPHY 153 TABLES FIGURES v i i TABLES 1. Positions and Elevations of Triangulation Stations. 2. I n i t i a l Positions and Elevations of Markers. 3. Longitudinal Surface Profile. 4. Transverse Surface Profiles. 5. Width of Glacier. 6. Ice Thickness. 7. Surface Slope. 8. Slope of Bed. 9. Curvature of Surface and Bed. 10. Horizontal Surface Velocity. 11. Horizontal Direction of Surface Velocity. 12. Horizontal Velocity at Edge of Glacier. 13. Mean Velocity on Transverse Lines. 14. Vertical Velocity and Normal Velocity. 15. Rate of Thinning of Glacier. 16. Longitudinal Surface Strain Rate. 17. Transverse Surface Strain Rate (measured). 18. Transverse Surface Strain Rate (calculated). 19. Surface Strain Rates at Boreholes. 20. Borehole Data. 21. Slopes, Velocities, Strain Rates at Boreholes. 22. Borehole Results. v i i i 23. Comparison of Mean and Surface Strain Rates. 24. Comparison of Measured and Calculated Strain Rates. 25. Strain Rates and Stresses on Transverse Lines. 26. Strain Rates and Stresses at Boreholes. 27. Ablation and Accumulation. ix FIGURES 1. Map to show location of glacier. 2. Map of glacier (2 sheets) (in pocket at back). 3. General view of glacier. 4. Airphoto mosaic of glacier (in pocket at back). 5. Map of bedrock (in pocket at back). 6. Recession of glacier terminus. 7. Profiles of surface and bed - longitudinal line. 8. Profiles of surface and bed and horizontal surface velocity - line A 9. Profiles of surface and bed and horizontal surface velocity - line B 10. Profiles of surface and bed and horizontal surface velocity - line C 11. Profiles of surface and bed and horizontal surface velocity - line D 12. Profiles of surface and bed and horizontal surface velocity - line E 13. Profiles of surface and bed and horizontal surface velocity - line F 14. Profiles of surface and bed and horizontal surface velocity - line G 15. Horizontal surface velocity - longitudinal line. 16. Vertical velocity and velocity normal to surface -longitudinal line. 17. Longitudinal and transverse strain rates - longitudinal line. 18. Change in configuration - hole 322. 19. Change in configuration - hole 209. 20. Change in configuration - holes 116 and 314. 21. Variation of ^ with depth - hole 322. ty 22. Variation of ^ with depth - hole 209. 23. Variation of with depth - holes 116 and 314. ty 24. Comparison of velocities of pipe and markers. 25. Effect of "shape factor". 26. Strain rate and stress - hole 322. 27. Strain rate and stress - hole 209. 28. Strain rate and stress - transverse lines. 29. Correlation between surface velocity and ice thickness. 30. Comparison of data with flow laws. 31. Ablation on longitudinal line. x i ACKNOWLEDGEMENTS I am particularly indebted to Professor J.A. Jacobs for overall encouragement, to Dr. J.C. Savage for numerous help-f u l suggestions and discussions and for permission to make use of his seismic data and borehole data in advance of pub-lication, and to Mr. J. Fairley for able assistance in the f i e l d . Mr. J.S. Stacey and Mr. J. Fairley were responsible for the overall organization in 1959 and 1960 respectively. The various projects on the glacier have involved some fifteen people from the Universities of Br i t i s h Columbia and Alberta, from practically a l l of whom I have received assistance at one time or another. The work could not have been carried out without the cooperation of the Superintendent of Jasper National Park. Mr. W. Ruddy of Snowmobile Tours deserves special mention for the provision of transport whenever requested and for assistance in numerous other ways. The work was financed by grants from the National Research Council. I am also grateful to the Director, Water Resources Branch, Department of Northern Affairs and National Resources, for survey data and copies of the map. I should also like to thank Mr. J. Allard for writing the computer programme for the reduction of the surface movement data. 1. INTRODUCTION 1.1. The Flow of Ice In Glaciers The f i r s t systematic measurements of glacier flow were made in the Alps about 1830. Several detailed studies followed in the 1840's and 1850's. These showed that the flow of a glacier resembles that of a highly viscous f l u i d . Considerable controversy developed over the actual mechanism of ice flow however. One of the f i r s t attempts at a theoretical interpretation of observations was due to S. Finsterwalder (1897). His "streamline theory" correlated accumulation and ablation with glacier flow, and explained glacier advances and recessions. This theory was mainly qualitative. In 1921 Somigliana put forward a quantitative theory. He assumed that ice behaved like a f l u i d of constant viscosity, and considered i t s stationary flow under gravity in an evenly inclined cylindrical channel. Lagally (1934) extended this theory and predicted the depth of the Pasterze Glacier. This was subsequently confirmed by seismic measurements. The "extrusion flow" theory, put forward independently by Streiff-Becker (1938) and Demorest (1941, 1943), postulated a property of ice quite different from any assumed in previous theories. It was assumed that, under high hydrostatic pressure, the stress required to produce a given deformation in ice was reduced. Thus ice at depth in a glacier should be -2-squeezed out by the pressure of the overlying ice. The velocity near the bed should be considerably greater than the surface velocity. This theory was never expressed in mathematical form. But Streiff-Becker made some f i e l d measurements which could have been explained on this hypothesis (Seligman, 1947). Both these theories proved unsatisfactory. Polyerystalline solids, such as ice, do not behave like Newtonian liquids. The velocity of a glacier i s much more sensitive to small changes in i t s thickness than would be expected on the theory of constant viscosity. Laboratory experiments have shown that hydrostatic pressures higher than any encountered in a glacier, have no effect on the creep behaviour of ice (Rigsby, 1958). Extrusion flow was never observed in several f i e l d experiments where i t would have been expected. (Gerrard and others, 1952; Sharp, 1953a, b; Mathews, 1959; Meier, 1960). The flow law of ice and i t s application to glaciers have received extensive study during the past decade. Laboratory measurements include those of Glen (1952, 1955, 1958a), Steinemann (1954, 1958a), Griggs and Coles (1954), Hansen and Landauer (1958), Butkovich and Landauer (1958), Rigsby (1958), and Mellor (1959). These experiments indicate that the strain rate i s proportional to the third or fourth power of the stress. The constant of proportionality depends on the temperature. The flow law i s not affected by the hydrostatic pressure. There i s no indication of a yield stress below which ice does not deform. At the same time, theoretical calculations of stress and i velocity in glaciers have been made by Nye (1951,; 1952a, 1952b, 1957, 1958a, 1959c, 1960) and others (Weertman, 1958, 1961; Shoumskiy, 1961a, b). In his earliest papers Nye made the simplifying assumption that ice behaves as a perfectly plastic substance. A l l subsequent work has been based on! the laboratory flow law however. i Many simplifying assumptions are of course necessary before results of laboratory experiments can be extended to the much more complex stress systems which exist in glaciers. In spite of this, Nye's theory has been conspicuously success-f u l in explaining many observed features. Rates of deformation of boreholes (Gerrard and others, 1952j Nye, 1957; Mathews, 1959j Shreve, 1961), rates of closure of tunnels (Nye, 1953, 1959a; Glen, 1956), the occurrence of surges of increased flow (Weertman, 1958; Nye, 1958a, 1960), the profile of the Greenland ice cap (Nye, 1959c; Weertman, 1961) are four examples in which there has been substantial agreement between theory and observation. On the other hand, i t has been shown that there are places in a glacier to which the theory does not apply, because the underlying assumptions break down (Nye, 1959a, Glen, 1961). In addition, Nye's formulation of the flow law for complex stress systems has been questioned (Glen, 1958c). At present, theory has tended to outrun the making of detailed f i e l d measurements. The extent to which the 4-assumptions can be expected to hold in a real glacier appeared to need further investigation. The object of the present study was to obtain data for this purpose. It was also hoped to suggest modifications to the theory in the event of i t s being found inadequate. The Athabaska Glacier was selected as a suitable location. It has a simple geometrical shape and i s easy of accesss. The study was modelled on a similar one carried out on the neighbouring Saskatchewan Glacier (Meier, 1960). Particular emphasis was placed on measuring velocity at depth. Con-siderable importance was also attached to measurements of strain rate and vertical velocity at the surface along the centreline of the glacier. The shape of the glacier bed was to be determined in as much detail as possible. 1.2. The Athabaska Glacier The Athabaska Glacier i s one of the main outlet glaciers from the Columbia Icefield. The Icefield l i e s on the Continental Divide and i s surrounded by some of the highest peaks in the Canadian Rocky Mountains. Figure 1 shows the general location of the glacier while Figure 2 i s a detailed map. The Icefield has an area of over 200 sq.Km. and a mean elevation of about 3000 m. The accumulation area of the Athabaska Glacier comprises some 7 sq.Km. of this and attains a maximum elevation of 3456 m. The glacier flows in a north-easterly direction in a steep-sided valley of f a i r l y constant -5-width. The terminus (latitude 52°12*N., longitude 117°14*W.) is within 2 Km. of the Banff-Jasper highway some 100 Km. from Jasper. A branch road runs to the edge of the glacier and snowmobiles for tourists are operated on the ice. This ease of access; made the Athabaska Glacier a very convenient place for glaciological work. Figure 3 i s a general view of the glacier and Figure 4 is an air photo mosaic. The rim of the Icefield l i e s at an elevation of about 2700 m. From here the glacier descends in a series of three i c e f a l l s over a distance of 2 Km. The elevation of the ice surface at the foot of the lowest i c e f a l l is 2300 m. At the terminus i t i s 1920 m. This i s about 120 m. higher than the terminus of the neighbouring Saskatchewan Glacier. The section from the foot of the i c e f a l l s to the terminus w i l l be referred to as the lower section. Almost a l l the glaciological work was done here. The length of this section is 3.8 Km.; i t s width 1.1 Km. The width varies only slightly, and the only bend i s a slight one about 1 Km. from the terminus. The slope of the lower section i s generally between 3° and 5° but steepens to about 15° at the terminus. The western half of the terminus ends in a glacial lake and is somewhat steeper. The ice velocity decreases from about 80 m./yr. just below the lowest i c e f a l l to 15 m./yr. at the terminus. The time taken for ice to travel from the Icefield to the terminus i s of the order of 150 years. L i t t l e information i s available about the thickness of -6~ ice in the upper section of the glacier. Isolated seismic measurements give 220 m. in the accumulation area, 92 m. near the centreline below the head wall (marker S5 in Figure 2), and 195 m. on the centreline below the second i c e f a l l (Ll). For 2 Km. down glacier from the lowest i c e f a l l , ice thicknesses on the centreline are in the range 250 m. to 320 m. Bedrock over almost 1 Km. of this section i s below the level of the glacial lake at the terminus. Down glacier from this there are two rises in the bed. The ice then thins rapidly towards the terminus. Seismic measurements show that the cross-section of the valley i s roughly parabolic. But there are suggestions of a shelf on the south-eastern side for a distance of about 1 Km. below the i c e f a l l s . Figure 5 i s a map of the bedrock. The f i r n limit l i e s about half-way up the highest i c e f a l l at an elevation of about 2500 m. The ablation area i s about 6 sq.Km. in extent. The lower section of the glacier i s clear of snow by early July in an average year. Annual ablation in this section averages about 4 m. of ice. Glaciers in this area are generally assumed to be temperate9 but no measurements are available to confirm this. The fact that the ice i s broken up in three i c e f a l l s should help to bring i t to the melting temperature i f i t were originally colder. Copious amounts of meltwater flow from under the terminus throughout the summer. Both sides of the glacier are heavily covered with -7-debris. The zone on the south-east side i s about 100 m. wide, that on the north-west side about 200 m. There are conspicuous old lateral moraines on each side of the lowest kilometre of the glacier. They rise up to 100 m. above present ice surface. There are a series of terminal moraines between the terminus and the highway. There are two tributary glaciers on the south-east side of the lower section. Neither i s joined to the main glacier although the upper one contributes a small quantity of avalanche debris. The lowest point of the bed of the main glacier l i e s a very short distance down from the upper tributary. The lower tributary also seems to correspond with a hollow in the bed. The f i r s t recorded v i s i t to the glacier was that of Stutfield and Collie in 1897 (Stutfield and Collie, 1903, p. 103-122). Fluctuations of the glacier over the past two centuries have been deduced by study of moraines and growth rings on nearby trees (Field and Heusser, 1954, p. 135$ Heusser, 1956, p. 282). Since 1945, parties from the Water Resources Branch, Department of Northern Affairs and National Resources, have made regular surveys of the terminus (Collier, 1960). These studies indicate that the glacier advanced during the early 18 t h century. By 1714 the terminus was further forward than at any time for at least the previous 350 years. This position corresponds very roughly with the -8= line of the present highway. Retreat started about 1720, but another advance in the f i r s t half of the 19* n century brought the glacier back almost to i t s maximum position. Recession began between 1841 and 1873 at different parts of the front, and s t i l l continues. Terminal moraines indicate temporary halts about 1900, 1908, 1925, and 1935. Recession since 1873 has totalled 1150 m. or an average rate of 12.5 m./yr. Since 1945 the rate has averaged 27 m./yr. There i s no sign of any halt. The behaviour of the Athabaska appears to be typical of glaciers in the area. Recession data are summarized in Figure 6. -9-2. FIELD METHODS. 2.1. General Research on Athabaska Glacier was a joint undertaking of the University of Brit i s h Columbia and the University of Alberta. The work was under the overall supervision of Professor J.A. Jacobs and Professor G.D. Garland, and was financed by grants from the National Research Council. Mr. J.S. Stacey was responsible for organization and was in charge in the f i e l d in 1959. These tasks were the responsi-b i l i t y of Mr. J. Fairley in 1960. Stacey was responsible for deep d r i l l i n g in 1959. Dr. J.C. Savage was in charge of d r i l l i n g and inclinometer measurements in subsequent seasons. The present author was responsible for surface movement studies. The observations were made by him and Fairley. 2.2. Surface Movement 2.2,1 General The surface movement survey employed conventional methods. The positions of markers set in the ice were determined periodically by triangulation from stations on bedrock. When a large part of a glacier has to be covered, triangulation i s much more rapid and convenient in the f i e l d than taping and levelling. The latter method is suitable for detailed study of a small area. Provided that stations are sited on bedrock and the observers are experienced, the =10 accuracy of t r i a n g u l a t i o n i s at least equal to that of taping and l e v e l l i n g . 2.2.2. Triangulation Stations The survey was c a r r i e d out from a network of 21 stations around the perimeter of the lower section of the g l a c i e r . These were set up during the summer of 1959 by a party from the Water Resources Branch, Department of Northern A f f a i r s and National Resources. The stations served as ground control f o r the making of a map from a i r photos. This work has been described by Reid (1961). Most stations were on bedrock but a few had to be s i t e d on moraine. Only one proved unsatis-factory due to lack of s t a b i l i t y . A stake supported i n a c a i r n was erected over the s t a t i o n mark when i t was used as a reference object i n the survey. 2.2.3. Observing Procedure A l l observations were made with a Wild T2 theodolite. Angles were read to the nearest second. Markers were observed i n rounds of s i x or seven, and each round was closed on the reference object. Horizontal angles were observed once on each face and further readings taken i f the f i r s t two d i f f e r e d by more than 5 seconds. V e r t i c a l angles were observed twice on each face. The two observers generally took alternate rounds. Each marker was observed from three stations and a few remote markers from four or f i v e . Except i n a very few instances, a l l observations to "11-one marker were made on the same day. There were intervals of at most 5 or 6 hours between observations from different stations. The survey was planned to obtain as satisfactory inter-sections as possible on each marker (ideally three rays intersecting each other at 120°). But the time factor made i t necessary to keep the number of stations visited on any one day to a minimum (generally 4). Observations were made between 7 a.m. and 7 p.m. Height of instrument above station mark was measured to the nearest 5 mm. 2.2.4. Markers Choice of markers followed the recommendations of Ward (1958). The markers were wooden stakes of 15 mm. square cross-section, 2.6 m. long. The wood was ramin, a S.E. Asian hardwood. A cloth flag was attached to the top of each stake. The markers were set in holes of circular cross-section d r i l l e d in the ice with a modified Ward-type auger of 32 mm. diameter (Ward, 1958). The holes were about 2.3 m. deep. This depth meant that the stakes had to be reset several times each season. But the start of the movement survey would have been considerably delayed i f 5 m. holes had been d r i l l e d i n i t i a l l y . Resetting consisted of deepening the existing hole. The horizontal position of the marker was thus unchanged. The stakes had to be trimmed to f i t the holes and were hammered into position. Observations were =12= always made to-the centre of the top of each marker. These markers were not entirely satisfactory. I n i t i a l l y they f i t t e d tightly in the holes. After a few days however the upper parts of the holes tended to enlarge and the markers became loose. The only solution was to keep d r i l l i n g the holes deeper, and loose markers were always reset before the start of each survey. The t i l t of any marker which was leaning during a survey was determined by measuring the length of stake above the ice surface and the vertical distance from the top to the surface. The approximate direction was determined in the few cases where i t differed from the direction of maximum surface slope. 2.2.5. Siting of Markers No prior information about velocity was available. The survey thus had the i n i t i a l object of obtaining an overall idea of the surface movement. In i c e f a l l s access, maintainance of markers, and interpretation of movement measurements are a l l d i f f i c u l t . Nor can ice thickness be measured with any degree of accuracy in heavily crevassed areas. Work was therefore largely confined to the lower part of the glacier. The general arrangement of markers is shown in Figure 2. Markers in the lower part of the glacier were arranged in one longitudinal and six transverse lines. The longitudinal line consisted of 30 markers (numbered L10 to L39) and extended from the foot of the i c e f a l l to the terminus. -13 Markers were either about 90 or 150 m. apart. The wider spacing was used where surface curvature was small. The aim was to place the longitudinal line along the centreline, i.e. line of greatest velocity. Its position was determined from velocity measurements over a period of one week. Subsequent measure-ments showed that the longitudinal line was about 50 m. towards the north-west edge of the glacier from the centreline. The six transverse lines, labelled B to G, contained from 6 to 10 markers each, a total of 42. Spacing between the lines was of the order of 500 m. Spacing between individual markers varied between 40 and 200 m. The lines extended into the debris-covered ice at each side of the glacier. Nine markers were placed in the relatively f l a t area between the lower two i c e f a l l s . They were arranged in a line extending about half-way across the glacier from the south-east side (markers Al to A7), plus two markers (LI, L2) near the centreline. Large crevasses prevented the extension of this line right across the glacier. Fifteen markers (Jl to J12, "B", "D", "E") were placed in the upper part of the glacier to obtain a rough idea of the movement in this area. Boreholes, which w i l l be described later, were also included in the movement survey. -14-2.2.6. Observation Periods A l l markers except J l to J12 were positioned in the f i r s t half of July 1959. Surveys were made between the following dates; July 19 - July 22 August 13 - August 18 August 30 - September 2 Markers were reset as necessary and a l l were reset immediately before the f i n a l survey. About 90% of the markers were s t i l l in position the following summer. Those lost were in the A line and the G and L lines near the terminus. The markers were reset, and surveyed between July 22 and July 25. Thereafter attention was concentrated on the O and L lines and a few other markers. The D line was surveyed on July 31, August 7, and August 13. The other markers were surveyed between August 12 and 14. The markers in the upper part of the glacier were set up during summer 1960 and surveyed twice over a period of 11 days. The glacier was visited again on November 13-14, 1960 and a further survey of the D and L lines was made. Further v i s i t s were made in January and April 1961. Bad weather prevented any useful observations in January, but 10 of the L line markers were resurveyed on April 10. The remainder were buried by snow. Most surveys included observations of surface movement at the boreholes. -15-Surface movement observations in July 1961 were confined to a single survey of the positions of the boreholes, the three J markers which survived the winter, and marker A7 which had travelled intact down the lowest i c e f a l l . 2.2.7. Strain Rate The three independent components of the strain rate tensor at the surface in the region surrounding each Worehole were measured by the method of Nye (1959b). Pour stakes were arranged in a square with one diagonal along the centreline and the borehole at the centre. The length of the sides was roughly 150 m. Ideally, the length should be made equal to the ice thickness. The surface was not uniform over an area of this size however, so a smaller square was chosen. The lengths of the sides and diagonals of each square were measured with a 200 foot (61 m.) steel tape. A nail was inserted in the head of each stake to serve as an accurate mark. To form intermediate markers an ice auger was d r i l l e d into the ice and i t s handle then removed. Each leg was measured once in each direction, and further measurements made i f the f i r s t two differed by more than 0.05 foot (1.5 cm.). It was not considered necessary to correct for temperature or sag. A spring balance was not used, so correct tension was merely estimated. This method measures strain rate with reference to axes parallel to the ice surface. Strain rates deduced from the -16-triangulation refer to horizontal axes. Strain rates around borehole 314 were measured between August 1959 and July 1960. For the other boreholes the period was July 1960 to July 1961. 2.2.8. Slip past Side Walls The rate at which the glacier sli p s past i t s side walls was measured by a method similar to that of Glen (1958b). Two stakes, some 12 m. apart, were d r i l l e d into the ice about 1 m. from the edge of the glacier. A mark was painted on bedrock about half-way between the stakes. The three sides of the triangle were measured with a steel tape. Measurements were made twice in 1959 and once in 1960. In Glen's method, one stake i s placed in the ice and two marks on the rock. The present method has the advantage that longitudinal strain rate can be measured in addition to velocity. But the direction of movement is undetermined. The stakes must be placed in the estimated direction of flow. Unfortunately there are very few places at the sides of the glacier where the ice-rock interface i s exposed. The only suitable locations were on either side of the lowest i c e f a l l . 17-2.3. Ablation and Accumulation 2.3.1. Ablation As the mass balance of the glacier was not being studied, accurate or extensive measurements of ablation and accumulation were not required. Ablation was measured by periodic measurements of the length of each movement stake protruding above the ice surface. Any snow lying on the ice was removed. Measurements were made to the nearest 0.5 inch (1.25 cm.). Measurements were made five times, at intervals of about 10 days, during summer 1959. These figures show the variations in ablation during the season. They do not give a total figure however, as ablation had started before a l l the markers had been set up. There was at least a foot of fresh snow on the glacier when the last set of measurements were made. It i s unlikely that any further ablation of ice took place in 1959. Measurements were also made to the ice surface at those markers which were visib l e above the snow in April 1961. This represents the level of the ice surface at the end of the 1960 ablation season. Total ablation data for the 1960 season are thus available for these markers. The length of each stake above the surface was also measured immediately before and after each r e d r i l l i n g , so that survey and ablation results could be corrected. -18-2.3.2. Accumulation Few accumulation data were obtained. The net winter's accumulation up to April 1961 was measured by digging down to the ice surface at the fifteen markers which were visi b l e . A l l other markers were completely buried by snow. This sample w i l l therefore give a low estimate of the mean accumulation. Three markers (J6, J8, J9), situated on the rim of the Columbia Icefield just above the highest i c e f a l l on the glacier, survived the winter. The net accumulation over the period August 1960 to July 1961 i s known for them. In August 1960, a snow pit was dug on the Icefield at the crest between the drainage basins of the Athabaska and Saskatchewan Glaciers. The previous summers' layer was distinguished by a faint dirt band. So the net accumulation for the previous year i s known at this point. 