2. General features of subsurface reflectors

ANITA reports 33 events with phases consistent with the expectations from CR-induced EAS events (Gorham, Reference Gorham2016, Reference Gorham2018; Hoover, Reference Hoover2010; Allison and others, Reference Allison2018). We compute the number of events above detection threshold E _thr in an observing time, T, reflecting from either surface of subsurface features with area coverage, A _eff, as

(1)$$\eqalign{N& \simeq A_{{\rm eff}}T\times \int_{E_{{\rm thr}}}^{\infty} \Phi\lpar E\rpar \, {\rm d}E \cr & = A_{{\rm eff}}T \times {\Phi_{0}\over \lpar \gamma-1\rpar E_{0}} \left({E_{{\rm thr}}\over E_{0}}\right)^{1-\gamma}\comma\; }$$

where we take the CR flux to be a power-law, Φ(E) ≃ Φ₀(E/E ₀)^−γ with γ ≃ 2.7 (Blasi, Reference Blasi2013). This allows us to estimate the total number of ordinary EAS events from the surface reflection (with phase inversion):

(2)$$N_{{\rm CR}} \simeq A_{{\rm eff}} T \times R_{{\rm surf.}} \times \alpha_{{\rm CR}}\comma\; $$

where we define α_CR ≡ Φ₀/(γ − 1) E ₀ ~ (E _thr/E ₀)^1−γ, R _surf. is the surface reflection coefficient, and A _eff is the effective area surveyed by ANITA in flight time T.

Similarly, we can estimate the anomalous uninverted radio events from subsurface reflection of CR-induced events. Recall that such events would be labeled by ANITA as ‘up-going’ based on their lack of phase reversal. The subsurface reflections may occur only for incidence angles small enough, so that the power transmitted into the ice at the air–ice interface is significant. For the firn index of refraction, the power transmitted downward exceeds 80% if the incidence angle is less than 70°. We note that the incident angles of the anomalous ANITA events are well below this upper bound and, in fact, these angles are smaller than the average incidence angle of the CR events (Schoorlemmer, Reference Schoorlemmer2016). Therefore, one can estimate the rate of anomalous events as

(3)$$N_{{\rm anom.}} \simeq \delta A_{{\rm eff}} T \times \lpar 1-R_{{\rm surf.}}\rpar ^{2} R_{{\rm sub.}} \times \alpha_{{\rm CR}}$$

where R _sub. is the subsurface reflection coefficient, and δA _eff is the area of the reflecting subsurface. A priori, there is no reason to assume that δA _eff is small. In general, δA _eff could exceed A _eff, especially if several layers at different depths are contributing to the subsurface reflections. However, to explain the ANITA events, only a small fraction of ice needs to host the reflecting features. The estimates in Eqs. (2) and (3) imply that the subsurface features should occupy an area

(4)$$\left({\delta A_{{\rm eff}}\over A_{{\rm eff}}} \right)\simeq {R_{{\rm surf.}}\over \lpar 1-R_{{\rm surf.}}\rpar ^{2} R_{{\rm sub.}}} \left({N_{{\rm anom.}}\over N_{{\rm CR}}}\right)\comma\; $$

where in order to account for ANITA's observations one needs, N _CR = 33 and N _anom. = 2. We plot the requisite area estimate from Eq. (4) in Fig. 2 as a function of the subsurface index of refraction assuming that the top layer has n = 1.3 (Kravchenko and others, Reference Kravchenko, Besson and Meyers2004). See the Appendix for a more detailed discussion of the assumptions going into this calculation.

Fig. 2. For a two-layer model, we plot the required area coverage of a subsurface reflector as a function of the subsurface index of refraction, n _sub. Three incidence angles (incidence angle is defined as θ _inc = 180 − θ _z, where θ _z is the zenith angle) are shown: incident angles of 55° (roughly corresponding to the anomalous event (Gorham, Reference Gorham2018)), 70° (roughly the average angle for ANITA CRs) and 80° (which is on the high end of incident angles for the ANITA CR events). Here it is assumed that the surface has n = 1.3.

