Instrumentation

The radar system comprised a GSSI Model SIR8 transmitter-receiver coupled to a single 80 MHz broadband transmit receive antenna with a 30 m cable. Navigation was provided by a combination of a bicycle-wheel odometer, a motion detector (used for sensing passage past grid poles accurately positioned by investigators at The Ohio State University (Reference Hulbe and WhillansHulbe and Whillans, 1994)) and an event-marking button. The data were logged directly from the CSSI receiver and navigation systems using a digital data-logging system designed and built at the University of Wisconsin. The sampling rate was 969 ± 12 MHz (1.032 ± 0.013ns sample⁻¹). The data logger was controlled by an IBM lap-top computer and the entire system was powered from a 12V automobile battery. The relatively light weight and portability of the system allowed it to be mounted on a single Nansen sled; the antenna was towed on another Nansen sled 30 m behind. The sled train was towed by a Skidoo Alpine snowmobile. Where surface crevasses posed a danger, a second Skidoo was added 20 m behind the antenna sled for safety purposes. The snowmobile sled-sled-snowmobile arrangement was roped together; personnel were tied together on a separate rope system.

Radar-wave speed in firn

Investigators from the California Institute of Technology (Caltech) drilled several holes into and through the ice at Upstream B Camp (Fig. 1) during the 1992–92 field season (e.g. Reference Engelhardt, Hymphrey, Kamb and FahnestockEngelhardt and others, 1990). We used the radar system to record echoes from the descending (or ascending) metallic drill tip; from the echo times, the electromagnetic wave speed in the firn was calculated.

Two types of tips were used by Caltech, one 4.2 m long and one 1.6m long. Each hung on the end of a radar transparent nylon-reinforced hose. Our antenna was placed about 7 m from the top of the borehole. The data-acquisition rate varied between one and four traces per second, while drilling progressed at about 100 m h⁻¹. The stem was tracked 12 times going up or down three different holes all within 100 m of each other. The depth of the stem was determined by calibrated meter wheels. The radar record was marked every 1 or 2 m in depth by a brief electronic pulse.

Velocity (v) versus depth (z) was calculated every 1 or 2 m from pulse travel-lime differences over 3–5 m depth intervals. A correction was made for the phase reversal from the conducting drill tip. Ray paths were assumed to be straight, an assumption that causes an error in wave speed of less than 1% at all depths.

Water introduced into the firn by Caltech’s hot-water drilling technique apparently had little effect on the measured wave speeds. As drilling progressed, large increases in echo strength were observed at specific stratigraphic levels. These changes were interpreted to mean that water was pooling at these particular levels. As long as the water content and thickness of the layer(s) did not change appreciably over the 1–3 min it took to penetrate (or withdraw through) 3–5 m of firn, and as long as that 3–5 m of firn did not include one of these wet layers, the effect of the water was removed by the travel time differencing.

The speeds are shown in raw form (Fig. 2a) and after a 5 m running mean has been applied (Fig. 2b). Also shown are least-squares regression lines of the form v(z) = a + bz fitted to the unsmoothed 0–50 m section. For the raw data (Fig. 2a), a= 213 ± l0 m µs⁻¹, b = -0.87 ± 0.06 µs⁻¹ and the square of the correlation coefficient is 0.63. For the smoothed data (Fig. 2b:, the correlation coefficient is 0.96; a and b are almost unchanged. Below 50 m, the wave speed is constant at 170 ± 4 µs⁻¹.

Fig. 2.

Plots of wave speed versus depth at UpB. In each, the vertical line is at 170 m µs⁻¹, the speed below 50 m. The sloping line is a least-squares regression-line fit to the data, and the curved line shows the wave speed expected from the known variation of density with depth. (a.) Individual measurements. (b.) .5 m running means.

