Introduction
The ice sheets of Greenland and Antarctica play an important role in the Earth’s climatic balance, Of particular importance is the possibility of a significant rise in sea level brought on by a change in the mass balance of. or the collapse of. a Iarye ice sheet (Reference BindschadlerBindschadlcr. 1991). An understanding of the processes that could lead to such change is hindered by the inability to determine even the current state pi the ice sheets. For example, it is not clear whether the mass of the Greenland ice sheet is increasing or decreasing (Reference Van der Veenvan der Veen, 1991), Ground-based measurements are capable of sampling only small areas of an ice sheet. Therefore, they may indicate only regional Change. Only in the past two decades has satellite remote sensing offered the potential for wide-scale monitoring of ice sheets.
Surface elevation observations are important to many aspects of the study of ice sheets. Topographic data are useful for mapping and detecting changes in the boundaries of the individual drainage basins that make up an ice sheet (Reference BindschadlerBindschadler. 1984). Short-scale i.e. a few ice thicknesses undulations in the topography are often caused by obstructions to flow created by the basal topography (Reference PatersonPaterson, 1981; Reference JohannessonJohanncssoh, 1992). Therefore, surface topography can be used to help infer conditions at the bed (Reference ThomasThomas and others. 1985). Furthermore, densely sampled surface topography is important for modeling glacier dynamics.
Although radar altimeters have been used to measure absolute elevations for ice sheets. they do not have sufficient resolution to measure short-settle topography. photoclinometry can measure the detailed surface topography of ice sheets (Reference Bindschadler and VornbergerBindschadler and Vornbegrr. 1994). This method is limited, however, by the need fot an extensive. linearly continuous grid of ground-control points. Thus, widespread application of this technique awaits the necessary control data, which must be provided by ground-based or airborne altimeiry surveys.
Interferometric synthetic aperture radar (SAR) provides a means to measure both detailed topography (Reference Zebker and GoldsteingZebker and Goldstein, 1986; Reference Li and GoldsteinLi and Goldstein, 1990) and surface displacement motion (Reference Gabriel, Goldstein and ZebkerGabriel and others 1989), using the phase difference between pairs of complex SAR images. When the images are acquired at different times from nearly repeating orbits repeal pass. interferumetric phase is affected by both topography und surface displacement. If the images are acquired simultaneously (single pass) with dual SARs, phase is affected only by topography. Thus repeat-pass interferograms provide motion and topographic data, while single-pass interferogrames yield only topographic information. The repeat-pass approach has been applied to Spaceborne ami airborne SARs. while the single-pass approach has been used only for airborne systems. This paper examines the repeat-pass approach with images from the first European Remote Sensing (ERS-1) SAR.
The topography of ice sheets is characterized by minor undulations with small surface slopes. This type of topography is well suited to interferometrie measurement. The Canatlian Centre for Remote Sensing (CCRS) airborne SAR has acquired single-pass interferograms to map the topography of glaciers on Bylot Island in the Canadian Arctic (Reference Mattar, Gray, van der Kooij, Farris-Manning and SteinMattar and Others, 1994). Motion-induced fringes (modulo-2π phase contours. however, hinder interferometric SAR estimation of topograph) from repeat-pass (i.e. ERS-1) interferograms. These additional fringes, which may be greater in number than die topographic Eringes, make it impossible to estimate topography directly from a single-difference, repeat-pass intrefergram. Kwok and others (in press) have shown that when the velocity field does not change with time, interferograms can be differenced to cancel ice-sheet motion and. thus, obtain a topography-only interferogram. This paper examines the use of double-difference interferograms to measure the detailed topography of ice sheets. Comparison with laser-altimeter data indicates that high-resolution (i.e.80 m pixels) digital elevation models (DEMs) with relative accuracies on the order of 2.5 m can be achieved. These results eslablish that repeat-pass satellite-radar interferometry can be used to map the topography of ice sheets with an accuracy suitable for many research goals.
This paper begins with a review of iterferometry, followed by a description of the double-differencing technique used to cancel the effects of ice-sheet motion (Reference Kwok and FahnestockKwok and Fahnestock, 1996). The next section describes the generation of DEMs for a single SAR frame in West Greenland. Airborne laser-altimeter data are used in establish accuracy. Interferometrically derived DEMs from adjacent frames are also examined. Finally, one of the DEMs is compared with the motion-dependent phase variation of an ice-sheet tnterferogram.
