The research article “Assimilation of radar freeboard and snow altimetry observations in the Arctic and Antarctic with a coupled ocean/sea ice modelling system” introduces a multivariate assimilation system using LEGOS radar freeboard and KaKu altimetry snow depth in addition to sea ice concentration observations and compares the performance of this novel model run to a free model run and one using only sea ice concentration observations. The paper is well structured, provides a good literature overview and clear motivation for the work. Many comparisons against independent validation data are presented and the results are discussed appropriately. Furthermore, the figures are neat, and the results prove a clear overall improvement compared to the other model runs. Therefore, I recommend this article for publication after minor revisions:

General comments:

1. As most plots are only for one specific month, it would be nice to have the same plots for all months in the appendix or to publish them as supplementary dataset.
   
   Moreover, there are several instances, where other months are discussed with a comment “not shown”, making it hard to follow.
   
   I also wonder how you chose the months for each plot, as the choices are not consistent (Fig 2 shows July 2017 and September 2017, Fig 3 and 8 show April 2017 for the Arctic, Fig 4 shows some plots for May in addition to October 2017 for the Antarctic, but Fig 9 shows September 2017 instead of October and Figure 5 and 6 show October 2018 and Jan-Feb 2019). There might be good reasons for the choices, but to prove that the plots weren’t cherry-picked to support the author’s arguments most, all plots should be published somewhere alongside the manuscript.
2. The appendix adds a valuable comparison to completely independent in situ data and I wonder why you did not include these in the main text.
   
   It would also be helpful to add RMSE values to these scatter plots and discussion.
3. Throughout the paper, English grammar is not always used correctly, but the text is generally understandable. I, therefore, trust that copy editing will deal with this.

Specific comments:

L.110: SSMIS is not explained. Here, I would maybe just talk about SIC.

L.200 onwards: In the methods section, SSMIS should be explained and mentioned in the text, also to make the difference to the OSISAF AMSR2 product, which is used later on, clearer.

L.207: How was the 40% value for Antarctica chosen?

L.213: ‘measure’ rather than ‘detect’

L.225 and 230: Which months are counted as winter? Please specify.

L.226: Do you average all altimetry observations within a model grid cell or within a radius from the grid cell? Any weighting? Please specify.

L.232: Why do you scale the uncertainties and how do you decide on this range?

L.245: How were these dates chosen?

L.252: referred to ‘as’ leads

L.254: I would call it ‘lead fraction’ rather than ‘lead content’

L.257: explain CDR

Figure 1 caption: Do you mean ‘range’ rather than ‘surface’ covered by them?

L.259-262 and L.285-314 + Fig 2: I suggest a separate section for SIC e.g. before the lead section. Especially the first paragraph on sea ice concentration (L259-262) currently sits between two paragraphs on lead fraction and interrupts the flow.

Figure 2: I would stick to SIC rather than SICONC in the titles and colourmap legend;

In the caption ‘experiences’ should be ‘experiments’?!

L.318: ..but there are more unobserved polynyas for MULTIVAR according to Fig.2 ?!

Figure 3 and 4: Shouldn’t the RMS have the same unit as SNV and RFBV?

I really like the distribution plots in a). Maybe these could also be added to the plots for SIC (Fig.2), total fb (Fig. 5, 6) and SIV (Fig. 8,9)?

Figure 4: It would be nice to see the full first peak around 0. Maybe add an inset with a higher y axis.

L.354: FREE diverges the most, but also matches best with the observations in May

L.460: ‘excludes’ rather than ‘includes’?

L.472: Stick to SIV instead of SIVOLU (I think this is what you mean?!)

L.486: I am missing a paragraph on the Antarctica plot in Figure 7b and specifically also a comment/explanation why the timings of sea ice volume decrease are offset between observations and model. In 2018, the observations clearly drop between September and October, whereas the models are still increasing.

Figure 8: I find greener colours in the table to mark worse results counterintuitive and would suggest using another colour like yellow.

Figure 9: Explain the white areas in the figure caption

L. 601: You say most of the analysis in Antarctica was done in summer when no data is assimilated, however, most plots for the Antarctic are shifted by 6 months compared to the Arctic and if not, why don’t you show those plots to make it a fairer comparison? Ideally, as mentioned above, all plots should be available to the reader anyway.

L.678: Explain VP and EVP

L.742: CRISTAL will also have a higher inclination orbit and hence provide these measurements with a much smaller hole (data gap) around the poles.

