Digital elevation models (DEMs) derived from across-track interferometric synthetic aperture radar (InSAR) data are a main data source for mapping the surface elevation and its temporal change over glaciers and ice sheets. Single-pass InSAR systems, such as the TanDEM-X (TDM) mission, are of particular interest for this task as they are not affected by variations in the atmospheric phase delay, ice motion and temporal decorrelation. For the analysis and interpretation of InSAR elevation over snow and ice, the effects of signal penetration have to be taken into account. The surface inferred from uncorrected InSAR elevation data refers to the position of the scattering phase centre in the snow–firn medium, resulting in an elevation bias versus the actual surface (Dall, 2007). The position and strength of scattering sources in the snow volume and the absorption and scattering losses are main factors defining the depth of the phase centre below the snow surface. Backscatter contributions from sources at different depths within a volume-scattering medium, observed under different incidence angles, are causing decorrelation, depending on the interferometric baseline and incidence angle (Bamler and Hartl, 1998).

Hoen and Zebker (2000, 2001) derived a formulation for estimating the power penetration depth in dry snow from the interferometric coherence, applying a radiative transfer model for estimating spatial decorrelation in a volume of uniformly distributed and uncorrelated scatterers characterised by exponential extinction. They applied this formulation to derive the C-band penetration depth for different sites in Greenland from the coherence of 3 d repeat-pass InSAR data of the ERS-1 synthetic aperture radar (SAR) mission. Forsberg et al. (2000) and Dall et al. (2001) compared surface elevation measured by airborne laser altimetry and C-band single-pass SAR interferometry on the Geiki ice cap in Greenland. They report zero InSAR elevation bias for wet snow and an average bias of about 10 m for dry snow and firn. Dall (2007) studied relations between the InSAR elevation bias and the power penetration depth in uniform volumes. He shows that the depth of the mean phase centre in a volume-scattering medium is approximately equal to the two-way penetration depth if the latter is smaller than about 10 % of the height of ambiguity (Ha), the height difference for a phase shift of 2π. Fischer et al. (2019a, b, 2020) studied various concepts for characterising and modelling the vertical backscatter distribution and retrieving the InSAR penetration bias in the percolation zone of Greenland based on airborne polarimetric multi-baseline InSAR data and in situ measurements of snow structural properties.

In recent years single-pass InSAR data of the TDM mission were widely applied for mapping surface elevation and elevation change on glaciers and ice sheets. The TDM mission employs a bistatic interferometric configuration of the two satellites TerraSAR-X and TanDEM-X flying in close formation in order to form a single-pass SAR interferometer (Krieger et al., 2013). Rizzoli et al. (2017b) compared surface elevation over Greenland measured by NASA's Ice, Cloud and land Elevation Satellite (ICESat) laser altimeter with the TanDEM-X global DEM. They report for frozen snow and firn in the wet snow zone, the lower and upper percolation zone, and the dry-snow zone mean values of the X-band InSAR penetration bias of 3.7, 3.9, 4.7 and 5.4 m, respectively. Abdullahi et al. (2019) use a linear regression model for estimating the elevation bias in TDM DEMs of northern Greenland. The model is based on empirical relations between coherence and backscatter intensity with the difference between the uncorrected TDM DEMs and airborne laser altimeter surface heights.

The complex layered structure of polar snow and firn has a major impact on radar signal propagation and interferometric coherence, an obstacle for establishing a generally applicable, physically based method for estimating the elevation bias of InSAR products. The work presented in this paper takes on this open issue, exploring relations between interferometric parameters and physical snow properties and investigating the feasibility of deducing the elevation bias from the interferometric correlation. The study is based on interferometric data of the TDM mission, data from optical satellite sensors and field measurements undertaken in December 2016 on Union Glacier in the Ellsworth Mountains, Antarctica. Logistic support was provided by the private company Antarctic Logistics & Expeditions LLC (ALE), which conducts aircraft flights to Union Glacier and operates a field station in summer. The study area comprises ice-free surfaces, bare ice, dry snow and firn, exhibiting a diversity of structural features attributed to local differences in wind exposure and snow accumulation. Time series of ICESat laser measurements from 2003 to 2009 and ICESat-2 data show near-steady-state surface topography, facilitating the intercomparison of TDM and optical elevation data.

