In this study, we used ice sheet SE observations from four European Space Agency (ESA) satellite radar altimeter missions: ERS-1, ERS-2, Envisat, and CryoSat-2. Since the launch of ERS-1 in 1991, satellite radar altimeters have continuously collected SE observations for 30 years using similar Ku-band altimeters. Currently, following the retirement of the first three missions, only CryoSat-2 remains in operation.

For ERS-1 and ERS-2, we used version RP01 of the Level 2 (L2) Geophysical Data Record (GDR) product from the REAPER (REprocessing of Altimeter Products for ERS) project, which has been reprocessed to align both the ERS and Envisat datasets (Brockley et al., 2017). For Envisat, we downloaded the latest L2 GDR product, version 3.0, from ESA, which is better than the previous version (version 2.1) in terms of coverage and performance at crossovers. For CryoSat-2, we used the latest Baseline D L2 GDR data provided by ESA. Over land ice, Baseline D improves the ascending–descending crossover statistics to 0.1 m from 1.9 m achieved with the previous version, Baseline C (Meloni et al., 2020). Before performing combined calculations, all erroneous height records were eliminated using standard quality flags.

To bridge the gap in observations between the ICESat (Ice, Cloud and land Elevation Satellite) and ICESat-2 laser altimeter missions, the Operation IceBridge (OIB) project implemented more than 1000 airborne surveys during 2009–2020. During the OIB campaign, the airborne laser altimeter payload (i.e., the Airborne Topographic Mapper (ATM)) recorded a large number of high-precision ice sheet SE observations. Additionally, prior to OIB, several Pre-IceBridge airborne ATM surveys were conducted between 1993 and 2008. The OIB and Pre-IceBridge ATM SE datasets can both be download from the National Snow and Ice Data Center. Because their accuracy is 10 cm or better (Krabill et al., 2004), we used these ATM elevation measurements (i.e., ATM L2 product) to validate our merged SE time series. Additionally, SECs derived from OIB and Pre-IceBridge ATM measurements (i.e., ATM L4 product) were also used.

Our previous study (Zhang et al., 2020) demonstrated that using a large amount of data to fine-tune the correction of intermission bias and ascending–descending bias can develop a more self-consistent and reliable combined elevation time series. However, use of the updated plane-fitting least-squares regression model of Zhang et al. (2020) to merge data from three or more satellite missions is not straightforward. As the number of satellite altimetry missions involved in the calculation increases, additional terms of system bias and the increasingly complex topological relationships between them must be considered in the least-squares regression model, making the model overly complicated and ultimately incomprehensible. However, it is possible to divide the calculation into several individual steps, reducing the complexity of the model while retaining its advantages.

First, the intramission ascending–descending bias at a grid cell for each radar altimeter mission can be estimated as follows: 1hlongi,lati,ti=h0+a0longi-long0+a1lati-lat0+a2longi-long02+a3lati-lat02+a4longi-long0lati-lat0+a5ti-t0+a6cos⁡2πti-t0+a7sin⁡2πti-t0+a8bsi-bs‾+bAD-1AD+reslongi,lati,ti, where hlongi,lati,ti denotes the surface height measured at longitude (longi), latitude (lati), and time (ti), re-tracked by the ICE-1 re-tracker (Bamber, 1994) for ERS-1, ERS-2, and Envisat; the OCOG (Offset Centre of Gravity) re-tracker (Wingham et al., 1986) for the CryoSat-2 LRM (Low Resolution Mode); and the Wingham–Wallis model fit re-tracker (Wingham et al., 2006) for the CryoSat-2 SARIn (Synthetic Aperture Radar Interferometric). As a threshold re-tracker, ICE-1 and OCOG are less sensitive to fluctuations in penetration, thereby being more precise in terms of ice sheet elevation measurements (Nilsson et al., 2016; Schröder et al., 2017; Slater et al., 2019). The reference epoch t0 was set to 2010.0 in this study; long0, lat0, and h0 represent the longitude, latitude, and height (at t0), respectively, of the center of a grid cell; a0–a4 are the quadratic expansion for surface topography; a5–a7 denote the linear and seasonal signals for temporal changes of SE; a8 is a parameter to mitigate the time-variable penetration effects of the radar signal using the anomaly of backscattered power (bsi-bs‾); bAD is for the intramission ascending–descending bias; AD is assigned a value of 1 for ascending tracks or 0 for descending tracks; and reslongi,lati,ti denotes the residuals of the regression. Note that the SARIn and LRM observations of CryoSat-2 should be calculated separately here. The heights with the intramission ascending–descending bias corrected can be derived from 2hclongi,lati,ti=hlongi,lati,ti-bAD-1AD. Second, the intermission bias between Envisat and SARIn (or LRM) of CryoSat-2 can be calculated from the corrected heights hclongi,lati,ti: 3hclongi,lati,ti=h0+a0longi-long0+a1lati-lat0+a2longi-long02+a3lati-lat02+a4longi-long0lati-lat0+a5ti-t0+a6cos⁡2πti-t0+a7sin⁡2πti-t0+a8bsi-bs‾+bim-1im+reslongi,lati,ti, where bim is for the intermission bias; im is 1 for Envisat observations or 0 for CryoSat-2 observations. The correction of the intermission bias can be applied by 4hcclongi,lati,ti=hclongi,lati,ti-bim-1im.

