Manuscript prepared for Earth Syst. Dynam. with version 2015/04/24 7.83 Copernicus papers of the L A T E X class coperni- cus.cls. Date: 7 April 2016 Reply to Lovejoy and Varotsos - I K. Rypdal 1 and M. Rypdal 1 1 Department of Mathematics and Statistics, UiT The Arctic University of Norway, Norway Correspondence to: Kristoffer Rypdal (kristoffer.rypdal@uit.no) Science is comprised of the creative process of formulat- ing new hypotheses and the systematic attempt to refute these hypotheses by testing them against observation. For the latter it is not sufficient to demonstrate that observations are con- sistent with the hypothesis according to some prescribed test, 5 one also must make sure that the observations are inconsis- tent with other plausible (null-) hypotheses. In the present context this is particularly clear, because the statement that a response is nonlinear in itself is a negation. The main propo- sition in the paper by Lovejoy and Varotsos (L&V) is that the 10 response is not linear. Thus, the only valid way of testing this statement against the data is to demonstrate that the linearity hypothesis is rejected by the data. In section 2.4 (Fig. 2) and section 3.3 (Fig. 4) of our com- ment (R&R-C) we demonstrate that a linear response is con- 15 sistent with the data. The reply of Lovejoy and Varotsos (L&V-R) only deals with section 3 of R&R-C, so we shall restrict ourselves here to the question of linearity and inter- mittencies. The issue of multifractal (clustered) intermittency versus 20 non-Gaussian Lévy processes and their long-memory deriva- tives was discussed at depth in a paper we recently published in Earth System Dynamics, with Shaun Lovejoy as a very ac- tive referee. 1 We find it strange that L&V-R do not refer to this paper and the associated discussion. 25 Our test presented in Fig. 4 is deliberately extremely sim- ple. It can be appreciated by anyone, without understand- ing of the analysis method. We have just employed exactly the same analysis method as L&V (using Lovejoy’s com- puter routines) on the data from a very simple linear response 30 model, a damped harmonic oscillator. The results of the anal- ysis are indistinguishable from L&V’s results from the same 1 M. Rypdal and K. Rypdal, Late Quaternary temperature vari- ability described as abrupt transitions on a 1/f noise background, Eart Syst. Dynam., 7, 281-293, 2016. doi: 10.5294/esd-7-281-2016. http://www.earth-syst-dynam.net/7/281/2016/esd-7-281-2016.pdf. Discussion: http://www.earth-syst-dynam.net/7/281/2016/esd-7- 281-2016-discussion.html analysis of output from the ZC-model. This implies that the L&V results do not reject the linear response hypothesis. L&V-R does not present any arguments against 35 the validity of this test. Page 1, paragraph 2: L&V-R cite two papers that are sup- posed to demonstrate the nonlinearity of responses to "spiky" forcing in some climate models, and state that their contribu- 40 tion has been to quantify this. There are many other papers that find very weak nonlinearities (see the paper by Andrews et al. and references therein). However, we do not claim that the response to such forcing impulses is linear - it seems quite plausible that they are not. Our claim is that the analysis by 45 L&V does not reject such a claim, and by no means repre- sents a “quantification of the nonlinearity.” They also write “The L&V method. . . simply exploits the known fact that linear transformations of a time series can only make linear changes to the exponent scaling function 50 ξ (q), they cannot affect the nonlinear part that is associated with the intermittency." In section 3.1-3.2 of our comment (R&R-C) we show that this is true only if the following three conditions hold: (I) The linear transformation can be repre- sented by a power-law response function. (II) The structure 55 functions are power-laws for all q. (III) Internal variability is negligible. Moreover, in section 3.4 we demonstrate by an example (the damped, harmonic oscillator) that the scaling function changes radically under this linear transformation when it is computed from the trace-moment analysis. 60 Page 1, paragraph 3: Here L&V-R defends the trace mo- ment analysis. The method is supposed to be effective par- ticularly because “. . . it removes the linear qH term in the structure function so that the linear K(q) part can be studied 65 directly." We can’t understand that subtraction of a straight line from a curved graph represents anything significant. The method is based on implicit assumptions that the underly- ing process is multifractal (that the structure functions are power-laws) with a distinct “outer scale" which defines the 70