We thank the reviewer for their helpful comments and the work they put into the review. Our answers are embedded in the original reviewers comments. Our answers always start by “Answer by the authors:” in bold and the answers are written italic.

Summary and Decision:

The manuscript “Assimilating CryoSat-2 freeboard to improve Arctic sea ice thickness estimates” by Sievers et al. presents a new study in which satellite-derived radar freeboard (FB) from the Alfred Wegener Institute (AWI), and sea ice concentration (SIC) from the Ocean and Sea Ice Satellite Application Facility (OSI-SAF) are assimilated into the CICE sea ice model in the Arctic, between the period 2018-2020. To benchmark the improvements gained from assimilating FB, comparisons are made to an experiment which assimilates only SIC, and another experiment in which no assimilation is performed. RMSE validation across the three experiments show that modelled FB is improved by assimilating FB and SIC observations, while no improvement in FB is obtained by only assimilating SIC. Comparing observations of sea ice thickness to thickness from the FB assimilation experiment shows that the representation of thicker ice is improved for a test case in March 2020. On the other hand, a snapshot example over the same period suggests that sea ice thickness after FB assimilation is now too low in the Canada basin. Comparisons of sea ice draft are also made with 3 separate moorings from the Beaufort Gyre Exploration Project (BGEP), where sea ice draft in the FB assimilation experiment is consistently improved over the 2018-2020 period, relative to the SIC only assimilation and reference experiments.

The notion of assimilating radar FB, as opposed to sea ice thickness, is well motivated, given the large uncertainties involved when converting FB to thickness, and the authors provide a good overview of this topic in the opening sections. I do have some concerns however relating to the clarity of the methods and the rigor of the validation, which I feel need to be addressed. The methods section in particular is difficult to follow, and the lack of details on the model experiments mean that reproducibility is an issue. Relating to the validation, at present it is difficult to say how well the assimilation is performing in a) different regions of the Arctic, and b) different times of the year. For example, is the thin ice in the Canada basin after FB assimilation a systematic feature throughout the 2018-2020 period? Or does this just occur in the one snapshot? I like the comparison to BGEP moorings as this shows a clear win for the FB assimilation at these locations. It would also be useful however to see e.g., monthly-mean spatial RMSE plots and time series comparisons (see some of my suggestions below). On this note, I’m also unsure why the authors have limited themselves to such a short period (2018-2020), when both CryoSat-2 and BGEP data are available back to 2010. I would strongly encourage the authors to extend their study to this full period in order to give more confidence that modelled thickness is indeed improved by assimilation of FB. I realise that this would create significantly more work and so may be an unrealistic request. Perhaps if some additional analysis shows convincingly that the assimilation is doing a good job between 2018-2020, then extending to 2010 will not be necessary.

In any case, I feel there is a fair bit of work needed before I can recommend this manuscript for publication. Therefore, I recommend major revisions for this article. My thanks to the authors for their work and I look forward to reading the next version!

Answer by the authors: We thank the reviewer for their insightful comments and the work they put into the review. The method section will be rewritten and restructured taken your suggestions into account. In regards to the validation, we feel the need to clarify that the AWI sea ice thickness should not be seen as data set use to verify the quality of the SIT resulting from the assimilation. The comparison is included to illustrate the difference between the classical approach used to generate the AWI CS2 SIT and the assimilated SIT, as both data sets initially used the same freeboard data. Both data sets are than verified against the BGEP mooring data. As this was not clear to the reviewer after reading the paper, we see the need to clarify this in the revised version of the paper. Further we agree with the reviewer that validation in other regions than the Beaufort gyre is needed. The lack of validation in different areas of the Arctic will be addressed in the revised version of the paper. For this validation recently released ice mass balance buoy data from the MOSAiC campaign will be used in similar fashion as the BGEP data in figure 8. In regards to the reviewer’s comment that validation in other periods than the winter would be desirable we see the need to point out that the AWI SIT data only covers the month October to April the simultaneous comparison in figure 8 cannot be done at other times of the year. However, the SIT from the assimilation is verified also during the summers of 2019 and 2020 in figure 7. As figure 7 shows that the assimilation improves the sea ice thickness at all 3 BGEP mooring locations throughout the full 2 year period displayed and the comparison with the MOSAiC data shows similar results, we would like to refrain from running the assimilation for the entire CryoSat2 period, as this would be very costly.

General Comments:

Introduction

• The authors have done a good job at summarising the various uncertainties/assumptions related to deriving sea ice thickness estimates (L26-86), however one key piece of missing information is the choice of retracking algorithm. The roughness characteristics of the sea ice cause different degrees of scattering of the radar echo, which are then convolved to produce an average height of the snow ice interface. A retracking algorithm which does not account for changes in scattering due to roughness may therefore produce a freeboard which is too high when sea ice roughness is high, and vice versa. Landy et al., 2019 for example have shown how the use of a ‘physically-based’ retracker can help mitigate these effects. I think the introduction section here should include a few sentences to highlight this as a source of uncertainty in sea ice thickness estimates.