2.4 Ice Thickness 2.4.1. Seismic Method The seismic work was carried out by Savage and Chisholm using standard exploration procedures. A 12-trace high-resolution seismograph manufactured by Houston Technical Laboratories (now Texas Instruments) was used. Each of the 12 traces was recorded twice, once with mixing (50% to the outside) and once without. The records therefore showed 24 traces. Three 27 c.p.s. geophones were used on each trace. The interval between traces on the cable was 15.25 m. The -19= standard shot was a half stick of 60% Forcite set in a water-f i l l e d d r i l l hole at a depth of 3.5 m. Occasionally a pattern of two or three such shots was used to overcome surface wave interference. The most satisfactory f i l t e r setting was found to consist of passing the 70 to 140 c.p.s. band. Thirty-six reflection sites were occupied. At each, at least one spread transverse to the glacier and one longitudinal spread were shot. Usable records were obtained at a l l but four sites. The usable sites included 16 points (including two boreholes) on the longitudinal line, 8 markers in trans-verse lines, and one other borehole. The remaining 7 sites did not correspond to points in the movement survey. As a general rule high quality records could be obtained for ice thicknesses in excess of 200 m. The records became marginal at thicknesses less than 130 m. Reflections were obtained from ice thicknesses less than 100 m. only with the greatest d i f f i c u l t y . A 2 Km. refraction line was also shot. The velocity of the P wave was determined to be 3660 + 60 m./sec. with no indication of a variation of velocity with depth. The velocity in the glacier bed was determined to be 4500 m./sec. This latter velocity i s typical of a competent bedrock. It should be noted however that the presence of a thin low velocity layer of morainal material i s not excluded by refraction results. -20-The reflection records were interpreted for both depth and dip of the reflecting section by standard procedures. In general, a l l readings were made at the centre of the f i r s t well-developed trough in the reflection group. A study of the best reflection records indicated that this trough followed the beginning of the reflection group by 7 m i l l i -seconds. Several reflection records were shot with the geophone spread doubled back on i t s e l f so that the geophones of traces 1 and 12, 2 and 11, 3 and 10, etc. were side by side. Traces 1 to 6 were unfiltered and 7 to 12 f i l t e r e d at the standard setting. In this manner the f i l t e r delay was determined to be 7 milliseconds. Thus the actual travel time was then the time read from the record, reading at the f i r s t trough of the reflection group, less 14 milliseconds. 2.4.2. Gravity Method This work was carried out by Mr. E.R. Kanasewich and has been f u l l y described (Kanasewich, 1960). Observations were made with a Worden gravimeter at 127 stations. These included a l l the surface movement markers. A three-dimensional analysis was used, and terrain corrections were applied out to 12.4 Km. The estimated accuracy of these ice thickness measurements i s -10% +15%. 2.4.3. Boreholes As w i l l be described in the next section, several boreholes were d r i l l e d in order to measure velocity-depth -21-profiles. Four of these reached bedrock and so provided accurate measurements of ice thickness. The approximate locations were as follows (the number of each borehole i s i t s depth in metres). Hole 314 - midway between L17 and L18 Hole 322 - 150 m. down glacier from C7 Hole 209 - at L29 Hole 194 - at L30 . In August 1960 three boreholes were d r i l l e d to bedrock by a we l l - d r i l l i n g crew under contract to the Alberta Research Council. The approximate locations were: Hole 250 - midway between C3 and C4 and 20 m. down glacier Hole 235 -5m. down glacier from D3 Hole 73 - 180 m. up glacier from G2. In July 1961 a borehole (Hole 248) was d r i l l e d near the position occupied by L27 in 1959 to check the interpretation of the seismic records at this point. It was not certain whether this hole reached bedrock. The depth of the hole was therefore regarded as a minimum value of ice thickness. Further examination of the seismic records then indicated a thickness somewhat greater than the borehole depth. The locations of a l l boreholes are shown in Figure 2. -22-2.5. Borehole Measurements 2.5.1. D r i l l i n g Technique The holes were d r i l l e d by electrically-heated thermal boring elements called "hotpoints". The design was developed by Stacey and has been described by him (Stacey, 1960). Power was supplied by a portable "Homelite" gasoline-driven motor generator. It supplied 2.5 Kw. at 230 volts and weighed 64 Kg. The output voltage could be varied between 0 and 230 volts by a rheostat in the f i e l d c i r c u i t of the generator. D r i l l i n g speeds averaged 6 to 7 m./hour. Speeds of 20 to 25 m./hour have been reported with another type of thermal d r i l l (Nizery, 1951), but this required 10 Kw. of power. The hole was lined with aluminium pipe. The hotpoint was attached to the lower end of the pipe. The external diameters of pipe and hotpoint were 4.2 and 5.1 cm. respectively. The pipe served as one conductor; an insulated cable inside the pipe as the other. A weak joint at i t s lower end enabled the power cable to be withdrawn when d r i l l i n g was completed. The hotpoint i t s e l f was lost however. The holes were d r i l l e d vertically, rather than normal to the glacier surface. An estimate of the ice thickness at each d r i l l site was available from the seismic data. When the hotpoint stopped d r i l l i n g at a depth greater than this, i t was kept running for several hours. If no further progress were made, i t was concluded that bedrock had been -23-reached. The pipe was then l i f t e d one or two metres off the bed. One problem encountered was that the pipe tended to become seized by the ice about 7 m. below the surface. This corresponds to the depth of the previous winters* "cold wave" in the ice. This d i f f i c u l t y has also been reported by Meier (1960, p. 32). It was overcome by d r i l l i n g a preliminary hole to this depth, lining i t with a pipe open at the bottom, and pouring down antifreeze periodically. The hole was then d r i l l e d beside this pipe. The f i r s t holes were lined with aluminium pipe of 3.5 cm. internal diameter. This l e f t a clearance of only 1.5 mm. between pipe and inclinometer. Thus i t proved d i f f i c u l t to lower the inclinometer down the pipe. Pipe of 4.1 cm. internal, 4.8 cm. external diameter was used after the f i r s t season, apart from one hole which was lined with the remaining narrow pipe. Increase in pipe diameter necessitated modifications in hotpoint design, and the external diameter was increased to 5.7 cm. The pipe was supplied in 3.05 m. lengths. These were joined by screw couplings. The external diameter of the couplings exceeded that of the pipe by 0.8 cm. Various methods of sealing the couplings were tried. None of these proved completely satisfactory. The pipes were generally kept f i l l e d with a solution of antifreeze and water. Water in the pipe appeared to cause failure of hotpoints however, -24-so attempts were made to keep two of the later holes (holes 209 and 194) dry. At the end of the season several gallons of antifreeze were poured down each pipe and the pipes were capped to prevent the entry of snow. These precautions proved sufficient to prevent the pipes from being badly frozen up by the following summer. Plugs of ice serveral metres thick were encountered, generally near the surface or near the bottom. This ice was melted out by a hotpoint designed by Savage for use inside the pipe. This hotpoint did not perform as expected and was barely adequate to clear the small amount of ice encountered. 2.5.2. Inclinometer The inclinometer was rented from the Parsons Survey Company. It was of the single-shot type. The positions of a compass needle and a pendulum inside the instrument are recorded photographically after a preset interval of time. Inclinations to the vertical of up to 4° can be measured to an accuracy of 0.1°. By using shorter pendulums, inclinations up to 10° or 26° can be measured with the same relative accuracy. Azimuths can be read to 1°. During the f i r s t two summers water tended to leak into the instrument and cause damage. But a leather washer which proved to be watertight was f i n a l l y f i t t e d . The inclinometer was lowered down the pipe on a stainless steel wire. Distance was determined by measuring the length 25-of wire paid out. This was done along a base line with marks at 25 foot (7.62 m.) intervals which was la i d out along the glacier surface from the borehole. 2.5.3. Acid Bottles After one year, one pipe had become so badly bent near the bottom that the inclinometer would not pass. Acid bottles were improvised to make the measurements. Glass bottles about 10 cm. long were half f i l l e d with hydrofluoric acid and f i l l e d up with o i l . This prevented the hydrostatic pressure of water in the pipe from breaking the glass. No appreciable etching of the glass should occur during the few minutes necessary for lowering the bottle. A suitable acid concentration had therefore to be determined by experiment. The bottles were l e f t stationary in the pipe for 20 to 30 minutes. The accuracy of this method was tested by making measurements at depths where readings could also be made with the inclinometer. Readings agreed within 1°. Acid bottles do not provide azimuth measurements. 2.5.4. Boreholes In 1959 holes were d r i l l e d to depths of 198, 228, and 314 m. Only the last reached bedrock. The others fa i l e d because hotpoints burnt put. Six hotpoints were used during the season. The aluminium pipe was removed from the 198 and 228 m. holes. An inclinometer survey was attempted in the 314 m. hole but results were unreliable. During this survey -26-the pipe became permanently blocked at a depth of 50 m. by a chain which was used to add weight to the inclinometer. In 1960 four holes were d r i l l e d to depths of 322, 209, 194 and 116 m. The f i r s t three reached bedrock. Hole 322 required four hotpoints, hole 116 two, and the other two holes, which were kept dry during d r i l l i n g , one each. The locations of these holes have been given in section 2.4.3. and are shown in Figure 2. Inclinometer surveys were made in holes 322, 209, and 116, and to a depth of 50 m. in hole 314. The spacing between readings was generally 15.24 m. The spacing was 7.62 m. in the top 50 m. of hole 322 and in some sections of hole 116, however. Within five days after completion of the d r i l l i n g of hole 194, ice formed within the pipe. An attempt to clear i t with a hotpoint supplied by W.H. Mathews fa i l e d after 50 m. had been cleared at an average rate 0.5 m./hr. Cause of failure was the formation of ice above the hotpoint which prevented i t s removal. In 1961 only one hole (hole 248) was d r i l l e d . Its sole purpose was to determine ice thickness, so the hole was not cased. Inclinometer surveys of holes 322, 209, 116 and 314 (down to 50 m.) were made at intervals of 7.62 m. Sections of pipe had to be removed to compensate for the lowering of the ice surface by ablation. But observations were made at the same points in each pipe in successive years. It i s hoped to repeat the surveys in 1962. 27-2.6. Other Data 2.6.1. Map The Water Resources Branch of the Department of Northern Affairs and National Resources has produced a map from their survey and air photographs made in 1959. This i s the map used for Figure 2. The map has a scale of 1:4800 and a contour interval of 10 feet (3.05 m.) below 7800 feet (2377 m.) and 40 feet (12.2 m.) above. A copy i s included in the report of Reid (1961). The Branch plans to make a new map every three years. The slope of the glacier surface at the survey markers and boreholes has been determined from this map. 2.6.2. Recession Data The Water Resources Branch made surveys of the terminus every year between 1945 and 1950, and every second year there-after. The surveys consist of a plane table survey of the position of the terminus, a vertical profile and a horizontal velocity profile across a line some 400 m. from the present terminus. The report of the 1960 survey also contains data from a l l previous ones (Collier, 1960). 2.6.3. Streamflow Records The Water Resources Branch also have flow records for the Sunwapta River at i t s exit from the lake at the terminus of the glacier. These records date back to 1948. -28-3. COMPUTATIONS 3.1. Position 3.1.1. Horizontal Position Horizontal positions were computed in terms of a rectangular coordinate system. The origin was taken at Triangulation Station 1 of the Water Resources Branch's survey. Its position was determined from a large scale topographic map to be 52°13'12"N., 117°13'30"W. One axis was taken along the survey base line which ran from station 1 to station 3. The azimuth of this direction was 189°25' measured clockwise from true north. The magnetic declination in 1960 was 24° £. The unit of coordinates was taken as 1 mm. The origin was given false coordinates £ » 2 000 000 N - 2 000 000. The coordinates of each of the 21 stations, and a l l azimuths between stations which were required, were computed from the results of the Water Resources Branch's survey. The f i e l d observations from each station were reduced as follows. For each marker, the mean of the face l e f t reading, and the face right reading plus 180°, were taken. This angle was converted to an azimuth by the relation: Azimuth of marker = (Azimuth of reference station) + (observed angle to marker) -(observed angle to reference station). -29-Subsequent calculations were carr i e d out on the University of B r i t i s h Columbia's Alwac III C computer. The programme was written by Mr. J. A l l a r d . Each marker was observed from three st a t i o n s . The stations were taken i n pa i r s i n the three possible ways. The coordinates of the marker were determined from each pair of observations by the formulae: Suffixes A, B, P r e f e r to the two observation stations and the marker respectively. The stations must be taken i n such an order that P l i e s to the r i g h t of the d i r e c t i o n A to B. A, B i n the formulae are the azimuths to P from stations A and B respectively. E, N are coordinates. Alternative formulae are E coi- A - E cof- B - NA -f MR coh / \ - col" B N -r £ B Ian A - Ian B &A - ( MP " N j fan A 30-It i s generally more accurate to use the tangent formula for angles between 0° and 45° and the cotangent formula for angles between 45° and 90°. But for ease of programming only one formula, the cotangent one, was used. The accuracy of the computer calculations was s t i l l more than adequate. The computer output consisted of three pairs of coordinates for each marker. If the spread of the three values was less than 10 cm., their mean was taken. Otherwise the three points were plotted and the centre of the inscribed c i r c l e of the triangle was taken as the position. A position was rejected i f the radius of the c i r c l e represented a distance of more than 25 cm., unless there was a sound reason for rejecting the observations from one station. In this latter case the intersection of the rays from the other two stations was taken as the position. In two or three instances, observations to a marker from one station were made a day later than the observations from the other two stations. In this case the ray from the station in question in the intersection diagram was displaced parallel to i t s e l f by an amount equal to the movement of the marker in one day. This movement was determined for another observation period. The computer output consisted of 10 figures. Station coordinates were 7 figure numbers (mm.), but coordinates of markers were rounded to 6 figures (cm.) for subsequent calculations. -31-Where necessary, coordinates were corrected for the amount the marker was leaning at the time of the survey. The computer also calculated the distance from each station to the marker, for each pair of rays. These distances were used in the height computations. In a few instances a marker was only observed from two stations. No estimate of the accuracy can be obtained in this case. The height calculations were therefore used as a check. The heights of the marker as computed separately from the two stations were compared. If they did not agree within 10 cm. i t was considered that the calculated position might be inaccurate, and i t was rejected. In November 1960, some markers were only observed from one station. It was necessary to assume a direction of movement in order to calculate the amount which the marker had moved since the previous observation period. The direction assumed was that for the period August 1959 to July I960. The amount of movement d was calculated from f, ( e A - e,) -*'n ( A - <9a) where O^, %^ a r e t n e azimuths of the marker from the station as observed in August 1960 and November 1960 respectively. A i s the azimuth of the assumed direction of movement, r^ i s the distance from station to marker -32-in August 1960. For the height calculations i t was also necessary to determine r 2 , the distance in November 1960. This was given by r 2 = r-^  + &t with 6 t = r, ( 0 a - £ , ) Co/ " ( 0 a - A ) ' Velocities determined by this method were naturally less accurate than the others. 3.1.2. Vertical Position Calculations were carried out in millimetres and results were rounded to centimetres. To reduce the number of figures, 6000 feet (182880 cm.) was taken as datum. A l l heights quoted are in centimetres above datum. The theodolite readings were converted to angles from the horizontal. The four readings (two on each face) were meaned. The computer input consisted of this mean and the height of instrument above datum. Distance from station to marker was computed as explained in the previous section. A standard correction for atmospheric refraction and earth's curvature was used. The value was 6.755 x 10" 1 1 r 2 where r is the distance in millimetres from station to marker. This corresponds to the value of 13.6" per Km. which i s generally used by surveyors. The formula used was h = h ' - r 0 + 6 • 755 « lo'"-r2 -33-where h » height in mm. of top of marker above datum h f «• height in mm. of instrument above datum 0 - observed angle of depression (positive) or elevation (negative). The horizontal observations from the three stations were taken in pairs as explained previously. Each pair yielded two heights (one from each station). A total of six heights for each marker was thus obtained. If the spread of the six was not greater than 10 cm. the mean was taken. If the spread was greater than 10 cm. the separate heights were examined. If the observations from any station were made under conditions which might suggest anomalous refraction (observations late in the day from a station near the ice for example), these observations were rejected. This happened on a very few occasions. Otherwise a l l the measure-ments were rejected and the height was l e f t undetermined. Heights of markers which were leaning were corrected. Heights were also corrected for r e - d r i l l i n g of holes. Movement studies thus refer to the element of ice which was at the foot of each marker at the start of the 1959 season. Observations to any one marker from three stations were regarded as simultaneous i f they were made on the same day. In fact they were separated by several hours. The errors introduced by this procedure are insignificant. -34-3.2. Velocity 3.2.1. Horizontal Velocity The magnitude (U) and direction (A) of the mean velocity of a marker were calculated from i t s coordinates at successive observation periods. This velocity i s measured in a horizontal plane. Horizontal velocities are much greater than vertical ones, and the slope of the glacier seldom exceeds 10°. Thus horizontal velocities do not dif f e r significantly from velocities measured in the plane of the glacier surface. 3.2.2. Vertical Velocity Two velocities must be distinguished. 1. Velocity V relative to a vertical axis. 2. Velocity v relative to an axis normal to the glacier surface at the point in question. The downward direction w i l l always be taken as positive. Velocities refer to a particle of ice just below the surface of the glacier (at the foot of each marker), v i s not the velocity of ri s e or f a l l of the surface, because the surface level changes as a result of ablation as well as of ice flow. The mean value of V between successive observations was calculated directly from the observed change in height above datum of the top of the marker. The mean value of v was calculated from the formula: = V cf - U cos A A 0^ 35 a g i s the angle of maximum surface slope at the marker, taken positive. AA is the angle between the directions of movement of the marker and of the corresponding point on the centreline. The angle i s measured in a horizontal plane. 3.2.3. Velocity of Slip past Side Walls In the diagram, P i s a fixed mark on rock; A l * A2 a r e * h e Positions of one marker in the ice at successive times; B-^ , B 2 are corresponding positions of the other marker. It i s assumed that A^ A 2 B^ B 2 i s a straight line. al> a2» b^, b 2, A1 B1» A2^2 a r e m easured. Angles A^ and B 2 are calculated from triangles A^B-jP, and AgBgP. A 1A 2 (=A^ 1 ) , B-jB2(= A-^) > t h e distances moved by the stakes, are then calculated. 3.2.4. Velocity Averaged over Width of Glacier It was necessary to calculate this quantity in order to use one method of estimating the transverse strain rate. The mean velocity was calculated for each transverse line of markers. Measurement of the width of the glacier i s treated -36-i n section 3.7.2. The v e l o c i t y i s comparatively small near the edges of the g l a c i e r . Small errors i n width therefore have l i t t l e e f f e c t on the mean v e l o c i t y . For each transverse l i n e , observed v e l o c i t i e s were plotted against distance across the g l a c i e r . V e l o c i t i e s at points 100 m. apart were determined by interpolation. The mean v e l o c i t y was calculated by numerical integration by the repeated Simpson Rule. For 11 points t h i s i s i n question. 3.2.5. V e l o c i t y Averaged over Thickness of Glacier To make an accurate c a l c u l a t i o n of t h i s quantity, the d i s t r i b u t i o n of v e l o c i t y with depth must be known. This i s only known at boreholes. However, the mean v e l o c i t y i s not very s e n s i t i v e to the v e l o c i t y d i s t r i b u t i o n . If i t i s assumed that the flow i s laminar and obeys the flow law determined i n laboratory experiments, i t can be shown (Nye, 1952b, p. 84) that where u Q , u^, u 2 • • are the v e l o c i t i e s at the points = B TI + I •a -u where u i s v e l o c i t y at depth y u, s i s surface v e l o c i t y B i s a constant n = 3 or 4 i s the index i n the flow law. -37-It follows that U - = 7" n 3 n where u i s the v e l o c i t y averaged over the t o t a l thickness h. rv + I = K h Also ^ - ia« Bwhere u^ i s the v e l o c i t y at the bed. Thus ^ s " <*b 7 1 + 2 For 71-3, i f u b «• 0 u » 0,8 u g u b " i u s " = °» 9 u s u b - u g u - u s u = 0.9 u g was taken as a representative value. 3.3. Slope 3.3.1. Surface Slope There were two possible methods of determining the slope of the i c e surface at markers on the long i t u d i n a l l i n e . 1. The height of the top of each marker above datum and above the i c e surface, and the distances between markers were a l l known. The slope at a marker was taken as the mean of the slopes between i t and i t s two adjacent markers. 2. The slope was measured from the map. With t h i s -38-method, the distance over which the slope was measured could be varied* In addition, this i s the only method available for markers off the centreline. For these reasons map measure-ment was used throughout. For purposes of analysis the mean slope over a distance comparable with the ice thickness was required. Distances were thus about 250 m. Shorter distances were taken near the sides of the glacier and where the surface slope was changing rapidly. The map distance was always measured between an integral number of (3.05 m.) contours. Slope was always measured in the direction of maximum slope. The map was made in 1959. Slopes at the positions of the markers in a l l three seasons were measured from i t . This procedure can be jus t i f i e d because the survey showed l i t t l e change in the surface configuration of the glacier, along the centreline, over the three seasons. 3.3.2. Slope of Glacier Bed The slope of the bed in both longitudinal and transverse directions was determined from the seismic records at some 30 points. These included holes 322, 314, and 2097and 14 other points on the longitudinal line. The slope could have been calculated by numerical differentiation at other points on the longitudinal and transverse lines. These data were not required however. -39-3.4. Stra i n Rates 3.4.1. Stra i n Rates around Boreholes The lo n g i t u d i n a l s t r a i n rate £.x , the transverse s t r a i n rate £ z , and the shear s t r a i n rate £ x z were determined from measurements of the squares of stakes around each bore-hole. These s t r a i n rates a l l r e f e r to the plane of the g l a c i e r surface. The method i s due to Nye (1959b). X. / 2. / " X It \ / ^ lengths Of the sides dl feehon Let x x, x 2, z i , z 2 be the lengths of the up-glacier, down-glacier, and two transverse diagonals respective-l y , and € x , i z > Let A denote the change i n length between two observation periods. The quantities 90 at A xA i a t A z , , A z : £ = 4 I3S a t A fa I, A I, were calculated (t years i s the time between observations). -40-The best estimates (in a least squares sense) of strain rates are then I3i" J 1 1 c _ -L P _i [_ c fe = X % J A 135* X 2. The standard errors of the estimates are 7 3 | r I , J3 IR \> JI | R | respectively, where R = J- ( £ o + ^  - ^  - £ | 3 y ) Measurements were made over a period of approximately one year. The sides of the squares were between 120 and 180 m. long. The measurements were not quite complete as some stakes were lost by ablation. 