Dear authors,

referee #1 has still substantial comments which, however, could be addressed rather quickly in my opinion. E.g. the first point concerning the term "scattering phase function": I agree that "scattering phase function" is a clearly defined term and its use in the abstract tends to raise wrong expectations.

Also, I agree with the general comment of the referee, that questions of the referee should usually be answered by modifying the manuscript, rather than by lengthy replies to the referee (unless there is good reason not to consider a point). The published manuscript should be understandable to the reader without requiring him to read the discussion (even if the discussion is publically available). This misunderstanding seems also to be the main reason for the long discussion after the quick review: While the referee expected changes to the manuscript, the author replied to many points offline.

As I said above, I think that the open points should be doable without too much effort, but the open points are important ones.

Sincerely,
Bernhard Mayer.

Dear authors,

thank you a lot for considering the referee's comments!

There is only one issue which has been discussed many times before by both referees and with the editor, and this is the assumption that the sky radiance distribution corrected for the airmass is a good approximation of the scattering phase function. This is an unproven hypothesis, and manuscripts should not contain unproven hypotheses, in particular not in the abstract. The referenced paper by Hoyningen-Huene deals with aerosol, and as outlined in the attached document they did not use "scattering phase function" as a synonym for normalised sky brightness. In the contrary, they used an iterative technique to derive the scattering phase function from the sky brightness.

If you want to keep this term, there is basically two ways to go: Either reference a paper that really shows that the single scattering approximation can be applied for transmission through typical cirrus clouds. Alternatively, show it in a sensitivity study, such as outlined in the attached document. My calculation illustrates that even for an optical thickness of only 0.5 single scattering does not apply. This is in line with text books which state that the single scattering approximation is only valid for optical thickness "much smaller than 1" and the "much" is relevant in this context.

We certainly agree that the second option is probably well beyond the scope of this paper. I would suggest to remove the assumption and use something like "airmass-corrected sky brightness" or similar. This is essentially a quick technical correction, since - as far as I can see - none of the findings of your paper depend on the fact that you observe the scattering phase function.

We are really close and the fact that I write this on the 24th of December demonstrates my willingness to get that quickly done!

Merry Christmas, and best wishes,
Bernhard Mayer.

Dear authors,

we could go on like this forever. However, it seems that is is your explicit wish to use imprecise terminology, so be it!

You are perfectly right in stating that the use of the term "scattering phase function" is of minor importance for your work. I would go even further: It has zero relevance for your work. Hence, why introduce an approximation if it is not needed? Why would one state that PI could be approximated by 4 in the introduction if PI wasn't used at all in a publication?

In your response you clearly state that the topic of your paper is the "retrieval of the halo ratio from measured sky brightness". I couldn't agree more! Two lines down you "state emphatically (again) that the proposed alternative term of “sky brightness” is not appropriate in our case". Why not? Dividing the measured quantity by the cosine of the viewing angle doesn't automagically transform it into the scattering phase function. Since we agree that the approximation is imperfect (your words), why not simply avoid it? My main concern is future papers citing Dandini et al as proof that one can use the single scattering approximation for cirrus clouds of optical thickness around 1. And of course my name as an editor being associated with it.

Thank you for providing (for the first time) numbers for the optical thickness. They are indeed smaller than 1 most of the time, but the slant optical thickness is definitely not MUCH smaller than 1 which is the requirement for the single scattering approximation. One also needs to consider that AERONET severely underestimates the optical thickness of cirrus by a factor between 1 and 2 (see e.g. https://www.atmos-meas-tech.net/12/169/2019/amt-12-169-2019.pdf page 178). Considering this correction (factor 1 .. 2) plus the slant path (factor 1.5) one can conclude that the slant optical thickness is between 1 and 2 most of the time. Even if you quietly replaced the "<<" which is the real requirement for the single scattering approximation by a simple "<", this condition would not be fulfilled most of the time.

In my view, this is evidence enough to NOT use the terms "single scattering approximation" and "scattering phase function" in that context. In particular, since it is not needed at all for the paper.

Go ahead as you please!