Answer by the authors: A discussion of the freeboard errors introduced by retrackers will be added to the revised version of the manuscript.

• L81: I’m a bit wary of saying that by assimilating FB, the effects of snow thickness and density errors are eliminated. Sea ice radar FB assumes that the radar echo is returned from the snow-ice interface, and this generally is not the case (e.g., Willatt et al., 2011;Nab et al., 2023). To appropriately model the scattering surface of the radar echo (and hence reduce uncertainty in FB) we need to account for snow thickness, density and other dieletric properties of the snow. Maybe just worth highlighting this as another source of uncertainty in satellite-derived sea ice thickness.

Answer by the authors: We will add a discussion of this point in the revised version of the paper.

• Figure 1 (and others throughout the manuscript): I suggest changing colours from red

and green (in this case, the BGEP locations) to something more colour-blind friendly.

Answer by the authors: The color scheme of all figure will be revised.

Methods and data

• I find section 2.2 a little hard to follow and am also struggling to relate it to section 2.6.

Is it essential to have these as separate sections? Can section 2.2 not be merged in with section 2.6? In any case, it would be useful to provide more details about the various model runs and how they were initialised etc, and potentially updating figure 2 with more information. For example, what are ‘VAR’ and ‘VARI’? I will summarise what I think I understand, and please correct me if I am wrong:

An initial experiment was run between 1995-2020. This experiment was run as an 80-member ensemble in coupled ice-ocean mode, and forced by ERA5 atmospheric reanalysis

Answer by the authors: We will combine section 2.2 and 2.6 in the revised version. It is correct that the initial experiment was run between 1995-2020 and forced by ERA5 atmospheric reanalysis, but only one experiment was run for this time. The ensemble is static, meaning, that the model background error is calculated from a selection of dates from the initial 25 year run.

- What were the initial ice/ocean conditions for this run?

Answer by the authors: ORAS5

-How do you e.g., perturb the ice/ocean model parameters to create the ensemble?

Answer by the authors: We do not run a full ensemble of 80 models. Instead we are running one ensemble and calculate the model covarriance matrix from the 20 years initial run. This is what we call a static ensemble.

- I’m not sure what is meant by increasing the variance to “account for biases” (L138). Are you not just increasing the variance to prevent ensemble collapse? Ultimately, you’re hoping that the data assimilation itself will reduce the biases

Answer by the authors: We use an static ensemble to reduce the computational costs. By a static ensemble we mean that we calculate the model background error from the initial run instead of running 80 ensembles and calculating the model background error from the ensemble members. Normally the model error is determined based on the ensemble members covariance matrix. In this study the covariance matrix is calculate based on model states from past dates in the initial run. For this, 80 days are selected from the years 2010-2020 from the initial run. The mean field is calculated from this model states and subtracted from all members, there after the variations are subtracted from the the current model state (described in section 2.2). This means that the bias can not be reduced as efficiently as with a full ensemble, why we add model states from other month.

o The 2018-2020 period of the initial experiment corresponds to the refRun

Answer by the authors: Yes

o The initial experiment at 2018-01-01 was used as initial conditions for both the sicRun and the fbRun. Assimilation over the 2018-2020 period is performed every 7 days.

Answer by the authors: Yes

o The increments from the assimilation runs are then saved, and then you effectively re-run the sicRun and fbRun experiments over the 2018-2020 period, except that the previously saved increments are now updating the model at every time step (through linear interp of the increments from 7 days to 600 sec).

Answer by the authors: sicRun and fbRun used the same initial conditions as the Reference run. That is correct. The increments are however not saved from the refRun. They are calculates after each assimilation step during the assimilation periode. So all three runs differ from 2018 to 2020.

- This is to prevent model shock after each assimilation cycle?

Answer by the authors: Yes

o I’m not sure where the ‘static ensemble’ fits into all of this? Could you explain? More generally, could you explain the motivation for only focusing on the 2018-2020 period rather than the entire CryoSat-2 period (2010-present)? By utilising the entire record I feel that you would be able to derive more rigorous statistics related to the improved FB and thickness from assimilation. For example, time series comparisons of monthly-mean FB and thickness between AWI and the assimilation run over the 2010- present period, for different Arctic regions.

Answer by the authors: For an explanation of the static ensemble see above answers. The motivation to focus on the years 2018-2020 is that this study is thought to be a first proof of concept of the developed FB assimilation. The SIT improvements seen in figure 7 in fbRun in contrast to sicRun and refRun are not drifting, or changing significantly over the 2 years displayed. This is the reason we believe that the period 2018-2020 is sufficient to show that the developed FB assimilation method improves the SIT.

L135: Could you provide more details on why you choose observational error estimates of 15% for SIC and 0.15m for FB, given that on L175 and L166 you state that the observational uncertainties are 10% and <0.07m, respectively?

