General Comments



The manuscript describes the addition of an Ensemble Kalman Smoother

(EnSRF) to the CIF inversion model and tests them with the ICON-ART

atmospheric tracer model.



The manuscript is well written and structured and overall in very good

shape. Its descriptions of Kalman methods are great and the topic is

relevant. The experimental setup and benchmark are also

appropriate for a new implementation. The evaluation of the results

could be a bit more extensive, though.



Figures and tables are correct and informative, though some of the

figures would benefit from added data in tables to be able to compare

numerical values.



The ease of comparing different methods with exactly the same setup

except for the inverse method is important:

https://doi.org/10.5194/acp-15-9747-2015 (own work) showed that

implementation details can have a bigger impact on the fluxes than the

method itself and using a unified assimilation system should allow to

avoid that.



I recommend publishing with minor changes and am willing to review

again.



Specific comments



Abstract:



- L 5: "mere efficient implementation": Table D1 shows a benchmark of

  the new code. Is there a benchmark of the old code to support this

  description?

- L 15: what does ICON-ART bring as advantage? Why does this use

  ICON-ART? It is in the title, so it should be in the abstract.



Introduction:



- L 39: very nice description! (nothing to change, just a note)

- L 74: please add a short overview of numerical weather prediction

  (NWP), tracer models, ICON, and ART. The reports on ICON may be a

  good resource; either to cite relevant reports, or to add a link in

  a footnote:

  https://www.dwd.de/DE/leistungen/reports_on_icon/archiv/archiv_reports_on_icon.html

  Maybe note that more details are provided in 4.1.1.

 



2.2



- L 178: what happens in case of nonlinear effects like OH-destruction

  of CH₄? This is not a problem, but should be noted.



2.3.2:



- L 218: “R matrix is diagonal”: for which observations is this true?

  What’s the scale of weather patterns?



3.1.1:



- L 293: nlag 2: please note below this the reason for using nlag 2.

  Also maybe reference the results from Table 3. Maybe also the

  benchmark in Table D1.

- L 311: “eigenvalues and eigenvectors are already computed”: how well

  does this scale? Is there a practical limit to the model resolution

  due to this inversion? Also maybe reference Table D1 (it shows that

  CIF performance is not the bottleneck).

- L 338: it is not clear whether the ensemble in the runs in the

  publication is generated from the B matrix. Please note explicitly

  whether it is or why it is not.



3.1.2:



- L 356 to 358: this part is hard to understand. Suggestion: “for

  window 2 of the full period, this is the second time it is optimized

  from the observations.”

- Figure 1 is beautiful and clear!

- L 361: why “nlag + 1”? The diagram only seems to show nlag.

- L 362 to 363: this note may be better placed at the examples, 3.1.1

  at line 294.



3.2



- L 405: “a wealth of” this is marketing language. Please give a short

  abstract of the features instead.



3.2.1:



- L 417: is it possible to quantify how much was possible before and

  how big simulations could be used before and how big they can be

  now? Maybe rough, like “Factor X”?



3.3.3:



- eq 39 (line 479): if J(x^a) is 0, this is 1. Are higher values better?



4.1.1:



- line 569: are higher resolution possible with this implementation?

  For example 6.5km? This might significantly improve the possible

  gains from high frequency column measurements. Are there boundary

  conditions for the Europe area?



4.2.1:



- Figure 3: please note that this is the best result from the LEVEL 1 comparison.

- Figure 3: please add more explanation to these results. Why are the

  neighboring cells so different in the error reduction (c)? Also

  please provide a table with the values of d (the clearest result) so

  readers can quantitatively compare the error reduction per country.



4.2.2:



- L 641: reference the results of this experiment in 3.1.1 to motivate

  why nlag 2 is used in the example.

- Table 2: why is the maximum lag 3? Would lag 5 be possible? Which

  lags are used in current CTDAS runs? Reference Table D1.



4.3:



- Figure 5, d: why does this show France, when the maps in Figure 4

  show little change in MUR in France? It could be good to split out

  MUR from Figure 5 and add a set of plots for MUR in different

  countries. And give possible explanations of the differences.

