3.1. Geometric calibration

Geometric calibration (geocoding accuracy) is not addressed in the present paper. In Reference Mohr and MadsenMohr and Madsen (2001) it is shown that the European Remote-sensing Satellite (ERS-1/-2) mapping system can achieve a geometric accuracy corresponding to 10 m rms horizontally on the ground. This is not the final geocoding accuracy, as phase errors (static or noise) decrease the accuracy of the interferometrically derived elevations, which again cause a horizontal shift of the image in the cross-track direction. The horizontal error is 1/tan θ times the elevation error, where θ is the angle of incidence. Thus, a, typical elevation error of 10 m rms corresponds to a horizontal error of 25 m rms for ERS-1/-2 where θ ≈ 23°. This will degrade the co-registration of ascending and descending data. This is, however, negligible considering that ice velocities on glaciers in Greenland and Antarctica typically vary slowly spatially.

3.2. Interferometric path-length distortions

Both the troposphere and the ionosphere cause path-length distortions equivalent to path-length changes from surface displacements. Perturbations with magnitudes and spatial appearance similar to those of atmospheric path-length errors might additionally be caused by a dry uneven snow layer, which could potentially change the interferometric phase if superimposed between the two acquisitions (Reference Mohr, Reeh and MadsenMohr and others, 1998). Recently, Reference Guneriussen, Høgda, Johnsen and LauknesGuneriussen and others (2001) investigated this phenomenon, which is not described in detail in the literature. However, for our Storstrømmen glacier (northeast Greenland) example we disregard the effect (see section 6.3.2).

In Reference Mohr and MadsenMohr and Madsen (1996), a simplified interferometric measurement model is described and analyzed. In the following the resulting key equations for error analysis are summarized.

For convenience we use interferometric path lengh, φ′ for the error analysis, not interferometric phase, φ We define φ′ = λφ/(4π)+ B _||, where λ denotes radar wavelength and B _|| is the parallel baseline (corresponding to the flat-Earth phase). The direct effect of path-length distortions on the measured line-of-sight velocity, v, and elevation, h, depends on the orthogonal baselines B _⊥,1 and B _⊥,2 and temporal baselines T ₁ and T ₂ for the two interferograms denoted 1 and 2. For interferogram 1, the sensitivities are found to be

(2)

(3)

where R denotes slant range and θ the angle of incidence. Similar equations can be derived for interferogram 2. We here assume that the errors in the two interferograms acquired at different epochs are independent.

The sensitivity of the equation system (1a–1b) to errors in the line-of-sight velocities depends on the line-of-sight unit vectors. The line-of-sight unit vectors are parameterized by their angle of incidence (with respect to a level surface), θ, and angle of the ground swath track, ψ, with respect to north. An approximate expression for ψ, as function of the geographic latitude of the point of interest and its ground-track distance, is given in Appendix A. To simplify the error analysis, we assume that the point of interest is mapped at the same θ in both the ascending and descending pass, which again assures that ψ are numerically equal in both passes. Solving Equations (1a) and (1b) for v _e, v _n and v _u yields

(4)

and

(5)

This shows that the east velocity, v _e, can be found unambiguously from interferometric radar data alone. Thus, the surface-parallel flow assumption, i.e. Equation (1c), possibly modified with the FD term or another constraint, is required only to separate the north, v _n, and up, v _u, components. For the further error analysis we assume no vertical motion, which leads to

(6)

To summarize, the velocity errors caused by a path-length distortion can be assessed, by using Equation (3) or a similar equation for each of the two pairs of interferograms, followed by an application of Equations (4) and (6).

