Dear authors,

Thank you for your responses to the referees' comments and sending your important manuscript to AMT. It is my great pleasure to proceed towards publication, but since I have struggled with details and notation of the formalism that you present, I would like to ask you for some additional clarification and comments. Consequently, I request some minor and technical corrections (see below).


Minor corrections:
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[0] Please check definitions, abbreviations, maths and units in your manuscript.
The choice of variable names makes equation (1) seeming somewhat unrelated to equations (2) and (3). Is this so ? The naming of the variable dP_frac is unfortunate and confusing, because unlike dP_sCO2, dP_frac does not designate an absolute pressure difference but a relative pressure deviation multiplied by XCO2 (eq. (A6)). Please consider renaming it. The other confusion is on p. 39, where c once is a number defined in Eq. (A3), and then is used throughout equations (A4) to (A7), but in eq (A7), it suddenly denotes another variable with dimensions of 1/pressure. It would probably be easier to use another variable name (eg a) for the dimensionless constant in equations (A3 and subsequent) and then identify coefficient c in Eq. (1) with a in the case of Eq (A6) and with X_CO2,raw/P_ret*a in the case of eq. (A7). Moreover, it is not quite clear what the symbol C_P in Eq. (1) stands for (correction term with respect to parameter P ?). Wouldn't it be more straightforward to use X_CO2, or DeltaX_CO2 ?

[1] If we identify Eqs. (A6) and (1), and use the results from Table 4, it seems that c = -0.9. In this case the Taylor expansion around c=0 made to arrive at eq. (A6) is likely not valid.

[2] Is it possible that the surface pressure is larger (or smaller) than both, P_ap and P_ret ? Then c should not be called fractional weight for which the constraint 0<=c<=1 would apply. Again, the fact that c = -0.9 is reported in Table 4 seems to indicate that this is the case, however.

[3] When going from Eq (A5) to (A6), you write "Taylor expansion in c around c=0 leads to:". Strictly speaking, this is not true, because (A5) is a proportionality, but eq (A6) an equality relation. Please correct.

[4] What is the significance of comparing rel. variations of X_CO2,raw/Pret with rel. variations in (P_ret-P_ap) (just before writing down Eq (A7)) ? Could you motivate any further ?

[5] On p. 10 you write "The dP_frac coefficient over land is close to 1". In Table 4, you give a value of -0.9. Why don't you use that value in the text ? Intuitively, the minus sign does seem to be correct (an increased P reduces X_CO2). Does the departure from 1 come from the water in the atmosphere ?

Technical corrections:
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p 1, abstract : consider to drop the word "currently" in "Currently all measurements of ..."
p 1, abstract : "Relative errors in the surface pressure estimates, however, propagate nearly 1:1 into relative error in bias-corrected XCO2" -> "Relative errors in the surface pressure estimates, however, propagate nearly 1:1 into relative errors in bias-corrected XCO2"
p 1, abstract : "are also caused by an coding error" -> "are also caused by a coding error"
p 1, introduction : "The O2 optical depth is observed at the so-called ..." -> "The O2 optical depth is observed in the so-called ..."
p 2, top. What is nearly similar and what would exactly similar mean ? Is it similar path or similar paths ? Please correct.
p 2, 2nd paragraph : "the spectroscopy of the oxygen" -> "the spectroscopy of oxygen"
p 5, last paragraph before section 3.1: I guess there is no such thing as a -x-axis. Could you use positive or negative x-direction instead ? Similarly, "rotate about the x-axis" instead of "-x-axis"
p 5, section 3.1 : Please clarify first phrase which begins with "The analysis" and ends with "z-axis". It is very long and its logic is not very clear.
p 6, upper part : "Using the preprocessors over the L2FP algorithm". Does this mean "Using the preprocessors after the L2FP algorithm" ?
p 6, "1.800 soundings" -> "1800 soundings"
p 6, "1.000 soundings" -> "1000 soundings"
p 8, 4.1 section title : Use singular as in text.
p 8, eq. (1) : index p on left hand side should be in small letters as it (probably) does not refer to pressure
p 8, eq. (1) : Introduce the symbol C_P at the left hand side in the text, and even better, replace it (see above).
p 8, eq. (1) c_i : please be more specific about the definition of the c_i. For instance, you could write c_i are regression coefficients which express the sensitivity of X_CO2 (C_P) to the selected parameter p_i.
p 9, Third sentence after eq (3): "The definition of co2_grad_del and DWS remain the same in v9." -> "The definitions of co2_grad_del and of DWS remain the same in v9."
p 13, References : Please adopt the formatting recommendations for references and prefer the doi.org link over any other, eg:
Porter, J. G., De Bruyn, W., and Saltzman, E. S.: Eddy flux measurements of sulfur dioxide deposition to the sea surface, Atmos. Chem. Phys., 18, 15291–15305, https://doi.org/10.5194/acp-18-15291-2018, 2018. (attention, there are links beginning with "http:https:" (eg Cogan et al))
p 22, Table 4 : units for coefficients and reference values are missing
p 23, Table 5, lines 3 and 4 : the meaning is the same even though variables are different. Please correct.
p 23, Table 5, line 10 : wco2 should correspond to the weak and not to the strong band
p 23, Table 5, line 11 : replace star by multiplication dot
p 24, Table 6 : Te et al. (2014)(2014) -> Te et al. (2014)
p 35, Figure 11 : please indicate in figure caption which data refers to the y-axis to the left and which to the scale on the right. The current choice of colors doesn't help.
p 39, right after eq (A2) : "mean molecular weight of water vapor" -> "molecular weight of water vapor"

Dear Authors,

Thank you for clarifications you made to the manuscript.

I point out that there is a misspelling in the reference to Rödenbeck (umlaut problem) and that the word "nearly" likely does not make any sense on top of p 2. It would be meaningful to speak about things that are nearly equal or very similar, but "nearly similar" sounds like very different. Do you intend to say that ?

I recommend that the manuscript should go into publication stage after these technical corrections.