3.1. Synthetic aperture radar

To map ice-shelf flow we used SAR data acquired during the second ice phase of ERS-1 (Table 1). The SAR data were obtained in raw format and processed using the Gamma software package (Reference Werner, Wegmuller, Strozzi and WiesmannWerner and others, 2001). We used a 5 km Antarctic-wide DEM (Reference Bamber and BindschadlerBamber and Bindschadler, 1997) to remove the topographic component of the interferometric phase. To refine the ERS-1 interferometric geometry, we used point (60 m diameter) measurements of elevation from the Geoscience Laser Altimeter System (GLAS), on board the Ice, Cloud and land Elevation Satellite (ICESat) (Reference ZwallyZwally and others, 2002). These data were acquired during the period 2003-07. We used GLAS Level 1B elevation data (GLA06), which include corrections for atmospheric propagation delays and the effect of solid Earth tides (Reference BrennerBrenner and others, 2003). Data points with no saturation elevation correction or large receiver gain values (>50) were discarded. Saturation correction was added to the elevations. Geolocations in the GLA06 dataset were used without additional corrections.

Table 1. SAR data used to form interferometric solutions. e1 signifies ERS-1 satellite; B⊥ specifies the perpendicular baseline of the SAR image pair

3.2. Tide model

To model the tidal motion of the Dotson Ice Shelf, we used the finite-element solution model FES2004 (Lyard and others, 2004), which performs well in the Amundsen Sea (Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others, 2011). FES2004 is a global tide model, with 1/8° resolution, which utilizes sparse Antarctic tide-gauge data (<10 records), together with TOPEX/Poseidon and ERS altimetry. The model was used for two purposes: (1) to generate tidal predictions coincident with the acquisition of our SAR data and (2) to compute the distribution of tidal signals from a year-long, hourly-resolution model run. Tide heights were obtained from the model just seaward of the ice front, at 74.1° S, 247.5° E, as we believe they are more accurate at that location (Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others, 2011) than under the ice shelf itself, where water column thicknesses are poorly known. The modelled tide height during the period of SAR data acquisition is shown in Figure 2a.

Fig. 2. (a) Modelled tide height and (b) modelled surface level atmospheric pressure at the Dotson Ice Shelf during the period of SAR data acquisition. Tide heights were estimated at 74.1° S, 247.5° E using the FES2004 tide model (Reference Lyard, Lefevre, Letellier and FrancisLyard and others, 2006). Atmospheric pressure was estimated at 74° S, 247° E using the ERA-40 reanalysis (Reference UppalaUppala and others, 2005). Shaded areas indicate periods over which interferograms were formed.

3.3. Surface-level atmospheric pressure

In the absence of in situ meteorological data, model data from the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40 reanalysis (Reference UppalaUppala and others, 2005) were used to determine surface-level atmospheric pressure changes at the Dotson Ice Shelf. Data at 74 ^° S, 247° E were extracted from a 1° × 1° regularly spaced grid (derived from an N80 reduced Gaussian grid), which was acquired from the British Atmospheric Data Centre at 6 hourly temporal resolution. These model data were used in two ways: (1) to generate predictions of surface-level atmospheric pressure coincident with the acquisition of our SAR data (via a linear interpolation between the two closest 6 hourly model records) and (2) to compute the distribution of pressure signals from a year-long model record. The modelled atmospheric pressure during the period of SAR data acquisition is shown in Figure 2b.

4.1. Conventional InSAR stacking

The conventional InSAR processing techniques used to map ice motion are well documented (Reference Goldstein, Engelhardt, Kamb and FrolichGoldstein and others, 1993; Reference Joughin, Winebrenner and FahnestockJoughin and others, 1995, Reference Joughin, Kwok and Fahnestock1996; Reference Kwok and FahnestockKwok and Fahnestock, 1996; Reference RignotRignot, 1996), so here we provide only a short overview of the methods used in this study. Interferograms were formed from co-registered SAR image pairs, with 3 days separating each acquisition (Table 1). This acquisition configuration was chosen to (1) minimize temporal decorrelation by keeping the interferometric temporal baseline short and (2) assess the method with respect to a simple and regular acquisition cycle. The sensitivity of our method to the temporal sampling provided by this acquisition cycle is examined below in Section 6.

