We validate different altimeter missions to reveal characteristic effects of their respective techniques and approaches to derive optimum results. Conventional altimeter systems usually use signals in Ku-band and have a beam limited footprint of 10 to 20 km. Over ice sheets their signals penetrate into the upper firn layers. As an example for a conventional pulse-limited radar we validate the Envisat mission, operated by the European Space Agency (ESA). We use the Ku-band measurements of its altimeter system RA-2 acquired during the entire operation period (May 2002 to April 2012). In the Level 2 product (SGDR V2.1) the slope-induced error is corrected for using the relocation method . This algorithm is designed to locate the measurement to the position where the first return signal comes from. The ESA data set contains results from two types of radar waveform retrackers applicable over ice sheets, ICE1 based on the offset centre of gravity (OCOG) retracker by with a threshold of 30 % and ICE2 a functional fit developed by. We use both retrackers and compare their performance.

The Goddard Space Flight Center (GSFC) developed an in-house processing chain for radar altimetry with slightly different approaches for deriving surface elevations. Here, the direct method of slope correction was applied, which corrects the measurement at the nadir position. This reprocessed Level 2 data set is called the Ice Data Record (IDR) and also contains different retrackers. We use the GSFC V4 β-retracker as summarize that this algorithm provides more accurate absolute elevations than threshold-based methods.

To remove potentially corrupted observations from the data, we used the measurement confidence flags (which are identical in the ESA SGDR and GSFCs IDR data sets) to find recorded distances out of range and to identify problems in the onboard processing and data handling, the ultra stable oscillator, the automatic gain control (AGC) or in the waveform samples. In addition to these instrumental errors we removed shots where the GSFC retracking algorithm failed as indicated by the retracking problem flag. In the SGDR data we furthermore used the fault identifier and the flag indicating that the ICE1 retracking in Ku-band was not successful.

Compared to the conventional SRA, ESA's CryoSat-2 has an improved resolution and accuracy due to its innovative design. In the smooth interior of the ice sheets the altimeter operates in the low-resolution mode (LRM) which is a conventional pulse-limited observation mode as in the missions before. Above steeper terrain, the altimeter is switched to SARIn mode. In this mode, the synthetic aperture radar (SAR) processing considerably improves the along track resolution utilizing the Doppler/delay shift. Hence, the beam-limited footprint is subdivided in flight direction into stripes of roughly 250 m in length. The interferometric processing of the reception times at the two antennas allows the determination of the across-track angle to the point of closest approach .

We compare two different processing versions, Baseline B and C, of ESA's L2I data set. The “I” in the product identifier stands for the in-depth data set. It provides more parameters and flags and, over land, offers an additional feature relevant for our study. In the basic L2 product the SARIn ambiguity flag indicates an elevation difference between altimetry and a DEM exceeding 50 m. In this case the interferometric angle is considered erroneous and the measurement position is set to nadir. In the L2I product, however, this is not applied. This product allows us, therefore, to also validate the data at the margins where the a priori DEM itself is prone to large uncertainties (see Sect. ). As an alternative approach, to identify outliers in the interferometric angle, we used the coherence flag and additionally excluded all measurements with a across-track angle exceeding 1∘ (corresponding to the very edge of the antenna beam). Furthermore, we exclude all data where the respective retracker height error flag indicates problems in the determination of the retracking point. For Baseline B, the waveform is processed using the CFI retracker . In Baseline C two additional retrackers have been applied to the LRM data: a threshold-based OCOG-retracker and another functional fit retracker called UCL, which is based on the Brown-model .

In contrast to radar altimeters, the ICESat laser altimeter mission has a ground footprint of only 65 m and does not penetrate into the snowpack. Hence, surface elevation accuracies at the decimetre level can be achieved which are almost comparable to our kinematic GNSS profiles. We use GLA12 elevation data (Zwally et al., 2014) from Release 34 (R34). We apply the saturation correction to the elevations and exclude all data where flags indicate off-nadir operation, orbit manoeuvres or any other factors degrading the orbit accuracy. We also remove data where the attitude flag indicates any problem with star trackers, gyro or the laser reference sensor. In order to exclude data affected by forward scattering in clouds or drifting snow e.g., we reject all returns with a gain value exceeding 200, with a reflectivity below 10 %, with a misfit between the received waveform and the Gaussian model exceeding 0.03 V or where more than one waveform is detected .

