By Andy May
In my previous post on multiple regression of known solar cycles versus HadCRUT5, I simply threw the solar cycles, ENSO, and sunspots into the regression blender and compared the result to various models that included CO2. Before reading this post, it is a good idea to read the previous one, since much of this post relies on the information in it. It was a very simple statistical analysis designed to show that the IPCC conclusion that rising CO2 and other greenhouse gases are “responsible” for “1.1°C of warming since 18501900” is probably erroneous. The difference between the HadCRUT5 18501900 average and the 20182023 (through all of 2022) is 1.18°C, so they are saying that essentially all the warming since the 19th century is due to humans. The analyses described in this post show they cannot be certain of their conclusion because they have ignored persuasive evidence that changes in the Sun caused at least some of the warming.
We have shown that various statistical combinations of known solar cycles correlated with HadCRUT5 as well as, or sometimes better than, changes in CO2 concentration. The way that the Sun might affect our climate is unknown. The IPCC only considers the direct effect of changing total solar irradiance (or TSI) directly on the Earth, as if the Sun were an incandescent light bulb over a piece of paper, but this cannot be correct. The climate effect of solar changes during a single 11year solar cycle is nearly an order of magnitude larger than the change in solar radiation can account for.
Recently great strides have been made in modeling and understanding the solar dynamo. However, modeling many important elements of the generation of solar cycles remains beyond our grasp. We only know their effect on Earth’s climate is much larger than the change in power received from the Sun during the cycle. We can examine the correlation of known (but poorly understood) solar cycles and climate change, but we cannot explain the mechanisms involved.
How additional CO2 can warm Earth’s surface is understood, but the climate sensitivity to CO2 is not known. Recent published estimates of the sensitivity, range from near zero to over 5°C/2xCO2 (2xCO2 means doubling of the CO2 concentration). The IPCC claims that human generated CO2 and other human activities have caused all (or essentially all) recent warming. This is speculation. We do not know how much changing CO2 can affect climate, and we can’t explain the large observed effects due to solar changes, so how can we know all the observed warming is due to CO2 and human activities? The advantage of the CO2 hypothesis is that the mechanism is known, but since the magnitude of the effect cannot be calculated accurately, quantitatively it is just as unknown as the solar effect, which the IPCC is clearly underestimating.
In this post we will take a closer look at the correlation between solar activity and HadCRUT5, and address some of the many comments to my previous post. First overfitting.
Overfitting
Solar cycles are not understood but can be observed in cosmogenic isotope studies that have been used to document the very long Hallstatt (or Bray 2400year, ±200 years) and Eddy (1000year ±30 years) cycles. These two long cycles correlate with the most significant climate events in history, the Bray Cycle correlates with the Greek Dark Age (~ 1200 to 800BC) and the early part of the Little Ice Age (~ 1300 to 1600, we target 1470 as the Hallstatt low). The Eddy Cycle correlates with the Medieval Warm Period (~ 950 to 1250), the latter part of the Little Ice Age (~1500 to 1816, we target 1680 for the Eddy low), and the Modern Warm Period (~1940 to ~2005).
The shorter cycles are not as climatically significant but noticeable. Both the “Pause” in warming and the cool period around 1910 correlate well with the Feynman Cycle, and the cooler period from 1945 to 1976 in the early part of the Modern Solar Maximum correlates with the Pentadecadal cycle. All these cycles are plotted for the instrumental period in Figure 1 along with HadCRUT5.
Figure 1. The known solar cycles plotted for instrumental era along with the HadCRUT5 global surface temperature record.
As some pointed out in comments on my last post, with this many cycles, multiple regression will always find a reasonable fit to almost anything trending upward. Further all the time series, including HadCRUT5, are strongly autocorrelated. The cycles are anchored to the solar lows or highs as specified in Ilya Usoskin’s 2016 and 2017 papers or Joan Feynman’s 2014 paper. The 22.1year Hale Cycle is anchored to early 2020 during the solar cycle 24 minimum. It has been proposed that the de Vries Cycle is a beat period between the Hale Cycle and the 19.86year orbit of the Sun around the solar system barycenter, this configuration is consistent with this hypothesis.
As can be seen in figure 2, this regression relies mostly on the quasilinear Hallstatt and Eddy Cycles. Frank Stefani does not like this idea and believes that only the better documented Feynman and de Vries Cycles and Log(CO2) are needed to model the period 1850 to the present. This is possible, Log(CO2) is also a quasilinear series and is similar to the Eddy and Hallstatt series (see the first post), so all three can substitute for one another, this is an argument that will not be settled by observations soon.
