Dear Authors:
First I would like to thank the Reviewers and especially you the Authors for the lively exchange of ideas. The timelines in the new online review process are a bit blurred now. My editorial decision, "major revisions", strictly speaking pertains to the original version of the manuscript, based on the 2 expert reviews and my cursory look at your manuscript. The Reviewers indicated they were willing to reexamine a revised manuscript so with this decision I am asking them to do exactly that.

I understand from your initial reply to the Reviewers and from your revision notes that you attempted to address many if not all of the Reviewers' comments. In any case, for the record and for your consideration for the next round of revision, here I provide a synopsis of the reviews and a few additional comments:

1) Both Reviewers asked for clarifications on a number of fronts. First, difference between annealing & 4dvar (Rev 2, 1st specific comment).

2) In this regard, both Revs specifically asked for an experimental comparison with 4dvar and/or ensemble filters (Rev 1, point 6, and Rev 2, specific comment #2).

Editor's Notes:
i) The addition of a cost/benefit assessment as to simple 4dvar vs the annealing approach, if not already in the revised version, would be very beneficial for the readers (i.e., compare 4dvar vs annealing results and discuss the added cost due to annealing - is it worth the extra cost?)
ii) I am sympathetic to your arguments about the inclusion or not of a comparison with ensemble filter results. I would like to hear from the Reviewers again on this front though.

3) Additional clarification requests include Rev 2's "other comments" #2, 3, and 4, as well as Rev 1's points 2, 3, 4, and 5. Of particular interest is whether your annealing method uses stochastic elements (comment #2 of Rev 1).

4) As to the global minimum question from Rev 1 (her/his point #1) - please make sure you clearly define in the main body of the paper what you mean by "consistent minimum".

5) As to Rev 2's "other comment" #5, use of a 2 or 3D model for demonstration purposes - while this is an interesting suggestion I am not sure if this is required for the current manuscript

6) As to the title of the paper (Rev. 1's point 7) - please note the word "precision" is a noun, while "precise" is an adjective. I am not a native English speaker (while Rev. 1 is) but I think it is grammatically not correct to place a noun ("precision") in front of another noun ("approximations") with an intent to describe the second noun with the first one. I understand your aversion to using "precise". Suggest you search for alternatives.

Additional note from Editor:
iii) The first sentence of your abstract is a nice concise description of data assimilation. I am not sure what made you change "observations of a complex system" to "an observed system" in the first sentence of the Abstract? I liked the original text here. The word "observations" has some useful connotations (system may not be completely observed, etc). This is lost when you use "observed system". The latter may even imply to some that the system may be completely observed. Perhaps you could consider this alternative (which is a combination of your original and revised first sentences):

"Data assimilation transfers information from observations of a (natural) system to a physically-based model system with state variables x(t)."

Thank you again for your diligence in responding to the Revs' comments. Please make sure you provide a point-by-point response to any new Rev comments if further revisions are requested by the Revs or myself. With regards
Zoltan Toth

I'm asking the Referees to review your revised manuscript as discussed earlier.

Dear Authors,
Thank you for your careful revisions. Both reviewers recommend publication of the revised manuscript. Rev. 1 has two minor comments (title and list of acronyms) - can you please look at these recommendations before the paper is published? I will take a quick look at your response before acceptance for publication. With regards
Zoltan Toth

Dear Authors,
Thank you for your latest revision and congratulations! ! I am glad to accept your manuscript for publication in NPG as is. Thank you for choosing NPG for publishing your research results. With regards
Zoltan Toth