Interferometry Background
Single-difference interferometry
The geometry of an interferometric SAR is shown in Figure 1. Ihe interferometer acquires two images of the same scene with SARs Located at S1 and S2. The first SAR is at altitude H. From S1 , the range, r0 . and look angle, θ, to a point on the surface are determined by the ground range, y, and elevation z. The range to the same pohu from the SAR at S2 . differs from r0 by Δ. Kora single-pass system, such as TOPSAR (Reference ZenkerZebker and Otbers 1992), two images are acquired simultaneously using separate antennas. In contrast, a repeat-pass interferometer acquires a single image of the same area twice from two nearly repeating orbits or flight lines. The baseline separating the SARs can be expressed In terms of components perpendicular to. B11. and parallel to. Bp. a reference-look direction. A Convenient choice is to let the nominal center-look angle, θc. deline the reference-look direction.
Geometry of an interferometric SAR.
The additional information gained by using an interferometer over a simde SAR is ibe range difference, Δ The range resolution of a typical SAR is not sufficient to measure this difference accurately for topographic mapping, Instead, the range difference is determined from the phase-difference information from the pair of complex SAR images. For a distributed target a pixel in a complex image can be represented as
Where k is the wave number andW1 is a complex, circular Gaussian random variable (RV) with amplitude A1 and phase ϕ1 (Reference Rodroguez and MartinRodriguez and Martin. 1992). Beiause the the modulo-2π phase is uiuformly distributed, a single complex image cannot be used to determine range information.
A complex inlerlcrograin is formed as the product of one complex SAR ¡muge witb lhe complex COiijugule of u Second. A pixel in the inlerferogrtim can be expressed its
The phase of this product is given by
Although ϕ1 and ϕ2 are both uniformly distributed, if W1 and W2 are correlated their difference, (ϕ1 — ϕ2). is not uniformaly distributed (Reference JoughinJoughin. 1995). In fact, the distribution of the phase difference can be quite sharply peaked if the complex images are well correlated.
Even with a narrow phase distribution, the phase difference is still only known modulo 2π A phase-unwrapping algorithm (Reference Golstein, Zebker and WernerGoldstein and others. 1988) must be used to remove the modulo-2π ambiguity. The range difference for repeat-pass iiiterferometry can then be determined using
where ϕunwarap denotes the unwrapped interlerometric-phase difference.
For a repeat-pass interferometer, Δ is affected by both topography and any movement of the target between orbits that is directed toward or away from the look direction of the radar. The interferometric phase can be expressed as the sum of motion- and topographs -dependent terms.
The displacement-dependent term is given by
where the subscripts are used to denote the coordinates at the center of the SAR resolution cell during first and second satellite passes. When the effects of motion and topography are separated, ϕ;displacement can be used to estimate ice-sheet motion (Reference Goldstein, Englhardt, Frolich, Kamb and FrolichGoldstein and others. 1993; Reference Joughin, Winbrenner and FahnestockJouglun and others. 1995; Reference Rignot, Jezek and SohnRigiiol and others, 1995).
In order to examine the effect of the baseline on the interferometric phase, the topography-dependent term can be approximated as
Variation in ground range, y. introduces a phase ramp across the interferogram. This ramp can be removed by subtracting the interferogram corresponding to a zero-height surface. For a fixed range. r0,y and Z are not independent variables and are related by
Applying equation (8) io compute the phase ramp corresponding to a zero-height surface, the elevation-dependent phase variation is given by
This indicates that the sensitivity of the interferometer to topography is proportional to the length of Bn .
Double differencing; to cancel ice-sheet motion
Motion-dependenl phase variation often dominates topography-dependent phase variation in ice-sheet inter-ferograms. The effect of motion must be removed from the interferogram before topography can be estimated. Kwoit and others in press have shown dial die motion dependent phase variation can be cancelled be cancelled interferograms. This technique relies on the assumption dial ice is displaced by the same amount over the periods spanned by two interferograms. In this Case the baseline-independent ϕcisplacement terms are identical for each interferogram. If phase ramps are removed from the interferograms, then, applying Equation (5) and (9). these interferograms can be approximated as
and
The two interferograms are differenced to form a double-difference interferogram.
This result is equivalent to a topography-only interferogram with a baseline equal to the difference of the baselines for the single-difference interferograms.
