Introduction
Jakobshavns Isbræ is one of the largest and fastest outlet glaciers draining the Greenland ice sheet. For 70 km inland it has the surface expression of an ice stream (Fig. 1), with heavily crevassed shear margins between the ice stream and the slower-moving ice sheet in which it is embedded.. No bedrock is exposed along its margins except near the 10–14 km long floating terminal area. The ice tongue is 600–1100 m thick. floats in a 1000–1500 m deep fjord and is bounded by gneissic walls extending up to 300 m a.s.l.. Speeds of up to 7 km a−1 are reached near the terminus of this glacier, leading to a calving flux of approximately 60 Mt d−1. At 75 km inland the ice is still moving 800 m a−1. and the overall surface slope is relatively steep, dropping 1600 m in 100 km (Reference Echelmeyer and HarrisonEchelmeyer and Harrison. 1990: Reference Echelmeyer, Wade and Iken.Echelmeyer and others, 1991a).
Location map of Jakobshavns Isbræ. Ice-stream boundaries are shown as heavy solid lines-with labels that are referred to in the text; small dots represent point seismic measurements, and large dots indicate satellite-derived surface velocity measurement locations described in Reference Echelmeyer and HarrisonEchelmeyer and Harrison (1990). Elevation contours in meters are derived from H. Brecher and T. Hughes (personal communication, 1988). Bedrock margins are stippled, and the calving front is shown as a hashed line. Surface elevation and ice-stream boundaries near the grounding zone are approximate.
An understanding of the mechanisms behind this fast flow is important because much of the Greenland ice sheet is drained by outlet glaciers of similar surface expression. While there has been considerable investigation of the long, comparatively flat ice streams draining the West Antarctic ice sheet into the Ross Ice Shelf (e.g. Reference BentleyBentley, 1987; Reference Shabtaie, Whilland and BentleyShabtaie and others. 1987; Reference Alley and WhillansAlley and Whillans. 1991), the unbuttressed ice streams draining into the Amundsen Sea, such as Pine Island and Thwaites Glaciers, are relatively unknown. These latter glaciers and many of the large outlet glaciers of East Antarctica have Surface characteristics which resemble those of Jakobshavns Isbæ, (Reference McIntyreMcIntyre. 1985). Similarly, model studies of the Laurentide ice sheet show that there may have existed outlet glaciers which flowed from the central ice sheet through the deep fjords to the east; these may have played an important role in the rapid demise of that ice sheet in the past (Reference Denton and HughesDenton and Hughes, 1981). Thus, an understanding of the fast-flow mechanism and dynamics of Jakobshavns Isbræ may he helpful in understanding the dynamics of both past and present-day ice streams and ice sheets.
To understand the flow mechanism of Jakobshavns Isbræ, knowledge of its ice thickness, basal topography and temperature distribution is required. Fieldwork and modeling by Reference Iken, Echelmeyer, Harrison and FunkIken and others (1993) and Reference Funk, Echelmeyer and IkenFunk and others (1994) give evidence for а 200–400 thick basal layer of temperate ice overlain by a cold core of ice which has been carried down from the upper drainage basin by the rapid flow. In this paper we describe seismic measurements of ice thickness in the Jakobshavns drainage basin, and provide a comparison with the airborne radar measurements made previously (Reference Gudmandsen and GudmandsenGudmandsen, 1970, Reference Gudmandsen and Peel1977) by the Technical University of Denmark (TUD). The seismic experiments were conducted in part to provide detailed ice-thickness information needed for the planning of hot-water drilling and associated ice-temperature measurements (Iken and others, 1989: Reference Humphrey and KHumphrey and Echclmeyer. 1990).
The method of pointwise migration of seismic-reflection data developed here, to our knowledge, has not been described previously in the glaciological literature. Simple formulae are developed which give depth, horizontal position and slope of the basal reflector from a linear geophone array and a single explosive source, The method has proven useful in other cases where radio-echo sounding has yielded unsatisfactory results due to a large thickness of temperate ice or complex geometry, or both (e.g. Ruth Glacier, Alaska (K. Echelmeyer, unpublished), and Taku Glacier, Alaska (Reference Nolan, Motkya, Echelmeyer. and Trabant.Nolan and others, 1995)).
We first describe the radio-echo-sounding data for the Jakobshavns region which existed prior to our seismic measurements. The seismic methods and analysis are discussed next, followed by a presentation of the seismic results with a synthesis of the radio-echo and seismic data. The measured ice thicknesses are then used to estimate the basal shear stress along the ice stream. Lastly, reflection coefficients are used to investigate the nature of the glacier bed.
