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1. At the close of the 19th century Svante Arrhenius and his co-workers expressed the heat trapping effect of the atmosphere in terms of its composition, specifically with respect to trace quantities of water vapor and carbon dioxide. They were working on very long time scales to explain the earth’s history of glaciation cycles (Arrhenius, 1896) (Hogbom, 1895) (Langley, 1889) (Tyndall,1861). Although his theory of glaciation cycles has long been discredited (Ewing, 1958) (Martinson, 1987), the warming effect of atmospheric CO2 put into the scientific literature by Arrhenius survived.
2. It was applied to a much shorter time scale of a century or less by Guy Callendar in 1938 (callendar_1938). Callendar was concerned about carbon dioxide emissions from the combustion of “large quantities” of fossil fuels by the industrial economy. His concern was that the carbon in fossil fuels dug up from deep under the ground by humans had been sequestered from the atmosphere for millions of years. He reasoned that this carbon does not belong in the current account of the carbon cycle and that therefore the injection of such external carbon into a delicately balanced carbon cycle is an artificial perturbation that could upset both the carbon cycle and climate system and cause runaway artificial warming with unknown and possibly catastrophic consequences (Callendar, 1938).
3. Soon after the publication of a series of global warming papers by Callendar, the world plunged into thirty-five years of cooling that lasted until at least 1975. During that time the Arrhenius warming effect of atmospheric composition and the related Callendar concern about fossil fuel combustion subsided. They were replaced by concerns about the cooling effects of aerosols in fossil fuel emissions (Rasool&Schneider, 1971) (Schneider, 1975) (Kukla, 1972).
4. That changed again in the late 1970s when the temperature curve reversed to a strong warming trend and two landmark papers by James Hansen and Andrew Lacis changed the narrative from cooling to man-made catastrophic global warming (Hansen, 1981) (Hansen-Lacis, 1984). These papers form the blueprint of the modern version of the theory of anthropogenic global warming and climate change (AGW) as the Arrhenius effect of carbon dioxide produced by the combustion of fossil fuels by humans.
5. The theory of surface warming due to absorption and re-radiation in the infrared band by atmospheric carbon dioxide (Pierrehumbert, 2010) yields the testable implication first proposed by Manabe, Wetherald, and Charney that surface temperature should be proportional to the logarithm of atmospheric carbon dioxide concentration (Charney, 1979). Called the Equilibrium Climate Sensitivity (ECS), it was computed as the increase in temperature in Celsius units for each doubling of atmospheric carbon dioxide concentration and that convention has persisted.
6. Jule Charney consolidated results from five climate models to report Equilibrium Climate Sensitivity variously as ECS=[2.0-3.5], [2.6-4.1], and [1.5-4.5]. Charney then declared without elaborating that the most likely value of the ECS = 3 with its uncertainty indicated by the range ECS=[1.5-4.5] (Charney, 1979). This range was adopted by the IPCC and has since become dogma in climate science.
7. However, the first significant paper on AGW in the modern era by James Hansen cited the first Charney estimate and reported ECS = [2.0-3.5] (Hansen, 1981). The IPCC uses the dogma Charney estimate and consistently reports climate sensitivity as ECS = [1.5-4.5] (IPCC, 2013) (IPCC, 2007).
8. Significant works on the estimation of the ECS with climate models and also from observations and paleo reconstructions have been reported in the last two decades (Andronova, 2000) (Gregory, 2002) (Forest, 2002) (Knutti, 2002) (Frame, 2005) (Murphy, 2004) (Stainforth, 2005) (Hegerl, 2006) (Kummer, 2017) (Johansson, 2015) (Stevens, 2016) (Aldrin, 2012) (Dressler, 2018). Their results are summarized here.
9. (Andronova, 2000) used 142 years of observations constrained by climate models and reported that ECS = [2.0-5.0] but found that more than half of that figure can be explained by solar variability with residual CO2 sensitivity ECS=[0.94-2.35]. Although not directly acknowledged by the climate science literature, this finding weakens the AGW argument that warming is human caused by way of the
   Arrhenius effect of fossil fuel emissions.