The apparent linear variation with depth is unexpected. The density-depth (ρ(z)) data presented by Reference Alley and BentleyAlley and Bentley (1988) show the expected decrease in dρ/ dz with depth downward to 50 m, where a solid-ice density of 0.90 Mg m⁻³ is reached; since v varies linearly with ρ (Bogorodsky and others, 1985), v(z) should show a similar form. We have plotted the expected v(z), calculated from ρ(z) using the linear relation of Reference Robin, Evans and BaileyRobin and others (1969), in Figure 2. Between 0 and 20 m, the measured and calculated wave speeds agree reasonably well, but between 25 and 50 m our measured velocities are consistently higher. (A further indication that water had little effect on the measurements.) Closely space gridded traverse lines in the vicinity of the drillholes showed buried crevasses very near two of the three drill locations. In both cases, the crevasse tops were located at about 25 m depth. We believe that propagation through crevasses is the likely cause of the unusually large wave speeds between 2.5 and 50 m. We note, however, that the data presented here cannot really differentiate between the two curves: the linear v(z) and the expected v(z) fall within 100 and 90%, respectively, of the standard deviations of the 5 m running means (not shown), For simplicity, we have used the linear relation for determining diffractor depths. The errors associated with the difference between the two curves are everywhere less than 1 m in depth and 10 years in age.

Shapes of theoretical diffraction hyperbolae

Many macroscopic strain features found on glaciers and ice sheets are linear in form. It is well known that, if the features are sharp enough, they appear as diffraction hyperbolae on radar records. Discontinuities associated with crevasses and faults obviously can cause diffraction hyperbolae: so can folding, if it is sharp compared to the curvature of the incident wave front.

We derive a relation between hyperbola shape and crossing angle, assuming the diffractor is a straight line at a constant depth. From the geometry (Fig. 3), s ² = x ² + d ², where s, x and d are slant, horizontal and vertical distances, respectively, from the antenna to the diffractor (Reference Jezek, Bentley and CloughJezek and others, 1979). Also, and where t″ is the one-way travel time along s, v̄ is the average wave speed to depth d, ℓ is the distance along the traverse line from the antenna to the crossing point, θ is the crossing angle and t′ is the one-way travel time along d. For v = a + bz, we find by integration v̄ as a function of Then

Fig. 3.

Sketches showing the assumed geometry, the parameters used in the determination of crevasse orientation and how they relate to an observed hyperbola.

From Equation (1), sin² θ can be calculated from the shape of the hyperbola; the sign of θ is not determined. Also, the ray path, s, is assumed to be straight. This introduces an error of less than 1% for incident angles of less than 45°, which constitute the majority of incident angles for any given hyperbola.

Several hyperbolae calculated from Equation (1) for different values of θ are shown in Figure 4. Calculated and measured hyperbolae from several crevasses that had known orientations (e.g. Fig. 5) agree within 5° or 10°, which is good considering that crevasses are not perfectly linear features.

Fig. 4.

Theoretical hyperbolae for linear diffractors at depths of 10 m (upper set) and 48 m (lower set) crossed at angles of 15° 30°, 60° and 90°.

Fig .5.

A calculated hyperbola (dashed line) superimposed on an observed hyperbola. The crossing angle is 45°.

Description of buried features

A number of different types of features produce hyperbolae and it is important to understand these differences if the hyperbolae are to be interpreted accurately. One type of near-surface crevasse produced what can be best described as single-sided hyperbolae (Fig. 6). While the surface expression of these crevasses was essentially linear, their edges were very irregular, and attempts to lower weights into them for measurement of depth were invariably thwarted by multiple snow bridges. Each side of these crevasses is characterized on the radargram by a separate set of single-armed hyperbolae with apices stacked from the surface to a depth of many meters. We interpret this as diffractions from a rough-walled crevasse, perhaps caused by simultaneous growth and burial.

Fig. 6.

Radargram we interpret as showing active crevasses with multiple snow bridges. Stacks of one-sided hyperbolae are particularly well shown in the features around 30 m from each end of the profile.