3.4.2. Longitudinal Strain Rate The longitudinal strain rate at the surface between each pair of markers on the centreline was calculated from • -a • 3Q2b j £ i.x i s the strain rate averaged over the time t years between successive observations. ^2 a r e distances between the markers at the beginning and end of the period. This strain rate refers to a horizontal axis. When | £ x | was less than 0.02 per year the following -41-formula was sufficiently accurate: * 4 - 4 3.4.3. Transverse Strain Rate Transverse strain rates £. between adjacent markers were calculated for each transverse line by the formula of section 3 o 4 o 2 • The transverse strain rate, averaged over the width of the glacier, was determined at each marker in the longitudinal line by Nye's formula (Nye, 1959c, p. 506), ^ U ctuj z ul) doc w i s the width of the glacier x i s distance measured along the centreline U i s velocity averaged over depth and width . A value of U - 0.75 U s was taken, with U s the surface velocity at the centreline. This combines a velocity of 0.85 U s averaged over the width (the average value for the six transverse lines), and 0.9 U g averaged over the thickness. This last value was derived in section 3.2.5. {JL iaJ To calculate - — , w was tabulated at intervals of dx. 100 m. of x . at each of these points was calculated diX. by the formula =42-where yiS , jjJh denote f i r s t and third mean central differences respectively. This formula i s obtained by differentiating the Newton-Stirling interpolation formula. Values of — at 100 m. intervals were then plotted, and values at each of the longitudinal markers found by inter-polation. 3.4.4. Rate of Change of Normal Velocity G>-y" Values of — (rate of change with distance down the Doc glacier of the velocity normal to the surface) were calculated by the same method of numerical differentiation as was used to calculate in section 3.4.3. 3.5. Ice Thickness The error of seismic measurements of ice thickness was considered to be smaller than that of gravity measure-ments. Seismic data have therefore been preferred to gravity. At boreholes, the depth of the borehole has been used in preference to the seismic value. The discrepancy between seismic and borehole measurements was very small however. Seismic measurements were made along the longitudinal line and transverse lines C and D. Gravity measurements have been used for the other transverse lines. Consistency of -43= results of the two methods was obtained as follows. Each gravity value on a transverse line was multiplied by the constant factor necessary to make seismic and gravity values agree at the point where the transverse line intersects the longitudinal one. For this reason, ice thicknesses quoted here sometimes differ from those given by Kanasewich (1960). It should be noted that gravity measures the vertical thick-ness whereas seismic measurements are normal to the bed. The difference between the two is within the standard error of the seismic data for a l l points on the longitudinal line. 3.6. Borehole Measurements Inclination to the vertical, <y, of the axis of the pipe, and i t s azimuth, A, projected on to a horizontal plane, at various depths, were measured in two successive years. The spacing between observations was generally 15.24 m. in 1960 and half this figure in 1961. The quantity to be determined is ^ , the gradient of velocity parallel to the surface in the direction normal to the surface. Various methods of analysis were used to see whether they led to appreciably different results. The formulae are summarized in this section. Symbols have the same meaning throughout and are only defined once. The various methods are discussed in section 5.3. -44= 1. Nye's formula (equation 22 of section 5 . 3 . ) , For each point at which inclinations were measured, 0 =. arc Tan (fan Y c o s AA) - &s Dashes refer to second set of observations. y = measured inclination A A = difference between measured azimuth and azimuth of flow direction as measured at the surface. cts =» surface slope (positive) t - time in years between observations. Three cases were taken. (a) ££ _ o "laminar flow". (b) — ^ the surface value. doc U * 4 (In Nye's theory of glacier flow i t i s assumed that ^ dec does not change with depth y.) (c) Variable — 45-K i s given by a - slope, subscripts s, b refer to surface and bed. h =• ice thickness. This formula i s derived in section 5.1. (equation 15). 2. Formula for curved pipe (section 5.3., equation 23). *!t = Q ' ~ 9 .sec* 9 - 2 ^ Van Q + ^ ^ 9 *a t " . — + (u fan § + u ) ( ^ c ^ ) ( ~ ^ J 0 = - f B) The last three terms on the right hand side are relatively small correction terms. Accurate values of the separate quantities are not needed. If inclination i s measured at distances (measured along the pipe) A s , 2 A s , • • • from the top, at the n* n point TV - I A s / 2 n AG] I i s the mean for the two years. A-s/ 71 -46. The value of -tt was obtained from a preliminary laminar flow analysis. v was determined by equation 13 of section 5.1. This gives a quadratic variation of v with depth and satisfied the boundary condition at the bed The term f- a n 2 @ i s negligible. dot Two cases were taken: (a) — zero, (b) variable doc with depth as in the last section. 3. , Integrated Method This method is due to Savage. Inclinations were measured at points As, 2As, * * • from the top of the pipe. (The zero point for measurements was the top in the f i r s t year. Sections of pipe had to be removed later to compensate for ablation. However, readings were taken at the same points in the pipe each year, except for the reading at the top.) The configuration of the borehole in each year was computed in terms of right-handed rectangular coordinate systems X, 7, Z and x, y, z. The X axis was horizontal, the x axis in the surface. The down-glacier direction was taken as positive. -47-The Y axis was vertical, the y axis normal to the surface. Both were taken positive downwards. The origin was the ice surface at the time of the f i r s t year's observations. The surface slope did not change significantly from year to year. Measurements of surface velocity gave Xq and Y Q in the second year. Z 0 was again taken to be zero. Coordinates were calculated by the formulae: X - X = - r Y - cos A A ^ + .sin Yn+, cos A A n + 1 ) As V - Y-n = " f ( C 0 S Y n C o 5 Yn-M ) A 6 • n + i 1 ^  2. n -7 _ 7 = J - / . s i n Y_ A A u + V n +, s i n A A r ) + I ) £ 6 ~ (*-. + Y~- ^ * s ) C 0 S ^ ( - X n h>- ^ + Y n ) ^ Z 0 = arc Tan ( fa * Y u COS A A^) The configuration of the pipe in the xy plane in each year was plotted. Displacement parallel to the x axis was measured from the graph, at equal intervals of y. When -48-converted to velocity, this quantity equals "D sec 8 where T) i s velocity normal to the pipe and 8 i s defined above. -j) - m, cos 9 + v sin 9 where u, v are velocities in the x, y directions respectively. The value of u at point n was determined from 6^ i s the mean value of 9 at point n for the two observation periods, v^ was found by linear interpolation between i t s measured value at the surface and i t s value u b tan (a^-a s) at the bed. Values of were found by numerical differentiation. Reduction of the data by this method was carried out by Savage. Calculation of coordinates and values of 0 and 0 was performed on a Bendix computer at the Institute of Geophysics and Planetary Science, University of California at Los Angeles. The interpretation of this data in terms of current theories of glacier flow, given in subsequent sections of this thesis, i s however the responsibility of the present author. -49-3.7. Miscellaneous Quantities 3.7.1. Curvature of Surface and Bed The rate of change of slope a with distance X down the glacier was determined, for markers in the longitudinal line, by numerical differentiation. Slopes relative to the horizontal were used. The curvature k was then calculated from In most cases the denominator could be taken as unity. 3.7.2. Width of Glacier This quantity i s required both for the drawing of transverse profiles and the calculation of transverse strain rate (section 3.4.3.). The large amount of debris on each side of the glacier made location of the edge d i f f i c u l t . It was however determined with f a i r accuracy by Kanasewich (1960) for his gravity survey. The width of the glacier at different points as measured between the zero ice thick-ness contours on Kanasewich*s Figure 4 has been used in the present study. 3.7.3. Long-period Change in Ice Thickness A rough estimate was obtained of the rate at which the lower reaches of the glacier have been thinning during -i = -50-the past century. Two triangulation stations were located on the crest of the prominent south-east lateral moraine. The heights of these points above the present ice surface were known. Evidence discussed in section 1.2. indicates that the glacier was at a maximum about 130 years ago. The level of the moraine was taken to correspond to the level of the ice at this maximum. The average rate of thinning over the past 130 years was thus calculated. The rate of thinning over the past 15 years i s known from the surveys of the Water Resources Branch. Every second year the height of the ice surface was measured across a transverse line about 350 m. from the present terminus. Agreement between rates given by these two methods was surprisingly good. 3.7.4. Travel Time for Ice A rough estimate was obtained of the time which ice takes to travel from the accumulation area on the Columbia Icefield to the terminus of the glacier. The integral centreline, was evaluated numerically by the Euler-Maclaurin formula. Ordinates were taken at 100 m. intervals in the lower part of the glacier. Above the lowest i c e f a l l , ordinates were taken at the 10 points near the centreline where surface velocity was measured. It was assumed in this calculation that a particle of , where u i s the surface velocity along the -51-ice travels parallel to the surface. This i s not of course the case. Thus the distance and the time were underestimated. On the other hand, the glacier appears to be thinner now than at any time during the past 100 years. (The travel time i s of this order.) Hence the present velocity i s li k e l y to be considerably less than the mean velocity over this period. This w i l l to some extent offset the other source of error. -52-4. ACCURACY 4.1 General The accuracy of the various measurements w i l l be dis-cussed in this section. The analysis does not apply to markers J l to J12. Measurements on these markers were of a lower standard of accuracy than measurements to the others. Throughout this discussion "error" means standard error. In the analysis, any doubtful or substandard observations were invariably rejected. The standard of accuracy of the data used should thus be f a i r l y uniform. Apart from inaccuracies in measurement, there i s also the possibility of errors in computation. A number of extra figures sufficient to make rounding errors negligible was always carried. A l l computations, except those carried out on the electronic computer, were done twice, independently. It i s hoped that this procedure eliminated any gross errors. Calculations on the computer were only done once, but the data input was double checked. 4.2. Position 4.2.1. Horizontal Position The basis for determination of position^ i s the survey of the Water Resources Branch which established triangulation stations around the perimeter of the glacier. The accuracy of the survey i s not discussed in the report (Reid, 1961). -53-Distances are quoted to 0.01 foot (3 mm.) however. Results should therefore be accurate to 1 cm. at least. This i s about ten times the degree of accuracy expected in the glacier movement survey. Any errors in positions of triangulation stations have therefore been ignored. Accurate location of stations presented no d i f f i c u l t y , as each was marked by a brass plug set in rock. Their s t a b i l i t y must be considered however, as a few were situated on moraines. Positions and elevations of markers as determined from station 9 showed a consistent difference from their positions and elevations as determined from other stations. Moreover, the discrepancy increased with time. Slumping of the station was the most plausible explanation. This was quite l i k e l y as the station was located on the crest of the south-east lateral moraine at a point where the crest i s extremely narrow. A l l observations from station 9 were therefore rejected. None of the observations from any other station gave any indication of instability. Each marker was observed from three stations. Ob-servations from each station provide a line on which the position of the marker must l i e . The mean of the three intersections of the three lines was taken as the position. The fact that the three intersections do not coincide can be attributed to inaccuracies in the following: alignment of theodolite and reference mark over their station marks; sighting on the marker; lateral refraction; -54- I and the fact that observations from the three stations were not simultaneous. The effects of these inaccuracies are reduced by taking the mean of the three intersections. There are, however, other sources of error which produce a consistent effect on observations from a l l three stations. These are slight differences between the horizontal positions of a marker before and after the hole had been redri l l e d , and inaccuracies in measuring the amount by which a marker was leaning. Over a l l the data, the spread (greatest distance between any two of the three intersections) had a mean value of approximately 10 cm. About 65% of the spreads did not exceed this value. This spread might be interpreted as three or four times the standard error of the mean position of the three intersections. However, i t has been taken as the standard error to allow for the consistent errors as well. During summer 1960 short term variations in velocity were studied by weekly observations of the positions of certain markers. An attempt was made to attain a higher standard of accuracy in these measurements and this was reflected in smaller spreads of the intersections. The standard error of these positions i s estimated to be 5 cm. No direct estimate of error can be obtained when markers were only observed from two stations. As was explained in section 3.1.1. however, such observations were -55-rejected i f the heights from the two stations did not agree closely. The normal standard error of 10 cm. has therefore been taken. As explained in section 3.1.1., some markers were only observed from one station in November 1960. The formula used was s I n ( A - 92 ) Thus = £ 5 + S(SA - 9,) + S(*-9A) <*- r< 9A - Q, l-an ( A - 0 J where £ denotes the error in a quantity. Typical values were S r - /0cm. £ ( e , - 0 = ^" S ( A " © J SA = /' o -f- = 0-6 K m = 3o' A - &x >/ 35 Hence o"d 20 cm. This i s the standard error of position in this case. The standard error of positions of markers J l to J12 is estimated from the spreads in the intersections to be 50 cm. To sum up, the standard error of horizontal position of a marker i s 5 cm. for the weekly observations in 1960, 20 cm. for some markers at the November 1960 observation period, 50 cm. for markers J l to J12, and 10 cm. in a l l -56-other cases. The standard error of the distance between two markers or between the positions of one marker at two different times i s J2~ times the standard error of position, i.e. 15 cm. in most cases. An independent check on this accuracy i s available. Distances between markers were measured by steel tape, for strain rate measurements, in a few instances. In four cases the same distances were determined by triangulation at about the same time. Taped distances were measured along the slope and so have been reduced to the horizontal. Distances have also been corrected for the time difference (up to 3 days) between the measurements, using measured values of longitudinal strain rate. Sources of error are different in the two methods. Errors in measurements by tape are due to incorrect tension, neglect of corrections for sag and temperature, and the fact that the tape was probably resting on the ice in places. Results are given in the following table: Distance (cm.) Markers Date Triangulation Tape L30 - L31 12/7/60 17802 17801 Hole 322 - Hole 116 23/7/60 16847 16847 Hole 322 - Hole 116 19/7/61 16473 16478 Hole 209 - Hole 186 19/7/61 15027 15021 -57-The differences between the distances as measured by the two methods are a l l well within the estimated standard error of 15 cm. This i s a satisfactory check on the accuracy of both methods. 4.2.2. Vertical Position The report on the survey of the Water Resources Branch states that the closure error in the vertical plane i s of the order of 1 in 77,000 (Reid, 1961, p. 7). Station elevations are quoted to 0.01 foot (3 mm.). It has been assumed that, to the standard of accuracy of the glacier movement survey, errors in station elevation are negligible. The mean of the measurements from three stations was taken as the elevation of each marker above datum. This process reduces the effect of errors in individual measure-ments produced by inaccuracies in sighting on the marker and in measuring the height of the instrument, and because observations from the three stations were not simultaneous. Diurnal variation in refraction makes this time difference important. The standard refraction correction which was used is equivalent to a height difference of 7 cm. between points 1 Km. apart. (The average distance from station to marker was 0.8 Km.; the maximum was 1.8 Km.). The difference between making no correction for refraction and refraction of double the standard value i s thus -58-equivalent to a height difference of 11 cm. over the average path length. It i s unlikely that variations of this magnitude were encountered. Most stations were 50 to 100 m. above the ice and the average vertical angle was a depression of 5°. Thus very few of the rays were close to the ice for appreciable distances. To test variations of refraction, vertical angles were measured to two markers at intervals of half an hour through-out one day. The spreads of angles were 8" and 11" for path lengths of 1 and 2.4 Km. respectively. The greater of these corresponds to a spread of height of 12 cm. To be of much value this test should have been carried out on several days under widely different weather conditions. However, i t may give some indication of the extent of diurnal variations of refraction. Moreover, anomalous conditions would have to persist throughout the day for the mean elevation to be affected by the same amount as the measure-ment from an individual station. These considerations suggest that refraction conditions would have to be very exceptional to produce an error of more than 10 cm. in the mean value of the elevation of a marker. There are factors which w i l l produce a consistent error in elevation measurements from a l l three stations. There are inaccuracies in measuring the change in elevation when the hole was redrill e d and the amount by which a -59 marker was leaning. Errors may also be introduced i f the marker was not gripped firmly by the ice at a l l times. The mean spread (difference between greatest and least values) of the elevation of each marker as calculated separately from the three stations was 5 cm. Readings were rejected i f the spread was greater than 10 cm. The mean spread should be greater than the standard error of the mean elevation. However, to allow for the effects of refraction and the other factors list e d above, the standard error of the elevation of each marker above datum was taken to be 10 cm. For reasons explained in the preceding section, the same standard error has been taken when the markers were only observed from two stations. In November 1960 certain markers were observed from only one station. The standard error of these elevations has been taken to be 20 cm. For the purpose of his gravity survey, Kanasewich made an independent determination of the elevations of a l l markers by stadia traverses. This provided a gross check on the accuracy. Kanasewich (1960, p. 16) quotes his results to the nearest foot (30.48 cm.). A l l elevations determined by the two methods agreed to this order of accuracy. Markers J l to J12 were further away from the triangu-lation stations than were other markers. Distances were -60-up to 3 Km., so the effect of variations in refraction was greater. The standard error of the elevations of these markers may be as great as 50 cm. 4.3. Velocity 4.3.1. Horizontal Velocity When the period between observations exceeded a week, errors in time measurement (not more than 1/4 day) can be neglected in comparison with those in distance. The error in distance measurement was estimated in section 4.2.1. to be 7 cm. for weekly observations and 15 cm. for the others (except for a few cases). Measured velocities ranged from 7 m./yr. to 270 m./yr.; 30 m./yr. was a typical value. The errors in this last velocity for measurements made over four time periods were as follows: Time interval 1 week 3 weeks 3 months 1 year Error in velocity (m./yr.) 4 2.5 0.6 0.15 A £1 Direction of velocity A i s given by tan A =» AN It follows that —: — - = — + - s i n 2 A A E A N/ If sin 2A i s given i t s maximum value, this reduces to A E since proportional errors in E and N coordinates are roughly equal. S(AE) i s not more than 15 cm.; 30 m./yr. i s a typical velocity. So A E i s about 15 m. for observations -61-over a period of a year. In t h i s case £A ^  1% or 1° say. For periods of 3 weeks and 3 months the errors are 15° and 4° respectively. 4.3.2. V e r t i c a l V e l o c i t y V e l o c i t y V ( v e r t i c a l ) . Values of V were generally between 0 and 4 m./yr. The standard error of the difference between two height measure-ments was 0.15 m. The standard error of V when measured over a period of one year was thus 0.15 m./yr. For measurements made over a period of 3 months the standard error was 0.6 m./yr. Measurements made over periods of less than 3 months were too inaccurate to be of any value. V e l o c i t y v (normal to surface). As i n section 3.2.2. OT = V cos o< - U cos A A i^-o Hence S-xr a SV-cosc^ + V sin of- + S U cos A A-5 rn •+• 0 sin AA • -si^  ^  + Sc*-Uco& AA cos The following were t y p i c a l values: 6* = 0-3° 6"(AA)=/° SU = ois m . / r . cos cf ~ I > Cos A A - / U = 3 0 s i n * = 0-/ - 0 0 2 * V = m h r ' -62-Hence Sv^0.35 m./yr. for measurements made over a period of a year. Most values of v were between 3 and 4 m./yr. 4.3.3. Velocity of Slip Past Side Walls In the notation of section 3.2.3., Z - - A 2 The error in estimating the direction of flow, and hence A the error in APA, greatly exceeded errors in measuring the distances a^, a 2. Hence Typical values were al> a2 ~ 8 m. A-£ = 12 m. sin APA = 0.5 £ (APA) = 20° 1 m. So the error in velocity was about 1 m./yr. or about 10%. -63-4.4. Slope 4.4.1. Surface Slope This was obtained by measuring the distance between an integral number of contours on the map. A distance comparable with the ice thickness at the point was chosen. A h A L S ~ ( A k ) S ( A L ) A h A L The distance was generally of the order of 200 m. This represented a map distance A L of 40 mm. The error 8 (A L) should not have exceeded 1 mm. For an average slope, A h was of the order of 20 m. The contour interval was 10 f t . (3.048 m.). The error has been estimated at 2 f t . (0.6 m.). An average slope was 5°. Hence 6 < £ , by the above formula, was 0.3°. 4.4.2. Slope of Bed The standard error of the slope of the bed as determined from the seismic records was estimated to be 1° for the best records and 20% for others. 4.5. Strain Rates 4.5.1. Strain Rates Measured by Taping The least squares analysis of Nye (1959b), described I in section 3.4.1., provides estimates of the standard it -64-errors of the s t r a i n rates. If R = — ( j . + £. - i - £ ) the standard errors of £ x , £ are ^ f l B l a n d of £,x ^  i s 721 Rl. 4.5.2. Stra i n Rate from Triangulation The following analysis applies equally well to longitudinal and transverse s t r a i n rates. Percentage errors i n measurement of t, the time i n t e r v a l , and t, , the distance between the markers, are small compared with the error i n the change i n distance. £i « ^ i tt I t The standard error of po s i t i o n i s 10 cm., that of distance J2 x 10 cm., and that of change of distance /2~ x 7 2 x 10 =» 20 cm. A t y p i c a l value of •£ was 100 m. It follows from the formula that the standard error of s t r a i n rate measured over a year was .002 per year, and .008 per year i f the period were 3 months. Over a period of less than 3 months r e s u l t s are too inaccurate to have much value. Smaller errors could have been obtained by using more widely spaced markers. However, i t must be assumed that s t r a i n -65-rate does not vary significantly over the distance over which i t i s measured. This would seldom be the case for marker spacings of more than 300 or 400 metres. Calculations based on the change of width of the glacier provide at best a rough estimate of transverse strain rate. It i s hard to quote a standard error. Strain rates calculated in this way are compared with direct measurements in section 6.5. 4.6. Ice Thickness 4.6.1. Seismic Method Seismic measurements made at borehole sites enabled the accuracy of the method to be tested. The comparison i s as follows: Depths in metres Hole Number Seismic Borehole 322 316 321 314 312 317 250 247 250 209 202 209 194 177 194 For holes 209 and 194 the seismic points were displaced from the boreholes by a horizontal distance of 60 m. The seismic value which appears in the above table has been extrapolated to the borehole location by using the indicated dip at the reflection site. -66-The above table indicates that the seismic measurements of ice thickness are too low by about 5 m. This could be accounted for by assuming that the f i l t e r correction and the estimates of the time lapse between the beginning of the reflection and the f i r s t well developed trough were each in error by about one millisecond. In as much as the smallest time division on the seismic record i s 5 milliseconds this assumption seems jus t i f i e d . It i s to be emphasized that the differences in ice thickness between various reflection sites are probably accurate to within 1 m. The information required, however, is not ice thickness and dip at isolated points but rather a continuous profile of the bed along several sections transverse to the glacier and a profile along a longitudinal section containing the centreline of the glacier. In the latter section important errors of representation arise for the following reason. The seismic reflection comes from the section of the bed closest to the shot point. Because of the high transverse curvature of the bed the section closest to the shot point is often displaced to one side. An extreme case of this type occurred for the reflection at L16 (see sketch). -67-The high quality record obtained at L16 indicated that the reflecting surface was 321 m. from the shot point and that i t dipped (relative to the ice surface) 30° + 1 . 