L.744: CIMR will also provide thin ice estimates like SMOS from L-band radiometry.

Review of "Assimilation of radar freeboard and snow altimetry observations in the Arctic and Antarctic with a coupled ocean/sea ice modelling system" by Chenal and co-authors. 

I am mostly an expert of sea ice data assimilation, but no so much of altimeter remote sensing. 

The manuscript presents "the first implementation of a multivariate sea ice assimilation scheme in both the Arctic and

Antarctica within a global ¼° modelling and analysis system", as stated by the authors, which is correct to my knowledge, other comparable studies are cited in the text but, at much lower resolution. 

The study takes a data assimilation system as used operationally and develops both a multivariate (multiple sea ice variables, though not ocean variables) assimilation scheme assimilating two new data sources on top of sea ice concentrations: radar freeboard and a novel snow thickness product. This represents an ambitious piece of work, of high technical complexity, associated with expensive computations. 

The choice of satellite observations is of high standards and makes the experiments very relevant and timely in the present literature. 

There are however some weaknesses that the authors should address before the paper is accepted for publication. 

The two assimilation experiments do not allow to evaluate separately the multivariate assimilation scheme from the effect of the assimilated data (multidata), as intermediate combinations of multivariate-monodata experiments would have allowed. The only exceptions are the locations and times when one or two input data types are missing. The paper is therefore shy on practical take-home messages. Rather than expensive additional experiments, the discussion of the benefits or weaknesses of the multivariate scheme could exploit better these special cases. In the present state of the manuscript, the two issues are mixed and hard to disentangle. 

Second, some explanations are omitted, or left implicit, which hampers the fluidity of the arguments: I am for example missing a paragraph in the introduction that sets upfront what the experiments are expected to deliver (which improvements and which possible degradations) and what are yet open questions. In practical terms, this means that the research questions should be coming two paragraphs earlier in the introduction and should be linked more concretely to the results. The physical instability mechanisms of the Southern Ocean polynyas are also exposed too late in the paper and should have been given earlier in the introduction. 

The experimental setup may have omitted some information that is important to understand the results later on. I was missing a clear indication of where the RFB and KaKu snow data are assimilated until I found Figures 3 and 4. More importantly, I am missing an explanation of what the assimilation does in the absence of one or two of these datasets. Perhaps introducing the observations before the data assimilation method could make the logic more fluid. The authors do not lay out the limitations of the data assimilation method upfront, so the reader discovers them by surprise as the negative results appear. For example when the MULTIVAR scheme deteriorates the SIC in Figure 2. I would expect - by elimination - that the assimilation of RFB is responsible for this bias. Since the multivariate (negative) correlation between RFB and SIC stems from a long model simulation, it could be that this long simulation is to be blamed. See for example Counillon and Bertino (2009) although in an ocean-only application (as an example, you do not have to cite this reference).  

Some of the minor design choices could have been better justified, for example the threshold on snow depths, which is not justified and comes back as a limitation later on. 

The snow thickess is the difference between AltiKa and CryoSAT-2 RFBs, so it is not independent from the CryoSAT-2 RFB, whereas the assimilation system assumes observations errors are independent. Although I do not see an immediate solution for that, this may be acknowledged in the Section 2.2.3.

The text puts the same emphasis on results that are trivial (i.e. that assimilation runs agree better with the assimilated data) and those that are less obvious. The discussion could flow more easily if the description of the experiments had a few sentences with the a priori expectations from each run. 

The case for assimilating RFB rather than SIT could be more conclusive. As long as the auxiliary data used in the RFB-to-SIT conversion are the same, there is no obvious benefit of assimilating RFB rather than a converted SIT product. The ice and snow density do not seem to take an active role in the sea ice model so belong in the observation operator. However the snow depth is a proper state variable, so I would expect that more accurate (satellite) snow depths justifies alone the assimilation of RFB. By opposition, the assimilation may lead to deteriorations of SIT in areas where the KaKu snow depths are missing, but I could not find any evidence of that in the manuscript. 

The manuscript overall structure is good, with a few exceptions noted above. However some repetitions could be avoided across different sections. 

The writing style is generally good and very careful, sometimes too careful to the point of becoming confusing. The statements are often too neutral, not indicating whether the results are as expected, good or bad. This hampers the reading of the paper. 

The graphics are very clear and highlight well the main messages. 