In Sect. 2 we describe the study area, present details on the satellite data, and give an account of the structure and morphology of snow and firn at different sites. Section 3 explains the basic concept relating the elevation bias and interferometric coherence in a uniform random volume. Section 4 deals with vertical co-registration of the different DEMs, including an analysis of the temporal stability of surface elevation, and describes the observed spatial pattern of the backscatter signals, coherence and elevation bias. Section 5 presents results of the inversion of the volumetric coherence in terms of the InSAR elevation bias and compares the retrieved bias with elevation differences between TDM DEMs and optical data. Section 6 includes the discussion, and Sect. 7 presents conclusions. The Appendix shows simulations for vertical backscatter distributions at snow pit sites and compares these with exponential backscatter functions.

The procedure for estimating the interferometric elevation bias is based on the inversion of the volumetric correlation factor, which can be derived from total coherence products (γtot) generated during InSAR processing. The total interferometric complex correlation coefficient (coherence) of a random medium is made up of the following contributions (Krieger et al., 2007): 1γtot=γtherm⋅γQuant⋅γAmb⋅γRg⋅γAz⋅γVol⋅γtemp. The terms on the right-hand side refer to the interferometric correlation coefficient related to the signal-to-noise ratio (γtherm), quantisation (γQuant), azimuth and range ambiguities (γAmb), baseline decorrelation (γRg), relative shift in the Doppler spectra (γAz), volumetric decorrelation (γVol), and temporal decorrelation (γtemp). Temporal decorrelation is not relevant for single-pass InSAR data over ground, including snow and ice (γtemp=1.0).

The thermal interferometric correlation component is related to the signal-to-noise ratio (SNR) of the two SAR images by 2γtherm=1/1+SNR1-11+SNR2-1. For single-pass InSAR the volumetric correlation coefficient can be derived from the total coherence by 3γVol=γtotγthermγQuantγAmbγRgγAz. The phase noise due to γAmb, γQuant and γAz of advanced SAR systems is small. For TDM single-pass InSAR interferograms Krieger et al. (2007) estimate the typical loss of coherence for each of the terms γAmb, γQuant and γAz at <2 %. Baseline decorrelation, γRg, is avoided by applying common bandwidth filtering.

Hoen and Zebker (2000) specify a formulation for the correlation factor in a uniform volume with exponential extinction in which the interferometric phase is proportional to the penetration length, dl: 4γVol=1/1+pπεdlBnr0λtan⁡θi2=1/1+πεdlcos⁡θiHa2, where λ is the radar wavelength, r0 is the slant range distance, θi is the incidence angle at the air–snow interface, Bn is the effective interferometric baseline, and ε is the dielectric permittivity; p=1 is valid for the combination of one monostatic and one bistatic SAR image forming an interferogram and p=2 for the combination of two monostatic images. Ha is the height of ambiguity in free space: 5Ha=λr0sin⁡θipBn. According to the radiative transfer approach the one-way power penetration length dl [m], where the intensity of the signal is attenuated to 1/e of the incident signal, is given by 6dl=1ke=1ks+ka, where ka and ks [m-1] are the absorption and the scattering coefficients. The one-way power penetration depth, dp, referring to vertical direction, is obtained by accounting for the refraction angle θr: 7dp=dlcos⁡θr. The vertical interferometric wavenumber, kz [rad m-1], relates the phase of the interferometric correlation to the geometric configuration of the interferometer, providing phase (φ) to height conversion: 8kz=∂φ∂z=2πHa. The wavenumber in a lossy volume accounts for the change in the propagation constant and refraction (Lei at al., 2016), yielding the following formulation for the height of ambiguity in the volume: 9HaVol=2πkzVol, where kzVol=kzεcos⁡θicos⁡θr. For dry snow and ice the absorption, losses are very small so that the real part of the permittivity can be used (Mätzler, 1996). In Table 1 the values for Ha and for kzVol (assuming a snow density of 400 kg m-3) are specified for the TDM scenes.