Radio-Echo Sounding of Jakobshavns Drainage Basin
Prior to the joint Eidgenssische Technische Hochschule (ΕΤΗ) and University of Alaska hot-water drilling program on Jakobshavns Isbræ. a compilation was made of pre-existing data on ice thickness and basal topography in the region. No description of ice thickness of sufficient detail for planning a drilling program had been published. Maps of the entire Greenland ice sheet were presented by Reference Radok, Barry, Jenssen, Keen, Kiladis and McInnesRadok and others (1982). Some coarse details of the Jakobshavns drainage basin were tabulated by Reference BindschadlerBindschadler 1984; using the maps of Radok and others. The source of these ice-thickness data was the airborne radio-echo soundings made by TUD in the 1970s (Reference OvergaardOvergaard, 1981. Reference Overgaard1982, Reference Overgaard1984a Reference Overgaard.b).
The TUD system was a 10 kW ice radar operating at a frequency of 60 MHz. The radar was mounted on an LC-130 aircraft and flown over much of Greenland at altitude of 500–1000 m above the surface. Some of the flights passed over the ocean, sea ice or land, in addition to the ice sheet. Horizontal navigation was done by an inertial navigation system of modest accuracy, and aircraft altitude was determined by pressure altimetry. Z-scope traces were preserved on a photographic film recorder with timing lines Reference Gudmandsen and PeelGudmandsen. 1977).
Because the drilling program required a much more detailed picture of ice-stream thickness than was available from the existing maps, we initiated a search for the actual TUD ice-radar data (Reference OvergaardOvergaard, 1981. Reference Overgaard1982, Reference Overgaard1984a Reference Overgaard.b). Such data were obtained from the World Data Center A — Glaciology (WDCA) in Boulder, Colorado, U.S.A., in the form of tabulations for each flight path over Greenland. The tabulations consisted of latitude, longitude (both to 0.1′), surface elevation and bed elevation at about 1.6 km intervals along parts of each flight. The data were entered into a database, from which contour maps of Jakobshavns drainage basin were constructed. Transverse and longitudinal profiles of ice thickness and surface and bed topography were also constructed. Coverage over the basin was limited, as is shown by the flight paths depicted in Figure 2.
Map showing TUD ice-radar sounding flight paths and smoothed contours of ice thickness defined from these data, in meters. Dots denote locations where basal returns were recorded. Ice-stream boundaries, calving front and bedrock are shown.
Two features of the radar data are apparent from Figure 2. First, there exist large sections of most flights in this region where no basal returns were identified. Many of the data gaps are over the ice stream or neighboring shear margins; presumably, heavy Crevassing caused scattering loss of the radio signals transmitted from the relatively high flying aircraft. Secondly. there were a few crossing points where basal returns were recorded on both profiles. Examination of these crossing points gives an estimate of the errors in the TUD data, both in the positioning system and in the radar itself. We have analyzed the crossing points for which data points from the two flight paths are within 1 km or less of each other. In some cases the bed and surface elevations were interpolated along the intersecting paths to the exact crossing point. The mean difference in elevation of the basal reflector was about 150 m; similar differences existed between the two surface elevations. In addition, some elevations for the ocean surface and the neighboring ice sheet were recorded as 100–200 m below sea. Thus, we take 150–200 m to be a reasonable estimate of the standard error in the elevations and a conservative estimate of the error in ice thicknesses derived from the radar data. A smoothed contour map of ice thickness is shown in Figure 2.
Overlaying the ice-stream boundaries onto the ice thickness map shows no apparent subglacial expression of Jakobshavns ice stream (Fig. 2). This apparent lack of a strong subgiacial expression for this ice stream was believed to be similar to what was found beneath some of the West Antarctic ice streams, from both ice radar and seismic measurements (e.g. Reference Shabtaie, Whilland and BentleyShabtaie and others. 1987; Reference Atre and BentelyAtre and Bentley, 1993; Reference Retzlaff, Lord and BentleyRetzlaff and others, 1993). For planning purposes, an ice depth of 1250±200 m was estimated from Figure 2 for the drill sites near and on the ice stream at about 48.4 °W longitude. No depths in excess of 1500 m were expected.
Seismic Methods
Seismic-reflection data were collected during the summers of 1986, 1988, 1989 and 1990. In 1986 a few spot measurements were taken as time allowed between ice-motion surveys. The 1988 season consisted of a single 7 km traverse across the ice stream. The traverse was oriented perpendicular to flow at a location downstream of where the ice stream is clearly defined by intense shear margins. This location was profiled in an effort to determine where the ice stream bed could be reached with the hot-water drilling system. The 1989 season consisted of several partial transverse profiles on the lower ice stream, again in an effort to determine where the base of the ice could be reached by hot-water drilling. The 1990 efforts were aimed at determining the inland extent of the subgiacial trough which had been discovered in 1988.
