Ice crystallizes in the shape of hexagons, and the direction normal to the basal plane is described with the c axis . Ice crystals are strongly anisotropic and 60 times softer along the basal plane than perpendicular to it . In a given strain regime, individual ice crystals deform preferentially along the basal plane and orient themselves so that the bulk c-axis orientation forms a distinct pattern, which we refer to as ice fabric. Elsewhere it is also described as crystal-orientation fabric (COF) or lattice-preferred orientation (LPO) . The radio waves are sensitive to the dielectric anisotropy, which follows the mechanical anisotropy described by the second-order orientation tensor . The bulk ice fabric pattern is described with a second-order orientation tensor (we refer to this as orientation tensor) using the eigenvectors v1,v2, and v3​​​​​​​ and eigenvalues λ1,λ2, and λ3 of an ellipsoid that best represents the average c-axis orientation of all ice crystals in the sample. The eigenvalues are normalized 1λ1+λ2+λ3=1, and to be consistent with the past polarimetric-radar studies, we assume 2λ1<λ2<λ3. Combination of Eqs. () and () set bounds on the eigenvalues (0≤λ1≤0.33, 0≤λ2≤0.5, and 0.33≤λ3≤1). The eigenvalues can be used to distinguish the ice fabric types such as isotropic (λ1≈λ2≈λ3), girdle (λ1≪λ2≈λ3), and single maximum (λ1≈λ2≪λ3) . The eigenvalues and eigenvectors can be used to describe the dielectric permittivity tensor ε′, containing the bulk permittivities εx′,εy′, and εz′ relevant for radio wave propagation (Sect. ).

The radar data in this study were collected using a phase-sensitive frequency-modulated continuous-wave radar system with a 200 MHz bandwidth and fc=300 MHz center frequency. This radar emits linearly polarized electromagnetic waves using a slot antenna where the direction of the polarization plane is aligned with the direction of the electric field vector (E) in the antenna as shown in Fig. a.

We use terminology from satellite radar polarimetry to distinguish the directions of the polarization with H and V, although, in a nadir-looking geometry, these are arbitrarily determined because H and V both have horizontal polarization plane at depth. Here, we name the horizontal (H) and vertical (V) polarization plane consistent with . However, we want to point out that this definition is different to the one applicable to seismic shear waves, where vertical receivers have a vertical component upon reflection at depth for non-vertical angles of incidence, and vice versa.

The model coordinate system is shown in Fig. c. The aerial line (TR) connects transmitter (Tx) and receiver (Rx), and by convention, we assume that H is parallel to TR; v1 and v2 are the horizontal eigenvectors which align with the direction of the smallest (εx′) and largest (εy′) horizontal principal permittivity, respectively . Hence, θ=0∘ if H is aligned with v1. The angle α is measured by compass with ±15∘ uncertainty for georeferencing the data. Here, we use polar stereographic coordinates, where counterclockwise rotation is positive.

Radar data at all the sites were collected at a fixed α, obtained from different antenna orientation in co-polarization (HH, VV) and cross-polarization (HV, VH) configurations as shown in Fig. b. We refer to these measurements as quad-polarimetric measurement. Radar data at Dome C were collected at 20 sites in January 2014. One of the sites is located within walking distance of the ice core site EDC. The remaining 19 sites (termed E(ast)0-E18, and W(est)0.5-W18, with the numbers relating to the distance in kilometers away from the dome) are aligned in a profile which is approximately perpendicular to the long axis of the dome and parallel to the flowline (Fig. a). At EDML, data were collected in January 2017, approximately 2.7 km northeast of the ice core site EDML (Fig. b). More information related to the individual ApRES sites is shown in Appendix .

