1.!!- -. SUfiFA-‘CE VELOQW AND STRAIN-RAKE MEA$EJW ON SEVERAL AMSKAN Mflm‘, E964 1'wa “to Dunn «a? M. S. KECREMK SYHE WHY Theoéore Wimam Havana. Jr. E965 \\ T\_ U u t 1- ' “V‘CY'QL ‘ '\ ‘ .‘ hior‘. v-( ‘1 Co". ' 3 «a /2§W' ‘rl: .111 to SURFACE VELOCITY AND STRAIN—RATE MEASUREMENTS ON SEVERAL ALASKAN GLACIERS, 196A By Theodore William Havas, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Sanitary Engineering 1965 ABSTRACT SURFACE VELOCITY AND STRAIN-RATE MEASUREMENTS ON SEVERAL ALASKAN GLACIERS, 196A by Theodore William Havas, Jr. Stresses are calculated from measured strain-rates of several Alaskan glaciers. Also, measured surface velocities are compared with velocities obtained by Nye's plasticity solution. They are in reasonable agreement although important local deviations are noted. The calculations were performed by means of several computer programs. ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. M. M. Miller, Professor of Geology and director of the Juneau Icefield Research Program, for his leadership in mountaineering and his informative lectures on glaciology. Sincerely appreciated are the patient efforts of Dr. T. H. Nu, Professor of Civil Engineering and participant in J.I.R.P., for his assistance in conducting the surveys, guidance in writing the thesis, and keen interest in the subject of glacier mechanics. Valuable assistance in computer programming was provided by Mr. P. Britner, Computer Specialist. The staff of J.I.R.P. is thanked for their cooperation in col- lecting the data and the logistic and communication services. ii TABLE OF CONTENTS SECTION ACKNOWLEDGEMENTS . TABLE OF CONTENTS LIST OF TABLES . LIST OF FIGURES LIST OF SYMBOLS CHAPTER I, INTRODUCTION . . . . . CHAPTER II, ELow CHARACTERISTICS CHAPTER III, SURFACE VELOCITY MEASUREMENTS CHAPTER IV, STRAIN-RATE MEASUREMENTS CHAPTER v, SUMMARY BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . APPENDIX A, DETERMINATION OF‘PRINCIPAL STRAIN—RATES APPENDIX B, DESCRIPTION OF COMPUTER PROGRAMS . APPENDIX C, DETAILS OF COMPUTER SUBROUTINES AND PROGRAMS . iii PAGE ii . iii iv . vii 15 28 38 39 Lo AA 59 TABLE IIa IIb IIC IId IIe IIIa IIIb IIIC IV LIST OF TABLES Correlation of symbols . Measured surface valocities Yearly velocities extrapolated from measured surface velocities of 6July-3September . Measured surface velocities of 6July—22August and coordinate distances . Special listing of measured horizontal angles (6July—22August) . . . . . Yearly velocities extrapolated from measured surface velocities of 6July-22August . Measured strain—rates Measured principal strain-rates Stresses calculated from measured strain—rates . Calculated velocities iv PAGE 16 19 21 22 23 30 3o 3 1 3b LIST OF FIGURES FIGURE 1 Map of part of Juneau Icefield . . . . . . . . . 2 Results of l9b9—l953 glacier movement surveys . . 3 Results of 1963 Taku Glacier velocity measurements . ha Locations of 196A glacier movement surveys and strain—rate measurements on Taku Glacier and Icy Basin . . . . . . . . . . . . . . . . . . . . Ab Bedrock profile along AA . . . . . . . . . 5a Locations of 196A strain-rate measurements on' the Norris Glacier Terminus . . . . Sb Location of strain—rate measurements in an ice tunnel . . . . . . . . 6 Locations of 196A velocity and strain-rate measure- ments on the Vaughan—Lewis Glacier . 7 Relationship between applied stress and strain—rate 8 Stresses acting on an element of glacier ice . . . 9 Types of glacier flow . . . . . . . . . . . lO Limiting stresses in a glacier . ll Variation of velocity with depth y 12 Variation of velocity across a glacier . . . . 13a Computer notations for glacier movements . . . . 13b Sign convention for horizontal velocity vectors . 13c Sign convention for vertical velocity vectors lb A comparison of measured surface velocities (X' components) along Profile AA . . 15 Square for strain—rate measurements . . . . . . . \J 10 . IO ll 12 1b . 2b . 2h . 2h . 25 28 LIST OF FIGURES (Cont.) FIGURE PAGE 16 Calculated and measured velocities (l96h) . . . 35 17 Components of strain in the x"-y" plane . . . b0 18 Deformation of a strain diamond . . . . . . . . Al 19 Mohr's Circle for strain-rate . . . . . . . . . A3 20 Sequence for computer cards . . . . . . . . . . AS 21 Flow diagram for Subroutine JIRP . . . . . . . . b9 22 Flow diagram for Subroutine NORRIS . . . . . . . S2 23 Relationship between velocity and shear stress . 5h 2h Relationship between dimensionless shear stress and dimensionless distance Y . . . . . . . . SS 25 Flow diagram for Subroutine PROFILE . . . . . . 57 26 Example data for Subroutine JIRP . . . . . . . . 62 2? Notation for Special listing of Subroutine JIRP. 6A 28a Example data (Case I) for Subroutine NORRIS . . 65 28b Example data (Case II) for Subroutine NORRIS . . 65 29 Notations for stakes in strain diamond . . . . . 68 30 Example data for Subroutine PROFILE . . . . . . 71 vi LIST OF SYMBOLS a1,a2,bl,b2,c1,cz,dl,d2 lengths of sides and diagonals of strain no A,B,n,w U V X Y (I). TO CUT X’y’xy diamond. yield stress. constant of gravitation. total depth of a glacier perpendicular to the bed. unit of length. surface longitudinal strain—rate. emperical constants x",y" components of displacement. dimensionless distances. angle of inclination of glacier surface. shear strain. rate of strain. angle from y"-axis (positive clockwise). effective shear stress. shear stress corresponding to no. components of stress acting on an element. density. x,y components of velocity (m/yr.). maximum measured velocity in x' direction (NU&r.). maximum velocity (m/yr.). dimensionless shear stress. dimensionless velocities. vii LIST OF SYMBOLS (Cont.) For computer programming (Appendices B and C) the following no— tations are used (see Table I for correlation between these symbols and text symbols). €<2~3U12J BO RO TO group index of angles . . ng stake number. movements. values of 9 (same as O in text). time. distance from base line to a movement stake. horizontal velocity vector. velocity vector. velocity component along x'-axis. velocity component along y'-axis. strain. maximum value of 9. length. total movement. residual value of a function. Skip value. maximum value of t. 2'). location of movement stake (in x',y', interval of time. value of B at t = 2. value of B at t = l. x' coordinate at t = l. x' coordinate at t = 2. viii LIST OF SYMBOLS (Cont.) ym YO (no distance from base line velocity. distance from base line distance from base line strain—rate. angle from base line to to stake with maximum measured to location of maximum velocity. to location of zero shear stress. a movement stake. ix Table I. Correlation of symbols. Subroutine Symbols and Words Symbols in Text on Computer Output JIRP D u E V V CT NORRIS E é E(O),E(h5),E(9O),E(135) é09é453é903é135 E(1),E(2>,E(3) é12é2:é3 EX,EZ,EXZ éxn,éyn;éxnyn SIGMAl,SIGMA3,SIGMAX 01’03’Ox" T t TAU T THETA G TXZ Txnyn PROFILE BETA B DISTANCE Y' TXY (Tx,y,)y': CHAPTER I INTRODUCTION Glaciers are large masses of ice, snow, and firn1 which flow. Observations of glacier movements have led to studies of the stress conditions in glaciers. This study presents the results of surface strain—rate and ve— locity measurements made on several glaciers in the Juneau Icefield of Alaska. The icefield covers an area of approximately 1500 square miles and consists of a complex of glaciers. One of the major valley glaciers of the Juneau Icefield is the Taku Glacier (Figure 1). This glacier is about LO miles in length and extends from the present high plateau center of the icefield down to the tide water level of the Taku Inlet. In this district there is a general pattern of retreat of the glaciers with the exception of the main trunk of the Taku (Miller, 1963). Advance or retreat of a glacier is commonly determined by terrestrial or aerial photogrammetric surveys of its terminus. How- ever, the mechanisms that govern the movements Of a glacier may be better understood if surface velocities and strain-rates are known. The object of the 196A study is to obtain information on the velocity distribution and stress conditions in the Taku Glacier and 1Firm is an intermediate form of glacier ice that is composed of recrystallized snow that has been retained from the preceding year or years. It may also contain refrozen meltwater. 58°45’ ”58.50 "’7'! {NORTH 3” I” I". \oo' / \ T ‘ -\\\ (a [1" I, 8 V ”“1, a t,” m, :1“, D GLAC‘ER “ 8‘ I: 3 E A, - \ . \ I I, \ a Rock I’ 2-," 2 3 W: E \Y =’ [3' 53 WATER , ’i \‘ ‘ IOO {i CONTOUR$\N was? ‘ 304% M——‘. l I54'Jo’ .34‘00’ Figure 1. Map of part of Juneau Icefield. 