Answer by the authors: In the revised version of the paper this has been rerun and the product associated error estimate will be used.

L150-155: Is there any post-processing applied after assimilation to ensure that the updated SIC is bounded between 0 and 1? If so, how is this bounding applied? Particularly to the category terms.

Answer by the authors: Yes. A description will be added.

I believe currently in the fbRun you are updating SIC and FB sequentially. Out of curiosity, is sea ice thickness updated during the assimilation of SIC, and similarly is SIC updated in the assimilation of FB? Do you also expect your results to differ if you first assimilate FB and then SIC?

Answer by the authors: The assimilation of FB is done after the SIC assimilation because the limits of SIC determines where to assimilate FB, so it seemed natural to assimilate the SIC first. Assimilating first FB and than SIC might lead to differences, but we assume them to be minor especially since the assimilation is distributed over all time steps.

Results

• L269-270: Suggest clarifying here that by “assimilation period” you mean November- March, as opposed to the whole 2018-2020 period.

Answer by the authors: Better naming for the period will be used in the revised version of the paper.

• L287: Does “beginning of October” and “end of winter” refer to a single day? Or a weekly average? Please clarify.

Answer by the authors: Clarification will be added in the revised version of the paper.

• Figure 5: Suggest including either correlation or R 2 values for both refRun and fbRun on each plot to quantify the improvement. I also suggest making this a 4-panel figure, and show the equivalent density plots for freeboard as well.

Answer by the authors: In the revised version the correlation coefficient will be included and additionally a table with correlation coefficient and bias for each of the month and both SIT and FB.

• L288: Can you also speculate why the “thin bias” problem is not improved with the assimilation of FB? Is this because the FB increment is spread linearly across the ITD categories (Equation 1), whereas in reality more weight should be given to the thinner categories?

Answer by the authors: In the revised version of the paper a table with monthly bias and correlation coefficient will be included for both the SIT and FB values. From those values it is evident that the biases of the refRun values are already lower than the fbRun’s SIT and FB biases, suggesting that the thin bias originates from the initial model FB and SIT.

• Figure 6: I don’t think panels C) and D) are all that informative. Especially considering that our observational estimates of snow thickness and ice density are inherently wrong. I suggest replacing these panels with freeboard comparisons (AWI-refRun) and (AWI-fbRun). It would also be good on each panel to include a mean RMSE value, as it currently looks as if panels A) and B) might actually be similar in terms of RMSE.

Answer by the authors: We will consider changing/excluding figure 6 with the reviewers comments in mind.

• Figure 8: Could you speculate why the differences between AWI and fbRun are significantly large at the beginning of the 2018-2019 period? Is this a spin-up issue?

Answer by the authors: We do not see evidence that this is a spin up issue. Comparing the improvement of the assimilated draft and the reference runs draft in figure 7 shows clearly that the draft in winter season 2018/2019 differs on similar magnitudes from the reference run as in the following years. We will add a line about this is the discussion.

Discussion

• L400-410: Is the imprint of the FYI/MYI zone on the difference plot not just due to the fact that the AWI snow thickness and ice densities are assigned constant values depending on whether FYI or MYI? In your case you’ve replaced the density with a linear function after Mallett et al., 2020, but ultimately this imprint of a FYI/MYI mask is just a reflection of what the observations are (hence my suggestion above to remove).

Answer by the authors: We will consider removing figure 6 all together.

• L412: The panels in Figure 6 are errors in modelled sea ice thickness relative to AWI, not uncertainties in AWI sea ice thickness, right?

Answer by the authors: Yes, this is a typo we missed in the editing process and will be corrected in the revised version.

Minor suggestions:

Answer by the authors: All minor suggestions will be corrected in the reviewed version, if not commented below. We thank the editor for their work!

Abstract

L4: Define SIC and FB acronyms

L6: Define AWI acronym

L10-11: This last sentence is rather vague. I suggest something like “Modelled sea ice draft errors are in good accordance with that of CryoSat-2 errors at BGEP mooring locations, with mean error differences less than 3 cm over the 2018-2020 period.”

Introduction

L17: I’m not sure what is meant here by “need to affect the model variable that the assimilation aims to improve”. Does it mean that we need to use an appropriate observational operator to map the model state variable which is being updated, to the space of the observations? E.g., an operator to map thickness to freeboard?

L18: I would clarify here that this statement relates specifically to Arctic sea ice predictability on seasonal-to-interannual timescales.

L24: Suggest stating explicitly that initialising thickness is better for predicting Arctic sea ice on seasonal time scales or longer (sea ice area persistence is more important at short lead times).

L55-57: Would also make reference here to the recently developed SnowModel-LG (Liston et al., 2020; Stroeve et al., 2020), which is being adopted in sea ice thickness products from e.g., Landy et al., 2022.

L65: The phrasing “add up” here feels a little vague. Does this mean that the error in sea ice thickness is equal to a linear sum of the errors in snow/ice/water density and FB?