10. (Gregory, 2002) derived sensitivity from observations 1860-2000 constrained by models for ocean heat uptake and derived a probability distribution of ECS which shows that the symmetry of the ECS distribution assumed by Charney and the IPCC in reporting confidence intervals does not exist. Gregory’s results show a long tailed distribution that is skewed right. The author reports its properties as a median of ECS = 6 and a 90%CI of ECS = 1.1 to infinity. The high end is not bounded. However, it is shown that with certain assumptions and model constraints the range can be reduced to 90% CI of ECS = [1.7 – 2.3]. The motivation for these assumptions is described by the author as “a range as narrow as that would be a great improvement on the current state of knowledge”.
11. The ECS Probability distribution derived from climate model simulations constrained by recent observations (Gregory2002) (Forest, 2002) yield a fairly wide symmetrical distribution with a 90%CI for ECS = [1.4 – 7.7]. The spread of 6.3 is more than twice that assumed by Charney and institutionalized by the IPCC. Also the asymmetry with a bias toward higher values of sensitivity does not exist in the IPCC and in the institutionalized AGW narrative.
12. The (Knutti, 2002) authors use A climate model with both CO2 and aerosol forcing to generate Monte Carlo simulations and thereby to construct a probability distribution of climate sensitivity. They find that the IPCC has grossly underestimated the width and location of the 90%CI for ECS. with the underestimation described as “a 40 per cent probability that warming will exceed the rise predicted by the IPCC, and a 5 per cent probability that warming will fall below that range”. The 90%CI for climate sensitivity is reported as ECS = [2.7 – 8.7]. This distribution is twice as wide as the IPCC distribution and offset by 1.2C towards higher values.
13. (Frame, 2005) estimates model constrained climate sensitivity from observations and reports a 90%CI for ECS = [1.4–4.1] with the median skewed left at ECS = 2.4. The results are in good agreement with Charney and IPCC values.
14. (Murphy, 2004) used large ensemble climate model runs to estimate the climate sensitivity and its uncertainty. They find that the sensitivity is represented by a 90%CI range of ECS = [2.4–5.4] slightly higher than Charney/IPCC but with comparable uncertainty.
15. (Stainforth, 2005) is a unique paper. It presents results from the large ensemble model study carried out in the climateprediction.net event organized by Oxford University (climateprediction.net, 2004). Thousands of climateprediction.net members participated in a “grand ensemble of simulations” with a general circulation model. They discovered a surprisingly large uncertainty in sensitivity estimates. The authors report a 90%CI of ECS = [1.9 – 11.5]. The results indicate that both climate sensitivity and its uncertainty are much higher than the estimates presented by Charney and the IPCC.
16. (Hegerl, 2006) uses 700 years of paleo climate data constrained with models to report observed sensitivity as high as ECS = [7.7-9.0] but a 90% CI of ECS=[1.5-6.2]. The uncertainty is greater than the Charney/IPCC estimates and the range is skewed right toward higher values.
17. More recently, (Kummer, 2017) presented empirical estimates of observational climate sensitivity constrained by climate models in a doctoral dissertation and reported ECS = 2.3 with uncertainty given by the 90%CI of ECS = [ 1.6-4.1]. These results support the Charney/IPCC values although they are somewhat lower.
18. (Johansson, 2015) addresses the uncertainty issue in climate sensitivity and finds that uncertainty in ECS can be reduced with appropriate corrections for the ‘pause’ in warming 2000-2014. The pause was most likely due to ENSO effects or low solar activity. Its effect on the lower bound of the 90%CI for ECS has been incorrectly estimated by the IPCC. He finds that the lower bound of the 90%CI should remain at 2°C and that lowers uncertainty to ECS = [2.0 – 4.5].
19. (Aldrin, 2012) estimates model constrained ECS from observational data and reports an asymmetrical probability distribution for ECS that is skewed right similar to the Gregory 2002 distribution shown in Figure 1. The authors (co-authors include Gunnar Myhre) report 90%CI for observational sensitivity unconstrained by models as ECS ≈ [1.2 – 7.7] and model constrained intervals of ECS ≈ [1.0 – 2.7], [1.0 – 3.5], [1.0 – 4.2], [1.3 – 4.9], [1.5 – 7.8], [1.5 – 7.3], and [1.0 – 7.0].
