Geocoded SAR data

The geocoded ERS SAR amplitude product (geocoded ellipsoid corrected (GEC)) provided by the European Space Agency was used exclusively for all feature-tracking velocity measurements as well as for measuring the temporal evolution of the up-glacier boundary of the zone of arcuate crevasses. The physical resolution of the GECs is better than 20 m; their pixel spacing is 12.5 m. During the processing of the GEC, the corresponding SAR amplitude is projected to Universal Polar Stereographic (UPS) coordinates using precise orbit information with an accuracy of 50.2m. The projection uses a constant reference elevation h0 on the World Geodetic System 1984 (WGS84) ellipsoid. We used GECs processed to h₀ = 0 m for the longitudinal velocity profiles PIG1–PIG4 (Fig. 1). These profiles span an elevation range from sea level to about 700 m. For a detailed velocity study of the arcuate crevasse zone (profiles PIG__ARC1–PIG__ARC3 in Fig. 1) we used another set of GECs with h₀ = 700 m that corresponds to the mean elevation of this region (Reference Bamber and BindschadlerBamber and Bindschadler, 1997).

A deviation Δh = h - h₀ of true terrain elevation h with respect to h₀ shifts the horizontal coordinates of a corresponding pixel by Δxr = Δh(sine)-₁ in look direction, with 9 being the satellite look angle. For ERS with 9 ≈ 23.5° we have δX_r ≈ 2.5 Δh. In flight direction there is a further shift Δx_t, which arises during SAR focusing from the dependence of the Earth rotational velocity on terrain elevation (Reference SchreierSchreier, 1993). At 75° latitude we get Δx_t ≈ 0.04 Δ h, which is two orders of magnitude smaller than the shift in look direction. For a feature-tracking image pair not acquired from two identical sensor positions, the shifts Δx_r of the two images will not precisely cancel. This will cause a systematic topographic error of horizontal velocity in look direction,

Here, r is the range distance, and 9 is the look angle; b _⊥ and Δt are the spatial (look-perpendicular) and temporal baseline of the feature-tracking pair, respectively. Table 1 shows the expected systematic velocity errors due to geo-registration for all feature-tracking pairs used in this study. Conservatively we have estimated Δh = 1000 m and Δh = ± 100 m for pairs with h₀ = 0 m and h₀ = 700 m, respectively. The results show that with the exception of one pair (row four in Table 1) with a short 105 day temporal baseline, the topographic velocity error is always considerably smaller than the 25 ma⁻¹ random error that is estimated for the feature-tracking method later on.

Equation (1) equally applies to the repeat mapping of features in individual images as long as only changes with respect to a“master” scene are considered. For mapping the upper limit of the arcuate crevasse zone, we solely used GECs with h₀ = 700m (lower half of Table 1). Errors due to geo-registration are an order of magnitude smaller than the 100 m accuracy estimated for these measurements later on. For the detection of grounding-line position, we used complex ERS SAR data (single look complex (SLC)) from the tandem and ice phase. Acquisition time, orbit and frame of the SLC data are listed in Table 2. SLCs between solid lines were used to form differential interferograms. In alater step, these differential interferograms were geocoded to h₀ = 0 m. Table 2 shows that grounding-line location is mapped in differential interferograms of different frame and track. Nevertheless, topography-dependent biases of Equation (1) are negligible here at the 400 m accuracy claimed for these measurements later on. This is because deviations of ice surface elevation from h₀ = 0 m are typically 5100m for the grounding zone.

No tie points are used in GEC production. This allows us to check the validity of the presented error analysis using feature tracking and repeat mapping checks on the nunataks and ice rises shown in Figure 1 (+ symbols). For all GEC pairs analyzed (lower part of Table 1; h₀ = 700 m), these points come out stationary to better than half a pixel (6.25m) per year, in accordance with the expected errors listed in Table 1. At the same time, this result provides overall validation for our implementation of the feature-tracking technique. The technique is described in the following.

