Please see attached file.

Review of paper: “Potential Artifacts in Conservation Laws and Invariants Inferred from Sequential State Estimation” by Carl Wunsch, Sarah Williamson, and Patrick Heimbach.

The paper addresses an important topic that is often overlooked or ignored. The respective debate goes on for quite some time and is definitely picked up and addressed in the paper by Bengtson et al. The paper is cited, but primarily with respect to observational inhomogeneities. The paper itself makes a very clear statement in the same direction as is being done here: that estimates of the past and future climate needs to be dynamically consistent and preserve basic principles. Might be useful to stress this latter point more beyond the reference to the Dee paper.

Having this said, I do like the paper on the one hand because of its message. However, similar to a pre-curser of the paper, it has a very strong text-book or tutorial character and over large parts agrees with the text books of Carl Wunsch. I do understand the rational for including the material. But I also think that the paper can benefit substantially by moving much of this into the appendix and, instead expand on the messages that should be conveyed (see also below). In a nutshell, they could be summarized as:

Changes in important functions such as total mass and energy, or volumetric current transports, usually are impacted in non-physical ways by (sequential) data assimilation approached, hindering the process of inferring climate change related trends from reanalyzes. This is because entry of data leads to violation of conservation and other invariant rules. This holds for filter approaches, but even in finite interval smoothing method uncertainties in all such estimation results remain. This message is important. However, I wish a bit more emphasis would have been given on the cure so that the reader is not left with a hopeless feeling.

The bottom line is: I belief this can be a useful paper. But I strongly advise to revise the paper, taken into account the above comments, but also the following minor remarks. The authors might also think about a slightly different format than a usual science paper. As an example, I can think of a commentary, but am not familiar if EGUsphere does offer this.

Minor comments:

• The papers is written in places in ways that I think are bit cryptic and definitely can be improved. As an example, the first sentence of the abstract is sitting there and the reader wonders what is the connection to the rest of the abstract? Same with the last sentence: here I wish the authors would make a clear statement about what can be done and what cannot be done.
• While re-writing the abstract, I suggest deleting “equation” in line 5 and spell out what “many of the issues” are.
• The first sentence of the introduction is hard to read. I suggest to re-write. Same with the sentence starting line 27. In fact, I suggest re-writing the entire paragraph.
• Delete “specifically” on line 29 and identify what you mean by “system trends” on line 31.
• Line 50: “what is meant by “system failure”? I suggest dropping system here too. Also, this paragraph could use a re-write and definition of what is meant, e.g., by “long-duration forecasts with rigorous models”? Why are rigorous models only “likely to preserve”? Aren’t rigorous models defines as preserving quantities (besides numerical effects)?
• Around line 60 a reference to a textbook defining, e.g., a Kalman filter, would be useful.
• In Section 2, several symbols are either used and only later defined (e.g., delta t on line 67). Or they are being shown with the context coming later (the x tilde (t, -) or x tilde (t,+) on line 78).
• The E matrix on line 74 has a name and a meaning that should be explained.
• Line 91 and following: why do you need to go to time step t + delta t before you merge a model and observations? I think you can do the entire argumentation here with x(t,-). If not, I would not understand why.
• Section 3.1.1 and following: I don’t see where I_6 was introduced and why the equation shown does imply that there is no observational null space. In fact, “null space” was not introduced.
• With reference to Fig 4 several statements are being made; but I am at lost which panel, which line statements refer to and where I should see what is being said. I suggest to re-write and expand. Holds also for other figures. While you re-write: I have trouble finding line patter (e.g., dahs-dotted) and reding lines.
• Fig 11 shows in its middle panel a zoom of the top panel. But now you suddenly change line pattern. I suggest keeping them to avoid confusion. Yellow lines are always hard to red. In the bottom panel of Fig 11, I suggest to use two solid lines. You have different colors anyway.
• In the discussion reference is being made to ECCO, stating that identical results would be obtain in a linear case. This leaves the reader wondering: are all ECCO results equally impacted and corrupted as the once discussed here. The paper would benefit significantly by closing the circle and providing a crips discussion of what the authors belief can be accomplished at all and if in deed al results are the same or where in fact differences exist.