3 several adjoining glaciers. Reference is made to the previous sur- veys of l9b9—l953 (Miller, 1963) and 1963 (Wu and Christensen, 196A) at Camp 10 and other sites (Figures 2 and 3). Surface velocity measurements were made across transverse pro- file AA at Camp 10 (Figure A), transverse profile BB at Icy Basin (Figure A), transverse profile CC on the Norris Glacier Terminus (Figure 5), and along longitudinal profiles AAl, AAZ, AA3, AA4 at Camp 10 (Figure l) and longitudinal profile DD on the Vaughan—Lewis Glacier (Figure 6). Strain—rate measurements were made at locations designated as I, II near profile AA (Figures 1 and A); III, IV, V at the Norris Glacier Terminus (Figure 5); and VI, VII on the Vaughan-Lewis Glacier (Figure 6). Figure 2. Results of l9h9—l953 glacier movement surveys. UL 5/ 5/ ask/claw I .___9 Sea. e. 0J5" “4/qu Figure 3. Results of 1963 Taku Galcier velocity measurements. “in I I l (H H \ \ § '- / \H (“u \ 1.________..339° H Crevassc - IOOO M. ' E a Rock VEl‘ou‘l' vec'br _. M ‘-‘ 5“ ’3' :jeuuer Sch-es: vectors 4—H +Ib1r Locations of l96h glacier movement surveys and strain-rate measurements on Taku Glacier and Icy Basin. Figure ha. I: z _______________ L_______I__u”. 35a¥ HORIz.SCALE- VERT. SCALE Figure Ab. Bedrock profile along AA. I/ /\\\ (SKETCPD Norms Gl. \ \ (”I Thrus+ n1 \ ‘ " quH’\,\ D - I], Sfdanahl‘ 7 Ice ‘ like, Figure 5a. Locations of 196A strain-rate measurements on the Norris Glacier Terminus. T ACTIVE' ICE ‘x—axn srmnmuT ICE 1 0NA\LS \N |c€ (CROSS—SECTION) Figure Sb. Location of strain-rate measurements in an ice tunnel. NM}. BRANCH TAKu GL \ \ ‘ \ i \\6 . I - NORTH a : °°°‘\ : ’ E, |\ 3&3" é : T’ ‘ ,, \ . /3r. 2 I] ' vemcmv vec'roa 4, , I ,*q_,+unr ‘3 ' H~‘ ,” STRESS VECTORS \ ~ I I‘ \ I Figure 6. Locations of 196A velocity and strain- rate measurements on the Vaughan-Lewis Glacier. CHAPTER II FLOW CHARACTERISTICS The downhill movement of a glacier is the result of stresses in the ice. To study the distribution of stress and velocity, consider an ideal glacier that has a constant gradient and thickness over its entire length. The glacier has infinite width so that its movement is not restrained by valley walls. This can be represented by a sheet of ice sliding down an inclined plane. The glacier is assumed to have a constant yield stress (Figure 7, curve 3) and to obey the , T sheat Stgess (N rate of strain, a Figure 7. Relationship between applied shear stress and strain— rate (after Nye, 1951). Curves: 1, ice; 2. Newton- ian liquid; 3. Plastic relationship assumed by Nye, 1951. Levy—Mises yield criterion T2 = i <0. - v2 + A = c2 m in which c is the yield stress in pure shear (Nye, 1951). The ef— fective shear stress is T and Ox’ 0y, TXy are the components of stress acting on an element of glacier ice (Figure 8). lO {3, Figure 8. Stresses acting on an element of glacier ice. Two limiting states of flow are possible. They are the extensive and the compressive flows which correspond to the active and passive Rankine states in soil mechanics. For the perfectly plastic mater- ial, there is no flow until the yield stress is developed everywhere along the slip lines (Figure 9). > Direction of flow "WV //\ ~ compressive flow extensive flow Figure 9. Types of glacier flow. 11 The limiting stresses that can be attained for the idealized glacier are shown in Figure 10. The actual stresses in a glacier may however, lie anywhere between these limiting values. —2c L +2c ‘ I Ox compressive Ox extensive - - . OX w my TXY Figure 10. Limiting stresses in a glacier. Glen (1955) from his laboratory tests on polycrystalline ice under uniaxial compression, has shown that the rate of deformation of ice is not a case of perfect plasticity (Figure 7, curve 1). He defined the strain—rate e as an exponential function of shear stress, or é = (7'9“ = ET“ (2) T = Shear stress (bars) n = Empirical constant A, B = Empirical constants for a given temperature and water purity (yr1 bar n) 12 The borehole measurements of Miller (1958) have shown that values of B and n may vary over a considerable range. Nye (1957) has calculated values of B and n from results of field measure- ments and found the best values for temperate glaciers to be O.1b8 and h.2 respectively. The distribution of velocity in a glacier (Figure 11) can be approximated by using these values of B and n in the plasticity solution of Nye (1957[a]). Figure 11. Variation of velocity u with depth y. In Nye's plasticity solution, the dimensionless velocities: IT and 'V' are l/n 2 n+1 1+ . [1‘ _ “2(n + 1)"T'1-n + nw ] i Xw (3) and Y= 1'- w (Y - Ybed) (b) in which i is the dimensionless shear stress given by Tm = “ (5) ['T'2 _ Yz]l/2 13 The dimensionless variables are =1 =X :91 = H = V IT“ TO ’ X To ’ Y 10’ -U— nOlo ’ T[_ D010 (6) in which 1’1 To D0 = B(To) ) 1O ='—" (7) 99X The term no is the surface longitudinal strain—rate and pgX is the component of the gravity force along the bed (x—axis). The variables u and v are components of velocity along the x and y axis respectively and w is equal to unity for a solution in which the longitudinal strain-rate is non—zero. In general, if (a2 - a1)¢¥ O or if the glacier surface is nearly parallel to the bed, it can be assumed that the diSplacement 63V (see A endix A Fi re 1") and the veloCit are inde endent Egg PP , 9U I Y v P of x . The dimensionless velocity 'V‘ may therefore be considered to have a constant value along the x-axis. This solution can be simplified if laminar flow is assumed. Laminar flow occurs if an upper layer of ice slips over the lower layer so that each element of ice deforms in simple shear. The longitudinal strain-rate is assumed to be zero. The velocity solu- tion can then be obtained by allowing w to equal zero in equations 3, A, 5 which simplify to 2 n+1 a U-UB-T— Y Y-O (f) 01" 1h The assumptions made for the velocity distribution with depth can also be made for the velocity distribution across a valley glacier between parallel walls. The solution is symmetric about the center line (Figure 12). “WWW-“W I ‘\ ' \ l . \ I \ ' 1 _ F"*d{_ __l Ho - ______ _ i *: y ’I7 ' I7 I Figure 12. Variation of velocity across a glacier. In this case, pgX in equation 7 is B which is the rate of change of T with y . XV In real glaciers, there is also a boundary condition at some distance along the z-axis. However, Nye's solution is a good ap- proximation when the z dimension of the glacier is very large compared to the y dimension (Nye, l957[a]). CHAPTER III SURFACE VELOCITY Surface velocity measurements were made at various locations on the Taku, Vaughan-Lewis, and Norris Glaciers (Figures b, 5, and 6) by triangulation with Wild T-l and T-2 Theodolites. Readings were made to an accuracy of i 2.5 sec. and i 0.5 sec. reSpectively. Stakes were driven into the firn at approximately equal spacings across the glacier for transverse profiles and along the glacier for longitudinal profiles. Several base lines were established to serve as reference points. The base line is the x'-axis which is approximately parallel to the down valley direction (x-axis) of the glaciers. The y'—axis is acroSS the valley and perpendicular to the x'-axis. The z'-axis is vertical and is positive upward. From each base line, a coordinate system was established with the first station having (O, O, 2') as its coordinates. The first station is defined as the station farthest to the right if an observer stands on the base line facing the move— ment stakes. The movements were calculated from a computer program and are given in Tables LI. The general computer notations for movement are given in Figure 13. Transverse Profiles Icy Basin (Profile BB): The surface velocity surveys on the Taku Glacier during l9h9—l953 (Figure 2) have shown that the direction 15 16 Table IIa. Measured surface velocities. G S.ak B c D E v ZETA EPSL romp I“ e M. M/INT M/INT M/INT M/INT DEGREES DEGREES 1 1 228.2 .08’ .OO .08 85.29 1 2 337.5 -.07 -.O1 .06 81.33 1 3 878.6 .08 .03 .08 70.25 1 8 683.8 59.18 3.00 -59.10 —87.10 1 5 771.2 —.28 —.15 .28 56.82 1 6 920.2 -.18 -.12 .18 89.05 1 7 1090.0 -.85 —.18 .81 66.52 1 8 1261.8 -.13 -.O5 .12 68.55 1 9 1886.7 -.17 —.O5 -.17 —73.9O 1 10 1582.9 .50 .11 .89 77.83 1 1 228.2 .77 .69 -.38 -26.82 1 2 337.6 1.27 .98 -.81 -39.69 1 3 878.7 2.20 1.83 -1.68 -89.55 1 8 628.3 3.29 1.86 -2.72 -55.68 1 5 771.5 8.98 2.80 -8.37 -61.25 1 6 920.3 6.85 2.83 -6.28 —65.56 1 7 1090.8 9.51 3.87 -8.86 -68.60 1 8 1261.9 3.91 3.36 2.00 30.70 1 9 1886.5 8.32 3.92 1.81 28.79 1 10 1583.8 19.31 5.33 -18.56 -73.98 2 1 388.6 8.51 .26 —8.50 -86.68 2 2 575.0 -2.66 -.O8 —2.66 -89.07 2 3 NA NA NA NA NA 2 8 1068.8 8.98 7.31 -5.18 -35.18 2 5 1807.6 17.79 17.03 -5.13 -16.75 2 6 1839.5 26.22 28.38 —9.75 -21.83 2 7 2276.1 38.96 32.01 —22.21 —38.75 2 8 2530.0 35.23 29.69 -18.95 -32.55 2 9 2858.8 35.23 27.76 -21.68 -37.99 2 10 3162 8 100.55 27.51 96.72 78.12 2 11 NA NA NA NA NA 2 12 NA NA NA NA NA 2 13 NA NA NA NA NA 2 18 NA NA NA NA NA 2 15 NA NA NA NA NA 2 1 380.1 -.85 -.1O .88 77.65 2 2 572.3 1.08 1.05 .23 12.82 2 3 NA NA NA NA NA 2 8 1059.7 8.71 3.72 -2.89 -37.80 2 5 1802.5 222.38 8.88 -222 29 -88.75 2 6 ‘1829.7 5.16 8.03 -3.23 -38.71 2 7 2253.9 -5.73 -2 90 8.98 59 60 2 8 2511.1 -18.87 -.38 18.86 88.67 2 9 2833.1 15.97 1.70 15.88 83.89 2 10 3259.5 100.56 1.11 -100 56 -89.37 17 Table IIa (Cont.) Grou Stake