Because the solar dynamo is not fully understood, we have no choice but to choose the best regression of these cycles on HadCRUT5 as our solar model. I understand that regressions are possible with other configurations of the cycles, but we have a solid foundation for this configuration. The regression is shown in figure 2.
Figure 2. A multiple regression model of HadCRUT5 using only the wellknown solar cycles. The coefficients (weights) for each of the cycles are listed and the regression statistics are given in the boxes. The decrease in global temperatures from 1944 to 1976 is not modeled very well, otherwise the model does a good job.
Because the input cycles and HadCRUT5 are autocorrelated the regression statistics (especially R2) shown are inflated to reality. Experimentation shows that most cycle configurations would result in R2 values above 0.8, although some were far lower than this. This R2 value of 0.83 is not great, but it is the best that can be obtained with these cycles, which is what we were after.
In this way, we created a single solar cycle predictor variable. The underlying reason for the cycles is very poorly understood. This is a statistical exercise, and it is the best match of these predictors to HadCRUT5, but that is all we can say.
Next we add other variables that proved significant in our residual and partial regression study. They are the Nino 3.4 index and the logarithm to the base 2 of CO2 or “Log(CO2)” time series. Oddly, adding the Nino 3.4 series, at least statistically, caused the sunspot series to become an insignificant (about 1%) addition to the regression. As a result, the sunspot series did not make the cut to be added to the regression, and the Nino 3.4 series was always significant at over 10%. This might be explained by the observed effect of the solar cycle on upper ocean temperatures described by Warren White and his colleagues at Scripps. Figure 3 shows the regression with Nino 3.4 added.
Figure 3. Adding ENSO (Nino 3.4) to the composite solar function. Inputs are normalized to make the coefficients comparable. The cooling period during the early 1960s is still not modeled very well.
Adding Nino 3.4 to the composite solar series increases the R2 to 0.85, but the coefficients suggest that the addition of Nino 3.4 is significant, but small, at 15%. Nino 3.4 is about a 15% addition with or without sunspots. The normalized coefficients tell us that, statistically, 85% of the regression is from the combined solar series and 15% is from Nino 3.4.
The input series in these plots (figures 3, 4, and 5) are all normalized so that the coefficients are comparable and can be used to compare the relative impact of the input series on the model. Figure 4 shows the result when Log(CO2) is added.
Figure 4. The regression when Log (CO2) is added. Inputs are normalized to make the coefficients comparable.
Figure 4 tells us that adding Log(CO2) does not change the R2 significantly, and it barely changes the regression model. The coefficients tell us that, statistically, the combined solar series added 79% to the model, ENSO is unchanged with a 15% addition, and Log(CO2) contributed only 6%. Finally, figure 5 shows the model created from just Log (CO2) and the combined solar series.
Figure 5. The combined solar series and the Log(CO2) series.
In figure 5, the R2 has dropped to 0.83, the solar time series supplies 87% of the result, and Log(CO2) only supplies 13%. Figure 6 compares the regression using the combined solar and Nino 3.4 to a regression using combined solar, Nino 3.4, and Log(CO2). As you can see, they are not exactly the same, but nearly so.
Figure 6. The models with the combined solar curve, ENSO, and CO2 compared to Solar and ENSO only. Although they appear to be exact overlays, they are slightly different. The suggestion is that CO2 did not add to the regression.
Figure 7 adds the combined solar plus Log(CO2) series to the plot. It now becomes apparent that once the solar cycles are combined into one predictor, it and ENSO produce the best regression model to predict HadCRUT5. How the solar cycles were created in the solar dynamo is unknown, but if our combined solar cycle series is correct, the major solar cycles are the dominant force behind recent warming.
Figure 7. Comparing all the models, solar plus ENSO plus CO2, solar plus ENSO, and solar plus CO2.
This analysis is not evidence that solar variability is the dominant cause of recent climate change. It merely shows that a statistically significant model of HadCRUT5 global average temperature series can be created from a combination of wellknown and welldocumented solar cycles. The physical reason for these observed solar cycles is unknown, although there are many plausible hypotheses that might explain them.
All the current possible mechanisms show the Sun acting as an AC field generator with a period of about 22 years. The longer modulations are poorly understood. Observations and proxies show that the Sun varies over both short and long periods, which causes solar output to change, and results in climate changes on Earth. What is the driving force for the solar changes? They appear to depend on the complex fluid motions in the Sun’s interior which, in turn, might be influenced by the varying gravitational action of the orbiting planets, but all this is unclear. The model we describe ignores all this complexity and only deals with the observed cycles. We created a very simple statistical model, but more elaborate and creative multiple regression solar models have been published recently, a quick summary of some of them follows.