For repeat-pass SAR observations of an ice shelf, the interferometric phase, v, is due to a combination of terms:

(1)

This expression describes the spatial variation in phase, relative to a spatially constant reference or phase offset, φ_ref. Phase variations across the image are caused by (1) the changing viewing angle across the ground track, φ_flat (as described by the shape of Earth’s ellipsoid), (2) surface topography, φ_topo, (3) surface displacement due to ice flow, φ_flow, tidal forcing, φ_tide, and the IBE, φ_ibe, and (4) noise in the received signal, φ_noise. The noise term encompasses both spatially correlated (e.g. atmospheric phase distortions) and uncorrelated (e.g. signal decorrelation) effects.

Firstly, we used the interferometric geometry to simulate and remove the ‘flat Earth’ signal. Where possible we used ICESat surface elevations over non-moving terrain to further refine our interferometric baseline estimate, and thus improve our model of this component of the signal. Where this was not possible, due to a lack of interferometric coherence over stationary regions, precise orbit information acquired from the Technical University of Delft, the Netherlands, was used. Loss of coherence over the period of interferogram acquisition was probably due to wind- or precipitation-driven changes to the near-surface snowpack.

Next, we unwrapped each interferogram. Ignoring the noise term (see Section 5.2) and the constant phase offset (which we address at the end of this section), the remaining variation in the interferometric phase comprises contributions from (1) topography and (2) the line-of-sight component of the various modes of surface displacement:

(2)

where λ is the radar wavelength (5.7 cm for the ERS-1 used in this study), B± is the component of the interferometric baseline perpendicular to the radar line of sight, z is the elevation of the target pixel above Earth’s ellipsoid, r is the range from the satellite to the target pixel, θ is the radar look angle and ψ is the incidence angle of the radar beam relative to the normal to Earth’s ellipsoid. Over the time period, Δt days, of the interferometric observation, Δh _flow denotes the horizontal ground range component of surface displacement due to ice flow, and Δz _tide and Δz _ibe denote the vertical displacement of the ice shelf in response to the tide and IBE, respectively. Stacking n interferograms yields a stacked phase:

(3)

To isolate the surface displacement component, elevations from a DEM (Reference Bamber and BindschadlerBamber and Bindschadler, 1997) were scaled by the effective perpendicular baseline of the stacked interferogram, and used to simulate and remove the topographic phase. Dividing the remaining stacked phase through by the total observation period of the stack yields an estimate of the ice flow velocity during that period (md~¹), subject to error from residual tidal and IBE signals:

(4)

This expression forms the basis of this work, whereby we use the stacked phase to estimate horizontal ice flow, and model statistics to determine the magnitude of the associated tidal and IBE errors.

The method so far only determines relative displacement, i.e. how displacement varies across the image space. To determine absolute displacement values (by estimating the reference phase offset), we referenced our displacement map to a set of ~2700 displacement estimates acquired where ice was grounded. These reference displacements were determined using the technique of coherence tracking (mapping surface motion based upon optimizing coherence between patches of SAR image pairs; Reference Derauw and Saway-LacosteDerauw, 1999; Reference Pattyn and DerauwPattyn and Derauw, 2002; Reference Strozzi, Luckman, Murray, Wegmuller and WernerStrozzi and others, 2002). Specifically, we first identified grounded areas by differencing two interferograms (Reference Hartl, Thiel, Wu, Doake and SieversHartl and others, 1994; Reference RignotRignot, 1996, Reference Rignot2002). This removed the ice flow component of the phase signal common to both interferograms and isolated the non-steady motion associated with the ice-shelf response to tidal and atmospheric pressure forcing. Next, we adjusted our InSAR velocities so that the mean InSAR-derived velocity (over regions identified as being grounded) matched the equivalent velocity determined using coherence tracking. This approach allowed us to utilize a spatially extensive dataset to reference our InSAR-derived velocities.

