Probably the most frequently used method for obtaining topography from remote-sensing data (either space or airborne) is stereo photogrammetry. This approach, however, using visible sensors such as SPOT, has a number of limitations over ice sheets (and to a lesser extent glaciers). First, there is rarely sufficient contrast to carry out stereo matching in the visible part of the electromagnetic (EM) spectrum. Second, cloud is ubiquitous in the polar regions and difficult to discriminate from snow. Third, to obtain absolute height measurements, ground control points are required, which are rarely available. As a consequence, other approaches have often proved more valuable. As mentioned earlier, microwave sensors are particularly useful in the polar regions owing to their all-weather, day/night functioning. The two instruments that have been used most successfully for deriving topography are radar altimeters and synthetic aperture radars (SARs).
Topography is perhaps the most fundamental observation for an ice mass. Land ice, over distances that are ca. 10-20 times the ice thickness, flows downhill, under the force of gravity. Accurate topography provides, therefore, information on both the magnitude and direction of flow. It also allows the identification of ice divides that separate flow in one basin (or glacier) from that in another. Surprisingly, until the launch of ERS-1 in 1991, the topography of Antarctica and Greenland was very poorly constrained, with uncertainties exceeding 200 m over much of the former (Bamber, 1994a).
A number of groups have used combinations of Seasat, Geosat and ERS SRA data to derive digital elevation models (DEMs) of both Antarctica and Greenland (Zwally et al., 1983; Remy et al., 1989; Bamber, 1994a; Ekholm, 1996; Liu et al., 1999; Bamber et al., 2001). The limit of coverage in Antarctica by satellite altimeter data was, until the launch of ICESat, 81.5°S. This 'hole at the pole' (covering an area of almost 3 million km2) is still problematic for numerical modelling and mass-balance studies of the ice sheet, as little reliable elevation data exist for this region. The GLAS data has recently provided coverage to 86° and CryoSat-2 (aimed for launch in 2009) should go a further 2° closer to the pole.
The SRA data from ERS-1 covers almost the entirety of Greenland barring the steeper margins of the ice sheet and the most northerly tip. To produce a full coverage DEM of the whole ice sheet and surrounding bedrock, SRA data from ERS-1 and Geosat were supplemented with airborne stereo-photogrammetry, airborne LIDAR (light detecting and ranging) data, InSAR-derived topography and, where no other adequate data were available, digitized cartography (Bamber et al., 2001). The result was a 1km posting DEM with a root mean square error over the ice sheet of between 2 and 14 m. Over the bedrock the accuracy ranged from 20 to 200m, dependent on the data source available. Figure 73.5 is a planimetric shaded relief plot of the DEM. Certain large-scale characteristics of flow are clearly visible. Major drainage basins and ice divides (where only vertical motion takes places) can be seen. The latter appear as bright 'ridges' ranging from relatively broad features for much of the northern half of the ice sheet to narrower and 'sharper' in character in the south. Much of the southern half of the large basin in the northeast has a broken appearance, reflecting a 'disturbed' pattern of flow in this area, most probably associated with a rougher bedrock topography and the existence of a fast-flow feature in the basin (Fahnestock et al., 1993; Layberry & Bamber, 2001).
As is evident from Fig. 73.5, the Greenland ice sheet is comprised of a number of drainage basins separated by ice divides. This is also the case for Antarctica. Each drainage basin can be treated, for the purpose of mass-balance studies, as a separate, independent flow unit. Owing to the size of the ice sheets, their
non-linear, integrative response to different forcing fields and their long response time, different basins may behave in entirely different ways at any one time. In southern Greenland, for example, one basin, east of the main divide, was observed to be losing mass, and the adjacent basin to the west of the divide was gaining mass by almost the same significant amount (Thomas et al., 2000b). This was a surprising result, with no clear explanation, and highlights the importance of a regional interpretation of mass balance signals. The key to this regional approach is the accurate delineation of ice divides. This has been undertaken for both Antarctica and Greenland using a GIS-based approach by combining slope and aspect, obtained from satellite-derived DEMs (Vaughan et al., 1999a; Hardy et al., 2000). The drainage basin masks were used to estimate the area of each basin and the volume of ice deposited as snow within each basin.
