2.1. Data processing

The basic Seasat dataset is composed of waveform data from the Seasat SDR (sensor data record) from September to October 1978. ERS-1 waveform altimeter product (WAP) from the 35 day orbit phase delivered by the U.K. Processing and Archiving Facility is used. in order to ensure a maximum ERS coverage, we selected five ERS-1 35 day repeat cycles. The selected cycles (Nos. 83, 85, 87, 89, 97) from 14 April 1992 to 3 September 1993 are chosen with the ahi-meter working in ocean mode, to ensure a waveform resolution comparable to that for Seasat. Cycle 87 (14 August-18 September) is taken as the reference set to minimize potential seasonal effects.

Both datasets are reprocessed with the same algorithms, especially in terms of the retracking technique which consists of fitting each waveform to an error function (Reference Féménias, Rémy, Raizonville and MinsterFéménias and others, 1993). Comparison between both backscattering coefficients is also needed to correct for apparent change in height induced by change in snow-surface characteristics (Reference Ridley and BamberRidley and Bamber, 1995; Legrésy and Rémy, in press). Thus, in the same way, both backscattering coefficients are corrected for the shift of the waveform inside the received window (Rémy and others, 1990) with in the retracking algorithm. NASA Goddard Space Flight Center orbits are used for Seasat, and precise orbits from Delft University for ERS-1. in both cases, the heights are given with respect to the World Geodetic System 1984 ellipsoid.

It is noticeable that the estimated error after mapping the topography from altimetric data from a given satellite shows a strong correlation with the geographical sampling (Rémy and others, 1989; Reference Brisset and RémyBrisset and Rémy, 1996) because of the 10 km scale topographic signal that is due to the presence of undulations. The direct comparison of two independent maps derived from different satellite missions with different geographical samplings would then be com rolled by this important short-scale signal. We consider here the difference in height at the Seasat-ERS crossover locations (fig. 1a). We have about 1800 crossover points between Seasat data and each ERS-1 cycle, leading to a total of 9000 crossover points.

2.2. Statistical description of error term

In order to reduce the remaining error after reprocessing, we used a total inverse technique adapted to ice-sheet mapping (Rcmy and others, 1989). Each term of the error budget is taken into account by describing its statistics in terms of a covariance function and a decorrelation length. Knowledge of the statistics of the various components of the residual error helps separate the error from the signal. If is the difference between two quantities from track i of Seasat and track j of ERS, and if d is the signal we are looking for and h is the observed height difference, one can write:

where e₁ is the instrument noise, including retracking noise, assumed to be white with a 30 cm rms amplitude. The associated covariance is a Dirac-delta function for each instrument. e_tm, is the error due to atmospheric and ionospheric errors. The dominant tropospherie error, the dry atmospheric error, is not corrected for since it depends on the local ice-sheet height of the measurement and is the same for both missions. Because of the spatio-temporal sampling, two different Seasat ERS crossover points are not correlated; the covariance function is then assumed to be a Dirac-delta function. The variance of the residual error is set to (10 cm)².

e_orb is the residual orbit error after the interpolation of the precise orbit. The orbit error is dominated by frequencies of around 1 cycle per revolution mainly due to error in the gravity field and erroneous initial elements in the orbit. The arcs are very short and the orbit error for each one is nearly a bias; the orbit covariance error can thus be described by a cosine with an exponential damping (Chchón and Schlax, 1993). Because of its large geographical wavelength, the residual orbit error is probably one of the most critical errors. in order to ensure a good constraint on this error, we removed data for which either track i of Seasat or track j of ERS has less than three crossover points on the whole sector. Note that Seasat tracks cross the area from east to west and are thus quite long, while ERS tracks are oriented north—south and are thus quite short. The non-orbital differences of height between ERS ascending paths and descending paths are constant along each track and can be included with in the orbit error. The a priori variances for both orbits are thus (1 m)².

Note that the unresolved geographically correlated orbit errors (Reference SchramaSchrama, 1992) cannot be modeled and corrected for at the present time. To our knowledge, no study of this subject has been done over Antarctica with multiple satellites. This error is inherent in the problem of comparison between two satellites but should mostly be of long wavelength.