In summary, a relevant candidate subsurface feature needs to satisfy the following requirements:

1. 1. In accordance with the estimate in Fig. 2, ${\gtrsim} 7\percnt$ of the area should host a reflector at some depth.

2. 2. There should not be significant attenuation for the EAS radio pulse above the reflecting feature, since this would render the signal undetectable. Roughly, if the detected anomalous event was attenuated by $\lesssim 0.2$, the resulting field amplitude would drop below the trigger threshold (Romero-Wolf, Reference Romero-Wolf2019). Since the attenuation length for radio waves in ice is 1.2–1.5 km (with some temperature dependence (Matsuoka and others, Reference Matsuoka, MacGregor and Pattyn2012)) for the frequencies probed by ANITA, this requirement is satisfied by any features not obstructed by an overlying layer of liquid water (the attenuation length in liquid water is much shorter (Ray, Reference Ray1972)).

3. 3. The reflection must occur without a phase inversion. A subsurface interface with a higher index of refraction above and a lower index of refraction below can reflect a signal without a phase inversion. Likewise, multiple layers of variable index of refraction can reflect a signal without a phase shift (Tikhonravov and others, Reference Tikhonravov, Baumeister and Popov1997).

4. 4. Given the wavelengths ANITA is sensitive to, the subsurface layer above the reflecting interface needs to be sufficiently thick, although the layer below the interface can be quite thin (Cavitte and others, Reference Cavitte2016). Similarly, the features should be > 100 m in diameter in order to be of the first Fresnel zone radius. Lastly, these surfaces likely need to be tilted with respect to the surface, such that double reflections (coincident surface and subsurface reflections) are rare. Note that the relative tilt can produce total internal reflection and suppress the signal. Given random orientations, one can expect this to occur roughly half the time.

Given these requirements, we now proceed to investigate which glaciological candidates may have the correct distribution and reflective properties.

3. Subsurface Antarctic candidates

Subsurface features that may have the right properties to account for the anomalous ANITA events include several possibilities:

1. (a) Double layers: The work of Arcone and others (Reference Arcone, Spikes and Hamilton2005) finds direct evidence for reflective surfaces without phase inversions. In particular, they find evidence for ‘thin double layers of ice over hoar’ which have reflection without inversion, and conclude that they are ‘extensive’ throughout West Antarctica. They model the observed reflections as high-permittivity ice sitting above low-permittivity hoar. The modeling done indicates that hoar thickness fluctuation is a major driver of the phase of the wavelet. These results were obtained with 400 MHz short pulse radar (ANITA is in the 200–800 MHz range).

2. (b) Firn density inversions: Ligtenberg and others (Reference Ligtenberg, Helsen and van den Broeke2011) and Kuipers Munneke and others (Reference Kuipers Munneke, Ligtenberg, van den Broeke and Vaughan2014) estimate snow and firn density (in the top 100 m of depth) in Antarctica for the period 1979–2017 at a horizontal resolution of (27 × 27) km² and a temporal resolution of 10–15 days, using a firn model that includes not only compaction, but also firn hydrology including melt, percolation and refreezing. The model has not been evaluated at the two locations of the observed ANITA events. Although there are minor variations in density due of dependence of densification on temperature following the annual cycle, these variations are quite small, and are not large enough to explain the ANITA observations. However, there are additional firn features not accounted for in this model.

   For example, ice core samples show substantial density and permittivity variations. We show as an example in the left panel of Fig. 3 a density profile from the East Antarctic Plateau (Laepple and others, Reference Laepple2016). These density variations are quite substantial and may constitute a plausible candidate for the ANITA events, and they are rather common in the 500 m area sampled in Laepple and others (Reference Laepple2016).

