Dear authors,

I received the two reports of the referees.
Peter Ditlevsen advices publication, though observes than "it would have been useful if some of the points that [he] raised in [his] previous review were addressed in the revision rather than in the separate response from authors to reviewer.". Shaun Lovejoy still has extensive comments and argues against your rejection of multifractality.

The article will no longer be sent to these two reviewers, and I am in principle willing to lean towards acceptance, but on the other hand I cannot proceed without further assurances. Let me summarise how I view the situation:

(a) Multi-fractality vs mono-fractality
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On the one hand, you show that the tau(q) 'looks' linear; and on the other hand, Lovejoy says that if you try to fit a higher order polynomial, it will do a little bit better. This leaves me to wonder to what extent either you, or Lovejoy, actually perform a proper model selection test between mono-fractality and multi-fractality. This state of affairs presents two options for resolution:

(1) either your offer a model selection test to the readers (this may require extensive work)
(2) either you somehow find the argument to say that it looks linear enough to, at least, give support to the validity of your interpretation. In other words, does it really matter to have multi- or mono-fractality ?

(b) definition of tau(q) and of beta
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I believe you should somehow use the opportunity of the paper to clarify the different definitions. You answered Ditlevsen to his satisfaction but the elements of answer do not appear in the final version of the manuscript. Lovejoy further argues about the confusion produced by the coexistence of different definitions of tau(q).

I would appreciate at least an informal reply to these two aspects (with a discussion of implications on the manuscript). I reserve at this stage the possibility of discussing matters with the chief editor and/or to involve a third reviewer even if we probably all agree that we want to proceed swiftly enough.

Sincerely yours,

Michel Crucifix

Dear authors,

Thank you for your revised version, which I consider acceptable for ESD pending a few technical corrections (mostly editorial, and as an anticipation of copy-editing which will be done anyway)

abstract, l. 4: Quaternary : capitalize 'Q'
abstract, l. 6-7 : English need to be polished

l. 314 : word missing between 'present' and 'analysis'
l. 315 : 'are' rather than 'is' explained
l. 318 - 320 are in need of some english-polishing
l. 321 : although the case of the whole section is quite clear, that last sentence need to be made a bit more explicit.

l. 394 : $\rangle$ rather than $\langle$

Figure 4 : check indices (a) -> (k) (?) on Figure