The above implies that Envisat is taken as a reference, which means that the bias between SARIn and LRM will be corrected in this step. After the intermission bias is corrected, Envisat and CryoSat-2 data will be consistent, and subsequently ERS-2 and then ERS-1 data can also be corrected to be consistent with them.

Third, all the consistent heights hcclongi,lati,ti can be used in the final least-squares regression to construct the SE time series: 5hcclongi,lati,ti=h0+a0longi-long0+a1lati-lat0+a2longi-long02+a3lati-lat02+a4longi-long0lati-lat0+a5ti-t0+a6cos⁡2πti-t0+a7sin⁡2πti-t0+a8bsi-bs‾+reslongi,lati,ti.

The elevation anomaly can be derived as follows: 6Δhlongi,lati,ti=hcclongi,lati,ti-a0longi-long0+a1lati-lat0+a2longi-long02+a3lati-lat02+a4longi-long0lati-lat0+a5ti-t0+a6cos⁡2πti-t0+a7sin⁡2πti-t0+a8bsi-bs‾.

Note that the removal of h0 is to facilitate the study of EC and that h0 can be used to generate an independent digital elevation model (DEM). The DEM and the corresponding surface slope and azimuth are shown in Fig. 1. When necessary, h0 can be added back. Then, the monthly SE time series for a grid cell can be obtained as follows: 7Δh‾long0,lat0,tj=1n∑i=1nΔhlong0,lat0,ti, where n is the number of corrected elevations in month tj.

To generate a robust time series of 5 km gridded elevations, the above least-squares fitting is first performed on a 2 km polar-stereographic grid over the GrIS using the ice sheet mask in Zwally et al. (2012). For each grid node, all observations within 2.5 km of the center of the grid node are used for the iterative least-squares estimation under the constraints of 3σ outlier rejection criteria and the same thresholds as in Zhang et al. (2020). Then, a 40 km floating median low-pass filter, similar to Schröder et al. (2019), and the same spatiotemporal median filter as used by Zhang et al. (2020) are implemented to generate the final merged robust 5 km gridded time series.

Assuming that the spatial patterns of GrIS SECs are stationary in time, the three-dimensional GrIS SE anomaly time series ∇Hlong,lat,t can be represented as a linear combination of the EOF modes eofilong,lat and principal components pcit (Chambers et al., 2002; Jin et al., 2012): 8∇Hlong,lat,t=∑i=1Neofilong,latpcit, where N is the total number of EOF modes and long, lat, and t denote the temporal and spatial position of a certain SE anomaly. The purpose of solving EOF modes is to supplement the sparse monthly gridded data attributable to poor observations in the early years. The average proportion of the monthly grid cells that need interpolations to be filled is 24.9 % for ERS-1 and 7.4 % for ERS-2, which are much higher than 1.1 % for Envisat and 0.8 % for CryoSat-2. In particular, there are seven monthly grids with more than 63 % of cells that need to be interpolated during the ERS-1 period. Therefore, we use the gridded time series during 2003–2020 obtained from the higher-quality observations of Envisat and CryoSat-2 here. To mitigate errors caused by extrapolation, only grid cells with at least 100 elevation anomalies in the 216 months of the 2003–2020 period are retained. The missing values in the gridded time series during 2003–2020 are interpolated using ordinary kriging, a technique usually employed to generate a DEM (Bamber et al., 2009; Slater et al., 2018).