Stefani, 2021
Frank Stefani uses double regression to model global sea surface temperatures (HadSST.4) with the aa index of solar variability and Log(CO2). The aa index is a robust proxy of solar output and correlates well with the sunspot number (see here for more information). Stefani does a much more extensive check of regression parameters than we do here. He also uses his model to predict temperature into the next century. His predictions show a reduced warming rate over the coming century. He uses his model to compute a climate sensitivity of 0.6 to 1.6°C/2xCO2, much lower than reported in the IPCC’s latest report (AR6). However, Stefani’s values are in line with other observationbased estimates of climate sensitivity. (link)
Scafetta, 2023
Scafetta constructs multiple regression models that include solar forcing, volcanic eruption effects, and Log(CO2). He emulates the IPCC’s model results using their assumptions, although he computes a smaller climate sensitivity of 1.4 to 2.8°C/2xCO2. Using more realistic assumptions, the climate sensitivity is reduced to 0.9 to 1.8°C/2xCO2, consistent with Stefani’s estimate above. Scafetta regressed on HadSST4, HadCRUT4, and HadSST3 as well as HadCRUT5, all producing similar climate sensitivities. His model accounts for a delayed response due to ocean buffering of absorbed solar radiation. To account for the possibility of urban bias, some of Scafetta’s regression studies were done only on sea surface temperature datasets. His study shows that only 20% of the solar influence on global temperatures is due to increased radiation. Other factors such as modulation of cosmic rays, solar driven atmospheric/oceanic circulation changes, or other processes are probably more important. These latter processes, and other solar driven amplifiers, are not programmed into the IPCC climate models, which is possibly why they underestimate the climatic impact of the Sun. (link)
Soon, et al, 2023
Soon et al. did a regression study of solar, volcanic, and human forcings on two Northern Hemisphere datasets, one with rural temperatures and one with a blend of rural and urban datasets. This paper is an extension of Soon and colleague’s earlier solar/CO2 regression study. They used two solar forcing datasets, the TSI dataset recommended by the IPCC, and another that was ignored in AR6. They found that the choice of temperature and solar forcing datasets made a large difference in the study outcome. The temperature and TSI datasets are all possible, none are established as better or worse than the other, yet how much warming is attributable to human activities or nature depends on the datasets used. This casts doubt on the IPCC conclusion that humans have caused all, or nearly all, recent warming. (link)
It is important to realize that nearly everyone recognizes that urban areas are warmer than the surrounding countryside and urban areas have been growing rapidly globally over the past century, surrounding previously rural weather stations. This casts doubt on warming trends generated with combined rural/urban datasets. Further, there is no definitive record of solar radiation output (TSI), there are both low and high trend TSI datasets and no way to tell which is correct since proper records are too short and inaccurate. Thus, a proper study would use both, as Soon et al. do. Soon et al. found that 85% of the 18502018 warming, using their “ruralonly” dataset could be explained by solar and volcanic forcing.
Stefani et al. 2023
Regression isn’t used in this paper, but it is of interest here because the authors connect the solar (Schwabe) and Hale Cycles to the de Vries (or Suess) Cycle via a 193year beat period between the 22.14year Hale Solar Cycle and the 19.86year orbit of the Sun around the solar system barycenter. They note (as have many others) that the de Vries Cycle is probably responsible for the ~190210year spacing of Solar Grand Minima during HallstattBray Cycle lows. The most recent example being the WolfSpörerMaunder minima between about 1300 and 1715, with the related Bray low at about 1500 (these are very similar to the values used in my model above). They also note that in some fashion, the de Vries and BrayHallstatt Cycles are related, or at least the de Vries Cycle appears to be modulated by the HallstattBray Cycle. (link)
Conclusions
These various multiple regression studies don’t prove anything, they aren’t even proper evidence of anything. But they do show that the IPCC assumption that the Sun had no effect on observed warming since 1750 is questionable. It also shows that their chosen TSI dataset and their assumption that the only impact of a changing Sun is the amount of radiation Earth receives is questionable. Both White and Haigh have established that amplifiers exist in Earth’s climate system that increase the impact of solar changes by a factor of four, perhaps by a factor of ten, yet this is ignored by the IPCC. The IPCC needs to go back to school and redo AR6 including all the research they ignored the first time.
I acknowledge the generous help from Dr. Frank Stefani and Dr. Willie Soon, but any errors in the post are mine alone.
Download the bibliography here.
Download the supplementary data here, it includes R code, data, and Excel spreadsheets to make all the figures in this post.