Feature-tracking velocity measurements

To obtain velocity averages that are relevant on seasonal and interannual time-scales, we track surface features such as crevasses that are passively advected with glacier flow. Weuse a straightforward implementation of the feature-tracking method similar to that of Reference Scambos, Dutkiewicz, Wilson and BindschadlerScambos and others (1992). We input two GECs ofthe same scene with atemporal baseline of months or years. To measure the surface motion at a specified location, we cut out and normalize a surrounding square chip of side length 1.5 km in the first image. Within a specified search depth we shift the second amplitude image on a two-dimensional grid with search dimension D and search step d. For each individual shift we create another normalized chip from the second image. As a correlation measure, we evaluate the total squared difference between the two chips. The shift at the global minimum of the correlation measure is taken as the horizontal motion at the specified location. We iteratively reduce d from 50 m to 12.5 m and 6.25 m starting with the solution of the previous coarser search step; D is reduced accordingly. The ratio (contrast) of the global minimum with respect to the next largest local minimum is stored as a quality flag. The shortest time interval used for feature tracking in this study is 35 days. For this interval the maximum accuracy of the velocity measurement is about 70 m a⁻¹; for longer intervals it is proportionally better. In regions without visible surface features the accuracy of the method is severely limited by low contrast of the correlation measure. An exception is short time intervals because image speckle is still partially coherent and is tracked by the method even when no macroscopic surface features are present (e.g. Reference Gray, Mattar, Vachon and SteinGray and others, 1998).

In contrast to interferometric velocity measurements, the feature-tracking method represents a longer time interval (months vs days) and gives absolute velocities without requiring reference points of known velocity. Furthermore, no corrections due to tidal motion or topography are necessary. Disadvantages of the feature-tracking method are the lower spatial resolution (1km vs 50 m for interferometric SAR) as well as the presence of high random scatter in the results due to lack of surface features or feature confusion. Our experience is that this random scatter can reduce the accuracy of individual data points to about 200 ma⁻¹, which, particularly for longer time intervals, is considerably less than the before-mentioned maximum accuracy of the method. This can occur for particular regions of an image pair with a large time interval, where ice motion has led to changes in the orientation and appearance of the surface features, or for an entire image pair if the backscatter amplitude has changed significantly from one image to the next. The latter condition is a rare case for our study area; all of the scenes compiled in Table 1 have extremely similar backscatter, which is visibly dominated by the spatial distribution of surface roughness. The random scatter of the velocity measurements makes the creation of meaningful two-dimensional velocity maps difficult, requiring considerable algorithmic effort for outlier removal, interpolation and smoothing. A more robust alternative is the evaluation of a velocity profile along a given flowline for repeat datasets. In critical regions with low image contrast, additional velocity points can be picked interactively within a swath of the width of the search chip (1.5 km in our case) centered around the flowline. After removing obvious outliers via the before-mentioned quality flag, and averaging over profiles comprising on the order of 100 points, maximum random errors are reduced to less than 25 ma⁻¹, which we assume as a conservative error bar for our velocity time series.

Mapping the up-glacier limit of the arcuate crevasse zone

The up-glacier boundary of the zone of arcuate crevasses on PIG at a given image date is traced by the uppermost crevasse visible in the corresponding image. Because new crevasses form at the boundary over time, hereby replacing the previous uppermost crevasse, the boundary of the arcuate crevasse zone is a “feature” that is not passively advected with glacier flow. The boundary was digitized directly in individual GECs, with digitization errors being random and smaller than 100 m. When calculating a time series of the mean position of crevasse onset from 430 digitized points, the corresponding error is consequently 525 m.

Mapping of grounding-line location

Grounding-line positions are evaluated from the representation of variable vertical tidal motion in quadruple interferograms of PIG (presented before in Reference RignotRignot, 1998). We additionally consider theoretical complications if the tidal differences between these scenes are small compared to the tidal dynamics (Reference Rabus and LangRabus and Lang, 2002). Misplacement errors including the uncertainty of ephemeral migration of the grounding line with tide level are expected to be smaller than 400 m for PIG. This error is probably dominated by the sensitivity of the grounding line to tide-level variations, the former being a spatially non-random bias for a given quadruple interferogram. Consequently, no significant reduction of the error is achieved when we use a longer segment of the grounding line to determine a time series of the mean grounding-line position. For simplicity, we assume 400 m to be a (conservative) error estimate for our time series of grounding-line positions.