5 ° towards the right edge (looking down glacier) of the glacier. This information suffices to plot the reflecting surface on the transverse profile. It does not determine the depth beneath the shot point. In the absence of other information the depth has been taken as 321 m., although a l l that i s certain i s that i t i s not less than this value. This case is extreme, but reflecting surfaces of transverse dips of 10° were not uncommon for shot points on the centreline, (See Table 8). The down glacier dips were quite moderate so the transverse profiles are relatively free of this error. -68-4.6.2. Gravity Method The error of ice thickness as determined by gravity measurements was estimated by Kanasewich (I960, p. 57) to be -10%, +15%. 4.7. Borehole Measurements The f i r s t two methods of analysing the borehole data employ the formulae: (1) 5" , J - ( |-ari Q' _ Tan 6 ) - ( fan 9* + d) (2) « °' ~ 9 0 - 2 ^ fan £ + ^ l a n * 0 + (u Tan t9 + n r ) ( s e c © ) ( | | ) These formulae consist e s s e n t i a l l y of a f i r s t term plus small correction terms. Large errors i n the correction terms w i l l not a f f e c t the r e s u l t appreciably. For the purpose of an error analysis only the f i r s t term w i l l be considered. To a s u f f i c i e n t approximation, both formulae can be written as: Errors i n measurement of t, the time i n t e r v a l , are n e g l i g i b l e compared with errors i n 6. -69 9 - arc l"an, ( fan. Y cos A A ) -Hence 0 = V' cos A A ' - V c o * A A i ( T ^ S G t ~ 1 year £ 0 = ^ ( ) f cwAA) = cos A A • ^  + Y .sin A A • 5" ( A A) <r £ Y + Y- r ( - A ) Errors depend on the type of d i s c used i n the inclinometer, The three cases are l i s t e d below. Disc 4° 10° 26° Ymax 4° 10° 20° & Y 0.1° 0.25° 1° o ( A A ) 1° 1° 1° 0.004 0.Q1 0.035 It i s d i f f i c u l t to estimate the error i n c a l c u l a t i n g ^~ by the t h i r d method described i n section 3.6. So i t has been assumed that the errors are the same as i n the other methods. The magnitudes of the terms i n formulae (1) and (2) above w i l l now be compared with these errors. -70-The f i r s t terms are l-an d' - fan 9 ^ 9 - 9 + ± ( 9^- 0*) ( 9 - e) szSs ~ (*'- *)[' + Difference <C -^ (a'-eXs'-*"*)* In hole 322 the maximum values occur at the bottom of the hole. They are 9' - 8 - 0.11 and 8' + 8 - 0.35. The difference term i s 0.0015 which i s negligible compared with the experimental error. In hole 209, the maximum values of the difference term are 0.0002 and 0.0075 when the 4° and 26° inclinometer discs are used respectively. In both cases then differences are less than the experimental error. The difference in the second terms in formulae (1) and (2) i s that between tan 9 and tan 8. The greatest difference occurs at the foot of hole 209. Its value is 0.01. When this i s multiplied by (=0.02), i t is negligible compared with the experimental error. As was explained in section 3.6., three alternative values can be taken for — . These are zero (corresponding to laminar flow), the surface value (corresponding to Nye's assumption that ^ i s constant with depth), and a value which varies linearly with depth according to the relation obtained by differentiation of equation 14 of section 5.1. Relevant values at the surface and bed are -71-given in Table 21. The greatest difference between any of these values i s 0.02. For hole 322, tan 6 does not exceed 0.15 except for the bottom reading where i t i s 0.25. These values correspond to differences of 0.003 and 0.005 in computed values of — . These differences are slightly less than the standard errors for the appropriate inclinometer discs. In hole 209, tan 0 does not exceed 0.2, except for the bottom value of 0.4. Corresponding differences in calculated values of — for a difference of 0.02 in — are 0.004 and 0.008. The latter value is negligible compared with the error for the 26° disc which was used. The other value i s comparable with the error for the 4° disc. The third term in formula (2) i s negligible in a l l cases. The fourth term on the right hand side of formula (2) is the correction for curvature of the pipe. Sections of borehole where inclinations were greater than and less than 10° w i l l be considered separately. The maximum value of the term (u tan 0 + v) sec 0 in any borehole was 5 m./yr. The maximum change in 0 over a section of borehole of length 15.2 m. was 7.8°. This corresponds to a value of of A S 0.009. So the maximum value of the correction term was 0.045. This occurred in a section where the 26° disc was used in the inclinometer. The maximum value of the correction term in sections where 4° or 10° discs were used, was 0.0075. These correction terms are larger than -72-corresponding errors in measurement. The conclusions to be drawn from this analysis are that the only difference between formulae (1) and (2) i s the correction term for curvature of the pipe. This correction term should be applied, although there are many places in each borehole where i t s effect i s negligible. The difference between a"laminar flow "analysis and one which makes allowance for the longitudinal strain rate i s of l i t t l e importance. 4,8. Ablation and Accumulation The error in measuring the distance from the top of a marker to the ice surface should not exceed 2 cm. This i s negligible compared with the sampling error inherent in the fact that ablation and accumulation can vary appreciably at points on the glacier only a few metres apart. 4.8.1. Ablation Ablation was measured over a period of 5 weeks at 22 markers arranged in a longitudinal line of 200 m. Values varied between 95 and 133 cm. There was no systematic variation with position, and only 50 m. separated the markers at which greatest and least values were recorded. If these markers are regarded as a sample from an area where ablation was uniform, the standard error of the distribution i s 9 cm. or about 10% of the mean. It has been assumed therefore that ablation measurements on single markers have a standard error of 10%. -73-4.8.2. Accumulation Similar d i f f i c u l t i e s occurred in measurements of accumu-lation. Measurements at 15 markers in the lower part of the glacier ranged from 0 to 2.15 m. The range of values must have been greater than this because the majority of markers were completely buried by snow. The great variation can be largely attributed to the effect of wind. It was concluded that the measurements were insufficient to give any estimate of mean accumulation. -74-5. THEORY 5.1. Distribution of Stress and Velocity The following theory i s a slig h t l y generalized form of that of Nye (1957). It is restated here from the beginning in order to make clear the assumptions which are involved at each stage. Assumptions: I. The ice i s isotropic. This i s probably a valid assumption for glacier ice. II. The rate of flow depends only on the stress and not on the time for which i t has been acting. In other words, a piece of ice remains under the same stress long enough for a steady state to be reached. III. Hydrostatic pressure does not affect the flow law. This i s an experimental result (Rigsby, 1958). Consider a rectangular coordinate system fixed in space. Let u, v, w be the velocity components. Components of stress w i l l be denoted by cn. or by Components of the stress deviator w i l l be denoted by <T: .- or c r etc. J 0 0 Components of strain rate w i l l be denoted by £, . or Let E 2, £^ be the second invariants of the strain rate -75-and stress deviator tensors respectively. Let j> be the density. Relations of the following types therefore hold: <r/. = <n • i ¥* i 2 £ Further Assumptions: IV. The ice i s incompressible. This implies that ir = o. V. The components of strain rate are proportional to the components of the stress deviator. The stress deviator is taken, not the stress, because of III. This i s also consistent with IV. Let A be the constant of proportionality. Then E 2 - 7\2 £.2 • It is shown by Glen (1958c, p. 174) that, under assumptions I to IV, the most general form of the relation between stress and strain rate i s iL] = - f Z C & . + B 07. + C T' cr'. where B, C are functions of <. > Assumption V implies that C - 0 . -76 VI. The glacier i s regarded as a block of ice of i n f i n i t e extent resting on a rough bed. The upper surface i s a plane inclined at angle as to the horizontal. Take x axis in the surface, pointing down the line of greatest slope. Take y axis normal to the surface, positive downwards. The z axis i s horizontal. VII. Consider plane strain, in which movement i s confined to the xy plane, i.e. w =• 0, ~ =» 0. Hence cr' - 0 ? T X I = 0 ? = O by V. Hence <r. = -L ( 0^ + cr ) and < = -<r; . ± (<rx - <r ) (1) Also < 3 = ° The equations of equilibrium are -77-(g^ = component of acceleration along x^ axis due to gravity). In the present case these reduce to + 3 V + ^ " 0 (2) lot :J + L_ + j><^  cos % - 0 ( 3 ) To obtain a solution of these equations i t i s necessary to make another assumption. VIII. £ and T x ^ are functions of y only. By a) < - i («* - < r a ) - + T x * ( O i - ^ X ^ - If) - O If <rx = <ra , <rx' . . o , ^ ' Tx* . This i s a special case. Thus . 2S (4) doc (2) and (3) have to be solved subject to the boundary conditions. <r = TT (atmospheric pressure) V 0 (5) for y " 0 and a l l x. For this condition to be consistent with VIII, the x axis must be taken in the surface, rather than horizontal or parallel to the bed for example. -78-By VIII, (3) reduces to 0"u whence where It follows that Thus by (4) Hence (2) becomes + J>J cos ^ = o ! f i « o (6) Now •sin o\ — ( cr - <r ) 4- r Hence . - IT - j* j y cos ^  + *]l[-(f3X s , n *s)' (7) (8) (6), (7), and (8) constitute the stress solution. It involves the function £^ which i s at present undetermined. It is possible to relax the assumption of plane strain to a small extent by putting T =» c, z so that - S > a constant. C i s added to the l e f t hand side of equation (2) and the stress solution (equation 7) becomes (7«) - 7 9 -Similarly one can put X , = c 2. . j It has been assumed that £ i s independent of x, and i t was shown that t! ~ 0 for a l l x. So A is independent of x, •3 The velocity solution can now be obtained. where -ir = -\r0 al" x = 0 ; j = 0 and r, s are constants. ( 9 ) 2f-Similarly — = O (10) By IV 7"" T " ~ ox. 0 y Hence (11) becomes — = —-, (12) ^ x «N (10) has general solution v = XU) + Y ( j ) By (12) X W ! T = - K a constant. So V = v. + soc - T^ y - f K ( x* + <^ ) (13) - 8 0 -7) a W ^ ,— =- - — = T + K u ^ X l>u ^ <t = <i0 + roc + KOCLJ + f(y) ( 1 4 ) where <t « -a0 x = o , j = O O.JT X =• O / <j = O The boundary condition at the surface (y = 0) i s "C^ = O. By assumption V, this implies that i = o The boundary condition on the bed is where m - tan (a^ - <zg) where a^ , is the slope of the bed at the point in question. It must be assumed that IX. changes only slowly with x. The f i r s t boundary condition leads to f'lo) - - * If h i s the ice thickness at x = 0, the value of y on the bed can be approximated in the form y =» h + mx + • • • • Thus i r = v 0 + 5 X - f ( h + mx. + • • • ) - 5 K £ ( H f m x. + • • )* + * * ] -81-where the subscript b denotes a value on the bed. The second boundary condition gives i r G - rh a m u 0 •t-But by (14) f(M - •a 0 ah X - O Hence U 0 - rh _ i or K = ^ ( - in u b - r h ) (15) Again, by (14) •f- + Hence / tic { -bx ) - T = ^ + £ Kh So (15) can be written h = nr0 - %<ib (16) where -afc } refer to x = 0. Equation (16) also follows directly from IV VII, for ^X ^ x J ~ 0 J ^ o ° as before. -82-Another derivation of equation (16) i s given in Appendix 1, The function j-(y) in (14) can be expressed in terms of A as follows: • - V J J s ' n ° ' t h X = 1/ f(y)~ f3 •*"><*•! }y*<*-i (17) since f (o) - 0 (see equation 14). For the solutions to be determinate, the flow law must be known. Moreover, i t must be assumed that X. The flow law does not involve the third invariant £ , / In other words, A is a function of £ only. If the flow law i s E A = F ( ) > X Now E 0 - X £ = — 2 * If r, K, /°(y), and the form of F( 1 A ) are known, this last relation enables £ to be determined as a function of y. (It is not a function of x because £^ i s not.) The velocity and stress solutions can then be determined. -83-The essential difference between the foregoing analysis and that of Nye i s that Nye assumes i n i t i a l l y that «b 5 3 a s * The boundary conditions for the velocity solution are then £ Xy » 0 on y «• 0 as before and v = 0 on y — h for a l l x. Substitution of this second condition in (13) gives s = 0, K => 0, v Q =• rh v =» r(h - y) which i s Nye's solution. The corresponding solution for u i s (putting K = 0 in (14)) -u, = - a Q + r o c + f C y ) Two consequences of Nye's solution should be noted. " S i r _ 1. — - 0 ~bx. Nye (1957, p. 119) makes i t appear that this i s an additional assumption. But the preceding analysis shows that i t follows from the other assumptions. Specifically, i t i s a consequence of assuming that , TK are independent of x and that the slopes of surface and bed are equal. It follows that i = 4 ^ . The solution (7) shows that i s always negative. But the rate of work -84-E X must be positive. Hence _ must be negative. 2. — - a constant. In other words, the longitudinal strain rate does not vary with depth. For the case when a b ^ ctg, Nye (1957, p. 126) adjusts the velocity solution to This does not quite agree with equation (13), which for s = 0, K - 0, i s Hence "V"b = u 0 - t h V - f ( h - ^ ) + v b = <r ( h - -r <*fa fan (* b -This expression involves u^, the velocity at the bed, not u, the velocity at depth y. The expression can also be derived from the condition of incompressibility which, for ^ «• a constant, i s Integrate both sides with respect to y between the limits y and h, and the expression i s obtained. I . a further development, Nye assumes that ^ i s constant on the bed. The following equation for the -85-longitudinal strain rate i s then derived. r = — (a + X cot- o O (18) where a = rate of accumulation or ablation. q = flow rate = volume, per unit thickness in z direction, which flows in unit time through a cross-section perpendicular to OJC. Rg - radius of curvature of surface, taken positive i f convex. The two respects in which the preceding theory differs from that of Nye, namely (1) 2— always negative and (2) — independent of depth, can be tested by f i e l d "dx. measurements. Equation (18) can also be tested in this way. Some further attempts to generalize the theory of this section are given in Appendix 2. 5.2. Effect of Valley Sides The assumptions of i n f i n i t e width (VI) and plane strain (VII) are questionable. An alternative approach i s to regard the glacier as one half of a circular cylinder. It is convenient to use cylindrical polar coordinates (r, 9, x) with origin at the surface on the centreline and x axis in the surface pointing down the glacier. The equations of equilibrium are -86-+ r'JV + ^ + ~r~ + S3*- ' ° Make the following simplifying assumptions 1. = - ° 2. <J^  cr T.^ , do not depend on 0 . 3. < r , does not depend on x . The equations then reduce to 3r o^c r ^' JJ + P I . = O The last equation has solution ^ = -J 13 *• " n °*s (19) This can be compared with the corresponding solution in the rectangular coordinate system for the centreline of an i n f i n i t e l y wide glacier, -87-In the above derivation i t has not been shown that solutions for the four non-zero stress components can be obtained from the three equations combined with the flow law. While the derivation i s not rigorous, i t does, however, suggest that allowance can be made for the f i n i t e width of the glacier by inserting a factor £ in the appropriate stress solution. This i s for a semicircular cross-section. For arbitrary cross-section i t seems natural to replace \ by the factor — — where S i s the area of cross-section, p i s the ph perimeter (excluding the upper surface), and h i s the ice thickness on the centreline. The same result was obtained from st a t i c a l considerations by Nye (1952b, p. 85-86). 5.3. Reduction of Borehole Data The formulae used in section 3.6. w i l l be derived in this section. It i s assumed that —>—.>— are constant both xn time and over the distance moved by the pipe between ob-servations. It must also be assumed that ice s l i p s freely along the pipe. -88 Let x*, y* be axes perpendicular and parallel to the axis of the pipe, and x, y axes parallel and perpendicular to the glacier surface, x, x* are positive in the direction of glacier flow and y, y' are positive downwards. Let G be defined by „ n & = cos G = ±t ds where s i s distance measured along the pipe. The relations between the coordinate systems are cc = oc' cos 9 -+- <j' vSin 6 y = - OC.' «S I 0 £ + y' COS & •a' = -u cos 9 - v- .sir, & A section of pipe w i l l begin to t i l t in proportion to twice the shear strain rate, as defined, i.e. d B ^ 2^' -89-It i s assumed that ice slips freely along the pipe, hence — , i s zero. T h u s * » . i « ' . ^ . S ' * t j » s « s s | | f w ^ ( 2 0 ) Plane strain has been assumed so that by the assumption of incompressibility. If 9 - 9 Q at t - 0, and 8 - 6^ at t » t, then ct ( t o o . © ) t = ^X d : c Near the surface, and |^ are comparable in magnitude doc <>y but tan 6 i s small. At depth, where 9 may be 30°, ^ is large compared with . Thus Hence t = _ _ . loo - S — !_ ^x ^ * 2>x ° o r (21) This equation was derived by Nye (1957, p. 130). - 9 0 -The maximum value of — at the boreholes on the Athabaska Glacier was 0 . 023 . In this case (^) can be neglected and the formula reduces to l± = — ( fan &, - Tan $ ) - | ^ ( fan © + h a o & ) (22) It was suggested by Savage that this formula could be improved by taking account of the curvature of the pipe. This means that in equation (20) 6 i s a function of y*. A term . —- must therefore be added. be O L J ' _ / . . . . - A .w . . . fi^ d 6 ? and (-U. 610 6* + V* COS #) / d 0 _ dn1 As Addition of this extra term means that the equation can no longer be integrated. The formula i s therefore — = — s e c 6> — ^. — T a n © •+• — f a n # + ( - a fan <9 t v ) ( j « c (9J — dLS (23) Values averaged over the period are used for the various quantities. -91-6. RESULTS 6.1. General This section consists of a brief review of the observational results. Detailed interpretation of certain aspects of these in terms of current theories of glacier flow is postponed to section 7. 6.2. Configuration of Surface Positions and elevations of triangulation stations and markers are given in Tables 1 and 2. Table 3 and Figure 7 give the surface profile along the longitudinal line. Profiles along the transverse lines are given in Table 4 and Figures 8 to 14. A l l the data in Tables 3 and 4 and in Figure 7 were obtained from the present survey. In Figures 8 to 14 elevations of additional points have been added and the edges of the glacier and the extent of the debris cover at the sides have been indicated. This additional information has been taken from Kanasewich (1960). Any discrepancies between tabulated elevations and contours on the map can be ascribed to inaccuracies in plotting positions of markers on the map. A l l elevations measured in the survey check with those determined quite independently by Kanasewich. The central parts of the transverse profiles show the convex shape typical of an ablation area. The sides are -92-relatively high because the thickness of the cover of debris is sufficient to reduce ablation appreciably. The tables l i s t the level of the surface in both 1959 and 1960. The change of thickness of the glacier in one year cannot be deduced from this data, however. Most of the apparent change i s a result of the movement of the markers. In any case, to be of much value, such measurements of change of surface elevation should be made at the beginning or end of each ablation season. The surface level at the end of the 1959 season i s known. Corresponding figures are not available for 1960, however, as the party l e f t the glacier before the end of the ablation. Measurements were made during the winter but by this time the upward flow of ice had produced an appreciable change in surface level. Evidence regarding the change of thickness of the glacier over a long period i s discussed in section 6.4. The slope of the surface at each marker i s given in Table 7. Values of surface curvature at points on the longitudinal line are lis t e d in Table 9. Table 5 l i s t s the width of the lower part of the glacier at different points. The maximum width i s 1240 m. at LI7; the minimum 890 m. at L30. 93 6.3. Configuration of Bed Figure 5 i s a map of the bedrock. Profiles of the bed along the longitudinal and transverse lines are shown in Figures 7 to 14. These figures were drawn on the basis of the measurements of ice thickness l i s t e d in Table 6. Direct measurements, either by the seismic or gravity method, were made at every point l i s t e d . It i s noticeable that the gravity method gives a much smoother profile than does the seismic. Slope of the bed, as determined from the seismic records, i s given in Table 8, and the curvature, obtained by differentiating the slope, in Table 9. The configuration of the bed has already been described in section 1.2, and w i l l be discussed further in section 7. 6.4. Surface Velocity The horizontal component of surface velocity (U), i t s direction (A) measured in a horizontal plane, and the vertical component (V) are given in Tables 10, 11, and 14 respectively. Velocity normal to the ice surface (v) i s also given in Table 14. Horizontal velocity was determined at weekly intervals in some cases. Inaccuracies of measure-ment make calculations of direction and vertical velocity over such periods of l i t t l e value, however. Direction has not been calculated for periods of less than three weeks. For vertical velocity the minimum period was three months. The normal velocity v could not be calculated for a few -94-markers because they lay outside the area of the map. The surface slope was therefore unknown. The quantities U, V, A determine the velocity vector. The component U was so much greater than V that the difference between U and the magnitude of the velocity vector was less than the standard error of measurement in nearly a l l cases. The only markers at which vertical velocity was measured where this did not hold were A6, A7, L38, and r,D". The horizontal direction of the velocity vector at a l l markers on the longitudinal line never differed significantly from the direction of the line at that point. Horizontal velocities along the longitudinal line are shown in Figure 15, and vertical and normal velocities in Figure 16. Each point has been plotted to correspond to the mean position of each marker at each period. Differences between the curves thus represent genuine seasonal differences in velocity, and are not merely a result of changes in position of each marker. Horizontal velocity along the centreline decreases steadily from 74 m./yr. at L10 to 15 m./yr. at L39. This comparatively high value within 40 m. of the terminus can be explained by the high wastage of ice from this section as a result of calving into the terminal lake. Figures 8 to 14 show horizontal velocities on the transverse lines. The average velocity for each line, calculated by the method described in section 3.2.4., i s -95-given in Table 13. Velocities at the edge of the glacier on each transverse line are given in Table 12. These values were obtained by extending the curves in Figures 8 to 14 to the edges. This represents extrapolation of velocity over distances up to 300 m., so the results are very inaccurate. Velocities at the edge at the two locations where measure-ment was possible are also given in Table 12. These measure-ments were made 1 or 2 m. from the edge of the ice at points some 50 m. down glacier from the A line. The locations (AO, A10) are marked on Figure 2. Near the i c e f a l l , the velocity at the edge i s roughly 10% of the velocity on the centreline. Near the terminus (G li n e ) , the velocity at the edge i s zero. The situation in the intervening region i s not very clear. There are no great differences in velocity at the edge between lines C to F. The average velocity i s between 80 and 90% of the maximum velocity on each of these lines. The most noticeable feature of the velocity profiles i s their lack of symmetry. For lines C to F the velocity at the south-east edge i s small or zero. At the north-west edge the velocity i s about 60% of that on the centreline (except for line C). The reason for this asymmetry i s not clear, but i t could be a result of the shelf which appears to exist under the ice on this side of the valley (see Figures 5, 9, 10, 11). Alternatively, i t i s possible that an error was made in locating the edge of the glacier on one side. However, an -96-error of the order of 200 m. would be necessary to eliminate the effect. An error of this size seems improbable. Extra-polation of the velocity to the edge by a smooth curve may be unjustified, and i t i s not impossible that the velocity at the edge i s zero everywhere below the i c e f a l l . The measured velocities at the edge do not show the asymmetry. On a l l the transverse lines, the flow at each side diverges from the centreline. The divergence i s seldom more than a few degrees, however. The divergence rises to 12° in the case of the F line where the glacier widens appreciably. Flow on the G line i s complicated by the glacial lake at the north-west half of the terminus. Calving of ice into the lake produces a high wastage of ice and the markers above the lake (G5 to 7) flow towards i t . There i s no discontinuity in velocity across the boundary between the debris-covered and comparatively clean areas of ice. As i s expected in the ablation area, ice flows upwards through the surface (i.e. v i s negative) throughout the lower part of the glacier. In general, upward flow i s greater towards the sides of the glacier than near the centreline. (In other words, near the sides v i s more negative and U i s smaller, so that the velocity vector i s more steeply inclined to the surface.) But the effect i s not very marked and the B line seems to be an exception. The value of v shows very l i t t l e variation with distance -97-down the glacier. There i s no significant deviation from a mean of -3.4 m./yr. along the longitudinal line between L l l and L35. The absolute value decreases between L35 and the terminus. A similar trend was observed on the Saskatchewan Glacier (Meier, 1960, p. 25). The upward flow of ice compensates for the lowering of the surface by ablation. Comparison of v with the ablation rate thus determines whether the glacier i s in equilibrium. Measurements of total ablation for a season are only available for one year, 1960. The mean value on the centreline was 4.2 m./yr. with very l i t t l e variation over the lower part of the glacier (Table 27). If this ablation rate is typical one might