Then, for the monthly grid that needs interpolation, the following equation can be established: 9vt=∑i=1Meofilong,latPCit-Tlong,lat,t, where Tlong,lat,t denotes the values already in this monthly grid; M means choosing the first M EOF modes; vt is the interpolation (reconstruction) error; and PCit is the principal components to be estimated corresponding to each of the M modes for this monthly grid, which can be estimated to minimize vt using a linear least-squares estimator. To determine M, we experimented by adjusting it from 1 to 216 modes. We found that both the percentage of the explained variance and the root mean square (RMS) difference between the reconstructed time series of monthly EC and that of the observations become insensitive after 30 modes with only minor changes, as can be seen in Fig. 2. Thereby, the elevation anomalies missing from grid cells during the period of the ERS missions are interpolated using EOF reconstruction. Note that we first deduct the seasonal signals using a least-squares fitting model with a second-order polynomial and seasonal terms before performing the EOF reconstruction and then add them back to the EOF reconstruction results here. The performance of both EOF reconstruction and ordinary kriging is shown in Fig. 3, illustrating the superiority of the former in comparison with the latter. Especially in extrapolation, there are many obvious over-interpolations in the ordinary kriging result.

Volume change of an ice sheet is an important parameter for determining the response of the ice sheet to the effects of climate change. The altimetric volume time series can be derived from the gridded SE time series as described in Zhang et al. (2020). Firstly, the effects of glacial isostatic adjustment (GIA) and elastic solid-earth rebound should be corrected, for they do not reflect changes due to ice and snow. Then, the altimetric volume anomaly for each cell can be obtained by multiplying the corrected SE anomaly by the area of the cell. The altimetric volume anomalies for individual drainage basins and major sectors can be calculated by integrating the resulting altimetric volume anomalies over larger regions. In this study, the ICE-6G_D model (Peltier et al., 2018) and a scale factor (Groh et al., 2012) were used to correct for the vertical crustal deformation related to GIA and elastic solid-earth rebound.

As described in Sect. 2.3, the elevation anomaly in a grid cell of the merged SE time series is derived using a median estimator. Thus, to obtain a realistic estimation of error, we also use the scaled median absolute deviation MADS as a metric of its uncertainty following Ewert et al. (2012): 10MADS=kmedianH-median(H).

The scale factor k is set to 1.4826 to make MADS a consistent estimator similar to the standard deviation.

As for those interpolated elevation anomalies in the gridded time series, because of the complicated interpolation methods adopted, it is difficult to estimate their uncertainty using formal error propagation techniques. Here, we use the scaled RMS of the residuals ε derived from the elevation anomalies h in a grid cell as follows: 11h=b0+b1t+b2t2+b3cos⁡2πt-t0+b4sin⁡2πt-t0+ε, where b0 is a constant and b1–b4 denote the linear, quadratic, and seasonal signals of the temporal changes of SE, respectively. A scale factor of 1.05 is used to compensate for the reduced RMS error due to fitting (Wahr et al., 2006).

It should be noted that when using our elevation time series to estimate the volume change for individual drainage basins and major sectors, the correlated uncertainties between adjacent grid cells should also be considered. According to Schröder et al. (2019), applying a scaling factor to the squared uncertainties can account for the autocorrelation over an area.

The average rate of SEC in a certain time interval can be calculated from SE time series using a least-squares fitting model with a first-order polynomial and a sine wave with a 1-year period. The additional annual items are used to avoid the bias caused when the entire annual cycle is not fully covered. Figure 4 shows the climatological seasonal maps and the amplitude of annual cycle of SE anomaly over the GrIS. The spatial distribution patterns and magnitude of the seasonal changes in SE of the GrIS are clearly presented. The significant signals of seasonal variation are mainly concentrated in the ablation zone below the equilibrium line identified in McMillan et al. (2016). Thinning in autumn (July–August–September) and thickening in spring (January–February–March) are driven by the seasonal fluctuations in surface melting, snowfall, and ice dynamics (Bartholomew et al., 2011; Slater et al., 2021). Between May and August, surface melting and enhanced ice dynamics when the surface meltwater gains access to the ice–bed interface, lubricating basal motion, lower the surface in the ablation zone. Snowfall and slowing in ice dynamics in September–April thicken the ice sheet. No evident seasonal fluctuations are found in the elevation of the GrIS interior.

