The manuscript „The EarthCARE lidar cloud and aerosol profile processor (A-PRO): the A-AER, A-EBD, A-TC and A-ICE product“ by Donovan et al. Give a detailed description of the aerosol L2a processing chain for the upcoming ESA EarthExplorer space mission EarthCARE. In documenting the basic steps of processing of the data from the ATLID instrument it is expected to be a very valuable source of information for possible users of the resulting products. The manuscript is clearly written and detailed enough to sever this need. The only thing the reviewer finds to be at bit sub-optimal is the introduction of a novel scheme to handle multiple scattering in Appendix B. Ideally this should have been a separate paper. But it is understood that the need to get the presented information publicly available before the start of the mission is of higher priority.

Before being accepted the following points should be addressed, the majority of them only being of editorial nature. But at a few places some additional clarifying text should be added.

Page 2 Line 27: To call all particulate scattering Mie scattering is rather unfortune as also stated a bit further down in the text. It is recognized to reworking the paper to use particulate consistently is a lengthy and tedious task but non-the-less strongly recommended!!!

Page 5 Line 122: ATB is not defined before in the text. One could guess that it would probably mean Attenuated Backscatter. But the definition should be made at the first occurrence of the acronym.

Page 5 Line 125: “Rather, the highest index within…”. This refers to the feature mask definition, but to help the reader, the meanings of the numerical values should be given here it should be noted that the feature mask values are ordered in ascending order with potential scattering strength.

Page 6 Line 134: “… are set to > 8 …“ Please translate this to verbal feature type!

Page 6 Line 141: “The scattering ratio calculations…” This seems to be only a half-sentence with unclear meaning.

Page 6 Lines 150-151: Again, feature mask entries should be given, at least additionally, in plane words.

Page 7 Line 165-166: “provides also” or “enables”, one should be sufficient.

Page 7 Line185ff: The term “layer” here lets one think of straight horizontal layers, but in reality, the aerosol structure is often not stratiform but modulated for example by gravity waves with amplitudes of +-500 m or more. And even without gravity waves the layers are typically more wedge shaped, instead of being horizontal bands. Maybe the algorithm can cope with such cases, but this is not clear from the text. Please add more information on this.

Page 8 Line 198: Step 16 seems to be missing. Please add a note that this is on purpose – if it is…

Page 9 Line 227: Using the log-form a quantity that naturally can come close to zero, like the extinction coefficient, can be problematic numerically. This introduces a pole at 0 which easily misguides the optimisation routine. Further it skews the nearly Gaussian input error statistics heavily making the minimisation of Eq. (9) not an optimal estimate any more. And the skewed error statistics leads to noise induced biases. If the whole reason for the log-form is to keep positiveness than there are methods to do constrained optimal estimates which are more robust. It is clear to the reviewer that the algorithm is, as it is at the moment and will not be changed in short term. But a few sentences more about the choice and possible alternatives should be added.

Page 10 Line 242 and 246: “…the effective radii are specified a priory by type.” And “… is the a priori (linear)uncertainty assigned…” To what value? And why this value (Ref)? Maybe a table which collects all the a priory values would be appropriate.

Page 10 Line 244: “It is assumed that the a prior errors are uncorrelated…” While for the observational errors it makes perfect sense to assume that different vertical bins are uncorrelated (at least for the raw signals without some averaging), this is not obvious at all for the a priory values. The atmosphere is not “white noise” and the possible deviations of a first guess from the true value no less. It is clear that it is hard to come up with sensible values just starting from scratch, and it may have serious impacts on the algorithm used for the optimisation, but this point should be discussed in more detail and a route to future improvements should be outlined.

Page 13 Line 345: Above or between???

Page 20/21/31: The colour scales are rather unfortunate. It is practically not possible to distinguish vales in S between 0 and 50. The one chosen for Fig. 9 is much better. It would be good anyway to harmonize the colour scales of the different 2-d charts.

Page 34 Line 439: use? Or find?

Page 36 Line 490: “Fig ?” Reference is missing.

Page 37 Figure A1 caption: “… log-derivative approach (Grey-solid-line) and …” this should be Black-solid-line, shouldn’t it? It would help the reader greatly if line legends could be added to the figures, like e.g. in Fig B5-B9!

