Review of Krochin et al., Operational calibration of a fully polarimetric radiometer for stratospheric temperature retrievals.

This study shows that using polarimetric information can result in a small improvement in temperature retrievals form a ground-based radiometer.  As such, it provides an especially useful study for any group trying to determine the cost-benefit of developing a system with full polarization capability for making such measurements.  It is certainly appropriate for publication.  

Line 92 – “Overall, it was found that the ratio between the middle atmospheric signal and the tropospheric signal amplitude increases from 1.05 to 1.2.”  Does this sentence refer to the range of ratios over several months at Jungfraujoch?  I think that is the case, but what is confusing is that there is then a mention of observations at Bern.  Is there a separate range of amplitude ratio variations at Bern?  I think part of the idea of this paragraph is to show the advantage of measuring from Jungfraujoch, but it is strange that this is all based on two single days of measurements.

Line 109 –Is it obvious that these 5 measurements suggested are sufficiently orthogonal to determine the 20 unknowns?

Line 232 – Does “Δϕ = 0.65±0.01π”  mean “Δϕ = 0.65π ±0.01π”?

Figure 13 – Would I be correct in understanding that the Q-component is the small difference between the blue and red lines in the top panels.  If so, could this be plotted on the middle panels?

Line 282 – As I understand it the opacity is determined directly from the temperature, pressure, and humidity profiles obtained from other sensors, so no information directly from the measurements (e.g. a tipping curve) is used to determine the tropospheric opacity.  Do I understand this correctly?

The manuscript by Krochin et al. describes a polarimetric observation of two oxygen transitions around 53 GHz, with an emphasis on calibrating a receiver system that provides the full Stokes vector. A central claim is that the calibration of a full-Stokes system has been simplified. This is novel work and merits publication in AMT; however, the presentation can be significantly improved, and a more in-depth analysis is needed to resolve a remaining discrepancy.

General comments:

The manuscript is organized fairly, and the work is outlined in sufficient detail. However, it is partly difficult to follow the presentation, at least for a non-expert in these particular measurements, due to a lack of rigor in the details. The acronym RCP can serve as an example. Despite its central importance for the final analysis, the acronym is never defined. Presumably, it means right-hand circular polarization, but there is doubt as "rc" is used for right-hand circular polarization in Eq. 6, and RCP should logically mean something else. Further problems with the presentation are highlighted in the detailed comments provided below.

Even though the calibration process is the focus of the work, the problems encountered in Sec 8 for RCP are put to the side by the comment "an artefact from the calibration process". Understandably, an issue like this can be complicated to isolate, but it should be clarified that all possible efforts have been made to understand the problem. Firstly, what calibration value(s) could explain the problem noticed, and what shortcoming in the setup or measurements could cause the value(s) to be wrong? Further, the radiative transfer tool used is assumed to be perfect. Presumably, modelling the Zeeman effect is a complex matter. Could there be issues in the spectroscopic parameters used, or any simplification in the implementation? Or could the information in the full Stokes vector also be analyzed for some other polarization pair (such as V and H) to shed more light on what could have gone wrong in the calibration?

Related to the last comment above is a seeming inconsistency between different parts of the manuscript. The manuscript is about "a fully polarimetric radiometer", and the full Stokes vector is also treated in Sec. 5. However, while in Secs. 7 and 8 just two polarisation states (or just the total intensity) are considered, while the full Stokes vector corresponds to four independent values. If RCP and LCP actually represent the full Stokes vector (any element zero?), this must be clarified. If this is not the case, an extended analysis, as indicated above, should be conducted.

Detailed comments:

Title: Why include "Operational"? No operational measurements are discussed. Most critical calibration measurements are done in the lab.

Introduction: The start of the Introduction is unclear, jumping between a broad statement about atmospheric observations and details regarding 3rd and 4th Stokes to ocean wind measurements. Consider restructuring the entire Introduction to achieve a logically flowing text. In addition, the Introduction should clarify exactly what has been done regarding full Stokes, at least within atmospheric measurements, but preferably all passive Earth observations. Some references are provided, but it is unclear whether this is a comprehensive review or merely a selection of examples.

Line s25-26: "The coupling to ..." Is this meant to describe the Zeeman effect? Please, describe more carefully.

Line 31: (Krochin et al., 2022b) The same mistake is found at other places.