Dall (2007) shows that the penetration-related elevation bias, hb, is approximately equal to the two-way power penetration depth, dp2, if the latter is small compared to HaVol. For large relative penetration depths (dp2/HaVol) the elevation bias approaches one-quarter of the ambiguity height. Normalising the coherence phase, ∠γ, by the interferometric phase at the volume surface (∠γ=φ-φsurface) yields the following relation, in which the elevation bias is proportional to ∠γ (Dall, 2007): 10hb=∠γ/kzVol=∠γHaVol2π. For a uniform volume a direct relationship between the coherence magnitude and the relative penetration depth can be defined, from which the phase can be computed (Dall, 2007): 11∠γ=-sgnHaVolarctanγVol-2-1. As according to this relation the coherence phase is uniquely defined by the coherence magnitude, the following formulation can be used for estimating the elevation bias: 12hb=-HaVol2πarctanγVol-2-1. We apply this equation for estimating the elevation bias from the observed coherence, using the magnitude of the volumetric InSAR correlation factor as input. According to this formulation the actual InSAR elevation bias becomes progressively smaller than dp2 with increasing relative penetration depth.

This approach is based on the assumption of a uniform volume with exponential extinction, whereas dry polar firn is a density-stratified medium featuring distinct differences in scattering and extinction properties between individual layers as well as depth-dependent changes. However, for inverting the observed interferometric coherence in terms of the elevation bias the assumption of a simple model is needed for describing the vertical backscatter and extinction properties. We tested the applicability of the uniform volume approach for describing the observed backscatter intensity, performing forward computations with a multilayer radiative transfer model (see Appendix A).

In this section we show the spatial pattern of backscatter intensity and coherence in the study area and relations of these parameters to the elevation bias inferred from optical sensor data. We start with an account of topographic reference data and the procedures applied for vertical co-registration, a critical step for estimating the penetration-related elevation bias.

Building on the signature analysis reported in Sect. 4, we focus on the use of the volumetric coherence for estimating the interferometric elevation bias by inverting γVol according to Eq. (12). For computing the vertical wavenumber in the volume and the refraction angle, we assume a snow density of 400 kg m-3, resulting in ε′=1.763 (Mätzler, 1996). Figure 8 shows plots of the computed elevation bias, hbInv, at the snow pit sites derived from γVol vs. the elevation difference dh between the InSAR DEMs and the REMA. The T2013/14 data show a highly significant linear relation between dh and hbInv, with a coefficient of determination R2=0.86. The root mean square difference (RMSD) is 0.74 m, attributed to errors in the computed hbInv and the DEM difference product. The T2016/18 data (mean of HH and VV polarisation) show a linear relation with R2=0.59 and RMSD = 0.84 m.

Maps of dh and the computed TDM elevation bias are shown in Fig. 9 for scene T2013B and in Fig. 10 for scene T2016. These two scenes have almost the same vertical wavenumber but different incidence angles. The differences between HH- and VV-polarised data of the T2016 and T2018 scene, respectively, are insignificant. We use the mean value of the HH- and VV-based DEMs of the single dates for the comparison in order to reduce the impact of random phase noise. In Table 3 mean numbers over the AoI are specified for dh, hbInv, γVol, the coefficient of determination (R2) for linear correlation between dh and hbInv, and the RMSD. The numbers for R2 and RMSD refer to the maps resampled to 8 m grid size, low-pass filtered over 7×7 pixels windows using a Gaussian function. The dh value of the TDMgl DEM (dh = 5.61 m) differs only by 0.06 m from the mean dh of the T2013/14 data. The TDMgl DEM is based on several TDM scenes acquired in 2013 and 2014. The dh map for TDMgl vs. REMA shows a similar spatial pattern as dh of the individual DEMs (Fig. S2).

As for the snow pit sites, the mean values over the AoI show distinct differences in dh, hbInv and γVol between the data sets with different incidence angles. The magnitudes of the elevation bias of the T2013/14 data (mean dh =-5.55 m, hbInv=-5.22 m) are larger than the corresponding values of the T2016/18 data set (dh =-4.59 m, hbInv=-4.79 m). As for the snow pit sites, this is opposite to the expectation for a uniform isotropic scattering medium for which |hb| should be larger in scenes with smaller off-nadir angles in the case of the same value of Ha.