L71: Suggest rephrasing to “The OSI-SAF ice type product (Aaboe et al., 2021) is one observational data set which aims to distinguish between FYI, MYI and ambiguous ice types.”

L76-77: The wording is a bit confusing here. Does it mean that the errors are systematically overestimated in the MYI zone and underestimated in the FYI zone in the OSI-SAF product?

L77-78: As far as I’m aware, sea ice area isn’t required to generate thickness anyway, so this sentence seems redundant. Could you explain what you mean here? Unless you’re referring to CryoSat-2-derived sea ice volume?

Answer by the authors: This refers to the error of the ice type product, the sea ice area of FYI and MYI. A clarification will be added.

Methods and data

L96: First time using PDAF acronym, please define.

L99: Suggest rephrasing to “An increment is the amount of change in a model state variable after the assimilation of observational data.”

L106: First time using WMO acronym, please define.

L109: Suggest removing first sentence on L109 and changing L111 to “the key variables are snow thickness (h s ), snow density (rho s ), sea ice density (rho i ), and ocean water density (rho w ).” Or similarly just defining the terms explicitly on L109.

L142: 80 model states? Or 80 ensemble members?

Results

L299: Suggest clarifying that the date 03-30-2020 is actually a 7-day mean.

L315: change to “independent of the satellite-derived FB data”.

L330: change here and elsewhere in manuscript from “data is” to “data are”

Discussion

L410: I’m not sure on Crysophere reference guidelines, but you might need to change “Sievers et. al (in preparation)” to something like “Future work will analyse the effects of different variables...”

We thank the reviewer for their helpful comments and the work they put into the review. Our answers are embedded in the original reviewers comments. Our answers always start by “Answer by the authors:” in bold and the answers are written italic.

Review of "Assimilating CryoSat-2 freeboard to improve Arctic sea ice thickness estimates" by Sievers et al. (2022)

This paper describes the assimilation of sea ice freeboard observations into a coupled ocean and sea ice model. The authors have compared the results of an analysis assimilating freeboard and sea ice concentration observations with a control run, and a run assimilating only sea ice concentration. They have used the AWI weekly CryoSat-2 sea ice thickness product, which is derived from the same freeboard observations as have been assimilated, and independent data from BGEP upward-looking sonar observations to validate their results.

General comments

The authors have developed a new method for the assimilation of FB (freeboard) observations rather than SIT (sea ice thickness) in their model. The technical method has clearly been well thought through and competently implemented. However, one of the main motivations for assimilating FB is being able to more easily quantify the associated observation uncertainties. Here, the authors have used a constant FB uncertainty, citing technical issues, which unfortunately means that they are unable to demonstrate the potential benefits of FB over SIT assimilation. Further, they have chosen to compare their FB assimilation results with the AWI SIT product, when a second run assimilating SIT into their own system would have provided a much improved comparison. A comparison to the AWI data is of interest, but is unable to adequately demonstrate the benefits of assimilating FB over SIT. These flaws in the methodology unfortunately mean the impact of the paper as it stands is limited.

Answer by the authors: We agree that the constant error estimates are a major short come of our study and will include reruns with variable error estimates in the revised version of the paper.

We also see the point of the reviewer that assimilating SIT in comparison would strengthen our point. However we have serious doubts that the error estimate of the SIT product is realistic and suitable for the assimilation. A more detailed discussion of this will be added in the introduction: Error estimates of SIT products are difficult for many reasons. For one, the variables used for snow and ice density and snow thickness are based on climatologies and point measurements with high uncertanties. But the main reasons we argue that the error estimate of the SIT product is unrealistic is the ice type discrimination and the fact that the ice type is not considered in the error estimate. A discussion of ice type data sets has long been lacking. Ye et al 2023 has recently assessed different sea ice type products including the OSISAF ice type product used in the AWI CryoSat2 data product. Comparing ice type data to NSIDC sea ice age data (Tschudi et al., 2020) they find that OSISAF ice type data for FYI has a bias of 0.42-0.6 10⁶km² and for MYI of -0.54 – -0.35 10⁶km². The comparison of Ye et al 2023 only includes FYI and MYI area and compares it to satellite obtained ice age products, but no ambiguous areas are considered. NASAs ice-chart based sea ice type product G10033-V001 shows that the ambiguous area is significantly larger than the one accounted for in the OSISAF product, or any of the ice type products compared in Ye et. al. 2023. Errors from ice type are not accounted for in the SIT error estimate at all.

Apart from the issue, that the error of the SIT is most likely underestimated in places where the ice type is ambiguous the SIT errors are in the order of up to 40% of the SIT while the FB errors are in the order of up to 14% of the FB values. This is why we decided to compare the resulting SIT to the AWI SIT and independent measurements. In-situ observations in the Arctic are however hard to come by. An additional in-situ SIT data set will be used to compare both the freeboard assimilated SIT from the model and the AWI CryoSat2 SIT. First comparisons show that these observations are significantly closer to the assimilated SIT than to the AWI data set SIT, strengthening the point of developing the FB assimilation method.