The Schwabe Cycle. ↑

Various writers refer to equilibrium climate sensitivity (ECS), the transient climate response (TCR), effective climate sensitivity (ECS). There are a bewildering number of ways to measure the effect of CO2 on climate, see here and here for a discussion. To avoid this confusion, we will only refer to “climate sensitivity” in this post. ↑

(Lean, 2017) ↑

(White, Dettinger, & Cayan, 2003) ↑

(Usoskin I. , 2017) ↑

(Usoskin, Gallet, Lopes, Kovaltsov, & Hulot, 2016) and (Usoskin I. , 2017) ↑

(Feynman & Ruzmaikin, 2014) ↑

(Stefani, Stepanov, & Weier, 2021) and (Stefani, Horstmann, Klevs, Mamatsashvili, & Weier, 2023) ↑

(Stefani, Stepanov, & Weier, Shaken and Stirred: When Bond Meets Suess–de Vries and Gnevyshev–Ohl, 2021) ↑

(White, Dettinger, & Cayan, 2003) ↑

They are normalized by subtracting their respective means and dividing by their standard deviation. The model is not affected, but the coefficients become comparable when this is done. ↑

(Charbonneau, 2022) ↑

(Charbonneau, 2022) and (Stefani, Horstmann, Klevs, Mamatsashvili, & Weier, 2023) ↑

The aa index data used was from NOAA, the British Geological Survey, and from (Nevanlinna & Kataja, 1993) ↑

(IPCC, 2021) ↑

(Christy & McNider, 2017), (Wijngaarden & Happer, 2020), (Lewis & Curry, 2018), (Lewis N. , 2022), and other examples in (Stefani, Stepanov, & Weier, 2021). Also see Tables 1 & 2 here. ↑

(Soon W. , et al., 2023) ↑

(Soon, Connolly, & Connolly, 2015), see also the summary here. ↑

TSI is total solar irradiance. The IPCC assumes that the increase or decrease in solar output is the only warming or cooling effect the Sun has on Earth’s climate. This is hotly debated, as there are recognized amplifiers in the climate system (Haigh, 2011). ↑

(Hoyt & Schatten, 1993) ↑

When two waves with dissimilar frequency interact, they cause an alternating constructive and destructive interference that is called “beating.” More here. ↑

The solar system barycenter is the center of mass of the solar system, which moves with the planets. The Sun moves about this barycenter in a complex orbit. More here. ↑

(White, Dettinger, & Cayan, 2003) ↑

(Haigh, 2011) and (Lean, 2017) ↑
Like this:
Loading…
Comments are closed.