4.2. Modelling tide and IBE error

Our stacked InSAR velocity solution does not utilize model predictions of tide or IBE to isolate ice-shelf flow. Instead we have aimed to minimize these signals by stacking displacement estimates. We now use the tide and IBE models to simulate the statistics of the residual tide and IBE signals, in order to quantify the associated error in our stacked predictions of flow (Eqn (4)). We formed time series from both the FES2004 tide model and the ERA-40 atmospheric pressure reanalysis for the entirety of 1994, at hourly and 6hourly intervals, respectively. We converted atmospheric pressure changes into changes in ice-shelf height (IBE displacement) using an inverse barometer approximation, namely the ratio determined empirically by Reference Padman, King, Goring, Corr and ColemanPadman and others (2003) of —0.95 cm hPa~¹. Reference Padman, King, Goring, Corr and ColemanPadman and others (2003) found little variation in estimates of this ratio derived from data collected at three widely spaced and different-sized ice shelves, so we assume this estimate is valid at the Dotson Ice Shelf. Next, we calculated the tidal and IBE displacements that would occur in a stacked interferometric prediction acquired at every point along our time series. We converted these modelled vertical displacements into annual velocities in the satellite’s across-track direction and so determined the distribution of velocity errors, arising from residual tidal and IBE signals, that could be present in our stacked prediction of flow.

4.3. Isolating flow displacement using model predictions

To assess our stacked velocity solution we used the same InSAR data (Table 1) to form alternative displacement maps, using the standard technique of removing the tidal signal from each individual interferogram, based upon the coincident tide model predictions (Reference RignotRignot and Jacobs, 2002; Reference Joughin and PadmanJoughin and others, 2003; Reference Rignot, Casassa, Gogineni, Krabill, Rivera and ThomasRignot and others, 2004; Reference Vieli, Payne, Du and ShepherdVieli and others, 2006). We have extended this method to additionally remove the modelled IBE component of ice-shelf motion. To isolate flow, we followed the interferometric method described above to convert individual interferograms into maps of absolute across-track displacement. Next, we used a double difference technique (Reference Hartl, Thiel, Wu, Doake and SieversHartl and others, 1994; Reference RignotRignot, 1996, Reference Rignot2002) to determine the pattern of non-steady (tide and IBE) displacement of the Dotson Ice Shelf (Fig. 1). To simulate the tidal and IBE signals occurring within each interferogram, we scaled the double difference solution so that the mean displacement across the freely floating portion of the ice shelf matched that of the combined modelled tide and IBE displacement. This prediction was subtracted from each conventional InSAR displacement map, in order to produce multiple predictions of ice flow.

4.3. Multiple aperture InSAR stacking

Conventional InSAR only measures one dimension of the displacement field: along the satellite’s line of sight. To determine 2-D vectors we estimated displacement along the satellite track using multiple aperture InSAR (MAI) processing (Reference Bechor and ZebkerBechor and Zebker, 2006; Reference Jung, Won and KimJung and others, 2009; Reference Gourmelen, Kim, Shepherd, Park, Sundal and BjornssonGourmelen and others, 2011). Previous work has shown that MAI offers improvements in estimates of along-track displacement, compared with tracking methods, both for solid Earth (Reference Bechor and ZebkerBechor and Zebker, 2006) and glaciological (Reference Gourmelen, Kim, Shepherd, Park, Sundal and BjornssonGourmelen and others, 2011) applications. MAI processing splits the antenna beam to form two sub-aperture single-look complex (SLC) images, oneforward- looking and one backward-looking, from each conventional SAR image. Then, from a pair of SAR images, the forwardlooking SLCs and the backward-looking SLCs are each combined to form two interferograms which are sensitive to displacement in both the azimuth and range directions. Differencing the interferometric phase of these forward- and backward-looking interferograms cancels the common range component of the interferometric phase and isolates a phase difference that is proportional to the along-track component of the displacement field. Because MAI is less sensitive to surface displacement than conventional InSAR (a 2n phase change corresponds to a 10m change in displacement for MAI versus 2.8cm for InSAR), phase gradients are lower, allowing the phase over moving and stationary areas to be linked. These stationary areas provide an absolute reference for our displacement estimates. Although MAI is insensitive to vertical ice-shelf motion, we still stack MAI interferograms, so as to amplify the steady ice flow signal, relative to temporally uncorrelated noise sources. Finally we combine the along-track velocity component from MAI with the across-track velocity component from conventional InSAR, in order to determine a map of velocity magnitude.

5.1. Stacked velocity map

We used the stacked multiple aperture and conventional InSAR solutions to determine the along-track (Fig. 3a) and across-track (Fig. 3b) components of our flow velocity vectors. We then combined these components to give a map of the magnitude of the 2-D flow velocity vectors (Fig. 3c). We assumed ice flow was in the horizontal plane, i.e. that the vertical component of flow was negligible. Our solution shows that the Dotson Ice Shelf is primarily fed by fastflowing ice originating from Smith and Kohler Glaciers. Close to the grounding line, where the ice funnels through a narrow gap, velocities exceed 500 ma^-1. Further downstream, on the freely floating portion of the ice shelf, speeds drop to ~320ma^-1, before increasing again to ~500ma^-1 close to the calving front.