Fig. 1. Sequence of treatment of comparison between Seasat and ERS-1 data north of 72° S in Antarctica. The isoline represents the topography elevation from Brisset and Rémy (1996). (a) Distribution of Sensal FRS crossover points. (b) Inversion of residual change in height between both missions, (c) Estimated error, (d) Change in backscattering coefficient related to change in snow characteristics, (e) Change in height correctedfor change in snow characteristics.

e_slope is the slope-induced error. It is due to the shift of the impact point in the upslope direction, and linearly depends on the satellite altitude. Due to an approximately 20 km difference between the altitudes of the two satellites, this error must be corrected for. At each crossover point, we estimate both sin face slope and surface curvature from the precise ERS-1 topography (Brisset and Rémy, 1996) and apply a slope correction.

e_ret is the difference in retracking error depending on the antenna characteristics. Indeed, Legrésy and Rémy (1997) pointed out that the complex convolution between km-scale topographic features and subsurface signal leads to a nonlinear behavior of the waveform in front of subsurface signal intensity that depends on antenna gain characteristics. They showed that even when Seasat and ERS waveforms are re-tracked with the same algorithm, the resulting height error has a 10 cm rms km scale and a 5 cm bias. Because of its intrinsic complexity, this error cannot be corrected for but can be approximated using an appropriate covariance function.

Finally, e_snow is the error due to change in snow-surface characteristics. It is probably the most critical error; in order to control its effect, the correction is performed only after the data inversion process (see section 2.4).

2.3. Mapping through an inverse technique

The mean difference in height (ERS height minus Seasat height) for the 9000 crossover points is +9 cm. Even after careful checking for all sources of bias, this value must be used with caution (see section 3.1). The variance of the difference in height is 1.8 m². The output grid size is set to 35 km in order to smooth noise due to km-scale topography, and to retain a good geographical sampling. The change in height we are seeking is modeled with a Gaussian function with an a priori variance of (0.5 m) and a decorrelation-length scale of 50 km. After inversion of the covariance matrix, the map of residual difference in height is shown in Figure lb (the averaged value is removed), while the a posteriori error is shown in Figure lC. North of 70°S, except between 100 and 110 E, the error is quite large mostly because of the inadequate sampling, so the differences should be ignored for these areas. However, from 145 to 85°E and south of 70° S the differences should be significant; indeed the relative amplitude of the signal is around I in. which is more than twice the modeled estimate.

2.4. Change in snow-surface characteristics

For the final term of Equation (1), e_snow, the error due to change in snow-surface characteristics, is probably the most critical. Legrésy and Rémy (in press) show that a temporal change in surface microroughness yields an artificial change in altimetric height deduced from the waveform, because of the penetration of the radar wave into the snowpack. The induced error in the estimated height depends on the snow-pack characteristics and can be corrected for with the help of the difference between the two backscattering coefficients, if one assumes that the snowpack characteristics have not significantly changed since the time of Seasat. Figure Id displays the difference between Seasat and ERS back-scattering coefficients; it varies from 2 to 2 dB, and has for the most part a long wavelength signal of zero mean. The more the backscattering coefficient decreases, the more the surface echo is diminished and thus the lower the surface topography becomes. For the studied area, it yields a height correction of < 30 cm rms. Most of the features observed in figure 1b are still present on the corrected map (fig. 1e).

Comparison of Figure Id with the Seasat backscattcring-coefficient map (Rémy and others, 1990) suggests a decrease of katabatic wind intensity in the area of strong winds. No such trend is recorded in the meteorological data. The 3 months of Seasat data probably correspond to a period of strong katabatic wind. Note that the difficulty induced by this kind of error is not specific to the comparison between the two missions. It will probably set the limit in the interpretation of long-term surveys of ice-sheet topography, because of the difficulties in separating an actual volume change from an artificial change due to a long-term trend in snow characteristics.

3.1. Observations

Even if we are at the limit of precision required for proposing a quantitative interpretation, a qualitative interpretation can still be given over a 220 km x 2900 km belt located between 70° and 72° S, and between 150° and 80° E. Change in ice-sheet height can be due to one or several physical processes: accumulation rate enhancement due to climatic warming (Reference Ohmura, Wild and BengtssonOhmura and others. 1996), stochastic change in accumulation rate iReference OerlemansOerlemans, 1981; Reference Van der VeenVan der Veen, 1993), change in snow transport (Reference Takahashi, Naruse, Nakawo and MaeTakahashi and others. 1988), change in outlet ice-flow conditions (Reference Alley and WhillansAlley and Whillians, 1984; Rémy and Minster, 1997) or Holoccne warming response (Huybrccht and Ocrlcmans, 1988).

