Interferogram generation

The details of the method used to generate radar interferograms of Petermann Gletscher have been described by Reference RignotRignot (1996). In brief, we combine two passages of the ERS satellite coherently to form a radar interferogram, which is then corrected for surface topography using a prior-determined precision digital elevation model (DEM) of north Greenland assembled by Reference EkholmEkholm (1996). DEM> control points are used to refine the initial estimates of the orbit separation between the successive passages of the satellite (the so-called interferometric baselines) obtained from the ERS precise orbit data distributed by the German Archive and Processing Facility (DPAF).

Two topography-corrected interferograms spanning the same time interval are then differenced to yield a difference interferogram which measures only the tidal displacement of the glacier. We refer to this difference interferogram as a “tide interferogram” in the rest of the paper. The success of differencing relies on the assumption that the glacier creep flow remains steady and continuous during the period of observation so that the displacement signal associated with creep remains the same in both topography-corrected interferograms and thereby cancels out when computing their difference. When this assumption is invalid, motion fringes are detected on grounded ice, as for instance in the case of the mini-surge of Ryder Gletscher (Reference Fahnestock and KwokJoughin and others, 1996).

The list of the ERS data used in this study is given in Table 1, along with tides predicted at the time of passage of the satellite. The 1992 interferograms combined ERS-1 data acquired along the same orbit every 3 days. The 1996 inter-ERS-1 on 31 December 1995. The glacier flows to the north along the eastern flank of Washington Land.

Fig. 1.

Radar amplitude image (140x104 km), in a polar stereographic projection (50 m sample spacing) of Petermann Gletscher, north Greenland, acquired by ISR1 denotes the ice-sounding radar data collected along the main ice flow (Reference Allen, Gogineni, Jezek and ChuahAllen and others, 1997) (Fig. 5). 1SR2 denotes the ISR data collected in the transverse direction, approximately at the equilibrium-line altitude (ELA)of the glacier. The hinge-line profile inferred from the ERS interferometric data is shown as a thin white line, west of Porsild Gletscher. ERS was flying from right to left in the figure, illuminating from the bottom (descending, right-looking pass). (© ESA 1995.)

Table. 1.

ERS data used in this study and tides predicted at the time of passage of the satellite using the FES.95.2 Grenoble ocean-tidal model (Le Reference Provost, Lyard, Molines, Genco and RabilloudProvost and others, 1998) at 81.5° N, 63° E. In 1995-96, ERS-1 and ERS-2 images acquired 1 day apart were combined to form interferograms. E1 22373 was combined with E2 2700, El 23876 with E2 4203, El 23332 with E2 3659 and E1 23833 with E2 4160. In 1992, ERS-1 images acquired 3 days apart were combined to form interferograms. El2947 was combined with E2 2904, El 2947 with El 2990, El 3248 with El 3205 and El 3248 with El 3291

Comparison of image products

To compare tide interferograms acquired at different epochs and along slightly different orbits, the interferograms are projected on to a common Earth-fixed grid, here a polar stereographic (PS) grid, with a secant plane at 70°N, and a 50 m sample spacing (Fig. 1).The actual spatial resolution of the ERS data is 20 m on the ground in the cross-track (or range) direction and 4 m in the along track (or azimuth direction).

The automatic projection and regridding of the ERS data is not of sufficient precision to guarantee that the gridded image data overlap perfectly on top of each other due to uncertainties in absolute positioning of the satellite. To estimate the residual offsets between multi-date images, we calculate the cross-correlation peak of the radar signal intensity between a reference radar scene and the other scenes. Cross-correlation of the signal intensity is calculated over non-moving parts of the scene, here the mountainous regions bordering Petermann Fjord. The retrieved image offsets are filled through a plane, which is then used to re-sample the different scenes to the reference scene. The goodness of fit of the plane fitting, which measures the precision of registration of the data, is 5 m in the case of Petermann Gletscher.