Radio signal propagation through ice sheets is polarization-dependent because of the dielectric anisotropy of the ice fabric. If the direction of v3 is vertical, and the remaining two eigenvectors (v1,v2) are in the horizontal plane, then the relation between the depth profile of the dielectric permittivity tensor and the orientation tensor is given by : 3ε′(z)=εx′000εy′000εz′=ε⟂′+Δε′λ1000ε⟂′+Δε′λ2000ε⟂′+Δε′λ3. For the dielectric permittivity at radio frequencies perpendicular to c axes, we use ε⟂′=3.15 , which is slightly lower than the value found by . The value of a dielectric anisotropy for a single crystal is set to Δε′≈0.034 . The vertical v3 assumption in this study is justified through measurements at the EDC ice core, where the direction of v3 varies only by about 5∘ around the vertical . Elsewhere in ice sheets, this may not be the case, which will cause an additional source of horizontal birefringence .

We model radio wave propagation through birefringent ice using the method developed by . It includes transmission and reflection of initially linearly polarized waves emitted with two polarization modes (H and V, with direction defined in the previous section). If z is the depth from the surface (positive downward), it assumes stratified ice with i=1,…N layers predicting the radar response as a function of the emitted polarization plane and ice fabric parameters. Radar transmission (T) and reflection (Γ) are represented by 2×2 matrices only because radar signal propagation is insensitive to the vertically directed v3. The transmitted radar wave ET and the corresponding radar reflection ER are 2×1 vectors, with each component containing the electric field information of the H and V polarization components, respectively . Because only relative phase and amplitude variations are considered, all information about the radio wave transmission and reflection can be inferred from the scattering matrix (S) at layer N: 4ER=SNET, containing the complex scattering unit 5SN=sHHsVHsHVsVVN=D2(zN)×∏i=1N[R(θN+1-i)TN+1-iR′(θN+1-i)]×R(θi)ΓiR′(θi)∏i=1N[R(θi)TiR′(θi)], where D and R are the depth factor and rotation matrix, respectively. The four elements of the scattering matrix are described as co-polarized scattering signals (sHH and sVV) and cross-polarized scattering signals (sHV and sVH).

To consider the polarization dependence of the reflection boundary, we formed the reflection ratio 6r=ΓyΓx, where Γx and Γy are the elements of the reflection matrix Γ known as complex amplitude reflection coefficients . Here we only use the real part of Γx and Γy. Therefore, r is a scalar quantity. Further details about the radar forward model implementation and definition of all the parameters in Eq. () are described in Appendix  and .

The parameters of interest that we aim to infer from the radar observations for each layer are the horizontal anisotropy Δλ=λ2-λ1, the ice fabric orientation angle θ, and the reflection ratio r. All of these quantities may vary with depth. Much information is gained by interpreting the coherence phase difference between sHH and sVV, which is a crucial development in the works from , extended by . The coherence phase difference ϕHHVV is the argument of the complex polarimetric coherence CHHVV, estimated via a discrete approximation, 7CHHVV=∑b=1MsHH,b⋅sVV,b*∑b=1M|sHH,b|2∑b=1M|sVV,b|2, with * as a complex conjugate, 8ϕHHVV=arg⁡(CHHVV), where M is the number of range bins used for vertical averaging, and b is the summation index. The depth gradient of ϕHHVV provides a way to relate the local phase gradient to Δλ at the direction of the horizontal principal axes 9Ψ=2cε′4πfcΔε′dϕHHVVdz, with 10Δλ(z)=Ψ(z,θ=0∘,90∘). The coherence magnitude 0<|CHHVV|<1 also tracks phase errors so that unreliable regions with ϕHHVV can be avoided . Therefore, we restrict the analysis to the top 2000 m, where typically |CHHVV|>0.4.

The ApRES stores the deramped signal , which is not represented in Eqs. () and (). The deramping corresponds to a complex conjugation of CHHVV . Therefore, we use Eq. () for the models and the conjugate of Eq. () for the radar data to calculate the coherence phase. We simplified Eq. () to a single-layer case (Appendix ), showing that the polarity of Ψ can differentiate the direction of v1 and v2 (Appendix ). If the coherence phase is based on the received signal, v2 is in the direction of Ψ>0 (i.e., TR ∥v2), and v1 is in the direction of Ψ<0 (i.e., TR ∥v1). When using observations, the depth gradient calculation of ϕHHVV is inherently difficult because any differencing scheme amplifies noise . We follow and apply a 1D convolutional derivative on both real and imaginary components of the complex coherence, which also avoids phase unwrapping.