One or two EG&G Geometrics 1210F 12-channel seismographs were used during the various field programs. If two instruments were used, the arrays were occasionally placed at right-angles, with the first geophone of each array at the apex. The two-dimensional array was necessary for depth determinations on the lower ice stream where bottom topography was expected to be somewhat more complicated than that of the inland ice stream, and because the array-shot point arrangements were severely restricted by surface crevassing in the high-strain-rate environment of the lower ice stream. Geophone spacing was 15 m, except in 1990 when 30 m spacing was used. For the spot measurements in 1986, 14 Hz geophones were used; the 1988 and 1990 traverses employed 28 Hz geophones. and both 14 and 28 Hz geophones were used in 1989. The geophones were arranged one per lake-out and oriented vertically. The number of samples per channel on the Geometries receivers is fixed at 1024. All 12 channels were recorded at either 1000 or 2000 samples per second. The resulting sampling window was rather short and often had to be delayed by up to 1200 ms in order to coincide with the arrival of bottom or wall reflections, In the ablation area, small (0.15–0.5 kg) explosive charges were drilled 2 m into ice; larger charges (up to 2 kg) were placed 3–4 m into the firn of the accumulation area. An analog high-pass filter with a corner frequency of 35–60 Hz was applied to the incoming signal. Data were recorded on paper printouts directly from the seismographs. Examples are shown in Figure 3.
Sample Seismograms from near the center of the STR profile. Several returns are shown, including P-waves reflected from the north and south walls (PPn, PPs. respectively) and the P-wave reflection from the center of the trough (PPb). Timing lines are 25 ms apart. Numbers near the bottom of each seismogram indicate horizontal distance along the surface.
Whenever possible the array was oriented parallel to the dip direction of the reflector. If a single array was employed, this was usually accomplished by first visually estimating bottom topography from surface topography. The array was then oriented perpendicular to the estimated slope of the bed. (We refer to this type of array as a “strike array”.) A single shot was then detonated off the immediate end of the array, and deviations from the expected time delay across the array (for a surface-parallel reflector) were used to determine the true strike and dip directions. If the array was not parallel to the strike of the reflector, it was reoriented and another shot was detonated to be sure the strike direction was correct. The final array was then rotated 90° so it would be oriented parallel to true dip in a “dip array”. This array orientation was then maintained as the array was moved across the ice stream. The orientation of the dip array was occasionally checked with a strike array and associated shot. This ensured that the depth profile maintained a transverse orientation.
Once the dip array was properly oriented, a shot was detonated either 15 or 30 m from each end of the geophone spread, depending upon geophone spacing along the line of the array. After these initial shots, two different shooting techniques were employed. In 1988 the array and shots were moved across the ice stream one array length at a time, stepwise, after shooting off each end of the array. In 1989 and 1990 the array was left in place while the shot was moved across the ice stream in either direction up to 3 km from the array. After several bed echoes were obtained for a particular array, the array was then relocated 0.5–2 km farther across the ice stream, and several shots were again detonated up- and down-dip from the array. Where bed topography was relatively simple, this shooting technique was found to be expedient in defining the channel shape, because it involved minimum set-up time. Offsets were never large enough to acquire subgiacial refractions.
Triggering of the seismograph was accomplished by two methods. The first utilized a thin trigger wire wrapped around the explosive charge. Upon detonation the open circuit of the broken wire triggered the seismograph. While effective and reliable, management of long trigger wires was, at times, difficult. The second method utilized an FM radio trigger system, which, while less cumbersome than the trigger wire method, was much more prone to false triggers and failure.
The sample rates usually allowed first arrivals to he determined within 1 ms (about 2 m in depth). In some eases ambiguous returns resulted from high bed roughness, glacier seismic noise, noise resulting from surface wave energy trapped by crevasses, weak echoes, or interfering echoes from different reflectors. The reduced clarity of these echoes increased the error in picking the first arrival to 2 or 3 ms. The worst-case error occurred for two shots on the lowest profile. S8 in Figure 1, where several of these factors combined to create a picking error of about 5 ms (about 10 m in depth).