3.3. Interferometric phase noise

The Cramer–Rao bound for the interferogram phase standard deviation is

(7)

where N is the number of looks and γ the correlation coefficient (Reference Rodriguez and MartinRodriguez and Martin, 1992). The phase noise can be converted to an equivalent path-length noise by a division by 4π/λ, which subsequently can be used to assess velocity errors through the equations presented in section 3.2. The phase noise is not correlated from interferogram to interferogram. It is also uncorrelated from resolution cell to resolution cell (i.e. approximately from pixel to pixel), as opposed to the effects of the atmosphere and dry snow, which both are expected to exhibit large spatial correlations within one interferogram. A snow event between the two image acquisitions, though, would also increase phase noise since it inevitably causes decorrelation.

3.4. Other phase distortions

Phase unwrapping errors are not considered. Errors are usually detectable by inspection, as they appear as linearly fragmented features. In specific applications where unwrapping errors are expected, the error analysis could be carried out similar to that for atmospheric disturbances. Also, a possible clock drift (Reference Massonnet and VadonMassonnet and Vadon, 1995) in ERS-1/-2 is ignored. Again, the error analysis could be carried out similar to that for atmospheric disturbances. For small areas (e.g. one standard frame) clock drift is not expected to be significant.

3.5. Baseline calibration

In order to measure elevation and displacements using repeat-track interferometric data, the baselines have to be calibrated using GCPs. In the following, GCPs are assumed to be located on stationary terrain. The following sources of errors are considered:

When a double-difference approach is used, and the same GCPs (with known line-of-sight displacement (e.g. zero)) are used for both interferograms, the GCP elevation errors will be identical in the two interferograms. Using equations (1) and (3) from Reference Mohr and MadsenMohr and Madsen (1996) it can be shown that such a GCP elevation error common to interferograms 1 and 2 will cancel out when the velocity is calculated. The derived elevations will, of course, be affected by the error in the GCP elevation. This decoupling of height references and velocity measurements is fundamentally different from the approach where the topographic phase is removed from one interferogram by using an independent DEM. In that case, the velocity error depends on the spatial and temporal baselines and the quality of the applied DEM.

In the following, a double-difference approach is assumed, so the velocity error from GCP elevation errors is ignored, leaving the phase noise and propagation delays as the error sources. It can be shown that a linear baseline variation in azimuth is approximately equivalent to a phase variation of the form

(8)

where x is the along-track coordinate, y the across-track coordinate, and a, b, c and d the constants to be determined by the calibration. The coefficients may be found by a least-squares estimation in a general linear model,

(9)

where is the phase errors at the GCP pixels, the phase distortions at the GCP pixels originating from phase noise, atmosphere, dry snow, etc., and n is the number of GCPs used. If the covariance matrix for the phase distortions at the GCP pixels is denoted , the covariance matrix for the parameter estimate , is , where superscript T indicates transposition. Thus, the standard deviation of the estimate of the phase correction value at a point (x; y) with is

(10)

assuming there is no correlation between the phase distortion at any GCP pixel and the phase distortion at the point of interest (x, y).This is the case for phase noise, but is often also a reasonable assumption for phase distortions caused by the troposphere and dry snow, provided that the GCPs (on stationary terrain) used to calibrate the interferograms are located at some distance (e.g. at least 10 km) from the areas of interest on the glacier. If the assumption of independence is violated, our method will overestimate the errors since both the direct effect of a phase distortion and the indirect effect through the baseline calibration are taken into account.

After calculation of the standard deviations of the phase errors caused by inaccurate baseline calibrations, the impact can be calculated by using a method similar to that used for path-length changes. It is further noted that the baseline calibration errors described by Equation (10) are highly correlated spatially. This is also the case for the combined effect of baseline calibration errors, which is a linear combination of the error in each of the four interferograms used (ignoring the stretch of the interferograms introduced during ground range transformation).