The description of some of the methods needs further clarification (see comments below). The results section needs to include more of a critical assessment - a description of the results is given, but why might we be seeing these? Some insight appears in the discussion section, but to avoid the reader having to keep referring back to the earlier figures, it should be included and expanded on in the results section instead. A discussion section can then feature more general points to tie the paper together. This would also remove some of the repetition of earlier information that appears in the discussion section. Additionally, the conclusions presented are inconsistent and some of the statements require further justification (see comments below).

Answer by the authors: Separating the result and discussion section in the paper is a decision taken deliberately by the authors. We however are aware that others might prefer a combination of results and discussion. We will investigate if a restructuring of the sections lead to a more concise text.

Additionally, the paper would benefit from more discussion of the theory behind why we might expect the assimilation of FB to be an improvement over SIT, see e.g. Kaminski et al. (2018) as a starting point: Kaminski, T., Kauker, F., Toudal Pedersen, L., Voßbeck, M., Haak, H., Niederdrenk, L., Hendricks, S., Ricker, R., Karcher, M., Eicken, H., and Gråbak, O.: Arctic Mission Benefit Analysis: impact of sea ice thickness, freeboard, and snow depth products on sea ice forecast performance, The Cryosphere, 12, 2569–2594, https://doi.org/10.5194/tc-12-2569-2018, 2018.

Answer by the authors: Thank you for pointing us to this study. It will be included in the revised version of the paper together with a discussion of the benefits of assimilating freeboard over SIT.

More generally, as detailed below, there are some confusing and contradictory statements in places. There are also some missing citations, statements that need to be quantified, and confusing notation is used in figures and equations.

Answer by the authors: We thank the reviewer for pointing them out and will correct the revised manuscript accordingly.

Specific comments

Major comments:

Line 135: Elaborate on how the values for the SIC and FB observation errors were selected. As stated above, it seems strange to use a constant FB error, given the arguments for using FB over SIT.

Answer by the authors: Reruns with variable errors will be included in the revised version of the paper.

Line 166: Uncertainty in the FB dataset is given as 0-0.07 m, but the authors have chosen to use 0.15 m as the observation error. This needs further explanation here.

Line 175: Uncertainty in the SIC data is given as 10%, but the authors are using 15% as the (constant) observation error. This needs further explanation.

Answer by the authors: The variable product error will be used in the revised version of the paper.

Line 196, Figure 7: Why not average the 10-second data first to get a daily mean, and then calculate the difference to the model daily field? And actually, this contradicts the caption of

Figure 7. This needs clarification.

Answer by the authors: The STD in figure 7 is calculated based on the 10-second data. Including the STD showcases the variability of the observations. The mean to calculate this STD is also what is shown in figure7 for consistency. To calculate this the difference between the daily mean and the 10-second data was used. In the revised version we will make sure that the text is consistent with the figure caption.

Line 212: I think the authors are describing an incremental analysis update (IAU) method, suggest describing as this as such (citation Bloom et al. (1996): Bloom, S. C., Takacs, L. L., da Silva, A. M., and Ledvina, D.: Data assimilation using incremental analysis updates, Mon. Weather Rev., 124, 1256–1271, 1996.)

Answer by the authors: This is correct and will be clarified in the revised version.

Line 214: The authors say here that FB is converted to SIT before subtracting the increment from the model. This implies that the conversion is performed for the FB observations, and thus that the authors are actually assimilating SIT. However, from reading the method in the paper, it seems that FB increments are in fact produced, and applied to the model radar FB (which is the novel part of the paper) before this is converted back to model SIT in order to distribute the increment over the thickness categories. Therefore, this statement needs to be reworded as it's rather misleading. Also line 230: "Since FB is not a model state variable, it needs to be transformed into sea ice thickness before it can be treated..." needs explanation added along the lines of "Since FB is not a model state variable, model SIT needs to be transformed into FB before the FB increment can be applied..." etc.

Answer by the authors: This will be clarified in the revised version: The FB must be converted into SIT at the first time step of the assimilation time step and than linearly spread over the time steps. If the FB would be spread over the time steps in the assimilation time step, changes in snow fall and sea ice salinity would lead to a different result as for which the increment was calculated.

Line 227-230: Citation needed for the statement about FB measurement reliability in regions of low SIC. How were the values of 80% and 0.05 cm chosen?

Answer by the authors: A discussion of the reason will be included in the revised version of the paper: There are two main reasons the assimilation is only applied in regions with SIC > 80% and SIT below 0.05m. 1) Thin FB is not measured by CryoSat as accurately as thick FB (Wingham et al. 2006, Ricker et al. 2014). 2) the SIT and FB is calculate from the models ice volume per unit area of ice. In areas with lower concentrations this can lead to SIT and FB values that are unrealistically high. To ovoid over estimation of SIT and FB following this artifact, a high SIC threshold was chosen for the FB assimilation.