The AoI mean values of dh and hbInv show minor differences: -0.33 m for T2013/14 and 0.20 m for T2016/18. The spatial patterns of dh and hbInv are similar, but the mean slope of the 2D distribution deviates from the 1:1 correspondence (Fig. 11). The magnitude of the computed elevation bias is overestimated over the areas with coarse-grained firn and small penetration depth in the surroundings of the BIA and underestimated in areas of higher accumulation rate. These depth-dependent deviations can, at least partly, be attributed to the simplified assumption of the uniform volume approach.

A critical issue for inverting interferometric coherence in terms of the InSAR elevation bias is the description of the vertical backscattering profile in the snow volume. A simple model is required, in particular if only single-channel backscatter data are available. We apply the model of Dall (2007), in which the vertical backscatter function is defined by the extinction coefficient of a uniform random volume accounting for the combined effect of absorption and scattering. In order to check the suitability of this model for describing the backscattering profile of layered polar firn, we performed backscatter simulations for the snow pit sites with a multilayer radiative transfer model (Appendix A).

The computed total σ∘ values at θi=40∘ are matching the observed total backscatter intensities of the T2013/14 scenes for snow pit sites 2 to 5. The simulated vertical backscatter profiles and the exponential profiles of the uniform volume approach show close agreement for sites 2 to 4. The variations between individual layers, tracing back to accumulation and wind erosion events as well as to seasonal effects, are suppressed in the vertical profile of the cumulative backscatter contributions. At sites 2 to 4 the uniform volume approach shows minor overestimation of the backscatter contributions from the top snow layers due to the assumption of a constant scattering coefficient, whereas the actual grain size in the near-surface layers is below average. This effect is more pronounced at pit 5, where the layers with small grains reach down to 1.4 m depth.

On the other hand, the radiative transfer simulations at θi=22∘, referring to the T2016/18 scenes, yield underestimation of σ∘ by several dB. For pit 1 the simulated backscatter intensity is underestimated also at θi=40∘. The incidence angle differences (Fig. 5) are most pronounced in the glacier zones with comparatively high accumulation. The radiative transfer model used for the simulations computes volume scattering for bi-continuous, random structures applying the improved Born approximation and assumes plane-parallel, homogenous layers (Picard et al., 2018). As a matter of fact, in addition to incoherent volume scatter, the contributions of rough internal interfaces as well as interlayer interferences play a role (Tan et al., 2017; Fischer et al., 2019b).

The most likely explanation for the increased backscatter towards near-nadir incidence is the increased scattering at interfaces between layers of different density and wind-packed structures. In particular on the main section of Union Glacier the snow surfaces are wind-roughened, showing elongated sastrugi at the metre scale and vertical roughness on the order of several centimetres. The surface roughness related to wind packing and erosion is also evident in snow pits at interfaces between snow layers and at wind crusts. Ashcraft and Long (2006) attribute the backscatter anisotropy, observed in scatterometer data of katabatic wind zones, to the anisotropy in the preferential roughness direction of the snow surface and of internal interfaces. Such behaviour was also observed for density-stratified firn on the East Antarctic Plateau and reproduced by simulations with a layered-medium radiative transfer model (Rott et al., 1993; West et al., 1996).

Whereas differential propagation effects lead to distinct co-polarisation differences in the phase and magnitude of the complex co-polarised (HH–VV) coherence, the differences in the interferometric coherence and derived penetration bias between the individual polarisations are insignificant (Table 3). This implies that the vertical backscatter distributions of the co-polarisation channels are similar. Consequently, the data from HH and VV polarisation can be combined for estimating the elevation bias, reducing the impact of noise. The differences in σ∘ between HH and VV polarisation are also small, as to be expected for low-incidence-angle data (Fung, 1994). On average, over the AoI, σ∘HH is higher by 0.2 dB compared to σ∘VV, and there is no distinct spatial variability.