In Appendix , we show that the quad-polarimetric measurement (Fig. c) can be used to synthesize the full radar return from any antenna orientation using a matrix transformation 11SN(θ±γ)=R(θ±γ)SN(θ)R′(θ±γ), where γ is the angular offset from θ. Equation () is the mathematical equivalent to rotating the antennas in the field for each polarimetric configuration. As demonstrated in Fig. , we find no significant differences between the synthesized and the full azimuthal rotation dataset with 22.5∘ increments.

For a given depth profile of Δλ(z), θ(z), and r(z), the radar return can be simulated using the forward model described by Eqs. () and (). We show a seven-layer synthetic model in Fig.  to visualize features in the radar data, which can be linked to ice fabric parameters. The model parameters used to generate Fig.  are shown in Table  .

Power anomalies illustrate the effects of anisotropic ice 12δPxx(θ,z)=20log⁡10|sxx(θ,z)|1n∑b=1n|sxx(θb,z)|forxx=HH, VV, HV, VH, where |sxx| is the amplitude of the complex received signal, and n is number of angular increments for θ. In δPHH, a number of co-polarization nodes (CPNs) occur, which result from destructive superposition of ordinary and extraordinary waves (Fig. a). The number of nodes per layer is only a function of ice fabric anisotropy in that layer, with higher horizontal anisotropy resulting in more nodes. The nodes occur at a variable angular distance (termed AD in Fig. a) if anisotropic reflection is relevant (e.g., L2 and L3 in Fig. a). The angular dependency of the co-polarization nodes on anisotropic scattering can be identified using a depth-invariant ice fabric orientation (constant θ). Previously, approximated the correlation between AD and r with a linear regression. As detailed in Appendix  we improved this by finding the analytical solution 13r=1tan⁡2(AD2). Differences in both approaches are illustrated in Fig. . Two important features in δPHH are therefore the frequency of occurrence of co-polarization nodes with depth (a first-order proxy for the horizontal anisotropy) and their angular distance (a mixed proxy for anisotropic reflections or depth-variable ice fabric orientation). δPHH can be 90∘ (e.g., L2) or 180∘ (e.g., L3) symmetric if rdB=0 or rdB≠0, respectively.

In a depth-invariant ice fabric orientation, the minima in δPHV align with v1 and v2, termed cross-polarization extinction (CPE in Fig. c). Using the radar forward model, this can be derived analytically for a single-layer case as 14δPHV(θ,z)=20log⁡10sin⁡(θ,z)cos⁡(θ,z)1n∑b=1nsin⁡(θb,z)cos⁡(θb,z), where the solutions are at θ=0∘ and ±90∘. In multi-layer cases, where θ changes with depth (e.g., L6 and L7 in Fig. b), δPHV also depends on other parameters, making it difficult to infer θ using δPHV alone.

The co-polarization nodes in δPHH can also be observed in ϕHHVV (termed DN in Fig. b). The depth of the node can be automatically estimated at the zero-phase transition. Unlike δPHH, the nodes in ϕHHVV are 90∘ anti-symmetric, and their polarity is insensitive to r. This can be used to determine the directions of v1 and v2. The angular width of the nodes (termed AW in Fig. b) decreases when rdB≠0 (e.g., L3 or L4). The absolute value of Ψ at the principal axis's directions (v1 or v2) is a first-order proxy for Δλ at a given depth (Eq. , Fig. d). Since the scaled phase gradient (Fig. d) is anti-symmetric, and only the positive gradient is in the direction of v2, we mask negative parts of Ψ from now on.

focused on the power anomalies from co- and cross-polarized measurements (δPHH, δPHV). and included the coherence phase gradient (Ψ) to quantify the ice fabric horizontal anisotropy (Δλ). However, particularly for multi-layer cases where the ice fabric parameters vary with depth, there has not yet been an established procedure for how ice fabric parameters can be reliably inverted from observations. Here, we use the previous work from , , and and provide additional justification to infer all the ice fabric parameters in a continuous depth profile.