The registered tide interferograms, centered at the hinge line of Petermann Gletscher, are shown in Figure 2. The hinge line is located at the inward limit of the zone of tidal flexing or flexure zone of the glacier. The flexure zone is a region about 4-6 km in length, characterized by a high fringe rate in Figure 2, where the glacier ice adjusts rapidly to hydrostatic equilibrium as it reaches the ocean.

2. Systematic mapping of the hinge line

Systematic mapping of the hinge line In a prior study of Petermann Gletscher, the glacier hinge line was mapped by locating the minimum of one-dimensional tidal profiles successively across the entire glacier width (Reference RignotRignot, 1996). While this earlier procedure was sufficient to locate the position of the hinge line for mass-flux calculations, it is not optimal for change-detection applications. Here, we improve upon the mapping precision of the hinge line by using a model-fitting technique. Instead of using a few points to locate the hinge line, the method uses entire profiles and is thus more robust to noise.

The glacier floating tongue is represented as a thin elastic beam of ice, of unlimited length, clamped solidly at one end on bedrock (at the hinge line), freely floating on sea water along its base and unconstrained along its sides. Let x denote the beam horizontal axis oriented perpendicular to the glacier hinge line. The solution of the differential equation describing the vertical displacement of the beam in response to tide has been discussed by Holdsworth (1969, 1977). The beam vertical displacement, >w(x), along the x axis may be written as

(1)

where w_max and w_min represent, respectively, the maximum and minimum vertical displacement of the beam along the x axis, β is a damping factor describing the decay length of the beam displacement along the x axis, and X_H is the hinge-line position along the X axis.

The elastic-beam model described in Equation (1) is applied to tidal profiles extracted across the entire glacier width, in a direction perpendicular to the iso-contours of vertical displacement of the glacier tongue represented by the interferometric fringes displayed in Figure 2. For each tidal profile, we estimate four parameters of Equation (l) in the least-square sense; the damping factor, β, the maximum and minimum height of the profile, w_max, and w_min, and the hinge-line position,x_H . The parameter of importance is x_H, since it defines the hinge-line position along each profile. The damping factor, β, is expected to vary slightly across the glacier width due lo changes in ice thickness. The minimum displacement, w_min, in principle close to zero, is a free parameter which accounts for the presence of phase noise over grounded ice (meaning the interferometric phase is not identically zero over grounded ice in the tidal interferograms). A measure of the goodness of fit is provided by the rms difference between the model fit and the interferometric data.

An example of model fitting is shown in figure 3a and b. Over the more than 1000 profiles analyzed in this study, the goodness of fit averages 3±1 mm. At this level of precision, one can hardly notice the difference between model fit and real data (fig. 3b), which illustrates both the low noise level of the data and the relevance of the elastic-beam model.

Model fitting is less reliable along the glacier side margins. In that region, the glacier tidal displacement is the result of complex interactions between the hinge line of Petermann Gletscher,the glacier side margins and the hinge line of Porsild Gletscher (figs 1 and 2). A consequence of this interaction is a “pinching” of the zone of tidal flexure along the side margins that cannot be accounted for by our simple one-dimensional elastic-beam model. As a result, the rms error of the model fit is higher along the side margins compared to the glacier center and the inferred value of β deviates substantially for that retrieved along the glacier center.

Fig. 2.

Tidal displacement and hinge-line position of Petermann Gletscher measured from ERS radar interferometry in (a) 1992 (El 3205-El 3248-E1 3291 in tables 1 and 2); (b) 1992bis (El 2904-El 2947-El 2990 in tables 1 and 2); (c) 1996 (E123332-E2 3659-E1 23833-E2 4160 in tables 1 and 2) and (d) 1996bis (El 22373-E22700-El 23876-E2 4203 in tables 1 and 2). Each fringe or full cycle of grey-tone variation represents a 28 mm differential displacement of the glacier tongue along the radar line of sight, equivalent to a 31 mm vertical displacement of the glacier tongue induced by changes in ocean tide. The location of profile P1 in Figure 3 is indicated in the upper left quadrant (white thick line), interferogram 1992 and is parallel to the ISRI profile shown in Figure 1.