Figure 2 show s an example of double differencing to cancel ice-sheet moiion for interferograms from frame 2169 (see Figure 3 for location). Both single-differerence interferograms are formed using 6 d pairs. These interferograms have not been unwrapped, so die fringes due to the modulo-2π- ambiguity are still present. The interferometric pair shown in Figure 2a was acquired during orbits 3218 and 3304 and has Bn ≈ 2 m. The baseline is small, so this is essentially a motion-only interferogram. Figure 2b shows the interferogram for orbits 3132 and 3218. which has Bn ≈ 183 m. Phase ramps corresponding to zero-height surfaces have been removed from both interferograms, The double-difference interferogram is shown in Figure 2c. The motion-dependent fringes, which are die cause of most of the complexity seen in the single-difference interferograme. are cancelled, leaving only fringes due to topography, The baseline of this topography-only interferogram is approximately 185 m.
an example of double differencing single-difference iterferogrames from (a)orbits 3218/3304
orbit 3132/3218 to generate
a topography-only interferogram.
Location of frame 2169 and part of Ihr AOL flight Une from 27 July 1993.
Althought the interferograms in this example have die same temporal baseline, it is not neccssary that both interferograms span the same number of days to cancel motion. For example, the phase of a 3 d interferogram can be doubled. This doubles motion-induced phase variation so that the result can be differenced with a 6 d pair. The baseline of a doubled interferogram is also effectively doubled.
An interferometric pair of complex images must be Correlated to succeed in creating a siugle-dilfference interferogram. To form a double-difference interferogram. however, the constituent single-difference interferograms do not need to be correlated with each other, because much of lhe phase variation from speckle is cancelled when the single-difference interferograms are formed. This means that as long as the motion field does not change, interferograms can be differentced even when separated by long intervals. Nerertheless, correlation between interferograms are in their registration. When interferograms are correlated and die baseline is known, they can he registered with sub-pixel accuracy using the сross-corrclation function computed for one image from each interferometric pair (Reference JoughinJoughin, 1995).
Where dure is no correlation between interferograms the locations of the corner points computed by the SAR processor at the United Kingdom Processing and Archiving Facility (UK-PAF) tue used to register the interferograms (Reference JoughinJoughin. 1995). The accuracy of this method was determined by comparison with the more accurate cross-correlation method for several interferograms that could be registered with both methods. The interferograms used for this lest are separated by intervals of 3, 6 and 15 d. Erors for the corner-point registration method are estimated to be 7.5 m in range and 22.4 m in azimuth. These are relative errors. For comparison, the nominal absolute accuracy of the corner-poinl locations is 50 m (Reference Smith, Wilson and MeadowsSmith and others, 1994).
DEM Generation
After the double-difference interferograms are formed, die next step is to remove the modulo-2π ambiguity in the phase. This is done with a phase-unwrapping algorithm (Reference Golstein, Zebker and WernerGoldstein and others, 1988; Reference JoughinJoughin, 1995). Because integration of the differenced phase is used to remove the modulo-2π ambiguity, there is always an unknown constant associated with the unwrapped solution, and thus only the relative phase is known. This unknown constant is solved for as part of the baseline estimation procedure (Reference JoughinJoughin, 1995).
ERS-1 OrbitS are not known well enough to estimate baselines with the level of accuracy needed to generate DEMs. As a result, the baseline must be determined using tie points (points of known elevation; Reference Zebker, Werner, Rosen and HensleyZebker and others, 1994a). When the baseline is modelled as varying linearly along track, at least four tie poinis are needed. A greater number of tie points can be used in the least-squares solution to help reduce the effect of tie-point error on the baseline estimate (Reference Zebker, Werner, Rosen and HensleyZebker and others. 1994a).
Many of the existing topographic data for ice sheets were acquired using radar-altimetry data, which has coarse horizontal resolution (i.e. a few kilometers). Tо determine the baseline length, we use tie points extracted from the National Survey and Cadastre of Denmark (KMS) DEM (personal communication from S. Ekholm. 1994). which was derived principally from radar altimetry. Because the ice sheets are relativey smooth, this DEM yields tie points with errors of about 20 m in the presence of undulating topography. Simulations indicate that baselines estimated using 100 or more lie points extracted from the KMS DEM yield relative errors of about 2–3 m in the estimated topography (Reference JoughinJoughin, 1995).
After baseline estimation, the next step in creating a DEM is to add back the phase ramps that were removed prior to phase unwrapping. The range difference is then estimated by scaling the unwrapped phase using the relation given by Equation (4). The baseline and Δ are used to solve for the look angle. θ (Reference Li and GoldsteinLi and Goldstein, 1990: Reference Rodroguez and MartinRodriguez and Martin. 1992). Elevation and ground range, relative to a curved earth, are computed using (Reference JoughinJoughin, 1995)
and
The final step in DEM generation is to interpolate the unevenly spaced ground-range and elevation values to an evenly spaced grid.