conclude that the glacier i s becoming thinner at a rate of about 0.8 m./yr. As was explained previously, no direct measurements of change of ice thickness were made in the present study. Some measurements over a long period are however available (see section 3.7.3.). Results are given in Table 15 and indicate a rate of decrease of thickness of 0.7 m./yr. Agreement between the two figures i s surprisingly good but may be fortuitous. It i s by no means certain that 1960 was a typical ablation season. While ablation data for 1959 do not cover the whole season they certainly include the greater part of i t . Prom these data (Table 27) i t appears unlikely that the total ablation in 1959 exceeded 2.5 m. Ablation at similar altitudes on the Saskatchewan 98-Glacier averaged about 3.5 m. in 1953. This was considered to be a reasonably "normal" year (Meier, 1960, p. 8, 11). It thus seems quite l i k e l y that 4.2 m./yr. i s an overestimate of the average ablation rate. The measurements of change of thickness refer to the region below L32. Variations in thickness of glaciers with time tend to be greatest near the terminus. The data of Table 15 suggest a lower rate of thinning at L32 than at L37. Thus 0.7 m./yr. i s probably an overestimate of the mean value for the lower part of the glacier (i.e. below L10). While i t i s lik e l y that the glacier is becoming thinner, i t s condition i s probably considerably nearer to equilibrium than a change of 0.8 m./yr. would indicate. Accurate information on the state of the glacier w i l l eventually be available from the maps which the Water Resources Branch, Department of Northern Affairs and National Resources, plan to make every three years. Over most of the lower part of the glacier values of V are also negative. This means that ice flows upwards relative to the horizontal. A similar result was found on the lowest 4.5 Km. of the Saskatchewan Glacier (Meier, 1960, p. 25). Markers situated on debris-covered ice (e.g. B9, C2, C13, F9) have vertical velocities considerably smaller than those of other markers. Ablation i s likewise reduced. Horizontal velocities in the upper part of the glacier -99-are also given in Table 10. The maximum velocity recorded was 266 + 25 m./yr. for marker J4 on the headwall. D i f f i c u l t y of access prevented the setting of markers in the steeper parts of the headwall, where velocities are probably considerably greater than this. Marker A7 moved intact from top to bottom of the lowest i c e f a l l in two years. Measured velocity was about 130 m./yr., but this i s the mean value over one year. 6.5. Strain Rate Longitudinal strain rates along the centreline of the glacier are given in Table 16 and Figure 17. The strain rate changes steadily from about -0.1 per year below the i c e f a l l to approximately zero at L21. The strain rate i s effectively zero between L21 and L27 and below this i t becomes compressive again. Values below L35 were measured over a period of only six weeks and so are less accurate than the remainder. However, the positive value between L38 and L39 i s probably genuine. L39 was some 40 m. from the edge of the ice c l i f f above the glacial lake and there were transverse crevasses nearby (see Figure 4). This value apart, the strain rate is never extending at any point along the longitudinal line. Apart from oblique crevasses along the margins and in the neighbourhood of the £ line, there are no crevasses between L10 and L37. Transverse strain rates calculated from the change in width of the glacier are l i s t e d in Table 18 along with -100 comparative measurements at the transverse lines. These measurements refer to the change in distance between the markers at each end of the line. Except at the E line (L27) agreement i s surprisingly good. This method of estimating transverse strain rate in a valley glacier thus appears to be reasonably satisfactory. The only data required are the width of the glacier and the surface velocity along the centreline. It should be noted that estimation of trans-verse strain rates at intermediate points by linear inter-polation between values measured at the transverse lines would not be satisfactory. The discrepancy at the E line probably results from irregularity of the glacier bed. The variation of strain rates between individual markers i s much greater on the E line than on the other transverse lines (see Table 17). This fact and the presence of crevasses oblique to the direction of flow between E4 and E6 suggest irregularities in the bed. This cannot be verified from seismic measure-ments as only two were made on the E line. The gravity profile does not show any irregularity. However, the ice thickness as determined by gravity at the intersection of the E and L lines was 215 m. The value obtained by a combination of seismic and borehole data was 248 m. (215 m. is the value given by Kanasewich (1960). Values for the E line in Table 6 have been adjusted to give 248 m.). This suggests that the gravity method i s not very accurate in -101-this area. Calculated transverse strain rates are shown in Figure 17. 6.6. Borehole Results The inclinometer measurements are given in f u l l in Table 20. Table 21 l i s t s slope, velocity and other relevant information for each borehole. The configuration of each borehole in each year i s plotted in Figures 18 to 20. Figures 21 to 23 show the variation of — with depth y. . ty Values of — computed by three methods described in ty section 3.6. are given in Table 22. This table supports the conclusions drawn from the error analysis of section 4.7. • II The difference between a laminar flow analysis and one which makes allowance for longitudinal strain rate and curvature of the pipe i s not important in this instance. It could be very important of course for boreholes in regions where the longitudinal strain rate i s high, or even for these boreholes in subsequent years when the distortion has increased . The difference between analysis by these methods and by the integrated method (column C of Table 22) does appear to be significant, however, at least in the case of hole 322. The discussion of section 7.1. indicates that the integrated method has a sounder theoretical basis than the other. Values of ~ in column C of Table 22 have therefore been ty used in subsequent analyses. The broad conclusions drawn are however unaltered whether the values in columns A, B, -102-or C are used. The most conspicuous feature of the results i s that ^5 is positive in the upper parts of holes 322, 314, and 116. In hole 322, |^ i s positive down to a depth of about 100 m. and u i s greater than i t s surface value down to about 180 m. Ob-servations only extend down to 45 m. in hole 314 but xr- i s b d positive throughout. In hole 116, — changes sign several times. This i s presumably the result of experimental error which i s somewhat greater in this hole than the others. But the velocity i s greater than the surface velocity throughout (the borehole only extends to 116 m., but the ice thickness is 315 m.). The effect i s not a very large one. In hole 322, for example, the surface and maximum velocities are 38.9 m./yr. and 39.5 m./yr. In few cases in fact do the individual values of exceed zero by more than twice their standard error. However, the general trend, and the fact that the effect i s shown at three boreholes in the same area, seem to leave l i t t l e doubt that the effect i s a real one. A similar result for a borehole on the Blue Glacier i s shown by Sharp (1960, p. 40) and other boreholes on the same glacier confirm i t (Sharp, private communication). The theoretical implications of this are discussed in section 7.4. The other conspicuous feature of the data i s the much greater distortion in the lower part of hole 209 than in hole 322. The velocity at the bottom of hole 322 i s 31.7 m./yr. or about 80% of the surface velocity. (The ice -103 thickness at this point was confirmed by seismic means. So i t is considered virtually certain that this hole extends to bedrock.) In hole 209, the velocity at the bed i s 7 m./yr. or about 25% of the surface velocity. This is an extrapolated value as measurements could not be made in the last 10 m. of this hole. It i s possible that the velocity at the bed i s less than 7 m./yr. The difference between conditions at the two holes can perhaps be explained by the configuration of the glacier bed. Figure 7 shows a hollow in the bed near hole 209. This hollow might well be f i l l e d with ice which i s semi-stagnant. The depth of the hollow appears to be of the order of 35 m. The velocity in hole 209 at this distance from the bed i s about 21 m./yr. or 75% of the surface velocity. This i s comparable with the figure at the bottom of hole 322. The implications of the borehole data regarding the flow law of ice and current theories of glacier flow are discussed in detail in section 7. 6.7. Ablation and Accumulation Ablation and accumulation data are given in Table 27. The ablation data are also shown in Figure 31. Abnormally low values of ablation at C2, E l , F l , and G8 can be explained by the thick cover of debris on the ice at these points. At other markers where the cover was thinner (B9, C13, C14, D l l , F9) the ablation i s only -104-slightly less than the average for clean ice. The most prominent feature of the data i s the great difference in ablation between the two seasons. The data show no significant variation of ablation rate with distance along the glacier. Nor i s any variation in total ablation clearly established for the part of the glacier below L10. The markers for which total ablation figures are available can be divided into two groups. The difference between the means of the two groups indicates a reduction of ablation of 1 cm. for each 4 m. rise in elevation. But i t i s quite possible that the difference between the two groups is merely a sampling fluctuation. The value i s less than the gradient of 1 cm. per 3 m. measured at comparable elevations on the Saskatchewan Glacier (Meier, 1960, p. 11). The net accumulation for the year 1959-60 as measured in a single snow pit on the crest between the drainage basins of the Athabaska and Saskatchewan Glaciers was 3 m. of f i r n . As was explained in section 4.8.2. no figure can be given for the mean accumulation on the lower part of the glacier. The number of measurements was insufficient. -105-7. DISCUSSION 7.1. Analysis of Borehole Data Measurements in boreholes can be used to determine — , ty the gradient, in the direction normal to the surface, of the ice velocity parallel to the surface. Interpretation of measurements of inclination of the pipe in terms of ice velocity i s not straightforward, however, and involves assumptions regarding the behaviour of the pipe. The d i s t r i -bution of velocity with depth plays a most important part in the subsequent discussion of glacier flow. These problems of interpretation w i l l therefore be treated f i r s t . Methods used to analyse the data in various borehole experiments w i l l f i r s t be reviewed brie f l y . The f i r s t borehole experiment was that of Gerrard and others (1952). Their inclinometers did not measure azimuth, but were arranged to measure inclinations in a plane parallel to the direction of flow. This azimuth control was not satisfactory in the f i r s t year. This did not greatly affect the results, however, as the pipe was nearly vertical then. Gerrard and others discuss possible sources of error. The couplings which joined the pipe sections were 2.5 cm. wider than the pipe i t s e l f . The length of the borehole changes in time due to deformation of the ice. The pipe cannot change i t s length and so i t must slide in the borehole. - 106-Th e protruding couplings oppose this. Gerrard and others state: "After about two weeks the ice in the borehole closed in around the pipe and gripped i t firmly." Another source of error was the tendency of the pipe, which was of steel, to sag under i t s own weight. Aluminium pipe has been used in a l l subsequent borehole experiments. Gerrard and others analysed their data on the assumption that flow was laminar. In other words, i t was assumed that the pipe measured the deformation due to a simple shear stress acting parallel to the surface. Subsequent measure-ments showed that the longitudinal strain rate at the surface had the comparatively high value of 0.14 per year, however. The borehole experiment on the Malaspina Glacier (Sharp, 1953a, b) was solely designed to test for "extrusion flow". The borehole was too far from any points on bedrock to permit measurement of surface movement. Surface strain rates were not measured. Analysis was therefore restricted to a plot of the deformation of the pipe over the two years. Reduction of the borehole data from the Saskatchewan Glacier (Meier, 1960, p. 30) was also carried out under the assumption of laminar flow. Allowance for the longitudinal strain rate of -0.013 per year was made in calculation of the flow law, however. As this borehole only extended to a depth of 43 m., deformation was small. Errors due to ice -107-strearning past the pipe, sagging of the pipe, and resistance of the couplings to flow of ice along the pipe, were therefore regarded as negligible. A laminar flow analysis was also used by Shreve for two boreholes on the Blue Glacier (Shreve, 1961). Surface measurements indicated that the longitudinal strain rate at these boreholes was "negligible". The f i r s t analysis in which allowance was made for the longitudinal strain rate was the re-analysis of the data of Gerrard and others by Nye (1957, p. 128-132). Mathews (1959, p. 452) also took account of the longitudinal strain rate (0.07 per year) in his analysis of data from the Salmon Glacier. Nye calculated the value of — at different depths by means of formula (21) of section 5.3. This procedure is subject to the assumptions that the pipe is straight, that ~ and |^ are constant in time and doc 3 j over the distance moved by the pipe in the year between observations, and that the ice slips freely along the pipe. The need for the f i r s t assumption is eliminated i f formula (23) of section 5.3. i s used. A l l that can be determined from measurements of surface markers or in boreholes are values of velocity and strain rate meaned over the distance which the markers have travelled between observations. A l l measurements are also made under the assumption that the glacier i s sufficiently near to a steady state so that the value of any of these quantities -108-at any point in space does not change appreciably between observations. This assumption, which seems a reasonable one, thus underlies a l l the measurements. The statement of Gerrard and others, that the pipe was gripped firmly by the ice, casts doubts on the assumption that ice slips freely along the pipe. In the present study, horizontal and vertical movement of the top of each pipe was measured by triangulation. The following table l i s t s the vertical velocities and those of nearby surface markers, arranged in order down the centreline. Holes 322 and 116 were some 100 m. off the longitudinal line. However, measurements on transverse lines indicated that the velocities varied l i t t l e over this distance. These data indicate that the vertical velocity of the top of each pipe was the same as that of the ice near the glacier surface. It is concluded that the pipe was gripped near the top. D i f f i c u l t i e s experienced during d r i l l i n g , Vertical Velocities V (m./yr., + downwards) (standard error —0.15) L16 L17 Hole 314 Hole 322 L18 L19 Hole 116 L20 L29 Hole 209 L30 Hole 194 no data 0.24 0.19 -0.42 -0.33 -0.14 -1.49 -1.70 -2.34 1.56 -0.19 -0.15 -109-when the pipes were seized by the ice 5 to 10 m. below the surface, makes this supposition quite likely. It i s attributed to the previous winter's "cold wave" not having been eliminated from the ice. The pipe cannot, however, have been gripped along i t s whole length. The longitudinal strain rate at the surface was compressive at a l l boreholes. The transverse surface strain rate was in a l l cases numerically small compared with the longitudinal one. The strain rate £. in the direction normal to the surface was therefore extending. The length of the borehole also increases with time as a result of the shear deformation. The pipe cannot stretch. It is therefore concluded that the pipe was gripped by the ice near the surface and that the remainder of the pipe was dragged upwards along the line of the borehole. As was pointed out by Savage, this casts serious doubts on the validity of the analysis described previously. Some data are also available as to whether ice tends to flow past the pipe rather than carry i t along. Hole 314 was situated on the longitudinal line between markers L17 and L18. Figure 24 shows the distances moved by the pipe and adjacent markers over the period July 1959 to August 1960. The point on the graph for hole 314 does not deviate signi-ficantly from the curve for the markers. This indicates that there i s no tendency for ice to flow past the pipe. The foregoing discussion indicates that the basis of -110-the f i r s t two methods of analysis of borehole data, as described in section 3.6., i s dubious. The third (integrated) method is therefore preferred. Values of computed by the different methods are given in Table 22. The conclusion, already stated in section 6.6. i s that, for these boreholes and times of observation, values calculated by the integrated method are significantly different from values calculated by the other methods. The broad conclusions, drawn from the borehole data in subsequent sections, are, however, unaltered whichever method of analysis i s used. 7.2. Validity of Assumptions The theory of section 5.1. rests on many simplifying assumptions. There w i l l be many places in real glaciers where these would not be expected to hold. The situation for the Athabaska Glacier w i l l be examined in this section. In section 5.1., assumptions I to V and X are general assumptions regarding the properties of ice. The only indication of whether or not these are valid comes from an overall comparison of f i e l d measurements with theoretical predictions. Assumption VI, that the glacier i s i n f i n i t e l y wide, is a poor assumption for valley glaciers. In the present case the width is only 3 or 4 times the depth. An alternative model, described in section 5.2., i s to regard the glacier as half a circular cylinder. Comparison of these two models i s - I l l -deferred to section 7.7. The assumption of plane strain (VII) w i l l be considered next. This implies that the transverse velocity w, the strain rates, £ , £ , £ . and their f i r s t derivatives with respect to z (where z i s the cross-glacier coordinate) are zero. Trans-verse surface strain rates ( £ ) are liste d in Table 18. The transverse strain rate does not exceed 0.01 per year except between L12 and L15 and between L33 and L35 where i t rises to 0.02 per year. It i s less than 0.005 per year at most places. The transverse strain rate was measured directly at the four boreholes (Table 19). Values were 0.002 (holes 322, 314), -0.001 (hole 116) and -0.0006 (hole 209). To neglect £ x thus appears to be a valid f i r s t approximation. Table 11 shows that, along the centreline, the direction of flow varies slowly from about 205° below the i c e f a l l to about 215° at L22 and back to 195° at the terminus. Thus the valley i s effectively straight. Thus the transverse velocity w on the centreline should be zero by symmetry. Table 11 also shows that there i s a region, roughly 250 m. wide, at the centre of each transverse line, across which the direction of surface velocity varies by not more than 1 or 2 degrees. In this region i t i s perhaps legitimate to assume that w and a l l i t s f i r s t derivatives are zero. Tables 10 and 14 give longitudinal and normal velocities on the transverse lines and show that — and are small near the centreline. The shear strain rates £„ and £, , -112-should thus be small. £ was measured at the surface at holes 322 and 116. Values were small but, at least at hole 322, differed significantly from zero (Table 19). The implication of this i s discussed further in section 7.4. It i s not certain whether the f i r s t derivatives of the strain rates with respect to z are zero. The longitudinal line of markers i s everywhere within 100 m. of the centreline (i.e. line of greatest velocity). The two boreholes (322 and 116) which l i e off the longitudinal line are as near the centreline as the markers are. It i s concluded that there i s some doubt as to how good an approximation the assumption of plane strain i s . In this respect the theory of section 5.2., which does not involve this assumption, may be preferable to that of section 5.1. i s regards the assumption (nil) that < and r ^ a r e independent of x i t i s useful to consider how rapidly such quantities as ice thickness, strain rates, and slopes vary with distance down the glacier. Profiles of surface and bed are shown in Figure 7. Ice thicknesses, and slopes and curvatures of surface and bed, are given in Tables 6 to 9. At L10.5, a short distance below the i c e f a l l , the bed has a slope of 17°. Between L35 and the terminus the ice thickness i s 100 m. or less and the slopes change quite rapidly in distances of this order. These areas w i l l there-fore be excluded from further consideration. Between L12 and L34 the surface slope varies gradually in the range 113-1.7° to 8.6°. Surface curvature i s about 10" per m. at L25 and L27 and three times that amount at L34. Otherwise —5 i t i s 5 x 10 per m. or less. Between L12 and L21 the bed slope changes slowly. It never differs from the surface slope by more than 5°. -4 Curvature of the bed does not exceed 4 x 10 per m. over this region. The bed slopes uphill at L21 and L23 and becomes more undulating below L27. There appears to be a rise with crest near L28, a depression near L30, and then a gradual rise to L32. Curvature of the bed increases to about 10~3 per m. at L29. For this reason, and because surface and bed slopes di f f e r by 10°, hole 209 may not be very well sited. The surface curvature i s very small there however (IO™5 per m.). Thus, throughout most of the lower section of the glacier, surface and bed slopes differ by only a few degrees and vary only slowly with distance x. More speci-f i c a l l y , this i s the case between L12 and L35 with the possible exception of the region between L27 and L32. The assumption should be particularly good in the area between L16 and L20 where boreholes 314, 322, and 116 are located. Longitudinal strain rates at the surface are given in Table 16 and Figure 17. The strain rate is compressive throughout virt u a l l y the whole region and does not exceed -0.03 per year below L14. It varies only slowly with -114-distance x. In general, the lower part of the Athabaska Glacier appears to be a very favourable region for which to make simplifying assumptions about the flow of valley glaciers. If the theory i s found to be inapplicable in this case, one might conclude that the places in real glaciers to which i t would apply are comparatively rare. 7.3. Variation of Longitudinal Strain Rate with Depth As was explained in section 5.1., the longitudinal strain rate i s expected, on Nye's theory, to be constant throughout the thickness of the glacier. Whether this is the case can be tested by two methods. Method 1* The hypothesis to be tested i s that -—^— - 0. Equation (16) of section 5.1 i s The method consists of using this equation to calculate / rlr ], the longitudinal strain rate averaged over the thickness. This value i s compared with (z^L \ , the value measured at the surface. The quantities above refer to the coordinate system of section 5.1. in which the x and y axes are respectively parallel and perpendicular to the surface. Measurements were made in the X, Y coordinate system in which the axes -115-are horizontal and vertical . The transformation equations are (for plane strain) -a =• (J cos + V J in (f. 5 >» The last equation can be simplified because 5j a n d 77 + J " 0 Hence, to a sufficient approximation, To avoid the assumption that L - 0 , equation (16) was modified to h±\ _ ^ - - t t b l*an ( cf> b - c ^ ) - h The conclusions are unaltered whether or not L i s taken X as zero, however. The quantities h, <zg, o b, Us, V s, ^^~rj > a n d ^ 2 w e r e -116-measured. Results are given in Tables 6, 7, 8, 10, 14, 16, and 18 respectively. U b i s known at boreholes 322 and 209. UQ was taken as the same fraction of U s at holes 314 and 116 as at hole 322, as these three holes were in the same area. At other points the limiting values = 0 and Ub => U s were taken. The calculation was carried out for the four boreholes, and the 11 points on the longitudinal line between L12 and L35 at which seismic measurements of h and afc were made. Results are given in Table 23. Accuracy of measurement must be taken into account. Standard errors of the various quantities are calculated in the relevant parts of section 4. They are also l i s t e d in the appropriate tables. As two extreme values were taken for U b there is no need to consider further errors in the term U D tan(<zb - o t s ) . Thus, from equation (16), £V0 S n where m = l " a n ( c f - c ^ ) Errors and typical mean values are £v Q = 0.35 m./yr. u b - 30 m./yr. £h - 5 m. v Q = 3 m./yr. m - - 0.15 to +0.2 h - 250 m. -117-Substitution of these values in the last formula gives a standard error of (' $3\ of about 12%. Thus i f and UxJ [•bx.J (—) dif f e r by more than about 25% of their mean value, i t i s l i k e l y that the difference i s a genuine one. Inspection of Table 23 shows that only in two cases (L23 and L30.5) does | — \ f a l l between the two limiting \ loo's values of (^) . At L14 is not far from the value of corresponding to U b = 0. (L14 i s , however, only about 600 m. from hole 322 where U b i s about 80% of Ug.) At a l l other points the discrepancies seem to be too large to be explained by experimental error. From L12 to L21 the strain rate appears to be less compressive at depth than at the surface. At L25, L27, L32, L34, and hole 209, the reverse i s true. These data therefore show that there are few places along the centreline of the glacier where the longitudinal strain rate is constant with depth. The modified theory of section 5.1. allows the longi-tudinal strain rate to vary (linearly) with depth. This theory cannot be verified numerically because the mean strain rate, as calculated by equation (16), i s bound to agree with the value r + \ Kh, with K given by equation (15). However, by equation (13), K i s equal to - — . This last quantity i s constant with depth, according to the theory. It can in theory be calculated by numerical differentiation of the measured values of v. In practice, however, v i s -118-about 3+0.35 m./yr. and experimental errors are such that numerical values of i t s second derivative are of l i t t l e value. However, the sign of zS can be determined from the data. This i s tabulated in Table 23. Signs are such that the mean strain rate should be greater or less than the surface strain rate according as — , i s negative or positive. Table 23 shows that only at L21 are the signs opposite to that predicted. Method 2. Boreholes 322 and 116 are on approximately the same flowline and about 160 m. apart. The distance between the pipes each year at different depths, and hence the longi-tudinal strain rate, can be calculated. Reduction of the pipe data has been treated in another section. The laminar flow analysis has been used in this case . If, at a given depth, x Q, x0* are the x coordinates t of the pipes in 1960, X j , x-^ the coordinates in 1961, l0> ^ the surface distance between the pipes in the two years, and t the time interval («*1 year), then <5x {(i + * ; - x , ) + ( i 0 + * ; - x 0 ) j This method of analysis should be valid in the upper 100 m. of the pipe because ^ i s small there and the values in the two pipes appear to be comparable. -119-Results are shown in the following table: y (m.) ~ (per yr.) 0 -.022 15.2 -.020 30.5 -.014 45.7 -.014 61.0 -.016 76.2 -.014 91.4 -.017 106.7 -.017 The experimental error i s large. These results should therefore be treated with reserve u n t i l another year's observations have been obtained. Nevertheless they support the conclusion that longitudinal strain rate becomes less compressive with depth. The observations of Glen (1956, p. 738) in the Austerdalsbre tunnel are of interest in this connection. The longitudinal strain rate decreased from 0.5 per year at the tunnel mouth to 0.15 per year at the end. The tunnel was approximately horizontal and of length 46 m. However, i t was at the foot of an i c e f a l l and the surface slope was 26°. Nye's theory i s not expected to hold under these circumstances. To sum up, the data which have been presented in this section suggest that the modified theory of section 5.1. i s -120 an improvement over the earlier version. 