To estimate the mapping precision of the hinge-line position shown in Figure 2 and plotted in Figure 4, we smooth the inferred positions using a square box averaging filter about one ice thickness in width (or 600 m) and compare the smoothed curve to the original one. The result is a 30 m rms difference between the two curves. If we assume that variations in hinge-line position occurring over a length scale of less than one ice thickness are due to noise, this result means that ± 30 m represents the statistical noise of our relative determination of the hinge-line position at one epoch.

Using the same box filter, we estimate the precision of detecting hinge-line migration. We assume that any deviation in hinge-line migration from its mean value calculated within a square box about one ice thickness in width is due to noise. Comparing various hinge-line positions, we find a rms noise of 40 m. This value is consistent with the 30 m precision in hinge-line mapping for individual interferograms, combined with the 5m precision in registration of independent tidal profiles.

Hence, the relative precision with which hinge-line migration is detected in the case of Petermann Gletscher is 40m. In the absence of ground-control points of known latitude and longitude, the absolute precision with which we know the hinge-line position is in comparison probably no better than 100 m. In effect, the ERS data are registered to a DEM of north Greenland at a 500 m sample spacing (Reference EkholmEkholm, 1996) with a precision no better than ± 0.1 sample spacing or ±50 m. This is a limitation of the satellite-radar technique compared for instance to Global Positioning System (GPS) surveys which are capable of locating points on the ground to within 5-10 m. It is, however, important to note that the uncertainty in absolute location of the radar data has no influence on the precision of delecting hinge-line migration. As long as it is possible to cross-correlate common fixed-image features between multi-date image data, the radar data alone are sufficient to detect hinge-line migration with great precision.

No ground measurements of the hinge-line position are available on Petermann Gletscher to compare with our results. Laser-altimetry data and ice-sounding radar data can locate the hinge line of Petermann Gletscher within a few hundred meters to a kilometer at best. No GPS data have been collected on Petermann Gletscher. Yet, GPS data collected on Rutford Ice Stream, Antarctica, exhibit a rms noise one order of magnitude larger (cm) than that achieved with ERS radar interferometry (mm) over the same area (Reference RignotRignot, 1998), meaning that the hinge-line position may only be mapped with a precision of 100-200m with GPS (Reference VaughanVaughan, 1994) compared to 40 m with satelliteradar interferometry. Simply stated, the precision of hinge-line mapping achieved with radar interferometry is totally unprecedented.

Fig. 3.

Tidal profile P1 (Fig. 2) measured interferometrically by ERS (dots) and model fit from an elastic-beam theory (solid line) arid (b) difference between the model fit and the ERS data. The rms error of the model fit is 1.7 mm. The inferred damping factor of the ice is β=0.3 km⁻¹. The inferred hinge-line position is indicated by an arrow in panel (a).

Hinge-line migration

Four independent mappings of the hinge line of Petermann Gletscher are shown in Figure 4. Two main features are identified from the inferred profiles: (1) the hinge line migrates back and forth on a short-term (days to months) basis in both 1992 and 1996, which we interpret as a change in ocean tide; (2) the hinge Line retreats several hundred meters between 1992 and 1996, which we interpret as a change in glacier thickness.

Fig. 4.

Hinge-line position of Petermann Gletscher at four different epochs (tables 1 and 2) inferred from model fitting of tidal profiles (Fig. 2). (a) Hinge-line position in 1992 and 1992bis (thin black lines), average position in 1992 (thick black line), position in 1996 and 1996bis (thin grey lines) and average position in 1996 (thick grey line), (b) Hinge-line retreat between the 1992 mean position and the 1996 mean position.