Data Analysis
Methods have been developed to ascertain complex basal topography using different types of two-dimensional geophone arrays on a glacier; these have been described by Reference DoellDoell (1963) and Reference RöthlisbergerRöthlisberger (1972). However, if the geophone spreads Can be positioned in a linear array parallel to the dip of the reflecting bed, as we have done, then a simplified formulation for pointwise migration of data from a single shot and a single array can be given. In this section we develop this formulation. Inputs are travel time and shot-to-geophone offset for any two geophones in a dip array. The method yields analytical expressions for the horizontal position, depth and slope of the reflector, assuming a homogeneous isotropic medium over a sloping half-space. The formulation is easily programmed into a pocket electronic calculator which is useful for in-field migration of seismic data and subsequent shot array location planning.
If the vertical plane defined by the array is perpendicular to the reflecting surface, as for a dip array, then migration becomes a two-dimensional problem. An image point is located equal-distant below the reflector along a bed-perpendicular line (Fig. 4). This distance, h, from the shot point to die image point is given as the solution of
where Vp is the P-wave velocity, t is two-way travel time for a given geophone. s is horizontal distance along the surface from the shot to the geophone, and β is the bed slope. A similar equation can be written for a second geophone with the same image point, and the two equations can be solved simultaneously for h:
Subscripts “I” and “2” designate the two different geophones; they are usually taken to be at the ends of the array. Given h from Equation 2, β can be obtained from Equation 1. The angle of incidence for the ray arriving at geophone 1 is given by
and the vertical ice thickness above the point of reflection, z1 , for the ray arriving at geophone 1 can be determined from
Similarly, the horizontal distance from the shot to the point οf reflection, x1 , for the ray arriving at geophone 1 is given by
A sketch of ray-path geometry and image point for a homogeneous isotropic medium overlying a tilled half-space, showing the parameters used in the two-dimensional determination of the reflector location and slope.
With the shot point at the origin, x is positive toward the geophones, and z is positive downward. Equation (4) and (5) assume the array is horizontal. A positive bed slope implies shallower ice under the shot than beneath the geophones; negative β is the reverse. For steep bed slopes the unknown parameters are sensitive functions of the travel-time difference between geophones (t1− t2), and therefore care must be exercised when measuring travel times. The quantities z and x may of course be determined for any geophone in the array if a different s and corresponding time are used. Specifically, if the travel times and distances to the geophones at opposite ends of the array are used separately in Equation (3), (4) and (5), then the end-points of the line segment along the bed which is illuminated by the reflection can be located and plotted.
In our analysis. P-wave velocity was assumed to be that of ice at −15 °С: vp = 3830 ms−1 [Reference RöthlisbergerRöthlisberger, 1972; Reference KohnenKohnen, 1974). This is based upon the results of Iken and others (1993), who found the ice temperature near site STR (Fig. 1) to vary between about 0 °C at the surface, −22 °C at 1200 m depth, and melting point again near the base of the ice stream.
Except at the farthest inland seismic locations, shots were drilled directly into ice, and therefore migration of these data required no firn correction. The geophone spreads were almost always within 1 ° of horizontal, and never exceeded 4 ° from horizontal. As discussed previously, the arrays were always oriented as close as possible to the dip direction of the reflector.
The relative positions of the shot points and geophones were determined by optical surveys; they are accurate to about ±2 m. Absolute positions for most arrays were determined by Magnavox Geoceiver satellite Doppler positioning-system methods; they are known to about ±3 m. The absolute locations of the two short traverses on the lower ice stream (S8 and L15 in Figure 1) were determined by air-photo interpretation, and their bearings were determined by compass. These latter arrays have an estimated absolute horizontal accuracy of ±200m.
Seismic Depth Profiles
In this section we present the results of the measurements made along each of the traverses shown in Figure 1. The seismograms shown in Figure 3 illustrate the quality and varying amplitude of the basal reflections. Where possible we compare the seismic results with the TUD airborne radar data.
Description of Profiles
In 1988, seismic measurements were made along the profile designated STR (Fig. 1). The stepwise shooting procedure resulted in detailed coverage of most of the bed beneath the transect, with rays of near-normal incidence at all points along the bed. Shot points, and thus the ends of the array, are shown as X’s along the surface in Figure 5. where surface and bed topography are displayed with no vertical exaggeration. No basal returns were recorded from the dashed region of the bed profile due to what appears to be a very steep (>50 °) transverse bed slope. The profile shows a deep subgiacial trough directly beneath the ice stream, with a center-line thickness of 2560 m and a thickness of about 930 m along the margins. The base of the ice stream is approximately 1500 m below sea level at this location. The data shown in Figure 3 were collected from near the center of this profile. Also provided in Figure 5 is a profile of surface velocity, showing the correspondence of ice velocity and ice-stream geometry.