20.  (Lewis and Curry 2018)  carried out a detailed analysis of observational climate sensitivity constrained by models. They report low sensitivity with 90%CI given by ECS=[1.05-4.05]. The uncertainty closely matches that of the IPCC with the range shifted towards lower values. The so called “uncertainty monster” in climate sensitivity research (Curry/Webster, 2011) is apparent in these findings in terms of the possibility of ECS<1.5 and ECS>4.5.
21. The confusion and uncertainty in the value for the ECS is further explored in “UNCERTAINTY IN EMPIRICAL CLIMATE SENSITIVITY ESTIMATES” downloadable from SERVER#1  or SERVER#2 . In this study, unconstrained observational ECS values are presented for the HADCRU temperature reconstruction 1850-2017. The results show a large range of ECS in a moving 60-year window. The observed ECS values range from small negative values to ECS>6. In addition, a split half test shows an unacceptable difference in ECS values between first and second halves of the study period. These results are not consistent with the existence of a climate sensitivity parameter that determines surface temperature according to atmospheric CO2 concentration. They are indicative of an unstable regression coefficient and therefore of the absence of the assumed correlation.
22. Statistically, the ECS is a linear regression coefficient that describes the responsiveness of surface temperature to the logarithm of atmospheric CO2 and its reliability depends on a correlation between these variables. The reason for the instability of this regression coefficient is that the assumed correlation is not found in the data. The correlation seen in the source data is shown to be spurious when the two time series are detrended. The details of the instability issue are described in a related post SPURIOUS CORRELATIONS IN CLIMATE SCIENCE and a downloadable paper posted on THE VALIDITY AND RELIABILITY OF CHARNEY CLIMATE SENSITIVITY downloadable from SERVER#1 or SERVER#2 . The study demonstrates that although regression coefficients may be computed without the assumed correlation, these coefficients will tend to be unstable with a wide range of possible values that have no interpretation because they are based on a sensitivity that does not exist.
23. It is now recognized that uncertainty in climate sensitivity is a serious issue in climate science and that it must be resolved. In (Stevens, 2016) Bjorn Stevens (co-authors include Steven Sherwood and Mark Webb) attempts to rescue the ECS concept and proposes ways to address the uncertainty issue in climate sensitivity research. The authors show that the IPCC sensitivity range of 90%CI(ECS) = [1.5 – 4.5] can be maintained and additional uncertainty controlled by addressing causes for extreme values. For example, aerosol forcing can yield values lower than 1.5 ECS and higher solar irradiance forcing can show higher values than 4.5 ECS. They argue that when such known effects are factored out, the correct CO2 effect can be extracted from the extreme values reported by purely observational estimates of ECS and ECS can be shown to lie within the Charney/IPCC range of ECS = [1.5 – 4.5].
24. A more radical proposal is found in (Knutti, 2017). Here the authors propose to abandon the ECS concept saying that the effort by climate science to address the uncertainty issue in climate sensitivity has failed and that this failure implies that the ECS must be abandoned because it has not proven to be a useful metric for presenting the essential relationship between atmospheric carbon dioxide and temperature that is central to the theory of anthropogenic global warming or AGW.
25. Knutti et al (co-authors Rugenstein and Hegerl) suggest that the failed ECS parameter should be replaced with the more stable and reliable carbon climate response to cumulative emissions (CCR) described in (Matthews, 2009). The authors state that “estimates of the CCR are better constrained than those of the ECS in observed warming and are more relevant for predicting warming over the next decades”.
26. Three papers, all of them published in 2009, broke new ground for climate science along the lines described by Knutti et al 2017. Matthews et al 2009 (co-authors Gillett, Stott, Zickfield) demonstrated a stable linear relationship between cumulative emissions and surface temperature and argued that this proportionality, computed as the slope of the line and referred to as the Carbon Climate Response (CCR) is the appropriate metric to relate warming to carbon dioxide emissions; with the further claim that it is more stable and reliable than equilibrium climate sensitivity (ECS).
