Volume 7, Number 158
24 Sept 2007

In this issue

The Causes of Global Warming:

A Graphical Approach

by Loren Cobb

Note: This is an updated edition of this essay, dated 28 March 2008. To see the original essay, click here.

This is a brief note on how to evaluate the causes of global warming, without the assistance of super-computers running advanced three-dimensional geophysical models of atmospheric and oceanic dynamics. The latter style of modeling has provoked quite a substantial controversy around the world, as it is based on methods that are nearly impossible for ordinary citizens to use and evaluate.
Figure 1: Global mean temperatures, from the University of East Anglia.

Instead of joining this rhetorical debate, my goal in this essay is to show a simple way to evaluate the major hypothetical causes of global warming, such as solar sunspot activity and greenhouse gases produced by the use of fossil fuels (coal, petroleum, and natural gas).

Figure 1 shows the trend in global mean temperatures, from the mid-18th century to the present. The data come from the well-respected Climate Prediction Unit of the University of East Anglia. The curve clearly shows fluctuating global temperatures, culminating in a sustained rise in global mean temperature from about 1980 to 2006, with a dip in 2007.

The specific data series that we will use in the graphs and analyses to follow is known as the HadCrut3v series, up to date as of March, 2008. This series is based on historical land and sea measurements, which correlate very well with satellite measurements of temperatures in the lower atmosphere (but this area has been controversial — click here for a discussion).

Solar Activity as a Cause of Global Temperature Changes

Figure 2: Global warming (blue) and solar sunspot activity (red, smoothed, right-hand scale).

One contributing cause of changes in global mean temperature is fluctuations in solar activity, which changes not only the total amount of radiant energy received on earth, but also the amount of cosmic radiation absorbed by the atmosphere. The numbers of sunspots visible on the surface of the sun is a useful proxy for solar activity of all kinds. Sunspots follow a somewhat irregular eleven-year cycle that has been in place with varying amplitude for three centuries. Figure 2 shows annual midyear sunspot numbers, as reported by the Royal Observatory of Belgium, after smoothing with an eleven-year trailing average. The eleven-year solar cycle is only faintly visible in this smoothed data, but long-term fluctuations in overall solar activity are clearly visible.

From 1850 up to about 1990 the two curves move up and down in rough synchrony, indicating that solar activity is a powerful cause of fluctuations in global mean temperature. Then, after 1990, the two curves begin to diverge somewhat. The correlation between smoothed solar activity and global mean temperature over the time span from 1850 to 2007 is about 0.61, indicating that solar activity alone may account for as much as 37% of the variance in global mean temperatures.

The "anthropogenic hypothesis" of global warming is that the burning of fossil fuels by humanity has released enough greenhouse gases of various types (especially carbon dioxide and methane) into the atmosphere as to have caused a rise in global mean temperatures. The anthropogenic hypothesis implies that the ever-increasing use of fossil fuels will destabilize the temperature balance of the planet, with dramatic and extremely dangerous potential consequences for all life on earth. Is evidence for the anthropogenic hypothesis visible in data of this type shown above? To answer this, we need to add a new variable based on the burning of fossil fuels.

Fossil Fuel Consumption as a Cause of Global Warming

Figure 3: Global warming (blue) and fossil fuel consumption (red, right-hand scale).

Figure 3 displays the consumption of fossil fuels (coal, petroleum, natural gas), together with the curve of global mean temperatures. Fossil fuel consumption is measured in gigabarrels of oil equivalent per year, shown on the vertical scale on the right-hand side of the graph. The "elbow" in the curve that occurred in 1950 is due to the sudden post-World War II acceleration in the use of fossil fuels. If the anthropogenic hypothesis is correct, then the rise in global temperatures from about 1980 to present was in major part caused by greenhouse gases released by the combustion of fossil fuels. If real, this rise seems to occur after some time delay, somewhere between ten and thirty years.

Indeed, the relationship between fossil fuel consumption and global mean temperature is strongest* when the fossil fuel series is lagged behind temperature by 25 years. The two curves are shown in Figure 4. The red curve (fossil fuel consumption) has been shifted to the right by 25 years, so that one can see how the rise in temperatures could plausibly have been caused by the increase in fossil fuel use 25 years earlier. Observe that the elbow in the red line now coincides with the beginning of the recent upward trend in global mean temperature (about 1980).

Figure 4: Global warming (blue) and fossil fuel consumption delayed by 25 years (red, right-hand scale).

A comparison of Figures 2 and 4 raises some immediate questions: Which is the cause of global warming, solar activity or fossil fuels? If they both are, which has the stronger influence? How well can statistical predictions from these two hypothetical causes explain the observed changes in global warming?