Interferometrically Derived DEMs for an Area in West Greenland
Study area and interferograms
This section examines DEMs created for the area covered by frame 2169 from track 25 of the ERS-1 SAR during the first and second ice phases. The location of this approximately 100 km by 100 km frame is shown in Figure 3. Four single-difference interferograms were generated using data from six orbits, Table 1 gives the orbit pairs, temporal baselines (time between the acquisition of images in an inlerferometric pair) and ESA baseline estimates (Reference SolaasSolaas, 1994) for the single-difference interferograms. Each interferogram in the table is assigned a number, n that is used for subsequent identification. Interferograms are then denoted as In . These interferograms are created using ECRS-1 full-seene, complex images processed ai the UK-PAF. The complex interferograms are multilook-averaged 4 pixels in range by 20 pixels in azimuth to yield a pixel spacing of approximately 80 m. As part of lhe processing, phase ramps corresponding to a zero-height surface tire removed.
The interferograms listed in Table 1 are used m creale six double-difference interferograms, which are identified in Table 2. The sequence of operations and the single-difference interferograms used to from each double-difference result are included in the far leflhand column of Table 2. The baselines in this table are derived from the baseline estimates given in Table 1 There of the double-difference interferograms do not include I1. For these interferograms registration is performed In cross-correlating one image from each pair after compensation (or the baseline. Because they are separated by 22 months, there is no coherence between I1 and the other inlerferograms. Asa result., double-differenced pairs that include I1 are registered using the corner-point registration algorithm mentioned earlier. The complex interferograms are double-differenced by multiplexing one interferogram with die complex conjugate of the other. Phase doubling of a 3 d complex interferogram to match the moiton of a 6 d interferogram. denoted as 2 ×In is achieved by multiplying ii with itself. To estimate the baseline, 132 evenly spaced tie points zt , were extracted from the KMS DEM (personal communication from S. Ekholm. 1994).
Orbit number and ESA baseline estimates for interferograms generated for frame 2169.
Double-differences interferograms for frame 2169 and corresponding baselines derived from ESA baselines
DEMs generated directly from die double-difference interferograms have large errors (i.e. a few hundred meters) (Reference JoughinJoughin. 1995), which vary primarily in the azimuth direction. The length scale of these errors is greater than about 15 km. It is likely that these errors are the result of phase irregularities in the complex images from the UK-PAF. We estimate the error by low-pass filtering the difference between die interferogram and a synthetic interferogram created using the KMS DEM. Interfero-grams are corrected by subtracting this estimate of the error (Reference JoughinJoughin, 1995). In the resulting DEM the kilometer-scale information is provided by the interferogram, while the longer-scale information (greater than 16 km) is supplied by the KMS DEM. This is not the ideal solution, however. as it allows long-wavelength errors in the altimetry-derived KMS DEM to be propagated to the interferometrically derived DEM. Further research is needed to establish the cause of the underlying long-wavelength phase errors in the interferograms so that ultimately they can be eliminated without the aid of a supplementary DEM. For now. this correction allows us to proceed with our investigation of interferometrometric estimation of ice-sheet topography.
Analysis of interferometrically derived dems
After correction for long-wavelength errors, the interferograms are used to create a set of six DEMs. These DEMs are averaged together to create a composite DEM. za. This DEM is displayed in Figure 4a as a shaded surface with an illumination source directed from above, along the z axis. The topography consists of a gentle slope extending from about 2400m in the top righthand corner to about 1100 in the lower lefthand corner. Kilometer-scale undulations typical of ice-sheet terrain are clearly visible. To belter illustrate these features, Figure 4b shows the DEM as a shaded surface from a vantage-point diretly overhead. The light source is the same as that used in Figure 4a. Visually this enhances the appearance of short-scale undulations, while de-emphasizing the long-scale trend of the slope.
Shadded surface of the composite DEM for frame 2169. The vantage-point is (a) from the side and (b) from above. The light source is directe from above along the z axis
There are no detailed DEMs for die entire SAR frame to compare with the interferometric DEMs. Instead, differences between the individual DEMs and the composite DEM are examined. Because the error for Za is unknown, these differences are measures of DEM consistency rather than accuracy. Nevertheless, it is instructive of examine how the DEMs differ. Accuracy is evaluated by comparision with laser-altimeter data for a single flight line in the nexi sub-section.