27. However, the CCR, later called the TCR, also contains a statistical flaw in terms of a spurious correlation. The spuriousness of the correlation between temperature and cumulative emissions is described in a related post SPURIOUS CORRELATIONS IN CLIMATE SCIENCE . Additional information on this issue may be found in the corresponding downloadable papers LIMITATIONS OF THE TCRE  SERVER#1  SERVER#2 , SPURIOUSNESS OF CORRELATIONS OF CUMULATIVE VALUES  SERVER#1  SERVER#2 ,  and FROM EQUILIBRIUM CLIMATE SENSITIVITY TO TRANSIENT CLIMATE RESPONSE   SERVER#1 SERVER#2 .
28. A further statistical issue with the TCR is described in a related post [LINK]   as a “MATHMATICAL INCONSISTENCY”. Briefly, the ECS theory of anthropogenic global warming implies a logarithmic relationship between cumulative emissions and temperature while the TCR implies a linear relationship between cumulative emissions and temperature.
29. Establishing a functional correlation between time series data is fraught with complexity and traps for the unwary. Time series often contain long term trends in the data that, if allowed to remain, can generate spurious correlations at any given time scale that is smaller than the time span of the data SPURIOUS CORRELATIONS IN TIME SERIES DATA. In such cases, detrended correlation analysis is used to separate long term trend effects from fluctuations at the time scale of interest (Von-Storch, 1999) (Shumway, 2011) (Prodobnik, 2008)  ALEX TOLLEY’S LECTURE 
30. An additional issue in time series analysis is the use of moving windows or of cumulative values to derive a new time series from the source time series for further analysis. All such procedures require repeated use of the same data items from the source series to construct the target series. The repeated use of the same data items in the source series in the construction of the target series, reduces the effective sample size and the degrees of freedom in the target series. Reference: ILLUSORY STATISTICAL POWER IN TIME SERIES ANALYSIS (SERVER#1. SERVER#2
31. Cumulative value analysis can be described as the opposite of detrended fluctuation analysis because it does exactly the opposite. It overwhelms fluctuations at any short time scale with information about the overall drift in time so that fluctuations appear to vanish and an apparent high degree of correlation magically appears in the full span of the series. Yet it is the fluctuations at a given time scale that contain the correlation information relevant to a theory of causation at that time scale.
32. The repeated use of the same data values in the calculation of cumulative values erases both time scale and degrees of freedom from the time series by reducing the effective value of the sample size and the corresponding degrees of freedom. Source document:  SERVER#1 .  SERVER#2  . In the computation of cumulative values of a time series of length N, the first value in the series is used N times, the second value N-1 times and so on until the Nth value which is used once. In general, the Jth value is used N-J+1 times. Thus, in total, (N*(N+1)/2) numbers are used in the computation of cumulative values. Since there are only N distinct values in the source series, the average multiplicity of use is M = (N*(N+1)/2)/N or M = (N+1)/2.
33. Multiplicity reduces the effective sample size to EFFN = N/M. In the case of cumulative values, the effective samples size is EFFN = (2*N)/(N+1). For an infinitely large sample, the effective sample size of cumulative values will be EFFN ≈ 2 and for any finite sample size less than infinity, EFFN < 2. In the evaluation of the statistical significance of observed sample correlations and regression coefficients, the degrees of freedom parameter, DF = EFFN-2, has a maximum value of DF = 0 for an infinite series of cumulative values. This means that time series of cumulative values have no degrees of freedom and therefore no statistical power. This principle has been demonstrated with Monte Carlo simulation and also with examples drawn from climate data in the cited papers above SERVER#1 .  SERVER#2
34. These considerations imply that the apparent advantage the CCR and TCR over the Equilibrium Climate Sensitivity (ECS) in terms of stability, reliability, lower values, and tight confidence intervals is illusory. The CCR/TCR values have no interpretation because there is neither time scale nor degrees of freedom in the time series of cumulative values.
35. A complete list of my papers may be found at SSRN.COM . and also at ACADEMIA.EDU

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