Our approach involves data preprocessing, initializing the model parameters, and parameter optimization using a constrained multivariable non-linear least-square inverse approach . All the three eigenvalues are then estimated from the estimated Δλ and optimized r using a top-to-bottom, layer-by-layer approach assuming isotropic ice at the surface.

Once the radar forward model is optimized, we attempt to reconstruct all the three eigenvalues in a top-to-bottom approach. We use an additional assumption for the standard scattering model where the reflection coefficient can be described using the Fresnel equations . If anisotropic scattering is caused by depth-variable ice fabric, then the reflection ratio at the interfaces i and i+1 can be approximated by 19ri=±λ2i-λ2i+1λ1i-λ1i+12. Here, for the sake of simplicity, we only use the positive results for r. Solving Eq. () using the optimized r and Δλ can fully reconstruct λ1, λ2, and λ3 in a nadir geometry, which will resolve the ice fabric types' ambiguity, as explained in Appendix . At the surface, ice is assumed to be isotropic (an assumption that we discuss later in Sect. ) so that λ11≈0.33, allowing λ21 and λ31 to be inferred from the estimated Δλ1 20λ21=Δλ1+λ11,21λ31=1-λ21-λ11. The eigenvalues for the surface can be estimated by iterating through Eqs. () and () and decreasing the value of λ11 by 1.0×10-5 at each iteration until all the surface eigenvalues fulfill the requirements in Sect. . For deeper layers i+1, all three eigenvalues can be reconstructed analytically by solving 22λ1i+1=λ1i-Δλi-Δλi+1ri-1 for λ1i+1 and inferring λ2i+1 and λ3i+1 with 23λ2i+1=Δλi+1+λ1i+1,24λ3i+1=1-λ2i+1-λ1i+1, where Eq. () is a reformed version of Eq. (). However, errors during the optimization may result in a reconstruction of the three eigenvalues, which do not comply with limits inferred in Sect. . In that case, Δλ and r are varied in a systematic search to find eigenvalues within the permissible limits. Solutions, in this case, are not unique, and additional constraints on the vertical gradients are required. Here, we use the vertical gradient between the two largest adjacent eigenvalues, where -5.0×10-4<λ3i-λ3i+1zi-zi+1<1.5×10-3 and |λ3i-λ3i+1zi-zi+1|>1.0×10-6. This correction does not significantly alter the results from the previous section but assures that the inferred eigenvalues are internally consistent.

Polarimetric ApRES data collected at EDC are shown in Fig. a–d. A co-polarization node occurs at 1100 m depth, and a second node develops at about 2000 m depth (Fig. a, b). The existence of only one pair of nodes over 2000 m indicates comparatively small horizontal ice anisotropy (i.e., low Δλ), similar to what has been observed at Dome Fuji . The angular distance between the two co-polarization nodes is close to 90∘, consistent with r close to 0 dB (Fig. a). δPHV shows little depth variability (Fig. c), suggesting that the ice fabric orientation angle (θ) does not vary strongly with depth. The scaled phase derivative (Ψ, Fig. d) is unclear in terms of polarity for the top 150 m. Below that, the polarity more clearly indicates the orientation of the largest horizontal eigenvectors.

Optimized model results in Fig. e–h reproduce the principal patterns of the radar observations. The reconstructed eigenvalues (Fig. i) capture the observed transition from isotropic to a girdle-type ice fabric in the ice core data. The reconstructed horizontal anisotropy (Fig. j) captures the mean well (Δλ‾(z>150m)=0.037), albeit showing less depth variability than the observations. Note that there is no significant change in the eigenvalues and horizontal anisotropy at a depth of the node's occurrence since the node's depth depends on the integration of the horizontal anisotropy above that depth and not at that depth. The ice fabric orientation at the top 150 m is poorly constrained due to the low horizontal anisotropy (Fig. k). The mean orientation of v2​​​​​​​ below 150 m is 124∘ relative to true north, which is almost perpendicular to the surface flow direction towards 45∘. The orientation cannot be validated with ice core data, which is azimuthally unconstrained. The mean estimated reflection ratio below 150 m is low (r‾(z>150m)=-3 dB, Fig. l), indicating that the role of anisotropic reflections is small.