Mean and standard deviations of die difference, 
, are given in lhe second and third columns of Table 3 The mean differences 
, varies In about ±3 m. Because mean differences should be determined largely by tie-point bias. which is the same for all the DEMs. die variability in these differences should be smaller than observed.
Comparison of interferometrically derived elevations, 
, with composite DEM, Za
Standard deviations range from moderale to quite large. In general, smaller values correspond to the interferograms with the larger baselines. The differenees are too large to be explained by baseline-estimation error alone (Reference JoughinJoughin, 1995). it is also unlikely that random phase error due to speckle can explain these differences, because the interferogrames have been haviely multilook-averaged. Localized phase-unwrapping errors could increase the error. but only for those parts of the images where Correlation is low. Most of the unwrapping errors are confined to the lower lefthand corner of the frame, where motion is greatest. Elevations are less than 1400 m in this region of the DEM. Phase- unwrapping errors are eliminated from the comparison by computing the standard deviation only for that part of the DEM where elevation is greater than 1400 m. These values, reported in the last column of Table 4. do not show a reduction significant enough to attribute the large values of 
to errors in phase-unwarapping.
Comparsion of inlerfermetrically measured elevation, 
, with AOL elevations zAOL
The dffierences between the composite and individual DEMs indicate that there may still he some unanticipated phase error in the interferograms. Shaded surfaces (Fig 5) are useful for observing differences among DEMs. Random phase error due to speckle appears as a high-frequency pattern because it varies independently From pixel in pixel. This error tends to be larger for images with shorter baselines. For example, the shaded surface in Figure 5b is more textured than the surface of the DeM shown in Figure 5a, which corresponds to a much longer baseline. Errors that show up as horizontal lines can be seen in the DEM illustrated in Figure 5 as well as in the DEMs that are not shown. These errors are the result of minor phase discontinuities in the complex SAR images. Neither of these effects is large enough to explain the large Standard deviations.
Shaded surfaces of individual DEMs generated from interferogram (a) 2 × I1 — I3 and 2 × I2 - I4
The DEM shown in Figure 5b has structures that appear as long ridges or valleys and extend nearly horizontally most of the way across the frame. In contrast, the DEM in Figure 5a is relatively free of such structures. Many of the other individual DEMs not shown have features with shapes and orientation of the same kind as the features visible in Figure 5b. They are not topographic features, because they differ from DEM to DEM. these “streaks” tend to average out in the composite DEM so that little of this type of structure can be seen in Figure 4.
The phase fluctuations that cause streak error are estimated by subtracting an interferogram simulated using the composite DEM from an individual double-difference interferogram. Phase error estimated in this way for I2 - I1 and 2 × I2 - I4 are displayed modulo-2π in Figure 6. Althought the estimales of phase error in Figure 6 correspond to magnitudes that have similar baseline lengths, the magnitudes of the streaks in each interferogram differ greatly. Observations of similar estimates from several other interferograms also indicate that Streak intensity is independent of baseline length. The sensitivity ofa DEM to streak and other phase errors is inversely proportional to baseline length. Thus, lhe best DEMs are achieved for interferograms having both low streak error and long baselines. Because the widths of the streaks are of similar scale m ihe topographic undulations, it is difficult to esliniale and remove this error by means of filtering.
Residual phase after subtracting the effect of topography from phase-corrected interferngrams for (a) I1 - I1 (B11 ≈ -106 m), and (b) 2 × I2 - I4 (B11 ≈ 118 m).
There are similar streaks in the correlation images for the interferograms. Reference Jezek and RignotJezek and Rignoi (1994) have also obse observed streaks with similar orientation in correlation images from a nearby ERS-1 swath. They have shown that the streaks are related to high-frequency variations of the azimuth registration. The underlying cause of this streak "“noise” is still unknown.
Comparison with laser-altimeter data
The NASA Airborne Oceauographic Lidar (AOL) has made several flights over the Greenland ice sheet. This laser altimeter performs overlapping conical scans about 150 m in diameter. Lighly elevation measurements are made per scan. AOL measurements of ice-sheet elevations are repeatable to within 10 cm (Reference Krabill, Thomas, Martin, Swift and FrederickKrabill and others, 1995, Reference Krabill, Thomas, Jezek, Kuivinen and Manizadein press). Accuracy is on the order of a few centimeters, so comparison with AOL data provides a good estimate of the accuracy of interferometric DEMs.