3.6. Non-stationary flow — same orbit family

The double-difference approach relies on an assumption of identical velocities during the observation periods for the two interferograms. For the purpose of analyzing the impact of a flow change, the velocity projections onto slant range, v ₁ and v ₂, corresponding to each of the two interferograms, are expressed in terms of an average value, v _r, and a difference, v_ε , thus v ₁ = v _r − v_ε/2 and v ₂ = v _r + v_ε /2. Note that the velocity corresponding to an interferogram is the average velocity between the acquisition dates of the images composing the interferogram. If it is assumed, incorrectly, that v_ε is zero, the estimated velocity projection becomes

(11)

In the ERS-1/-2 tandem case, where T ₁ = T ₂ = 1 day, it is desirable to choose spatial baselines with opposite signs, as in that case |(B _⊥,1)+(B _⊥,2)/(B _⊥,1)+(B _⊥,2)| < 1. This implies that v _m will be in the range [v ₁ ; v ₂]. In other words, the measured velocity projection is between the velocities at the two intervals of acquisitions. If spatial baselines are of equal sign, the ratio is larger than 1 and the combined estimate of the line-of-sight velocity thus more sensitive to flow changes. Optimal baselines are further discussed in Reference Mohr and MadsenMohr and Madsen (1996). The elevation estimate is similarly

(12)

Again, spatial baselines of opposite sign minimize the sensitivity to flow variations.

3.7. Non-stationary flow — different orbit family

A non-stationary flow between the ascending- and descending-orbit acquisitions also affects the direction and magnitude of the derived flow. To analyze the effect, the flow vectors corresponding to each look direction are expressed in terms of the average velocity and a difference , i.e. and . The measured line-of-sight velocities thus become and for the descending and ascending datasets, respectively. Assuming a level flow, the error velocity is written as , where η is the direction of the change. The resulting difference between the derived velocity and the average value becomes

(13)

(see Mohr and Madsen,1996). For glaciers with a seasonally varying velocity, the direction of flow is often constant, and only the magnitude changes. From Equation (13) it is seen that even given a constant flow direction, the measured flow will not necessarily be aligned with the flow. This is illustrated in Figure 2. It is also noted that the magnitude of the measurement error vector is never zero but always between and .

Fig. 2.

Magnitude (a) and directional (b) deviation from the average flow with a ±5% flow magnitude change between ascending- and descending-orbit acquisitions. Track angles of 18°, 28° and 38° correspond to approximately 67°, 76° and 80° N for ERS-1/-2 at mid-swath.

5.1. Flux-divergence calculation

The FD, i.e. the submergence/emergence velocity, V, is calculated numerically, as

(16)

where E = Fhv _e, N = Fhv _n, and Δ_e and Δ_n denote the grid spacing in the east and north directions, respectively. It proved necessary to smooth the FD field (Reference Reeh, Mohr, Madsen, Oerter and GundestrupReeh and others, 2003) to obtain convergence of our solution. An analysis of the criteria for stability of the solution is beyond the scope of the present analysis. We use a (2m + 1) by (2m + 1) box filter, i.e.

(17)

For convenience we skip the (k, l) indices from , E and N, and assume a common grid spacing Δ _x . We then find

(18)

Thus, the FD at the centre of the box filter only depends on the E and N values at the boundary lines of the filter andthe values at the adjacent points immediately outside the boundary.

5.2. Flux divergence — (Fh) errors

This subsection concerns errors in the derived vertical velocity, v _u, originating from the Fh term in the FD, i.e. in Equation (1c). For the error analysis we again ignore the coupling between Equations (1c) and (5). The error sources considered are:

Assumptions and notation for h and F errors, respectively, are described below, followed by an analysis of the effect of the different errors after the applied grid averaging (Equation (18)).

5.2.2. F-value variations

Flow over basal irregularities causes variations of the F factor, from large values over basal highs to smaller values over basal lows. In the following it will be shown that this reduces the impact of an unknown basal relief.

