Review of “How well can satellite altimetry and firn models resolve Antarctic firn thickness variations?” by Maria T. Kappelsberger et al.

This study uses regression statistics and PCA to systematically extract and analyze the robust signals and residuals from combinations of satellite altimetry- and firn densification model-derived height changes over the last three decades over the Antarctic Ice Sheet. The final product is a set of ‘adjusted firn thickness variations’ which the authors demonstrate describes the spatial and temporal variations better than both the underlying altimetry and model products. Their careful analyses of spatial and temporal characteristics of the regression residuals allow for both an assessment of times and places where the final product has strengths and weakness and an estimation of the associated errors. I believe this is a creative solution to the general problem of how to draw robust information from two data sources which both have their own known and unknown spatio-temporal variations and errors.

The paper is mostly well-written in clear and concise language (with a few exceptions, see below), the notation is adequately introduced and the figures are clear and illustrative. The conclusions follow from the analyses. However, the paper is very long and as a reader I was not always clear on where it all was going.

Both the methodology and the final product of adjusted firn thickness variations should be of interest to the parts of The Cryosphere’s readership working with either satellite altimetry or firn densification modeling. I do have some concerns with respect to the presentation and I therefore suggest to accept the manuscript with some major revisions.

## Major concerns ##

The manuscript is very long and for a long time, I was unable to see where we were going and how all the notation, techniques, differences etc. were to be used. I realize that the authors want to be systematic by introducing all the methodology in Section 3 before using it in Section 4 and discussing it in Section 5. But this means that, until somewhere in Sections 4 and 5, I had been loaded with a large amount of notation, techniques etc. without really knowing why I needed to know this.

I suggest:

• Try to shorten the manuscript. I know you want to be thorough and systematic, but is there really no way of making it shorter? It is a very long read.
• Somewhere in the beginning make an overview of what the problem is, and the pathway to solve it. Try to include a cartoon or flowchart showing the data coming in, all the intermediate products, the residuals, the analyses done on the residuals, and the use you make of these things. Include also the notation in each box of the chart and the relevant section numbers - not just where they are derived in Section 3 (as in Table B1) but also where they are calculated in Section 4 and used in Section 5. That would provide a road map for the reader making the journey through it all easier to navigate in.

## Minor issues and typos ##

L53: SOME RCMs specialize…

L178: ERA5: What is the resolution of ERA5? 31 km, right? And this is “downscaled” to 27 km, right? Please argue why? Do you have reason to believe the 27 km RACMO data is better than the 31 km ERA5 data as input to the FDM?

L181: MERRA2 is downscaled from what resolution to 12.5 km? And how?

L204: The use of parentheses and the two short sentences “Rates…” and “The three…” is quite clumsy here. Please rephrase.

L206: “evaluated by comparing to”

L214: Try to think of a better title for this sub-section

L219 (and many other places, e.g., 401): The use of parenthesis after a full stop is not the usual way of doing it. Either a parenthesis refers to and is part of the sentence that is full-stopped, or maybe it should not be a parenthesis at all.

L262: The sentence starting with “Observations” is very difficult to read. Perhaps start with the fact that you find higher noise levels from the older sections and use this to motivate why you introduce a weighting. Also, say which variable (r^A?) represents this noise you talk of.

L265-269: This is very difficult to understand. Please see if you can rewrite it more clearly.

Eqn 1, 2, and 4: These equations all include a, b, c, d1, …, but they are different (and subject to different regressions) in the three equations, right? Either change the notation or write this out very clearly.

L294: “scale it SUCH that”

L321: “deterministic”: What do you mean by deterministic? And later on (L669+673), you talk of “stochastic”. Exactly what is stochastic? I cannot see any noise terms added anywhere in your methodology.

L371-375: Was very difficult to read. I think I understood it when the results were shown later on, but when reading it here I did not get it.

L380: You say that the fv^Ma are standardized prior to PCA, but in Fig 4 you say that the EOFs have units of m. How can the EOFs have units if the input is standardized and thereby non-dimensional?

Fig 7: Why are d-f not identical (or at least similar) to the EOFs in Fig 4? Are you not projecting the model signal on to the PCs that came out of a PCA on exactly that signal? Should that not be a way of recovering the EOFs, i.e., by projecting the signal onto the PCs? Or does it have to do with standardized vs non-standardized signals?

L455: What is R_s? I cannot remember having this introduced before.

L478: “underlying time series IS displayed”

L481: “HAS stronger”

L482: “IS closer to”

L601: includes

L635: The sentence starting with “Thus, the …” is difficult to understand.

L691: “in THE snow”

L712: The sentence starting with “We deliberately” does not read well. Particularly the word “deliberately” seems odd. Please try to rephrase the sentence.

L723: outperforms

L725+731: The sentences “However, one caveat should be noted.” are a bit odd and short. Suggest you combine them somehow with the sentences coming after.

L733: resolveS

L734: “evaluated AT grid cell level”

L735: Perhaps underline that the basin 5 and 8 numbers are also calculated at grid cell level as in the previous sentence.

L735-736: The sentence starting with “Across basin 8” makes it sound as if you only did this spatial analysis over basin 8, but didn’t you do it over all basins?

L736: Suggest to combine the sentence starting with “The large” with the one before.

L736: “are due” should perhaps be “are likely due” or some other modifier to weaken the claim.

L745: “TO the original”

L746: Do you not rather subtract the modeled firn thickness variations from the altimetric variations? That is what eqn A1 says, but your text says the opposite.

L791: Where can the IMAU-FDM data be found?