Fig. 3. Flow velocity of the Dotson Ice Shelf. (a) Along-track velocity component derived from stacked MAI; white arrow indicates the satellite along-track direction. (b) Across-track velocity component derived from stacked InSAR; white arrow indicates the satellite acrosstrack direction. (c) Velocity magnitude from combined along-track (a) and across-track (b) components. The background image is taken from the MODIS Mosaic of Antarctica (Reference Haran, Bohlander, Scambos, Painter and FahnestockHaran and others, 2006).

5.2. Error assessment

To assess the viability of our stacking method, we considered the error associated with our stacked velocity solution. We define the error, ?, associated with interferometric estimates of horizontal ice-shelf flow as

(5)

The error components refer, in left-to-right order, to errors from residual tidal displacement, residual IBE displacement, unmodelled topography, unmodelled baseline effects, atmospheric phase distortions, loss of coherence, error in determining an absolute reference and error in the phase-unwrapping process. To simplify this assessment, we assumed that all errors were independent. This provides a close approximation of the true error, since the dominant error components (namely tide and IBE, as outlined in this section) are independent. The extent to which these error sources affect the across- and along-track components of our displacement field varies as a consequence of the different techniques used. We address the error associated with each component in turn.

Conventional InSAR error

Firstly we considered the effect of each error component (Eqn (5)) on our conventional InSAR estimates of across- track displacement. To assess the tide and IBE errors (ε_tide and ε_ibe) we used 1 year’s worth of model data to estimate the distribution of these signals occurring over the timescales of our interferometric acquisitions, and converted these to equivalent errors in the across-track component of our velocity solution (Fig. 4). In each case we computed the signals corresponding to (1) a single interferogram (i.e. the 3 day difference in tide height), (2) two- stacked interferograms and (3) three-stacked interferograms. Both the tide and IBE contributions to the velocity error tend to diminish in size as the interferometric predictions are stacked, primarily as a consequence of the longer observation period over which displacement is measured. The distributions of the stacked tidal and IBE signals are roughly normal and centred close to zero. In contrast, the twin-peaked distribution of the single-interferogram tidal displacement indicates that, in a single interferogram, a much larger signal is likely to be present, and demonstrates the value of stacking to reduce the tidal contribution. To determine the contribution of these sources of error to our three-stack velocity estimate, we combined the standard deviations of the modelled tidal (20 cm vertical displacement over a 9 day period) and IBE (18cm vertical displacement over a 9 day period) signals. This yielded a combined residual vertical motion of 27cm, which equates to an error of 22ma^-1 in the across-track component of our stacked velocity solution.

Fig. 4. Modelled distribution of (a) tidal and (b) IBE contributions to conventional InSAR estimates of across-track flow velocity at the Dotson Ice Shelf. Each panel shows the expected distribution of across-track velocity errors arising from the tidal and IBE motion of the ice shelf within a single interferogram (3 day separation), and for two- and three-stacked interferograms. Tide was computed from hourly realizations of the FES2004 tide model, and the IBE from 6 hourly realizations of the ERA-40 reanalysis of surface-level atmospheric pressure, converted into changes in ice-shelf height using the empirical relationship determined by Reference Padman, King, Goring, Corr and ColemanPadman and others (2003). Both models were run for the entirety of 1994, and the resulting vertical displacements were converted into equivalent annual velocities in the satellite’s across-track direction.

To determine an absolute reference for our InSAR displacements, we matched the mean InSAR and mean tracking range displacements over grounded ice. Consequently, any bias in the tracking displacements is transferred to our InSAR solution. We estimated this referencing error, ɛ_ref, from the displacement values of pixels that were located on stationary areas. Over the 3 day period of our tracking measurements, the mean across-track displacement of these points was 5.9 cm, equivalent to a bias of 7ma^-1 in our velocity solution.