The InSAR has the capability of providing much higher resolution topography (ca. 25 m) than is possible with conventional
SRAs. Small-scale topography (roughly equal to the ice thickness) can provide information on flow features that are related to longitudinal stress gradients in the ice and basal topography. In fact, bed topography and lubrication have been inferred from surface topography and motion data using an inverse modelling approach (Thorsteinsson et al., 2004). As mentioned earlier, InSAR, in general, produces relative elevations only and the absolute height control must be provided from another source, such as a course resolution DEM derived from SRA data (Joughin et al., 2001).
Over an ice sheet, the gravitational driving force that makes the ice flow is a function of the surface slope and ice thickness. If the slope is estimated over an appropriate distance (typically ca. 20 times the ice thickness) it is reasonable to assume that the ice flows downhill (Paterson, 1994). It is therefore possible to trace particle paths downslope, from an ice divide to the coast. If the net mass balance (accumulation—ablation) is integrated along these flow lines, then the ice flux at any point can be estimated. The depth-averaged velocity at a point is simply this flux divided by the ice thickness. This estimate of velocity is what is required to keep the ice mass in steady-state. Hence the name: balance velocity. This quantity has been calculated for the grounded portions of the Antarctic and Greenland ice sheets using satellite-derived topography, combined with terrestrial measurements of accumulation, ablation and ice thickness (Joughin et al., 1997; Bamber et al., 2000a,b). The results provide the most spatially extensive depiction of flow over the entirety of the ice sheets. Although balance velocities/fluxes cannot be used on their own to determine mass balance, when combined with observations of the present-day velocity field from InSAR data, for example, they can highlight important changes in flow regime that may be related to changing state of balance (Bamber et al., 2000a; Bamber & Rignot, 2002).
Balance velocities provide a valuable qualitative picture of the pattern of flow but they are an estimated rather than observed quantity. There are two commonly used methods for measuring surface velocities directly. The first is known as feature tracking and uses the motion of 'features' from one or more images (usually from visible sensors) to determine velocity vectors (Scambos et al., 1992; Bindschadler et al., 1996). The main problem with this approach is that much of the interior of the ice sheets (and even some glaciers) is featureless and there is, therefore, nothing to track. In addition, errors are proportional to pixel size, the time interval separating image acquisitions and the actual velocity. As a consequence, for slow-moving ice (less than about 100myr-1) the errors can be unacceptably large.
The second approach, as mentioned earlier, uses SAR interfer-ometry. The advent of repeat-pass InSAR, after the launch of ERS-1, has resulted in a wealth of data on both ice sheet and glacier surface motion. InSAR works well over ice masses ranging in size from small valley glaciers to ice-sheet-basin scale, although there are a number of caveats. The method relies on maintaining coher ence between the two or more images used. This means that the scattering properties of the surface must not change significantly. If they do, due to surface melting, snowfall or strong winds, for example, it is difficult to produce an interferogram. This limits the use of InSAR largely to winter months for ice masses that experience melt during the summer. Given that even parts of the ice sheets appear to have seasonal variation in their velocity field (with higher values during the summer melt season) (Zwally et al., 2002a), winter-time data may not provide a representative estimate of the mean annual velocity or ice flux. Perhaps the most fundamental issue surrounding the use of InSAR is the relatively small archive of data and the dearth of planned future missions that might alleviate this. As a consequence using InSAR methods alone to investigate temporal changes in velocity is generally problematic and often impossible. In addition, interferometry only provides motion information in the look direction of the SAR and if this deviates from the flow direction of the ice by more than about 60° then it is unusable. If the ice is flowing too fast then a problem known as phase ambiguity can arise. This is where the relative displacement of a pixel with respect to a neighbour, in the time interval between the image acquisitions, is greater than one wavelength. This results in the fringes becoming disjointed and broken, as can be seen in the downstream portion of Ryder Glacier in Plate 74.2b of Joughin (this volume). A solution to the last two problems has been to apply the principles of feature tracking in visible imagery to SAR data (Gray et al., 1998, 2001). This approach is known variously as speckle or amplitude tracking and complements InSAR and visible feature tracking procedures (Joughin, 2002). It can be used, for example, in areas where no clearly identifiable 'features' exist, such as the interior of the ice sheets, and is well suited to fast flowing features such as ice streams. The method still requires coherence to be maintained between images and the relatively long orbital repeat period of satellites such as ENVISAT (35 days) can be problematic (shorter repeat periods, such as 3 days in the case of ERS-1, reduces the likelihood of changes to surface having taken place). It remains to be seen, therefore, whether the use of SAR data for land-ice motion studies will remain as fruitful as it has been over the past decade.