We validate the surface elevation data derived from satellite altimetry by applying the crossover method, outlined in Sect. , to the intersections of the along-track altimetric profiles with our kinematic GNSS profiles. Therefore we interpolate the elevation of both profiles linearly if the distance between the two adjacent data points is less than 500 m. The along-track data point spacing amounts to 172 m for ICESat , 300 m for CryoSat-2 and 400 m for Envisat . In fact, the altimetry measurements represent average elevations over the respective footprint area. As we did not measure 2D-grids but straight lines, this cannot be taken into account here. The smoothing of the profiles with a filter length of 180 m, however, resembles the ICESat footprint at least in profile direction.

The largest error source for radar altimetry over a distinctive topography results from the slope correction. found that crossover differences between ICESat and Envisat are less than 3 m for slopes below 0.1∘, but up to 50 m and more for slopes above 0.7∘. Hence, the validation of SRA needs to consider different surface slopes. and binned their elevation differences with ICESat with respect to the slope. The obtained quasi-continuous functions clearly depict the growing differences with increasing surface slope. The amount and spatial coverage of our crossovers does not allow a comparison in such a high sampling of slope. Instead, we investigate regions of different characteristic slope in separate histograms. This allows us not only to calculate a mean and standard deviation for each zone but also to identify deviations from a Gaussian distribution. Those histograms display the full range of results including potential outliers. In order to reduce their impact, an iterative 5σ filter is applied in the calculation of the mean and standard deviation.

We subdivide the region under investigation into four zones according to their mean surface slopes: > 0.5, 0.5–0.15, < 0.15∘ and, as a subset of the latter characterized by extremely little surface roughness, the hydrostatic equilibrium area of subglacial Lake Vostok. The crossover differences between Envisat data and the GNSS profiles (Fig. ) clearly demonstrate the relationship between surface slope and SRA errors and motivates our subdivision. The first zone comprises the coastal areas. Outliers and large errors in the SRA elevations are frequent there due to the rugged topography. On the other hand, it is the zone where the largest elevation changes would be expected. Hence, this zone introduces the largest uncertainties in ice-mass balance estimates based on SRA . The subsequent zone of intermediate slopes is still close to the coast and of low elevations. It may therefore also be subject to significant elevation changes. At the same time, SRA provides a higher accuracy there compared to the first zone. The third zone comprises the flat interior of the ice sheet. Here, elevation changes are generally small but, because of its vast areal extent, nevertheless important for mass balance studies. The ice above Lake Vostok, constituting the fourth zone, floats in hydrostatic equilibrium . Surface gradients are very small and homogeneous in this area. Thus, the influence of the slope-induced error vanishes, offering a unique opportunity to study other effects such as the surface penetration of the radar signal.

Due to its smaller footprint size, the ICESat surface elevations are less sensitive to surface slope compared to SRA. However, the Geoscience Laser Altimeter System (GLAS) altimeter was operated in several laser operation campaigns due to laser degradation. Between the campaigns systematic biases exist . If not accounted for carefully, any systematic biases between campaigns can corrupt the inference of temporal surface elevation changes h˙. To determine those biases, different surface types have been used, including the salt flat Salar de Uyuni , the global oceans Urban in, the ice surface above Lake Vostok , the Antarctic low-precipitation zone or leads and polynyas in sea ice areas . The estimated biases differ significantly between different data releases and, within the same release, depending on the surface type used for calibration.

Within the region around subglacial Lake Vostok (Fig. b, 100–108.5∘ E, 76–79∘ S) applied a least squares adjustment of crossover differences between elevation profiles of ascending and descending ICESat (I) orbits, acquired during laser campaigns i and j (ΔhijI-I). To cope with the lack of an absolute reference, these authors introduced a zero-sum condition. As a consequence, the laser campaign biases were determined as relative biases, relating to their overall average. This method relies essentially on the assumption of a stable surface throughout the ICESat observation period. This assumption is justified by the observational results of .

Using the same region of ice-surface around Lake Vostok we derive a new set of laser campaign biases b for release 34. In addition to the ICESat crossovers (I) used by we also include crossover differences between ICESat and our kinematic GNSS profiles (ΔhiqI-K) and crossover differences between different GNSS profiles (ΔhpqK-K). Including the unbiased GNSS profiles allows us to solve for the surface elevation change rate h˙ between the respective observation epochs t as an additional parameter and thus to overcome the assumption of a stable surface. We are hence able to separate real elevation trends h˙ from the apparent trend hb˙ implied by the laser campaign biases. Furthermore, the incorporation of the unbiased GNSS elevations allows us to avoid the zero-sum condition and thus to determine absolute laser campaign biases. Combining all crossover differences results in three different types of observation equations: ΔhijI-I=bi-bj+h˙Δtij+ϵΔhiqI-K=bi+h˙Δtiq+ϵΔhpqK-K=+h˙Δtpq+ϵ. To account for the individual uncertainty of each GNSS profile, we introduce the epoch-wise surface elevation uncertainty RMSS as weights for the elevations. As shown in Sect.  the empirical intra-expedition crossover differences RMSX are about 5 cm larger. For this reason we add 5 cm to each RMSS. For the ICESat elevations, we use campaign-specific average RMSX‾ (Table ) obtained from intra-campaign crossovers in the Lake Vostok region.