For the purpose of error analysis, we consider plane flow over an undulating base. The ice flux is assumed not to vary in the direction of the flow, i.e. flux changes due to surface mass balance are neglected. For ice-sheet conditions, which apply to Storstrømmen, the surface undulations are out of phase by π/2 and an order of magnitude smaller than the bottom undulations (Reference PatersonPaterson, 1994, p. 265). In that case, the ice thickness, h, can be assumed to vary as

(19)

where h ₀ is the mean ice thickness, d is the ratio between the amplitude of the basal undulations and the mean ice thickness, ω = 2π/L, where L is the wavelength of the basal undulations, and s is the distance in the direction of flow. Using Reference ReehReeh (1987) it can be shown that in the plane-flow case the Fh product is approximately

(20)

where F ₀ is the F factor of the mean flow and c a dimensionless parameter in the range [0; 1], depending on the h ₀ /L ratio, ice properties, etc. A typical value is c = 0.5. Equation (20) shows that the variation over the undulation of Fh is reduced by a factor c as compared to the h variation.

If the ice-thickness distribution was known in a dense grid, a glacier flow model could in principle be used to calculate c and thus an accurate Fh value for each gridpoint. For the less detailed ice-thickness distribution of our Storstrømmen example, the latter term of the righthand side of Equation (20) is not known at the gridpoints, and therefore must be treated as an error term.

Bias in the F value is denoted β_F . In order to simplify the error analysis, the bias is assumed constant for all gridpoints within the filter.

5.2.3. Effect of grid averaging

For the error analysis, the E and N terms in Equation (18) are treated separately but similarly. According to the discussion in section 5.2.2, we write F and h as

(21a)

(21b)

The ice-thickness noise, ε_h , is assumed independent from gridpoint to gridpoint. The phase of the unknown bottom undulations, φ, and ice-thickness bias, β_h , are assumed fully correlated for all gridpoints along one border of the filter, but values on the different sides of the filter are assumed uncorrelated. The bias in F, β_F , is assumed constant for all gridpoints.

Errors of v _e are neglected for the moment, but will be treated in section 5.3. Ignoring second- and higher-order terms, we may express an E_m,j value at a gridpoint (m, j) as

(22)

For clarity, indices of the error terms, i.e. φ, ε_h , β_h and β_F , are omitted. After some calculations (see Appendix B), we find

(23)

for the rms of the error in the vertical velocity, σ _V, caused by errors in the (Fh) term. In Equation (23), , and are the rms values of ε_h , β_h and β_F , respectively. To simplify Equation (23), the center-point (or average) values F ₀, h ₀ and v _H are introduced. The terms and are the emergence/submergence velocities originating from east–west and north–south FD, respectively. In Appendix B, alternative expressions for and are given.

5.3. Flux divergence — interferometric errors

This subsection addresses errors in the derived vertical velocity, v _u, originating from the interferometric velocity term, , in the flux divergence, i.e. in Equation (1c). For the error analysis we ignore again the coupling between Equations (1c) and (5). We consider:

(24a)

(24b)

where and are the rms values of the distortions, ω _e and ω _n the unknown spatial frequencies, and φ _e and φ _n the unknown phases. Undulations could, for example, be caused by atmospheric distortions or a dry snow layer.

Velocity biases are denoted and for the east and north components, respectively. In order to simplify the error analysis, the biases are assumed constant for all gridpoints. A bias could, for example, be caused by a baseline calibration error.

5.3.1. Effect of grid averaging

Similar to section 5.2.3, we write

(25)

for an E term. For clarity, indices of the error terms, i.e. , φ _e and , are omitted. The noise on the velocity, , is assumed independent from gridpoint to gridpoint. The phase of the unknown velocity undulations, φ_e , is assumed fully correlated for all gridpoints along one border of the filter, but values on the different sides of the filter are assumed uncorrelated. The velocity bias is assumed constant for all gridpoints. Carrying out the analysis similar to that for the Fh term yields

(26)

where is the rms value of the horizontal velocity noise, the rms of horizontal velocity undulations, and and the rms of the velocity bias in the east and north directions, respectively.