   It is well-known that the constructive interference effects from scattering on a medium consisting of multiple thin layers can produce a large reflection coefficient at some wavelengths (Vinogradov and Zeldovich, Reference Vinogradov and Zeldovich1977) (see also Pirozhkov and Ragozin (Reference Pirozhkov and Ragozin2015); Stearns (Reference Stearns1989) for generalizations). The ice core sample displayed in Fig. 3 has just this structure, consisting of a number of thin layers with small index of refraction differences. Motivated by this, we display two calculations of the reflection coefficients in Fig. 3. In one case (the red curve), we compute reflection from a medium consisting of regularly spaced layers of 10 cm thickness. This results in a sharp resonance feature, as expected (Vinogradov and Zeldovich, Reference Vinogradov and Zeldovich1977). In contrast, the blue curve assumes layers whose thickness is randomly chosen between 3 and 15 cm. In both instances, the layers have alternating refraction indices, chosen between n ₁ = 1.3 and n ₂ = 1.6 for a 60° incidence angle. We have explicitly computed the phase change in reflections from regular and random layers, and found that the phase shifts are close to zero for the range of wavelengths with maximal reflectivity, in agreement with the results of Tikhonravov and others (Reference Tikhonravov, Baumeister and Popov1997) for regular layers. We note that reflections from multiple layers will induce a time delay, impacting the measured time profile of the pulse. This makes a multi-layer reflection interpretation of the event reported in Gorham (Reference Gorham2018) unlikely, though it may be a candidate explanation for the event in Gorham (Reference Gorham2016). We stress that the specific ice core sample in the left panel of Fig. 3 and the modeled reflection coefficient in the right panel of Fig. 3 are to be understood merely as a proof-of-principle.

3. (c) Wind/ablation crusts and Sastrugi: These are abundant in megadune regions, and may create low-density regions with large grains above higher density snow. Sastrugi are essentially wind eroded snow, which make irregular ridges on the surface (Scambos and others, Reference Scambos2012). Given that these regions have a variety of slopes as well, they naturally help explain the lack of double-reflections (simultaneous surface and subsurface reflections). By some estimates, as much as 11$\percnt$ of the East Antarctic Ice Sheet is covered by the so-called ‘wind glaze’ (Scambos and others, Reference Scambos2012), forming a surface with a polished appearance with nearly zero accumulation due to persistent winds. This could produce both the needed reflection phase and range of angles. Since these wind crusts are denser than typical snow, they would naturally have larger indices of refraction than typical surface snow.

4. (d) Ice fabric layers: Ice-sheet fabrics are formed as a result of rheology and stress, leading to macroscopic ice crystal alignment. Some fabrics appear to have the right dielectric properties to produce a reflection without phase inversion even without the index of refraction contrasts (Matsuoka and others, Reference Matsuoka2003). In this case, it is the contrasts in crystal orientation fabric that source strong reflections (Matsuoka and others, Reference Matsuoka2003). The spatial distribution of ice-sheet fabric is not very well known since ice cores are restricted in number and distribution across Antarctica (Wang and others, Reference Wang2018). There is some indication that ice fabric layers are more widespread than originally believed (Wang and others, Reference Wang2018; Siegert and Fujita, Reference Siegert and Fujita2001; Siegert and Kwok, Reference Siegert and Kwok2000), which makes it plausible that the distribution is sufficiently common to produce the observed reflections.

5. (e) Subglacial lakes: Most lakes appear to be hydrostatically sealed, and therefore lack an air–water interface which would otherwise provide a useful reflecting surface without phase inversion. The bottom of the lake could in principle work, but only rather shallow and low conductivity subglacial lake regions (Schroeder and others, Reference Schroeder, Blankenship, Raney and Grima2015) would be able to produce a reflection without significant attenuation in water. Exploiting the time delay and amplitude attenuation in water, radio echo surveys provided the first direct evidence for that subglacial lakes were at least several meters deep (Gorman and Siegert, Reference Gorman and Siegert1999). Recent model estimates suggest that $\lpar 0.6\pm 0.2\rpar \percnt$ of the Antarctic ice/bed interface is covered by subglacial lakes (Goeller and others, Reference Goeller, Steinhage, Thoma and Grosfeld2016). Given that subglacial lakes lack a water–air boundary, and that they cover ${\lt} 1\percnt$ of the Antarctic area, we do not consider these especially promising candidates for ANITA. We note however the possibility that impurities in the accreted ice above a lake could in principle produce a higher index of refraction layer above a lower index layer, though is likely uncommon. We note that the ice above Lake Vostok has been found to contain ice fabric contrasts, which could source strong reflections (MacGregor and others, Reference MacGregor, Matsuoka and Studinger2009).