The effect of the remaining error terms upon our velocity solution is likely to be small in comparison to the aforementioned errors. Because of the minimal topographic relief in our study area, DEM errors are small (~5 m; Reference Bamber and Gomez-DansBamber and Gomez-Dans, 2005) relative to the 70 m altitude of ambiguity (the elevation change equivalent to one complete cycle of the interferometric phase) of our stacked interferogram. This topographic error, •_t0p₀, equates to a 0.9 cm across- track displacement error in our 9 day stacked interferogram, equivalent to a 0.4 m a^-1 error in our velocity estimate.

As a measure of the displacement error arising from loss of coherence, ɛ_coh, we calculated the errors associated with the average coherence of each interferogram (Reference Rodriguez and MartinRodriguez and Martin, 1992). The resulting displacement error in our 9 day stacked interferogram is 0.4 cm, equivalent to an error in the across-track component of our velocity solution of <0.2ma^-1. Atmospheric distortions of the interferometric phase, ɛ_atmos, occur because of spatial and temporal inhomogeneity in the troposphere and ionosphere. To quantify the tropospheric component, we followed the procedure outlined by Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others (2011). Based upon the length- and timescales appropriate to our study and the parameterization determined by Reference Emardson, Simons and WebbEmardson and others (2003), we estimated the error arising from the tropospheric variability to be at most 4.1cm in our 9 day stacked interferogram, equivalent to a 1.7 m a^-1 error in our velocity solution. We neglected ionospheric errors (Reference Gray, Mattar and SofkoGray and others, 2000), as these tend to manifest themselves as distinctive features and we do not see any evidence of these features in our dataset. We have aimed to minimize any unmodelled baseline effects by, where possible, refining our baseline estimates and checking for residual long-wavelength phase gradients over stationary areas. It is possible that some small residual baseline error may exist in our velocity solution, which we have not accounted for in our error budget. The process of stacking will, however, further reduce any such effect. During the phase-unwrapping process we unwrapped along a path that gave no visible unwrapping errors, so we assume no unwrapping errors in our velocity solution. The contributions from errors associated with our across- track displacement estimates are summarized in Table 2. Of these, the errors arising from atmospheric distortions and loss of coherence are specific to the C-band frequency of the ERS SAR and will vary for sensors operating at alternative frequencies. Combining all across-track error sources yields a 23 m a^-1 error in this component of our velocity solution. This error component is dominated by the errors from tidal and IBE effects.

Table 2. Summary of terms contributing to displacement error in the three-stack (9 day) InSAR estimate of across-track displacement

Velocity magnitude error

Commonly in studies of ice flow, it is estimates of velocity magnitude (Fig. 3c) that are of most interest. The simplest approach to determining the error associated with the velocity magnitude is to sum in quadrature the along- track and across-track error components, which yields an error of 31 ma^-1 in our velocity magnitude solution. This approach assumes independence of along-track and across-track errors. The MAI and InSAR observations are both derived from the same SAR data, so the coherence and referencing errors affecting both velocity components will not necessarily be independent. However, because these errors form only a tiny fraction of the total InSAR error (Table 2), we expect any increase in the total error due to the non-independence of these components to be small. The extent to which the error vector (defined by the along- and across-track error components) affects the velocity magnitude will vary according to the orientation and magnitude of the velocity vector relative to the error vector. For a 2-D flow velocity vector, v, with an associated error vector, e, the change (i.e. error) in the velocity magnitude, ɛ| _V |, resulting from the vector addition of e to v, can be described using the cosine rule:

(6)

If the vectors v and e are placed head to tail, then θ is the inner angle formed where they meet, such that when θ = 180° the vectors v and e point in the same direction, and θ = 0° indicates that v and e are orientated in opposite directions. The error in velocity magnitude peaks at 31 m a^-1 when the velocity and error vectors are orientated in the same, or exactly opposite, direction. The quadrature sum of our along- and across-track error components therefore represents an upper bound upon the error associated with our velocity magnitude estimate. In more favourable cases the error in velocity magnitude will be significantly smaller, tending to zero when the velocity and error vectors are close to orthogonal. Consequently, a simple quadrature sum of the error components will often overestimate the true velocity magnitude error.