The hinge-line migration across the glacier width (calculated excluding a region about 250 m wide along the side margins where hinge-line mapping is less precise) is 78±213 m between the two 1992 interferograms and 42±162 m between the two 1996 interferograms (where ± denotes the value of the standard deviation of the difference). The mean value of the migration is therefore above the 40 m precision level of detection of hinge-line migration discussed earlier and so is its standard deviat ion. This means that the hinge-line migration is real and that it is also spatially varying across the glacier width.

The reality of the hinge-line migration detected between the two 1992 interferograms and the two 1996 interferograms is discussed in more detail in the next section where we show that the measured migration is in good agreement with that predicted from ocean tides. The reality of the spatial variation in hinge-line migration across the glacier width indicates that the transition from grounded to floating ice is not a well-defined grounding line, but rather a “grounding zone”, which extends up to a couple of hundred meters in some areas and only a few tens of meters in others. Presumably, the width of the grounding zone varies as a result of spatial variations in surface and basal slope.

The hinge-line retreat measured between 1992 and 1996 using four positions of the hinge line varies from a lower value of 212 ± 230 m for interferogram 1992 minus interferogram 1996bis (figs 2 and 4) to a maximum value of 333 ± 127 m for interferogram 1992bis minus interferogram 1996. The retreat measured comparing the mean 1992 and 1996 positions (thick curves in Figure 4) is 272 ± 120m. The retreat is well above the noise level of detection (40m) and above the level of migration associated with tide. We attribute the 4 year hinge-line retreat to glacier thinning.

Glacier thinning

The glacier slope of Petermann Gletscher at the hinge line may be calculated from the glacier-elevation profile collected by the NASA/Wallops airborne laser altimeter (Reference Krabill, Thomas, Martin, Swift and FrederickKrabill and others, 1995) combined with ice-thickness data also collected in 1995 by the University of Kansas' ice-sounding radar (ISR) (Reference Allen, Gogineni, Jezek and ChuahAllen and others, 1997). The surface slope along the profile whice follows the glacier center line is 1% at the hinge line. The inferred basal slope is also 1 % (Fig. 5). The KMS DEM confirms that the glacier surface slope is 1% in that region.

Fig. 5.

Thickness profile of Petermann Gletscher, north Green-land, obtained from 1995 laser altimetry data for the surface (Reference Krabill, Thomas, Martin, Swift and FrederickKrabill and others, 1995) and ice-sounding radar data for the thickness (Reference Allen, Gogineni, Jezek and ChuahAllen and others, 1997). The precision in surface elevation is 10 cm. Ice thickness is known with 10m uncertainty. The hinge-line position inferred from radar-interferometry data in late 1995 is indicated by an arrow. The grounding line and the line of first hydrostatic equilibrium of the ice are 1-2 km below the hinge line (Reference Rignot, Gogineni, Krabill and EkholmRignot and others, 1997). The glacier surface and basal slopes, noted respectively ?_s and ?_b, in the text and shown by 10km long solid line fits in the figure, and are both equal to -1±0.1%.

The change in ice thickness, δh, produced by a grounding-line migration, δh,(here assumed to be equal to the hinge-line migration), was derived by Reference Thomas and BentleyThomas and Bentley (1978). We re-write their expression as

(3)

where p_w is the density of sea water, p_i is the column-averaged density of ice, ?_s is the basal slope counted positive upwards and ?_s is the surface slope also counted positive upwards. Thinning corresponds to δh < 0 and retreat corresponds to δx < 0.

Using ρ_w = 1030 kgm⁻³,ρ_i=917 kgm_-3 and ?_s = Ob = -0.01. we find that a hinge-line migration of δ_xH -272 ± 120 m in 3.87years (mean lime difference between the 1992 data and the 1996 data according to Table 1) corresponds to a change in glacier thickness δh of -78±35 cm ice a⁻¹, meaning ice thinning.