6.1. Summary of interferometric error budget

The error budget for the direct effects of interferometric errors in the example data used is presented in Table 2. Each of the contributions is described in separate subsections below.

Table 2.Effect of different error sources using the example data fromTable 1

For reasons also described below, we consider the atmosphere and baseline errors (closely related to the atmospheric phase delays) the major error sources. With a 0.3 cm rms atmospheric path-length error, the combined effect of atmospheric and baseline errors in the area of interest is typically 6.2 m a⁻¹ rms for the horizontal velocity and 16 m rms for the elevation. This is in line with the empirical results from Reference Mohr, Reeh and MadsenMohr and others (1998) and Reference Reeh, Mohr, Madsen, Oerter and GundestrupReeh and others (2003). It is noted that for larger areas of interest and for more humid climates, significantly larger atmospheric path-length distortions, and thus also baseline calibration errors, could be expected.

6.2. Geometric calibration

The effect of misregistration between ascending- and descending-orbit interferograms is ignored as glacier velocities in Greenland and Antarctica typically vary slowly. However, it should be noted that the processing system used in Reference Reeh, Mohr, Madsen, Oerter and GundestrupReeh and others (2003) was not fully calibrated geometrically, so the co-registration error is likely to be slightly larger than estimated here.

6.3. Interferometric path-length distortions

In this subsection, path-length distortions caused by the atmosphere and dry snow are assessed.

6.3.1. Atmosphere

Atmospheric path-length distortions may originate from both the ionosphere and the troposphere (Reference Tarayre and MassonnetTarayre and Massonnet,1996).

Ionospheric irregularities typically have greater spatial correlation lengths than tropospheric irregularities (Reference Hanssen, Weckwerth, Zebker and KleesHanssen and others, 1999). Over the small scales relevant in the present study, ionospheric distortions are only significant during unusual conditions, as the interferometric product is calibrated with GCPs spaced at most a few tens of kilometers from the study area (Reference Gray, Mattar and SofkoGray and others, 2000). Thus, in the present error analysis, ionospheric distortions are ignored as a separate error source, but are lumped together with the tropospheric path-length distortions.

The spatial characteristics of tropospheric path-length distortions are significantly different from day to day (Reference HanssenHanssen, 1998). Typical values for the spatially non-linear phase distortions (tropospheric and ionospheric) over a 21 km by 21 km arctic area are found to be ±0.5 cm peak to peak, corresponding to ±1 rad (Reference Mattar, Gray, Geudtner and VachonMattar and others, 1999). This is consistent with other reports. In the following analysis, we use 0.3 cm rms for the atmospheric path-length variation.

Given 0.3 cm path-length distortions in each of the four example interferograms, the effect on the decomposed line-of-sight velocities and elevations is calculated from Equations (2) and (3). The results are given in Table 3. The combined rms error is, using Equations (4) and (6) and thus assuming surface-parallel flow, found to be 2.1 m a⁻¹ in v _e and 3.9 m a⁻¹ in v _n. Thus, the total horizontal velocity error is 4.4 m a⁻¹ rms, as listed inTable 2. In our approach, the surface elevation is determined from the ascending-orbit interferograms only, and thus the total elevation error is 9 m rms as listed inTable 2.

Table 3.Effect of a 0.3 cm path-length error (i.e. 0.67 rad) applied successively to each of the Storstrømmen interferograms inTable 1

The small spatial baselines of the descending-orbit data of course reduce the accuracy of the elevations (Equation (2)). However, since the baseline ratio is close to zero, the accuracy of the velocities is not severely affected (Equation (3)). The geocoding accuracy, which is important when ascending- and descending-orbit data are combined, is degraded, but this could in principle be circumvented by using the ascending-orbit DEM as topography reference for the velocities derived from descending-orbit data.