The validation of the Envisat data (Fig. ) shows that in the flat interior all processing versions provide precise elevations. Nevertheless, the crossovers over Lake Vostok reveal significant differences between the three retracker versions. The standard deviations of the two functional fit retrackers yield similar results (51 cm for the β-retracker, 44 cm for ICE2). In contrast, the precision or the ICE1-retracked data is better by a factor of two (22 cm). This confirms the findings of , which argued for the superior precision of threshold retrackers. With respect to the kinematic GNSS profiles, the mean bias of all processing versions is negative. This can be explained by the penetration of the radar signal into the upper firn layers. However, significant differences between the retrackers are evident here. Compared to the ICE1-retracker, the mean reference surface of the ICE2 functional fit is 90 cm lower. Thus the influence of variations in firn pack properties is much stronger for this retracker. Between different data sets, a comparison of the biases of different retrackers should be treated with care. Here, elevation differences might also be caused by other factors as a different instrumental calibration value or alternative models for range correction.

Compared to Lake Vostok, where the slope effect is negligible, significant differences can already be observed in the zone of least slopes (< 0.15∘). Even the smooth topography there introduces additional uncertainties of about 30 cm for the relocated ESA data and 1.5 m for the GSFC data corrected by the direct method. Furthermore, the mean biases are shifted in the positive direction. The histograms reveal that this is a consequence of a deviation from the Gaussian distribution of the crossovers. The increased amount of positive differences means that the GNSS elevation is lower than the altimeter value. This is a consequence of the inability of the radar signal to observe depressions which are significantly smaller than the beam-limited footprint diameter . For all versions, this effect increases progressively as the slope and hence the magnitudes of the depressions also get larger. In the intermediate surface slope zone the standard deviations of the ESA data sets grow to 3 m and in the coastal zone they grow up to 10 m. Here, the differences between the two ESA retrackers become negligible. For the GSFC product these errors are significantly larger. Hence, our results support a clear preference for the relocation method. Nevertheless, in zones of larger slopes the error of the slope correction becomes the dominating uncertainty contribution.

To avoid loss of tracking, Envisat switched the tracking bandwidth from high resolution to two lower-resolution modes when approaching steeper terrain. We analysed the performance of modes separately. Table  shows the results of each mode in the different zones. The number of crossovers indicates that the majority of data, even close to the coasts, was acquired in high-resolution mode. In the central zone the accuracy of the lower-resolution data is, as expected, worse. In the coastal zone, however, the other modes yield better results (if their small number of crossovers is considered as representative). The variations of the mean biases demonstrate, in turn, that these mode switches induce offsets in the data. We agree therefore with not to use the sparse data of the lower-resolution modes for precise elevation change studies.

The results of the validation of CryoSat-2 elevations are shown in Fig. . A comparison with those from Envisat (Fig. ) clearly shows the advantage of the SARIn mode in zones of larger slopes. Furthermore, a comparison of the recent Baseline C version with the previous Baseline B documents the improvements made by solving several issues.

A primary issue solved from Baseline B to C was a range bias of 67 cm in SARIn and 20 cm in LRM data . In the SARIn data this bias reduction is clearly visible in all zones. Besides that, no major changes are evident and the standard deviations remain the same. Comparing the Baseline B LRM with the respective CFI-retracked version of Baseline C, we find a significant improvement in standard deviation. The refinements in the retracking procedure itself, described by , are probably responsible for those improvements. This holds true especially in the zones of stronger slopes where the retracking is more challenging. In the absence of slope-related effects on Lake Vostok, the correction of the range bias is also evident in the LRM data.