An AOL flight line from 27 July 1993 crosses the upper righlhand corner of frame 2169 Figure 3 shows the loctition of frame 2169 and the AOL flight line. The portion of the flight line that intersects the SAR frame is 76 km long. The AOL data are subsampled along the center of the scan to obtain measurements with 80 m spacing, which is approximately the pixel spacing of the interferometric DEMs. To allow comparison with interferomctric DEMs, die elevation. latitude and longitude of each point along the profile are converted to the ground-range coordinates of the SAR frame. It might have been possible to obtain more accurate baseline estimates by using lie points from the AOK data. This was not attempted, however, because the goal is to establish die accuracy of DLMs created using the radar-altimetry tie points alone.
The mean, 
, and standard deviation. 
, of the difference between the AOK elevations, zAOl
and the mterferoraetrkally derived elevations, 
, are reported in Table 4. The values of 
vary from about O m 5 m. The mean dilferenee between die AOK profile and the profile extracted from die KMS DEM is 3.43m. Thus, the mean difference of 3.81 m for the composite DEM is moslly attributable to differences between the AOK profile and KMS DEM. Streak noise prohably contributes to some of the variability in the mean differences directly, and also indirectly through an increase in baseline estimation error.
Standard deviations for the individual DEMs are all in rough agreement with the results from the comparison of composite and individual DEMs (Table 3). Only two of the individual DEMs differ from the AOK data by significantly more than 5 m. The relative accuracy of the composite DEM is 2.56 m. which is almost equal to that of the best individual DEM.
Relative differences in elevation Itır the AOL data, the composite DEM and the best individual DEM are identified by comparing the profiles after subtraction of the average elevation Erom each profile Fig 7. DEM error is largest (about 5 m) for the first 10 km of the profile, but stays below about 2.5 m for the rest of the profile. This comparison demonstrates that it is possible to generale ice-sheets DEMs with relative error of about 2.5 m. Elimination of streak noise and better tie points (i.e. AOL data) should further reduce elevation error.
Error from random-phase noise due to speckle varies nearly independently from pixel to pixel. Therefore, the fairly smooth interferometric profiles indicate that this type of phase noise is a relatively minor source of error. The AOK profile is smooth (i.e.. there is hule vertical variation greater than 1 m) on the scale of the 4 (range) by 20 (azimuth) multilook pixel size, so considerably more multilook averaging could be applied to further reduce what little phase noise there is from speckle.
There may be considerable penetration at C-band (i.e. up to 25 m for dry snow: Reference MätzlerMätzler. 1987), so that the measured surface differs from the actual surface. Any bias created by penetration to a uniform depth will he removed during the baseline estimation procedure, which tends to fis the average height to that of the tie points. spatially varying depth of penetration could cause relative differences between the measured topography and the aetual surface topography. Comparison with AOL data indicates that these differences are less than 2.5 m in the area we have examined, which is in the percolation zone. We have not done a comparison for other snow facies. We penetration is likely quite different. The effect of penetration on interferometric DEM accuracy is a topic that needs further study.
Comparison of AOL profiles with the composite DEM and with the DEM computed using 2 × I1 - I3. Tо illustrate relative differences the mean elevation is subtracted from each profile. Ίhe profile runs from each SE and corresponds to that part of the AOL. flight line illustrated that intersects frame 2169 (Fig,. 3).
DEMs for Other Frames
Interferometric DEMs are also derived for the frames adjacent to 2169: frames 2151 and 2187. The orbits used are the same as those given in Table 1 The baselines are similar to those lisled in Tables 1 and 2. Several of the interferograms for frame 2151 have many streaks and generali) low Correlation. Salisfactorv DEMs can only be created for I2 — I1, 2× I1 - I3 , and 2 × I1 - I4.. Figure 8 shows the composite DEM for frame 2151 formed using these three DEMs. Because even the interferometric pairs for these DEMs are not strongly correlated, topographic errors in these DEMs for this frame are greater than for the DEMs from frame 2169. Some of this additional error is seen in Figure 8 as streaks and minor discontinuities.
Shaded surface of the composite DEM for frame 2151. The vantage-point is (a) from the side and (b) from above. The light source is directed from above, along the z axis.
The DEM for frame 2151 shows a sharp transition from a gently undulating Surface like that seen in frame 2169, to much smoother topography at higher elevations. Because the scale of dynamically supported topography is determined by ice thickness, velocity and bedrock topography (Reference PatersonPaterson, 1981; Reference JohannessonJohannesson, 1992), this transition most likely reflects the effects of thicker ice and lower surface slopes toward the interior of the ice sheet.