Line 45: Not needed to also consider Doppler broadening?

Line 47: Is this really the first investigation of this broadening? In general, there seems to be a lack of older works.

Line 54: What 7? Figure, section or equation? In addition, no need to write "see" in these references, it is implied.

Figures: The descriptions of the figures are, in general, too short. Ideally, it should be possible to grasp the figure without reading the text. In general, there should not be a figure title; that information should be in the figure text. Take care of details.

Figure 1: As an example, write "Frequency offset [MHz]", instead of "fr offset [MHz]"

Figure 2: Explain the numbering of components in the figure text, not in the main text (neither repeat in the main text). It's a bit confusing to start the numbering of components in the first figure with 3.

Line 71: In what way are eight spectra obtained?

Lines 72-73: An incomplete sentence.

Line 77: Figures shall be cited in order.

Figure 4: This figure can be removed. It does not provide any new information.

Figure 7: What is displayed in the second panel must be defined.

Line 99-101: Please explain why.

Eq 5: The mathematical notation shall distinguish between scalars, vectors and matrices. Please refer to the journal's author instructions.

Line 111: What grid? It seems that a description of the measurement setup is missing. It is not sufficient to broadly refer to some references.

Line 120: "g33, g44 are functions of gvv, ghh" Clarify the nomenclature. Or does this just mean that: g33 and g44 are functions of gvv and ghh

Line 144-145: This needs further explanation. As a general note, if any quantity is not in SI units, this needs to be clarified.

Line 164: What is "principal brightness"? In any case, the meaning of T1 and T2 must be clarified.

Equations: Please, describe more clearly how different equations are related. Eq 39 is one such example. From what equation(s) can this be derived? For clarity, this review has not considered the details in the equations.

Line 206: T3 and T4 are presumably from Eq 5. However, it would be helpful to simply add (Eq. 5) after discussing T3 and T4.

Line 254: Here T_LCP and T_RCP are introduced. They are defined mathematically, but an explanation of their physical meaning must be added - especially their relationship to T_lc and T_rc.

Sec 6: Most of the content of Sec 6 is found in a long series of articles, and can be removed. To leave room for describing the core content more carefully. On the other hand, some parts of Appendix A, of less standard character, are vital to understand some results. There is no reference to Appendix A, and it was first during a second reading that some details became clear.

Line 271: Isn't the azimuth angle also of importance?

Line 283: What is a baseline correction retrieval?

Line 286-287: "no sensor characteristics were implemented" sounds wrong. At least angles and channels are mentioned.

Line 288: Treated as independent? In what way?

Line 292: Any motivation to set z_c to 1 km? Later, \sigma_a is set to 30 K. Fluctuations of 30 K with a vertical size of about 2 km do not sound like a reasonable assumption.

Line 309: Motivate why reporting results from all these combinations? One reason is a significantly non-linear situation, but as the results throughout are similar, this does not appear to be the case.

Line 311: There is a missing space between K, and \sigma_a.

Line 313: Please define all acronyms. Here an explanation of MR is missing.

Line 361: Presumably, the total intensity spectrum was fitted well, but please clarify.

Line 400-401: This refers to circular polarization, but the manuscript is about the full Stokes vector.

Lines 428-429: Agreed that this improves the temporal resolution, but can this not also be achieved by other polarization pairs, such as V and H?

Addition:

The text above was completed before the authors uploaded a reply to RC1. In the reply they write "Since we have never estimated the rotation of the polarization plane within the optics, our Q- and U-Stokes components are relative to a rotated polarization plane. Because of this, and because we never use the Q-Stokes component, we did not show it in the manuscript. A plot of the Q-Stokes component for this date is attached below this response."

This may be clear for an expert after reading the manuscript, but it is likely to be missed by most readers. In fact, some comments give a different impression. In the abstract, it is stated that "we present the full-rank Stokes vector". This is wrong; the Q and U components are not shown. Even if "full-rank Stokes" could be claimed to be correct in some manner, it is a misleading statement, as not all Stokes elements are determined in such a way that they can be used in the retrievals. Or any other meaningful manner?

Accordingly, the manuscript must be carefully rewritten to more appropriately describe the work and the achievements, in line with the comments above. To this can be added, is the simplification of the calibration only a result of the lower ambition in the determination of the Stokes elements; is there any novelty at all in the approach?