Overall I am very positive that the study makes an important and interesting contribution to the field and is worthy of publication in The Cryosphere, provided that the issues pointed out above are corrected in a revised version. 

Detailed comments. 

- L40 Chen et al. 2024 is not in the Arctic but in an idealised square model. 

- L100 Since only RFB_mD is used in the experiments, it could be mentioned that RFB_og is only shown indicatively for users of the original product. 

- L120 The section on the sea ice model does not mention the treatment of submerged sea ice. Since negative RFBs occur in the results it is necessary to know if the assimilative model will freeze submerged snow into more saline ice.  

- L135 The data assimilation scheme does not mention that the ensemble is static and indirectly indicates a "long simulation without assimilation". The - strong - assumption that the long simulation is representative of the model errors during the assimilation period should be stated explicitly, including an indication of the years of the long simulation. 

- L148 Why the reduced grid for the ocean? Maybe this point is unimportant for the paper and should be omitted to simplify this section. 

- L162 Indicate that assimilating the OSTIA freezing point temperature is intended to make the sea ice and ocean assimilation consistent with each other.  If the freezing point temperatures in OSTIA and NEMO differ due to sea surface salinity, can the difference induce sea ice melt or freeze? 

- L178 Why a Gamma distribution? What are its parameters? 

- L183 Are there any cases when the SIT exceeds the bounds of its thickness category? How is that handled by the model? 

- L189 This threshold should be explained since I did not expect it to become important later. This is especially mysterious since it reduces snow depths that are otherwise allowed in the forward model. 

- L235 I am not sure what this statement refers to. By construction the SNV depends directly on SIC and on SNT so the Kalman filter should estimate it well, unless the long free simulation has a pathological behaviour. 

- Tables 1 and 2 are very welcome to summarise the complex information, but Table 2 could include a little more information like the seasons and latitudes at which the different observations are available, which would prepare the reader to what happens when and where RFB or snow are not assimilated. 

- L255 Ivanova et al. compared 11 algorithms. Coming down to 2 is not really quantifying the uncertainty. There are weaknesses of PMW observations in the summer and different choices of tie points made at OSI-SAF and NSIDC to cope with them. Maybe they provide a lower and upper bound rather than an uncertainty estimate. 

- L270 The SSMIS data is assimilated, not the NSIDC, so it should be expected that the results agree better with SSMIS. 

- L307 Between 0.04 and 0.13 is rather vague, can you provide the number? 

- L318 "Unobserved polynyas" This sounds like the blame is on observations, unless you mean "too large polynyas"? An oceanographic discussion of the ice-ocean conditions for polynyas should precede this sentence but is only coming near the end of the paper. 

- L324 Shouldn't it be expected that the assimilated run matches the assimilated data? 

- Comparing Figure 3 to Figure 2, I see that the deterioration of SIC in the summer happens outside of the KaKu snow data coverage, so it appears to be caused by the assimilation of RFB. Since the free model underestimates the RFB, I expect that the positive innovation of RFB turns into a negative increment of SIC. Is it so that the RFB is somehow negatively correlated to the SIC in the summer covariances, and is that model-based correlation trustworthy? 

- L355-360 If the FREE run has excessive sea ice cover, does it also receive snow that should have fallen in open water? Does that explain why UNIVAR does better? 

- L365 Conversely, is the missing snow falling into too large simulated polynyas? 

- L370 With local data assimilation, one would expect that the spatial distribution is respected. 

- L376 The small bias should be expected for the assimilated observation. 

- L380 The observation errors should be given earlier in Section 2.2.2. 

- L383 More positively skewed. Explain why the values of a sea ice variable like RFB should be skewed. 

- L385 Biases cancel off if you average enough of them. I don't think that biases can be compared to observation error standard deviations. 

- L392 Is the ice submerged when the RFB is negative? 

- L410 The comparison to ICESat-2 is interesting but could be better introduced. For example is the ICESat-2 total freeboard physically the same as the LEGOS (AltiKa) total freeboard? Indicate upfront that ICESat-2 is also available in the summer. 

- L410 Can the validation of ICESat-2 separate cases of bare ice from snow-covered ice? Could this validate the RFB alone instead of the more ambiguous RFB+snow? 

- L414 "the constant densities": do you mean the model densities or other constants? 

- L416, 418, 422: should we expect that MULTIVAR yields better statistics? 

- L420 Why does UNIVAR have lower freeboard than the free run? 

- L425 Do you have insights whether this is thanks to the update of SIV or to the snow that insulates the ice? 

