I should start by saying, that this is an excellent and deep research. Calculating the the aerosol extinction coefficient is a challenging task, particularly in the presence of noise. The authors convincingly demonstrate that the Optimal Estimation (OE) technique reduces uncertainty in the calculations compared to traditional analytical methods. Additionally, the approach enables the determination of the optimal height resolution for different altitude ranges. Another key strength is its ability to account for both random and systematic uncertainties, including calibration errors.

Overall, this is a high-quality scientific manuscript that is certainly publishable in its current form. However, while certain aspects may seem self-evident to the authors, readers less familiar with OE may benefit from additional explanations.

Technical comments

Eq.4. Symbol “C” is the same as in Eq.1?

p.7 ln.18 “If the residual is approximately one or less, then the solution agrees well…”. Would be good to explain or provide a reference.

p.9 ln 19. “the prior profile of backscatter is taken to be zero”. Probably needs explanation, why it is zero. Just wonder, if a prior profile can be calculated from standard analytical approach.

Fig.5. What are the units? Are measurements normalized?

Fig.11. I have difficulty to understand this figure. Why effective resolution in OE method increases in  non-monotonic way? For grid of 50 m the uncertainty is the highest, but effective resolution is also high (700 m). Would be good to provide more explanations.

Fig.14f. At altitude of ~0.5 km uncertainty of OE is higher than for analytical solution. Can it be explained?

The article describes an algorithm for an improved extinction retrieval from high spectral resolution lidar (HSRL) measurements. An analytical solution exists for the extinction coefficient derived from HSRL measurements, which often suffers from low signal to noise ratios. The presented optimal estimation solution has the benefit of less noisy extinction coefficient and lidar ratio profiles, which is desirable for aerosol research with lidar. Similar approaches were already reported for Raman lidar measurements and spaceborne HSRL observations. Here, the approach is developed for the airborne HSRL of NASA Langley Research Center. Another highlight is the characterisation of measurement uncertainties.  The article is clearly written and describes the algorithm and applies it to simulated and real measurement data. The topic fits in the scope of AMT and should be published after minor revisions which mostly concern the figures.

Minor comments:

1. While the article focusses on the application to airborne HSRL systems, the possibility of the application to ground-based HSR lidars should be considered. New ground-based HSRL systems were developed, e.g., Jin et al., 2020.
   
   Jin, Y.; Nishizawa, T.; Sugimoto, N.; Takakura, S.; Aoki, M.; Ishii, S.; Yamazaki, A.; Kudo, R.; Yumimoto, K.; Sato, K. & Okamoto, H.: Demonstration of aerosol profile measurement with a dual-wavelength high-spectral-resolution lidar using a scanning interferometer, Appl. Opt., Optica Publishing Group, 2022, 61, 3523-3532
2. P4L20: Here, you call the lidar system HSRL-2, later just HSRL2. It is up to you to decide how to name your lidar system.
3. Eq 11 is not really an equation (no =). Furthermore, the journal standards expect a vector to be in bold italic.
4. In nearly all figures, the units of the optical properties are missing.
5. Furthermore, the figures are often quite small. Sometimes, the minor ticks are not resolved well.
6. Fig 2 + 12: The perpendicular signal (in green) is multiplied by a factor of 10, not the particle signal. The label inside the figure is wrong.
7. Fig 2: There are no pink circles as mentioned in the caption.
8. You use the terms “particle/particulate” and “aerosol” synonymously in the description of the optical properties and signals, e.g., particulate depolarization ratio and aerosol depolarization ratio or particle(-dominated) signal or aerosol signal. It is ok, if you focus on aerosol and exclude clouds. However, it would be nice to harmonize the terms throughout the manuscript and especially the axis labels in the figures. Sometimes, the particle dominated channel is referred to as “para” – parallel in the axis labels (e.g., Fig 5+18).
9. The green lines in all your figures look almost blue to me. Maybe you find a different type of green to be clear.
10. P12L16-17 Why the systematic uncertainties are not included in the analytic retrieval?
11. P14L13-14: Probably, this is the reason why the analytic retrieval has the lowest residual in the topmost kilometer for all three signals.
12. P14L17 Fig 1 does not contain uncertainty estimates. Probably, you want to refer to a different figure here.
13. Fig 6 + 15: I would add a, b, c … to the subplots.
14. Please estimate how important are Fig 7+8 and Fig 16 + 17. Maybe a description in the text is sufficient. It is up to you to decide.
15. May I suggest, furthermore, to combine Fig 10 and 11 as Fig 10a and 10b?
16. Fig 18: The exponents 10^x are hard to read.
17. Please add a section about the code availability.

Overall, it is a well-written manuscript which presents an important improvement in the analysis of HSRL extinction data. The method provides great benefit for the research community and should be published after some minor revisions. Well done.

Review of "Aerosol extinction and backscatter optimal estimation retrieval for high spectral resolution lidar" by S. P. Burton, J. W. Hair, C. A. Hostetler, M. A. Fenn, J. A. Smith, and R. A. Ferrare, proposed for publication in Atmospheric Measurement Techniques.