6. (f) Snow-covered crevasses/hollow caves/ice bridges: An ice-to-air boundary would have the correct properties for reflection without phase inversion. However, fumarolic and volcanic ice caves do not seem to be sufficiently common for the ANITA events. Crevasses are common in regions of fast flow, but are not common in the middle of the ice sheet where the ANITA events are observed. Note however that drained subglacial lakes (e.g. dolines) are more widespread than previously, and thereby present an additional mechanism for the generation of air cavities within the ice (Lenaerts and others, Reference Lenaerts2017).

7. (g) Englacial layers of dielectric and/or density contrasts: At depths beyond the firn (though still ${\lesssim} 1$ km), dielectric contrasts in the ice may be sufficiently common to explain the events (Peters and others, Reference Peters, Blankenship and Morse2005; Barnes and others, Reference Barnes, Wolff and Mulvaney2006). Moreover, in principle, density contrasts in deep englacial layers qualitatively similar to what is displayed in Fig. 3 may also source strong reflection coefficients. Recently, radar has been used to detect layers of sediment within the ice (Winter and others, Reference Winter, Woodward, Ross, Dunning, Hein, Westoby, Culberg, Marrero, Schroeder, Sugden and Siegert2019), which could also form a strong englacial dielectric.

Fig. 3. Left panel: An example Ice Core sample from the East Antarctic Plateau (Laepple and others, Reference Laepple2016). Here the black curve shows the density from a dielectric profiling (DEP) technique, while the red shows the result from high-resolution X-ray computer tomography (CT). Right panel: Power reflection coefficients (in power) for scattering from a multilayered medium as a function of wavelength. The red curve is calculated assuming regularly spaced layers of 10 cm thickness, whereas the blue curve assumes layers whose thickness is randomly chosen between 3 and 15 cm.

4. Future probes

We summarize the status of potential candidates in Table I. Future probes of subsurface reflections can be designed to definitively test the origin of ANITA events and to learn about the properties of Antarctic ice. One can identify the subsurface structures causing reflections by following up with a radar observation at the sites of anomalous events. Although this capability does not yet exist, it may be possible to design a future ANITA-like neutrino experiment using two spatially separated detectors for simultaneous observations of the same reflection, which could provide additional information about the subsurface layers. We leave a dedicated analysis of these possibilities for future work.

Table 1. Summary table of candidates and their ability to satisfy the requirements of the ANITA observations

If subsurface features are ultimately responsible for the ANITA events, the distribution and extent of such features will be important for ANITA going forward. Moreover, a dedicated effort to determine if the ANITA events originate from a subsurface reflector may be relevant for glaciology by providing additional information such as the extent, spatial distribution and reflective properties of these features.

In addition, it is possible that ANITA's ultra-high-energy CR waveforms may contain signs (or telling absences) of sub-surface reflections, which may produce multiple or spatially overlapping pulses. Such overlapping pulses can be expected to occur when both surface and sub-surface reflection occurs from the same originating event. Although the two up-going ANITA events do not show evidence of pulse overlap, future data may help elucidate the viability of this hypothesis if overlapping pulses are observed.

Moreover, there are additional relevant data sets that can be exploited in order to provide a comprehensive understanding of surface and sub-surface reflection characteristics. For example, digital echo models (with near total continental coverage) could be utilized for a surface roughness analysis for possible phase inversion (Howat and others, Reference Howat, Porter, Smith, Noh and Morin2019). Second, The Center for Remote Sensing of Ice Sheets (CReSIS) has used ultra-wideband microwave radar to map near-surface internal layers in polar firn, which may contain information on near surface (<1 m depths) reflections (Panzer and others, Reference Panzer2013). Further, the High-Altitude Calibration (HiCal) instrument collected data which specifically targeted surface reflection characteristics (Gorham, Reference Gorham2019).

Additional candidates may come from subsurface features originating from pond refreezing although these may only be limited to ice shelves (Hubbard, Reference Hubbard2016).