Reply to Benestad's comment on "Discussions on common errors in analyzing sea level accelerations, solar trends and global warming" by Scafetta (2013)

I found Benestad (2013, hereafter B13)’s criticism of Scafetta (2013a, hereafter S13) misleading. More significantly, B13 did not find any physical nor mathematical error in S13. Thus, S13’s scientific results remain fully confirmed. B13 complained that S13 would have misrepresented Benestad and Schmidt (2009, hereafter BS09) because BS09 too argued that the regression models studied in BS09 are flawed because of multicollinearity of the constructors. However, BS09 did conclude that “the most likely contribution from solar forcing a global warming is 7±1 % for the 20th century” using regression models of the global surface temperature. S13 questioned this conclusion and demonstrated that their result is consistent only with outdated hockey-stick paleoclimatic temperature reconstructions (e.g. Mann et al., 1999). BS09 studied two regression models for the 1900–2000 global mean temperature. Equation (1) uses as constructors the solar and GHG forcings, which are collinear because both trend upward: 〈T 〉 = α0+ 0.7 4 α1S +5.35α2×ln(ρ)+μ (e.g. Lean and Rind, 2008). Equation (2) uses the 10 GISS forcings, and 9 functions out of 10 are collinear to each other: 〈T 〉 = β0 + β1FS+β2FGHG+β3FO3+β4FH2O+β5Fland+β6Fsnow+β7FAer+ β8FBC + β9FRefl + β10FAIE + μ. The solar contribution to the 20th century warming is estimated to be ∼ 10 % using Eq. (1) and ∼ 7 % using Eq. (2). The latter result is consistent also with the GISS ModelE prediction (S13, Fig. 6d). B13 complained that BS09 compared “climate models and observations” that “only included two co-variates”, not 10 as shown in BS09 Eq. (2) and as S13 would have claimed. However, S13 (Table 1) demonstrated that during the 20th century solar forcing is collinear (r ≈ 0.7) with other 8 constructors taken singularly. Volcano forcing was the only exception. A regression model is misleading also if it is based on just two collinear constructors, as B13 claimed to have done. At the end, BS09’s “7 %” claim is only supported by the GISS ModelE prediction; the result remained not validated by robust data analysis and, therefore, BS09’s argument falls into circular reasoning. Indeed, demonstrating the multicollinearity flaws of the regression models studied in BS09 was not the ultimate goal of S13. S13 proposed a solution using the regression methodology in a situation where the solar forcing is not collinear (|r| / 0.2) with other forcings (S13, Table 3). Thus, B13 misunderstood S13’s argument that was based on two major results: (1) S13 (Fig. 6) demonstrates that GISS ModelE underestimates the empirical solar signature by a factor varying from 3 to 8; (2) S13 (Fig. 7) demonstrates that BS09’s conclusion that the sun contributed only ∼ 7 % of the 20th century warming is compatible only with outdated hockeystick paleoclimatic temperature reconstructions. For example, Mann et al. (1999) temperature reconstruction shows a preindustrial climatic variability of ∼ 0.2 ◦C that implies a very small climatic solar effect. Yet, more recent paleoclimatic temperature reconstructions (e.g. Moberg et al., 2005) show a far greater preindustrial climate variability (∼ 0.7 ◦C), implying a strong climatic solar effect yielding a solar contribution to the 20th century warming comparable with the anthropogenic one. These results fully confirm Scafetta and West earlier works (2005, 2006) that BS09 criticized. S13 (Sect. 4) demonstrated that BS09 misapplied the Maximum Overlap Discrete Wavelet Transform (MODWT) by erroneously adopting the periodic padding instead of the reflection one yielding Gibbs artifacts. B13 acknowledged BS09’s math error, but surprisingly complained that Scafetta and West (2006) provided insufficient analysis