The main improvement in the performance of the Baseline C LRM data has been introduced by adding two additional retrackers (UCL and OCOG). The OCOG retracker shows standard deviations of about 20 cm over Lake Vostok, which is similar to the corresponding ICE1 retracker for Envisat. In contrast, the functional fit models show standard deviations of ∼ 50 cm which is similar to the results of the ICE2 retracker of Envisat. For the entire low-slope zone (< 0.15∘) we obtain similar results when comparing CryoSat-2 LRM to Envisat. In the intermediate zone (0.15–0.5∘ ) the two missions cannot be compared directly as the statistics for CryoSat-2 only relate to the subzone where the LRM is applied, which covers only the gently sloping areas. It should be noted that, even though SARIn mode is usually applied in coastal regions only, there are still some SARIn data available over Lake Vostok. On 28 July 2010 and the first week of June 2013 CryoSat-2 observed whole profiles across Antarctica in SARIn mode and also passed our region under investigation (including Lake Vostok) several times. These profiles allow us to directly compare the different modes and, in the case of Lake Vostok, the performance of their retrackers. The first column of Fig.  shows that the accuracy of SARIn is quite similar to the CFI retracker in LRM.

Different observational techniques have substantiated the stability of the surface elevation above Lake Vostok over timescales of typical satellite altimeter mission life times . This stability, together with the low precipitation and the continuous monitoring of relevant parameters at Vostok Station, makes Lake Vostok an ideal area to examine apparent elevation variations in the altimetric time series. Spurious variations can be related to changes in surface backscatter and thus the backscattered power σ0 of the altimetric signal . Commonly, the relationship is determined as a regression coefficient and its influence is removed from the elevation time series . Figure displays monthly averages of the crossover differences between the kinematic GNSS profiles and different CryoSat-2 LRM data sets. The Baseline B product (panel a) yields a high correlation as well as trends of the opposite sign in the elevations and the backscatter values. A trend derived from this elevation data set suggests a surface increase of 9.8 cm yr-1 which clearly contradicts all results from other studies. For the three retrackers of the Baseline C product (Fig. b–d) none of the backscatter curves show a significant trend any more. Nevertheless, the retracking methods based on functional fits (CFI in b, UCL in c) still exhibit a high correlation between Δh and σ0. Here, the retracking point is defined by the fit of the functional model to the whole waveform. Hence, it is more affected by volume scattering. In contrast the OCOG retracker uses a 25 % threshold of the OCOG amplitude and thus locates the retracking point much closer to the first radar return. The results in Fig.  indicate that threshold retrackers (panel d) produce the most precise elevations, especially in terms of repeatability. This confirms similar findings by . The seasonal variation of the signal almost vanishes. However, there is still a very small remaining amplitude, which correlates quite well with σ0. This indicates that there might still be some remaining effects of the snowpack properties superimposed on the elevation time series. Once the large variations disappeared, some jumps of a few decimetres are revealed. Apparent elevation jumps in two winters (2011/2012 and 2013/2014) correspond to abrupt backscatter increase at the same time. detected a similar jump in Envisat data and referred it to changes in snowpack properties due to strong wind. However, the meteorological records from Vostok Station (wind, precipitation, temperature; not shown here) do not show any significant peak at the times of the jumps. Inconsistencies in the applied correction models of the ionosphere, troposphere and tides can be ruled out as the origin of these jumps. None of the time series of these features show variations exceeding a few centimetres. Future studies including additional data sets will hopefully show whether these jumps are related to remaining processing issues or physical processes.

Prior to the validation of ICESat elevation data, we first determine the ICESat laser campaign biases as described in Sect. . The resulting biases are given in Table . The simultaneously derived surface elevation change rate h˙ from Eq. () amounts to 0.0 ± 0.2 cm yr-1. This is new, independent evidence for the stability of the surface elevation above Lake Vostok. It confirms our results in Sect.  and those of previous studies . It also justifies the assumption of a stable surface made by as a precondition for the campaign bias determination. It is, therefore, not surprising that our biases are very similar to the updated set of for R33 including the Gaussian-Centroid (G-C) correction, presented by . The major difference is an offset of about 5 cm. This arises from the fact that in this study, we perform an absolute calibration.

The chronological sequence of the laser campaign biases implies a trend h˙b which distorts any determination of surface elevation rates if the biases are not applied. This trend over the entire ICESat operational period amounts to 1.17 ± 0.34 cm yr-1. Table  and Fig. a compare the results of our new set of laser campaign biases with those of recent publications. For consistency, we limit this comparison to publications using either R33 data including G-C correction or R34. used an internal crossover adjustment over the low-precipitation zone of the East Antarctic ice sheet (EAIS) as well as absolute calibration using an ICEBridge lidar profile along the maximum latitudinal extent of the ICESat mission (86∘ S). To account for any possible elevation changes, the authors apply corrections for glacial isostatic adjustment (GIA) and firn densification. It is interesting to note that, in principle, both sets of their biases are very similar to our results but their artificial trends h˙b are 0.2–0.6 cm yr-1 larger than ours (Fig. a, column h˙b in Table ). This might be due to some unmodelled effects in the firn densification model and/or GIA correction of . Nevertheless, both of their sets of laser campaign biases (obtained in different regions of Antarctica and with different methods) agree with our results within their stated accuracies.