B C D E v ZETA EPSL p M. M/INT M/INT M/INT M/INT DEGREES DEGREES 2 11 NA NA NA NA NA 2 12 NA NA NA NA NA 2 13 NA NA NA NA NA 2 18 NA NA NA NA NA 2 15 NA NA NA NA NA 2 1 380.5 5.56 5.88 -.99 -10.25 2 2 572.6 108.70 6.91 -108.88 -86.22 2 3 838.6 13.58 12.82 -5.50 —23.90 2 8 1056.8 21.17 16.69 -13.02 -37.96 2 5 1180.2 200.02 28.31 198.53 83.02 2 6 1826.5 36.51 30.70 —19.76 -32.77 2 7 2258.8 57.10 36.38 -88.01 -50.82 2 8 2525.5 67.03 80.36 -53.51 —52.97 2 9 2889.0 98.06 51.55 —78.67 -56 77 2 10 3158.9 113.05 56.83 -97.72 -59.82 2 11 NA NA NA NA NA 2 12 NA NA NA NA NA 2 13 NA NA NA NA NA 2 18 NA NA NA NA NA 2 15 NA NA NA NA NA 3 1 NA NA NA NA NA 3 2 1187.2 9.28 8.80 -3.88 -28.58 3 3 1191.0 8.98 7.36 5.16 35.02 3 8 1213.8 7.27 6.68 -2.95 —23.97 3 5 NA NA NA NA NA 3 6 NA NA NA NA NA 3 7 NA NA NA NA NA 3 8 NA NA NA NA NA 3 9 NA NA NA NA . NA 3 10 NA NA NA NA NA 3 11 2500.1 78.78 88.09 -62.35 -52.36 3 12 2519.2 68.28 88.83 -86.85 —86.27 3 13 2581.6 118.82 37.89 -107.97 -70.66 3 18 2616.1 139.83 35.13 -135 38 —75 85 3 15 2659.2 183.27 83.31 -163 28 -62.96 3 16 NA NA NA NA NA 3 17 2880.3 89.87 56.08 —69.71 -51.19 3 18 2855.1 80.63 85.85 -66.60 -55.69 3 19 NA NA NA NA NA 3 20 NA NA NA NA NA 18 Table Ila (Cont.) Grou Stake B c D E v ZETA EPSL p M. M/INT M/INT M/INT M/INT DEGREES DEGREES 8 1 NA NA NA NA NA 8 2 NA NA NA NA NA 8 3 861.5 80.78 77.09 —28.16 -17.80 8 8 518.7 9.58 9.22 2.88 18.80 8 5 586.0 8.18 7.78 2.81 17.22 8 6 615.6 7.91 7.63 2.09 15.31 8 7 686.0 8.55 8.50 .90 6.06 8 8 681.2 9.30 9.30 .36 2.19 Group 1 = ICY BASIN Group 2 = C-lO, TAKU GLACIER Group 3 = LONGTUDINAL STAKES, C—lO, 1-5, ABOUT STAKE 8*6-lO,5*1l,8*l6-20,9 Group 8 VAUGHAN-LEWIS 19 Table IIb. Yearly velocities extrapolated from measured surface velocities of 6July - 3September. ' C D E v ZETA EPSL GROUP STAKE M/YR. M/YR, M/YR. M/YR. DEGREES DEGREES 1 1 9.51 8.72 -3.79 -23.88 1 2 15.37 12.17 -9.39 -37.65 1 3 27.25 18.37 —20.13 -87.61 1 8 780.99 61.11 -778 59 -85.51 1 5 59.23 28.25 -52.06 —61.52 1 6 88.09 38.20 -76.83 -66.01 1 7 118.11 81.85 -1O6.31 -68.70 1 8 89.82 81.66 26.60 32.56 1 9 53.03 88.81 20.78 23.02 1 10 237.59 68.83 -227.53 —73.26 2 1 86.12 38.36 -30.77 —81.88 2 2 652.58 88.20 -650.80 -85.76 2 3 117.66 100.11 -61.82 -31.69 2 8 211.92 168.77 -128 17 —37.21 2 5 331.57 281.13 -175.80 -32.02 2 6 811.08 359.55 -199.29 -29.00 2 7 586.01 398.69 —373.05 -83.10 2 8 552.08 828.83 —353.05 -39.75 2 9 712.52 893.19 -518.25 -86.20 2 10 807.97 520.17 -618.25 -89.92 2 11 889.07 863.09 -758.98 -58.61 2 12 1101.98 866.60 -998.31 -68.95 2 13 1272.85 890.29 *1178.63 -67.38 2 18 NA NA NA NA 2 15 NA NA NA NA 3_ 1 NA NA NA NA 3 2 138.98 122.78 -56.07 -28.58 3 3 131.23 107.86 75.32 35.02 3 8 106.17 97.02 -83 13 -23.97 3 5 NA NA NA NA 3 6 NA NA NA NA 3 7 NA NA - NA NA 3 8 NA NA NA NA 3 9 NA NA NA NA 3 10 NA NA NA NA 3 11 1150.82 702.59 —910.96 ' -52.36 3 12 939.12 689.15 —678.68 -86.27 3 13 1671.72 553.59 *1577.80 -70 66 3 18 2082.89 513.26 a1977.36 -75.85 3 15 2677.57 1217.23 *2388.89 -62.96 3 16 NA NA NA NA Table IIb. (Cont.) 2O c D E ZETA EPSL GROUP STAKE M/YR. M /YR. M/YR. M/‘YR. DEGREES DEGREES 3 17 1307.17 819.38 *1018.51 -51.19 3 18 1178.03 668.08 -973.01 -55.69 3 19 NA NA NA NA 3 20 NA NA NA NA 8 1 NA NA NA NA 8 2 NA NA NA NA 8 3 1381.19 1279.81 —801 07 -17.80 8 8 158.36 153.11 80.85 18.80 8 5 135.16 129.11 80.00 17.22 8 6 131.32 126.66 38.67 15.31 8 7 181.92 181.13 18.97 6.06 8 8 158.86 158.35 5.90 2.19 21 .oa-o.:eaH u a aaouo aw.au- oa.ma- aq.mmmm ma.meam ca ma.aa ma.oa am.am mm.ooa w.moam 6a a wo.oa- pa.wm- ca.mmwm ma.4m©m a am.am- wo.am- oa.am mm.mm p.4mpm a a aa.m- aw.am- oo.aamm ao.6mmm m mm.mm- ma.wa- mo.mm mm.mm 6.6mmm w a am.m Ha.6m- am.mmmm @o.oamm a ma.am- am.mm- Ho.mm em.wm H.6amm a a mm.a ma.mm- ob.mmwa ma.mmwa o mm.am- ma.m- am.am mm.om m.mmmfi o a am.a- mm.wa- Hm.m64a mo.aoaa m ma.oa- ma.m- mo.aa ma.aa o.aoaa m a CA- 83.07 opdmg $332 4 Adam- 8:.m- Hm.” gap @462 a H <2 a2 a2 a2 a2 m a aa.m- oo.m- am.mam oo.mam m No.8w- oo.m- 46.- op.m- o.mam m a ma.m- ao.m- 86.6mm. mm.4wm A a ,;ao.om- om.a- om. Hm.4 6.4mm a a Began 3386 as} :5: :5: :5: .2 ca om oa om p amam aamm > m a o m meaem enema .moocmpmmp mpmcmppooo 6cm pm3m3 oommuSm Umpdmmoz .OHH ofinmh 22 Table IId. Special Listing of Measured Horizontal Angles (6July - 22 August). (g) STAKE ANGLE HHI—‘l—‘HHHHHH O\OOO\]O\\J'LJZ"\.AJI\JH H H \JNNNNNNNNN \J'IU'IWUIWU‘IU'LWWU'I wwwwwwwmww 1...; O\ocn\10\\nI:-w my» O\o CO\]O\U'11?‘\.A)I\)H O\o CD\10\U'IIZ"LQ NH H 23 Table IIe. Yearly velocities extrapolated from measured surface velocities of 6Ju1y - 22August. C * D E ZETA EPSL mpm> SHWE MAR. MAR. MAR. Dflmflfi mmmms 1 1 33.62 1.97 -33.56 -86.68 1 2 -19.88 -u32 -19.88 -89.07 1 3 NA NA NA NA 1 8 66.61 58.87 -38.38 -3S.l8 1 5 132.59 126.96 —38.22 -16.75 1 6 195.83 181.82 —72.66 -21.83 1 7 290.39 238.59 -165.53 -38.75 1 8 262.57 221.38 -181.25 -32.55 1 9 262.59 206.96 -161.62 -37.99 1 10 789.53 205.03 720.98 78.12 \ \ 75:,j——J [—_ ii— First station 71.! 73.1 lj fiyll,J—__‘ “——‘7J5,j Figure 13a. Computer notations for glacier movements. 1 +E + ZETA + ZETA W -V +EPSL +EPSL +93+D { O \ -C’-D (?\\\\\////j§ ‘._horizontal K } -EPSLW-EPSL 9a?“ ‘ZETA " ZETA +1] ‘V SUI” ace -E V ‘i O original position Figure 13b. Sign convention for Figure 13c. Sign convention for horizontal velocity vectors. vertical velocity vectors. 25 of the velocity vector may deviate appreciably from that Of the val— ley. At several points the velocity vectors Show large components towards the east (y'). This easterly movement has also been observed in the bore—hole measurements at site lOB (Miller, 1958). These observations suggest that a part of the glacier flow may be diverted into Icy Basin (Miller, 1963). However, the 1968 survey of Icy Basin (Figure 8) shows that the flow into this basin is negligible. Taku Glacier (Profile AA): The down valley direction of the glacier at this profile is approximately S82°E (Figure 8). The average direction of the velocity vectors is S780E. These vectors show large y' components particularly in an area 2000 to 8000 meters west of site lOB. Similar y' components of lesser magnitudes were also observed along this profile during 1952 in the vicinity of stakes 5 and 6 (Figure 2). In addition, the movements between two consecutive time intervals in 1968 show a change of 37 degrees in direction while in 1952, Miller (1963) Observed a change of about 80 degrees. These variations in the velocity vectors indicate that the flow conditions may be quite different from that of the ideal glacier. Since this behavior was not observed in 1963 (Wu and Chrintensen, 1968), the unusual flow direction cannot be attributed to bedrock con- figurations. The Change in direction of flow on the Taku Glacier may have been influenced by the flow from some of the tributaries or basins on the west side of the valley. Also, one may question how accurately these firn measurements have reflected the actual ice movements. 26 The firn measurements of the Taku Glacier show that the x' component of the maximum velocity vector (Profile AA), during the interval 6Ju1y - 22August , is 238 Mfiyn (2.18 ft/Uay). This is in reasonable agreement with 1963 value of 250 M/yr. (2.28 ft/Uay). The average velocity in this interval is 155 M/yr. (1.8O ft/day) which compares favorably with the 1963 value of 160 M/yr. (1.88 ft/ day). For the entire measurement period (6July - 3September) , a maximum velocity of 520 M/yr. (8.7 ft/day) was recorded. The only other record of comparable high velocity is 865 M/yr. (8.2 ft/day) which was obtained during a "short period" in 1952 (Miller, 1963). These records show that the velocities flucuate yearly and also between individual measurement periods. A comparison of the measured velocities is given in Figure 18. lboO‘b . Z A 5,5.0w "O 13 \ )4 . .08- o E: [3 Maximum Velocity " 01 for entire season. >’3' on A .. 