Figure 9 shows the composite DEM for frame 2187. A large region corresponding to an outlet glacier is almost completely decorrelated, This region could not be unwrapped for any of the interferogrames. The missing elevations in this area have been replaced with elevations from the kMS DEM. The interferograms for this frame have more streak errors than those for frame 2169. As a result, DEM accuracy is likely to be less than that for frame 2169. Nevertheless, a lack of visible streaks on the right hand side of Figure 9 indicates that a reasonable level of accuracy (i.e. a few meters) may have been achieved.
Shaded surface of the composite DEM for frame 2187. The vantage-point is (a) from lhe side and (b) from above. The light source is directed from above, along the z axis.
The DEMs in earlier Figures are presented in the ground-range coordinates of the SAR frame. For most applications it is desirable tO represent the DEM in some other geographic coordinate system, such as the polar stereographic coordinates of the SSM/I grid. This coordinate transformation can be made using a geocoding procedure (Reference JoughinJoughin. 1995). Because the interferometric equations yield both elevation and ground range, terrain-correction is accomplished automatically (Reference Li and GoldsteinLi and Goldstein, 1990), Transforming the DEMs into a common coordinate system makes it is possible to combine and compare them with other elevation data sets. For example, Figure 10 shows the shaded surface of the interferometric DEMs for frames 2151. 2I69 and 2187 combined with a section of the KMS DEM. The increase in topographic detail offered by interferometry is striking.
Shaded surface of the combination of interferometric DEMs for frames 2151,2169 and 2187 and part of the KMS DEM. The light source is directed from above, along the z axis.
Topography and Phase Due to Ice Motion
Motion causes complex fringe patterns in ice-sheets interferograms. For example, the interferograms in Figure 2a and b have many closed-phase contours du to motion that resemble bull’s-eyes. An understanding of the complexity seen in these fringer patterns is needed before estimates of velocity can he made. In particular, it is important to determine the relative contributions of vertical and horizontal motion to the interferometric phase. Reference Joughin, Winbrenner and FahnestockJoughin and others (1995) used a simple model to demonstrate that bull’s-eye patterns are caused by vertical displacement as ice flows over bumps in the surface topography. Their model indicates that, over length scales of less than a few ice thicknesses, phase variation is dominated by vertical velocity. This is because the steep (23°) incidence angle of the ERS-1 SAR makes phase more sensitive to longitudinal gradients in vertical velocity than to longitudinal gradients in horizontal velocity, by a factor of roughly 2.4. Over longer length scales, where the magnitude of the slope variation is not as great, phase variation is dominated by horizontal displacement. In ihis section we confirm this analysis by comparing phase variation due to motion with the surface topography of our interferometrically derived DEM.
We begin by using die DEM from frame 2169 to simulate and remove the direct effect of topography (i.e.ϕtopography ) from the interferogram formed with images from orbits 3089 and 3132, The phase, (ϕdisplacement , is unwrapped and then high-pass filtered to remove variation with wavelengths longer than 7 km. which corresponds to at least three ice thicknesses. Topography and high-pass filltered ϕdisplacement for a 96 km by 86 km subsection of frame 2169 are compared in Figure 11. The DEM is shown as a shaded surface with a view from above. The light source is directed downslope to illuminate slopes in the approximate direction of flow. Uphill sides of bumps are illuminated, while downhill sides appear dark. The contours of the high-pass filltered phase are plotted over the shaded surface. A variable contour interval is used. For magnitudes less than or equal to 2π. red and blue contours with an interval π/2 are used, for larger magnitudes green and Cyan correspond to contours with а 2π interval. Red and cyan are used for contours with negative values, and green and blue for those with positive values.
countours of the high-pass filter phase plotted over the shaded surface of the DEM for a subsection of frame 2169. The light source is directed dowmslope from top right to bottom left.
Positive contours correspond strongly with downhill sides of undulations, while negative contours correspond with uphill sides. This is exactly what would be expected if the short-scale phase variation were dominated by the effects of vertical displacement. On the uphill side of a bump, deviation from the average slope is positive, so that vertical motion is upwards causing the phase to decrease. On the downhill side, downward ice motion results in positive-phase variation. The changes from positive to negative phases (from blue to red contours in Figure 11) generally Follow the areas of dark-to-light transition. These transitions correspond to peaks and valleys where the slope magnitudes are at a minimum. This close agreement between the high-pass filtered phase and surface slope is strong support for the conclusion that for ice sheets, small-scale phase variation (i.e. bull’s-eyes) is dominated by the effect of vertical rather than horizontal motion.