7.4. Variation of Velocity with Depth As was mentioned in section 6.6., ^ i s positive down to a depth of roughly 100 m. in holes 322, 314, and 116. The data are given in Table 22. The very high value of 0.14 at 15.2 m. in hole 116 i s probably the result of an original kink in the pipe. It i s thus questionable. Other values are generally in the range 0.005 to 0.010. The standard error i s 0.004. Few of the values differ from zero by more than twice the standard error. However, the general trend, and the fact that the effect i s shown at three boreholes in the same area, seem to leave l i t t l e doubt that the effect i s genuine. A similar effect has been observed on the Blue Glacier (Sharp, private communication). The stress solution (equation 7 of section 5.1) shows that T i s always negative. The rate of work £. T must be positive. Hence h± must be negative. Positive values of |^ are thus inconsistent with Nye's theory, in which ^ i s zero. On the revised theory (section 5.1.), however, i s not zero. Hence |^ may be positive provided that i S negative and numerically greater than jfj • If i t i s assumed that components of strain rate and stress deviator tensors are proportional, +• z~ must be zero at the surface because X U i s . He -121-a negative value of — at the surface implies a positive ^x value of ^ there. Values of — at the surface, calculated *J *x by numerical differentiation of measured values of v (see section 3.4.4.), are -0.0045, -0.0005, -0.0055, and -0.0011 at holes 322, 314, 116, and 209 respectively. The standard error i s about 0.005. The values are a l l negative and are comparable in magnitude with the values of — in the upper ty 100 m. of each borehole. The magnitude of the errors makes i t impossible to verify whether ^ +l^ r i S i n fact negative at a l l depths. But the 3j ox. results are not inconsistent with the revised theory. There i s also another factor which might explain this effect. According to equation (7*) of section 5.1. the stress solution i s where C = at" Z = O b Z. 6 was measured at the surface at holes 322 and 116. Values were .0034 and .0017 respectively (Table 19). Thus is expected to be positive at these points. Axes were chosen so that holes 322 and 116 l i e slightly on that side of the centreline where z i s negative. T* should be zero on the centreline by symmetry. Thus — ^ z i s negative at the boi holes. Its value i s unknown, but i t w i l l decrease the -122-numerical value of f and may even make T positive. In the latter case, values of would be positive even i f were zero. In practice f X 2, may vary with depth y, although the theory given above does not permit this. Thus i t i s possible that the effect i s only important near the surface. It i s hoped that a further series of measurements in the boreholes, combined with measurements of k at hole 209 (where £ x i s always negative), w i l l provide more information about this effect. 7.5. Numerical Check of Equation 18 Equation 18 of section 5.1. enables the longitudinal strain rate to be calculated from the ablation rate, ice thickness, flow, surface slope, and curvature. In Table 24, calculated and measured values are compared at the two boreholes and eleven other points on the longitudinal line where ice thickness was measured by seismic means. The flow q was calculated as the product of ice thickness and mean velocity. The latter was taken as 90% of the surface velocity, as explained in section 3.2.5. Surface curvature was calculated by the method of section 3.7.1. Except at L21, there i s no agreement. Equation 18 rests on the assumption that the shear stress on the bed of the glacier i s constant, in addition to a l l the previous assumptions (I to X in section 5.1.). It has been shown in preceding sections that some of these assumptions are -123-questionable. The lack of agreement in this case i s thus hardly surprising. 7.6. The Flow Law of Ice The present study i s concerned not so much with the flow law of ice i t s e l f as with i t s application to glaciers. However, the form of the flow law i s important in the sub-sequent discussion. Work on this subject w i l l therefore be reviewed b r i e f l y in this section. Numerous laboratory experiments have been made to determine the flow law of ice, both in single crystals and polycrystalline aggregates. While some details remain uncertain, the general properties are now well established. Results have been summarized by Glen (1958a, c). A single crystal deforms pl a s t i c a l l y only by gliding on the basal plane (McConnel, 1891; Mttgge, 1895; Glen and Perutz, 1954; Steinemann, 1954; Nakaya, 1958). It has long been uncertain whether there was any preferred glide direction (Glen and Perutz, 1954; Steinemann, 1954). This question has recently been discussed by Kamb (1961). No yield stress, below which no deformation occurs, has been found (Glen, 1958c, p. 171). The relation most frequently studied i s that between strain and time for constant load. Experiments with single crystals have been carried out by Griggs and Coles (1954), Steinemann (1954), Jellinek and B r i l l (1956), Butkovich and -124-Landauer (1958, 1959), and Kamb (1961), among others. For a given stress, the strain rate increases with time. Eventually, however (after perhaps 10 or 20 hours), the strain rate reaches a steady value. Resting of the crystal in the middle of the test does not restore the original part of the curve. Nor does a change of glide direction. Similar behaviour i s ob-served irrespective of whether the stress i s applied as a tension, compression, or shear. The flow law i s of the form i = A T where L i s the effective shear strain rate and f i s the effective shear stress, (Thus for experiments with uni-axial compression or tension £. i s J~3 x uniaxial strain rate and T i s (\//j3)x stress.) k and n are constants. Hydrostatic pressure does not affect the flow law provided that the difference between the temperature of the experiment and the melting point i s kept constant (Rigsby, 1958). This experiment was carried out at pressures up to 350 bars. The pressure never reaches this value in any glacier or ice-sheet except in Antarctica. The value of k depends on the temperature; that of n does not. Values of n obtained in the experiments li s t e d above range from 1.5 to 3,9 with a mean of about 2.5. Experiments with randomly oriented polycrystalline ice have been carried out by Glen (1952, 1955), Steinemann -125-(1958a, b), Butkovich and Landauer (1958, 1959), Vialov (1958), and Mellor (1959), among others. In contrast to the case of single crystals, the curve of strain against time under constant stress for polycrystals has an i n i t i a l transient stage in which the strain rate decreases. Thereafter strain rate attains a steady value. However, i f the stress i s greater than 4 bars the strain rate f i n a l l y increases again (this last figure refers to Glen's experiment, carried out near the melting point). A l l details of the relation between the properties of polycrystals and single crystals are not yet clear. However, the i n i t i a l decrease of strain rate can be ascribed to interference between crystals with different orientations. Production, by recrystallissation, of crystals more favourably oriented for glide in the direction of the stress produces the subsequent increase of strain rate (Glen, 1958a, c). Formation of complex interlocking grains makes intergranular s l i p very d i f f i c u l t . The flow law of a randomly oriented polycrystalline aggregate appears to be of the same form as that of single crystals, and with about the same value of index n. The following values of n refer to the steady part of the creep curve. They exclude the i n i t i a l transient part and also any possible f i n a l reacceleration under high stress. Glen (1954) obtained values of 3.2 or 4.2. Steinemann's (1958a, p. 25) values ranged from 1.9 at 1 bar to 4.2 at 15 bars. These values were for laboratory ice. Mellor (1959) obtained a -126-value of 4.2 for Antarctic glacial ice at -30°C for stresses between 2 and 15 bars. Butkovich and Landauer (1958) carried out two series of experiments, one with shear stresses in the range of 0.5 to 3 bars, the other with uniaxial compressions from 7 to 28 bars. The temperature was -5°C. Both commercial and glacial ice were used. A simple power law with n = 2.96 f i t t e d a l l the data. Two other proposed laws proved less satisfactory. These were i = A s\n h — To and t - a T + # - T J These results at high stresses do not agree with those of Glen, quoted above, or of Vialov (1958, p. 389), who observed a great increase of strain rate at stresses above about 5 to 7 bars (at -8°C). This effect i s not relevant to the present study because stresses on the glacier are certainly less than 2 bars. The possibility of a change in the flow law at a stress of about 1 bar is of considerable importance to glacier flow however. The shear stress i s less than this value throughout the greater part of a glacier. Glen (1955, p. 536) states that there may be a bend in the stress-strain rate curve at about 1 bar. Steinemann, as quoted above, obtained a value of 1.9 for n at 1 bar. Vialov (1958, p. 389) states that n 127-is reduced to 1.5 below 1 bar. Shoumskiy (1958, p. 248) states that only in the lowest layers of a glacier i s n equal to 3 or 4. It has a lower value throughout the bulk of a glacier. He identifies the transition point as that at which primary recrystalligation becomes the predominant mechanism of flow. Mellor (1959) also states that n i s 1.5 below 1 bar, but gives no evidence in support. Glen (1958a) c r i t i c i s e d Steinemann*s results on the ground that, at low stresses, the stress was not maintained for a sufficiently long period. The steady state was thus never reached. It i s not clear whether Vialov's results are free from this objection. The experiments of Jellinek and B r i l l (1956) showed Newtonian viscous flow. The strains were very much smaller than those in any other experiment, however, and the time period was shorter (2 or 3 hours). Weertman (1955, 1957a, b) has considered different dislocation mechanisms and derived theoretical creep laws. These are power laws with indices between 2.5 and 4.5 As was stated earlier, the constant k in the flow law depends on the temperature. Glen (1955, p. 532) showed Q that his data f i t t e d a variation of the form exp ( - — ). RT R is the gas constant and T the absolute temperature. Glen obtained a value of 31.8 K cal./mole for Q, the creep activation energy. Lliboutry (1959) obtained a value of 35.4 K cal./mole from various data. More precise measure-ments (Jellinek and B r i l l , 1956j Raraty and Tabor, 1958) 128-indicate that the correct value i s about 14 K cal./mole, however. From a study of phase equilibrium within polycrystalline ice, Steinemann (1958c) has concluded that the behaviour of ice at the melting point can be obtained by extrapolation of results for cold ice. There are no specific mechanisms to modify the flow law near the melting point. Certain d i f f i c u l t i e s arise in the application of these results to glacier flow. It must be assumed that crystals in glacier ice are randomly oriented so that the ice i s isotropic. It must also be assumed that each piece of ice remains under the same stress long enough for a steady state to be reached (these are assumptions I and II of section 5.1.). Glen (1958c, p. 180) has shown that the f i r s t condition may be relaxed. The ice may become anisotropic as a result of the stress but the direction of stress must not change. The complex stress systems which exist in a real glacier present greater d i f f i c u l t i e s . In a l l but one of the experiments described above the stress consisted either of uniaxial tension or compression, or of a simple shear. The sole exception was the experiment of Steinemann (1958a), in which uniaxial compression and shear were superimposed. Results have been interpreted by Glen (1958a, c). They appear to contradict the assumption (V and X of section 5.1.) that the relation between strain rates and stress deviators is of the form £ t • = X ( £ ^ ) <r[j . -129-It i s perhaps premature to reject this relation u n t i l further experiments have been carried out to confirm Steinemann's result. It seems unlikely that a general solution of the stress equations could be found for the case of a more general flow law which involves the third invariant £ of the stress deviator 3 tensor. Derivation of a flow law from measurements on glaciers w i l l be discussed in more detail in following sections. Previous studies have shown broad agreement between f i e l d and laboratory results. Nye (1957, p. 130) concluded that the results of the Jungfraujoch borehole experiment (Gerrard and others, 1952) were consistent with a flow law of the form £. = O ' 14.6 t Here -2- £• = EA > -2- t - £ A ., strains are measured in yr , stresses in bars. The constants have the values determined by Glen (1955). Mathews (1959) calculated a value of 2.8 for n from borehole data from the Salmon Glacier. Shreve (1960) obtained a value of 2.6 for two boreholes on the Blue Glacier. Hansen and Landauer (1958) obtained a value of 3.77 from the rate of closure of a borehole in the Greenland ice-cap. The temperature in this case was -25°C. Measurements of the rate of closure of tunnels dug in glaciers also give information on the flow law. Nye (1953) has analysed these data. He concluded that data from three tunnels (on Z'mutt Glacier, Vesl Skautbreen, and at -130-Jungfraujoeh) were consistent with a power law with n = 3.07. Data from another tunnel (on Arolla Glacier) were not. Sub-sequent measurements in a tunnel on Austerdalsbre did not agree with the law either, (Glen, 1956). However, these last two tunnels were each located at the foot of an i c e f a l l , where the simplifying assumptions of the theory are not expected to hold. Landauer (1959) obtained a value of n of 2.8 from measurements of shear in two tunnels at the edge of the Greenland ice-cap. The shear stresses were considerably less than 1 bar. He obtained a value of 3.2 from measurements of the rates of closure of the same tunnels. The tunnel data cover a lower range of stresses than do most other measure-ments. They lend no support to the suggestion that n i s about 1.5 for stresses below 1 bar. Meier (1960, p. 43) has proposed a flow law which includes a viscous term. He analysed a l l available data from boreholes and tunnels and also included Glen's laboratory results. He concluded that the flow law deviated significantly from a simple power law, and f i t t e d the relation Here Units of stress and strain rate are bars and yr respectively. Meier plotted the data from the Saskatchewan Glacier -131-twice, once f o r a laminar flow analysis and once with the longitudinal s t r a i n rate taken into account. He also included laminar flow analyses of data from boreholes at the Jungfraujoch and on the Malaspina Glaci e r . The measured s t r a i n rate at the Jungfraujoch was +0.14 per yr. Meier states that the Malaspina borehole was i n a region of strong compressive flow. Elimination from Meier's graph of a l l points f o r which laminar flow was erroneously assumed, removes a l l points from the "viscous" part of his curve ( i . e . a l l points f o r stresses le s s than 0.2 bars and most points f o r stresses l e s s than 0.8 bars). The remaining points (except f o r the Austerdalsbre tunnel data which l i e f a r from any other points on the graph) do not deviate s i g n i f i c a n t l y from a simple power law. The conclusions from t h i s section can be summarized as follows. The flow law f o r ice i s of the form £ = ft f . This applies to sing l e c r y s t a l s or randomly oriented poly-c r y s t a l l i n e aggregates, under u n i a x i a l compression or tension or simple shear, k but not n depends on the temperature. Neither depend on the hydrostatic pressure. Measured values of n vary between 2 and 4.5 with a mean of about 3. The value of k appears to be about 0.3 or 0.4 at 0°C, for 6- i n y r " 1 and T i n bars. This r e l a t i o n holds f o r stresses between 1 and 5 bars and possibly over a wider range. It i s also possible that n f a l l s to about 1.5 f o r stresses below 1 bar. The extension of the equation to complex stress systems i n the form i s doubtful. -132-7.7. Effect of Valley Sides In section 5.1. the shear stress was shown to be = - jo J J sin c* (equation 7) It was assumed, among other things, that the width of the glacier was i n f i n i t e . An alternative approach was adopted in section 5.2. The glacier was regarded as half of a circular cylinder. The shear stress in this case i s T x r. = - j f J * s m <*s (equation 19) For arbitrary cross-section i t i s approximately T « r = ~ Ff J r <*s (19a) where the "shape factor" F » — r • A i s the area of cross-ph section of the glacier, p i s the perimeter excluding the upper surface, and h i s the ice thickness at the centreline. On a vertical plane along the centreline r equals the depth y. If a flow law, based on laboratory results, i s assumed to be applicable in a glacier, the relative merits of these two assumptions can be assessed* Curves of — against y ty are calculated from the flow law. They are then compared with the observed results from the two deep boreholes. Glen's flow law for quasi-viscous creep was adopted, (Glen, 1955). Reasons for this choice w i l l be given in section 7.9. In the present notation the law i s -133-Units of ylTd and yi ^  are yr. 1 and bars respectively. It i s sufficiently accurate for the present purpose to use an analysis in which the longitudinal strain rate i s zero. This simplifies calculation of the theoretical curves. In this case V i i -2. oy The surface slope a s was measured at each borehole. Values of F for each transverse line were determined from the profiles of Figures 10 to 13. Values at the boreholes were found by linear interpolation between these. The values were 0.58 and 0.62 for holes 322 and 209 respectively. Curves were also drawn for F - 1, the case of an i n f i n i t e l y wide glacier. Calculated and observed curves are compared in Figure 25. Strain rates calculated without the shape factor are 10 to 20 times greater than those observed. The assumption that the glacier i s i n f i n i t e l y wide i s thus invalid in this case. Adoption of a flow law with a lower index than 4.2 would have reduced the separation between the theoretical curves. The conclusion would not have been altered, however. -134-A more complicated calculation which makes allowance for the longitudinal strain rates also leaves the conclusion unchanged. The general solution of the stress equations in cylindrical coordinates has not been obtained. The solution for the case of the in f i n i t e sheet, as given in section 5.1., has therefore been adopted, with the stress solution modified by the shape factor. The shear stress i s T X y = - F ^ j j - s m c ^ . In effect, every component of the stress deviator tensor i s multiplied by F , because the measured strain rates are unchanged whether F i s inserted or not, and i t i s assumed that components of strain rate and stress deviator tensors are proportional. The constant of proportionality i s thus effectively multiplied When the glacier i s regarded as a cylinder, measurements of change of surface velocity on transverse lines provide information about the flow law. In equation (19a) r can be taken as z, the distance measured across the glacier surface from the centreline. It must be assumed that the longitudinal and transverse strain rates and the velocity normal to the surface are zero, and that the surface i s a plane. In this case, by V F . -135-A l l quantities on the right-hand sides of these equations are known. Thus the relation between §Eg and \t!x can be derived. For the D and E lines, — at the surface on the centre-line has values -0.004 and +0.005 respectively (Table 16). Corresponding values of |^ are -0.002 and +0.003 (Table 18). These are small compared with values of ^ , except near the centreline. The case of the E line i s slightly doubtful because there are diagonal crevasses a short distance down glacier from i t . The assumption that the glacier surface i s a plane i s reasonable, but the assumption that v i s zero i s at best a rough approximation. One practical d i f f i c u l t y i s accurate location of the centreline. Its position was determined by interpolation between the two markers in each line where the velocity was greatest. Its position and thus the values of z may be in error by perhaps 25 metres. Values of for markers near the centreline are thus inaccurate. Values of — and T on the D and E lines are given in Table 25 and their logarithms are plotted in Figure 28. Agreement with the linear relation predicted by the flow law i s considered to be satisfactory in view of the inaccuracy of the data. If on the other hand the glacier i s regarded as an i n f i n i t e sheet, and i f the flow i s laminar (i.e. i f the surface and bed are parallel planes and i f the velocity vector i s at a l l points parallel to the surface and directed down the direction of greatest surface slope) the velocity -136-at any point on the surface i s The index m depends on the r e l a t i v e proportions of d i f f e r e n t i a l movement i n the ice and s l i p on the bed (Nye, 1959c, p. 497). The i c e thickness at the point i s h. A l i n e a r r e l a t i o n between log u and log(h s i n as) i s thus expected. These values fo r the D and £ l i n e s are plotted i n Figure 29. No r e l a t i o n -ship i s apparent. However, f a i l u r e of the assumption of laminar flow may account f o r t h i s . Figures 28 and 29 can be regarded as a d d i t i o n a l evidence fo r regarding the g l a c i e r as a cylinder, rather than an i n f i n i t e sheet. 7.8. Stress and St r a i n Rate at Boreholes The second invariants of the s t r a i n rate and stress deviator tensors were evaluated f o r each point at which i n c l i n a t i o n s were measured i n the two deep boreholes. The formulae are given below. Their derivation and the meanings of the symbols are given i n section 5.1. tx. J ^ = 5 - K x ^x -137-y T ~ [ Xx.) > ^ = ( bo.y were measured at the surface. — was measured at various depths y. h, u b, and the value of x corresponding to a given value of y were also derived from the borehole data. F was obtained by interpolation between values on the trans-verse lines. J> was taken as 0.91. The upper part of hole 322 (above 150 m.), in which — differs from zero by less than twice i t s standard error, has been excluded from the analysis. Values of §E 2 and £ £ are given in Table 26. Their logarithms are plotted in Figures 26 and 27. It should be noted that the data have not been smoothed. It is d i f f i c u l t to make a precise error analysis of these data. The errors may be roughly assessed as follows. In the upper part of each borehole the major part of -138-iE« I s f — \ . The standard error of h± i s about 10% (see 2 2 I 3xJ section 4.5.). Hence the standard error of ^ E 2 i s about 20%. In the lower part of each borehole the major part of £ E 2 i s (x^^ • T h e error of ^ i s again of the order of 10% in hole 209, but somewhat greater than this in hole 322 (see Table 22). A 20% error in | E 2 w i l l cause an error of 0.2 in log ^ E 2 . The error in y i s small. Although F and as are subject to errors of the order of 5%, these w i l l affect a l l values for the one borehole equally. The error in f i s thus negligible. There w i l l be an error in \ i!^ however, because this quantity i s calculated from £ E 2 • A linear relation between log ^ E 2 and log ^ i! i s regarded as a satisfactory f i t at each borehole, within the accuracy of the present data. This does not, of course, exclude the possibility that measurements over a period of several years w i l l reveal significant deviations from a linear relationship. The data from both boreholes and the transverse lines are a l l plotted on Figure 30. Within the experimental error, a l l the points for the boreholes and some of those for the transverse lines appear to l i e on the same straight line. This i s interpreted as an indication that the basic assumptions of Nye's theory are reasonable approximations. The important assumptions are that T A^ varies only slowly with x, that components of strain rate and stress deviator tensors are proportional, and that the constant of proportion--139-a l i t y depends only on £ . The question of the optimum value of the index in the power law i s discussed in the next section. 