The average SEC rates and their uncertainties over the periods 1991–2000, 2001–2010, 2011–2020, and 1991–2020 from our merged elevation anomaly time series are shown in Fig. 5. As reported by Shepherd et al. (2020), the GrIS has been losing ice throughout most of the intervening period. Thus, maps of these long-term average SEC rates show signals of continuous thinning in many areas along the coast. Overall, the most notable signals of GrIS thinning are concentrated on the western coast of Greenland, especially along Melville Bay and near Jakobshavn Isbræ. Comparison of the average rates in the different periods reveals significantly accelerated and expanded thinning in many outlet glaciers, e.g., Jakobshavn Isbræ and Upernavik Isstrøm on the western coast of the GrIS, Zachariæ Isstrøm and Nioghalvfjerdsfjorden Glacier in the northeast of the GrIS, Kangerdlugssuaq Gletscher and Helheimgletscher in the southeast of the GrIS, and Petermann Gletscher and Humboldt Gletscher in the northwest of the GrIS. The main contributor to the significant thinning detected in these regions is ice dynamics. The volume of solid ice being discharged into the ocean is increasing because of warmer air and ocean temperatures (Mouginot et al., 2015; Aschwanden et al., 2016; Shepherd et al., 2020; Wood et al., 2021). Signs of thickening are evident mainly in accumulation areas with higher elevation in central and northwestern parts of the GrIS, e.g., the area near King Christian X Land. These weak signals of thickening mainly reflect the increase in SMB caused by a combination of high snowfall and low surface melting (Simonsen et al., 2021). These thinning and thickening spatial patterns are also confirmed from ICESat and ICESat-2 (Smith et al., 2020; Ewert et al., 2012).

Irrespective of whether thickening or thinning, it can be seen that the rates of SEC in different periods vary. To gain insight into the spatiotemporal changes of average SEC rates, the average SEC rates and their uncertainties at 5-year intervals during 1991–2020 for our time series are illustrated in Fig. 6. Considerable variation in the mean SEC rates is evident, e.g., abnormal accumulation during 1996–2000, gradually increasing loss from 2000–2005 to 2011–2015, and deceleration of thinning during 2015–2020. Benefitting from the higher temporal and spatial resolution of our combined time series, the small-scale spatiotemporal evolution of the average rates of SEC can be analyzed in detail. Taking Jakobshavn Isbræ as an example, Fig. 6 clearly reveals its evolution from thinning in the early 1990s, to equilibrium in the late 1990s, to accelerated thinning in the first decade of the 2000s, and then to decelerated thinning in recent years. Similarly, the evolution of other glaciers can be obtained from our time series. It should be noted that due to the natural defect of radar altimeter, our time series is not suitable for glaciers that are too small or too steep. Sørensen et al. (2018) has arbitrarily excluded all grid cells which are located on slopes exceeding 1.5∘ to avoid the possible large uncertainty.