Conclusions
Ice-sheet DEMs can be generated successfully from interferograms that have been differenced to cancel motion. Comparison of a laser-altimeter-measured topographic profile with data from the interferometrically derived DEMs yields relative errors as low as 2.55 m. Absolute elevation accuracy of about 4 m is achieved using radar-abimetry tie points. Thus, interferometric DEMs can provide elevation data with the accuracy needed for many glaciological investigations (a goal of 5 m is suggested by Reference ThomasThomas and others (1985)). It is likely that these accuracies can be improved with better processing and more accurate tie-point information.
Most of the topographie error in the interferometrically derived DEMs is caused by variations in the registration offsets in the azimuth direction, which show-up as streaks in the correlation and phase images. The cause of these streaks is unknown. Because streak error is a major limitation to DEM accuracy, further research is needed to determine its cause and to find ways to eliminate it. If streak error can be eliminated, it may be possible to achieve accuracy below 2 m. This is because the small surface slopes of the ice sheets allow extensive multilook averaging to reduce random-phase noise due to Speckle, which is often the limiting factor to interferometric DEM accuracy. The relatively small slopes of the ice sheets also mean that large slope-related errors (i.e. layover regions) that affect interferometric accuracy are not a factor.
There are unexplained low-frequency errors in ihe clouble-dillerence interferograms. These errors can he removed using a radar-altimetry-derived DEM. This is not an ideal solution, however, because any long-wavelength errors in the altimeiry -deriv ed DEM are propagated to the interferometrically derived DEM. More research is needed to determine the cause of this type of error. Then it may he possible to eliminate the error without the aid of an altimeiry-derived DEM.
Our results are important to the consideration of future interferometers. The accuracies reported here are for DEMs created from double-difference, repeat-pass interferograms. These inlerferograms are subject to far more noise than single-difference. single-pass inlerferograms. significaently better topographic accuracy could be achieved if a single-pass, spaceborne interferometer, such as TOPSAT (Reference Zenker, Farr, Salazar and DixonZebker and Others, 1994b). were built.
We have demonstrated that double-difference interferograms can be used to cancel motion even when the interferograms are acquired nearly 2 years apart. This is significant because it means that interferograms acquired during the- ERS-1/ERS-2 tandem mission can be used to create ice-sheet DEMs. Those interferograms will have temporal baselines of Only 1 d, so they should yield better results than the 3 and 6d interferograms used in this work.
Interferometry has been shown capable of providing much more detailed topographic data than radar altimeiry. Nevertheless, interferometry is not a Substitute for altimetry. Interferometiy depends on the points for absolute accuracy. In contrast, radar altimetry measures absolute elevation, but provides limited topographic detail. Interferometric DEMs may allow improvements in the accuracy of radar-altimeter estimates of elevation (personal communication from Reference Bindschadler and VornbergerR. A. Bindschadler, 1994). Undulations in the ice-sheet topography cause errors in radar-altimetry data. Interferometric measurements of the relative topography, which include these undulations, potentially could be used to remove surface-induced errors in radar altimetric measurements.
While other studies (Reference Goldstein, Englhardt, Frolich, Kamb and FrolichGoldstein and others, 1993; Reference Joughin, Winbrenner and FahnestockJoughin and others. 1995; Reference Rignot, Jezek and SohnRignot and other, 1995) have used satellite-radar interferometry Io examine ice-sheet motion, we have applied this technique for measuring the detailed topography of ice sheets. We have compared our high-resolution estimates of topography with motion-only interferograms to confirm an earlier analysis (Reference Joughin, Winbrenner and FahnestockJoughin and others, 1955) that short-scale phase variation is dominated by the effect of vertical displacement. It is important to note that a powerful aspect of interferometry is its ability to collect coincident motion and topography data. Combined high-resolution motion and topographic data should aid in lhe understanding of ice-sheet flow dynamics and help glaciologists to test and refine their models (personal communication from H. Gudmundsson, 1994).
Acknowledgements
I. Joughin and D. Winebrenner carried out this research under NASA grant NAGW-1526. M. Fahnestock was supported by NMTPE grant 5121-PP-0129. R. Kwok performed this work at the Jet Propulsion Laboratory, Califonia Institute of Technology, under contract with NASA. W. Krabill conducted this reasearch at NASA Goddard Space Flight Center, Wallops Flight Facility.
We thank S. Ekholm of the National Survey and Cadastre, Copenhagen, Denmark, for providing us with the DEM. We also thank the anonymous reviewers for their many helpful comments.