The techniques described above lend themselves to two distinct approaches to determining mass balance. The first, known as the integrative or geodetic approach, is used to infer a mass change from estimates of a volume change. The latter quantity is derived from measurements of dh/dt, often, but not exclusively, from either laser or radar altimeter data. Airborne laser data have been used, for example, to infer the mass balance of the whole of the Greenland ice sheet (Krabill et al., 1999,2000) and SRA data, covering a different and slightly longer time interval (1978-1988), were used for elevations above the equilibrium line (Davis et al., 2001). The ERS-1 radar altimeter data, obtained between 1991 and 1996, have been used to assess the mass balance of about two-thirds of the Antarctic ice sheet (Wingham et al., 1998). Although these results are of great value, they do have their problems. Perhaps the most profound of these is the relatively short time interval of the measurements, which, when combined with the relatively high natural variability in accumulation rates in Greenland, means that most of the dh/dt signal observed in the SRA data has been attributed to variations in accumulation rate rather than any long-term trend in mass balance (McConnell et al., 2000). Another problem is that variations in climatic conditions, principally precipitation, temperature and wind, affect (i) the ratio of surface to volume scattering of the radar pulse and, hence, the characteristics of the returned signal and (ii) the densification rate of the surface firn layer, which can be some 100-120 m deep. Elevation changes may, therefore, not be related to a change in mass but merely a change in density of the firn layer. With new SRA missions providing an extended time series, and the possibility of combining dh/dt data with gravity measurements from satellites such as GRACE (Wahr et al., 2000), it is hoped that the error in mass-balance determination from this approach will be considerably reduced in the near future (Velicogna & Wahr, 2002; Wahr et al, 2004).
The second approach to mass-balance determination is known as the component or flux divergence approach (Rignot & Thomas, 2002). Here, the flux of ice crossing some line or 'gate', perpendicular to the direction of flow, is compared with the accumulation of ice upstream of this line. A convenient location to determine ice flux, for glaciers with a floating tongue, is the grounding line, GL. The data required for this approach comprise: location of the GL, ice thickness and surface velocity at this location and net mass balance (the accumulation minus ablation, integrated over the upstream catchment area). The location of the GL has been successfully identified using InSAR techniques, which have also been used to derive the velocity field (Rignot, 1996). Ice thickness can be obtained either from in situ observations or assuming hydrostatic equilibrium for the floating ice and inverting elevation to give thickness, if the ice and seawater densities are known. Accumulation rates must, in general, be determined from sparse in situ data, possibly combined with PMR brightness temperatures (Vaughan et al., 1999a). At present, accumulation rates are one of the main error sources for the flux divergence approach. Nonetheless, the approach has been used on a number of outlet glaciers in Greenland and Antarctica, providing improved estimates of mass balance on an extensive, regional basis (Rignot et al., 1997; Joughin & Tulaczyk, 2002; Rignot, 2002b; Rignot & Thomas, 2002).
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