Our 5 km gridded time series can also provide a more detailed evolution of SEC characteristics on temporal scales of up to 30 years. The four examples presented in Fig. 7 illustrate that our results have the capability to pinpoint such signals of GrIS SEC. Jakobshavn Isbræ is the largest and fastest outlet glacier on the western coast of Greenland; however, its thinning throughout the observational period since 1991 is not continuous. For example, short-term deceleration of thinning and thickening during 1996–2001 and since 2014, caused by the deceleration of the ice flow (Joughin et al., 2004; Khazendar et al., 2019), can be seen in Fig. 7a. A rapid drop in the surface elevation of Jakobshavn Isbræ is evident during 2003–2013. The rate of surface lowering increases with increasing distance from the grounding line. During this period, the mean rate at position A (Fig. 7a) is up to -2.85 ± 0.04 m yr-1. Upernavik Isstrøm consists of five glaciers, all of which flow into the same fjord. Zachariæ Isstrøm and Nioghalvfjerdsfjorden drain the majority of the Northeast Greenland Ice Stream. Unlike Jakobshavn Isbræ, their surfaces have lowered consistently since 1991 (Fig. 7b and c). The average rates of thinning at A, B, and C in Upernavik Isstrøm are -1.34 ± 0.03, -1.01 ± 0.01, and -0.60 ± 0.01 m yr-1, respectively. The thinning rates of Zachariæ Isstrøm and Nioghalvfjerdsfjorden are slower, i.e., mean SEC rates of -0.67 ± 0.02, -0.28 ± 0.01, and -0.21 ± 0.01 m yr-1 at A, B, and C, respectively. Similarly, the closer it is to the grounding line, the faster the rate of thinning of the ice of those glaciers is. King Christian X Land, located in the northeast of the GrIS, is a highly representative accumulation area. Ice velocity in this area is very small, and there is no outflow glacier, and its change is driven mainly by SMB (Aschwanden et al., 2016; Velicogna et al., 2014). It shows weak continuous thickening over the entire observational period since 1991 (Fig. 7d). The rate of thickening at A, B, and C is 0.13 ± 0.01, 0.11 ± 0.01, and 0.06 ± 0.01 m yr-1, respectively. Whether continuous thinning or thickening, the rates do not remain constant; i.e., there are always periods of acceleration or deceleration that are evident traces of the driving force of climate change on ice sheet change. For example, abnormal melting in 2012 and accumulation since 2016–2017, both driven by the North Atlantic Oscillation (NAO), are clearly visible in the time series of the above regions (Wood et al., 2021; Simonsen et al., 2021).

The volume time series of the entire GrIS and certain sub-regions estimated from our time series are shown in Figs. 7 and 8, respectively. Linear- and quadratic-trend estimates can be inferred from the volume time series using a least-squares fitting model with a second-order polynomial and a sine wave with a 1-year period.

Over the entire GrIS, we detect an overall volume loss of 53.8 ± 4.5 km3 yr-1 with an acceleration in loss of 2.2 ± 0.3 km3 yr-2 during 1991–2020 (Fig. 8). Six of eight ice drainage systems show volume loss (Fig. 9). The largest volume loss (19 ± 1.4 km3 yr-1) and greatest acceleration in loss (0.9 ± 0.1 km3 yr-2) are both from the ice sheet along the northwestern coast (Fig. 9h). Drainage systems located in central-western and southwestern parts of the GrIS are the other two largest contributors to ice loss with volumes and rates of acceleration of -10.2 ± 1.3 and -0.5 ± 0.1 km3 yr-2 (Fig. 9g) and -10.2 ± 1.9 and -0.6 ± 0.1 km3 yr-2 (Fig. 9f), respectively. The only two drainage systems to show volume accumulation are located in central-eastern (Fig. 9c) and northeastern (Fig. 9b) parts of the GrIS. However, their trends of volume accumulation are very weak, i.e., 1.3 ± 0.8 and 0.1 ± 0.1 and -1.6 ± 2.6 and 0.5 ± 0.2 km3 yr-2, respectively.

