Ice masses, such as glaciers and ice sheets, are made up of many individual ice crystals of varying size, shape, and orientation, together forming the bulk crystal orientation fabric (COF) [1]. On Earth's surface, and for the planetary water ice masses that we know of, the individual crystals have a hexagonal basal plane and orthogonal c axis, a molecular structure called ice Ih [2]. The electrical properties of the ice-Ih crystal are unique depending on its orientation [3]; for example, at radio wavelengths, the real permittivity is ∼1% larger (slower wave speed) when the wave is polarized in line with the c axis than across [4,5]. In other words, the crystal NSF grant #2317927 is electrically anisotropic or "birefringent", and an ice mass can inherit a bulk anisotropy if its COF consists of aligned crystals. Ice masses are commonly anisotropic since COFs evolve toward preferred directions as ice flows [6]. Measured COFs can therefore be used to interpret ice-flow history [7,8] and can help predict future ice-flow behavior caused by COFinduced mechanical anisotropy [9,10].
COF in glaciers and ice sheets can be measured indirectly and remotely with radar sounding [11]. A radar wave depolarizes as it propagates through anisotropic ice, so a measured polarization difference between the transmitted and received waveforms can be attributed to birefringence in the ice column. There are specific polarimetric survey strategies that target birefringence [11,12], but the signals can can also manifest as a radar power loss [13] even in the conventional nonpolarimetric radar surveys that cover much of the Greenland and Antarctic ice sheets [14,15]. Although birefringent power losses can theoretically be corrected as a term in the radar power equation [16], they are rarely considered in other analyses of radar power such as attenuation or reflectivity. Whether in a polarimetric or conventional survey, the birefringent signatures depend on frequency of the transmitted signal. Radar sounders are generally designed in the broad frequency band from HF to UHF (∼10 6 -10 9 Hz) for which ice is nondispersive and non-absorptive [3,17]. Higher frequency radars with a large bandwidth give superior range resolution for distinguishing annual stratigraphy in the near-surface snow/firn [18], and lower frequency radars propagate to deeper depths in the ice with less attenuation.
Here, we describe the frequency dependence of birefringent signatures in non-polarimetric radar sounding surveys and provide a power correction for the associated power losses which can be used in multi-frequency systems. We first provide the theoretical background on how birefringent signatures manifest in non-polarimetric surveys, then show an example with a dual-frequency system in East Antarctica that expresses different birefringent signatures for each transmitted frequency. We argue that, for co-deployed multi-frequency
TABLE I LIST OF ALL ALL VARIABLE DEFINITIONS USED IN THIS WORK. x, y, z spatial dimensions t time ψ azimuth angle ϵr relative permittivity ∆ϵr change in permittivity (effective anisotropy) fc center frequency v fs wave velocity in free space ϕ phase γ rotation of polarization ellipse Ω beat frequency L beat wavelength H horizontally polarized wave component V vertically polarized wave component
radars, we can use the differences between birefringent signatures to increase our confidence in the interpretation of anisotropy and correct for birefringent losses in each system. We propose our own algorithm for birefringence corrections and discuss how it may be used in both terrestrial and planetary applications.
To discuss birefringence and the associated power losses in non-polarimetric radar surveys, we first give a brief background on the nature of polarized wave propagation through anisotropic media. For a plane wave, the complex electric field amplitude can be described as a Jones vector [19] for which the wave components, H and V , can be combined to yield the equation for an ellipse [20] traced out over time, t = 1/f c , where f c is the instrument center frequency. The wave is therefore elliptically polarized. The shape of this polarization ellipse can be described with two variables, a phase delay
and a rotation
where * is the complex conjugate. At |H| = 1 the wave polarization is unrotated (linear polarization in the x direction) and at |V | = 1 it is fully rotated (linear polarization in the y direction). If the receive antenna is also polarized, it will selectively measure the electric field from the wave component that aligns with it. When a polarized wave propagates through anisotropic media, the polarization ellipse changes shape with propagated distance, and the component rotated out of view of the antenna appears to go missing as a birefringent power loss.