  Portugal has no direct observations, but changes. MUR in UK also

  seems to change much more than in France. There are also countries

  in eastern Europe with bigger changes. Are these effects from being

  at the border of the simulated area?

- Line 693: as above, see the maps in Figure 4. Other countries seem

  to show bigger changes than France.

- line 751: the note about the heaviside function seems to be evading

  a result. Maybe compare with the other functions at larger

  correlation length?

- line 760: “to the best of our knowledge” sounds like this sentence

  should not be stated. Remove?



4.5:



- line 777: “consistent with established literature”: please reference

  appendix B.



Appendix D:



- Table D1: please reference this table from table 2, page 26. Please

  give some additional information, how this benchmark data was gathered.







Technical corrections



- L 18: add a comma: (N₂O), or

- L 231: the note duplicates parts of line 218. Please disentangle the

  two sentences to avoid the duplication.

- L 249: wrong hyphenation: anal-ysis — might need explicit hyphenation via \hyphenation{a-na-ly-sis}



- L 283: which format is used? Where is it described? Citation, or a

  footnote with a link?



- L 314 to 122: a long discussion for a simple point. Suggestion: “the

  default behavior of CIF is to erase the scaling factors after

  conversion. But some models (including ICON-ART) need scaling

  factors as import, so they are preserved here.” Feel free to shorten

  further.



- L 333: “and a scaling” this seems to be missing the word “factor”.



- line 635: please add a note that the results are discussed in section 4.3.



Appendix B:



- Figure B1: please label the colorbars (scaling factor / unitless)

- Figure B2: please label the colorbars with the variable shown (uncertainty reduction / unitless)



Appendix C:



- Figure C1: please show the lines with different line styles to be recognizable in grayscale.

In this study, the authors improved and upgraded the initial version of the EnSRF method in CIF, including enhancements to the batch optimization and localization methods, as well as improvements in technical details to increase assimilation speed, post-processing speed, and overall performance.

The manuscript is well-written and clear in structure. Especially, the description of the theoretical process is very comprehensive and detailed. Additionally, it includes the technical implementation and the characteristics of both the initial and current versions. The synthetic experiments demonstrate the performance of the CIF-EnSRF method and test the impact of different parameter schemes. An additional comparison with CTDAS is provided to prove that the current version of CIF-EnSRF is almost equivalent in effect to CTDAS. I recommend that this manuscript be accepted after the following clarifications and revisions.

1. P6, L160: The dimensions [n×N] should follow Z, not B, as B has dimensions[n×n].
2. P24, Figures 3a and 3b show the map of the true and posterior scaling factors. However, I have a question: why not include a comparison of the maps for the prior, posterior, and true fluxes, as flux maps could provide a more direct view of their spatial distribution? Additionally, are the values of the prior scaling factors equal to 1?
3. Section 4.2.2 and Table 2: In LEVEL2  experiment (F), the authors investigated the impact of adjusting the mean and variance of the ensemble on the inversion results. Please provide the values for the mean and variance of the ensemble in both the adjusted and unadjusted experiments, i.e., what were the mean and variance values used in Table 2 for EXP_F_f1* and EXP_F_t1*? Only by providing the specific data parameters and comparing the extent of their changes can the importance of this parameter be better demonstrated.
4. In Section 4, a series of synthetic experiments for CO₂ flux inversion were conducted to demonstrate the performance of the CIF-EnSRF method. In Appendix B, the authors conduct a further comparison with CTDAS, but this time they selected the synthetic inversion of CH₄. There are some notable differences between CH₄ and CO₂ flux inversions; for example, the assimilation module for CH₄ flux inversion involves generating separate perturbations for natural and anthropogenic emission categories from the B matrix, which is not the case for CO₂. Could you clarify why CO₂ flux inversion experiments were used earlier, but in Appendix B, CH₄ flux inversion experiments were chosen for comparison instead of continuing with CO₂ flux inversion?
5. P37, L844-L858: What are the differences between the CTDAS configuration and the CIF-EnSRF configuration here? It would be helpful to provide a comparison of their configurations and explain the similarities and differences.
6. While the ICON-ART model is highlighted in the manuscript title, it is not mentioned in the abstract or introduction. It might be helpful to include at least a brief mention of ICON-ART in these sections.