Comparison with predicted tides

Tidal predictions at the time of passage of the ERS satellites were made using the Grenoble finite-element ocean-tide model FES.95.2, which predicts through hydrodynamic modeling and altimeter-data assimilation the eight major tidal constituents on a complex finite-element grid of variable resolution, ranging from 200 km in the deep ocean to 10 km along the coastlines. The solutions have been widely distributed on a 0.5 x 0.5 degree grid for convenience (Le Reference Provost, Lyard, Molines, Genco and RabilloudProvost and others, 1998).

For verification of the model, we compared the four major constituents (M2, Kl, Ol and S2) predicted at Thank God Harbor (north of the Petermann Gletscher front) to those measured by the U.S. Polaris expedition in 1871 (Reference BesselsBessels, 1876). The root-mcan-squarc error of the predicted constituents is 1.7 cm, which is consistent with the estimated global precision of 2.8 cm derived by Le Reference Provost, Lyard, Molines, Genco and RabilloudProvost and others (1998) for the FES.95.2 model. Hence, the ocean tides predicted by the tide model are accurate to within a few centimeters.

The predicted tides Tor Petermann Gletscher (81.5° N, 63° W) at the time of passage of the ERS satellite are listed in Table 1. Comparison of these data with the tidal displacements measured in the ERS tide interferograms is shown in Table 2. The difference between predicted tide differences and those measured by ERS is 1.5 cm on average with a 3.6 cm standard deviation. The agreement between model predictions and measured tides is remarkable given that the model solution does not include the detailed geometric characteristics of Petermann Fjord and is made for an area 80 km north of the hinge line, at the mouth of Petermann Fjord in Hall Basin. The result confirms both the relevance of the tidal model and the precision of the interferometric measurements of tidal differences.

Table. 2.

Tidal differences measured by ERS compared with tidal differences predicted by the Grenoble FES.95.2 ocean-tidal model (table 1). Interferogram 1996bis is the différence between pair El 22373-E2 2700 and pair El 23876-E24203. Its predicted tidal difference is (90.2-86.5) (63.8 49.9)= -10.2 cm. Interferogram 1996 is the différence between pair El 23332 =E2 3659 and pair El 23833-E2 4160. Interferogram 1992 is the difference between pair El 3248-E13205 and El 3291-El 3248. Interferogram 1992bis is the difference between pair El 2947-El 2904 and pair El 2990 El 2947. Its tidal difference is calculated as (7.6+ 11.5) + (7.6-52.2) = -25.5cm

Tide predictions are useful for interpreting interferometric measurements. Radar interferometry only measures differences in tidal displacement and hence does not provide any information on absolute tide. As the tide changes, the glacier hinge line is expected to migrate back and forth to maintain hydrostatic equilibrium, usually retreating when the tide is high and advancing when the tide is low. If the tidal amplitude and the glacier slopes are known, the bias in hinge-line position induced by tide can be estimated to deduce the mean sea-level hinge-line position.

Using the calculated tidal differences in Table 2 and a 1% glacier slope, we predict that the 1992 hinge-line position should migrate 50 m between the two 1992 tide interferograms vs 78m measured (the sign of the migration is correctly predicted). Similarly, the hinge-line migration should migrate 5 m between the two 1996 tide interferograms vs 42 m measured. The differences between calculated and real migration are within the noise level of the hinge-line migration detection (40m). We conclude that short-term variations in hinge-line position are due to changes in ocean tide.

Analysis of mass fluxes

The hinge-line flux of Petermann Gletscher is 12.0±0.5 km³ ice a⁻¹ (Reference RignotRignot, 1996). We now compare this result to the balance flux calculated from the glacier accumulation and ablation above the hinge line. To calculate surface ablation, we use Reference Huybrechts, Letr and ReehReeh's (1991) degree-day model, following the implementation of Reference Huybrechts, Letr and ReehHuybrecht and others (1991). A degree-day factor of 9.8 mm deg⁻¹ d⁻¹ is used since this is the median value measured in north Greenland by past experiments (9.6 mm deg⁻¹d⁻¹ on Storstrommen Gletscher in the northeast measured by Reference Boggild, Reeh and OerterBoggild and others (1994), 9.8 and 5.9 mm deg⁻¹d⁻¹ measured by Braithwaite and others (1998a, b) and Reference KonzelmannKonzelmann and Braithwaite (1995) at two sites near the Hans Tausen ice cap in north Greenland). The degree-day factor of 8 mm deg^-1d⁻¹ commonly used by ice-sheet modellers in the remainder of Greenland, and which is based on studies conducted at lower latitudes along the west coast, underpredicts ablation in the north, as pointed out by Reference BraithwaiteBraithwaite (1995). We estimate the accuracy of our ablation estimates to be 10% assuming that the degree-day factor we use is correct and that the only source of error in the calculation is associated with the 1km spatial resolution of the Greenland DEM used lo compute the glacier area above the hinge line.