As already mentioned, we found an average bias between the two datasets of +9 cm. On a technical level, confidence in this value is very low because of the lack of calibration between both heights and backscattering coefficients. in any case, on a geophysical level, a tentative interpretation is also very poor. in fact, the great limitation of such a study is the short duration of the observations, which does not allow us to separate short- and long-term changes in surface elevation. Errors in the short term can be due to the stochastic fluctuations of accumulation rate Ocrlcmans, 1981), or induced firn densification that leads to differences between changes in elevation and changes in ice volume (Van der Veen, 1993). We will assume with Oerle-mans (1981) that these random fluctuations of accumulation conditions over Antarctica have a large spatial wav elength and thus hide or obscure any other global-scale signal.

The relative rise of inland areas with respect to areas of lower altitude is clearly observed (sec fig. 1e; 105-130° and 90-100° E). These regions exhibit a rise rate of up to 3.5 cm a⁻¹ for an average accumulation rate of 15 cm a⁻¹ (Ohmura and others, 1996), leading to a relative positive imbalance of 25% locally. This value is consistent with the results of Reference Budd and WarnerBudd and Warner (1996) who compared observed and computed fluxes. Conversely, a substantial decrease of the elevation of about 3 cm a⁻¹ occurs at lower altitude, for instance directly over the Aurora subglaciol basin (see 71°S, 110° E) or above the Astrolabe basin (70°S, 138° E). Note that one can hav e some confidence in the amplitude of the signal between rise and decrease, given the estimated error.

Our inverse method allows the estimation of the a posteriori error. The mapped error suggests that we are at the limit of the altimeter precision, but it should be remembered that the two previously discussed errors cannot be taken into account in the a posteriori estimation: the change in snow characteristics and the geographically correlated orbit error.We must therefore be very guarded in the following discussion and only suggest some hypotheses. At this time, we do not have corroborating in situ observations to support our hypotheses.

3.2. Some hypotheses

The observation of a relative rise of the high-altitude region with respect to the lower-altitude region goes against the idea of a large-scale signal due to a proportional and systematic increase in accumulation rate. However, the onlv area that has a different behavior, i.e. that exhibits a rise at a low altitude, of the order of 20-25% of the accumulation rate, is in the eastern part between 140 and 150 E and between 70 and 72°S. Indeed global climate models indicate that this area is the most sensitive to change and is subject to a rise of up to 5 cm a⁻¹ in the case of climate warming (Ohmura and others, 1996). On the other hand, field measurements in Wilkes Land suggest recent increases in snow accumulation at GD03 (69° S, 115.5° E) and GD15 (69° S, 130°E) (Morgan and others, 1991) and from D12 to D60 (north of 68°21'S around 138°Ε) (Reference Pourchet, Pinglot and LoriusPourchet and others, 1983). Both field measurements are at too low an altitude to be compared with our results, and furthermore no correlation can exist between the trends of a coastal region and a 200 km inland region (Reference Isaksson, Karlén, Gundestrup, Mayewski, Whitlow and TwicklerIsaksson and others, 1996).

Some areas where elevation decreases seem to coincide with areas of significant depression of the bedrock. Bedrock topography is displayed in Figure 2, where the superimposed arrows represent the along-slope integrated change in height. Negative values corresponding to surface lowering are represented by arrows directed outwards towards the coast; positive values are represented by arrows directed inland.

Fig. 2. Bedrock topography map (m) from Reference Drewry and DrewryDrewry (1983). Superimposed arrows represent intensity of along slope averaged in upslope direction. Arrows indicate direction of coast for negative values, and direction ofpole fur positive values.

One can imagine that the observed lowering signal is due to the Holocene warming that took place at the end of the last glaciation, about 16 000 years ago. But the effect of that warming propagating into the ice is likely to be confined to the upper part of the ice thickness and to have had minimal impact on the deeper basal layers of the ice sheet in this region which is generally above 2500 m (Drewry, 1983; Young and others, 1989).

Part of this signal may be the signature of the sea-level rise induced at the end of the last glaciation. Sea-level rise acts on outlet ice-flow conditions that are diffused upslope (Alley and Whillans, 1984; Rémy and Minster, 1997). This signal, which is exhibited 400 km inland from the coast, could have arisen from a perturbation on the profile at the terminus that happened a few millennia ago (Alley and Whillans, 1984) and whose effect depends of the drainage path. This may have had an influence near the area upslope of Law Dome (68°S, 112°E) and near the area upslope of Dumont d'Urville (67° S, 140° E), that only numerical modelling could verify.