7.9. Comparison of Data with Different Flow Laws Two sli g h t l y different approaches can be used to obtain information about the flow law from glacier data. Different forms of flow law can be assumed, and the optimum values of the numerical constants in these laws can be deduced from the data. Alternatively, one can test whether the data deviate significantly from a flow law determined in the laboratory. Both approaches w i l l be considered here. A simple power law was assumed and the value of the index n calculated as the regression coefficient of log £ E 2 on log \ £^ . Results for the four sets of data were as follows: Data n number of points D line 1.4 5 E line 2.2 4 Hole 322 2.3 12 Hole 209 5.2 12 It w i l l be explained later why values of n determined from the transverse lines might be less than those determined from boreholes. The standard error of n, as determined from the regression -140 analysis, i s about 0.8 for the two boreholes. Errors of the other data w i l l be considerably greater. As was explained in section 6.6, conditions in the lowest 35 m. of hole 209 may be unusual as a result of the configuration of the bed. If the last three points are omitted from the regression, the value of n for this hole i s reduced to 2.8. The fact that the value can be almost halved by omitting three points reinforces the view, suggested by the large standard error, that values of n derived by this method are unreliable. The inclinometer used in these experiments reads to 0.1°. Considerable care was taken in making the measure-ments. The overall accuracy could be improved sli g h t l y by reduction of the spacing between measurement points in the pipe. This w i l l be done in future measurements. However, i t i s considered unlikely that the standard of accuracy can be improved to any great extent. The accuracy of the data from the transverse lines could be improved considerably by a considerable increase in the number of markers. Apart from inaccuracies in measurement, the complex stress systems which exist in glaciers make interpretation of observations d i f f i c u l t . The most that one can expect to do i s to demonstrate whether glacier measurements are consistent with a flow law determined in the laboratory. It i s desirable to know the value of k, the multiplying constant in the flow law, as well as the index n. The constant k depends on the temperature. A value determined -141-at the same temperature as the glacier, or else one adjusted for the temperature difference, must therefore be used. Two of the most extensive series of laboratory experiments are those of Glen (1955) and Butkovich and Landauer (1958). Their values have been taken. It i s not absolutely certain that the Athabaska Glacier is at the pressure melting temperature. Flow laws for -1.5°C as well as 0°C have therefore been used. The constants are given below. They refer to a flow law of the form / j E A = \ ( J i i'^ ) ^  . Units of /JT a ? yiT; , and k are yr."*, bars, and yr" 1 bar"* respectively. Source k n T°C Glen ("quasi-viscous creep") 0.148 4.2 0 0.020 4.2 -1.5 Glen ("minimum observed creep") 0.854 3.2 0 0.116 3.2 -1.5 Butkovich and Landauer 0.435 2.96 0 0.372 2.96 -1.5 The temperature correction i s somewhat uncertain. Glen measured the "minimum observed creep" at four different temperatures. He assumed that k varied with temperature according to the Boltzmann law exp. ( — — ) and obtained RT the value of Q which best f i t t e d his data. The point corresponding to measurements at -0.02°C did not l i e on the curve, however. Thus the change in the value of k, as -142-tabulated, for a temperature change from -1.5°C to 0°C i s much greater than predicted by the Boltzmann law with Q = 31.8 K cal./mole. Moreover, t h i s value of Q i s now con-sidered to be too high (see section 7.6.). In the table above, the values of k for the minimum observed creep at the two temperatures are Glen's values. Glen only gives a value of k at 0°C f o r the case of quasi-viscous creep. The value at -1.5°C i n the table was calculated so that the r a t i o s of k at the two temperatures were the same for the two creep laws. The experiments of Butkovich and Landauer were c a r r i e d out at -5°C. The value of k at 0°C has been calculated by the Boltzmann law with a value of Q of 14 K cal./mole, as has been done by Weertman (1961, p. 960). The value at -1.5°C has been calculated i n the same way. The change of k with temperature i s very much smaller i n t h i s case than i n Glen's r e s u l t s . However, i t has not been c l e a r l y established that k does vary with temperature according to the Boltzmann Law, e s p e c i a l l y near the melting point. Thus Glen's determinations of k which were made at the two temperatures i n question are to be preferred. These s i x flow laws are plotted i n the form log § E 2 against log § z.^ i n Figure 30. A l l points from the two deep boreholes and the two transverse l i n e s are also shown. If the points at the lower end of the curve ( i . e . f o r stresses less than 1 bar) are disregarded, Glen's law f o r quasi-viscous creep (n = 4.2) provides the best f i t to the -143-data. A l l the borehole points l i e between the two lines which correspond to this law. The f i t would be very satisfactory i f the glacier had a mean temperature of about -0.75°C rather than the pressure melting temperature. Unfortunately, no temperature measurements are available. It i s to be hoped that some w i l l be made at some future date. The possibility that n may be reduced for stresses below 1 bar was discussed in section 7.6. The deviations from the straight lines at low stresses in Figure 30 might be taken to support this theory. In addition, the values of n calculated from the D and £ lines are 1.4 and 2.2. The mean stresses are 0.33 and 0.66 bars respectively. The inter-pretation i s doubtful however. The points which show this trend are a l l derived from the transverse surface profiles and at points within 200 m. of the centreline. The position of the centreline was not determined to better than +25 m. Thus the calculated shear stresses at the points are not very reliable. Also the shear strain rate in this region is l i t t l e greater than the longitudinal and transverse strain rates. Thus the assumption on which the calculation rests i s dubious. In addition, measurements of surface velocity include both differential motion within the ice and slipping of the glacier on i t s bed. At hole 322, which i s about 500 m. up glacier from the D line, slipping represents about 80% of the total velocity. Weertman (1957c) has put forward a -144-theory of the slipping of a glacier on i t s bed. There i s l i t t l e experimental evidence for or against i t . According to this theory the velocity of slipping i s proportional to RTi + l + w n e r e i s t h e shear stress on the bed, R measures the roughness of the bed, and n i s the index in the flow law. When the glacier i s regarded as a cylinder, as here, X should be constant along a transverse line. In this case only differential motion within the ice would contribute to — , the rate of change of velocity with distance from the centreline. But the theory may be an oversimplification. In addition, the roughness of the bed might be smaller near the sides of the glacier than near the centre. This might result either from the nature of the bed or because meltwater was more plentiful at the sides than near the centre. This would reduce the value of n derived from measurements of variation of ^ with z. The question might be resolved by measurements of surface velocity at several points near the middle of a transverse line in a region where longitudinal and transverse strain rates were small and the glacier was not slipping on i t s bed. An alternative method would be to make measurements in the upper part of a borehole where longitudinal and trans-verse strain rates were less than about 0.005 per yr. None of the present boreholes meet this requirement. -145-APPENDIX 1 Alternative Derivation of Equation (16) The following alternative derivation of equation (16) in section 5*1. i s due to Savage. o Take x, y axes in directions of tangent and normal to the surface at the origin. Let y = y s (x,t) be the surface y = y D (x) be the bed. Consider a particle of ice just below the surface. Let D denote differentiation following the particle. Pec Du Let <JL = ) if — —2. (Velocities are measured D t P t following the particle.) Let ] denote distance below ice surface, and h ice thickness. - 1 4 6 -o*JC C<X ox - 0 l i b = m = f a n (d- b-<^) where d>b - * - slope of bed relative to x axis. The continuity equation can be written as }x * t 1 where "u/ i s the rate of accumulation (or ablation). The flow = I it J k J b otx I -147-" D t > x ty at dec. But* p t Dt and H = ^  The p a r t i c l e i n question i s just below the surface. So u = u s and v - v g i n the l a s t two equations. Substitution i n the continuity equation y i e l d s which i s equation (16). It i s important to note that the ablation (or accumulation) rate does not occur i n the f i n a l equation (16). -148-APPENDIX 2 Generalizations of the Theory Some possible generalizations of the theory of section 5 . 1 . w i l l be outlined here. The results appear to be of l i t t l e practical value, however. F i r s t , a three-dimensional treatment w i l l be considered. In this, assumptions I to VI and X of the previous section stand. VII (plane strain) i s not made. VIII i s replaced by the assumption that every component of the stress deviator tensor i s independent of x. IX i s replaced by the assumption that the glacier i s of constant thickness h. The equations for the velocities are obtained as follows. The incompressibility condition gives The stress deviator components being independent of x give 'hup = 0 O 0 o o - 1 4 9 -= 0 2. The boundary conditions are that v =• 0 on y = h for and z. It i s also necessary to make the s i m p l i f y i n g assumption that the v a r i a t i o n of each v e l o c i t y with z i s l i n e a r . The following v e l o c i t y solutions are obtained u = <"0 + a x + ax 2 + a , a 4 x z + f, (j) The constants are the values of the following quantities at the o r i g i n . a l l x and z. Also £ = t = 0 , on y = 0 for a l l x V = - ( a , +• a 0 + a , a 4 z)( j - h) a Yx tpt(o) = <pt'(o) = (fx(o) = tpA'(o) = 0 The functions can be evaluated i f the flow law i s known. The stress equations reduce to -150-» o >3 Hz « O The boundary conditions are IT (atmospheric pressure) r = r = o on y = 0 for a l l x, z« The solutions are T r u tot 1 1 ~ f3i3 da One objection to this solution i s that in i t r ~ , i s independent of y. An alternative approach i s to consider plane strain but to attempt to eliminate the assumptions that £. and X* are functions of y only. If i t i s assumed instead that cr. i s independent of x (i.e. = ), the equations for -151 the stresses become Hence 2>x* fy* 6) Txy _ ^ Yxvf dx Solutions are ^ - f. <* + 3. 1 + The boundary conditions for a plane surface are ^ - ° } on y =» 0 for a l l x . To satisfy these h " f a , 3> = -3* Hence dx - ° > __L*y = o and thus ax 0 , l £ a * 0 >x Thus the assumption that <r^ i s independent of x implies that £ and T v , are also. The single assumption that T i s independent of x leads to a more general case. In this case - 1 5 2 -V = ^ ' a s before, but T i s an arbitrary function of y, and i s a function of both x and y. Further simplifying assumptions are necessary. An alternative approach i s to consider plane strain and assume that <r ' - rrj - o , without assuming that any of the stress components are independent of x. Since <rx . cr - O , ^ = ^ > - ^ and this case i s included in one which was treated before. -153-BIBLIOGRAPHY The following abbreviations are used. I.A.S.H. 46 - International Association of Scientific Hydrology, Publication 46; Proceedings of General Assembly at Toronto, 1957, Vol. IV, Snow and Ice. Similarly, Publication 47; Proceedings of Symposium at Chamonix, 1958, Physics of the movement of ice. Publication 54; Proceedings of General Assembly at Helsinki, 1960, Snow and Ice. Publication 55; Proceedings of Symposium on Antarctica, Helsinki, 1960. J. Glac. - Journal of Glaciology. P.R.S. - Proceedings of Royal Society (London), Series A. Butkovich, T.R., and Landauer, J.K. 1958. The Flow Law for Ice. I.A.S.H. 47, p. 318-25. Butkovich, T.R., and Landauer, J.K. 1959. The Flow Law for Ice. U.S. Snow, Ice and Permafrost Research Establishment, Research Report 56. Collier, E.P. 1960. Study of Glaciers in Banff and Jasper National Parks 1960. Water Resources Branch, Department of Northern Affairs and National Resources. Demorest, M. 1941. Glacier Flow and i t s Bearing on the Classification of Glaciers. Bulletin of Geological Society of America, Vol 52, p. 2024-25. Demorest, M. 1943. Ice Sheets. Bulletin of Geological Society of America, Vol. 54, p. 363-400. Field, W.O., and Heusser, C.J. 1953. Glacier and Botanical Studies in the Canadian Rockies 1953. Canadian Alpine Journal, Vol. 37, p. 128-40. -154-Finsterwalder, S. 1897. Der Vernagtferner. Wissenschaftliche Erganzungshefte zur Zeitschrift der deutsch und Gsterreich. Alpenvereine, Vol. 1, No. 1, p. 1-112. Gerrard, J. A.F., and others. 1952. Measurement of the Velocity Distribution along a Vertical Line through a Glacier, by J.A.F. Gerrard, M.F. Perutz and A. Roch. P.R.S., Vol. 213, No. 1115, p. 546-58. Glen, J.W. 1952. Experiments on the Deformation of Ice. J. Glac., Vol. 2, No. 12, p. 111-15. Glen, J.W. 1955. The Creep of Polycrystalline Ice. P.R.S., Vol. 228, No. 1175, p. 519-38. Glen, J.W. 1956. Measurement of the Deformation of Ice in a Tunnel at the Foot of an Ice f a l l . J. G l a c , Vol. 2, No. 20, p. 735-45. Glen, J.W. 1958a. The Mechanical Properties of Ice. 1. The Plastic Properties of Ice. Advances in Physics, Vol. 7, No. 26, p. 254-65. Glen, J.W. 1958b. The Slip of a Glacier past i t s Side Walls. J. G l a c , Vol. 3, No. 23, p. 188-92. Glen, J.W. 1958c. The Flow Law of Ice. I.A.S.H. 47, p. 171-83. Glen, J.W. 1961. Measurement of the Strain of a Glacier Snout. I.A.S.H. 54, p. 562-67. Glen, J.W., and Perutz, M.F. 1954. The Growth and Deformation of Ice Crystals. J. G l a c , Vol. 2, No. 16, p. 397-403. Griggs, D.T., and Coles, N.E. 1954. Creep of Single Crystals of Ice. U.S. Snow, Ice and Permafrost Research Establish-ment. Report II. Hansen, B.L., and Landauer, J.K. 1958. Some Results of Ice-Cap D r i l l Hole Measurements. I.A.S.H. 47, p. 313-17. Heusser, C.J. 1956. Postglacial Environments in the Canadian Rocky Mountains. Ecological Monographs, Vol. 26, p. 263-302. Jellinek, H.H.G., and B r i l l , R. 1956. Viscoelastic Properties of Ice. Journal of Applied Physics, Vol. 27, No. 10, p. 1198-1209. -155-Kamb, W.B. 1961. The Glide Direction in Ice. J. Glac., Vol. 3, No. 30, p. 1097-1106. Kanasewich, E.R. 1960. An Interpretation of Some Gravity Measurements in the Canadian Cordillera. Thesis, University of Alberta. Lagally, M. 1934. Mechanik und Thermodynamik des stationaren Gletschers. Leipzig. Landauer, J.K. 1959. Some Preliminary Observations on the Pla s t i c i t y of Greenland Glaciers. J. G l a c , Vol. 3, No. 26, p. 468-74. Lliboutry, L. 1961. Temperature des couches inferieures et Vitesse d'un inlandsis. Comptes Rendus, Vol. 252, No. 12, p. 1818-20. Mathews, W.H. 1959. Vertical Distribution of Velocity in Salmon Glacier, B r i t i s h Columbia. J. G l a c , Vol. 3, No. 26, p. 448-54. McConnel, J.C. 1891. On the Pla s t i c i t y of an Ice Crystal. P.R.S., Vol. 49, No. 299, p. 323-43. Meier, M.F. 1960. Mode of flow of Saskatchewan Glacier, Alberta, Canada. U.S. Geological Survey Professional Paper No. 351. Mellor, M. 1959. Creep Tests on Antarctic Glacier Ice. Nature, Vol. 184, No. 4687, p. 717. MUgge, O. 1895. Uber die Plasticitat der Eiskrystalle. Neues Jahrbuch flir Mineralogie, Geologie und Palaeontologie, Vol. 2, p. 211-28. Nakaya, U. 1958. The Deformation of Single Crystals of Ice. I.A.S.H. 47, p. 229-40. Nizery, A. 1951. Electrothermic Rig for the Boring of Glaciers. Transactions of the American Geophysical Union, Vol. 32, No. 1, p. 66-72. Nye, J.F. 1951. The Flow of Glaciers and Ice-Sheets as a Problem in Plasticity. P.R.S., Vol. 207, No. 1091, p. 554-72. Nye, J.F. 1952a. A Method of Calculating the Thickness of the Ice-Sheets. Nature, Vol. 169, No. 4300, p. 528-30. -156 Nye, J.F. 1952b. The Mechanics of Glacier Flow. J. G l a c , Vol. 2, No. 12, p. 82-93. Nye, J.F. 1953. The Flow Law of Ice from Measurements in Glacier Tunnels, Laboratory Experiments, and the Jungfraufirn Borehole Experiment. P.R.S., Vol. 219, No. 1139, p. 477-89. Nye, J.F. 1957. The Distribution of Stress and Velocity in Glaciers and Ice-Sheets. P.R.S., Vol. 239, No. 1216, p. 113-33. Nye, J.F. 1958a. Surges in Glaciers. Nature, Vol. 181, No. 4621, p. 1450-51. Nye, J.F. 1958b. A Theory of Wave Formation in Glaciers. I.A.S.H. 47, p. 139-54. Nye, J.F. 1959a. The Deformation of a Glacier below an Icefa l l . J. G l a c , Vol. 3, No. 25, p. 387-408. Nye, J.F. 1959b. A Method of Determining the Strain Rate Tensor at the Surface of a Glacier. J. G l a c , Vol. 3, No. 25, p. 409-19. Nye, J.F. 1959c. The Motion of Ice-Sheets and Glaciers. J. G l a c , Vol. 3, No. 26, p. 493-507. Nye, J.F. 1960. The Response of Glaciers and Ice-Sheets to Seasonal and Climatic Changes. P.R.S., Vol. 256, No. 1287, p. 559-84. Raraty, L.E., and Tabor, D. 1958. The Adhesion and Strength Properties of Ice. P.R.S., Vol. 245, No. 1241, p. 184-201. Reid, I.A. 1961. Triangulation Survey of the Athabaska Glacier July 1959. Water Resources Branch, Department of Northern Affairs and National Resources. Rigsby, G.P. 1958. Effect of Hydrostatic Pressure on Velocity of Shear Deformation of Single Ice Crystals. J. G l a c , Vol. 3, No. 24, p. 273-78. Seligman, G. 1947. Extrusion Flow in Glaciers. J. G l a c , Vol. 1, No. 1, p. 12-21. Sharp, R.P. 1953a. Deformation of a Vertical Borehole in a Piedmont Glacier. J. G l a c , Vol. 2, No. 13, p. 182-84. -157-Sharp, R.P. 1953b. Deformation of Borehole in Malaspina Glacier, Alaska. Bulletin of the Geological Society of America, Vol. 64, No. 1, p. 97-99. Sharp, R.P. 1960. Glaciers. Eugene, Oregon, University of Oregon Press. Shoumskiy, P.A. 1958. The Mechanism of Ice Straining and i t s Recrystallization. I.A.S.H. 47, p.244-48. Shoumskiy, P.A. 1961a. On the Theory of Glacier Motion. I.A.S.H. 55, p. 142-49. Shoumskiy, P.A. 1961b. The Dynamics and Morphology of Glaciers. I.A.S.H. 55, p. 152-61. Shreve, R. 1961. The Borehole Experiment on Blue Glacier, Washington. I.A.S.H. 54, p. 530-31. Somigliana, C. 1921. Sulla profondita dei ghiacciai. Rendiconti della accademia nazionale dei Lincei, No. 30. Stacey, J.S. 1960. A Prototype Hotpoint for Thermal Boring on the Athabaska Glacier. J. Glac., Vol. 3, No. 28, p. 783-86. Steinemann, S. 1954. Results of Preliminary Experiments on the Plastic i t y of Ice Crystals. J. G l a c , Vol. 2, No. 16, p. 404-13. Steinemann, S. 1958a. Experimentelle Untersuchungen zur Plastizitat von Eis. Beitrage zur Geologie der Schweiz, Geotechnische Serie, Hydrologie, No. 10. Steinemann, S. 1958b. Resultats experimentaux sur l a dynamique de l a glace et leurs correlations avec le mouvement et l a petrographie des glaciers. I.A.S.H. 47, p. 184-98. Steinemann, S. 1958c. Thermodynamics and Mechanics of Ice at the Melting Point. I.A.S.H. 47, p. 254-65. Streiff-Becker, R. 1938. Zur Dynamik des Firneises. Zeit-schrift fur Gletscherkunde, Vol. 26, p. 1-21. Stutfield, H.E.M., and Collie, J.N. 1903. Climbs and Explorations in the Canadian Rockies. London, Longmans, Green and Co. -158-Vialov, S.S. 1958. Regularities of Ice Deformation. I.A.S.H. 47, p. 383-91. Ward, W.H. 1958. Surface Markers for Ice Movement Surveys. I.A.S.H. 47, p. 105-10. Weertman, J. 1955. Theory of Steady State Creep based on Dislocation Climb. Journal of Applied Physics, Vol. 26, No. 10, p. 1213-17. Weertman, J. 1957a. Steady State Creep through Dislocation Climb. Journal of Applied Physics, Vol. 28, No. 3, p. 362-64. Weertman, J. 1957b. Steady State Creep of Crystals. Journal of Applied Physics, Vol. 28, No. 10, p. 1185-89. Weertman, J. 1957c. On the Sliding of Glaciers. J. G l a c , Vol. 3, No. 21, p. 33-38. Weertman, J. 1958. Travelling Waves on Glaciers. I.A.S.H. 47, p. 162-68. Weertman, J. 1961. Equilibrium Profile of Ice-Caps. J. Glac., Vol. 3, No. 30, p. 953-64. TABLE 1 POSITIONS AND ELEVATIONS OF TRIANGULATION STATIONS. Units of coordinates are cm. Elevation i s cm above datum of 6000 feet (« 182880 cm). Station E N Elevation 1 200000 200000 14266 2 not used 3 200000 249276 14942 4 294909 246037 11658 5 191724 288699 21020 6 313580 317056 25677 7 201720 347205 28749 8 313822 365583 32921 9 218963 409146 32850 10 314319 383070 34092 11 241414 466335 42414 12 330010 401102 33193 13 264965 510239 47076 14 364695 439354 40326 15 285971 549786 58838 16 412254 504918 45952 17 325728 591070 42191 18 453176 583609 52152 19 337211 616872 51617 20 468119 626003 56232 21 356578 660399 57652 TABLE 2 INITIAL POSITIONS AND ELEVATIONS OF MARKERS. Units of coordinates are cm. Elevation is cm above datum of 6000 feet (=• 182880 cm). Reference point i s top of marker. Results are corrected i f marker was leaning. Standard errors of position and elevation are 10 cm except for markers with an asterisk. The standard errors of these are 50 cm. Marker Date E N Elevat; A 1 17.8.59 376159 669537 53996 2 21.7.59 382423 668065 54346 3 II 388080 666335 54562 4 II 399644 663342 54593 5 II 410393 659676 54391 6 17.8.59 421059 656290 53774 7 21.7.59 426304 655238 53878 B 1 t i 362237 625508 44122 2 t i 368327 622594 43840 4 it 381878 616212 44271 6 i t 397065 609073 44684 7 II 406433 604664 44683 9 II 423536 596233 44060 C 2 II 327198 577054 40814 3 it 335941 573617 40425 5 it 350967 567929 40920 7 tt 365048 562631 41241 9(L17) 22.7.59 378699 557469 41388 11 21.7.59 392803 552240 41274 13 n 408996 546219 41158 D 2 14.8.59 307593 524141 37123 3 20.7.59 315271 520687 37986 4 tt 324501 516392 38135 5 23.7.60 332347 509167 38187 6 20.7.59 343303 507661 38276 7(L21) tt 349235 505024 38212 8 tt 352641 503345 38243 9 23.7.60 359389 496568 38191 10 20.7.59 369902 495414 38331 11 23.7.60 378427 489019 38583 Table 2 continued E 1 2 0 . 7 . 5 9 2 6 8 8 1 4 4 7 2 3 1 9 3 4 4 2 9 2 2 0 . 7 . 5 9 2 7 4 9 2 1 4 6 7 9 6 2 3 3 5 8 4 3 II 2 8 5 1 0 6 4 6 0 7 8 9 3 4 3 5 2 4 i» 2 9 3 5 2 9 4 5 4 8 8 9 3 4 4 6 4 5 1 4 . 8 . 5 9 3 0 2 2 4 5 4 4 8 4 8 0 3 4 4 1 4 6 2 0 . 7 . 5 9 3 0 9 7 0 3 4 4 3 5 4 4 3 4 4 8 9 7 it 3 1 7 9 3 6 4 3 7 8 3 3 3 4 3 3 2 F 1 1 9 . 7 . 5 9 2 2 8 1 1 2 3 9 4 7 3 2 2 5 8 7 8 2 it 2 3 6 1 3 1 3 9 1 2 3 4 2 5 6 7 6 3 it 2 4 3 6 2 2 3 8 8 0 5 3 2 6 1 8 3 4 tt 2 5 2 6 7 3 3 8 4 2 3 3 2 6 5 7 6 6 tt 2 7 1 0 9 8 3 7 6 4 7 3 2 7 3 6 0 7 ( L 3 2 ) tt 2 7 6 0 3 0 3 7 4 4 3 3 2 7 2 2 1 8 tt 2 7 9 6 3 8 3 7 2 9 0 3 2 7 1 6 8 9 tt 2 9 4 7 0 2 3 6 6 6 3 9 2 6 7 1 2 G 1 tt 2 2 1 0 0 9 3 3 8 4 2 3 1 7 9 7 0 2 tt 2 2 9 6 0 5 3 3 4 0 4 6 1 8 6 4 0 3 tt 2 3 7 9 1 7 3 2 9 8 7 8 1 8 6 5 7 4 rt 2 4 7 0 3 2 3 2 5 3 3 2 1 8 6 1 1 5 tt 2 5 6 1 4 5 3 2 0 8 1 1 1 8 5 3 6 6 ( L 3 7 ) tt 2 5 9 8 1 6 3 1 9 0 0 1 1 8 1 3 0 7 tt 2 7 0 0 8 2 3 1 3 9 1 3 1 6 8 6 3 8 ti 2 8 4 2 7 9 3 0 6 9 0 5 1 7 4 3 3 L 1 1 . 9 . 5 9 4 0 5 5 9 4 6 7 8 0 7 0 5 4 8 7 7 2 2 1 . 7 . 5 9 4 0 8 8 4 5 6 6 7 8 2 7 5 4 9 5 3 1 0 it 4 0 6 8 1 5 6 2 7 7 7 0 4 6 0 5 4 1 1 tt 4 0 4 3 8 0 6 1 9 9 4 7 4 5 3 8 5 1 2 tt 4 0 1 4 2 7 6 1 0 5 7 2 4 4 8 9 5 1 3 tt 3 9 8 4 8 0 6 0 1 1 6 1 4 4 3 8 2 1 4 tt 3 9 5 4 5 9 5 9 1 5 0 0 4 3 8 4 1 1 5 tt 3 9 2 5 6 6 5 8 2 2 0 8 4 3 3 0 9 1 6 2 2 . 7 . 5 9 3 8 5 7 1 5 5 6 9 9 7 9 4 2 4 2 1 1 7 ( C 9 ) it 3 7 8 6 9 9 5 5 7 4 6 9 4 1 3 8 8 1 8 tt 3 7 1 4 1 5 5 4 4 4 9 4 4 0 2 6 2 1 9 tt 3 6 4 7 8 3 5 3 2 6 3 8 3 9 3 9 1 2 0 tt 3 5 7 2 5 5 5 1 9 2 6 1 3 8 7 1 6 2 1 ( D 7 ) 2 0 . 7 . 5 9 3 4 9 2 3 5 5 0 5 0 2 4 3 8 2 1 2 2 2 2 2 . 7 . 5 9 3 4 1 6 1 5 4 9 1 5 2 2 3 7 7 4 0 2 3 1 8 . 8 . 5 9 3 3 4 0 9 2 4 7 8 1 4 3 3 6 9 4 0 2 4 tt 3 2 7 1 2 3 4 6 5 7 6 3 3 6 1 6 9 2 5 2 2 . 7 . 5 9 3 2 0 1 6 2 4 5 3 2 3 4 3 5 4 9 8 2 6 tt 3 1 5 9 7 4 4 4 5 7 8 7 3 4 9 7 0 2 7 tt 3 1 2 4 5 1 4 3 9 4 5 5 3 4 3 2 4 2 8 2 . 9 . 5 9 3 0 7 5 5 2 4 3 0 7 2 0 3 3 0 3 0 2 9 2 2 . 7 . 5 9 3 0 0 7 2 1 4 1 8 5 2 4 3 1 6 3 5 3 0 1 9 . 7 . 5 9 2 9 2 0 6 0 4 0 3 0 4 6 2 9 7 3 1 3 1 it 2 8 3 2 7 4 3 8 7 3 7 1 2 8 2 6 2 3 2 ( F 7 ) tt 2 7 6 0 3 0 3 7 4 4 3 3 2 7 2 2 1 Table 2 continued L33 19.7.59 271369 366109 26373 34 266754 357879 25177 35 262771 350773 23896 36 260782 331803 20760 37(G6) 259816 319001 18130 38 260356 307726 14520 39 261054 303763 13236 J 2* 3.8.60 467773 668030 60561 3* 466827 761382 72920 4* 474135 778249 79804 5* 484673 794154 84501 6* 426469 812361 90947 7* 441050 815973 88588 9* 502652 805554 87370 10* 432071 770318 69436 11* 392464 727440 64750 12* t» 394704 697026 57886 "B"* 5.8.59 475513 768687 77221 »D"* 424019 717000 63649 460635 691757 63392 Hole 314 23.7.60 373519 547471 40664 Hole 10 31.7.60 360917 544838 40286 Hole 322 23.7.60 359034 548077 40289 Hole 116 25.7.60 352884 532393 39318 Hole 209 19.7.61 297176 412612 30647 Hole 194 »i 288849 400103 29306 Hole 73 12.8,60 236365 350206 21861 TABLE 3 LONGITUDINAL SURFACE PROFILE. Distances are in metres from the 1959 position of L10. Elevations refer to ice surface measured in cm above datum. Standard error of distance 15 cm, of elevation 10 cm. Date 27.7.59 24.7. 60 Marker Distance Elevation Distance Elevat; L10 0 45897 75 45239 11 82 45208 149 44860 12 180 44727 240 44445 13 279 44214 335 43969 14 380 43691 432 43434 15 477 43134 527 42863 16 618 42243 664 41932 17 761 41236 804 40927 Hole 314 832 873 40408 18 910 40107 950 39846 19 1046 39211 1083 39052 20 1199 38566 1234 38486 21 1362 38055 1396 38002 22 1517 37598 1551 37518 23 1669 36922 1701 36798 24 1811 36164 1843 36069 25 1951 35313 1989 35151 26 2042 34818 2074 34586 27 2115 34151 2146 33868 28 2211 33096 2243 32758 29 2354 31495 2385 31092 30 2531 29589 2560 29303 31 2711 28103 2738 27937 32 2859 27023 2886 26830 33 2954 26213 2981 25942 34 3049 24994 3074 24629 35 3130 23718 3155 23305 36 3321 20611 37 3450 17993 38 3562 14371 39 3603 13067 TABLE 4 TRANSVERSE SURFACE PROFILES. Distances are between adjacent markers. Units are metres. Elevations refer to ice surface. Units are cm above datum. Standard error of distance 15 cm, of elevation 10 cm. Date 27.7.59 24.7.60 Marker Distance Elevation Distance Elevation A 1 54006 53656 2 64 54167 53636 3 59 54413 4 119 54392 5 114 54195 6 112 53563 51198 7 53 53663 51146 B 1 43939 43408 2 67 43636 68 43201 4 150 44084 151 43713 6 168 44469 170 44195 7 104 44486 105 44259 9 191 43875 193 43521 C 2 40698 40690 3 94 40171 93 40047 5 161 40749 159 40611 7 150 41059 151 40830 9 146 41236 147 40927 11 150 41119 151 40866 13 173 41012 173 40856 D 2 37112 37094 3 84 37822 83 37729 4 102 37958 101 37865 5 107 107 38037 6 100 38088 100 38020 7 65 38055 65 38002 8 38 38077 38 37990 9 95 95 38013 10 95 38152 95 38162 11 110 110 38391 c Table 4 continued E 1 34305 34070 2 75 33413 77 33234 3 125 34141 124 33879 4 103 34280 101 34066 5 108 34358 110 34152 6 89 34305 90 33998 7 100 34187 101 33918 F 1 25807 2 88 25468 3 81 26000 4 98 26398 26164 6 200 27177 201 26895 7 53 27023 54 26830 8 39 26988 40 26765 9 163 26598 168 26304 G 1 17787 2 96 18442 18098 3 93 18505 4 102 18445 5 102 18404 6 41 17993 7 115 16708 8 158 17308 17130 TABLE 5 WIDTH OF GLACIER. Units are metres. Marker Width L 10 1060 11 1050 12 1045 13 1080 14 1175 15 1180 16 1220 17 1240 18 1235 19 1220 Marker Width L 20 1190 21 1175 22 1165 23 1140 24 1125 25 1110 26 1080 27 1075 28 1040 29 955 Marker Width L 30 890 31 915 32 955 33 1015 34 1080 35 1115 36 1210 37 1210 38 1210 39 1210 TABLE 6 ICE THICKNESS. Units are metres. Standard errors are 5 m for seismic measurements, -10% +15% for gravity. Direct measurements were made at a l l points list e d . Positions of markers are those of July 1959. Location of points SI to S6 and H5 are shown in Figure 2. L10.5 i s half-way between L10 and L l l . L30.5 i s half-way between LSO and L31. Marker Thickness Method A 1 38 Gravity 2 71 »t 3 97 ti 4 134 t» 5 160 ?» 