It has been well recognized that a significant drawback of the optimal estimation method is that a high retrieval time is required, especially for the high-resolution spectrometers. The manuscript “Ground-based Temperature and Humidity Profile Retrieval Using Infrared Hyperspectrum Based on Adaptive Fast Iterative Algorithm” proposes a new method to improve the retrieval speed of the optimal estimation algorithm for retrieving temperature and water vapor profiles from AERI data. I would like to remark the good experimental design to fully evaluate the fast retrieval algorithm, which is important to justify the superiority of the proposed method. The manuscript is well organized and figures are presented in a concise manner and easy to follow. It is interesting and well suited to the audience of the journal and worth being published after a minor revision. The specific comments are as follows:

Major comments:

1. The value of K_Index determines the iterative process of Jacobians. However, the threshold of K_Index is chosen by the distributions of the K_Index values for each iteration, which is dependent on the datasets used in the experiment. This affects the suitability of the fast retrieval algorithm. The authors should point this out. More discussions on this inadequacy of the proposed algorithm should be provided in Section 3.3.3 or in the conclusions.
2. Figure 3: I am confused by the X-axis in the two panels. The authors said that IC and DFS change with K_Index are denoted with black lines, while the X-axis represents K_index is red. The illustrations of Figure 3 seems elusive to me and thus further clarification is needed in the figure caption or in the main text.
3. 4.2.3 Accuracy:The smoothing error cannot be ignored when retrieved profiles are compared directly to radiosondes. Thus, the radiosonde observations should be smoothed with the averaging kernel to minimize the vertical representativeness error.
4. One subject where the manuscript lacks is the discussion on the comparison between the retrieval time and the temporal resolution of AERI spectrum. If most of the AERIoe's retrieval time exceeds the temporal resolution, then the importance of the fast retrieval algorithm will be highlighted and vice versa. Please discuss this issue.

Minor comments:

For the title, may be “Ground-based infrared hyperspectral retrievals of temperature and humidity profile based on Adaptive Fast Iterative Algorithm” is better.

Line10: “due to” is usually not placed at the beginning of a sentence

Line12: “part” -> “step”

Line15: “is” -> “was”

Line17: suggest revising to “resulting in an average retrieval time reduction from 8.96 min to 3.69 min” instead of “with the average retrieval time reduced from 8.96 min to 3.69 min”

Line41: “FTIR” -> “The FTIR instrument”

Line45: “which is more advantageous” can be revised to “which makes it more advantageous”

Line57: this sentence should be reworked

Overall:

By reducing the computation time of the AERI-OE process in half, this seems like a major improvement that represents a solid contribution to a measurement technique, and is therefore appropriate for the journal. However, I have two main problems with the manuscript:

1) The authors state that the new method results in equivalent retrieval accuracy. However, for water vapor, the bias increase appears to be up to about 40% (0.7 to 0.9), and the RMS increase up to ~12%. This does not seem to me to be comparable retrieval accuracy.

2) The authors need to clarify the scope of the work and fix the errors, typos and unclear parts of the paper.  Is the novely of this work just in implementing the k_index and updating the Jacobian less often, or did they also introduce different formulations, methodology, etc?  Given that this work follows closely from Turner and Lohnert (2014), the authors should make it clear what is the same as in that prior work by referencing it as needed, avoiding repeating details from it except as necessary, avoiding typos/errors when they do paraphrase from that work, and discussing clearly what is novel in this work. For example, Eqn (1) differs from prior work (and from Rodgers (2000)) in that Xa is replaced with X0.  Is this intentional, and if so, why? More examples of specific issues follow.

Other comments:

- Did the authors modify the AERIoe code itself or did they develop a new codebase from scratch? Please state in the Data availability if/where/how the fast AERIoe code is available. (Proprietary or open source? How does one obtain it?).

- Given that the main goal is to reduce the computation time, specifics in that regard are needed. Has the code timing been analyzed and what are the bottlenecks? I assume calculation of the Jacobian is the main bottleneck; is that the case?

Abstract: 

The abstract requires significant revision.  Examples:

- Please begin with a sentence that more clearly gives the background - something like: “Two methods for retrieving … are physical and statistical retrieval algorithms …”

- Line 12: Begins with “Further analysis showed…” but no analysis has yet been discussed. What changes were made to the Jacobians and why was that expected to speed up performance (but didn’t)?

- The time estimates are not useful without knowing what type of computing platform was used.  Perhaps just give the percent improvement. Also, are 3 significant figures warranted here?

- What is meant by “certain impact”?

- What is meant by “to some extent”? Why not state the convergence rate of the traditional algorithm?

- The authors say that “The retrieval accuracy of the fast retrieval model is equivalent to that of the traditional algorithm.” However, on lines 346-348 differences indicate that the accuracies are not equivalent.

- How is the convergence criteria adjusted to give reliable retrieval results?  It was previously stated that the results were equally accurate. Do you mean they are equally accurate when they both converge?

Lines 115-124: 

This section is a paraphrasing of Turner and Lohnert (2014), which should thus be referenced here. The section also has a number of errors/typos:

Line 115: If the authors are using X0 = Xa, they should replace X0 with Xa in Eqn (1) so it is consistent with Turner and Lohnert. If not, they should explain this change.

Line 116 says “Y is the observed radiance vector, F(X) is the AERI observed spectrum…”  Is it rather that Y is the observed radiance vector (from the observed AERI spectrum) and F(X) is the estimate of Y from the forward model calculation?  It would also be helpful to define that the background refers to the a priori atmospheric state, if that is the case.