We also checked the information content of the phase, ϕHH-VV, and magnitude, |γHH-VV|, of the complex HH–VV-polarised correlation coefficient regarding potential contributions in support of elevation bias retrievals. Measurements with a ground-based (Jordan et al., 2019) and an airborne (Dall, 2009) polarimetric ice sounder in the dry-snow zone of Greenland show major variations in ϕHH-VV related to anisotropic scattering at internal interfaces and to the crystal orientation of ice fabrics. Leinss et al. (2016) derived ϕHH-VV from TerraSAR-X and ground-based scatterometer data in order to determine the dielectric and structural anisotropy of seasonal snow, showing distinct differences in ϕHH-VV related to the snow microstructure. We computed the complex co-polarised correlation coefficient from data of the T2016 scene using estimation windows comprising about 125 independent samples (Fig. S3). The co-polarised coherence, |γHHVV|, reflects to some degree the spatial pattern of the elevation bias. In the BIA |γHH-VV| is 0.92, in the low-accumulation areas near the BIA 0.6 to 0.7, and in the vicinity of the ALE camp 0.3 to 0.5. In the AoI the linear correlation coefficient, R2, between |γHH-VV| and dh amounts to 0.29. Reduced coherence can be attributed to decorrelation by the non-coherent scattering components in the volume.

The mean co-polarised phase difference, ϕHH-VV, of the BIA is close to zero (0.05 rad), as to be expected for backscatter from a comparatively smooth surface. In the AoI (snow and firn) the mean value of ϕHH-VV amounts to 0.50 rad with a high standard deviation (0.39 rad) because the ensemble comprises positive as well as negative values. Positive phase differences, an indication of anisotropic scattering at horizontal structures such as internal interfaces, are dominating on the main glacier. A substantial part of Driscoll Glacier shows negative ϕHH-VV values, the reason for which is unclear. In the AoI the correlation coefficient R2 between ϕHH-VV (comprising positive and negative values) and dh is close to zero, whereas R2 for the magnitude of the phase difference, |ϕHH-VV|, is 0.19. These observations are an indication of differential propagation effects between HH- and VV-polarised waves in the snow–firn volume, but the sources of the anisotropy are not uniquely defined. In order to assess the value of the complex co-polarised correlation coefficient for supporting retrievals of the penetration-related elevation bias, dedicated studies are needed, employing preferably multi-incidence angle observations.

According to theory the penetration bias and phase centre depth at a given frequency and polarisation change with the baseline and the incidence angle (Dall, 2007). This was verified with airborne data by Fischer et al. (2020), showing that the changes are significant in particular at small volumetric wavenumbers (long baselines). This is evident comparing two scenes with almost the same incidence angle: T2016 (θi=21.6∘, kzVol=0.120) and T2018 (θi=22.1∘, kzVol=0.072). Both the dh and hbInv values indicate deeper penetration for T2018 compared to T2016, amounting in the AoI to dh = -4.80 m for T2018 versus -4.38 m for T2016 and to hbInv=-5.18 m for T2018 versus -4.40 m for T2016 (Table 3).

For a uniform scattering medium, deeper penetration is expected for a steeper propagation path (lower incidence angle). This can be checked by comparing two scenes with almost the same vertical wavenumber and different incidence angles: T2013B (θi=38.6∘, kzVol=0.121, dh(AoI) = -5.63 m) and T2016 (θi=21.6∘, kzVol=0.120, dh(AoI) = -4.38 m). However, assuming for T2016 a volume with the same scattering and absorption coefficients as for T2013B, the expected elevation bias for T2016 is -6.04 m. The same behaviour is evident for the mean dh values of the snow pit sites: 〈dh〉=-5.70 m for T2013B, 〈dh〉=-4.35 m for T2016. This mismatch also points to increased scattering at internal interfaces at low incidence angles, as concluded from backscatter modelling (see Appendix A).

