My recommendation for the paper is to proceed and publish as is.

The topic of the paper is very interesting because the determination of the overlap function in lidar systems is still a crucial point for the retrieval of the aerosol optical properties at very low range in tropospheric studies, in particular at very low range. Therefore, papers on this specific topic are always welcome. The error estimation related to the lidar ratio value selection is very interesting too. The paper is well written and clear, but requires some clarifications.

- Some comments related to the scientific content:

Line 64: The expression of the molecular lidar ratio S_m = 8π/3 sr is an approximation, whereas it depends on depolarization and refractive index at each wavelength (see Bucholtz A., Rayleigh-scattering calculations for the terrestrial atmosphere, APPLIED OPTICS, Vol. 34, No. 15, 1995). Which is the influence of this assumption on the overlap estimation?

Line 98-99: About the sentence “we neglect the difference between the aerosol extinction coefficients at λ₀ and λ_R”. This assumption could be acceptable in the case of the pure rotational Raman, but not in general, like for 532 nm in this paper, where the vibro-rotational Raman at 607 nm is used. The authors should explain better.

Line 164-165: see previous comment.

Lines 169-173: The authors say that “the 355-nm has a sudden fall below approximately 400m. For this reason, in this particular case of optical alignment we should distrust the overlap function retrieval below that height”. Why did not the authors try to improve the alignment and use a better measurement?

Figs. 2 and 3:

Did the authors performed a comparison with the overlap functions obtained from the Wandinger and Ansmann method?

The overlap function should not be higher than 1, contrary to what the figures show, suggesting that the overlap function contains further kind of corrections. This behaviour is present also in low aerosol load conditions that should be used to retrieve the overlap function, so affecting the extinction and backscatter retrieval that, as known, is particularly critical at low range. In this context, how the authors use the overlap function, especially at low range, if the overlap function presents the reported dependence on the lidar ratio?

Which is the full overlap altitude of the system used by the authors in the paper?

- Some suggestions related to the text:

Line 31: I suggest replacing “To overcome” with “To reduce”, because the problem is not removed.

Line 71: In eq. (2), maybe, it should be O(R_m) instead of O(R).

The manuscript includes a rather interesting and novel concept for the calculation of the overlap function in lidar systems and provide an application of the technique. I would recommend it for publication after major revisions that address the following points:

-- The authors use an aerosol free region as a reference point and then start calculating the overlap below using an analytical approach. The problem is that according to Eq 18, the error due to the lidar ratio is cumulative with height. The point b, c, and d on page 5 are not correct and should be update in order to highlight this. By starting the overlap calculation at an aerosol free region, one carries along a systematic error accumulated from all the previous altitudes until the actual full overlap is reached. But then it doesn’t make sense to use this technique because the accumulated error can be simply too large. This effect can be partly seen in figures 2 and 3 where the overlap function is constantly and significantly > 1. even above the full overlap range (2-3km).

To circumvent this issue, I recommend one of the following alternatives.:

• The authors could use a measured aerosol lidar ratio profile between the aerosol free region and the full overlap range. In that way they avoid the accumulation of uncertainty from the previous layers. The lidar ratio above the full overlap region should be known from the Raman inversion (no iterations needed)

• It could be possible to apply this technique (perhaps without too critical adaptations) by starting from a height where the aerosol extinction and backscatter coefficients are known. Such values are already available from the Raman inversion (above the full overlap range).

-- The authors assume that the backscatter and extinction cross sections for both molecules and aerosols do not change significantly between the elastic and the vibrational Raman wavelength and use this assumption to simplify a term to go from Eq 9 to Eq 10. This is not correct. For molecules, relative differences of the Raman with respect to the elastic cross section are in the order of 30% (40%) for 355nm (532nm). For aerosols the difference are indeed smaller ~10% (13%) for an Angstrom of 1 but can generally range from zero to 23% (33%) for an Angstrom ranging from 0 to 3. The authors should:

• include at least the molecular wavelength dependency of the backscattering/extinction cross sections in their formulas because it is known and well parameterized (proportional to λ^-4).

• either include the aerosol Angstrom exponent as an additional parameter to the equations along with its uncertainty, similar to how the treated the lidar ratio uncertainty (ideal solution)

• or, if this makes things impossible to solve analytically, provide a paragraph with a theoretical analysis of the expected uncertainty due to a varying Angstrom exponent

-- The experimental application faces many challenges, such as unrealistic overlap values above the full overlap range and sharp drops to the Raman backscatter profiles below 400m even though they should be, in theory, overlap independent. From my point of view the paper stands just fine with the theoretical part and an experimental application is not necessary. However, if the authors want to include it then they should make sure that:

• the overlap between the Raman and the elastic channels is indeed the same. Different interference filters (IFF) are applied per channel that can lead to overlap-like effects in the signals as the angle of incidence (AOI) of the collected beam on the IFF changes from the near range to infinity. This is especially important for the Rotational Raman channel because changes in the AOI create an effect equivalent to shifting the IFF transmission with respect to the incident wavelength. Such overlap-like effects are expected to be pronounced in a Rotational Raman channel due to the proximity of the central wavelength of the IFF to the Cabannes line and due to the temperature dependence of the Rotational Raman cross sections. A good experimental way of verifying this is to check whether the 355-387 derived overlap is the same as the 355-354 one, or wether the Raman backscatter at 355-354nm is the same in the near range as the 355-387 one. For 532-607nm the authors could prove that by using an IFF with a different bandwidth in subsequent measurements they get the same overlap function. Without such a verification I wouldn’t recommend publishing the experimental part.

• They provide information of the expected full overlap of the system per case. This can be done with a telecover test or with a Ray-tracing simulation.

More specific comments can be found in the uploaded pdf file with inline comments.