Page 38 Figure B1: Here also in-plot legends would be beneficial! And the plotting style very much reminds of the pen-plotters used until the early 80s of the last centuries. The authors are strongly advised to use a homogeneous and up-to-date style for their plots!

Page 40 Line 546: “… the divergence of the forward scattered light will also be Gaussian with a divergence…” This is a simplifying assumption since the foreword scattering peak is not exactly a Gaussian! Typically, this assumption is good enough, but the phrasing here suggest that this is a mathematical truth, which is not the case.

Page 41 Eq. B8: “H(Theta_sc(z)  >  0)” It is obvious what the authors try to express here, but formally this makes no sense. H is defined for the real number, but Theta_sc(z)  >  0 is a comparison with a Boolean result. Even if one assigns 0 for false and 1 for true this does not work as for the common definition of H, H(0) = 1 and H(x) = 0 only for strictly negative x.

Page 43: Again, horrible figure style. See comment above.

Page 45 Line 628: “calculate” -> calculated

Page 45 Line 649: Here and from the preceding text one could get the impression that Platts eta is some sort of system constant which can be “calibrated” using some higher accuracy MS-algorithms like e.g. Monte-Carlo simulations (as done for B4). But according to Platts papers neta also depends on penetration depth and optical thickness of the cloud and the variation are much larger then +- 10%. Maybe the introduction of the additional tail-function f_e takes over some of the effects of eta and stabilises its value. This should be discussed in more detail.

Appendix: The reviewer had no time check that the equations for the partial derivatives of the foreword model are correct. He only hopes that they have been checked by a second person or a computer algebra program.

This paper provides a detailed description of one of the key sets of algorithms to be applied to the forthcoming EarthCARE lidar, along with rigorous testing on three test scenes. There are numerous innovative aspects to the algorithms that are covered in the appendix.  I recommend publishing subject to (a large number of) minor corrections.

MAIN COMMENTS

1. Introduction: please cite Wehr et al. (2023) and/or Eisinger et al. (2024) to put the products described in this paper in the full context of the mission and the other products.

2. Introduction: the coverage of the relevant previous literature is inadequate. Please cite papers on the existing HSRL lidars in space (Aeolus and ACDL), e.g. Flament et al. (AMT 2021), Ehlers et al. (AMT 2022) and Liu et al. (AMT 2024), discussing briefly how this work differs from your own. Regarding the algorithm, you could also cite Mason et al. (AMT 2023 "A unified synergistic...") which builds on Delanoe & Hogan (2008) and also uses an optimal estimation approach to invert the lidar signal accounting for multiple scattering. 

3. L211: my understanding is that chi-squared is not the same as a cost function, since it normally only captures the deviations of the model to the observations (i.e. only the first term in your Eq. 9). If you must use chi-squared then the chi in L211 should definitely be squared.

4. Eq. 9: The use of two different types of "x" (one linear and one logarithmic) in the same equation is inconsistent with previous literature. You should define your state vector (boldface "x") as whatever is returned by your minimization algorithm.  In your case (and also in Delanoe & Hogan 2008) it happens to contain the logarithm of physical quantities.  So there is no need for a separate x^l vector - just define x appropriately. The forward model can still be written as F(x) as it is obvious that the first step of F will be to convert the logarithmic input values back to linear. The a-priori vector (x_a) also then does not need the "l" superscript.

5. Eq. 15: this doesn't make sense to me and I can't find it in Kliewer et al. An obvious problem is that the errror variance (sigma^2) of a linear variable has the units of the square of the units of that variable. But it is added to "1", which has no units! If sigma^2_xlin is the error variance of a linear quantity xlin, then the error variance of log10(xlin) is sigma^2_xlin / (xlin^2 * ln(10)^2) by the rules of error propagation, so why not use this? Since the a-priori error used in OE algorithms is often just chosen without rigorous derivation, an alternative is to specify that the error variance of the a-priori estimates of the state variables is a constant for each type of variable, indicating a constant fractional error.  For example, a "factor of 2" error in xlin implies that x=log10(xlin) has an error standard deviation of log10(2)=0.3 and hence a constant error variance of 0.3^2=0.09.

6. Eq. 16: this also seems incorrect, but for a different reason. As far as I can tell your observations and forward-modelled observations in the cost function are kept in linear space, so surely the observational error covariance matrix should be kept in linear space? It is not relevant here whether the state vector is linear or logarithmic with respect to underlying physical quantities. Also, the definition in the text immediately after Eq. 16 refers to the sigma of x, not y.