The impact of the incidence angle on the mean discrepancy between the observed dh and the computed hbInv is comparatively small. For T2103/14 the mean value for dh - hbInv amounts to -0.33 m, for T2016/18 to 0.20 m (Table 3). This shows that the uniform volume approach, based on the volumetric coherence, delivers on average reasonable penetration corrections. However, biases for deep and shallow penetration, respectively, indicate systematic deviations from the uniform volume approach. In the first case the smaller grain size of the top snow layers causes a shift in the scattering phase centre to larger depth compared to exponential extinction. The second case, an overestimation of hb in wind-exposed low-accumulation zones, can be attributed to the larger size of the scattering elements and a denser sequence of internal interfaces.

In this study we investigated the feasibility for estimating the penetration-related elevation bias of interferometric topographic products over snow and ice by inverting the volumetric coherence. Single-pass across-track SAR interferometry has been widely applied for comprehensive, spatially detailed measurements of glacier and ice sheet topography as the measurements are not impaired by temporal decorrelation of the interferometric signal, variations in atmospheric propagation conditions, cloudiness and variable illumination. A main concern for the use of InSAR DEMs over glaciers and ice sheets is the correction of the elevation bias. A common approach is the use of laser altimetry data as a reference for vertical co-registration (e.g. Abdullahi et al., 2019; Rizzoli et al., 2017a, b; Wessel at al., 2016). However, altimetry data often lack the required temporal coincidence and coverage for comprehensive corrections.

We applied and evaluated the method of Dall (2007) for deriving the elevation bias of dry polar snow and firn by inverting the volumetric coherence of X-band InSAR data of the TanDEM-X mission. This method is based on the assumption of a uniform volume with constant scattering and absorption properties, whereas actual snow–firn volumes show depth-dependent changes in the density and the scattering elements as well as variations at a small vertical scale related to stratification. The use of a simple model is required because the inversion of a single parameter does not allow the representation of the complex layered structure. The study area, Union Glacier in Antarctica, comprises ice-free surfaces, bare ice, dry snow and firn with different structural properties depending on wind exposure and accumulation, a suitable environment for studying the performance of the inversion algorithm. For the statistical analysis we focussed on level glacier areas, including sites of field measurements, in order to minimise the impact of possible errors in SAR image co-registration with optical reference data.

The TanDEM-X data set comprises interferometric pairs with different interferometric baselines as well as with two distinctly different incidence angle ranges. This enables us to study the impact of these parameters on the computed elevation bias. In spite of the simplified representation of the vertical backscatter profile, the inversion according to the model of Dall (2007) provides reasonable estimates. The mean values of the computed elevation bias over the level glacier area, derived from data of the different TDM scenes, range from -4.4 to -5.8 m, varying with the baseline and incidence angle. The mean differences between the computed penetration bias and the estimates based on differencing of optical and TDM DEMs range from 0.38 to -0.64 m for the different scenes. There is a trend for overestimation of the elevation bias in areas that are subject to high wind exposure and low accumulation rates and for underestimation in areas with higher accumulation. In both cases deviations from the uniform volume structure are the main reason. In the first case the dense sequence of horizontal structures related to internal wind crusts, ice layers and density stratification causes increased scattering in the near-surface layers. In the second case the smaller grain size of the top snow layers causes a downward shift in the scattering phase centre compared to the uniform volume approach.

Advancements for the estimation of the InSAR elevation bias can be expected from progress in the representation of snow–firn structural properties in models for radar signal propagation. After all, the derivation of the interferometric elevation bias from volumetric coherence is a promising option which should be carried forward as it delivers spatially detailed information coinciding in both space and time with the topographic products. Fischer et al. (2020) analysed deviations from the uniform volume approach for airborne multi-frequency polarimetric SAR data. They explored sources and interaction mechanisms responsible for these deviations and tested different models for vertical backscatter contributions, concluding that in the case of single-polarisation data the inversion based on the uniform volume model is a preferred approach. Beyond that, Fischer at al. (2019a) demonstrated the added value of multi-baseline polarimetric InSAR data for deriving the depth of dominant scattering layers in polar firn, key information for locating the position of the scattering phase centre within the volume. Consequently, further progress on InSAR signal penetration in layered media can be expected from the use of polarimetric data as well as from multi-baseline and multi-angle observations.