The central borehole shown in Figure 5. drilled in 1988, did not reach the bed owing to the large ice thickness there. The boreholes on the north and south margins of the ice stream were drilled in 1989. after the drilling equipment on the surface had advected down-stream about 1 km from the original 1988 profile location. Short seismic profiles were made at each marginal zone beneath these 1989 borehole locations to determine the expected depth for drilling. These results are indicated by the heavy lines and the term “Bed down-glacier” in Figure 5. The boreholes reached the bed at both the north and south margins, and the seismic depths agreed with their actual depths to within ±30m (2%), thus providing an estimate of the combined errors due to timing, estimated wave speed and migration methods of the survey.
Lower panel shows the seismically derived cross-section at the STR profile. Shots are shown by X’s. and flags represent motion survey poles. The dashed line indicates the area where no basal returns were acquired. Three borehole locations are shown. The two marginal sites were drilled in 1989, after the drilling equipment had advected downstream about 1 km from the STR profile. “Bed down-glacier” refers to reflection data from these two areas. No vertical exaggeration. The upper panel shows annual surface velocity vectors along this transect. Numbers provide distance north and south from the ice-stream center line.
A comparison of this seismic profile with, a TUD ice-radar profile is shown in Figure 6. Outside the ice stream the ice thicknesses agree within the error where they can be compared. However, only one radar return is indicated within the boundaries of the ice stream, and this depth is less than half that determined seismically. A Doppler positioning-system control point was surveyed along the radar profile as part of the motion surveys; the TUD surface elevation was elevation was found to be about 150 m lower than the control-point elevation and surface elevations at the STR profile. The radar-derived surface and bed elevations were therefore adjusted by this amount.
Comparison оf seismic data along STR profile with nearby TUD ice-radar date. The inset shows the location of the radar flight path and the seismic transect. The + marks location of a surface control point surveyed in 1988. Vertical exaggeration is about 6 to 1.
A 21 km long section was profiled in 1990 at L25, located 12 km upstream from STR (Fig. 1). This profile was acquired both to determine whether the trough was unique to the ice stream, and to determine the influx of ice into the STR profile where the boreholes were drilled. The shear margins of the ice stream were not as intensely crevassed here as they were farther downstream, and yet a trough was found beneath the ice stream which was similar in depth (2580 m) and width to that found at STR (Fig. 7). Ice thickness outside the trough was found to be fairly constant at about 1000 m.
Seismic profile at L25, with distance from ice-stream center line. Shot points are shown as X’s: geophone arrays are short heavy lines along surface; dots with line segments through them indicate a reflection point and associated reflector slope; the length of the associated line segment indicates the section of the bed illuminated by the reflection. No vertical exaggeration. Ice-Stream boundaries are located at about 2.6 km south and 3.5 km north of the center line.
An additional seismic profile was made at L27, about 24 km upstream of STR (Fig. 1). The reflections were of poorer quality, and no reflections were obtained from the channel center line at this location even after several attempts. Shooting near the center of the ice stream at this location resulted in an audible caving sound. These shots were detonated in what was probably a several-meter-thick layer of old lake ice that bad been advected downstream from a nearby surface lake, which may partially explain the unusual sounds and lack of returned energy. The estimated basal topography still shows a deep sub-ice-stream trough (Figs 8 and 9a). The estimated center-line ice thickness is about 2800 m, which is greater than that measured anywhere downstream.
Comparison of seismic data along the L25 and L27 transects and the TUD ice-radar data along nearby flights. Inset shows location of seismic profiles (heavy lines) and radar flight paths with dots where radar returns were recorded; the two seismic profiles are about 12 km apart. Vertical exaggeration is about 6 to 1.
Four seismic cross-sections of Jakobshavns ice stream from L27 (a) downstream S8 (d). Shot points are shown as X’s; dots with line segments through them indicate a refection point and associated reflector slope; the length of the associated line segment indicates the section of the bed illuminated by the reflection; dashed lines indicate areas where no reflections were acquired. Ice flow is into the page; north is to the right. All profiles are to the same scale with no vertical exaggeration.
Two ТUD radar profiles crossed the ice stream between L25 and L27, as indicated in the inset in Figure 8. Considering that the radar flight lines were not as close to the seismic line as those described in Figure 6. we find reasonable agreement between seismic and radar ice thicknesses (within 400 m) outside of the ice stream. However, there is again only one radar reflection indicated over the ice stream itself, and this return shows no subgiacial channel.