Line 238-241: The description of how the work of Mallett et al. (2020) relates to the method used by the authors is confusing and needs clarification. For example, is it the calculation of c_s in equation 3 that uses a seasonal snow density? If the same snow density is used throughout, why mention the linear function? Additionally, 10 cm is quoted on line 60, and 15 cm on line 239, for the improvement in SIT from the method of Mallett et al. (2020).

Answer by the authors: This will be clarified in the revised version.

Line 269: Why might the RMSE increase?

Answer by the authors: This is discussed from line 355 onward. The area covered with sea ice increases and the area where the error is zero (the open ocean) decreases.

Line 272: Why? And could this indicate an issue with the assimilation?

Answer by the authors: Why is discussed from line 351 onward in the discussion. The error of the assimilation is lower than the error of the reference run (except in the first weeks of SIC assimilation), showing that the assimilation improves the FB and SIC.

Line 284: "differences can illustrate the impact of changing the method of converting FB to ice thickness". Since there are model uncertainties for the runs and observation uncertainties in the AWI data, this is not a clean comparison of the impact of assimilating FB over SIT. A better choice would have been to compare a run assimilating FB with a run assimilating SIT. As discussed, the AWI data has different characteristics in the snow thickness and sea ice density, so the comparison is not an assessment of the benefits (or otherwise) of assimilating FB over SIT.

Answer by the authors: The aim of comparing the two SITs is not to asses if FB assimilation is better than SIT assimilation, but to show that the FB assimilation gives SIT results in a comparable range as the direct conversion. With the inclusion of the MOSAiC SIT observations this will be more evident.

Section 3.2: Why might we be seeing these differences in results?

Section 3.3: What do the results indicate?

Answer by the authors: This is discussed in the discussion.

Line 340-341: Should show mean difference in Table 1 here too, especially as bias is discussed

Answer by the authors: A table including monthly biases and correlation calculated to compare the AWI data and the fbRun data will be added in the revised version.

Line 360: Why does SIC improve?

Answer by the authors: Because it’s assimilated.

Line 369: What about 2019-2020?

Answer by the authors: We will include a discussion of the first weeks of assimilation SIC errors in the revised version of the paper: The higher errors in SIC accrue after the model run free for 7 month. The assimilated runs run from a different initial state than the refRun. This naturally will lead to some differences in SIC. In some years this is closer to the OSISAF SIC in others it is not. An other reason for the differences are that all assimilation disturb the models physical equilibrium. In the fbRun two variables are changed and thicker sea ice melts slower which likely lead to the higher SIC RMSE in the fbRun in the beginning of the assimilation period 2019-2020.

Line 372: Quantify this, show mean difference and RMSE.

Answer by the authors: Will be added in the revised version.

Line 378: Why might the assimilation run be worse than the reference run?

Answer by the authors: See answer above (comment regarding Line 369).

Line 397: Would the "week following the 30th March" be better described as April? It is stated earlier in the paper that only FB observations between November and March are assimilated due to melt pond issues. Why was this week chosen?

Answer by the authors: The week 30.03-05.04.20219 includes 2 days of the month March and is there for included in the assimilation.

Line 401: Would be helpful to show an ice type figure as this is referred to in the discussion. Needs more information on how ice type is used in the assimilation runs and the AWI product for interpretation of the differences described.

Answer by the authors: The ice type is not used in the assimilation. The assimilation uses model values instead of constant values used by the AWI sea ice product. This model values are described in the method section (section 2.1) and are depending on the amount of salt in the sea ice and on the snow fall in the forcing. Figure 6 shows two examples of AWI data values and model values. The AWI data differs so much that it is difficult to show the variability of the model in comparison.

Line 412-415: Figure 6 shows differences between the AWI product and the assimilation runs, rather than specifically uncertainties in the AWI dataset. This part is a bit confusing. Why is this a prerequisite for comparison to BGEP?

Answer by the authors: Figure 6 only compares two FB SIT products. Figure 6 is thought to give a general overview how the differences look, but not as a validation. To validate the SIT an independent measurement is needed. Since the BGEP measurements are the only measurements we decided to discuss how the difference between the AWI SIT and fbRun SIT looks first. Since the MOSAiC data was made available in the beginning of this year the revised version will also compare to SIT measurements from Ice mass balance buoys deployed during the MOSAiC campaign. We see that figure 6 might confuse more than showcase the differences, and are considering leaving it out.

Line 418, 420-422, 440, abstract: Contradiction in conclusions - e.g. in Section 3.3 it is stated that the fbRun and AWI results are "almost equal" and there's "no clear bias", but lines 420-422 state that FB assimilation leads to improvement. Suggest calculating statistical significance of the mean and RMS differences to determine a definitive and justifiable conclusion, and be consistent with this throughout paper. Additionally, line 444: "comparable SIT as the AWI SIT product", but e.g. figures 5, 6 show differently... (also line 453)

Answer by the authors: Not all improvements reffer to improfments of the AWI data and the assimilated data. In some cases it is meant that the assimilation improved the SIT compared to the non assimilated run. This will be clearifyed. We will also add more SIT in-situ observations to compare to both the AWI SIT and the assimilated SIT to strengthen our point.