6 164 7 155 tt B 1 112 tt 2 194 tt 4 282 tt 6 308 tt 7 301 tt 9 238 « C 2 115 tt 3 141 Seismic 4 257 Gravity 5 271 Seismic 6 302 Gravity 7 312 Seismic 9(L17) 314 Gravity 10 300 tt 11 262 Seismic 12 242 Gravity 13 206 tt D 2 162 Seismic 3 235 Borehole 4 276 Gravity 6 311 Seismic 7(L21) 310 tt Marker Thickness Method D 8 290 Gravity 10 192 Seismic E l 87 Gravity 2 130 " 3 197 4 231 " 5 249 Seismic 6 249 Gravity 7 246 " 8 178 " 9 87 " F 1 56 2 90 3 103 " 4 113 5 118 " 6 122 " 7 122 " 8 113 9 85 10 30 " G 1 35 " 2 46 3 54 4 54 " 5 54 6 50 7(L37) 42 8 50 " Table 6 continued L 1 1 9 9 Seismic H 5 1 8 7 1 0 . 5 3 1 2 tt S 1 3 3 4 1 2 3 2 3 M 2 2 9 9 1 4 3 1 7 tt 3 2 7 8 1 6 3 2 2 tl 4 2 7 9 3 1 4 3 1 7 Borehole 5 9 2 L 1 9 3 1 0 Seismic 6 1 8 7 2 1 3 1 0 tt 2 3 2 7 3 n 2 5 2 4 8 ti 2 7 2 4 8 tt 2 9 1 8 6 it 2 0 9 2 0 9 Borehole 1 9 4 1 9 4 ti L 3 0 . 5 1 6 7 Seismic 3 2 1 1 8 tt 3 4 1 1 3 ti Seismic TABLE 7 SURFACE SLOPE. Units are degrees. Standard error i s 0.3°. Measurements always made in direction of maximum slope. Positions refer to July each year. Slope Slope Marker 1959 1960 Marker 1959 1960 A 1 6.1 7.2 E 5 3.8 4.5 2 5.4 6.1 6 3.9 3.9 6 5.7 18.7 7 5.2 5.3 7 5.7 18.7 F 2 8.6 B 1 9.8 9.0 3 7.0 2 6.1 4.9 4 5.8 6.1 4 4.0 4.2 6 5.4 5.8 6 3.2 3.5 7 5.0 5.0 7 2.9 3.0 8 5.4 6.1 9 4.4 3.0 9 7.2 9.5 C 2 1.7 1.7 G 1 14.9 3 3.4 3.4 2 13.6 13.6 5 2.6 2.8 3 14.8 7 3.7 3.7 4 15.0 9 4.3 4.4 5 14.7 11 4.0 4.0 6 14.6 13 2.5 2.5 7 25.5 D 2 1.8 1.8 L 1 3.0 3.7 3 2.6 2.8 2 3.7 4 2.5 2.5 10 4.5 4.0 5 2.5 11 4.0 2.6 6 2.0 2.0 12 2.8 2.8 7 1.7 1.7 13 2.8 3.0 8 1.7 1.7 14 3.1 3.3 9 1.6 15 3.6 3.7 10 1.6 1.6 16 4.1 4.1 11 1.4 17 4.3 4.4 E 1 7.6 8.0 18 4.2 4.1 2 5.2 4.8 19 3.2 2.9 3 5.2 5.4 20 1.9 1.9 4 4.5 4.8 21 1.7 1.7 Table 7 continued L22 2.2 2.2 23 2.8 2.8 24 3.2 3.4 25 3.8 3.8 26 4.2 4.8 27 5.2 5.8 28 6.2 6.2 29 6.4 6.4 30 5.9 5.9 31 4.4 4.4 32 5.0 5.0 33 6.8 7.2 34 8.6 9.0 35 9.8 9.8 36 9.8 37 14.6 38 21.2 21.2 39 23.0 Hole 314 4.3 4.3 Hole 10 3.6 Hole 322 3.6 Hole 116 3.0 Hole 209 6.4 Hole 194 5.9 "B" 19.3 "D" 9.3 17.6 **E" 13.0 J 2 13.6 3 23.0 4 20.0 5 9.6 10 19.9 11 10.7 12 15.0 TABLE 8 SLOPE OF BED. These slopes were determined from seismic records. Slope measured relative to surface. Units are degrees. Standard error is 20%. U, D, L, R indicate that ice i s getting thicker in that direction (up, down, l e f t , right, looking down glacier). E, G, F, P indicate grade of record (excellent, good, f a i r , poor). Spread - trans, and long, indicate that s p l i t spread cable was used and shot point centred. up, down, l e f t , right indicate that shot point was at one end of cable and spread was in indicated direction from shot point. Positions of markers refer to July 1959. Location of points SI to S6 and H5 are shown in Figure 2. L10.5 i s half-way between L10 and L l l . L30.5 i s half-way between L30 and L31. Marker Spread C 3 up trans C 5 long trans C 7 long trans C l l trans D 2 long trans D 6 up l e f t right D10 up l e f t E 5 up right L 1 up down l e f t right L10.5 up down le f t right Slope Grade 0 P 13 L P 2 D G 24 L G 0 G 16 L G 30 R G 9 U P 6 L P 6 U E 15 R G 1 R G 1 D F 33 R F 3 U P 8 L G 14.5D P 1. 5U G 6 L P 10 R P 13.5D E 11 D G 10. 5L G 8 R G Table 8 continued L12 long 4 U G trans 5 R G L14 long 5 U £ trans 22 R E L16 long 1 U G l e f t 30 R G Hole 314 up 1. 5U £ down 2. 5D £ l e f t 22 R £ right 5 R E L19 long 0 £ trans 4 R E L21 long 4.5U G trans 13 R G L23 long 7 U G trans 7 R G L25 up 2 D G trans 26 R F L27 long 6 D P trans 6 R G L29 long 10. 5D G trans 6 R G L30.5 long 4 U G trans 9 R F L32 up 1 D P long 1 0 G right 5 L P L34 long 3 D G trans 5 L P L37 long 1 U F trans 0 F Hole 322 long 2 D E trans 2 L E H 5 up 7 D G down 7 D P l e f t 2 L P right 15 L P S 1 l e f t 1 R G right 2 L E S 2 down 9 D G le f t 28 R G right 16 R G S 3 down 4 D E l e f t 17 L E right 9 R E Table 8 continued S 4 up 12 D G down 9 D G l e f t 18 L G right 18 L P S 5 long 0 G trans 3 R F S 6 long 0 F trans 35 L G TABLE 9 CURVATURE OF SURFACE AND BED. These were obtained by numerical differentiation of slope data in Tables 7 and 8. (Table 8 data were transformed to slopes relative to horizontal.) Positions of markers refer to July 1959. Units are per 10$ m. Minus sign means concave surface. Curvatures Marker Surface Bed L 10.5 - 19 - 232 12 12 6 14 7 37 16 1.5 18 Hole 314 - 10 - 18 19 - 8 - 33 21 6 8 23 6 63 25 15 58 27 9 41 29 - 11 - I l l 30.5 2 39 32 33 52 34 26 9 37 Table 10 HORIZONTAL SURFACE VELOCITY (U). Units are m./yr. Periods are A 21. 7.59 — 15. 8.59 Standard error (m./yr.) 2.5 B 15. 8.59 — 1. 9.59 3 C 1. 9.59 — 24. 7.60 0.15 D 24. 7.60 — 31. 7.60 4 E 31. 7.60 - 7. 8.60 4 F 7. 8.60 — 13. 8.60 4 G 13. 8.60 — 14.11.60 1 H 14.11.60 o 10. 4.61 0.6 J 10. 4.61 19. 7.61 0.6 This i s the horizontal component of velocity. Its direction i s given in Table 11. The vertical velocity component i s in Table 14. The difference between U and the magnitude of the velocity vector i s negligible for nearly a l l markers. There are variations of two or three days from these periods in individual C£LS6»5 • Positions are the mean positions of the marker at each period. A horizontal line means that the velocity i s an average value for the periods covered by the line. Marker A B C D E F G H A 1 49.0 46.1 2 69.0 69.3 — 3 94.3 85.5 4 126.5 119.2 5 135.8 131.8 6 125.9 134.4 129.0 -7 128.0 131.7 127.1 129.2 Table 10 continued B 1 38.2 40.0 2 45.5 47.4 4 55.3 55.7 6 57.2 57.8 7 55.5 57.3 9 48.8 50.3 C 2 13.7 13.4 3 29.7 5 34.0 38.8 7 38.0 42.7 9 see L17 11 42.5 23.8 — 33.0-29.1 — 57.7 28.9 25.5 25.6 21.6 32.8 42.8 24.4 30.5 27.1 66.4 30.8 29.0 29.0 26.7 34.2 63.6 32.5 28.6 27.3 19.3 34.6 54.4 26.6 28.0 28.8 46.6 32.4 30.4 25.9 32.0 32.0 28.9 25.1 27.5 19.7 13.3 18.7 13 33.1 35.3 D 2 3 29.0 26.5 4 32.6 5 6 30.2 7 see L21 8 35.5 9 10 27.8-11 E l 18.2 2 27.7 24.5 3 31.6 4 34.0 24.8 28.9 35.8 5 24.0 33.2 31.0 6 30.1 32.8 31.4 7 30.8 32.6 32.9 F 1 7.9 2 18.2 12.8 3 23.6 17.6 4 27.1 21.7 26.5 6 25.2 23.5 27.4 7 see L32 Table 10 continued i1 8 24.4 20.4 25.8 9 22.8 17.5 21.9 i 1 4.8 2.8 2 10.0 10.1 11.4 3 21.0 13.3 4 22.9 16.6 5 20.8 16.1 6 see L37 7 19.9 11.0 8 7.9 6.8 7.2 , 1 112.0 2 116.3 117.1 10 78.8 73.4 74.0 72.0 66.0 63.1 11 68.5 63.5 66.2 65.8 58.6 58.7 12 63.1 53.5 59.3 61.4 55.0 53.9 13 59.4 45.8 55.3 53.7 50.9 50.8 14 52.8 46.2 51.9 50.5 48.0 15 46.5 39.1 49.6 16 46.4 35.8 46.5 46.1 17 37.8 32.9 43.6 47.4 18 37.9 33.0 40.7 43.4 37.0 19 35.7 37.5 39.4 31.6 20 32.8 21.9 35.2 36.2 30.3 21 32.7 25.2 34.4 35.3 25.5 28.6 22 32.3 25.4 34.4 30.2 30.4 29.7 23 33.4 28.7 24 33.0 31.6 27.8 29.4 25 34.1 23.8 33.2 31.8 29.9 29.6 26 27.8 33.2 30.9 27.6 28.9 27 32.5 32.3 28 32.8 31.7 30.6 29 26.4 31.7 32.9 31.3 30 29.9 22.0 28.5 32.7 31 27.2 21.6 27.0 32 29.8 26.1 33 26.8 22.1 26.8 27.0 32.2 24.6 32.5 Table 10 continued L34 26.8 22.3 26.5 27.2 35 26.9 21.3 25.6 26.1 36 22.6 19.0 37 22.6 13.9 38 16.7 13.7 15.7 39 15.1 15.7 Hole 314 41.5 47.1 38.3 39.7 37.1 38.6 47.7 Hole 10 38.3 42.6 39.9 37.1 Hole 322 45.1 37.6 33.2 35.7 40.0 Hole 116 44.7 34.3 36.5 33.0 32.4 41.4 Hole 209 28.6-Hole 194 31.9 "D" 128.7 147.7 154.0 ME" 100.8 Period August 3-14, 1960 Standard error = 25 m./yr. J 2 52 J 7 11 3 200 9 171 4 266 10 91 5 188 11 30 6 11 12 88 Period 14.8.60 - 19.7.61 Standard error « .7 m./yr. J 6 11.9 9 142.1 TABLE 11 HORIZONTAL DIRECTION OF SURFACE VELOCITY. Periods are AB 21. 7.59 - 1. 9.59 Standard errors 10° C 1. 9.59 - 24. 7.60 1° DEF 24. 7.60 - 13. 8.60 15° G 13. 8.60 - 14.11.60 4° H 14.11.60 - 10. 4.61 2° J 10. 4.61 - 19. 7.61 4° There are variations of 2 or ,3 days from these periods in individual cases. These directions are measured clockwise from the base line which had azimuth 189.4° relative to true north. A horizontal line means that the azimuth is an average value for the periods covered by the line. ixker AB C A 1 204 2 200 3 199 4 195 5 195 6 197 7 196 198 B 1 214 212 2 211 211 4 209 209 6 207 206 7 204 205 9 202 202 C 2 204 205 3 207 DEF G H 197 199 224 Table 11 continued C 5 211 208 7 208 207 9 see L17 11 206 13 206 207 D 2 212 207 3 212 214 208 4 220 213 212 208 5 216 211 6 216 214 207 210 7 see L21 8 218 213 209 212 9 221 10 218 214 209 11 E 1 217 220 2 216 3 216 4 215 220 215 5 215 224 6 213 211 7 212 212 F 1 2 207 3 201 4 202 205 6 202 202 7 see L32 8 201 201 9 190 189 G 1 209 2 198 203 3 196 4 196 5 184 6 see L37 Table 11 continued G 7 182 8 199 201 L 1 197 2 196 10 203 203 206 11 204 205 205 12 204 206 206 13 205 206 207 14 207 206 205 15 209 206 16 210 206 208 17 205 207 206 18 209 208 208 19 209 199 20 213 210 212 21 215 213 211 22 217 214 214 215 23 216 213 24 216 208 214 25 219 214 211 214 26 212 213 214 213 27 214 205 28 - 213 210 213 29 210 212 218 209 30 210 210 211 31 204 205 207 204 32 202 204 33 199 199 34 197 196 193 35 195 194 192 36 192 37 185 38 189 190 39 183 Table 11 continued Hole 314 206 Hole 10 Hole 322 Hole 116 Hole 209 Hole 194 "D" 203 "E" 204 J 6 9 207 206 208 208 209 208 207 208 209 208 209 210 2io 211 199 196 228 TABLE 12 HORIZONTAL VELOCITY AT EDGE OF GLACIER. Units are m./yr. These are rough values obtained by extrapolation. Velocities Line S.W. Edge Centre N.E. Edge A 20 134 B 17 58 36 C 0 44 15 D 6 35 21 E 7 33 22 F 0 27 16 G 0 20 0 The following values were measured directly. The location was about 50 m. down glacier from the A line. Units are m./yr. The standard error i s 1 m./yr. S.W. Edge 10.7 N.E. Edge 10.8 TABLE 13 MEAN VELOCITY ON TRANSVERSE LINES. The ratio i s that of the mean surface velocity to the maximum surface velocity. Line Ratio B 0.86 C 0.81 D 0.87 E 0.85 F 0.87 G 0.69 TABLE 14 VERTICAL VELOCITY AND NORMAL VELOCITY. V » vertical velocity (measured directly). v - velocity normal to ice surface (calculated from V, horizontal velocity, and surface slope). Positive direction i s downwards. Units are m./yr. Periods are Standard errors are C G H J 1. 9.59 13. 8.60 14.11.60 10. 4.61 24. 7.60 14.11.60 10. 4.61 19. 7.61 0.15 0.6 0.4 0.6 0.35 1.4 0.9 1.4 There are minor variations from these periods in individual cases. Other periods are too short for results to be of sufficient accuracy. A horizontal line means that the velocity i s an average for the periods covered by the line. Positions are those of the marker at each period. Velocities are those of a particle of ice just below the surface (i.e. at foot of marker), except at boreholes where the velocity is that of the top of the pipe. C G H J Marker V V A 1 0.85 -4.46 2 2.60 -4.37 6 26.20 -2.86 7 24.81 -3.66 B 1 2.46 -4.15 2 1.27 -3.26 4 0.10 -3.90 6 -0.99 -4.34 7 -1.05 -4.02 9 0.97 -2.26 C 2 -1.28 -1.67 3 -3.83 -5.63 5 -2.14 -3.97 7 -1.15 -3.86 Table 14 continued C 9 see L17 11 0.43 -2.50 13 -1.09 -2.64 D 3 -3.24 -4.61 -3.29 -4.54 4 -2.61 -4.06 -3.57 -4.77 5 -2.66 -3.92 6 -1.94 -3.12 -3.25 -4.19 7 see L21 8 -2.66 -3.67 -3.29 -4.13 9 -2.86 -3.56 10 -2.94 -3.80 £ 2 -2.43 -4.57 3 -1.45 -4.37 4 -1.56 -3.90 5 -1.57 -3.99 6 -1.12 -3.35 7 -0.15 -3.15 F 4 -1.49 -4.25 6 -1.22 -3.93 7 see L32 8 -1.87 -4.45 9 1.41 -1.72 G 2 0.66 -1.95 L 1 0.49 -6.06 10 -1.93 -6.51 11 0.14 -3.68 -2.69 -5.39 12 -0.80 -3.62 -2.53 -5.15 13 -0.77 -3.58 -1.85 -4.57 14 -0.34 -3.24 -1.52 -4.31 15 0.04 -3.07 16 0.24 -3.08 17 0.19 -3.13 18 -0. 14 -3.11 -0.48 -3.15 19 -1.49 -3.48 -0.87 -2.48 20 -2.34 -3.49 -0.60 -1.59 21 -3.16 -4.20 -2.98 -3.84 22 -2.55 -3.89 -2.42 -3.60 23 -1.82 -3.45 24 -2.01 -3.93 -2.47 -4.15 25 -1.94 -4.14 -0.93 -2.93 26 -1.21 -3.83 -0.08 -2.42 27 -0.18 -3.32 28 0.11 -3.48 0.82 -2.53 29 1.56 -1.99 2.67 -0.84 30 -0.19 -3.13 31 -2.23 -4.32 -0.94 -2.86 32 -2.08 -4.37 -3.21 -4.55 •1.41 -5.29 •1.97 -4.73 •2.05 -4.65 •1.60 -4.34 -2.49 -3.68 -1.46 -3.95 Table 14 continued L33 -0.88 -4.17 34 0.53 -3.58 35 1.57 -2.86 38 5.44 -0.65 Hole 314 -0.40 -0.44 -0.40 Hole 10 -0.53 Hole 322 -2.18 0 Hole 116 -1.58 -1.97 Hole 209 Hole 194 -0.49 -0.15 "D" 29.8 -5.6 J 6 2.8* j 9 21.6* * Standard error =0.7 m./yr. TABLE 15 RATE OF THINNING. See section 3.7.3 for details of method of calculation. Location Thinning(m) Period(yr) Rate(ra./yr.) Source L36 9.5 13 0.73 Water Resources survey. L32 76 130 0.58 Height of stn.9 above ice. L37 107 130 0.83 Height of stn.7 above ice. TABLE 16 LONGITUDINAL SURFACE STRAIN RATE. Strain rates are measured in a horizontal plane. Positions are the mean positions of the markers at each period. Units are per year. Periods are Standard errors are AB 21. 7.59 C 1. 9.59 G 13. 8.60 H 14.11.60 1. 9.59 24. 7.60 14.11.60 10. 4.61 .015 .002 .008 .005 There are variations of 2 or 3 days in individual cases Markers AB C G H L10-11 -.103 -.103 -.062 11-12 -.074 -.040 -.054 12-13 -.042 -.043 -.034 13-14 -.034 -.030 14-15 -.023 15-16 -.022 16-17 -.021 17-18 -.019 18-19 -.022 -.041 19-20 -.018 -.008 20-21 -.007 -.010 21-22 -.001 +.012 22-23 -.004 f-,008 -.002 23-24 -.003 \ 24-25 -.001 + .016 0 25-26 +.001 -.027 -.008 26-27 -.002 /+.016 27-28 + .003 1 28-29 -.007 +.004 29-30 -.018 30-31 -.009 31-32 -.004 32-33 -.002 33-34 -.006 34-35 -.014 35-36 -.017 36-37 -.016 37-38 -.031 38-39 +.010 TABLE 17 TRANSVERSE SURFACE STRAIN RATE (MEASURED). Strain rates are measured in a horizontal plane. Positions are the mean positions of the markers at each period. Units are per year. Periods are Standard errors are AB 21.7.59 - 1.9.59 .015 C 1.9.59 - 24.7.60 .002 There are variations of 2 or 3 days in individual cases. Markers AB C Markers AB B 1- 2 +.003 G 1-2 +.02 2- 4 +.010 2-3 +.02 4- 6 +.016 3-4 +.01 6- 7 +.016 4-5 + .03 7- 9 +.014 5-6 0 C 3- 5 -.009 6-7 0 5- 7 0 7- 9 +.004 9-11 +.001 11-13 +.004 D 2- 3 -.008 3- 4 -.010 4- 6 -.003 6- 7 + .005 7- 8 -.006 8-10 0 E 2- 3 -.002 3- 4 -.014 4- 5 +.022 5- 6 +.011 6- 7 +.007 F 2- 3 + .02 3- 4 0 4- 6 +.006 6- 7 +.003 7- 8 + .014 8- 9 + .02 c TABLE 18 TRANSVERSE SURFACE STRAIN RATE (CALCULATED). Units are per year. Measured values are between the two end markers in each transverse line. Calculated values are obtained from the change in width of the glacier. They are thus average values across the whole glacier. Markers refer to their mean position over winter 1959-60. Standard error of measured values i s .002. Marker Transverse strain rate. Calculated Measured L 10 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 11 12 13 14 15 16 ) -.008 -.002 + .017 + .024 +.010 + .006 + .004 + .001 -.002 -.003 -.003 -.002 -.003 -.003 -.002 -.004 -.004 -.007 -.012 -.011 -.002 +.012 +.003 0 0 + .004 + .009 + .013 + .011 + .008 +.003 + .014 0 0 0 TABLE 19 SURFACE STRAIN RATES AT BOREHOLES. Units are per year. Measurements refer to the plane of the glacier surface. x, z directions are down and across glacier respectively. Strain rates Borehole v 7 314 -.0196 +.0015 322 -.0229 +.0019 +.0034 116 -.0168 -.0012 +.0017 209 -.0114 -.0006 Standard errors 314 undetermined 322 .0007 .0007 .0006 116 .0017 .0017 .0014 209 undetermined TABLE 20 BOREHOLE DATA. Standard errors - Inclination 0 . 1 ° , 0 . 2 5 ° , 1 ° for 4 ° , 1 0 ° , 2 6 ° discs respectively. Azimuth 1 ° . "Distance" measured along pipe from top in 1960. Hole 322 Direction of surface flow N 12 E 18-20 July, 1960. Distance(m) Inclination Azimuth Disc (if not 0 3.96 7.62 0.9 S 26 W 15.24 0.3 S 82 W 22.86 0.9 N 65 W 30.48 0.5 N 31 W 38.10 0.0 45.72 1.4 S 62 W 53.34 60.96 0.7 S 11 W 68.58 76.20 1.7 N 73 W 83.82 91.44 6.0 S 20 W 99.06 106.68 4.1 S 28 W 114.30 7-12 July, 1961. Inclination Azimuth Disc 4 ° ) ( i f not 4 ° ) 0.4 S 09 W 0.1 N 0.3 N 06 W 0.7 N 30 W 0.8 N 30 W 0.3 N 08 W 1.0 S 61 W 0.9 N 52 W 0.4 S 30 E 1.9 S 44 W 1.3 N 73 W(?) 0.2 N 80 W(?) 4.8 S 28 w 5.6 s 33 w 4.3 s 28 w 4.4 s 26 w Table 20 continued 121.92 4.1 S 26 W 10 4.9 S 27 W 129.54 4.1 S 09 W 10 137.16 4.0 S 21 W 10 4.8 S 20 w 144.78 3.5 S 26 w 10 152.40 3.8 S 16 W 4.0 S 21 w 160.02 5.2 S 12 w 10 167.64 2.9 S 43 W 4.1 S 33 w 175.26 3.0 S 23 w 10 182.88 4.4 S 07 £ 4.9 S 10 £ 190.50 5.0 S 20 W 10 198.12 1.6 s 63 W 3.5 S 43 W 205.74 2.0 S 22 W 10 213.36 2.8 s 44 W 4.1 S 27 W 220.98 3.2 S 56 w 10 228.60 2.9 s 75 W 4.7 S 47 w 236.22 4.2 S 52 w 10 243.84 2.1 s 74 W 5.0 S 39 w 10 251.46 4.3 S 56 w 10 259.08 3.2 N 71 W 3.7 s 73 w 10 266.70 4.9 s 46 w 10 274.32 1.6 N 62 W 3.9 s 43 w 10 281.94 5.0 s 47 w 10 289.56 6.0 N 87 W 10 8.9 s 73 w 10 297.18 6.8 s 63 w 10 304.80 11.3* S 73 W 10 13.5 s 60 w 26 312.42 13.0 s 48 w 26 320.04 11.9* s 42 W 10 20.0 s 31 w 26 320.95 20.0 s 31 w 26 * Standard error = 0 . 5 ° Table 20 continued Hole 209 Direction of flow at surface N 14 E. 9-10 August, 1960. 10-11 July, 1961. Distance(m) Inclination Azimuth Disc Inclination Azimuth Disc (if not 4°) ( i f not 4°) 0 0.6 S 48 W 1.83 0.7 S 40 W 7.62 0.4 S 40 W 15.24 0.2 N 43 W 0.3 N 20 E 22.86 1.0 S 57 W 30.48 1.0 S 56 W 1.0 s 46 W 38.10 1.0 s 42 W 45.72 0.6 S 84 W 1.0 s 52 W 53.34 1.0 s 40 W 60.96 0.6 s 72 W 1.1 s 50 W 68.58 1.3 s 41 W 76.20 0.7 s 73 W 1.3 s 45 W 83.82 1.7 s 35 W 91.44 0.2 s 43 W 1.5 s 33 W 99.06 1.5 s 28 W 106.68 1.7 N 83 W 1.7 s 38 W 114.30 2.1 s 40 W 121.92 2.2 N 68 W 3.0 s 54 W 129.54 4.0 s 43 W 137.16 0.7 N 40 W 4.6 s 37 W 144.78 4.6 s 40 W 10 152.40 1.7 N 75 W 6.2 s 28 W 10 160.02 8.9 s 17 W 10 167.64 2.0 S 55 W 10.0 s 21 W 10 175.26 13.0 s 21 W 26 182.88 1.5 N 89 W 17.0 s 19 W 26 183.79 17.0 s 20 W 26 Table 20 continued 195.07 198.12 1.4 S 56 W 201.47 207.26 1.9 S 36 W * Readings with acid bottle S.E. -Hole 116 Direction of flow at surface N 13 E 24 July, 1960. Distance(m) Inclination Azimuth 0 1.0 N 7.62 15.24 3.5 S 16 W 22.86 30.48 5.0 N 55 W 38.10 45.72 6.2 N 44 W 53.34 6.0 N 52 W 60.96 3.0 N 22 W 68.58 76.20 4.2 N 36 w 83.82 91.44 4.3 N 27 E 99.06 106.68 3.4 N 04 W 114.30 3.3 N 08 E 32 .5 * 24 .8* Disc = 1 0 ° throughout 13 July, 1961. Inclination Azimuth 1.7 N 05 E 3.1 N 46 W 4.3 N 40 W 5.0 N 46 W 5.2 N 40 W 5.0 N 52 W 5.9 N 47 W 3.4 N 28 W 3.8 N 09 W 4.7 N 41 W 4.0 N 36 W 4.5 N 36 E 4.0 N 08 E 4.0 N 03 W 3.1 N 08 E Table 20 continued  Hole 314 Direction of flow at surface N 12 E Disc =• 4° throughout Distance(m) 0 7.62 15.24 22.86 30.48 38.10 45.72 3 August, Inclination 1.0 1.3 0.9 1.0 1960. Azimuth E N 87 E N 68 E S 85 E 14 July, 1961. Inclination Azimuth 1.4 1.6 1.1 1.0 0.9 1.1 N 86 E N 71 E N 45 E N 55 E S 79 E S 59 E TABLE 21 SLOPES, VELOCITIES, STRAIN RATES AT BOREHOLES. See section 5.1 for definition of symbols. Symbol Units Hole 322 Hole 209 Hole 314 Hole 116 h as k v b Vb m. degrees •t m./yr. tt tt tt tt tt 321 3.5 5.3 38.9 31.7 -0.46 3.02 -2.84 1.03 209 6.3 16.9 28.8 7.0 1.25 2.12 -1.94 1.29 314 4.2 5.3 40.8 0 -2.98 315 2.9 2.9 35.2 -1.60 -3.36 (-) \ ax As per yr. -0.0229 -0.0114 -0.0196 -0.0168 (h) U x / b ti -0.0012 -0.0213 (•hr\ U x J s it -0.0045 -0.0011 -0.0005 -0.0055 fit). ii 0.0019 -0.0006 0.0015 -0.0012 K aw1 yrT 1 0.000068 -0.000048 TABLE 22 BOREHOLE RESULTS. The tabulated quantity is jg the rate at which the velocity parallel to the surface changes with depth. Units are per year. Results of 3 different methods of analysis are given: A "Laminar flow" (see part l a of section 3.6.) B Addition of corrections for longitudinal strain rate (variable with depth) and curvature of pipe (part 2 of section 3.6.) C Integrated method (part 3 of section 3.6.) Hole 322 Standard y(m.) A B C error 4.0 + .006 + .007 7.6 18 20 + .009 15.2 7 10 16 22.8 5 8 6 30.5 4 6 5 38.1 5 8 5 45.7 4 6 9 .004 61.0 + .007 + .009 0 76.2 -.024* -.022* + .003 91.4 +.028* +.023* -.013 106.6 -.004 -.004 4 121.8 14 16 8 137.0 14 15 6 152.2 4 5 15 .01 167.4 23 23 16 182.6 7 8 19 197.8 36 36 27 .004 213.0 29 28 21 228.2 47 47 35 243.4 63 63 47 258.7 40 38 58 273.9 69 70 69 .01 289.0 60 63 55 304.1 65 57 46 .035 318.9 -.117 -.076 76 * These values are doubtful. Table 22 continued. Hole 209 y(m.) A B C 1.8 -.002 + .001 15.2 + .004 5 +.003 30.5 -.004 -.002 -.006 45.7 12 8 9 61.0 12 9 13 76.2 13 11 18 91.4 23 20 24 106.7 25 22 31 121.9 50 49 55 137.1 90 90 85 152.3 115 117 121 167.4 160 163 189 182.5 320 320 335 196.9 -.638 -.623 526 Hole 116 y(m.) A B 15.2 +.181* +.140* 30.4 -.012 -.012 45.6 + .023 + .023 53.2 -.009 -.009 60.8 0 0 76.0 0 0 91.2 -.002 -.002 106.4 -.009 -.009 114.0 + .003 + .003 Standard error .004 .01 035 Standard error .01 * These values are questionable. Hole 314 y(m. ) 15.2 30.4 45.6 + .008 2 10 Standard error .004 TABLE 23 COMPARISON OF MEAN AND SURFACE STRAIN RATES. measured calculated sign of U4 u b - 0 u b measured ub = u s L12 -.068 -.019 -.005 14 -.032 -.029 -.015 — 16 -.021 -.015 -.012 — Hole 314 -.0196 -.013 _ Hole 322 -.0229 -.014 L19 -.021 -.007 -.007 Hole 116 -.0168 -.010 L21 -.021 -.013 -.004 + 23 -.003 -.009 + .006 — 25 + .002 -.013 -.017 + 27 -.001 -.020 -.038 + Hole 209 -.0114 -.016 + L306 -.011 -.014 -.003 + 32 -.004 -.046 -.050 + 34 -.011 -.048 -.059 + TABLE 24 COMPARISON OF MEASURED AND CALCULATED STRAIN RATES. These values of ^ were calculated by equation 18 of section 5.1. For details of the comparison see section 7.5. calculated measured L12 -.105 -.068 14 +.031 -.032 16 +.011 -.021 314 -.038 -.020 L19 -.070 -.021 21 -.024 -.021 23 +.023 -.003 25 +.029 + .002 27 +.023 -.001 209 -.022 -.011 L30.5 -.038 -.011 32 + .007 -.004 34 + .008 -.011 TABLE 25 STRAIN RATES AND STRESSES ON TRANSVERSE LINES. See section 7.7. for explanation of symbols and method of calculation. Units are z - metres u - m./yr. Iz T - bars F = 0.56, 0.61 for D and E lines respectively. Marker u ^± z T Iz * z D 2 23.82 3 29.10 4 32.78 6 34.22 7 34.38 8 34.56 10 31.99 E 1 18.66 2 24.49 3 31.62 5 33.22 6 32.82 7 32.55 -0.0627 434.5 -0.642 -0.0363 341.7 -0.504 -0.0070 187.6 -0.276 -0.0025 51.4 -0.076 0 0 -0.0135 113.9 -0.168 •0.0770 318.5 -1.174 •0.0572 218.4 -0.805 •0.0144 100.5 -0.370 0 0 •0.0027 95.1 -0.350 TABLE 26 STRAIN RATES AND STRESSES AT BOREHOLES. See sections 5.1. and 7.8. for explanation of symbols and method of calculation. Units of strain rate and stress are yr." 1 and bars respectively. O'V —- - -.004 and -.001 for holes 322 and 209 respectively. Hole 322 y S T *S u 152.2 -0.013 -0.010 -0.481 2.69xl0~4 0.621 167.4 12 10 .530 2.44 .684 182.6 11 12 .577 2.65 .612 197.8 10 16 .625 3.56 .544 213.0 8 13 .673 2.33 .626 228.2 7 20 .723 4.49 .586 243.4 6 26 .769 7.12 .588 258.7 5 31 .819 9.86 .691 273.9 4 37 .866 13.85 .760 289.0 3 30 .914 9.09 .845 304.1 2 25 .962 6.29 .931 318.9 01 40 -1.008 16.01 1.015 Hole 209 y S T * 3 30.5 -0.013 -0.004 -0.185 1.85xl0~4 0.396 45.7 14 5 .278 2.21 .682 61.0 14 7 .370 2.45 .694 76.2 15 10 .462 3.25 .696 91.4 16 13 .554 4.25 .770 106.7 17 16 .646 5.45 .886 121.9 17 28 .739 10.73 .749 137.1 18 43 .831 21.73 .814 152.3 19 61 .924 40.82 .935 167.4 19 95 1.016 93.86 1.071 182.5 20 .168 1.108 286.24 1.245 196.9 21 .263 1.195 696.10 1.439 TABLE 27 ABLATION AND ACCUMULATION. Ablation measured in cm. of ice, accumulation in cm. of snow. Positions are mean position of each marker at each period. Periods: Ablation 1959 17 July - 27 August (end of season) 1960(a) Start of season - 4 August 1960(b) Start - end of season Accumulation 1960-1 End of summer 1960 - 10 April, 1961. Standard error i s about 10%. Ablation Accumulation Marker 1959 1960a 1960b 1960-1 A 2 174 3 152 4 175 5 182 6 146 7 217 116 B 1 160 324 2 175 260 3 201 257 4 210 5 192 338 6 207 309 7 214 265 9 171 C 2 74 3 409 4 138 300 5 145 292 6 161 260 7 157 296 8 173 191 9(L17) 131 229 10 146 156 11 165 175 12 144 173 13 127 213 14 121 Table 27 continued. D 3 166 306 438 4 147 273 405 5 154 262 406 6 133 7(L21) 187 218 8 167 288 9 189 261 10 245 11 147 187 E 1 88 2 181 315 3 207 363 4 179 5 210 273 6 181 316 7 168 211 8 144 126 F 1 47 122 2 322 3 182 324 4 170 302 5 193 326 6 162 326 7(L32) 173 331 8 156 334 9 108 127 G 1 217 3 222 4 162 5 152 6(L37) 166 7 194 8 74 66 L 1 142 284 2 154 10 150 317 437 11 196 289 418 12 196 321 441 13 217 272 396 14 218 262 364 15 173 241 16 168 220 17 131 229 18 184 194 19 187 184 Table 27 continued. L20 138 267 21 187 218 22 142 272 411 81 23 165 260 24 158 258 434 131 25 188 271 425 160 26 176 282 451 76 27 190 253 28 220 269 29 152 191 30 182 226 114 31 185 304 102 32 147 331 122 33 178 292 34 234 35 201 184 36 171 37 166 38 166 39 217 Hole 314 215 Hole 116 168 Hole 194 142 ATHABASKA GLACIER Distance (metres) ahead of I960 position metres above datum metres above datum A Line 200m — CROSS-SECTION FIGURE 8 HORIZONTAL SURFACE VELOCITY B Line CROSS-SECTION FIGURE 9 HORIZONTAL SURFACE VELOCITY C Line CROSS - SECTION FIGURE 10 HORIZONTAL SURFACE VELOCITY D Line CROSS - SECTION FIGURE II HORIZONTAL SURFACE VELOCITY E Line FIGURE 12 HORIZONTAL SURFACE VELOCITY F Line 200 m-CROSS - SECTION 40H FIGURE 13 HORIZONTAL SURFACE VELOCITY G Line Distance (m) FIGURE 14 HORIZONTAL SURFACE VELOCITY h + 5 3 4°0 800 1200 1600 Distance (metres) Distance ( metres) FIGURE 16 VERTICAL VELOCITY AND VELOCITY NORMAL TO SURFACE ON LONGITUDINAL LINE CO Q 5' 3D Q h O - 0 5 L 2 0 L 2 4 - X -e — K — o — x -+0-05 400 6 0 0 co 3 3D o h -0-05 L 2 4 Distance (metres) L 3 0 1 2 0 0 1600 L 3 5 L 3 9 Longitudinal _ _ _ 0 -©-" CD +0-05 — I 2000 2400 I 3 0 0 0 3200 Distance (metres) FIGURE' 17 LONGITUDINAL AND TRANSVERSE STRAIN RATES ON LONGITUDINAL LINE 3600 I 9 6 0 1961 Displacement parallel to surface (m ) FIGURE 18 CHANGE IN CONFIGURATION - HOLE 322 Bedrock Displacement parallel to surface (m) FIGURE 19 CHANGE IN CONFIGURATION - HOLE 209 FIGURE .20 CHANGE IN CONFIGURATION - HOLES H6s3l4 Bedrock HOLE 209 H O L E S 116 a 314 FIGURE 2 3 VARIATION OF 4^ WITH D E P T H 8 y 5 0 -5 0 0 Distance ( m ) FIGURE 24 COMPARISON OF VELOCITIES OF PIPE AND MARKERS ( bars ) + oo LJ _|co g o 0-05 h 0-025 0-01 ,09.o i K +HOLE 322 FIGURE 26 STRAIN RATE AND STRESS 0-5 0-75 1-0 2-0 + CM UJ — CM C P o o o 0 o o o © o 0 (?) o § 0 i 2 . + i I J2 hO-2 r-0-1 h0O5 CM UJ l-ICM -0025 -001 HOLE 209 FIGURE 27 STRAIN RATE AND STRESS + UJ - OJ O /ill (bQrS) 0-5 1-0 2-0 -2-0 M>25 h-005 F I G U R E 2 8 S T R A I N R A T E A N D S T R E S S T R A N S V E R S E L I N E S 1-4 a> CL O co CD o CO to CO CD c II -C I-3H I-2H i-H c 'co o l<H 0-9 H 0-8H 0-7-X = D line o = E line o © o o x X i o g | 0 u 1-3 1-4 (u= horizontal velocity in 1 •5 m./yr. ) FIGURE 29 CORRELATION BETWEEN SURFACE VELOCITY AND ICE THICKNESS S/Y £ 2' (bars) 0-1 0-5 1-0 2-0 + Hole 209 n = 4-2 ( Glen) * Hole 322 n = 3 2 (Glen ) © D line n = 2-96 (Butkovich a Landauer ) • E line The lines of each pair refer to 0 ° C <* — l*5°C FIGURE 30 COMPARISON OF DATA WITH FLOW LAWS LIO LI5 |_20 L24 ~l 1 l r 2000 2400 , 3200 3600 Distance (metres) FIGURE 31 ABLATION ON LONGITUDINAL LINE 


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