6.1. Comparison of velocity predictions

Residual tidal and IBE displacements contribute the greatest error to our across-track velocity solution. To assess the impact of these two error sources upon our across-track velocity estimates, we compared our stacked map of across- track displacement to solutions obtained by a commonly used method (e.g. Reference RignotRignot and Jacobs, 2002; Reference Joughin and PadmanJoughin and others, 2003; Reference Rignot, Casassa, Gogineni, Krabill, Rivera and ThomasRignot and others, 2004; Reference Vieli, Payne, Du and ShepherdVieli and others, 2006), whereby model predictions are used to remove the tidal and IBE signals. We formed three velocity estimates by removing modelled tidal and IBE signals from each of the individual interferograms used in our stacking solution. Figure 5 shows a comparison of these solutions extracted from a transect along the primary line of ice flow (transect location shown in Fig. 1). We have plotted only the across- track component of velocity, as only this component is sensitive to vertical motion. Where ice is grounded (indicated by P in Figs 1 and 5) all velocity estimates converge. Because the InSAR solutions are tied down using all grounded locations (red in Fig. 1), and not just at point P, this convergence confirms that errors arising from unmodelled topography and baseline effects are small, and that flow remains steady over the period of data acquisition. Where the ice is freely floating (greater than ~10 km along the transect), each velocity estimate exhibits a roughly constant offset, which we interpret as resulting from vertical displacement due to the combined effect of the tide and IBE. The fact that this offset displays little spatial variation over the floating ice shelf indicates that our velocity estimates are not greatly affected by spatial variability in the tidal and IBE signals over the length scales of the ice shelf. This is consistent with the modelled IBE’s spatial variability over the ice shelf at the times of data acquisition, which produces on average 3 ma~¹ variation in the velocity signal. We do not perform an equivalent assessment for the tidal signal because the accuracy of the FES2004 model under the ice shelf is uncertain (Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others, 2011). Assuming that these spatial patterns are uncorrelated in time, then stacking will further diminish the effects of any such spatial variability.

Fig. 5. Across-track component of the Dotson Ice Shelf flow speed (transect location marked in Fig. 1). P indicates a pinning point where the ice is grounded. Black curves indicate the maximum and minimum displacements of three interferograms (I1, I2 and I3; Table 1) which include tidal and IBE signals. Crosses indicate the range of these interferometric predictions of displacement, after modelled tide and IBE have been removed. Red curve indicates stacked prediction of displacement, with no use of tide or IBE models. Red shading indicates uncertainty of stacked prediction, determined from tide and IBE model statistics.

We compared InSAR velocity predictions before and after modelled tide and IBE corrections. Using model predictions to remove the tidal and IBE signals from each of the interferograms decreases the variability between each of the interferograms (blue crosses in Fig. 5), compared to the original interferograms (black curves in Fig. 5), and indicates that models successfully account for some of the tidal and IBE motion. However, some variation still exists, probably as a result of residual unmodelled vertical ice- shelf motion. This variation is indicative of the accuracy limits of tide and IBE model predictions. In contrast, our stacked velocity solution does not rely upon model predictions and consequently is not limited by the accuracy of these estimates. By utilizing stacking, we have maintained independence from model data, while achieving a velocity prediction that falls within the range of model-dependent solutions (i.e. the region bounded by the blue crosses in Fig. 5). This gives us confidence that, for the data used here, stacking only a small number of interferograms (three in this case) provides a reasonable, and model-independent, estimate of flow velocity. This study is, however, only based upon a relatively limited set of SAR data and we have no independent means of determining the actual tidal and IBE displacement. Consequently, we cannot discount the possibility that, fortuitously, the tidal and IBE signals happened to cancel in our particular stacked prediction of flow, and in other instances our stacked solution may be more greatly affected by these signals. Our error model does, however, account for this uncertainty, because we have modelled the distribution of all potential tide and IBE errors. This provides a measure of the possible variation in our velocity solution, depending upon the particular tidal and IBE realization present in our data. The fact that we have only stacked relatively few interferograms is reflected in the spread of this distribution. We assess the magnitude of this error and the implications for our stacking method in Section 6.2.

6.2. Comparison of methods for error estimation

When ice-shelf flow velocities are determined from a single interferogram, together with a single model realization of tide and IBE displacement, then different image pairs produce varying velocity solutions (as shown by blue crosses in Fig. 5). This variation results from inaccuracies in model predictions of tidal and IBE displacements. Accordingly, the error assigned to such a velocity solution must account for the effect of these model inaccuracies. A direct model validation is difficult in remote areas, such as the Amundsen Sea where we know of only one attempt (Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others, 2011) to assess the accuracy with which tide models can specifically reproduce the interferometric tidal displacement (Reference RignotRignot (2002) assessed a model’s ability to simulate the difference between two displacements, but not the displacement itself). While Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others (2011) produced the first such accuracy estimate in the Amundsen Sea, imprecision inherent in the method used limited the estimate of the associated across-track velocity error to 22 ± 17ma~¹. Consequently, the error associated with flow velocities determined from a single interferogram (blue crosses in Fig. 5) is not well constrained.