- L427 I don't have an impression of how good a spatial correlation of 0.6 is. Maybe remove this statement. 

- L438 "unobserved ice-free zones" again implies observations are wrong, while the model obviously has excessive ice. 

- Figures 5 and 6: Should we expect that the summer freeboard is lower than the winter? 

- L493 Please clarify that the LEGOS and CS2SMOS data are not completely independent due to the use of CryoSat-2. 

- Figure 8: That the MULTIVAR is performing best in the polar hole is interesting. But I am confused by the notion of "LEGOS zone", is it the zone where RFB data or KaKu data is available? 

- L520 Clarify that LEGOS and SMOS data are independent. 

- Figure 9 has masked the areas where the SMOS data is saturated. So it is difficult to see where the "LEGOS zone" stops. Is it at 80S latitude? 

- L545 The section is inconclusive because the authors have not indicated which of LEGOS or SMOS data is more realistic. In the Antarctic, the SMOS thickness increases gradually into the ice, which seems more realistic to me, but I would appreciate the authors' view on this. 

- L552-557 This passage could be less contorted if the pros and cons of a threshold was stated upfront (see comment above). 

- L559 Even if the snow melts away completely in the summer, the timing of the snow melt also affects the melting of the sea ice below, which can hold a longer memory than the snow itself. 

- L566 Is it unrealistic that FREE and UNIVAR have submerged sea ice? 

- L586 This is the first time that the authors mention what happens in the absence of KaKu snow observations, and in indirect wording. See previous recommendation to lay the special cases in plain sight earlier in the description of experiments. 

- L587 The snow and RFB are indeed related both by Equation 1 and by the model dynamics, it would be interesting to see what this relationship becomes in the long model simulation. 

- L622 Why not assimilate SMOS data as well? 

- L650 "mostly the MYI" should be "only the MYI", right? 

- L655 Is 895 kg/m3 the model or observation density? 

- L666 the "physical accuracy" is misleading since the densities are only used for calculating the RFB in the output. 

- L675 is another instance when multivariate covariances are discretely mentioned. This sentence could imply for example that the SIV and SNT are incorrect in summer and then feedback negatively to the SIC, but the authors should clarify if this is the case. 

- L682 Indicate whether these open waters are correct or artificial. 

- L687 Indicate the inversion of ocean temperatures below the ice. 

- L699-705 These assimilation settings are repeated from earlier methods section. It would be more logical to indicate first there that these settings are different from the default in order to mitigate the appearance of polynyas. 

- L744 CIMR should also include SMOS-like measurements of thin ice thickness. 

- L796 Have the moorings data been used to calibrate the LEGOS product or are they strictly independent? 

- L811 Is an ice draft of 0 m synonymous of open water? 

Typos and grammar: 

The English can be improved by paying attention that verbs, adjectives and nouns are chosen carefully for clarity. 

- L21-23: This sentence is too contorted, avoid the "seem" for something that should be certain and rephrase the last clause. 

- L38 "The EnKF". 

- L62: Incomplete sentence, or remove "that". 

- L79: Replace "stems" by stem (according to "sources") and remove the bold 's'. 

- L84: Times disagreement between "stated" and "recommend". 

- L315 "higher presence of leads" -> "larger lead area" 

- L376 biase -> bias 

- L436 "mostly" here is misleading, I believe you mean "UNIVAR is underestimating [...] freeboard values the most". 

- Figure 5 caption has swapped the dotted and dashed lines. 

- Figure 7 caption: use SIV instead of SIVOLU. 

- Figure 7 caption: SIC "either from the supplier or from OSI-SAF", for which product? 

- L495 "coherent" is slightly misleading, do you mean "consistent" or "similar"? 

- L509 "fewer" should be "lower".  

- L552: "aims at keeping" -> "keeps"? 

- L581 "demarcation" is unclear: do you mean the edge of the observations domain? 

- L655 "a particularly pronounced discrepancy", is it much higher or much lower? 

- L697 ... in places where the equilibrium of the model ... 

- L699, 700 Enumerate "First,... Second, ..." for clarity. 

- L718 Iceat-2. Also capitalise ICESat-2 consistently throughout the paper. 

References

Counillon, F., & Bertino, L. (2009). Ensemble Optimal Interpolation: multivariate properties in the Gulf of Mexico. Tellus A, 61(2), 296–308. https://doi.org/10.3402/tellusa.v61i2.15549