Eqn. 1: I'm curious why this formulation is used instead of the Levenberg-Marquardt formulation (Rodgers 2000, Eqn 5.36). How is the behavior the same or different? Carissimo et al. 2005 state that their method is almost equivalent to Levenberg-Marqardt. In Levenberg-Marquardt, increasing gamma decreases the step size and makes the retrieval weighted more toward steepest descent. How is the formulation here the same or different?

Figure 1: This figure needs improvement and explanation. E.g. please define “iterative observations” and “iterative profiles”  in the caption. Use of the symbol “Sa” is inconsistent with use of “Jacobians” instead of “K”. K_Index has not yet been defined.

Line 118:  I don’t think n is the number of iterations, but rather the iteration number.

Line 120: The description of how gamma is used is not clear.

Line 122: Remove “progress”.  

Line 122: Please change “is not allowed to converge until…” to “Iterations are continued until…” if that is what is meant here.

Line 124: Use consistent symbols. You have superscript n sometimes and subscript n other times.

Line 214-215: It is not true that “what affects IC and DFS lies only in gamma and Jacobian”. In fact, when gamma = 1, IC and DFS are determined by Se and Sa, with the purpose of the Jacobian being to transform Se into the state space for Sa, so that they have the same units and size (rows and columns). I think what you mean is that IC and DFS only change with iteration due to changes in gamma and the Jacobian. (But see below).

Lines 214 - 232: I don’t understand the logic here. On line 224, it is strange to say that gamma changes with the adjustment of the profile, since gamma is prescribed. Figure 3 is confusing. The x-axis goes in the reverse direction as the retrieval proceeds, the figure caption description seems to be wrong (red is actually K_index), and it isn’t stated where K_index starts and ends (starts at the high end, ends at the low end?).  It is not surprising that the DFS and IC increase as gamma drops to 1, since gamma weights the retrieval away from the observation and toward the first-guess, which presumably has no information content at all. It is also not surprising that there is not much change in DFS and IC with the Jacobian, since, as stated previously, the purpose of the Jacobian here is to transform Se onto the dimensions of Sa. I don’t see how this shows that the change of the Jacobian has less influence on the retrieval ability than gamma. Gamma is not supposed to influence the retrieval ability, but only the retrieval stability. That is why iterations are continued until gamma is 1, whereupon the retrieval equation is equivalent to the Gauss-Newton formulation and the maximum information content is used. In fact, I don’t see the point of this paragraph or figure at all.  The authors could simply state that if X is not changing much, as evidenced by the K_Index, then the Jacobian is probably not changing much either, and therefore does not need to be recomputed.  (Note, however, that this is not necessarily true, and they need to show that it is an ok approximation).

Turner and Lohnert state that “Future versions of AERIoe will use the Carissimo et al. (2005) approach in order to more efficiently converge and reduce computational time.” Did the authors explore that approach, and how might that change their analysis?

Sections 3.1 and 3.2. are unclear. The description of the retrieval forms is confusing. Is the state vector comprised of the temperature and log of water vapor on the 37 atmospheric layers? Why isn’t it parametrized, given that there are far fewer degrees of freedom? It continues to be difficult to tell what is new here and what is the same as previous work.  Please avoid repeating details where you could reference the previous work. For example, you could say, “The forward model is the same as that described by Turner and Lohnert, except as follows…” Was LBLRTM used here to apply the spectral response function, in contrast to the previous work? It is stated that LBLRTM can be used to calculate the Jacobian.  Was it used for this purpose? Again, is this a departure from previous work?

Line 176: Please remove the statement that LBLRTM is the most accurate forward model or provide a reference for it.

Line 196-197: Please rephrase this: “…determined whether updating or not by monitoring the indicators that can reflect the changes of Jacobian in the iterative process “

Line 344: Please change, “with only slight differences in BIAS metrics between 500 m and 1.5 km” to include a quantitative value, such as “with differences within x% to y%”

Lines 346 - 348: It seems like “a maximum increase of 0.29 g/kg in BIAS and a maximum of 0.32 g/kg in RMSE” are significant.  The bias increase appears to be up to about 40% (0.7 to 0.9), and the RMS increase up to ~12%. This does not seem to me to be comparable retrieval accuracy. Please clarify.

Line 350: More detail is needed about how you calculated “Pearson’s correlation coefficient between two datasets on the y-axis and the ratio of the standard deviation on the x-axis”, and the caption of Fig. 10 needs to be improved.

References: 

- For use of FTIR viewing solar spectra, you could also reference the work of Kimberly Strong’s group; e.g.: https://amt.copernicus.org/articles/7/1547/2014/

- For retrievals from AERI, please add a reference Rowe et al. 2006, which used constrained linear inversion to retrieve temperature profiles from an AERI instrument: Rowe, P.M., Walden, V.P. and Warren, S.G., 2006. Measurements of the foreign-broadened continuum of water vapor in the 6.3 μm band at− 30° C. Applied optics, 45(18), pp.4366-4382.

English grammar and clarity:  If possible, please have a native English speaker edit your paper throughout, including use of “the” in English, which is very challenging to get right. Examples:

Line 25 and throughout: When you are talking about something in general, omit the word “the” and change the noun to the plural.  Examples: change “the observation network” to “observation networks”. Change “the convective scale numerical weather prediction system”  to “convective scale numerical weather prediction systems”.  Change “the radiosonde profiles” to “radiosonde profiles”.  Only use “the” if you are talking about a specific thing, and if you have made it clear which one you are talking about. For example, on line 73, add “the” before “ARM program” since it is clear which program you are talking about (ARM).

Line 26: Please define all acronyms (e.g. NWP model)

Line 32: Change “shows” to “has” or “demonstrates”.

Line 57: remove “to boot”