In contrast, because velocities derived using the stacking method are independent of model predictions, we avoid the problem of quantifying the associated model accuracy. In our method we do not use models to simulate and remove specific tidal and IBE displacements, but only to estimate the statistics of these signals, in order to assess the distribution of possible errors within our stacked velocity prediction. As such, we only require that the statistics of the model are sufficiently accurate; in other words, that the model simulates lifelike behaviour. A previous study (Reference Shepherd and PeacockShepherd and Peacock, 2003) found that tide model error was dominated by inaccuracies in predicting the tidal phase. Consequently, model statistics should adequately reflect the set of all tidal amplitudes, even if they do not correctly predict the timing at which a given amplitude occurs. The combined tidal and IBE error determined from model statistics for the stacked velocity solution is plotted in Figure 5. This assessment of error is consistent with the variability of solutions that utilize model predictions to remove the tidal and IBE signals, as would be expected, given that both the error associated with our stacked solution and the model error in predicting tidal and IBE displacement (Reference McMillan, Shepherd, Nienow and LeesonMcMillan and others, 2011) are ~20ma^-1. By stacking only three interferograms, we have achieved an accuracy comparable to that provided by current tide models. Future satellite missions, offering the prospect of forming larger image stacks, have the potential to further reduce this stacked velocity error.

6.3. Generalization of stacking method

The results of this study indicate that, for the dataset considered here, stacking provides an effective method for reducing tidal and IBE errors, without using model predictions to remove these signals. Here we assess the wider applicability of this method by considering alternative sampling regimes, resultingfrom satellites with different orbit configurations. We also assess the potential for a further reduction of errors by stacking a larger number of images.

The motivation for this assessment is provided by the prospect of new data, from satellites which will make regular, short-repeat acquisitions that are well suited to this stacking method. In the first half of 2011, ERS-2 was placed into a 3 day acquisition cycle, thus providing a new set of data with the potential to create larger image stacks. Looking to the future, the planned launch of the Sentinel-1 and RADARSAT Constellation satellites offers the prospect of regular SAR acquisitions, albeit with different temporal sampling regimes, and with it the potential to create larger image stacks and further reduce tidal and IBE errors.

Sensitivity of tide and IBE to temporal sampling

This work has focused on a specific acquisition configuration: a series of interferograms formed from a regular 3 day acquisition cycle, such that the reference images of each interferogram are separated by 6 days (Table 1). Using this configuration we have modelled the distribution of tidal and IBE signals that could occur within our stacked interferogram. If, however, the tidal and IBE signals exhibit temporal structure, then the magnitude of these signals will vary according to the temporal sampling offered by the satellite. To assess this variability, we modelled the tidal and IBE signals for a range of sampling configurations. For each configuration, we modelled the signals in both a stack of three and a stack of five interferograms to investigate how the sensitivity was affected by the number of interferograms stacked. We varied both the temporal baseline of the interferogram (i.e. the separation between the two SAR images forming each interferogram) and the interferogram separation (i.e. the separation between each master image in the interferogram stack). Repeating the procedure described in Section 4, tide and IBE model statistics were calculated for each sampling regime. For every distribution, the standard deviation of the modelled signal associated with the image stack was converted into an equivalent error in the annual flow velocity. Results are shown in Figure 6.

In both stacks, the tidal and IBE signals tend to decrease as the temporal baseline increases. This is because of the decreasing influence of the tide and IBE, relative to the steady flow signal, as the observation period increases. With respect to the period of separation between interferograms in the stack, the tidal signal displays a clear structure (Fig. 6a and b), and exhibits considerable sensitivity to the time separation of interferograms. In both the stack of three interferograms (Fig. 6a) and the stack of five interferograms (Fig. 6b) the tidal signal peaks at a contribution of ~70ma^-1 when interferograms are separated by a multiple of ~14days. This is a consequence of the strong beating of the fortnightly (Mf) tidal constituent at our study site, which is clearly evident in the modelled tidal record (Fig. 2a), and which dominates the long period (greater than diurnal) tidal signal.

