Final decision of editor based on comments of two reviewers, major revisions undertaken and a re-review of the revised version by one of the first reviewers.

Dear Johannes Fürst,
This paper received two in depth reviews and one additional comment that were overall positive and clearly pointed out the relevance and importance and also quality of the work, but identified also some substantial issues and some substantial suggestions for revisions. Besides specific more technical points these main points concerned:
1) streamline/focus and slightly restructure manuscript
2) term ‘apparent mass balance’
3) label figures
4) error estimates (observational, formal, results discussion, etc…)
5) flow direction computations
6) limitations from model physic, sliding,…
7) discussion of wider use of methods/data requirements

In the revised version the authors very carefully addressed the substantial and minor technical points and explained or justified their revisions very well. Referee 1 (Brinkerhoff) also made this point in his re-review of the revised version and only has very few and mostly minor additional points which are listed.
I myself also spotted some very few additional technical corrections that I listed below.

Thus the manuscript is very close to final acceptance for publication in TC but before I ask the authors to address the short list of points below.


***Re-review D. Brinerhoff

In this study, Furst and others present an updated method for solving the problem of inferring ice thickness from sparse observations coupled with surface data. They then apply this method to three glacial systems on the Svalbard Archipelago. The validity of the resulting estimations of ice thickness are established with a detailed error analysis.
The paper is clearly relevant and within the appropriate scope for the journal. Contemporary interest in methods for inferring ice thickness are of great interest to the community, as evinced by numerous recent publications on the subject, including a comprehensive intercomparison (Farinotti et al., 2016).
This papers contribution to the field stems primarily from its presentation of a way to circumvent some of the arduous data requirements required by the method on which it is based (Morlighem et al., 2010), and also provides an interesting real world application of the method, providing an important window into the performance of a promising algorithm with the various nuances of real observations. In my review of the discussion paper I raised a number of points, primarily about inconsistencies in the mathematical methods and in the interpretation error. The authors have done an excellent job of correcting their methods where appropriate, or at least providing substantive discussion of the method's limitations when my requests were felt outside the scope of work. The organization of the paper has also been adapted, and the result is very clear and easy to follow. Outside of a few very minor points, I suggest publication of this manuscript.

On temporal errors
My comment on non-contemporaneous measurements producing an additional source of error was in- tended to suggest to the authors that the error bounds that they are using might generally be too small, and that they should be made larger because of this effect. My intent was not to provoke a reference to my paper, since the effect I mentioned is a pretty general principle and not a result. Instead of the reference, I'd like to see a short discussion of how (or if) non-contemporaneous measurements might increase uncertainty estimates.

Divergence operator notation
\nabla is the gradient operator, \nabla \cdot is the flux divergence operator. Section 2.3.2 uses \nabla [: : : ] as the divergence operator. Please ensure consistency and correct usage, or make the appropriate definitions.

Typos
P2 L14 loose ! lose
P7 L30 Eq. 10 appears to be missing some things.
P9 L11 to ! too


Additional corrections from editor A. Vieli
**************************************************
p 10 line 29: it should be singular: ‘ice cap’ (not plural)

p. 17 line 6: should it not say: in SMB, and surface geometry and to a flux…’

p. 23 line 12: Do you not refer to Fig 6c and d here? and then in the next sentence you refer to VIC only so maybe continue with: ‘The ice-cap setup (VIC) shows…’

Fig. 2: figure pairs ab, cd and ef all show the same unit and scale range (m/y) but to me a bit puzzling, the color-scheme used is for all different, but I can not see any reason why this should be (or is there?). Would it not be easier to compare/relate the different fields if the color scales would be the same?

P 22: Fig 6/Caption: 1) say that in a and b, black dots indicate points with used thickness measurements. And further ,I struggle a bit to distinguish between purple and black dots in c and d

P 24 Fig 7: I would rather plot a clear (thick line) for the ‘identity line’ as it is a LINE, not a area (grey area).

p. 30 line 18: ‘all over the glacier’ is awkward: better say ‘over the entire glacier area’ or ‘over the whole glacier area’.

p. 31 line 2, typo: ‘satellite’ (not ‘stellite’)

Appendix, p. 35, line 8: should here not be refered to Fig. A3a (fig A1 does not show ice thickness)?

Appendix, p. 35, line 14: similar, should here not be referred to Fig. A3b?

Fuerst paper TCD 2017

Decision of editor based on comments of two referees, major revisions undertaken, one review by one of the intial referees and a re-revised version of the manuscript.

Dear Johannes Fuerst,
After a positive re-review by one the initial referees, the authors addressed the very remaining issues of the referee and myself the editor very well and ai pleased to announced that the paper is now accepted for publication in TC. At this stage I would also like to thank the authors for their excellent handling of revisions and their patience.
Best regards

Andreas Vieli, the editor

14th July 2017