In 1989, measurements were made at S8. approximately 30 km downstream of STR (Fig. 1) again in preparation for possible drilling there. This location is about 6 km above the grounding zone and has a surface ice speed of about 3800 m a−1. Heavy crevassing in this region placed severe restrictions on the location and mobility of the array there. A right-angle 24-geophone (2 seismograph) array was located near the center of the ice stream, and several shots were detonated up to a few kilometers away with the use of a helicopter. The resulting surface and bed profiles are shown in Figure 9d. Basal returns are shown as dots, with short lines through them representing the corresponding bed slope. A prominent “transverse” icefall was observed to be flowing sub-perpendicularly into the ice stream about 1.5 km northeast of the array. This icefall overlies the step in the bed which defines the northern margin of the subgiacial trough. This basal step was well defined by reflections from the top and bottom of the step, although no returns from the face of the step itself were observed. Center-line thickness was found to be about 1760 m, with a surface elevation of about 500 m. No basal returns were obtained from the southern half of the channel, even after repeated attempts. Thus, the shape of the bed to the south was estimated from the transverse surface morphology of the ice stream at the S8 profile; this estimate is shown by the dashed line there.
Ice Thickness Distribution
In Figure 9 we plot, to the same stale, the surface and basal topography along each of the profiles described above. From the inland ice at L27 (60 km upstream of the grounding zone) toward S8 (6 km upstream of the grounding zone), the channel narrows markedly, and ice thickness in the center of the stream decreases from about 2800 m to 1760 m. The elevation of the base of the trough increases slightly from about 1500 m to 1260 m below sea level, with an entrenchment into the surrounding bedrock decreasing from 1600 m to 900 m.
The spot Seismic measurements made at various locations along the ice stream (Fig. 1) were combined with the profile data to obtain an ice-thickness map of the ice stream as a whole. We have assumed that the ice-radar data are reliable outside the ice stream (which our limited comparisons justify at the ±200m level) to produce a contour map of the ice thickness over much of the lower Jakobshavns drainage (Fig. 10).
The ice stream follows the subglacial trough along its length. Two branches of the trough coalesce at about 48 °W, and this is mimicked in the surface morphology, although well-defined shear margins arc not present at this location. The onset of obvious ice-stream motion begins at roughly the same position as the point of coalescence. Outside the ice stream, ice thickness is on the order of 900–1400 m, with a gradual thickening toward the interior of the ice sheet. The ice-sheet elevations shown in Figure 1 indicate that the bed of the ice stream is more than 1000 m below sea level along its entire length. No reliable ice-thickness data have been obtained along the northern branch of the ice stream which enters just above the fjord (Fig. 1), but surface morphology, flotation level and speed of this branch indicate that it is not likely to be as thick, nor is a subgiacial trough likely to extend very far inland to the north.
Map оf ice thickness compiled from seismic and TUD ice-radar data. Contours are dashed where poorly constrained.
Basal Shear Stress Along the Ice Stream
The large ice thicknesses along the ice stream, coupled with the relatively steep surface (Reference Echelmeyer, Wade and Iken.Echehmeyer and others 1991a), lead to large basal shear stresses and. consequently, a significant component of ice deformation within the ice stream. The deeply incised bedrock trough and overlying ice complicate the calculation of the basal shear stress,
The simple formula
where, now, H is the center-line thickness and
Sketch showing the geometer assumed in calculating the shape factors, (a) F1 (b) F2. Parameters are explained in the text.
where S is the total cross-section area of the channel plus the ice above (ABCDEA. Fig. 11b), and P is the “wetted” perimeter of the channel which provides drag on the ice stream (BCD, Fig 11b). In this case we assume that the ice alongside the ice stream does not produce additional drag on the ice stream. Assuming the trough is parabolic in shape, with depth hT and width WT , and denoting the thickness of the overlying ice by hj (Fig. 11b), we have
where
F1 will be a minimum value of the effective shape factor because ice alongside the ice stream is assumed (unrealistically) to he immobile along the parabolic shape, and F2 is a maximum in that it places all of the drag onto the “wetted” bedrock trough. The values so calculated are given in Table 1, along with basal shear stress calculated from Equation 6 using F1 and F2 . Surface slope has been estimated from Reference Echelmeyer, Wade and Iken.Echelmeyer and others (1991a).
where A is the flow-law parameter, and we have taken the flow-law exponent to be 3.