Full consideration of wave depolarization in anisotropic media uses transfer matrices to update the Jones vector [11,21].
Here though, we only consider power losses and in particular the "beat frequency" or periodicity of the birefringent loss signature. Rotation of the polarization ellipse stems from the differential wave speed between orientations which creates a phase delay between orthogonally polarized wave components. Following the assumptions of Jordan et al. [22], for small deviations about a mean (polarization-averaged) permittivity, the two-way phase delay is
where v fs is the free-space velocity, ϵ r is the bulk relative permittivity, and ∆ϵ r is the effective anisotropy that the wave is subjected to. Considering vertically propagating plane waves, ∆ϵ r can only change in z. The beat frequency is then calculated as the depth gradient in phase divided by 2π radians per wavelength. Within a layer of uniform anisotropy, the integral in equation (3) reduces to z∆ϵ r , so the beat frequency is
with units m -1 , where ∆ϵ r and ϵ r are the depth-averaged dielectric anisotropy and permittivity, respectively, and L = 1/Ω is the associated beat wavelength. The extent of rotation of the polarization ellipse, i.e., maximum change in γ, depends on the degree of misalignment from the principal axes. At maximum misalignment, ψ = (2n+1)π 4 , the wave is fully rotated and a power minimum is seen in the copolarized acquisition. At maximum alignment, ψ = nπ 2 , the wave is unrotated, no birefringent losses are observed, and a cross-polarized antenna receives no power (cross-polarized extinction). Unlike the magnitude, the frequency of the beat is independent of azimuth, so in cases where: i) the radar antennas are misaligned from the ice anisotropy, ii) the beat can be distinguished from repeated reflecting horizons, and iii) the ice is sufficiently thick for multiple beats to be observed, this phenomena can be quantified.
Equation ( 4) has a dependence on the instrument center frequency, so the beat signature manifests differently between radar systems. Multi-frequency systems could therefore be designed to potentially overcome requirements (ii) and/or (iii) or to correct the birefringent losses entirely.
As a demonstration of multi-frequency birefringence, we use data from the 2023-24 Antarctic airborne radar mission by the Center for Oldest Ice Exploration (COLDEX), which was flown with two radar instruments onboard the same aircraft to resolve different targets simultaneously [23].
The first instrument is the University of Texas Institute for Geophysics (UTIG) Multifrequency Airborne Radar Sounder with Full-phase Assessment (MARFA). MARFA has 60 MHz center frequency and 15 MHz bandwidth, targeting the icebed interface and deep englacial layers. The second is the Center for Remote Sensing and Integrated Systems (CReSIS) accumulation radar. This CReSIS radar has 717.5 MHz center frequency and 60 MHz bandwidth, targeting the shallower englacial layers and with finer range resolution. The transmit and receive antennas are co-polarized for each, but the two systems are perpendicular to one another.
The airborne survey broadly covers the region between South Pole and Dome A, with the objective to find a glaciological setting which may preserve old ice [24]. The selected flight track in Figure 2 is along 110 • E longitude, crossing the continental ice divide between Beardmore and Nimrod glaciers (which flow into the Ross Ice Shelf) and Academy Glacier (which flows into the Filchner Ice Shelf). Ice velocities are slow in this area (order 1 m/yr), so the anisotropy in presentday ice would have taken centuries or millennia to develop.
The birefringent beat is visible at both instrument frequencies. In places where anisotropy is weak (e.g., at ∼10 km along-track distance) the beat is not well resolved by the lower frequency system since the beat wavelength approaches the full ice thickness. On the other hand, when the anisotropy is strong (∼260 km along track) the low-frequency beat is better distinguished and the high-frequency beat is difficult to distinguish among the reflectivity variations. In this way, the two beats in a dual-frequency system are complementary measurements.