To calculate mass accumulation, we use Reference OhmuraOhmura and Reeh (1991). Since their map was not available digitally, it was recreated on the computer (personal communication from Reference Fahnestock and KwokFahnestock and Joughin, 1996) using the original ice-core data listed in their paper, interpolated on a regular grid to match best their manual interpolation. The precision of the data is difficult to establish, especially in the north where data sampling is coarse. We assume that accumulation is known with 10% accuracy based on the quality of the ice-core data (10-20 year records only) but the impact of the interpolation over large areas could arguably induce larger errors.

The result of the analysis is a total accumulation of 13.1km³ ice a⁻¹ above the hinge line, a surface ablation (or run-off) of 2.0 km³ ice a⁻¹, and a balance flux at the hinge line of 11.1 km³ ice a⁻¹ (Reference Rignot, Gogineni, Krabill and EkholmRignot and others, 1997). If the above estimates are correct, the hinge-line ice flux of Petermann Gletscher exceeds its balance flux by 0.88 ± 1.0 km³ ice a⁻¹. Although the uncertainty is large, the result suggests that Petermann Gletscher loses mass to the ocean.

Joughin (personal communication, 1997) calculated an ice flux of 12.4 km³ ice a-1 near the equilibrium-line elevation of Petermann Gleischer combining ERS imerferometry data and ice-sounding radar (ISR profile in Figure 1). The total area in between the ISR line and die hinge line is 1056km². In that region, accumulation is 0.28 km³ ice a⁻¹ and ablation is 1.42 km³ ice a⁻¹. The balance flux at the ISR line is therefore 12.2 km³ ice a⁻¹, which is only 0.2 km³ ice a⁻¹ lower than the measured ice flux. Hence, the glacier appears to be nearly in balance at the location ol the ISR line. If this is true, it means that the mass deficit of Petermann Gletscher is concentrated in its lower reaches, not in the interior, Averaged over the 1056 km² area in between the ISR profile and the hinge line, the 0.88 ± 1.0 km³ ice a⁻¹ mass imbalance translates into a glacier thinning of 83±91 cm ice a⁻¹.

The analysis of ice fluxes is therefore suggestive of ice thinning, nearly at the same rate as that deduced from hinge-line migration. The agreement between the two methods can probably be taken as a coincidence given the large uncertainty in the ice-flux estimates. Taken differently, however, it could also mean that our estimates of accumulation and ablation for Petermann Gletscher are reasonably accurate.

This study shows that it would be misleading to conclude that Petermann Gletscher is in balance based solely on an analysis of ice fluxes in the interior. The comparison of fluxes should be done at the coast if the overall objective of the study is to measure the contribution of this part of the ice sheet to sea-level rise. To improve the quality of the balance fluxes, more accumulation and ablation data are needed in the north. The net advantage of the hinge-line migration method is that it does not depend on our- knowledge of accumulation and ablation. Until more data on accumulation and ablation are collected in the north, detecting the hinge-line migration of northern outlet glaciers may be the most effective way of complementing the measurements of ice-volume changes to be conducted by laser-altimetry systems over the next decades. Of course, the technique is also applicable lo Antarctic glaciers since most of the Antarctic coastline is fringed with ice shelves and floating ice tongues.