King et al. describe a new, publicly-available toolbox for paleoclimate data assimilation. The manuscript provides a description of the key concepts underlying DA, a tutorial with direct reference to examples coded in MATLAB, and a discussion of some of the benefits and pitfalls of DA in the paleosciences.

This toolbox and its accompanying documentation are excellent resources for the paleoclimate community. I think that the manuscript is well deserving of publication in GMD. In particular, I appreciate its readability and accessibility, which hopefully will help reduce some of the mystique and opacity surrounding DA methods and make them more broadly usable.

I am suggesting major revisions given the number of comments, though I do not think that addressing them will be particularly onerous. I did not work my way through the examples, but I believe that they are adequately well documented to allow future researchers to retrace their steps. I also attended a workshop demonstrating the use of this code and found those (very similar) examples clear and well-executed.

Comments

Is DASH an acronym? If so, please spell it out.

l1: I would not refer to paleoclimate DA as novel; these approaches have been around for 20+ years

l39: I'm not sure about the context for "equilibrium" here... is it the disequilibrium of climate that is at issue, or the imprint of anthropogenic forcing?

l58: There is a substantial literature using DA approaches in ocean GCMs for paleoceanography not referenced here. Approaches using the adjoint of ocean models (4DVAR) include Winguth et al. 2013; Kurahashi-Nakamura et al. 2014; Dail and Wunsch 2014; Kurahashi-Nakamura et al. 2017, and Amrhein et al. 2018. See also the work of Jake Gebbie and Olivier Marchal using simplified transport models. An early example combining paleoceanographic data with a simple ocean model is Legrand and Wunsch (1995).

l63: Given the discussion in 5.2, a caveat seems in order here that while nearly any parameter can be reconstructed, there is not necessarily a guarantee of skill in doing so

l68: DA models can relax assumptions (they do not do so by default)

l112: typo, Finally,in -> Finally, in

l130: This expression describes offline approaches, but not necessarily online ones, where X_p is a function of observations at previous times

l138: I suggest ", a state vector might also..." since this is not always the convention elsewhere where a state vector represents a particular time.

l146: "time step": This is a bit confusing in reference to a proxy record; is this not another example of a time slice, just referenced to a model?

l166: An additional important category of errors are representativeness errors that arise from comparing proxy values at, e.g., points in space or time to model values representing time slices or large grid boxes

l172: "...into the climate models" -- I suggest rephrasing since this doesn't reflect what is happening in offline DA

l183: "help quantify" -- in fact, they do quantify the uncertainty, at least so far as the KF is concerned

l185: delete second )) on R

l190: I thought this was a nice description of particle filters.

ll216 and 218: These expressions could use some more context. What is k? What is s? Note s was recently used for particle filter weights. Often argmax has a subscript denoting which argument is being chosen to maximize an expression; that would be helpful here.

l220: For this discussion (and others, e.g. proxy error) I think it would be useful to introduce the equation y = hx + n connecting proxies and the state.

l409: For context you may wish to reference the meteorological approach of "super-observations" (e.g. Purser et al. 2020).

l411: I think that you mean that the error-covarying proxies are processed using a single block update? Also, I am unsure of the example of assimilating CFRs given here. Presumably one gets a non-diagonal R from the CFR, and then there is a SVD or similar super-obbing procedure prior to block assimilation? It could be helpful to spell this out more.

l630: Does the off-diagonal R change how DASH works under the hood? Does it use sequential observation processing for diagonal R?

l661: Variance adjustment methods may be common, but I think it's important to note here that there is no simple fix for the variance loss issue. What happens to uncertainties when those methods are used?

Sec. 5.2: Consider referencing Amrhein et al. (2020), which quantifies errors from model covariance bias in paleo DA using imperfect-model experiments. These biases can also lead to large errors in estimates of uncertainty in offline EnKF.

l673: distal -> distant

l674: I think that accuracy (more than underlying complexity) is the more immediate issue

l692: assume -> assumes

l745: Again, I encourage referencing the literature on adjoint approaches in paleoceanography, which conserve physical properties in the ocean.

l746: I think this is a nice articulation of an oft-overlooked point.

l763: I disagree that ensuring homogeneity of updates (which seems very unlikely!) is necessary for computing means. However, it is important to take into consideration the uncertainty of underlying grid boxes when computing means, so that regions that are less updated by data have larger uncertainties. These errors then propagate into the estimate of the index. See, e.g., Wunsch (2006), "Discrete Inverse and State Estimation Problems" (linked here: http://sites.science.oregonstate.edu/~restrepo/577/wunschbook.pdf), Section 2.7.5, for a discussion.

General comments

This manuscript was a pleasure to read and will be an excellent contribution to Geoscientific Model Development. King et al., present a Matlab toolbox, DASH, for data assimilation of paleoclimate data. While the concepts are not new, the implementation and presentation of a well-documented toolbox is novel and very welcome. Data assimilation is becoming increasingly relevant and important to the paleoclimate community. The DASH toolbox will greatly enable the use of DA by scientists who are interested in the techniques but might have a large barrier to entry due to the coding and data management required. As a proxy-paleoclimatologist who does not currently use DA but has been intrigued by it, I found this paper and the accompanying resources available through MATLAB to be clear, helpful, and well-executed. The documentation of the code and the demos are pleasantly thorough.

My only minor suggestion is that the introduction somewhat glosses over limitations to DA apart from the technical barriers to implementation. The paragraph from line 57-70 is a great description of the strengths of DA, but a naive reader might think that DA is a spotless approach to paleoclimate reconstructions. The limitations/common mistakes in DA are treated thoroughly in the discussion at the end. I wonder if it would be useful to mention these limitations/pitfalls in the introduction, and to point out that a well-documented toolbox like DASH might help a new user avoid pitfalls.

Minor Technical Corrections:

Line 86-87: define LMR and PHYDA

All figures: consider labeling the panels A, B, C, etc., for easy reference in the caption and text

Figure 3: missing axes labels (temperature, years)