In addition to studying the long-term trend of the altimetric volume change of the ice sheet, our merged time series also provides detailed insight into small-scale fluctuations in volume change that reflect the effects of climate change on a temporal scale of up to 30 years. The evolution of ice sheet volume for the entire GrIS and certain sub-regions can be divided into different processes, as shown in Figs. 7 and 8, respectively. Before 1997, because of the contribution of various drainage systems in western Greenland (Fig. 9a and e–h), the GrIS presented rapid volume loss. Simonsen et al. (2021) thought that these ice losses were attributable mainly to the main outflow glaciers along the western coast. Then, the overall volume of the GrIS was alleviated, as also confirmed by the changes in the 5-year average SEC rates (see Fig. 7). This is attributed to an increase of the SMB in the northeastern and central-eastern drainage systems (Fig. 9b and c) and to a reduction of ice discharge of outlet glaciers along the western coast (Fig. 9g and h). Subsequently, the GrIS entered a period of rapid ice loss because of the reduced SMB that was mostly attributable to meltwater runoff and increased ice discharge (Fig. 9a and e–h) (Simonsen et al., 2021; Shepherd et al., 2020; Velicogna et al., 2014). Then, all drainage systems entered another period of slowdown in ice loss. In fact, these processes are full of the traces of the effects of climate change. The rapid ice loss since 2003 was driven by the transition of the NAO from a high positive phase to a low-to-negative phase, which reduced SMB by enhancing melting and reducing snowfall and accelerated ice discharge of outlet glaciers by driving warmer subsurface waters on the continental shelf (Bevis et al., 2019; Wood et al., 2021). The subsequent slowdown was because the NAO transitioned back to a more positive phase. It came from the anomalous increase in snowfall and anomalously low surface melting due to NAO-driven shifts in atmospheric forcing since 2016–2017 (Shepherd et al., 2020; Simonsen et al., 2021) and the slowed ice discharge attributable to NAO-driven shifts in oceanic forcing since 2010 (Wood et al., 2021). The weak signal of volume accumulation of drainage systems located in central-eastern and northeastern parts of the GrIS (Fig. 9b and c) was also attributed to the two short-term abnormally increased snowfalls driven by the shift of the NAO, one in the early 2000s (Shepherd et al., 2020) and the other in the late 2010s (Simonsen et al., 2021). The volume of accumulated low-density snow exceeded that of lost high-density ice.

To validate our merged results, we first used the high-precision ATM L2 surface heights. Before performing a comparison, a 40 km floating median low-pass filter was applied to the ATM L2 data to eliminate outliers. Moreover, the mean height (at t0) of the center of each grid cell h0 was first added back into our merged GrIS SE anomaly time series to match the surface heights. Then, we searched for all ATM L2 observations located within 2.5 km and a 10 d interval of each of the grid nodes of our three-dimensional time series. The result of subtracting the elevation value of a grid node from the median of the ATM L2 observations represents the difference for that location. The results of the validation are shown in Fig. 10a. It can be seen that the larger differences are concentrated primarily in steeper areas at the margins of the GrIS. This might be due to the poor observation accuracy of radar altimeters in areas of complex terrain (Zhang et al., 2020). Another possible reason is that many of them are interpolations or extrapolations. Over the GrIS, the median, RMS error, and the 10th and 90th percentile ranges (P90–P10) are -2.82, 99.43, and 145.25 m, respectively (Table 1), i.e., better than obtained for the 5 km interpolated grid cells of the DEM of Slater et al. (2018), which are comparable to our calculations in terms of strategy and resolution (their median and RMS error values were 25.4 and 138.6 m, respectively).

We also used ATM L4 SECs to evaluate our merged results. Similarly, a 40 km floating median low-pass filter was also used to eliminate outliers in the ATM L4 SECs before performing the validation. The ATM L4 SECs are derived from every two coincident ATM elevation measurements. We compared the ATM L4 data points with grids in our merged gridded time series that lay within a 2.5 km radius and a 15 d interval of the observation instants of that point. Subsequently, the SE differences and SEC differences between the ATM L4 observations and our merged time series at the same epochs were obtained, as shown in Fig. 10b and c. The spatial distribution patterns of SE differences and SEC differences are similar to those of the SE differences mentioned in Sect. 4.1. The larger differences are distributed in areas with complex terrain at the margins of the GrIS. The median, RMS error, and P90–P10 over the GrIS are also listed in Table 1. Overall, the median values of these difference are both near 0, and the two P90–P10 values are both relatively small. Thus, although the RMS error of the SE differences is larger than that of both Schröder et al. (2019) and Zhang et al. (2020) for the Antarctic Ice Sheet, our result is still considered reliable. Furthermore, the integrity of ATM L4 data covering only the outlet glaciers of the West Antarctic Ice Sheet and the Antarctic Peninsula Ice Sheet is limited.

Although a series of processes to ensure the accuracy and reliability of the merged results have been proven to be effective by comparison with other independent datasets above, there still exist some limitations in the merged SE time series. These limitations mainly come from the natural defects of radar altimeter.