13 a A Average Veloc1ty 8 2.0“ a D a for entire season. r—l A A g I A A ° "Short term" maximum 1’0 velocity. If 1 341% O\O N m: 43U\Ln \o \o\o O\O\O\ O\ O\O\ u—«u—«I H H Hr—i Figure 18. A comparison of measured surface velocities (x' com- ponents) along Profile AA. 27 Norris Glacier (Profile CC): The stakes on the Norris Glacier Terminus melted out during the 7 day interval between measurements. Hence no results were Obtained. Longitudinal Profiles Taku Glacier (Profiles AAl, AAZ, AA3, AA4): The few records that are available here show general agreement with the measured velocities along Profile AA. Vaughan-Lewis Glacier (Profile DD): These stakes were placed on the crest of waves that originate at the foot of the Vaughan- Lewis Icefall. Starting downstream from the foot of the icefall, the velocity first decreases and then increases. The analysis of the velocity changes is given in Chapter IV. CHAPTER IV STRAIN—RATE MEASUREMENTS The purpose of the strain-rate measurements was to determine the strain—rate tensor. The procedures used were the same as those outlined by Nye (1959). Stakes were driven into the firn and set in a square pattern (Figure 15). The square had equal sides for measure- ment of the normal strains in four directions making angles of 0°, 85°, 90°, and 1350 with the y” axis. For each location, the K" Figure 15. Square for strain-rate measurements. stakes were oriented so that c1 and C2 (x"—axis) lie approximately along thelongitudinal direction (x—axis) of the glacier. Strain diamonds were constructed at locations on the Taku, Vaughan—Lewis, and Norris Glaciers (Figures 8, 5, and 6). On the Taku and Vaughan-Lewis Glaciers, the strain diamonds were marked by stakes which were driven into the firn. They had to be redriven after a 10 day interval because of ablation. On the Norris Glacier Terminus, holes 3 meters deep were drilled into the surface Of the glacier 28 29 in which no reference stakes were placed, It was found that all but 1—2 inches of the holes remained after ablation during a 7 day period. Strain diamonds were also located on the wall of an ice tunnel that was part of a thrust fault in the Norris Glacier Terminus. These strain diamonds were formed by nails driven into the ice. Measurements were made to an accuracy of t 1/8 in. for the surface measurements and i 1/16 in. for the ice tunnel measurements. The measured strain rates are calculated with a computer pro— gram (Appendices B and C) and are summarized in Tables III. The strain—rates ée are obtained by averaging the strain—rates of sides or diagonals of the square. The results are checked against the theoretical relationship G0 + 690 = é45 + é 135 (9) This equation holds if the material maintains a constant volume within the square. The measured strain—rates are used to calculate the best values of éx"’ éy"’ and éx"y" by the principal of least squares (Nye, 1959, 1957[b], see also Appendix B). They are given by e = - l é + é + é + 1 é X" 8 o 8 45 8 90 g 135 . _3. 1. 1. 1. eyn _ 8 6o + H 845 ' 3 690 + 8 6135 (10) G = - 1 é + 1 é X"y" Q 45 E 135 From éx"’ éy"’ and G x"y" the major and minor principal strain- rates G1 and B3 are calculated by means of Mohr's Circle (Appendix A). Assuming no volume change, the intermediate principal strain éz 3O Table IIIa. Measured strain—rates(*10HH—SHHRHH-1) STATION T E(O) E(85) E(9o) E(135) RE 1 1 -.555 .271 .033 -.222 -.571 1 2 -.O99 .088 .068 -.O91 .012 2 1 -.510 —1.351 .136 1.158 -.181 2 2 -.387 —1 817 .237 1.129 .137 3 l .595 —3.925 —8.138 -3.638 .017 3 2 -.218 7.093 12.891 5.577 .008 8 1 -37.O63 65.026 111.696 1.633 7.973 5 1 -58.837 —2.O28 -11 950 —92 268 27.905 6 1 -.888 .082 .800 -.O98 .368 7 1 -1.787 -1.816 .766 -.152 .587 RE )+E(90)-E(85)-E(135) H II E(O T SEQUENCE OF TIME Table IIIb. Measured principal strain—rates.(*lO**-5HR**-l) EX EZ EXZ E(l) E(2) E(3) THETA .176 -.813 -.286 .266 .236 -.502 70.08 .061 - 102 -.O68 .085 .081 -.126 70.06 .182 -.865 1.255 1.158 .283 -1 837 127.77 .202 -.821 1.273 1.201 .219 -1 820 128.12 —8.182 .591 .185 .593 7.551 —8.188 -.95 12.891 -.219 -.758 12.938 -12.672 -.262 86.70 109.702 —39.057 —31.696 116.178 -70.685 —85.529 78.86 -18.926 -61.813 -85.122 9.703 80.339 ~90.083 57.61 .709 -.579 -.O68 .713 —.130 —.583 86.99 .619 —1.893 .632 .769 1.278 —2.088 103.36 Group 1 = STRAIN DIAMOND -D- AT CACHE Group 2 = STRAIN DIAMOND —B- Group 3 = NORRIS STRAIN DIAMOND Group 8-5 = THRUST PLANE DIAMONDS, X AXIS UP AND TO LEFT Group 8 = UPPER SET, UPPER RIGHT Group 5 = UPPER SET, LOWER RIGHT Group 6 = STRAIN DIAMOND ON VAUGHAN—LEWIS NEVE AREA Group 7 = STRAIN DIAMOND ON NAVE#7, VAUGHAN—LEWIS 31 Ham.- mmm. mmm.a- ppm.- afio.a om.moa amfl. mafi.- NHH. woo. a cmo.H Nwo.- 0mm.- mmo.a 68a. am.om mmo. amo.- flao.- moo. o ama.m- mam.a- cao.m- m0a.m- mam.m Ho.am mam.a m06.a- amo.a amp. m AHA.4 amp.- amp. 6pm.: mmo.m oa.ma paw.m ooa.m- cem.o- mafi.ofl a maa.m mmfl.- ama.H mma.m amo.fi ma.mw mam. Mao. mNN.- NAN. m mao.- mem. OON.H- moo. Jam. mm.amfi oafl. mmH.- mmo. mofi. m pqo.- mam.- oao.a- awe. pap. ao.o~ amo. wmo.- mac. mflo. a xaonm NRA maaon Haaon Baa damma m Amvm Amvm Aan Hauoa .Ammdmv mmmupm «AHI**m>vmmpmu1:HMppw popdmmos Eopm popmasoamo mommoppm .OHHH manme is found by él + éz + és = O . (11) The stresses are calculated from the strain—rates by using Glen's flow law (Equation 2). Nye (1953) has defined é and T as 262 élz + e22 + 632 , (12) and 2T2 = 01'2 + 02,2 + 03'2 (13) in which Oi, is thechviatoric stress tensor and is given by I .. _ I ' _ I ' 1 C51 é61302 é62303 g e, (18) Then the principal stresses are I 01 = 201' + 03 , 02 = O , 03 01' + 203' (15) and Ox"’ 0y" , and Tx"y" are calculated by Mohr's circle assuming that the principal axis of strain—rate and stress coincide. The principal stresses at the individual sites are shown in Figures 8, 5, and 6. Site I (same as site D, 1963): The velocity vectors in this vicinity (movement stakes 9 anle) converge and show a transverse 5 (y‘) compressive strain-rate of 0.58 x lO_ x hr-l. This agrees with the compressive strain-rate of 0.15 X lO-5 x hr-1 at site I along the y'—axis. Nye (1952, 1959) has shown that crevasses are formed perpendicular to the direction of maximum tensile stresses. This is not obvious at this site since only small tensile stresses exist and only fine cracks appeared in the firn. 33 Site II (same as site B, 1963): The direction of the maximum tensile stress (major principal stress) is in good agreement with the observed crevasse patterns (Figure 8). The velocity vectors in this vicinity (movement stakes 3 and 5 -1 8) Show a transverse compressive strain-rate of 1.3 x 10’ x hr . There is also a compressive strain—rate (0.89 x 10.5 x hr_1) at site II along the y'—axis. The discrepancy in the strain-rates may be the result of comparing measurements made at locations that are not very close to each other (see Figure 8). The strain-rate measurements of 1963 show a tensile stress of l i bar for the Taku Glacier, while in 1968, the measurements made at nearly the same location Show a small compressive stress of 0.05 i bar. This may have resulted from an increased flow from an upper source area. The measured strain—rates from sites I and II Show that the average longitudinal (x"-axis) strain-rate (no) is +0.16 x 10_5 x hr-1 and the shear stress (Txnyu) between the glacier and the valley wall is between 1.5 to 1.9 bars. These values are applied to Nye's plasticity equation and the results are given in Table IV. These results are compared with the measured velocities in Figure 16. Because of the irregularities in the valley and the sensitivity of the velocity u to values of no and Tx"y"’ the approximate agreement with Nye's solution is considered satisfactory. The maximum transverse velocity v for stake 7 is 32 M/yr. by Nye's solution. The measured transverse velocity is 129 M/yr., at stake 7. These velocities are not in agreement. However, a more detailed analysis would be difficult since this glacier does not resemble the idealized glacier described in Chapter II. 38 Table IV. Calculated velocities TXY= 1.5BARS DISTANCE PLOTTER CALLED TmElu%ME DISTANCE PLOTTER CALLED TXY= 1.5BARS DISTANCE 388.59 575.00 839.28 1068.81 1807.63 1839.85 2276.08 2331.58 TXY= 1.9BARS DISTANCE 388.59 575.00 839.28 1068.81 1807.63 1839.85 2276.08 2331.58 BETA = .OOO627 \EMEUY,MVNR BETA = .OOO799 \EUENY,M¢NR BETA = .OOO627 VELOCITY, M./YR -66.66 81.52 132.0. 175.38 211.13 232.23 239.82 239.66 BETA = .OOO799 VELOCITY, M./YR —509.72 -215.75 23.08 127.98 198.61 230.05 239.39 239.66 35 Curve +, T XYI Curve ++, T XYz '1“ Measured velocity (Reproduced from computer plotter output) Figure 16. Calculated and measured velocities (1968). 36 Site III;'Phis site is shown to have a tensile stress in the longitudinal (x"—axis) direction. This is inconsistent with the usual concept of compressive stresses in a glacier terminus that is pushing against stagnant ice. However, if only the first time inter- val is considered, there is a compressive longitudinal (x"-axis) strain-rate. At the end of the first interval, the holes were only 2 inches deep and were redrilled. The distances were not remeasured after redrilling and therefore the measurements which showed large tensile strains may nOt be reliable. Sites IV and V: These sites are part of a series of strain dia- monds located on the wall of an ice tunnel. The lengths of the sides of the strain diamonds were measured between centers of nails. The movement of the ice caused the nails to rotate and therefore made the distances difficult to measure. For this reason, the results may not be reliable. This is also suggested by the extremely high values of RE(volume loss) in the results. The nails of the other strain diamonds in the ice tunnel fell out because of melting of the ice. Sites VI and VII: The strain-rates at these sites are in reason- able agreement with the stress conditions as indicated by the move- ment stakes. In general they indicate compressive stresses in the area immediately below the icefall and tensile stresses in areas further down glacier. These stress conditions can be associated with the formation and movement of the waves below the icefall. As these waves move down glacier, the amplitude decreases and the wave length increases. The velocity measurements show a faster movement of the 37 ice further down glacier where tensile stresses are produced. The strain diamond on wave #7(Eigure 6) shows a tensile strain- rate (x'—axis) of 0.68 x 10—5 x hr_1 and the movement stakes 7 and . . . . . . 