The birefringent beat is commonly used itself as a signal for interpretations of ice anisotropy [25,26], but it is also a loss term in the radar power equation [13]. In theory, one could correct for birefringent losses with a model of the ice anisotropy, but in practice such an effort is difficult since it requires a precise knowledge of the past and present ice flow. Otherwise, corrections are plausible where an ice core has been drilled and the ice sampled directly, but extending that correction would require an assumption for how the anisotropy changes away from the of observation. Instead, a more viable birefringence correction uses multiple independent geophysical measurements at distinct frequencies, removing the circularity of using a signal to correct itself. Knowing that each waveform interacts with the same ice, the only free variable in equation 4 is the center frequency, so any additional measurements at variable frequency add confidence for noisy signals.
We add our correction into the standard processing pipeline (Figure 3). To extract the cleanest beat signature, we first average traces incoherently within a moving aperture. This averaging decreases the coherent signal for reflecting horizons (englacial layers) while maintaining the incoherent magnitude which expresses the birefringent beat. We then use a highpass filter in the fast-time dimension to filter the radar power image, removing the low-frequency signal (attenuation).
With the two filtered images, we convolve their signals in the frequency domain, using an expected offset in their beat signature based on their different center frequencies in equation 4. Then, if there is a frequency band in which a highly correlated beat is observed in both signals, that band is artificially suppressed in each. Finally, the beat-suppressed signal is returned to the time domain and the net power difference is applied to the original, unfiltered image. In the COLDEX example (Figure 4), we observe clear beat frequencies at 1.7 and 8.5 km -1 in the low and high frequency systems, respectively. The power correction is applied to both signals and effectively removes the dominant beat signature.
We described the physical nature of radar birefringence in ice, the frequency dependence of birefringent power losses, and designed a new birefringence correction applicable for dual-or multi-frequency radar sounding. Since our method leverages two distinct measurements of the same material property, it is robust to noisy signals. In other words, multifrequency birefringent analyses are feasible among reflectivity variations (e.g., for the reflection at the ice-bed interface) whereas single-frequency analysis is generally not.
Applications of our correction include those for investigations of radar attenuation and interface reflectivity in glaciers and ice sheets which are commonly used to interpret englacial [27,28] and subglacial [29,30] properties, respectively. Past studies of attenuation and reflectivity have generally ignored birefringent losses since, in commonly used single-frequency radars, the beat signature cannot always be distinguished from reflectivity variations. Birefringent losses could bias those previous results in cases with anisotropic ice. For example, attenuation estimates which use reflected power from the ice-bed interface assume that the correlation between power and thickness is primarily a result of attenuation [31,32]; however, as we showed above, birefringent losses vary with thickness (range/depth) as well. In fact, for commonly used radar frequencies and with realistic ice anisotropies, the depthpower gradient is within the range of realistic attenuation rates ∼5-20 dB/km (e.g., solid black line in Figure 1A). Our method requires an assumption that the ice anisotropy does not change within the vertical depth window over which we infer the birefringent beat, so it may be limited in areas with a strong vertical gradient in crystal fabric.
Multi-frequency birefringence analysis and our power correction also have relevance for radar sounding in planetary environments, although less work has been done on ice anisotropy there. The two radar sounders currently orbiting Mars operate at different frequencies, MARSIS at 1.3-5.5 MHz [33] and SHARAD at 20 MHz [34]. Those two instruments are on separate platforms with different orbits, but it could still be feasible to search for birefringent loss correlations between the two in places where anisotropy might be expected to develop in the Martian polar layered deposits. Even more promising, the REASON instrument aboard the Europa Clipper is dual frequency itself, with a lowfrequency (6 MHz) and a high-frequency (60 MHz) band [35]. Anisotropy may be expected in Europa's ice shell, based on convection driving COF evolution [36]. If so, a birefringent correction will be necessary for any interpretation of reflected power from beneath the anisotropic ice.