Changes of surface velocity

Velocity magnitude measured with the feature-tracking method along profiles PIG1,2,3,4 is shown in Figure 2. Temporal coverage is November 1995–October 2000. According to these measurements, PIG undergoes temporal velocity variations on the order of 10–15% of the mean velocity, with velocity differences diminishing up-glacier. The mean velocity has increased monotonically throughout the observation period (Fig. 2b). Noticeable velocity differences reach 480km above the grounding line and about 50 kmpastthe up-glacierlimit of the weakly grounded ice plain discovered by Reference Corr, Doake, Jenkins and VaughanCorr and others (2001).

Fig. 2. Velocity magnitude along longitudinal feature-tracking profiles; labels refer to Figure 1. Left: spatio-temporal distribution of velocity magnitude. Right: velocity magnitude as a function of time averaged over profiles PIG3 + PIG4 (on floating PIG), resp ectively.

Compared to the case of the velocity variations found forTG (Reference Rabus, Lang and AdolphusRabus and others, 2003), the decision between external and internal forcing for PIG is less obvious. Here, the key question is whether the source of the temporal velocity variations is at the (i) down-glacier or (ii) up-glacier end of the region affected by the velocity variations. In the first case, the velocity variations are caused by the (external) dynamics of the floating part of the glacier that is dominated by ocean and sea-ice conditions, while in the second case they are caused by the (internal) dynamics of the grounded parts of the glacier. We are looking for constraints that allow one of the two possibilities to be ruled out.

The upper limit of the observed velocity changes suspected from Figure 2 lies in a region of arcuate crevasses that is visible in Figure 1. For a series of seasonal and interannualimage pairs in the period 1992–2000, we calculate feature-tracking velocities along three detailed profiles, PIG__ARC1,2,3, following approximate flowlines through the center and sides of the crevassed section of the glacier (see blow-up in Fig. 1).We are particularly interested in the spatial trend of the velocity profiles when they approach the (uncrevassed) region above the uppermost crevasse. Because PIG__ARC3 unfortunately crosses the frame boundary for several image pairs, velocity measurements are not available closer than 10km from the crevasse onset. PIG_ARC1 starts at the crevasse onset, but, due to the intense crevassing in this part of the glacier, valid measurements with the feature-tracking method are not obtained 58km down-glacier of the crevasse onset. For the central profile PIG_ARC2, valid measurements are available 5km down-glacier of the crevasse onset, and therefore this profile has the greatest potential for up-glacier extrapolation of the spatial trend of velocity change. No suitable surface features were found above the crevasse onset that would allow a reliable up-glacier extension of the feature-tracking velocity measurements.

Figure 3 shows velocity magnitude along the three profiles obtained for the different feature-tracking pairs. For PIG_ARC2 the detrended results immediately show that the velocity change decreases significantly when approaching the crevasse onset. The up-glacier decrease of the velocity variations along PIG_ARC1 and PIG_ARC3 at the glacier sides is considerably weaker than for PIG_ARC2 close to the center line. Looking at Figure 3 it is evident that quantitative extrapolation of the velocity change to the crevasse onset will not be reliable, even for PIG_ARC2. As the spatial trend of velocity change appears non-linear, any extrapolation would be dominated by the data points closest to crevasse onset. However, the accuracy of individual feature-tracking data points can be considerably less than the 25 ma⁻¹ accuracy that we estimated for an average over several points. As a result, the extrapolated velocity change over the 8 year observation period at the crevasse onset of PIG_ARC2 could be anywhere in the range 0–125ma⁻¹. (The corresponding range of velocity change per year is 0–16ma⁻¹.)

Fig. 3. Spatial and temporal velocity characteristics in the arcuate crevasse zone: (a) velocity magnitude; (b) velocity minus a linear spatial trend (black solid line in (a)).The x axis refers to path distance from the start of the profile; the location the crevasse onset averaged over the observation period is at the left border of the graphs.