1 - —1 8 Show a ten31le Strain-rate (x'—aXls) OI 0.88 x 10 5 x hr . CHAPTER V SUMMARY The measurements on the Taku Glacier in 1968 show surface velocities reaching a maximum of 520 M./yr. and a stress in the longitudinal di- rection of -0.05 bar with a shear stress of about 1.? bars at the valley walls. The measured strain—rates are in general agreement with the crevasse patterns and the measured surface velocities. However, there are variations in the velocities and stress conditions of the Taku Glacier from year to year and also between measurement periods. The change in direction of the measured velocities for 1968 is difficult to analyze with a simplified model of glacier flow. Another difficulty that still needs to be investigated is the relationship between measurements made in firn and the actual ice movements. 38 BIBLIOGRAPHY Glen, J. W. 1955. The creep of polycrystalline ice. Proc. Roy. Soc. A, 228, 519. Hill, R. 1950. Mathematical Theory of Plasticigy, Chp. 1. Oxford, Clarendon Press. Hoffmann and Sachs. 1953. Introduction Plasticity for Engineers, Chp. 1., MCGraw-Hill. Miller, M. M. 1958. Phenomena associated with the deformation of a glacier bore hole. Union Géodésique et Géophysique In- ternationale. Association Internationale d'Hydrologie Scientifique. Assemblée generale de Toronto, 1957. Tom. 8, 837. Tiller, M. M. 1963. Taku Glacier Evaluation Study. State of Alaska Department of Highways, 137. Nye, J. F. 1951. The flow Of Glaciers and ice sheets as a problem in pure plasticity. Proc. Roy. Soc. A, 291, 558. Nye 82. 3 , J. F. 1952. The mechanics of glacier flow. J. Glaciol., 2 Nye, J. F. 1953. The flow law of ice from measurements in glacier tunnels, laboratory experiments and the Jungfraufirn bore hole eXperiment. Proc. Roy. Soc. A, 229, 877. Nye, J. F. 1955. Comments on Dr. Loewe's letter and notes on crevasses. J. Glaciol. 2, 512. Nye, J. F. l957[a]. The distribution of stress and velocity in glaciers and ice—sheets. Proc. Roy. Soc. A, 239, 113. Nye, J. F. 1957[b]. Physical properties of crystals. Oxford, Clarendon Press. Nye, J. F. 1959. A method of determining the strain-rate tensor at the surface of a glacier.' J. Glaciol., 3, 809. Poulter, T. C., and others. 1989. Seismic measurements on the Taku Glacier, by T. C. Poulter, C. F. Allen and S. W. Miller. Standord, Calif., Stanford Research Institute. Wu, T. H. and R. W. Christensen, 1968. Measurement of surface strain-rate on Taku Glacier, Alaska, J. Glaciol., 5, 305. 39 APPENDIX A DETERMINATION OF PRINCIPAL STRAIN—RATES The deformation of two sides of an element AOB is shown in Figure 17. - dy" a T—O . U L I V" V , __________ dx" 0' B! V dX" 3.3 EX" Figure 17. Components of strain in the x”—y” plane. The component of strain in the x” direction is given by All (16) ex" = a X" and the change in angle AOB is defined by the shear strain Yy"x" -3U quxn " in + 3" (1") ax" ( av . . . . . . . .. If 3;?" 13 equal to zero, then the shear Strain ey"x" lS dellned as (18) direction of 81 The general notation of positive stress and strain for a strain diamond is shown in Figure 18. A H / \ ’00 y / \ / \ \ J / \\ 2 / 0 .9. / \ \ ‘H / \\ N / / \ \ / \ // Y nxn \\ H / ,2! \ \ T3 / \ 3 / 135° \ E? 85° H m ;> 1 X" 900 T Xiiy" 3 Figure 18. Deformation of a strain diamond. The strain-rates in the direction 0°, 85°, 90°, 135° to the y" axis may be computed from the following equations. éO = Oéxn + OYynXu + éyn G45 = éxnsin2 0 + yyux"cos 0 sin 0 + éyncos2 0 (19) 690 = EX" + OquXn + 06y" 6135 = exnsin2 0 - Yy"x"cos 0 sin 0 + éyncos2 0 These equations may be re~written in matrix notation as GO 0 0 1 ex" ‘ l l 1 0 645 E E E Yynxn ego = 1 0 O . (20) e . 1 l 1 y" [—6135 L7; ‘72 '9 L 83 and the strain—rate along the x" and y" axis are then obtained from equation 25 as 0 l O . 3 0 l I ex" = ‘ E 6o + E 545 + E 690 + E 61:55 o l o 1 o eynxn = '5 e45 - § 6135 (28) . _ 3 a l o l 0 1 0 Gym ‘ E 6O + E 545‘“ fi 690 + E 8135 The principal strain-rates may then be calculated from Mohr's Circle (Figure 19). Shear strain—rate _ 1 asStrain-rate I : exily" é +é ‘\\\ | ._____JL X" 2:" _ . I Figure 19. Mohr's Circle for strain—rate. APPENDIX B DESCRIPTION OF COMPUTER PROGRAMS The methods of computer programming make possible fast and re- liable computations. These methods were used to obtain the quantities in Tables II, III, and IV, and to plot Figure 16. Three main subroutinesl have been compiled on tape #800 (M.S.U. computer center). They are named JIRP, NORRIS, and PROFILE which consists of equations for calculations of surface velocities, strain- rates and stresses, and Nye's plasticity solution, respectively. When using a tape, programming procedures must be altered by the use of control cards in the user's program. There should be between 3 to 5 control cards in the user's program for tape #800 (Appendix C, p. 60). The most important one of these has a FORTBIN,2,6. statement. This converts a program, written in Fortran 60, into the Binary language and compiles it on tape unit 2. Any subroutines needed for the program are then located on the tape unit corresponding to the second character, 6. For a proper program execution, the computer cards should be arranged as shown in Figure 20. The data cards have a special for- mat depending on the Specific subroutine that is requiested (Appendix C). 1A subroutine is a sub—program which can be entered by a CALL statement with the subroutine's name and suitable variables for arguments. 88 1 Blank card Data Comment cards (JIRP,NORRIS) B,n Assigned Values fl 3) (PROFILE) 1 Blank UN' card “' END END CALL JIRP \\\ PROGRAM SAMPLE \\\( FORTBIN,2,6. \\\ - REWIND,6. \ CONTROL. \\\ REMARK,TAPE 800 ON UNIT 6,TYPE CONTROL,2. NCOOOOOO NAME Figure 20. Sequence for computer cards. Subroutine JIRP This subroutine computes surface velocities from the measured angles between the base line and the movement stakes (Figure 13). At the survey sites, certain readings were not made because of fog or the fact that a stake may have fallen over. For these con- ditions, a pseudo angle of 361° can be used in the data. The measured angles from the data cards are stored in the mem- ory. These angles give the stake locations based on the Cartesian Coordinate system. This location can be represented by V(x', y', z'). . t E,J 1, 2, 3, 8, ... 1, 2, 3, 8, J in which Vt j are the values Of x', y', 2' at time t for stake 3 j. The measured angles needed to calculate Vt j are denoted by ) 8t-3, 8t-2, 8t—l, 8t. 1, 2, 3, 8, 5, (29) 1) 23 33 h) 5) '1’9.J 9 J t in which the values of 9 denote the four measured angles required to determine the coordinates for each stake at a given time (t). The change in location or the stake movement (m) is given by m. . = V — V. . (3O) t,-J t+l3j C,J and the total movement (M) as t=T M1 - me (3° t=1 in which the maximum value of t, (T), is determined by the maximum value of g, (G), on the data cards (refer to equation 29). Since 87 there are four measured angles at each particular time, we have, G T = 8 (32) in which the corresponding integer values of T and G are T = l, T = 2 T = 3, ... and G=8,G=8,G 12, Whenever ”kg,j equals 361°, the time is denoted by t'. In the following example, '1’11’2 is equal to 361°, (9 = 11, and t = 3) V3)2 is indeterminate and t' is therefore equal to 3. The move- ments are represented by Vl,2, Vz’z’ Vt',2’ v‘32, 1,5,2, ... VTJZ 1 IL IL 1. ll I m m. c .0. 1,2: tl_1,2:mtt,22 m4,23 mT-l,2 The two letters NA (not available) are printed on the output for the indeterminate movements m., . and m., . t’ _1)J t’ )J In order to compute the total movement (M), this subroutine considers the movement (m) ; however, to account for missing angles we introduce the movement N defined Nt',1 = Vt+2,1 ‘ V9.1 (33) in which .2 is the increment of time between two known values of Vt j (not considering pseudo angles of 361°). Hence, mt j is a ) 3 Special case of Nt' j in which 2. equals 1. 5 The movements in this example can now be represented by L__ Jl I I II | ml 2 , N3 2 , M4 2 ’ on. HIT 3 3 3 -1,2 in which 1 equals 2 . The total movement (M) is then revised to t=T t=T / M. = ‘ . . N. . 