Line 452: Need to present the aims of the study in the introduction to the paper rather than just at the end.

Answer by the authors: This will be changed in the revised version of the paper.

Line 451: Being able to use FB uncertainty would really improve the paper, can these technical issues be overcome?

Answer by the authors: The revised version will replace the constant FB uncertainties with the variable ones that originate from the product itself.

Line 453: If the conclusion is that assimilating FB gives similar results to SIT, then what is the justification for implementing a new method?

Answer by the authors: With the additional observation data from the MOSAiC campaign it is evident that the FB assimilation gives better results than the classical approach.

Minor comments:

Answer by the authors: All minor comments will be corrected in the revised version, if not indicated otherwise below. We thank the reviewer for their comments.

Line 23: Add an extra sentence on what is meant by SIT and SIC having a memory

Line 26: SIT data is also available from other satellites, need citation for CryoSat-2 being the "most commonly used"

Line 31: Add "to reduce the uncertainty in the observations" to the end of this sentence for clarity. Also add explanation of why the uncertainties when the model converts back to SIT are less important (i.e. because they don't affect the assimilation).

Line 35: "to a large extent" - quantify this

Line 46: "most products" - citations needed

Line 65: Inconsistent - have just said the influence of sea water density is negligible

Answer by the authors: Since the ocean model calculates the water density and the water density is part of equation 2 it would be unnecessary to use a newly introduced constant value.

Line 70: OSI SAF ice type data is not the only product available. Is used in this paper what is meant? Or for CryoSat-2 products? Clarify.

Answer by the authors: The AWI data set uses the OSISAF ice type data and it is the most commonly used ice type data set in the CryoSat2 sea ice thickness products (Sallila, H. et. al. 2019). In the revised version a discussion of the accuracy of ice type data in general will be included in the introduction.

Line 77: "unambiguous" should be "ambiguous"

Line 95: sea ice thickness and freeboard datasets

Line 103: "more recent versions" - need to be more specific here, version numbers, citations.

Line 105: "Icepack" needs a citation, and "by default linked" needs clarification

Line 109-111: The shorthand "rho_i" etc are mentioned before the explanation of the variables on line 111.

Line 116: Suggest rewording, e.g.: "The density of fresh ice is set to 882 kg/m3 and the amount and density of brine are calculated to produce an estimate of ice density. The sea surface water density is calculated in NEMO." The detail of which functions were used is not needed.

Line 119: Not just the area in red? The blue part as well?

Figure 1: The coastline in the zoomed area does not match up (can see it's rotated, but it still is not the same area covered)

Figure 1 caption: Date is needed for the CryoSat-2 data shown.

Line 137: What is meant by "static ensemble"?

Line 138: How was the run analysed? What is the reference used to assess model biases? Needs more information.

Line 160: Define "DTU21"

Line 166-167, also line 176: What kind of interpolation? Bilinear? etc. Probably don't need to specify CDO was used.

Line 167-168: Define "per one assimilation time" - one week? Suggest referring to "orange lines" (or tracks) in Figure 1 rather than "orange area"

Line 171: SSMIS is the instrument, the satellite is the DMSP series

Line 174: Is this definitely Level 4 data? The 10 km daily product (OSI-401-b) is Level 3, which makes more sense for assimilation. I don't think there's a 10 km Level 4 product?

Line 179: "sea ice thickness output from the assimilation" should presumably be "sea ice thickness model output from the assimilation run" or similar, unless it's output specifically from the assimilation step only

Line 183: Need a citation for NSIDC AMSR2 snow depth product

Line 184: "following Mallett et al. (2020)" needs more information, add e.g. "as described in section..."

Line 188: Remove the word "completely" as it's not an independent dataset as it is derived from the same freeboard observations - which is the reason it was chosen presumably?

Line 189: Suggest replacing "based on" with "from the model, adjusted using" for clarity

Line 194: Expand on what is meant by "tilting errors"

Line 200: Are both SIC and SIT weekly assimilation?

Line 201: SIC observations are available over the summer months, so was this done to aid comparison? Clarify.

Line 220: I think data from the restart file should be described as data at t=0 (or t_0 as in figure 2). What is inc_icon?

Equation 1: What about t_n (as in figure 2)?

Figure 2: Equation 1 and Figure 2 should use consistent notation, and these should all be defined (e.g. VAR, VARI, cat are not defined)

Equation 2: Need a citation for this equation. What is r_0? Radar freeboard at t=0?

Line 237: Repetition of line 234, combine this information.

Line 255: Detail of function names is not needed, suggest reword to "After each assimilation, sea ice thickness in all categories is set within the range defined in Section 2.1."

Equations: Notation is quite confusing, especially use of subscripts such as r, 0, which are not clearly defined.

