| 08 May 2026
Modeling the refractive index profile n(z) of polar ice for ultra-high energy neutrino experiments
Abstract. We have developed an in-situ index of refraction profile n(z) for cold polar ice, using the transit times of radio signals broadcast from an englacial transmitter to 1–5 km distant radio-frequency receivers, deployed at depths up to 200 m. For propagation through a non-uniform medium, Maxwell’s equations generally admit two ray propagation solutions from a given transmitter, corresponding to a direct path (D) and a refracted or reflected path (R); the measured D vs. R timing differences (dt(D,R)) are determined by the refractive index profile. We constrain n(z) near the South Pole, where the Askaryan Radio Array (ARA) neutrino observatory is located, by simulating D and R ray paths via ray tracing and comparing simulations to measured dt(D,R) values. We demonstrate that our dt(D,R) timing data strongly favors a glaciologically-motivated three-phase densification model rather than the single exponential scale height models typically employed by in-ice radio neutrino detectors. Effective volume simulations for a detector of ARA station antenna depths yield a 14 % increase in neutrino sensitivity over a neutrino energy range of 1018 − 1021 eV using the three-phase model compared to the single exponential.
Review of: Modeling the refractive index profile n(z) of polar ice for ultra-high energy neutrino experiments
Summary
Couberly et al. present in-situ measurements of the englacial refractive index profile n(z) near the South Pole, using an experimental setup consisting of an englacial transmitter positioned at variable depths and receivers deployed within the ice surface with lateral offsets ranging from 1-5 km and depths up to 200 m. Direct (D) and Refracted (R) ray paths are detected by receivers and the difference in travel time as a function of transmitter-receiver distance, dt(D,R), is used infer the refractive index profile.
The central finding is that the observed dt(D,R) data support a three-phase densification model proposed by the authors as opposed to a 1- or 2-stage densification model. The authors demonstrate that the estimated effective neutrino sensitivity volume yields a 14% increase when 3-stage densification is used to model the refractive index profile.
This manuscript could be an interesting contribution to both the glaciological and neutrino detection communities. However, I think this work would be significantly improved by a clearer outline of the experimental design section to improve organization in order for the reader to have a better grasp of how these measurements were actually taken.
General Comments
Comment 1: At present I feel that it is difficult to get a clear grasp on the geometry/operating parameters of the receivers, the SPD transmit antenna, and calibration antennas. I think it would be useful to have a more detailed description of the entire array, the types of measurements taken (both with calibration antennas and the SPD transmit antenna), description of the SPICE borehole, as well as some of the dimensions upfront in section 2.2 Experimental Layout. As it stands this section is very brief and I think a lot of the pertinent survey geometries details (for instance frequency range, acquisition depths, antennas spacings) seem to be added sporadically throughout the next few subsections. Overall, I think the content of this manuscript will be of interest to the wider glaciological community, but description of the experiment should be a bit more accessible as this is a relatively unique setup.
Comment 2: I think the authors could also consider combining Figures 2 and 3 to create one singular survey geometry figure, again, to centralize some of this information. Similarly, I think Figures 4 and 5 could be combined, highlighting the concept of the D and R travel paths.
Comment 3: From what I understand the main analysis was performed using the VPol channels. Since the array also contains HPol channels has this data also been used to investigate the refractive index profiles? Any comments on anisotropy? Although it is possible the ice in this area does not have a preferred crystal orientation.
Comment 4: It seems to me that the discussion in section 6 ‘Shadowed Zone’ could be moved earlier in the paper. Especially since it references Figure 5 (and kind of answers one of my specific comments below). In general this section breaks the flow that would otherwise exist between Section 5 and Section 7.
Comment 5: The abstract mentions: “Effective volume simulations for a detector of ARA station antenna depths yield a 14 % increase in neutrino sensitivity over a neutrino energy range of 1018 − 1021 eV using the three-phase model compared to the single exponential.”
But I don’t see this discussed explicitly anywhere in the main text of the manuscript? I assume there should be some discussion of this in Section 7?
Specific Comments
Lines 28-32: I think it is worth explicitly stating that the dielectric constant of ice typically increases with depth, resulting in a decreasing radio wave velocity profile which is specifically the cause of the shadow zones for shallow receivers you mention.
Lines 42-44: This statement probably does not need to be a parenthetical.
Lines 46-48: The parameter b0 is not defined on its own here (it is just wrapped up in the definition of the surface snow density). Even if this is just an empirical constant it should be mentioned. Also, even though it is obvious, z is also not defined after this equation.
Line 108-109: “Nearby (30–50 m away) englacial pulsers are used to calibrate channel positions.” This seems a bit breezed over and it isn’t clear to me that the authors go into much detail later on about how these calibration antennas are used other than to say “nearby calibration pulsers only test n(z) over the limited depth range of the deployed ARA antennas” in Lines 116-117. Is the calibration process critical for the analysis? If so, maybe describe a bit more in detail.
Figure 5 caption: “Shadow zone refers to the area above and to the right of the dashed blue line where pulses cannot be detected.” I’m having a little trouble understanding this part. I get that the deep pulser is at -1300 m, but I don’t quite understand the shadow zone area. For instance if there were a receiver at (-200, 2000) you would still see a path from the transmitter at -1300 m. I guess being more explicit about what this dashed line indicating the shadow zone represents would be helpful. Do you mean that if the transmitter is above this line there would be no R path?
Figure 6: the x-axis says ‘SPUNK’ depth. Should this be SPD? But maybe I’m missing something.
Line 169-170: would be nice to explicitly state upfront that ∆ dt(D,R) is the difference between the measured and simulated values of dt(D,R)
Line 193-195: Is it possible the references to Figures 11 and 12 aren’t quite right? For instance, “Figure 12 compares effective volume simulations using the two ice models for neutrino detector stations with 170 m and 100 m deep antenna deployments” but don’t these plots in Figure 12 only consider the 170 m depth case? Also when discussing Figure 11 in the earlier lines, it seems only the left panel is mentioned (ie, the 100 m case)? Considering this part of the analysis seems to be one of the main motivations for characterizing the refractive index profile, I think more time can be spent making sure the discussion of the figures is clearer.