Next, we apply the inverse approach to ApRES data collected at the EDML drill site. Contrary to what has been observed at EDC, co-polarization nodes can barely be localized in δPHH as no 90∘ symmetry is apparent (Fig. a). This indicates that anisotropic scattering is relevant (r≠0 dB), as already noticed earlier . Moreover, the coherence phase shows many nodes (Fig. b), indicating a much stronger horizontal anisotropy (i.e., large Δλ). This is comparable to the ice core at Mizuho, equally located in a flank flow regime . Although δPHV shows almost no depth variability in ice fabric orientation (Fig. c), it is not straightforward to identify the direction of v1​​​​​​​ and v2 using the polarity of Ψ because of the strong ice anisotropy (Fig. d).

The optimized model (Fig. e–h) reproduces all basic features seen in the radar data. Inferred model parameters closely follow the ice core measurements both in terms of absolute eigenvalues (Fig. i) and horizontal anisotropy (Fig. j). A shallower development of the girdle ice fabric compared to EDC is detected. At this site, the mean ice fabric anisotropy at the top 200 m is weak, but in comparison to EDC it is strong enough to detect the ice fabric orientation. The mean estimated horizontal anisotropy below 200 m in EDML (Δλ‾(z>200m)=0.265) is more than 7 times stronger than EDC. The mean inferred orientation of v2 below 200 m is 174∘ relative to true north (Fig. k). Similar to EDC, this is near-perpendicular to the ice flow direction at the surface towards 90∘. The estimated reflection ratio in EDML (Fig. l) can be divided into two major zones (r‾(200m<z<850m)=16 dB, and r‾(z>850m)=-15 dB). Contrary to EDC, anisotropic reflections are more relevant, and the previously suggested existence of two anisotropic scattering zones above and below approx. 850 m appears in the observations and the optimized model output.

After investigating specific characteristics of a dome position (EDC) and a flank-flow regime (EDML), we next investigate a local dome-to-flank transition (36 km). At Dome C, 19 sites are located along a profile extending 18 km away to either side from the local ice dome (Fig. a), and a summary of the results is presented in Fig. . We focus on the upper 2000 m, where the signal-to-noise ratio and the coherence magnitude are sufficiently high. All stations yield coherent results, showing an isotropic ice fabric that gradually evolves into a weak girdle with depth. The depths of the first co-polarization nodes can be detected at all sites (dashed green line in Fig. b). It is shallowest beneath the dome and moves to larger depths further away from the dome in the flanks. The depth variability in the co-polarization nodes results in a Δλ that is most anisotropic beneath the dome and less anisotropic in the flanks (Fig. c). The orientation of the eigenvectors is poorly constrained in the upper 200 m. At larger depths, they are oriented parallel (v1) and perpendicular (v2) to the surface flow direction, in line with what is inferred in Sect. .

The method we developed in this study extracts the depth variability in ice fabric horizontal anisotropy (Δλ) and anisotropic reflection ratio (r), which leads to estimating all three eigenvalues required for the second-order ice fabric orientation tensor. We also estimate the georeferenced ice fabric orientation (v2) as a function of depth. The results of our method are comparable with laboratory measurements and could be integrated into anisotropic ice flow models . Our main assumption is that the strongest eigenvector (and with it the orientation tensor) is aligned in the vertical.

In terms of the data preprocessing, there are no structural differences in our data between synthesizing the polarization dependency out of a single set of quad-polarimetric measurements (Appendix ) and the more common polarimetric measurements in glaciology, where antennas are kept parallel or perpendicular while being rotated several increments between 0 and 180∘ . In addition to significantly reducing the field time for data acquisition, an advantage of these measurements is that the georeferencing error only occurs once during antenna setup and is not accumulated over multiple re-positioning cycles. However, it is required that the data have a sufficiently high signal-to-noise ratio (e.g., using |CHHVV|) in order to not synthesize misleading symmetries out of noise.