7. L500: it would be useful to state the difference in speed between your algorithm and the quasi-small-angle algorithm of Hogan (2008). 

8. L515: There are actually two causes of tails, but this section only describes the small-angle enhancement of the Rayleigh scattering below cloud. The other occurs only in optically thick clouds and is described in numerous places, e.g. the introduction of Hogan (2008). Since the reader might be more familiar with the other mechanism, please stress here which mechanism you are talking about.

9. Fig. B1: is this just a forward model or is a retrieval also involved? It would be better/simpler if this plot simply showed the forward model for a known profile.

10. Sections C1.x: are these really needed since you are simply taking the derivative of some expressions? Just say you implemented analytically-derived derivatives.

MINOR COMMENTS

11. L4: "will provide" -> "provides".

12. Abstract and introduction: mention that "L2a" means "level 2a", which means retrievals based on a single instrument.

13. L11: Surely "lidar extinction, backscatter and depolarization" only applies to the A-EBD product, not to the entire A-PRO processor? This definition appears to contradict how A-PRO is defined in the abstract.

14. L15: second comma should be a colon to introduce the list.

15. L29: I think the Eloranta reference should be bracketed, e.g. "...Resolution Lidar (HSRL; Eloranta 2005)".

16. There are a large number of places where words are incorrectly or inconsistently capitalized.  I guess these will be picked up at the copy-editing stage but I would point out L29 (etalon), L69 (signal-to-noise), captions of Figs. 1 & 6 etc (grey, black, blue and indeed all colours), L109 (microphysical, estimation), L207 (extinction), L301 (section), L315, L322, L324, L328, L334, L411, L436, L507.

17. Text describing Eqs. 1-3: specify that these are backscatter or extinction *coefficients* (otherwise could be cross-sections) and that the "b" terms are *attenuated* backscatter coefficients.

18. L61: No need to double the author names: write "...due to Platt (1981) and the approach of Hogan (2008)."

19. L67: This line needs to work with or without the bracket, so I assume you mean "...particularly when small (or even possibly negative) values..."

20. L74: "tens of kilometres".

21. L83: "Joint Standard Grid"'

22. L103: as a mathematical symbol, S should be in italics. Also L183.

23. Section 2.2 beginning L122: please be consistent in the way you refer to the steps of the algorithm in bold, including whether or not you capitalize "step", and please replace "first step" by "step 1" and "2nd step" by "step 2". Likewise section 2.3.1.

24. L122: define "ATB".

25. L123: "quadratically".

26. Eq. 5: the "(z)" is missing from the final denominator.

27. L140: the quote marks are inconsistent here. In fact there is no consistent use of single or double quotes through the paper.

28. L175: don't start a new paragraph until all the terms in the previous equation have been defined.

29. There are many places where quotation marks are used unnecessarily, e.g. L181, L301, L302, L316 (twice), L347.

30. L186: "insure" -> "ensure".

31. L187: "the each"?

32. L194: what is the difference between steps 14 and 15? Where is step 16? Perhaps a table or flowchart would help.

33. L219: ratio of what to what?

34. Eqs. 13 and 14 (and elsewhere) I think "a" should be a subscript of "R", not be written as Ra. Best to avoid two italic letters being used as one variable as it looks more like two variables being multiplied. Another example is "nl" in L228.

35. L276: where is step 5?

36. L345: "Section B" -> "Appendix B"?

37. Fig. 6 and others: the red boxing and arrows is the kind of thing you'd find in a presentation - wouldn't it be better to put the boxes in Fig. 5e and here have four properly labelled panels?

38. Fig. 7: I suggest you use white in the total-attenuation region to indicate no retrieval, rather than black which implies you have retrieved a lidar ratio of zero.

39. Figs. 4 & 11: it appears that you are using a coloured contour to render the feature mask, but when one class jumps abruptly to another, all the intermediate colours will be rendered between them, incorrectly.

40. L383: "lida".

41. L411: section A0.1 -> section A1?

42. L490: missing reference to a figure - did you mean Fig. B1?

43. L493: define ECSIM.

44. L538: "irregardless" -> "regardless" or "irrespective".

45. L550: definite "fov" and capitalize.

46. L556: the effective what?