Line 261: Isn't the orange area just an example track for a specific week? Reword this.

Figure 3: Is y-axis ice concentration (fraction) or ice area? Would also be clearer shown as a line graph rather than plotted as points.

Figure 4 (top panel): Clarify that the green SIC is underneath the red (and not the blue, or missing)

Figure 5: What are the lines on the outer right edges and tops of the plots? What is meant by "initially observed FB locations"? Also add the years to March and October.

Line 308: Better to use e.g. "notably" instead of "significantly" as implies statistical significance in this context.

Line 321: Does this mean there is no interpolation of the model data, just using the closest grid point to the observation location?

Line 325: Days rather than "time steps"?

Line 329: "indicates periods with no ice present" in the observations

Line 330: Explain why no data are available Figure 7 caption: Method should be in the text (and contradictory, see comment for line 196 above). What is the lighter/darker shading on the figure?

Line 333: Statistically significant?

Line 349: Suggest changing "assimilated SIC and FB and the..." to "assimilated SIC and FB observations and the..." for clarity.

Line 366: Can the model change SIT/add new ice?

Answer by the authors: The SIT is not change, but since the area is changed the volume is changed, so new ice is added in that sens. Moreover, when no ice exists in a specific grid cell and the SIC increment suggest formation of new ice, 10 cm new ice is added in the thinnest category as described in section 2.6.1.

Figure 8: Suggest choosing a different (non-filled) symbol for the fbRun as not possible to see the '+' symbol when the points are overlapping. Extend the x-axis as the symbols are cut off at the edge

Line 380-382: Confusing wording here.

Line 387: Is this mean sea ice thicknesses?

Line 389: Would be better if lines were overlaid to show this

Lines 389, 390: I think "dominate" should be "dominant", but what does this mean?

Line 407: Filtered how?

Line 416: "assimilated draft" implies this data has been assimilated, suggest changing to "draft from the assimilation runs" or similar. Also line 417, "assimilated data" should be "assimilation run data", or similar, and also line 419.

Line 417: Suggest "time steps" should just be "times".

Line 460: Figure 8 results are at the BGEP locations only (add this information to sentence)

Technical corrections

The paper could do with an edit to improve readability Acronyms in the abstract need to be defined (or not used)

Answer by the authors: We thank the reviser for the corrections and apologize for the missed errors. We will put extra focus on proofread the revised version.

Line 34: "climatology's" should be "climatologies"

Line 60: "Resent" should be "Recent"

Line 92: "on an assimilation method" should be "of an assimilation method"

Line 101: "assimulation" should be "assimilation", also "Nemo" should be "NEMO"

Line 129: "nor" should be "or"

Line 137: "from a initial" should be "from an initial"

Line 142: "where" should be "were"

Line 165: "where" should be "when"

Line 185: Suggest replacing "with the help of" with "using" (and similar elsewhere)

Line 190: Suggest changing "to validate against" to "to validate the model against"

Line 195 (and 320, 322, 328): "standard derivation" should be "standard deviation"

Line 211, 224: Write "not zero" rather than "!=0" in the text

Line 212: "was" should be "is"

Line 214: "it's" should be "its"

Line 217: I think "fractal" should be "fractional"

Line 218: "here after" should be "hereafter"

Line 224: "exist" should be "exists"

Line 258: "Consentration" should be "Concentration"

Line 259: Define "RMSE" acronym line 281: "comparable" should be "comparably"

Figure 3 caption: "location" should be locations"

Line 333: "in-sito" should be "in-situ"

Line 390: repetition of "the"

Line 424: "3 years assimilation run is" should be "3-year assimilation run are"

Lines 425-426: Suggest rewording: "The presented method calculates an increment using modelled FB and then converts the..."

Line 446: "does" should be "the"

Line 449: "spacial" should be "spatial"

Line 454: "recommendable" should be "recommended"

Answer by the authors:

References:

Sallila, H., Farrell, S. L., McCurry, J., and Rinne, E.: Assessment of contemporary satellite sea ice thickness products for Arctic sea ice, The Cryosphere, 13, 1187–1213, https://doi.org/10.5194/tc-13-1187-2019, 2019.

Wingham, D. J., et al. "CryoSat: A mission to determine the fluctuations in Earth’s land and marine ice fields." Advances in Space Research 37.4 (2006): 841-871.

Ricker, Robert, et al. "Sensitivity of CryoSat-2 Arctic sea-ice freeboard and thickness on radar-waveform interpretation." The Cryosphere 8.4 (2014): 1607-1622.

Ye, Yufang, et al. "Inter-comparison and evaluation of Arctic sea ice type products." The Cryosphere 17.1 (2023): 279-308.

Tschudi, Mark A., Walter N. Meier, and J. Scott Stewart. "An enhancement to sea ice motion and age products at the National Snow and Ice Data Center (NSIDC)." The Cryosphere 14.5 (2020): 1519-1536.