1 thm + 2 : to (38) t=1 t=1 in Which = . . = ., _ . = ., mt), 0 if t t l , t t (35) and N. . = 0 if t t' O t), i (3 ) In general, there may be any number of values of t' and Nt',j provided that 365 3 (3?) Limiting the value of ,2, is an inherent quality of this subroutine. When the value of ,2. is the limiting case (18: 3), the value of T is also limited by this subroutine to T = t; + 3 (38) in which t; is the first of the two successive values of t' . If at a given time t' the values of ”F g j are 361 for all ) stakes j for a given g , that group may be omitted from the data. If the group that is omitted is G (or consecutive groups such as G, G-l, ...), the terminal 9 becomes G' and therefore does not yield an integer value for T in equation 32. In this case, the correct value of G for a given T is determined when values of G are chosen to satisfy G-lfG'§G+2 (39) A flow diagram for Subroutine JIRP is given in Figure 21 and format Specifications are stated in Appendix C. 89 Data Print 8 Input cards YES Was the total velocity computed? N0 Are there any groups remaining? YES \ NO (Collect from memory) Are there any com- ponents of velocity? YES Sum all com— ponents and compute total movement Sum horizontal com— ponents and compute total horizontal movement 2/ [Print total movementS} Read all angles for a siteL NO EXIT Change all angle to radians and find reduced angles from base line to stakes. Make 361 angles equal 0. [For each stake: Are there direct and reverse readings? NO Only N0 Only N0 Average ‘ YES reverse direct rea ings YES YES Are all stakes accounted forfEENO Search arrays of angles ES Print NA set for each stake. Find a movement com- value #’0? N0 ponent = 0. YES Search and find a YES Is it in the right second value ¥’0? sequence of time? NO YES [Make it next sequenéfl+———- YES P ' ' NA . . . . rint C9TDUL€ dISCance to. NO Was it a "filled-in" but store Stakes and components 1 9 com onents of movement. va ue. . p 1n memor Are there enough I vertical angles to r———§E§—i00mpute tOtal vel NO . Were the horizontal find total velocity Figure 21. velocities found by i“filled—in" values?i ocitfl Y . . ... Print veloc1tles N0 and store com- Flow diagram for Subroutine JIRP. ponents in memory 50 Subroutine NORRIS This subroutine is designed to compute strain—rates and stresses from the strain diamond data described in Chapter IV. The calculations of strain—rates are based upon Changes in lengths of the sides or diagonals of the strain diamond. This subroutine considers these lengths as LL k k _ l: 2) 33 h) 5: 63 7: 8' t 1, 2, 3, ... in which Lt,1 to Lt,B are the lengths a1, a2, b1, b2, cl, c2, d1, d2 (Figure 15) respectively for time t. The values of strain (F) are Lt+1,k ' Lt,k > t,k F and can be schematically represented by L1,k ’ Lz’k ’ L3,k ) L4’k 900 l 11 J1 J F F F ... l’k 3 Z’k 3 3,k 3 However, if the stakes are redriven after a measurement, the dis- turbances effect the quantity (L. t+1,k — Lt k) for the tlme lnterval 9 At. To eliminate this, the distances are remeasured after the stakes are reset. Whenever the stakes are redriven and remeasured, the time is denoted by S1 . The values of Si (same as Skip value, Appendix C) are indicators to the computer that the strain between the interval t = Si and t = Si+l is not to be computed. This is represented by L1,k . LSl,k ’ LS,+1,R ’ LSZ,k ’ LS2+1,R 9 L83,k . Ll I I J L I Fl k 3 F2 k 2 F3 k 3 '°° ) 3 .9 51 in which the times for redriving are for i=l,i=2,i=3,... It is shown in Appendix C that if i = -l on the data cards, the values of Si are automatically taken as 2, h, 6, ... . How— ever, other values of Si can be obtained by Specifying them on the data cards (Appendix C). Once the values of strain (F) are determined, then the strain— rates (é) can be calculated by _ 1 _ et,n — gm: (Ft,k-l + Ft,k) 1'1 — 1,2,3,LL. (L11) k = 2n t = 1,2,3,... in which Wt is the interval of time between successive values of t. The index n corresponds to values of Q for 0°, L50, 90°, 135°, respectively. The calculations for stress are then carried out by Mohr's circle of stress and strain-rate. A flow diagram for NORRIS is given in Figure 22 and format Specifications are stated in Appendix C. 52 Data Print fi Read Numbe Input Cards of groups Is this a sur—,‘ NO lFinishedfi YES.aPrint all -veyed strain ‘ results in diamond? '0 " '-o ' v \ss Read lengths Read angles of diamond and and convert convert all to ,to radian feet. Compute dis- tance out to _siahe. 7 Compute strain Find length§\ and average to of sides. find 59. Store in memory. Save all result r in memory. Are there any moreg‘ readings in qroup? NO (Collect from memory) Compute stresses» Compute each, EX,EZ,B 01,03,0y,T,TX§__I and El,E2,E3. ’ Average each, EX,EZ,EXZ to find total El,E2,B3, and Theta. Figure 22. Flow diagram for subroutine NORRIS. Subroutine PROFILE This subroutine computes the solution to Nye's plasticity equa- tion and plots the results together with measured velocities. The results of Subroutines JIRP and NORRIS provide the data for Subroutine PROFILE. If the x' and x" axes coincide, then the necessary data consists of the x' component of velocity (same as D total movement, from JIRP), the shear stresses Txny" (same as ) TXZ from NORRIS), the x" component of strain—rate (same as EX from NORRIS), and the distances x', y' (same as R0 and BO from JIRP, Special listing). If the two axes (X' and x") do not coincide, then it is necessary to calculate the components of shear stress and strain-rate along the x'-axis. Subroutine PROFILE does not make a distinction between x' and x" , therefore in describing this subroutine, the x"-axis will be referred to as the x'—axis. Subroutine PROFILE computes the Shear stress at the valley wall (Tx,y,)y,=o from the Shear stresses at y' my. = W + (T,,,y.>y.=O (L2) in which I3 = % hwy.) (2.3) The shear stress and velocity diagrams are compared in Figures 23. The value yT is the distance from the valley wall to the point of zero shear stress. For the ideal glacier described in Chapter II, this point should coincide with the point of maximum velocity at distance yo from the valley wall. The distance (ym) from the valley wall to the movement stake showing a maximum velocity Sh .__,. u,TX,y, Ill/l (a \\\\\\\\\\ I////// \\\\\\\ ' I .\\\ VII/III \\\\ \ LV/I/I/ ;\m\\1// I ym+1 ym Yo ym-l I \\ ”77,, 1\\\\\\\ n III I \\\\\Y Ill Ii/;\\\\\\\ 171/; II! \\\\\\x\ x I” 07/! \\\ \\\ / '1 ll // \‘\ ‘\\‘ //’ l \ H3TXI I 7 “(fl \>\\\\ a I/ / \\\\\_\\\ II////// \\\\\\\\ I/I/fl/A \\ Ill y\\\\\\ / y\\\\\\ /////// \\\\\\§ .... h TxtyryT _____._.__JL_ x Measured velocity 0 Shear stress from measured strain-rates W777 \\\\\,,. mu \\\\\\ Ill/II \\V\ \ l/l7////\\“ “\ - -7” . " \\\\ \ \ \ \ ”an”. I \\\\\-“ ”’” ”f “ ‘ ““ Figure 23. Relationship between velocity and shear stress. 55 is given on a data card (Appendix C). The distance yT is computed from equation N2 and compared to the distances on the velocity profile. This subroutine considers yT to be an acceptable value if Yo or ym+l T T’ L 131.5 2 i i I I ‘: Figure 2b. Relationship between dimensionless shear stress rr‘and dimensionless distance Y. After Nye (1957[a]). 56 A function1 named ROOT was incorporated to solve equation 5 for values of VTjin 2T1 R = “I“ — YZ‘T'Zn-Z - 1 (AS) in which R is a residual value and is equal to zero when values of rT'('T;) and Y(Yp) satisfy equation 5. The procedure is to select a value for Yp and choose two values for O A , a new upper limit for rFu is taken as W‘A . The values of Yp and "fifi are obtained by repeating this process until (TU. " TA)(TA " TL) = 0 (LL?) which is limited to an accuracy of 10 decimal places on the computer. A flow diagram for subroutine PROFILE is given in Figure 25 and format specifications are stated in Appendix C. Apart from the calculations, there are extra features in each subroutine. General comments may be printed with the results in 1A function is Similar to a subroutine except that it is entered when there is an equation containing the function's name and suitable expressions or variables for arguments. 57 Data Find the Is there more than Input values of B,n. NO one strain diamond? Is plot out- Convert y" dis- ssume pt. ofFZero YES put required. tances to have shear at location YES NO origin at pt. of of maximum velocigy . X . O shear. .+ Is this the L Compute slope of first qroup?--NO shear diagram YES‘ C9mPULe shear Is distance to zero Print message Stress in V31“ shear pt. consistent to operator ley wal. an 4 YES with the location of subtraCt .2 . .. and ause. bar maX1mum velocity or ' .2 adjacent Stakes? Compute veloc ity at 20th of Print error NO inch of scaled( diagno tic distances on . oaper. YES IS data reliable. / 3 code) Plot velocity with pen down. NO NO IS compute NO_ IS computed EXIT veloc1ty = veloc1ty = vmax/2? v max? YES YES Draw line from raw 2nd half curve and label of curve symmetric. NO Are 2 curves dd ffl to valley drawn? wall shear stress YES ’4} Plot measured YES Are there SCRIPT velocities with d"? CompUte r NO velocities Plot letters by Subtract .h from zisiaizzs finding coordin- valley wall Shear ates based on a stress. é//// 9 pt. sq. //Ng///’//f Print distances 2 sets of Add .h to valley and velocities! NO computed wall shear stress velocities? YES tJMore groups. Figure 25. Flow diagram for Subroutine PROFILE 58 Subroutines JIRP and NORRIS. Subroutine JIRP also has a provision for a special listing of coordinates and angles. Subroutine PROFILE has two provisions. One is for assigning values to the constants in Glen's flow law other than those calculated by Nye. The other provision is a method of printing titles on the output from the plotter_by using Script cards (Appendix C). APPENDIX C DETAILS OF COMPUTER SUBROUTINES AND PROGRAMS A program for tape #hOO is correctly prepared when the computer cards are arranged as shown in Figure 20. The statements begin in column 7 on the third through the sixth cards. The statements on the other cards begin in column 1 except for the data cards. These have a Special format depending on which subroutine is requested. Identification Card The following symbols are punched in the appropriate columns of an ID Card. Symbol Columns % 1 C 2 (a Six digit problem number) 3,b,5,6,7,8 (last name of user beginning in ...) 