The value of the flow-law parameter in Equation (11) requires a choice of the mean ice temperature in the cross-section. This value of A is best determined by a weighted mean over the center-line ice thickness, with the weighting factor determined by the rate of deformation at each depth. This is because Equation 11 relies on the assumptions that
Using the ice temperature given by Reference Iken, Echelmeyer, Harrison and FunkIken and Others (1993) and the values of A as a function of temperature given by Reference PatersonPaterson (1981), we find that (
Parameters for calculation of effective shape factors and basal shear stress, as described in text and shown in Figure 11. Range of
These values agree well with those evaluated using Equation 6–(10), except at S8. If
Basal Reflection Coefficient
Several records showing multiple returns were encountered along the STR traverse, some of which are shown in Figure 3. Even a cursory glance at these seismograms reveals markedly stronger wall reflections than those from the center of the trough. When more than one return occurs on the same seismogram, the echo strengths are easily compared, since difficulties arising from variations in geophone coupling, shot coupling and shot size are eliminated; and because the Geometrics 1210 operates with a fixed gain setting, amplitude can be determined directly from the seismograms. In 1988, shots were never detonated farther than 15 m from the end of the array, which resulted in near perpendicular incidence for all reflections, so complications arising from refractions and S-wave conversions are also eliminated. Before the echoes can be compared quantitatively, however, the amplitudes of each return must be modified to account for spherical divergence, bed reflectivity and geophone directivity.
Corrections
If it is assumed that P-wave attenuation in cold ice is small (Reference RöthlisbergerRöthlisberger. 1972; Reference Bentley and KohnenBentley and Kohnen, 1976), and that density and wave speed are constant, then spherical divergence can be accounted for using the relation
where IU and IL are the acoustic impedances of the upper and lower media, respectively.(The acoustic impedance of a medium is the product of the compressional wave velocity and the density). Note that
The quantity Av is taken to he the maximum amplitude of the return pulse, averaged over 12 channels whenever possible. Mean two-way travel time across the spread is used for Δt.
In Equation 14, both Ao and R are unknown, but for more than one echo on the same record, Ao is common to both and may be eliminated using two different return amplitudes; specifically the amplitude of the wall reflection (subscript “w”) and that of the reflection from the center of the trough (subscript “c”):
The ratio of reflection coefficients,
From Figure 12 it is clear that the magnitude of the reflection coefficient of the walls is generally about 2–3 times that of the center. The two largest deviations, labeled S1 and N3 in Figure 12, were both caused by a change in echo strength from the wall rather than from the center of the trough. Although the possible explanations for variations in reflection strength are numerous, it may be that a convex and concave bed, respectively, contributed to these deviations. Note that bed slope of the south wall increases from 38 ° to 53 ° between the S2 and S1 reflection points (Fig. 12), and is then lost. Also, the reflection causing the large N3 ratio originates in the middle of four other reflection locations on the north wall (Fig. 12), all of which show about half the reflection strength of N3. However, because the sample population is small (n−1), the S1 and N3 ratios cannot be treated as statistical outliers. If they are kept, the mean of the ratios is 2.4; it is 2.3 if they arc not. The respective standard deviations are 1.1 and 0.2. We therefore take the possible range of
Upper panel shows several ratios of wall-reflection amplitude to the amplitude of the reflection from the center of the trough. Data are from the STR profile. Lower panel shows the locations of the reflection; C denotes the center of the trough.
To determine the sign of Rw/Rc , the phases of the reflections must be examined. If one reflection is reversed in phase with respect to the other. Rw/Rc is negative; otherwise it is positive. Unfortunately the signal/noise ratio was such that the first motion for the reflections from the center of the trough could not be accurately determined (Fig. 3). We therefore must carry the possibility that Rw/Rc may be either positive or negative.
Calculation of Acoustic Impedance
If the acoustic properties of the wall can be estimated, Rw/Rc can be used to investigate the nature of the ice-stream bed. Between 50 km north and 100 km south of the Jakobshavns fjord the exposed bedrock is comprised largely of granodiorite gneiss, and the large-scale structural fabric of southern Greenland trends east west (Geological Survey of Greenland, 1971). Also, a sample recovered from the bed during the 1989 drilling operation at a location about 1 km downstream of the S2 reflection point was composed of quartz, hornblende, plagioclase and biotite, which is consistent with a granodiorite gneiss source. We therefore make the assumption that the bedrock in the vicinity of STR is comprised of granodiorite gneiss or a similar material, and that the wall echoes are from a relatively clean interface.