In contrast, the trends obtained by Urban in and from calibrations over the ocean differ significantly from the results of calibrations over Antarctica. calculated offsets from open water and thin ice in leads and polynyas in polar sea ice and used them to determine elevation changes over Antarctica. They obtain an elevation change rate h˙ for Lake Vostok of 2.02 cm yr-1. This contradicts the results of this study, but also those of two independent data sets in , i.e. static GNSS observations and kinematic GNSS profiles using snow mobiles compare. It is interesting to note, however, that the trends implied by our laser campaign biases and those of differ by 2.11 cm yr-1. This explains the discrepancies of the elevation change rates obtained by over Lake Vostok as a result of the applied set of laser campaign biases. The choice of these biases influences the derivation of elevation change rates from ICESat over the entire Antarctic ice sheet. Hence this also explains the disparity between their ICESat-derived mass budget and the mass-balance estimates of many other studies (e.g. Shepherd et al., 2012; McMillan et al., 2014; Martin-Espanol et al., 2016), especially in East Antarctica as documented in Scambos and Shuman (2016) and Richter et al. (2016). Urban et al. (2013) (as updated in Scambos and Shuman, 2016) obtained significantly smaller biases using the global ocean as a reference surface. The 2016 paper also discusses that due to different GLAS sensor saturation levels, those ocean-derived biases are not applicable over a high-albedo ice sheet surface.

After applying the laser campaign biases as corrections to the ICESat surface elevations, we calculate the crossover differences with respect to the kinematic GNSS profiles. The results in Fig. b show the very high accuracy of the ICESat data, even in the coastal zone. The crossover differences in the less sloping regions indicate that both data sets have practically the same precision (compare Table ). Here, the bias-corrections have only a minor influence on the standard deviations (e.g. 11.1 cm for the uncorrected, 10.7 cm for the corrected data over Lake Vostok) but become much more important when the temporal distribution of the data is analysed. Close to the coast we observe a small increase in standard deviation (approx. 30 cm for slopes exceeding 0.5∘). This might be an effect of the increased surface roughness which affects the interpolation of the elevation to the crossover point. A part of this increased noise could also be explained by topographic effects within the 65 m laser footprint.

To avoid the influence of surface elevation changes, previous studies as or validated only laser campaigns in very close temporal proximity (within some months) to their profiles. With a standard deviation of 10–20 cm in the majority of the area, we can confirm the results of . However, due to the different ICESat release versions used, their mean offsets are not comparable to our results. use measurements on snowmobiles near Greenland summit for a similar validation. Their spread of ICESat – GPS differences is significantly lower than in our study. However, their assessment is limited to a single ICESat repeat pass section of 6 km length. In contrast to those studies, our profiles cover a much longer temporal range. This allows us to pinpoint the maximum range of elevation changes h˙ in this area to very tight limits. Hence, we do not correct for elevation changes in this comparison. Especially in the coastal zone, this could explain the increased offset as well as the larger noise of the crossovers. The obtained offset and standard deviation, in turn, constrain the magnitude of possible elevation changes in these zones in addition to the results in Sect. .

A deeper view into the temporal variability of the crossover differences in the Lake Vostok region is given in Fig. c. To show the variability during the different GNSS seasons, we selected a single ICESat campaign (L3C, which has the highest precision RMSX; see Table ) as reference and analysed the spread of the crossovers over time. The monthly averages vary by less than 10 cm and, within their standard deviations, nearly all of them can be considered to be zero. The spread of the values is very likely a result of the remaining uncertainties of the antenna height reduction in the GNSS profiles. Interannual variations in accumulation σ<5 mm yr-1 or a water discharge from Lake Vostok can be ruled out as possible causes. Due to the long temporal base, those data sets nevertheless allow a precise trend h˙ of -0.4 ± 0.4 cm yr-1 to be derived. Together with the result of Sect.  and the estimate for the surface elevation change rate h˙, obtained from Eq. () in the laser campaign bias estimation, those are three very consistent results from different methods of obtaining h˙ from our GNSS profiles.