In Figure 4a and b we show time series of mean velocity and velocity gradient obtained from linear least-squares fitting along the three profiles. The least-squares fit was not carried out on datasets with less than four data points along the profile. Figure 4a shows that mean velocity displays a near-linear increase over time. The total increase is about 0.15 km a⁻¹ for each of the profiles. In contrast, the evolution of the velocity gradients in Figure 4b is significantly different between the profiles. As expected, the gradient of PIG_ARC2 shows a significant temporal increase consistent with the up-glacier reduction ofthe velocity variation that is visible in Figure 3.The gradient of PIG_ARC3 also shows a monotonic increase, but the magnitude is much smaller than that of PIG_ARC2. The gradient of PIG_ARC1 does not show a net increase or decrease over time.

Fig. 4. Temporal velocity changes along profiles PIG_ARC1,2,3 and crevasse-onset migration. (a) Velocity magnitude at 15 km from average crevasse onset vs time; (b) gradient of velocity magnitude vs time; (c) average crevasse-onset location vs time.

The latter findings suggest that at the glacier sides the transition from near-uniform to decreasing velocity variations must occur up-glacierofthe first valid feature-tracking measurements (and closer to the crevasse onset). If we assume that the velocity increase goes to zero for all profiles at a similar distance up-glacier of the crevasse onset, this indirectly suggests a higher temporal increase of longitudinal extension at the glacier sides. This is compatible with crevasses at the center being considerably less frequent and more diffuse with respect to the sides (Fig. 1, blow-up). The highest density of crevasses occurs at the east side of the glacier, where PIG_ARC1is located.

Comparison with interferometrically measured velocity change

Table 3 compares our feature-tracking results with corresponding results from the recent interferometric study by Reference Rignot, Vaughan, Schmeltz, Dupont and MacAyealRignot and others (2002). The center-line profile used by Rignot and others parallels PIG3+4 and PIG__ARC2within 15 km distance on the floating part of PIG and in the arcuate crevasse zone, respectively. For our study, velocity change Δ v in Table 3 is taken from Figures 2b and 4a. To make Δ v values comparable between the studies, we have converted them to velocity change per year in all cases. Within expected error bounds, there is good agreement between the studies. Obtaining similar results with the long-term temporal averaging of the feature-tracking method demonstrates that the interferometric results (representing mere 1 – 3 day temporal averages) are not significantly biased by ephemeral velocity changes in this case.

We also compare rates of surface lowering for defined sections of PIG. Following Reference Rignot, Vaughan, Schmeltz, Dupont and MacAyealRignot and others (2002), we calculate surface lowering rate according to

Here, H denotes ice thickness; corresponding values are taken from Reference Lythe and VaughanLythe and Vaughan, (2001). The first and second terms in Equation (2) denote thickness change due to vertical strain and advection changes, respectively. As individual values Δv _1/2 at l₁ = 30 km and l₂ = 80 km closely agree between the studies, surface lowering rates will also be similar; for the period 1996–2000we obtain Δh/Δt = –0.17 and –0.18 m a⁻¹ for interferometry and feature tracking, respectively. This differs considerably from values of –1.1ma⁻¹ obtained by Reference Shepherd, Wingham, Mansley and CorrShepherd and others (2001) for this segment of PIG. The shorter segment between 50 and 70 km chosen by Reference Rignot, Vaughan, Schmeltz, Dupont and MacAyealRignot and others (2002) produces Δh/Δt = –0.94ma⁻¹, in closer agreement with Shepherd and others’ corresponding value of –1.5ma⁻¹. Close agreement with Shepherd and others’ measurements is not necessarily expected, as their values may include a background of ongoing thickness change unrelated to the recent velocity change. Equation (2) only describes the thickness change relative to this background.