10 (time limit in minutes) 31 (program name beginning in ...) 36 Control Cards The control cards immediately follow the ID Card. Statements are punched for the control cards beginning in column 1 and are as follows: 59 60 Card Statement 1 REMARK,TAPE boo ON UNIT6,TYPE CONTROL,2. 2 CONTROL. 3 OUTPUT,G,27. <——(use only if plot output is requested in Subroutine PROFILE) h REWIND,6. 5 FORTBIN,2,6. It is permissable to have several computer decks in a series. If more than one program at a time is to be executed, then a new ID Card followed by new control cards are necessary for each. Since the first two control cards listed above stop the computer in order that the operator may check tape units, they need not be included in any subsequent decks. However, one may choose to replace the first two cards by a single card punched as REMARK,TAPE LOO ON UNIT 6. for computer programs other than the first deck. Program Cards These statements follow the control cards and begin in column 7. They are Card Statement 1 PROGRAM (any name, 7 letters or less, except JIRP, NORRIS, or PROFILE) 2 CALL (choice of JIRP, NORRIS, or PROFILE) 3 END h END 5 Blank Card 6 RUN. <+— (begins in column 1). 61 Comment Cards (JIRP, NORRIS) These cards are designated by a dollar Sign ($) in the first column and immediately follow the RUN. card. Any statement punched in columns 2—80 will then be printed with the computer output. Subroutine JIRP A sample of data is given in Figure 26 in which the columns for the computer cards have been designated by arrows. The Index Card is the first data card on which there are three numbers. Column A is a number for a site (1,2,3, ...). Column 8 is the group index g to the angle 7’9), (see Figure 13). The last number ends in column 12 which is the total number of cards to follow for a given group. The cards following the Index Card are the Angle Cards. The first one contains the sighted angles to a base line station. The remaining cards in the group are the recorded angles to the move- ment stakes. The angles are given in degrees, minutes, and seconds respec— tively for each direct and reverse reading. These values appear on the Angle Cards ending in columns 9, 19, 29, 39, A9, S9 reSpectively (see Figure 26). If there are any fractional parts of an angle, they may be given as a decimal value appearing in the tenth column 10,20, 30,b0,50,60 (no decimal point). Angles that are in degrees, minutes, or seconds with a value of 00 need not be punched since blanks are read as zeros. 62 Columns 0 h 59 l2 l9 29 39 L9 59 Data Card f l 11 1 l 1 l l l 1 I 1 1 3 2 l 13 25 52 361 3 I 106 21 17 361 b | 106 22 26 361 S l 1 2 3 6 91 3b 269 35 7 l 112 20 10 2L8 2o 15 8 I 361 361 : (Etc.) | l l l I | I Direct Readings Reverse Readings I ‘ ‘ l -l 8 3 I 90’ 29 hO" ‘270 29 557 l 103 27 30 257 3b 20 l 100 uh ho 259 17 35 I 6 385.h O ' 2 1 3 : (Etc.) | T T T T T T T T T T 0 )4 89 12 19 29 39 D9 59 Figure 26. Example data dards for Subroutine JIRP. 63 The Index Card which precedes the last group (G) of Angle Cards must have a negative value for the site number. This indicates to the computer that this is the last group for that site. After the last group of Angle Cards for each Site, there are two remaining cards. They are the Base Line and Time Cards, reSpec- tively. The Base Line Card contains the base line length (meters) and is punched anywhere within the first 10 columns provided that a decimal point is also included. The Time card contains the time interval (days) between the first and last readings and should ap- pear on a card ending in column b (no decimal point). This card completes one site. The next cart to follow is an Index Card for another group of angles at a different site, or if there are no remaining sites, the next card is blank. A blank card after a Time card therefore term- inates the program. One may also obtain a special listing of all horizontal angles and distances of a coordinate system passing through the first sta- tion by punching a second dollar Sign (8) in column 2 of any comment card in JIRP. It should be noted that if there are any values of 3610 for WFg’j , will not be 361. They will be the values of 'fJg , for the next ) the angles that are printed for the special listing g for that angle measured from the same station if there are enough values to compute a movement. Otherwise they will be zero. For example if 7V%3 = 3610, 7pqs = N20 then the Special listing will give 7P§$ = L20, 7V%3 = L20. If 7Ps3 = 3610 , 1P%3 = 3610 and no values are obtained for 7V13,3 , then 7P53 = 0 , ’7U93 = O. J ) 6h The values of y' for times t = l and t = 2 are given by BO and A0 , reSpectively. The values of x' are given by R0 and TO reSpectively. These distances that are given in the special listing of JIRP are Shown in Figure 27. i=2 7F+ p |\ \ t=1 A0 : \ ;t + movement stake \ / . : \ / , B0 BASE 1106—1 , . \ ,/ I Effi' *EA aiI__’—':§ TO RO Figure 27. Notation for special listing of Subroutine JIRP In this figure, T0 is positive and R0 is negative. The final tabulated results are based upon the same coordinate system (Figure 13) in which B = BO D = TO — R0 (L18) E = A0 — BO C = (02 + 52)l/e Subroutine NORRIS Sample data are given in Figures 28. The first data card con- tains the total number of strain diamonds. This number ends in column b. The general order after the first data card depends on the method employed for the measurements. 65 .mHmmoz mchsoeosm eom HHH emmov meme eHaemxm .Hmm wesmHa H mmH H.uemv H mam H- m m m H H w Hem cm Nm oHH a Hem mo moH e Hem mH am moH m Hem NH Hm eoH H Hem mm mm mH m H- m H m H H m H H memo H H H H H H H H H H «3.8 mm mH mm mm Hm OmmH 0H NH m H o .mHmmoz eeHHsoeesm HOH HH emmov whee mHQmem .mwm mesmHa H.0Hmv H- H H m HH mmw as meH as mam me HOH me mH QHHH am cmoH as wwHH we Nmm me NH oH so HH so mwm mo 00m mo HH mmoH am mmHH am mmH me mmm me HeHHmH oH mmm Sm mm OOH mo omw mo . m HH mm meH mm omH mo mHm mo m mmHH mm mHHH mm 00m mo mmm mo m mmoH Sm msoH comm mom mo 00m me o m m H H mmH m Hema.oH .oo.am - HHH H N H H m m H echo H H H H H H H H H H H H H H memo cmma came comm ome oH mm Om mm Hm cm mH oH H H o 66 CASE I (Measurements by taping) Card (2) INDEX CARD (3) SKIP CARD (optional) (A) TIME CARD (5, ...) DATA The format of the Index Card is Column A strain diamond number (1,2,3, ...) 8 number of sets of readings 20 number of time intervals 2D skip index, in which the skip index is a number which indicates the number of time intervals when the only movement was due to the redriving of the reference stakes. This number may be from -l to 6 . A skip index with a value of zero (blank) indicates a comparison of data for consecutive intervals. A minus number indicates a compar- ison of data for every other interval. A positive number indicates the number of skipped intervals which appear on the next card. This next card is the Skip Card and contains these interval numbers (skip values). The numbers should end in columns h,8,l2,l6,20,2b. A maxi- mum of 6 numbers may therefore be used. This card should not appear unless a positive number is used for the skip index. The following cards contain the time intervals (weeks). One value is to appear on each card in columns 1 and 2 with decimal values in columns 3 and b (no decimal point). 67 The next card contains the measurement data. They may be given with decimal values of a foot or with inches and decimal parts of an inch. The lengths a1, a2, b1, b2 (Figure 15)(in feet) should end in columns 8,28,b8,68. The next card contains the last A lengths c1, c2, d1, d2 (in feet). The numbers should end in columns 8,28, D8,68. If there are decimal parts of a foot, they can be included as the 9th and 10th columns (9,10; 29,30; b9,50; 69,70), with no decimal points. If the measurements are also in inches, the number of inches should end in columns l8,38,58,78, and decimal parts of an inch in the 19th and 20th columns (19,20; 39,AO; 59,60; 79,80), reSpectively. If the square is very large (as on adjacent longitudinal profiles), then it is more convenient to use CASE II. CASE II (measurements by surveying). The format of the Index Card is Column A strain diamond number (1,2,3, ...) 