The first is the penetration of the signal into the surface snow, which causes a radar altimeter not to observe the actual surface height of the ice sheet. Furthermore, surface processes such as melting, refreezing, and firn compaction might produce a new reflecting surface that could result in errors. For example, the abrupt increase in the CryoSat-2 recorded elevation in the interior of the GrIS during the extreme melt event in July 2012 resulted from the change of penetration depth caused by surface melting (Nilsson et al., 2015; McMillan et al., 2016). This study used elevations re-tracked by the threshold offset center of gravity re-tracker (ICE-1 re-tracker and OCOG re-tracker) and the common strategy of including corrections for waveform parameters into the least-squares regression model (see Eq. 1) to mitigate the time-variable penetration effects of the radar signal. Because it is less sensitive to changes in volume scattering, the threshold offset center of gravity re-tracker has been used to reduce the effect of penetration (Nilsson et al., 2015; Schröder et al., 2017; Schröder et al., 2019). The latter has also been performed in many previous studies (Flament and Remy, 2012; Sørensen et al., 2018; Zhang et al., 2020). However, as presented by Slater et al. (2019), the influence of the time-variable penetration depth would not be completely eliminated, even when applying a waveform deconvolution procedure (McMillan et al., 2016). Thereby, a small residual signal caused by the 2012 melt event and manifesting as a surface elevation increase signal is found in the merged time series. In regions above 2000 m in altitude, the elevation increased by approximately 0.16 m on average between the months before (January–June 2012) and after (August–December 2012) the extreme melt event, consistent with pervious findings (Slater et al., 2019). In the future, with the accumulation of long-term continuous observations by satellite laser altimetry of ICESat-2, it seems feasible to obtain actual penetration depth and model predictions to better compensate for the fluctuations in penetration depth. On the bright side, surface penetration suppresses noise induced by seasonal snowfall, making radar altimetric measurements more relevant to mass change than those obtained from laser altimetry (Sørensen et al., 2018). Therefore, our SE time series of multiple radar altimetry missions is more suited to track dynamical processes and interannual or long-term surface processes (Zhang et al., 2020; Simonsen et al., 2021).

Complex terrain and drastic changes in elevation could bring extra uncertainty in the merged time series. The beam-limited footprint of a radar altimeter with a radius up to kilometers makes it difficult for the radar altimeter to accurately measure ice surface height in those areas. Terrain undulations on the kilometer scale or smaller might make the biquadratic surface polynomial approximate the local ice surface topography inaccurately and thereby introduce errors into the correction for existing topography-induced height differences between the individual shots. Surface elevation observations from data products have been relocated by the point of closest approach were used in this study to suppress the influence related to the excessive size of footprint. The possible terrain correction errors caused by small-scale relief can only be expected to be suppressed by the mean estimator. Thus, the uncertainties of average SEC rates (Figs. 4 and 5) for marginal areas with complex terrain are larger than those for the central ice sheet. This is also reflected in the estimation of intermission bias and ascending–descending bias (Frappart et al., 2016; Zhang et al., 2020), although we have used large amounts of data to fine-tune them for each grid cell, which has been proven to ensure better self-consistency and reliability of the combined elevation time series (Zhang et al., 2020).

Additionally, interpolation or extrapolation of unobserved cells might also introduce uncertainty into the merged results, especially in steep and very active areas at the margins of the GrIS. The limited number of valid elevations, along with the greater uncertainty of several of them, would inevitably cause interpolation (extrapolation) error. This study used the EOF reconstruction method to reduce the error, which can incorporate more temporal and spatial information to constrain the interpolation results. However, some interpolation with large uncertainty still exists in some steep or narrow glaciers at the margins of the GrIS. The first three outliers of the volumetric time series shown in Fig. 9c are caused by this error. To avoid the large uncertainty caused by interpolation, Sørensen et al. (2018) arbitrarily excluded all grid cells located on slopes exceeding 1.5∘, and Schröder et al. (2019) excluded all data prior to 14 April 1992 from ERS-1, while we provided the merged non-interpolated time series in the dataset.

Overall, the above factors might cause errors to our time series, but it is difficult to formally account for them. Thus, according to previous studies, a straightforward estimate of uncertainty was given in this study as described in Sect. 2.5. It is an empirical estimation, and there may exist some underestimation due to the errors from above sources which are difficult to quantify.