We also calculate Δh/Δt along the center line of the arcuate crevasse zone (profile PIG__ARC2) using Δv from Table 3 (numbers refer to the period 1992–2000 in this case). For the stretch between l ₁ = 0km (crevasse onset) and l₂ = 30km, ice thickness is approximately constant at H₁ = H₂ = 1900m, which renders thickness change by advection negligible here. To bracket Δh/Δt for the feature tracking, we substitute the estimated error limits 0 and 16ma⁻¹ of Δv₁ into Equation (2). Respectively, we obtain Δh/Δt = –1.7 and –0.7 ma⁻¹. Alternatively, we can calculate surface lowering using the linear spatial velocity gradient taken from Figure 4. For the period 1992–2000,this produces Δh/Δt = (0.020 – 0.025)61900/8ma⁻¹ = –1.2ma⁻¹, which as expected falls in between values calculated for the error range of Δv₁. Using the corresponding numbers of Reference Rignot, Vaughan, Schmeltz, Dupont and MacAyealRignot and others (2002) from Table 3, we obtain Δh/Δt = –0.6ma⁻¹. Finally, Reference Shepherd, Wingham, Mansley and CorrShepherd and others (2001) report surface lowering of –0.5 ma⁻¹ for this section of PIG. These results suggest that the velocity variations of PIG likely cannot go to zero at the up-glacier boundary of the arcuate crevasse zone, as this would produce unrealistically large rates of surface lowering in the arcuate crevasse zone. This implication is discussed later on.

Changes of crevasse-onsetlocation

For a segment in the vicinity of PIG_ARC1 (Fig 1, blow-up) we digitized the crevasse onset for all images involved in the feature tracking. The area enclosed by the uppermost crevasse at subsequent image dates was calculated, and divided by the length of the uppermost crevasse (mean of the two image dates). The resulting time series of spatially averaged crevasse-onset location is shown in Figure 4c.The succession of filled and open symbols representing (austral) spring and fall dates, respectively, shows that there are no systematic seasonal oscillations of crevasse-onset location. Nevertheless, short-term fluctuations like the one from austral fall 1992 to austral spring 1992 can occur. The apparent long-term trend of crevasse migration, however, suggests that new crevasses are spawned up-glacier at an average rate not compensated by glacier motion. This causes a net inland extension of the crevassed zone.

It is interesting to calculate how this extension compares quantitatively with the observed surface velocity increase. Total migration of the crevasse onset from 1992 to 2000 is on the order of 0.8 kmcorresponding to 100ma⁻¹. Extra displacement during the 8 year observation period due to the velocity increase was 0.56(8 years) 6(0.15kma⁻¹) = 0.6 km, corresponding to 75 ma⁻¹. If PIG were undergoing a true surge, the opening of just a few deep crevasses within the100 mincrement marking the advance of the up-glacier surge front would explain the observed extra displacement. Velocity change would be zero up-glacier of the crevasse onset and constant throughout the arcuate crevasse zone. In the discussion section we will see, however, that the scenario of a true surge conflicts with several other observations and constraints.

Changes of grounding-line position

Figure 5 shows changes in grounding-line position since 1992 across the central part of PIG. It is apparent that the initial 3.5 km retreat during the period 1992–94 that was first seen by Reference RignotRignot (1998) did not continue linearly but was followed by stagnation of the mean grounding-line position during the period 1994–2000. Even the large retreat of the grounding line during the earlier period could result from a mere ephemeral reduction in ice thickness because the lowermost grounded part of PIG is a near-isostatic ice plain (Reference Corr, Doake, Jenkins and VaughanCorr and others, 2001).

Fig. 5. Fluctuations of grounding-line position of PIG derived from differential interferometric measurements. Left: sketch maps showing reference grounding-line position (solid line) plus grounding-line position at other times along a common sub-segment. Right: mean grounding-line position with respect to the reference along the chosen sub-segment.

Interpretation of recent velocity change

We are faced with the question whether the velocity variations of PIG observed during the last decade are actively caused (i) by an internal flow instability, or alternatively (ii) by the external dynamics of the floating glacier parts. The first possibility involves variations of the back pressure that is caused by shear against the slow-moving marginal ice shelves surrounding the floating part of PIG as well as basal shear of the ice plain immediately above the grounding line. Calving of the ice front or retreat of the grounding line will cause negative back-pressure variations that propagate up-glacier, leading to increased crevassing far inland. The second possibility leads to positive longitudinal stress that then would have caused an enhanced down-glacier velocity increase. This enhanced velocity increase may then cause destabilization of the floating glacier parts, leading to dynamic thinning and grounding-line retreat as well as enhanced calving.