Fig. 6. Modelled sensitivity of (a, b) tidal and (c, d) IBE signals to the interferometric temporal sampling regime. Results are plotted for stacks of three interferograms ((a, c)) and five interferograms ((b, d)). Each plot shows the standard deviation of the modelled velocity error arising from the tide or IBE. Each standard deviation is calculated from the set of all modelled signals, obtained from a year-long model run, such as those shown in Figure 4. The temporal baseline specifies the time period separating the pair of SAR images used to form each interferogram; the interferogram separation indicates the elapsed time between the master images of consecutive interferograms in the stack. The white boxes mark the sampling regime used in this study. Interferogram separations shorter than the temporal baseline have been set to zero.

The sensitivity of the tidal signal to the temporal sampling regime implies that the repeat time of the satellite will influence the effectiveness of the stacking technique. Worst- case errors occur where the sampling frequency matches that of the dominant long-period tidal constituent. In this case, stacking is unlikely to achieve the accuracy offered by a method that uses model predictions to remove the tidal and IBE signals. For the majority of sampling configurations, however, our analysis (Fig. 6) indicates that stacking will be an effective technique. The temporal sampling provided by the ERS acquisitions used in this study (indicated by white boxes in Fig. 6) proves favourable in producing a relatively small tidal signal. Furthermore, this analysis provides an indication of the effectiveness of stacking given the 6 day revisit time planned for Sentinel-1. For a continuous stack of 6 day interferograms (where the slave image of the preceding interferogram becomes the master image of the next interferogram), this analysis suggests that stacking five interferograms, and hence observing a full monthlong period, would reduce the tidal error contribution to ~5m a^-1.

In contrast to the tidal signal, the period of separation between the stacked interferograms has little effect upon the magnitude of the IBE signal (Fig. 6c and d). This is due to the lack of temporal structure in the atmospheric pressure record. Comparing the three-stack and five-stack scenarios, both the tidal and IBE error contributions tend to be lower in the five- stack, as a consequence of the longer period of observation. In the case of the tidal signal, the temporal structure is less well defined in the 5-stack; again a consequence of the longer total period of observation, which captures more completely the full tidal cycle.

Benefits offered by a larger stack

In this study the data archive has limited us to stacking only three regularly spaced interferograms. The ERS-2 2011 3 day acquisition phase and the planned Sentinel-1 and RADARSAT Constellation satellites offer the prospect of regular SAR acquisitions and with it the potential to stack a greater number of interferograms. To investigate the possible benefits offered by stacking a greater number of images we used the tide and atmospheric models to estimate the magnitude of these signals in larger stacks (again following the procedure described in Section 4). In particular, we considered three sampling scenarios: (1) the configuration used in this study (3 day interferograms, 6 days between each reference image in the interferogram stack); (2) a continuous series of observations from 3 day data (i.e. the slave image of the previous interferogram becomes the master image of the following interferogram); and (3) as in (2) but with 6 day repeat data (as planned for Sentinel-1). Results for up to ten stacked interferograms are shown in Figure 7. In all three sampling scenarios, the tidal and IBE errors decrease as a larger stack is formed; a consequence of the longer period of observation. The greatest benefit is gained with the first few interferograms that are stacked, particularly in the cases where interferograms are separated by 6 days, when simply stacking two interferograms reduces the tidal signal by two-thirds. As more interferograms are stacked, there is a general trend towards diminishing improvements, although the availability of a larger number of interferograms has the benefit of providing the opportunity to selectively choose interferograms, based upon criteria such as coherence and baseline characteristics.

Fig. 7. Variation in the velocity error arising from modelled tidal and atmospheric pressure (IBE) signals, according to the number of interferograms stacked. Velocity error is dependent upon the temporal sampling regime (Fig. 6), so we show results for three configurations: (a) the configuration used in this study; (b) a continuous 3 day sampling configuration, where the slave image of each interferogram is used as the master image of the following interferogram; and (c) a continuous sampling configuration (as in (b)) but for a 6 day repeat cycle, as is planned for the Sentinel-1 satellites. Velocity errors are calculated from the standard deviation of the tidal and IBE signals, modelled over a year-long period.