The authors study the correlations between errors in XCO2 satellite retrievals, based on reference lidar measurements, and discuss various ways to account for them in atmospheric inversions. The paper looks a bit like the clean minutes of a brainstorming meeting: every sentence is well written but the logical flow is curvy and difficult to follow. The authors have not done enough to make their thoughts accessible and to take the text beyond elaborate speculation, perhaps simply because their thinking is not yet ripe for publication. Maybe it doesn’t matter, as the paper will be cited anyway given the role of this activity for the OCO-2 team, but for the few who will bother to read it, it may be a daunting task, perhaps in the end wasted. I am listing a number of comments here to help clarify the presentation.

• Footnote 1, p. 25: the disclaimer here is a bit hidden, but it is actually essential. Basically, if the “good reasons” listed here are correct, all results of the paper can be ignored. This observation could be fatal for the patient reader who painfully reaches this page… In the end, nothing is given to convince the reader that the MFLL-OCO-2 differences do indeed represent OCO-2 errors, that the two scales of correlation lengths found (10 and 20 km) should be used at all in OCO-2 error models. It’s embarrassing.
• There is good and interesting math here, but the authors belittle it by arbitrarily rejecting certain math results: why should the negative weights not be physical (l. 20) or considered undesirable (l.331)? They simply follow from the authors’ correlation model: if the authors are not satisfied with this consequence, they should change the model rather than fooling the math.
• l.35-37: the sentence seems general, but does not apply to GOSAT in practice.
• l.43: why would it make little sense to assimilate the measurements individually? From the text, it is obvious that it is so much easier than trying averages. So, this can make a lot of sense. Personally, I would still prefer the averaging but for reasons that are not discussed here (numerical stability, but this is only a feeling).
• l.48: interesting comment… The dependence of the correlations on the scene questions the representativeness of the ACT measurements used here. This key element is only briefly touched on in the warning in line 181.
• l.123: OCO-2 was already defined in l. 39. Same comment for ASCENDS a bit later.
• l.168: the motivation behind removing the linear trend is obscure. For the constant value removal, I do not see how this affects the calculation of the autocovariances.
• l.208 and 219: Kalman filters are used to control fluid state variables, but not boundary conditions such as surface fluxes. They are outside the scope of the discussion, unless the authors refer to simplifications like the Kalman smoother, but in this case the assimilation window covers periods much larger than the observation error correlations lengths which are discussed here.
• Section 3.1.3: Some main elements (the tridiagonal influence matrix, the statistically-optimal inflation factor) have been shown years ago by Chevallier et al (https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2007GL030463) in a short paper. It may have been too brief, but was much more accessible, I think.

General comments.

Authors propose an innovative approach to the construction of the error covariance matrixes for XCO2 observations by OCO-2 satellite. The method relies on comparison of OCO-2 retrievals to the collocated XCO2 observations by an airborne lidar made along ground tracks of several tenth of km in length, and thus avoids use of model simulations for estimating the amplitude and spatial statistics of the OCO-2 retrieval errors, which constitutes a major innovation presented in this work. Authors derive spatial error correlation lengths in the range of 10 to 20 km, which is a new result, based solely on observations. Authors consider 2 variants of the spatial error correlation models that are used to evaluate the errors and errors correlations for target 10 second averages derived from intermediate 2-second mean data, and propose ways to overcome related technical difficulties. The paper is well written and can be accepted after minor revisions.

Detailed comments

Although the problems and remaining deficiencies are extensively discussed in the Summary and conclusions, it is worth revisiting a couple of topics below:

1. The paper would benefit from deeper look at the problem with negative covariance elements, addressed in Sect. 3.2. Similar problem with spatial correlations would appear with covariance matrixes for surface fluxes errors in the same inverse problem for which the data uncertainty matrix is constructed. But the problem of negative elements for surface fluxes is not widely known, thus there is a possibility that the problem here is caused by an unrealistic design of the covariance matrix, but not by statistical properties of the data in hand.

2. Unfortunately authors do not provide information on the amplitude of the correlating component – what is a fraction of the (OCO-2 – lidar) difference that is correlated at 10 km or less scale. If that is only a fraction of the 1.5-2 ppm of error as found by comparison with TCCON, then the origin and spatial scales of remaining is unknown, thus it can be treated as random and uncorrelated. Bell et al (2020) notice that the correlations at local scale are pretty low, they write: “We conclude from these low correlations that for an average scene with no strong variability in the XCO2 field, OCO-2 and the MFLL do not typically “see” the same small-scale features”. In practice, the adopted level of correlation coefficients as shown on Eq. 15 do not appear justified by the comparison with MFLL and may possibly come from a separate source.

Minor comments, technical corrections:

L9  ‘Errors in the CO2 retrieval method have long been thought to be correlated at these fine scales’ – It would be more accurate/safe to say that data are correlated, rather than the errors.

L44 Alternatively, one can call this ‘summary value’ an ‘average value’

L48 Authors imply the errors here correspond to model-observation difference, and contributed by model errors related to smoothing due to coarse model resolution. Its better to define somewhere above this point what is implied by ‘errors’.

L86 Adding some MIP paper reference should be useful here (eg Crowell et al. 2019).

L167 Any rationale for detrending Y rather than X itself?

L203 As the MFLL data are first aggregated to 7-9 km blocks, it appears that one needs to clarify here on how the analysis would become useful for finer scales.

L221 Although temporally uncorrelated errors are convenient for Kalman filters, it doest seem to be excessive requirement, need to add a reference to appropriate text, if exists.

L627 Mistype: correct ‘Zendolo’ to ‘Zenodo’

L660 For web document, need to give url.

Correction to RC2 comment of May 3, 2021, there was a mistype on my comment, revised line:

L221 Although temporally uncorrelated errors are convenient for Kalman filters, it does seem to be an excessive requirement, need to add a reference to the appropriate text, if exists.