The signal quality and noise level, particularly in the HHVV coherence phase, are important. In areas with high horizontal anisotropy and consequently densely spaced co- and cross-polarization nodes (i.e., the EDML case), care needs to be taken that the denoising does not average over multiple nodes. Derivation of the initial guess for the inverse approach depends on the data quality and is guided by characteristic features in synthetic forward models, some of which can be analytically described for one-layer cases. Multi-layer cases, however, are difficult to interpret, particularly if the ice fabric orientation (v2) changes strongly (by several tens of degrees) with depth (e.g., ice shelves and glaciers). Fortunately, this does not appear to be the case for the data presented here (Figs. k, k) so that the initial guess already results in a forward model that adequately captures characteristic features in the data. The optimization improves the model–data misfit but does not lead to significant differences with our first informed guess. Nevertheless, this step is required to predict the depth variability in all the three eigenvalues (Sect. ).

The reconstruction of the eigenvalues assumes isotropic ice and firn for the surface (λ1=0.33). This is reasonable for the dome and flank-flow settings considered here but may need to be revisited in other settings where ice fabric can develop near the surface as ice streams and ice shelves. More critical is the reflection ratio itself, which is ill-constrained in magnitude and amplifies small changes in the eigenvalues across the reflection boundaries. This is mitigated by the range of allowed eigenvalues (Sect. ), and it is those constraints that facilitate the derivation of all eigenvalues from the anisotropic reflection ratio. The predicted eigenvalues (λ1, λ2, and λ3) in this method show a good match to the ice core observations in both cases.

The azimuthal constraints that radar polarimetry provides can, in general, not be validated by ice core measurements, with few exceptions (e.g., ). However, the alignment of the ice fabric principal axes with the surface-flow direction detailed below adds credibility to our inferences and shows advantages of this approach over previous attempts focusing on the power anomalies only . The underlying reason for this is that the polarity of the depth gradient of the polarimetric coherence phase is independent of anisotropic scattering.

The inversion requires an initial guess (Sect. ) that is based on experience from synthetic test cases. In our experience with radar polarimetry and the explored ice dynamic context, this grants a robust solution, also because a wrong initial guess results in a large model–data misfit that can be identified easily. In the future, this can be improved by using gradient-free optimization schemes (e.g., in a Bayesian framework) that can correct for a poor initial guess by exploring the parameter space more systematically.

Our strongest assumption is that the strongest eigenvector (v3​​​​​​​) should be close to vertical. While this assumption is justified here, as flow at domes is dominated by vertical compression, and the crystal c axis tends to be aligned in the vertical, it may not apply elsewhere in ice sheets and cause an additional source of horizontal birefringence . While it is possible to explore other effects than that of the largest eigenvector being vertical (, p. 13), it is impossible to circumvent the fact that the radio wave propagation is vertical and hence insensitive to changes along that direction. In the future, we envision the use of wide-angle surveys with curved ray paths (e.g., ) to overcome this limitation.

With the assumptions mentioned above, radar polarimetry is now a step closer to constrain the second-order orientation tensor. However, this is still not the full representation required to characterize all ice fabric types, for example because a strong vertical girdle and weak horizontal cones will have a similar second-order orientation tensor. A combination with seismic studies recovering the fourth-order elasticity tensor is therefore still warranted.

We now investigate our inferred characteristics of ice fabric variation at the dome, where flow is dominated by vertical compression, compared with the flanks, where flow is dominated by vertical shear. Our inverse approach shows higher horizontal ice anisotropy at EDML compared to Dome C throughout the ice column. This increase from the dome to the flank supports earlier inferences that ice anisotropy is larger in areas with significant horizontal strain compared to settings where vertical compression is dominant . This is in contrast, however, with the observed decrease in ice anisotropy in the Dome C transect (Fig. c), where the ice fabric is more anisotropic at the Dome compared to the flanks. Our hypothesis is that this near-field anomaly reflects ice dynamic modification of ice fabric through the Raymond effect . predict local ice-dynamically induced ice fabric variability up to approximately five ice thicknesses to either side of ice divides. The 36 km long Dome C transect images an ice thickness of about 3000 m and hence approximately covers this domain. The absence of Raymond arches in the radar stratigraphy beneath Dome C suggests that these need a longer time to evolve, whereas the ice fabric pattern likely reflects the instantaneous operation of the Raymond effect. We acknowledge that there are other explanations for the ice fabric pattern under Dome C, such as across-profile flow or bedrock influence. In any case, we want to highlight here how, due to the spatial extension of our observations, our inferred ice fabric distributions combined with an anisotropic flow model can be used to test these and other hypotheses.