6.4. Interferometric phase noise

For the Storstrømmen example, the number of looks is approximately 20, large enough for Equation (7) to be used to represent phase noise. Observed correlation values for rock and ice are provided in Table 1 for each of the four interferograms. Converting the correlation values to phase noise, combined with the results presented in Table 3, yields the estimated phase-noise error terms presented in Table 2. It should be noted that these values are applicable for the comparison with pole measurements presented in Reference Mohr, Reeh and MadsenMohr and others (1998), but during the derivation of the vertical emergence/submergence velocity presented in Reference Reeh, Mohr, Madsen, Oerter and GundestrupReeh and others (2003) a further averaging was applied, which reduced the phase variance with a factor on the order of 10.

6.5. Baseline calibration

For the example data, the number of GCPs is 6. Although two GCPs are located within 8 km of each other and two within 11 km, the phase distortions are assumed uncorrelated between the pixels where GCP calibration is performed, i.e. , where is the unity matrix of order n. Note that for a denser GCP network, it is important to include the co-variance structure of the (atmospheric) phase noise in Σ. This structure is discussed, for example, in Reference Williams, Bock and FangWilliams and others (1998). In Figure 3, the spatial variation of is shown assuming σ = 1 rad at the six GCPs in the example data. It is observed that in the area enclosed by GCPs the standard deviation of the calibration residual is typically 1 rad. In line with Reference Joughin, Kwok and FahnestockJoughin and others (1996), it is also seen that outside the area enclosed by GCPs the errors increase rapidly.

Fig. 3.

Storstrømmen glacier, northeast Greenland. Phase (baseline) calibration errors in radians in one example interferogram assuming a 1 rad uncorrelated rms noise at the GCPs. All four example interferograms are calibrated using the six GCPs indicated by the grey “+”symbols. The location of the area is indicated by the insert. The light-gray area is covered by SAR interferometry. The dark gray area is additionally covered by ice-sounding radar. Arrows indicate line-of-sight directions from ascending and descending orbits at the center of the area.

For the evaluation of the baseline contribution to the velocity errors, only phase noise and atmospheric disturbances are included. Disturbances due to a dry snow layer are unlikely, as argued above. From Figure 3, it is seen that within the area in the example data enclosed by GCPs, the baseline error is typically equal to the error at the GCPs. A typical value for the rms velocity errors is thus estimated to be equal to the velocity errors originating from the atmosphere and phase noise (Table 2). In most cases, though, we use a spatially varying velocity error (of baseline calibration origin), based on the factor times the combined error originating from the atmosphere and phase noise.

With respect to elevation, the elevation errors on the GCPs also have to be included. Assuming a 10 m rms error on the GCPs, the corresponding phase error terms are, using the baselines from Table 1, found to be 0.92 rad and 0.13 rad for A₁ and A₂, respectively. Again ignoring the dry-snow contributions, adding the phase noise and atmospheric disturbances and using the results fromTable 3 yields an elevation rms error of 13 m originating from improper calibration of baselines (Table 2).

6.6. Non-stationary flow — same orbit family

For the example data, the baseline values for the ascending-orbit data have different signs (Table 1). This implies that in the case of a non-stationary flow, the derived line-of-sight velocity (displacement rate) will be a weighted average of the velocities at the two interferogram observation periods (Equation (11)). Thus, the derived velocity has been present at least at one instant in the period from the acquisition of the first to that of the last SAR image. In other words, the line-of-sight velocities derived from the ascending-orbit interferograms are correct for at least one specific but unknown epoch in the period between the acquisitions. The same argument applies to the descending-orbit data. The baseline ratios of 1/7 and 1/19 for ascending- and descending-orbit data, respectively, though, give large weights to the velocities at the acquisitions with short spatial baselines.