8 number of sets of readings l2 "5" (a constant) 16 group number (g) 20 number of time intervals 2D skip index. Apart from the Index and Skip Cards, the formats for the angle and the Base Line Cards are the same as in Subroutine JIRP. The format for the Time Cards are the same as in Case I only they appear at the end of the group G for each site. 68 Case II is designed for D movementstakes with the order of angles as shown in Figure 29. i=3 \ \70 ,m /2\ 1,4 Figure 29. Notations for stakes in strain diamond. Subroutine PROFILE No Comment Cards are allowed in this subroutine. To change the constants B and n (equation 2), a card with a dollar Sign (8) in column 1 is inserted after the RUN. card. The new values for B and n end in columns 7 and lb (with decimal points), of this card. If this card is absent, the values of 0.1b8 and h.2 are automatically taken for B and n. The next (or first) card is the Index Card. It contains the following: 69 Numbers Ending in Columns 3 total number of strain diamonds 6 total number of measured velocity points 9 number of stake at which maximum velocity occurs 12 a code for the type of output. 0 = no plot, 1 = plot. 13 a code to by pass an error. 0 = stop, 8 = continue. 17 number of distance units/inch along y'-axis 21 number of velocity units/inch along x'—axis 30 number of the strain diamond (STRAIN-RATE,EX) to be used in the calculations. A zero or blank indicates an average EX to be used. If column 12 contains a one, then the OUTPUT,G,27. control card must be used (see p. 60). If this column contains a zero or a blank, then the OUTPUT,G,27. control card may be omitted as well as the scale factors in columns 17 and 21 of the Index Card. The scale factors give the Scales in the horizontal direction (same as x' and is limited to 11 inches, the width of the paper) and the vertical direction (same as y' and is not limited) of the plot output. The next cards are the Strain Diamond Cards. These contain the following: Numbers Ending in Columns 7 value of TXZ in bars (with decimal point) lb distance (M.) from base line to strain diamond in decreasing order (no decimal pt. if 2 deci- mal places are used) 21 average EX (bars) for an individual strain diamond (with decimal pt.). 70 Following these are the Measured Velocity Cards. They contain the following: Numbers Ending in Columns 7 X' distance (M.) of the movement stakes (R0 from JIRP) 1A Y' distance (M.) (B or B0 from JIRP) 21 x' components of velocity in m/yr. (D from total movement, JIRP). N0 decimal points are needed in the above values provided that 2 decimal places are used in each. An example of the data cards is given in Figure 30. The next card may be any one of the following three. If it is blank, the program terminates. It may be an Index Card for another group of data. Finally, it may be a card which puts a title on the plot output. The last type of card is called a Script Card because it must contain the word SCRIPT beginning in Column 1. The format of a Script Card is as follows: Ending in Column 6 the word SCRIPT 9 size of block letters (ie. .25 is 1/A inch) with a decimal point. 10 number of inches above the maximum scaled y' value to start printing. 11—80 any title or name. Column 11 is the first letter to be printed. A dollar Sign (8) or an asterisk (*) must be punched immediately after a title on a Script Card. This marks the end of the title. The dollar Sign 71 01 3 67 9 121A 17 21 Data 11 1 11 1 1 1 1 1 Card 1 2 6 3 1 A50 100 2 -.373 285A70 .119 3 .903 106881 .192 A -1832 180763 12696 S -2279 1839AS 181A? 6 -2971 227608 23859 7 -3h87 253001 22138 8 —38DA 285b79 20696 9 ~A2AO 316275 20503 10 SCRIPT.SO2VELOCITY PR0EILE8 11 SCRIPT.2S NAME* 12 2 10 7 0 (Etc.) Figure 30. Example data for Subroutine PROFILE. 72 indicates that the printing begins with the contents of column 11 at a distance of 10.5 inches to the left of the right margin of the paper. The asterisk causes the plotter to print the title so that a blank Space after the last letter ends at the right margin. More than one Script Card may be used to print as many titles as desired. Once a number in column 10 has been Specified, another need not be given in additional Script Cards. A blank in column 10 of sub- sequent Script Cards will automatically indicate that the Spacing between the lines of print is 1/2 the size of the letters of the previous title. However, if different spacings are desired then different values may be Specified on each Script Card. A blank card after the Script Cards terminates the program. The subroutines, functions, and programs, as they were written, are given by the following: .051712 HAVAS , 7 3 COMPILE~TAPE 400 :MARK.TAPE 400/wRITE RING :MARKQTAPE 400 ON UNIT 6‘TYPE CONTROLOEO ’NTROL. IWINDoéo ’RTBIN020L06930 SUBROUTINE JIRP DOL=5320202020202020 DIMENSION ANGLEA(SO)oANGLEB(SO)oMP(SO)oNP(SO)vDEGS(SOo50)9 00002 18ASE<20).NONE150.50)oH(12)oFI13.25.?)o6113.25o7).DA(13.25o7)oFOGIE 00003 20).GOF120).TIME120).KPARISO).MOPcso).KOM(30) 000 4 4 FORMATIBOX.15HSPECIAL LISTING) 5 5 FORMAT12A1.A6.9A8) 6 6 FORMATIIH0.A1.A6.9A8) 7 73 FORMAT181x.IHJ.6x.2HBO.8x.2HAo.8x.2HRO.8x.2HTO) 00008 74 FORMATIEOX.12.zx.F8.2.ax.F8.2.2x.F8.2o2X.F8.2) 00009 75 FORMATIBOX.12.4X.12.2x.F7.2) 10 77 F0RMAT(4X01204X01207X92HNA07X02HNA97X02HNA914X02HNA) 00011 780FORMAT12x.5HGROUP.2X.5H$TAKE.4X.1HC.8x.IHD.8x.IHE.7x.1HV.5X.4HZETA 00012 1.4x.4HEPSL) 13 79 FORMAT(//.2x.39H8=PERPENDICULAR DISTANCE FROM BASE LINE) 00014 80 FORMATI//.25X.14HTOTAL MOVEMENT) 15 810FORMAT116x.5HM/YR..5x.5HM/YR..4X.5HM/YR..3x.5HM/YR..1x.7HDEGREEso1 00016 1x.7HOEGREES) I7 82 FORMATI2X.47HC=MAGNITUDE OF TOTAL HORIZONTAL VELOCITY VECTOR) 00018 830FORMAT12x.54HD=MAGNITUDE 0F HORIZONTAL VELOCITY VECTOR IN DIRECTIO 00019 IN) 20 84 FORMATI15x.18HPERPENDICULAR To 8) 21 850FORMAT(2x.59HE=MAGNITUDE OF HORIZONTAL VELOCITY VECTOR IN DIRECTIO 00022 IN OF 8) 23 86 FORMATIEX.36HV:MAGNITUDE OF TOTAL VELOCITY VECTOR) 00024 870FORMAT12x.57HZETAaPOSITIVE ANGLE OF V. BETWEEN AND AwAY FROM BASE 00025 1LINE) 26 880FORMAT(2X.53HEPSL=ANGLE BETwEEN V AND A HORIZONTAL GLACIER SURFACE 00027 1) 28 89 FORMATIZXo46HM/INT2METERS PER EFFECTIVE MEASUREMENT PERIODS) 00029 900FORMAT(4XOIZO4X0IZOIXOF76102X0F76292X0F70202X0F76201X9F76202X9F662 00030 192X9F602) 31 910FORMAT12x.5HGROUP.2X.5HSTAKE.3X.IH8.9X.IHC.8X.IHD.8xo1HEo7xo1HVo4x 00032 1.4HZETA.4X.4HEPSL) 33 92 F0RMAT(4X01204X01291X0F7.102X9F7.292X9F76202X0F762010X0F662) 00034 930FORMAT<17X.2HM..6x.5HM/INT.4x.5HM/INTo4x.5HM/INT.3X.SHM/INTc1xo7HD 00035 lEGREESolXo7HDEGREES) 36 94 FORMATIF10.3) 37 95 FORMATIF4.0) 38 960FORMAT(4X61294X012o2XoF76202X9F762o2XoF76201X9F76202X6F66202X9F602 00039 1) 40 97 FORMATI4X.12.4X.IE.2X.F7.2.2x.F7.2.2x.F7.2.10x.F6.2) 00041 98 FORMAT(4X91204X¢12oéXoZHNAo7Xo2HNAo7XoZHNA67X92HNA614X02HNA) 00042 99 FORMAT(//) 43 100 FORMATIBI4) 44 COMMON ANGLEA.ANGLEB.MP.NP.DEGS.NONE.F.G.DA.FOG.GOF.TIME.KPAR.MOP. 00045 IKOM.8A5E.H 46 PRINT 99 47 KEYaO 48 49 10 798 799 7000 101 102 103 1017 3018 $019 3020 1021 419 420 5000 800 1000 801 802 899 HEAD 59H IF(DOL*H(1)) 89798 PRINT 69 (H(J)9J=2912) IF(DOL-H(2)) 991099 KEY=1 GO TO 9 BACKSPACE 2 IF(KEY) 79897999798 PRINT 4 PRINT 99 PRINT 73 PRINT 91 PRINT 93 60 T0 101 READ 949 EXIT READ 949 EXIT READ 1009(JCOUNT9LC0UNT9MC0UNT) IF(MCOUNT) 10291159102 CALL DEGRAD(LCOUNT9MCOUNT1 [E(JCOUNT) 10391019101 00 4020 J=8980894 IF£ .960 10 11 12 13 11 12 13 14 15 16 17 18 19 20 51 52 53 54 55 56 57 58 59 70 71 72 73 74 75 76 77 GO TO 10 961 YA=Y*.5*SO*CY 962 XA=X+SQ*CX 963 GO TO 10 964 YA=Y 965 XA=X+SQ*CX 966 GO TO 10 967 YA=Y 968 XA=X+95*SO*CX 969 GO TO 10 970 YA3Y‘95*SQ*CY 971 XA2X+95*SQ*CX 972 CALL pLOTIYA9XA9MX9SY9SX) 973 MX=I 974 CONTINUE 975 X=X+1925*SQ*CX 976 IF(KM(M+111 13913912 977 CONTINUE 978 XfiTX 979 Y3Y‘195*SQ*CY 980 END 981 SUBROUTINE NAME(KRT9KM9SQ9F9SY9SX9CY9CX1 00982 DIMENSION KMI6019KRTI6O) 983 D0 42 N=1960 984 NASN 985 TIC3N 986 IF(8HA *KRT(N)) 1191911 987 IF(8H8 ‘KRT(N11 1292912 988 IFIBHC -KRTCN11 1393913 989 IF(8HD ‘KRTIN11 1494914 990 IF(8HE '“KRTIN11 1595915 991 IF‘8HF -KRTIN11 1696916 992 IF18HG “KRTCN11 1797917 993 IF(8HH "KRT(N)) 1898918 994 1F(8HI *KRTIN)’ 1999919 995 IF(8HJ -KRT(N)1 20910920 996 IF‘BHK ~KRT(N11 51961951 997 IFIBHL ‘KRT(N1) 52962952 998 IF(8HM ~KRT1N11 53963953 999 1F(8HN *KRT(N)) 54964954 01000 IF(8HO *KRT(N)) 55965955 01001 IF(8HP ~KRT1N11 56966956 01002 IF(8HQ ‘KRT(N)1 57967957 01003 IFIBHR ‘KRTIN” 58968958 01004 IF(8HS ‘KRTIN11 59969959 01005 IF(8HT ‘KRT(N)) 70980970 01006 1F(8HU -KRT(N1) 71981971 01007 IF(8HV “KRTIN11 72982972 01008 IF(8HW -KRT(N)) 73983973 01009 IF(8HX ‘KRT(N)) 74984974 01010 IFCBHY -KRT(N)) 75985975 01011 1F18HZ “KRTINI’ 76986976 01012 IF(8H+ -KRT(N11 77987977 01013 IFC8H1 ‘KRTIN11 78988978 01014 78 79 90 92 10 61 62 63 64 65 66 67 68 69 80 81 82 83 84 85 86 IF¢8H2 ~KRT(N1) 79.89979 1F(8H ~KRT