Focusing on the top 200 m of the inferred Δλ and v2 reveals a significant difference between the two sites. At EDML, the ice fabric anisotropy is stronger in the top 200 m, resulting in a better-constrained ice fabric orientation, whereas at EDC it is entirely unconstrained (Figs. d and d). It appears that the ice fabric orientation develops more rapidly in areas with significant horizontal flow compared to areas with essentially vertical compression only.

In both the EDML and Dome C areas, the inferred ice fabric orientation varies little over the depth intervals considered, and in both cases, the inferred orientations line up with the surface flow field. More specifically, v1 is approximately oriented along-flow, and v2 is approximately oriented across-flow. Those directions also align with the principal strain rate components (Fig. ) in Dome C and EDML . In both cases, v2 is approximately parallel to the direction of the maximal principal strain rate component, whereas v1 is aligned with the along-flow minimal principal strain rate component. At Dome C, where ice flow velocities are low, derivation of the strain rate field is not trivial and adds additional assumptions of the surface topography . Note that ice is compressing in the direction of flow and not extending, as is often assumed in simplified theoretical examples, which is why it is important to reference the ice fabric to the direction of extension and compression and not the flow.

The origin of the difference in radar polarimetry between EDC and EDML is the degree of ice fabric alignment in the horizontal, which can be quantified as the difference between the horizontal eigenvalues of the orientation tensor. This difference is larger for EDML than for EDC. Our study adds to the body of evidence that ice fabric is induced by flow because the preferred direction for horizontal ice fabric aligns with the direction of compression . In addition, the stronger horizontal alignment of the ice fabric at EDML, compared to EDC, corresponds to a stronger compression that can be observed by comparing the strain ellipses in Fig. . It is interesting to notice how sensitive radar polarimetry is to horizontal ice fabric alignment, the main observable for downward-looking radar. Despite the small differences in horizontal ice fabric eigenvalues at EDC (Δλ<0.05 in Fig. j) our technique is able to recover ice fabric in most of the column. This is of particular interest as the ice fabric could contain a record of past changes in ice flow conditions .

More theoretical work is required to understand the vertical variability in horizontal anisotropy, which is picked up in radar polarimetry through the strength of the anisotropic reflection ratio. At EDML, the reflection ratio is a dominant and required factor to explain the radar signatures, while at Dome C, it is close to negligible. have observed a similar increase in anisotropic scattering between Dome F and Mizuho, suggesting that this may be a generic feature in ice sheets that requires more investigation. Contrary to EDML, the signal at Dome C is dominated by birefringence, and the contribution of anisotropic reflection is small. Yet, it appears that it leaves a small signature in the data that can be detected. Moreover, our analysis suggests that there are no other mechanisms (e.g., a directional interface roughness) contributing to anisotropic reflections. This point requires confirmation from other ice core sites because the recovery of all three eigenvalues (and their corresponding directions) offers significant possibilities of constraining ice fabric in ice sheets in general.

Although anisotropic reflections at Dome C are small, there is a noticeable change in the δPHH of direction in the depth interval from 1500–1700 m, which coincides with the transition from Holocene to glacial ice, as is also the case for the EDML site . The inversion does not pick up this feature in r as it is at the boundary of the domain (Fig. l), and we do not have a complete understanding why glacial–interglacial transitions should be accompanied by changing reflection ratios. Nevertheless, this may provide us with an additional tool to explore age–depth relationships at future ice core sites.