A non-stationary flow will, on the other hand, change the measured elevations. The change of line-of-sight velocity being known, the impact on the measured elevation can be calculated from Equation (12). For the Storstrømmen example, it is unknown whether or not the velocity changes during the winter, since there are no global positioning system (GPS) measurements from the winter. It is known that the summer velocities over much of the glacier are larger than the annual means.We believe that the increased summer velocity is caused by meltwater penetrating to the bottom and thereby enhancing sliding. During winter (September–May) there is no meltwater, so we assume a negligible seasonal velocity variation during the observation period. GPS measurements of surface velocities during the observation period (1992–95) show a general decrease on the order of 10 m a⁻¹ per year (Reference Reeh, Mohr, Madsen, Oerter and GundestrupReeh and others, 2003). If assumed to be uniform, this is a decrease of 2 m a⁻¹ during the observation period.

Using Equation (12), and the interferogram parameters from Table 1, it is found that a 1 m a⁻¹ change in the across-track component of the horizontal velocity causes a −2.3 m change in the measured elevation. Changes in the along-track direction have no effect, as these do not change the line-of-sight velocity. If the flow change is averaged over all directions, the rms error is reduced to . Thus, a 2 m a⁻¹ rms horizontal across-track velocity change corresponds to 3.2 m rms (Table 2).

6.7. Non-stationary flow — different orbit family

From Equation (13) it is seen that the ratio of the flow measurement error and the flow change is between tan ψ and 1 / tan ψ. The direction of the flow measurement error is not necessarily aligned with the direction of the flow change. Thus, for our study area, where ψ ≈ 28° (see Equation (27)), a 1 m a⁻¹ flow change between the ascending- and descending-orbit acquisitions will at most result in a 0.9 m a⁻¹ deviation from the average value.

For Storstrømmen, flow changes are expected to be primarily in the direction of the flow. This is supported by repeated measurements of positions of poles drilled into the ice. The pole measurements also suggest that the potential flow changes are rather independent of flow magnitude, except for the areas with very small or zero velocities. Since the flow directions are varying over a large azimuth interval, the correct approach would be to compile a map with the effect of a potential flow change. However, as shown below, the effect is expected to be less important than atmospheric distortions and baseline calibration errors. Therefore, we simplify the error analysis to provide one number for the effect instead of a map. This is done by estimating a value for the standard deviation of the potential flow change and assuming an unknown direction. For other glaciers, with a more unidirectional flow, it might be more appropriate to assume a fixed direction of the flow change.

If a flow change, , is assumed having a magnitude of 1 m a⁻¹ and averaging over all directions, the east and north errors are found to be tan and cot , respectively,for the Storstrømmen example (Equation (13)). As described above, no direct information is available on the flow changes during winter. Again, a 10 m a⁻¹ change per year is assumed, corresponding to 2.8 m a⁻¹ in the period between the ascending- and descending-orbit interferograms with the shortest spatial baselines. Thus, the errors due to flow changes are estimated to be 0.5 m a⁻¹ in the east direction and 1.8 m a⁻¹ in the north direction, i.e. 1.9 m a⁻¹ horizontally (Table 2).

Since the non-stationary flow contribution is small compared to the atmosphere and baseline errors and since we have no clear evidence for the phenomena, we do not consider the non-stationary flow a major error source for our example data.

8.1. Flux-divergence error assessment — (Fh) errors

In section 5.2 above, five types of errors are considered: noise, bias and grid interpolation errors in ice thickness h, errors due to undulations in F, and bias in F. The resulting error in the vertical velocity can be calculated by using Equation (23). In this subsection, the magnitude of each contribution, required for an application of Equation (23) is assessed, and the results for our Storstrømmen data shown in Figure 4b.

8.2. Flux-divergence error assessment — interferometric errors

In section 5.3 above, three types of errors are considered: noise, undulations, and bias of the horizontal velocity, . The resulting error in the vertical velocity can be calculated by using Equation (26). The magnitude of each contribution required for an application of Equation (26) is assessed in this subsection, and the results for our Storstrømmen data shown in Figure 4c.