Although somewhat variable. 5.0 km s−1 and 2.7 Mg m−3, are reasonable estimates of velocity and density, respectively, for granodiorite gneiss (Reference CharmichaelCharmichael, 1982). It is important to note that, because
Diagram of acoustic impedance as contours in the P-wave-velocity, density domain. Shaded regions indicate possible velocity-density pairs for Ic. Light-and dark-gray areas indicate the range of possibilities if
The loci of velocity-density pairs which lie in the possible ranges of Ic are shown in Figure 13 as shaded areas. If Rw/Rc is positive, then the most reasonable range for the P-wave velocity of the bed is about 2–4 km s−1, with a corresponding bulk density range of 2.5–2.0 Mg m−3. within this range lie common velocity-density pairs for lodged tills and other compacted sediments (Reference CharmichaelCharmichael, 1982). The range is further restricted if the S1 and N3 ratios are excluded, but the result is still a compacted sediment. If Rw/Rc is negative, then the material at the bed in the central part of the channel has a velocity and density only slightly greater than that of water, implying that the bed is fluidized.
The bed beneath the central part of the ice stream is more likely to be composed of lodged till or some similar material than to be fluidized, because the results of Reference Echelmeyer, Wade and Iken.Echelmeyer and others (1991b), and the close agreement between
For comparison, the estimated density and seismic velocity for the material found beneath a considerable area of Ice Stream B. West Antarctica, are also shown in Figure 13. (Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others, 1987); Rooney and others, 1987; Reference Kamb and Engelhardt.Kamb and Engelhardt, 1991). These values lie between the two estimates for material beneath Jakobshavns ice stream.
Conclusions
Seismic-reflection measurements show that a deep subglacial trough underlies Jakobshavns Isbræ for most of its length (Fig. 10). Using the migration method developed here, we find center-line ice thicknesses to be about 2500 m, while the surrounding ice sheet is about 1000 m deep. The ice thicknesses within the stream vary markedly from those shown in Figure 2. Outside the stream, the thicknesses are in agreement with those derived by Reference Fastook, Brecher and HughesFastook and others (1995) but are not in agreement with those derived within the stream. The onset of obvious ice-stream motion corresponds to the beginning of the main subgiacial channel. The bed of the ice stream is more ıhan 1000 m below sea level along most of its length.
Although we have not examined the original records, this subglacial feature was not observed in die radar data available to us. These data show only a few returns over the ice stream, and those present were less than half the seismically derived ice depth. This is due in part to the energy loss at the highly crevassed surface, in part to the absorption loss in the thick ice of the stream, and in part to the broad spread of the wave from after passage through 2000 m of air and ice, which can result in the first reflection coming from the upper edge of the trough. Clearly airborne radio-echo sounding, when flown at an altitude of 500–1000 m above a heavily crevassed surface, must be interpreted with care. Similar conclusions have been reached in Antarctica by C. Swithinbank (personal communication, 1993).
Reflection coefficients indicate that a layer of lodged till or similar sediment is present near the bottom of the trough. Unlike the relatively thin and flat ice streams that drain through the Siple Coast in West Antartica, a sediment layer need not play a critical role in the dynamics of Jakobshavns Isbræ (Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others, 1987). This is because the large center-line ice thickness and relatively steep surface slope along the ice stream lead to basal shear stresses on the order of 2.0-2.6 × 105 kPa. Consequently, we expect rapid internal deformation of the ice. Indeed, the close agreement (Table 1) between the basal shear stress calculated using channel geometry and surface slope,
Acknowledgements
We wish to thank all those who helped in the fieldwork, including D. Cosgrove. A. Iken, P. Moore. D. Roberts, and especially K. Petersen, and K. Swanson of PICO. We also would like to thank W. Harrison and S. Shabtaie for many helpful comments on the manuscript, and C. Bentley and T. Hughes for their reviews of the manuscript. A. Barrow helped prepare the manuscript.
This work was funded under the U.S. National Science Foundation grant DPP-8722003.
Appendix Discussion of Approximation in Equation 14
In estimating acoustic impedance we have assumed that the vertical component of a wave incident on a geophone at the surface can he approximated by a simple cosine function. The validity of using a cosine can he shown as follows. For a non-vertically incident P-wave, the surface displacement is the sum of the displacements caused by the incident P-wave and by P- and S-waves reflected at the free surface. For a plane-wave approximation, the time-harmonic displacement (frequcney ω) can be written
where uz is the vertical displacement, As are the amplitudes of the incident and reflected P-Waves, Br is the amplitude of the reflected S-wave. i is the angle of incidence and reflection of the P-wave, r is the reflected angle of the S-wave, and
where k is the ratio of vp to the S-wave speed (Reference Aki and RichardsAki and Richards. 1980. sec. 5.2). Combining Equations (A1) and (A2) gives
The righthand side of Equation (A3) is the factor that our simple cosine correction in Equation 14 ignores. This factor is plotted in Figure 14, where we have assumed that k = 2.0 for ice and we have made use of the relation
The correction factor for non-vertical incidence as calculated from the righthand side of Equation (A3).