Another type of signature was observed from freshly opened, straight-edged, smooth-walled crevasses with burial depths of less than 1 m (Fig. 7). These show a single strong diffraction hyperbola at depth but almost no surface or wall signature. We associate the strong diffractions with the bottoms of the crevasses because in several cases the diffractor depths agreed with the measured depth of the corresponding crevasses. We attribute the lack of a signature from the top edges of these crevasses to their proximity to the surface and the antenna-beam angle, and the weak returns from the walls to a combination of the beam angle, the absence of snow bridges and the smoothness of the crevasse Walls.

Fig. 7.

Radargram from a very smooth-walled freshly opened crevasse. Hyperbolae near the surface can be seen only very faintly. This crevasse is about 0.7 m wide at the surface, about 37 m deep and covered by a 0.3 m thick Snow bridge.

A third type of return was characterized by relatively weak hyperbolae originating from discontinuities associated with apparent faults (Fig. 8). The layering of the firn associated with these hyperbolae shows onsets of about 1 m, and these hyperbolae are only observed in areas of broad, gentle folding where there are few or no crevasses.

The fourth and most common type of hyperbola, however, we interpret as originating from buried snow bridges, because they are associated with distinct sags or synclines in the strata often bounded on each side by horizontal layers for 100m or more (Fig. 9). These hyperbolae occur in stacks that begin at the depth where the curvature of the syncline first exceeds the curvature of the wave front. Note that these hyperbolae provide a minimum depth to the original crevasse, since the first hyperbola in a stack is not necessarily correlated with the top of the crevasse.

Fig. 8.

Radargram showing weak hyperbolae originating from possible faults. The figure on the right is an interpreted section were hyperbolae associated with faults are dashed Lines. Offsets are about 1 m. The hyperbolae centered around 60 m across and 0.35µs depth probably have all off-nadir origin. The vertical disturbance that originates at 60 m along the surface is caused by a metallic grid pole.

Fig. 9.

Radargram showing hyperbolae associated with sagging bridges over buried crevasses. Note how the returns from sagging beds change downward into intersecting hyperbolae.

Orientation and distribution of buried features

Regardless of the nature of the sources for the hyperbolae, crossing angles can be determined on the assumption of linearity. The shapes of the hyperbolae are more sensitive to change in the crossing angle at shallow angles than at near-normal angles (Fig. 4). Shallow angles thus provide the most accurate orientations, and are encountered primarily on transverse transects (those oriented perpendicular to the flow). Longitudinal transects (oriented parallel to flow) generally provided larger numbers of crossings and thus provide a better estimate of crevasse density. Transverse and longitudinal transects were interpreted separately; the results agree to 10° or better. The ambiguity in the sign of the crossing angles was removed by using correlations between closely spaced or crossing grid lines where they existed, and then by assuming that the orientations vary smoothly between those locations.

It is clear from a histogram of the distribution of hyperbolae with depth (Fig. 10) that there are two distinct crevasse populations. The shallow, young crevasses are currently active and are found in the grid northeastern part of the survey area (Fig. 11a). The orientations of the crevasses along the upstream northern edge of the grid are in good agreement with those mapped previously by Reference Whillans and van der VeenWhillans and others (1993). The crevasses downstream of that area tended to be buried slightly (hence the tail on the upper population), and therefore were not mapped by the photogrammetric technique employed by Whillans and others. The other crevasses, centered at a depth of about 30 m, have orientations that appear to vary rapidly or abruptly in the central part of the survey area (Fig. 11b). In reality, the two sets of crevasses probably intersect in this area. Figure 11b shows crevasses on either side of this area crossing the traverselines at approximately equal but opposite angles; since our method of determining orientation cannot distinguish positive from negative angles from a single traverse line, the two sets could easily be mixed.

Fig. 10.

Histogram of the number of diffractors versus depth. Only the shallowest hyperbola of a stack was counted.