There are several arguments that persuade us that an internal flow instability is causing the recent velocity variations of PIG. The active region of this internal forcing must be in or up-glacier of the arcuate crevasse zone.

First, the time series of velocity increase and crevasse-onset migration both show significant linear trends, while this is clearly not the case for the time series of groundingline retreat (Fig. 5) or that of ice-front calving (the ice front was remarkably stable up to the great calving event of 2001). Secondly, if grounding-line retreat had caused the velocity increase, one would expect the back-pressure anomaly caused by the 1.3 km 1992–94 grounding-line retreat propagating up-glacier to be reflected in the later velocity data. This is, however, not the case. Thirdly, strong spatial gradients of velocity change caused by a flow instability originating in the arcuate crevasse region are expected to become increasingly subdued while propagating down-glacier. This is consistent with observed small gradients of velocity change near the grounding line, as well as indirectly with the lack of more dramatic grounding-line retreat.

That the entire floating part of PIG (including the ice plain) is currently passive is independently confirmed by Reference Rignot, Vaughan, Schmeltz, Dupont and MacAyealRignot and others (2002) who find near-constant velocity change in this region (their fig. 2). In contrast, active external forcing (e.g. through recent back-stress variations by calving at the ice front or by unstable friction at the shear margins) should cause significant spatial gradients of velocity change on the floating part of PIG. Velocity increase between 1996 and 2000 at the lower boundary of the arcuate crevasse region (0.15 kma⁻¹) is two times smaller than at the upper boundary of the ice plain (0.3kma⁻¹), but the relative velocity increase is remarkably similar at aboutθ%. Because the floating part appears passive, this again suggests forcing by longitudinal stress changes up-glacierof the region considered, i.e. in or above the arcuate crevasse zone.

The up-glacier boundary of the suspected internal flow instability is not well constrained by the feature-tracking data. The boundary must be several kilometers above the onset of arcuate crevasses. This is because zero velocity change at the crevasse has been shown to produce unrealistically high rates of surface lowering. Further, zero velocity change at the glacier center (compatible with the data from profile PIG__ARC2) would imply a velocity discontinuity at the glacier sides because variations appear uniform for profiles PIG_ARC1 and PIG_ARC3 as close as 10 km down-glacier of the crevasse onset. Such a discontinuity would require the formation of surge-type crevasses reaching deep into the interior of the glacier. Such crevasses would need to be water-filled to form and stay open, a condition which is incompatible with the cold climatic conditions in this region of PIG (Reference Vaughan, Alley and BindschadlerVaughan and others, 2001). Finally, no velocity discontinuity is visible in the one differential interferogram of Reference Rignot, Vaughan, Schmeltz, Dupont and MacAyealRignot and others (2002) that has sufficient-quality coverage of the arcuate crevasse zone. Judged from this interferogram, which represents a comparison of 1day velocity averages in 1996 and 2000, the region of velocity change seems to extend about 25 km up-glacier of the crevasse onset. Velocity change per year along the crevasse onset is on the order of 15 ma⁻¹ across the central part of the main glacier trunk.

Origin of the flow instability

While a flow instability associated with the arcuate crevasse region likely causes the current down-glacier velocity variations of PIG, we can only speculate on what caused the flow instability in the first place. This is because the strong temporal trend of the variations implies that the velocity of PIG probably increased many years before observations began in 1992. Both internal and external forcings could have been the original trigger. Mass-balance fluctuations in the upper PIG drainage, thought by Reference Wingham, Ridout, Scharroo, Arthern and ShumWingham and others (1998) to explain their observations of surface elevation change, are an example of possible internal forcing, while adjustment due to past changes of ice-shelf configuration in Pine Island Bay could be a potential external reason.