We can provide at least preliminary answers to these questions with nothing more complicated than multiple regression. The statistical results of this procedure are quite clear, though of course subject to interpretation.

Predicting Global Temperatures from both Solar Activity and Fossil Fuels

When global mean temperature is statistically fitted to a simple linear combination of solar activity and lagged fossil fuel consumption, then the predicted temperatures fit the observed data much better than for either solar activity or fossil fuels alone. If we truncate the data at 1985, i.e. before global warming began in earnest, then fossil fuel consumption has an influence on global temperature that is about twice as strong as solar activity. If, however, we include all the data from 1850 through 2007, then the influence of fossil fuel consumption is almost three times stronger than solar activity, though the latter remains a powerful influence.

Figure 5: Actual global mean temperatures (blue) and as recreated by an autoregressive statistical model (orange), using solar activity, fossil fuel consumption, and previous year's global mean temperature.

Figure 5 shows how well the two causes acting together predict annual global mean temperatures. The red curve of predicted temperatures based on:

  1. solar activity,
  2. fossil fuel consumption, and
  3. the previous year's temperature,

explains fully 87% of the variance in global mean temperatures over the time span 1850–2007. As statistical models go, that is an extremely good fit! The details of the model and its input data are given here.

My conclusion from this little exercise in statistical curve fitting is that both solar activity and fossil fuels are powerful causes of fluctuations in global mean temperatures. There is indeed an anthropogenic component, and it is very strong.

Note: The predicted line (orange) and the observed line (blue) in Figure 5 display a very high degree of conformance in part due to the presence in the prediction equation of the previous year's temperature. Statisticians call this model auto-regressive, because it includes this term. While there are excellent physical and statistical reasons for making the model autoregressive, it does have one unfortunate side-effect: it makes the graph of predicted and observed values look deceptively good. If we were to use a model that omits the auto-regressive term — less physically realistic but also less deceptive to the eye — then the comparable graph of predicted and observed temperatures would look like this (click here). This simpler model explains 78% of the total variance in global mean temperatures from 1850-2007.

What Happens Next?

For the graph in Figure 6, I have extended the prediction into the future by using the known fossil fuel consumption figures for the last 25 years, i.e. out to 2032. Beyond 2032 the prediction uses future fossil fuel consumption rates predicted by me in TQE 155, based on Hubbert analyses of coal, natural gas, and petroleum (including extra-heavy oil from Canada and Venezuela). In TQE 155, I presented my best guess on the future of human energy consumption. If those guesses are correct, then fossil fuel consumption will rise to about 77 billion barrels of oil equivalent per year in about 2025. That would imply a further rise in global warming from the present until about 2055, when temperatures will be about 1.2°C (that is, about 2.1°F) higher than they were in 1950, followed by a decline thereafter.

Figure 6: Predicted global temperatures (orange) out to 2100.

Of course, all of the above analysis is purely statistical, using no physical principles whatsoever. That is a weakness if one believes in the geophysical models developed by climate scientists, but it may be a strength if one prefers instead to see the raw phenomenon displayed in simple graphs and analyzed with elementary and easily reproducible statistical methods.

Feedback Effects

If the actual rise in temperature causes more greenhouse gases to be released, for example from methane hydrates on the ocean floor or from arctic tundra, or the loss of the reflectivity of melting arctic ice, then the actual rise in temperature will be much greater. Self-reinforcing effects of this nature are not included in the simple statistical model used here.

"If nothing else were changed by warming, a doubling of carbon dioxide would ultimately lead to a temperature change of about 1.2 degrees C," says atmospheric physicist Gerard Roe of the University of Washington (quoted in Scientific American, October 2007). "In fact, because of internal processes within the climate system, such as changing snow cover, clouds and water vapor in the atmosphere, our best estimate is that the actual warming would be two to four times larger than that." In other words, the earth may warm between 2.4 and 4.8 degrees C.

And therein lies the problem. Any increase of more than 1.5 degrees C may result in the mass extinction of up to a third of all biological species known to science. At this level of extinction, what will happen to the food chain that supports human existence? Worse, this is just one of the dire consequences envisioned in "State of the Science: Beyond the Worst Case Climate Change Scenario" (Scientific American, November 2007).

* Note 1: When global mean temperature is statistically predicted from solar activity and lagged fossil fuel consumption, then the fit of the model is best when fossil fuel consumption is lagged by 25 years. To see a graph of model fit versus lag, click here.

** Note 2: The temperature data series display some serial dependence, which implies that the standard measures of statistical significance given by a multiple regression package will not be accurate unless some form of "autoregressive" model (or equivalent) is used. To see details of the actual model and its statistics, including data, click here.

This essay first appeared 24 September 2007.
evised on 4 December 2007 to include a final section on feedback effects.
Revised on 14 December 2007 to include a clarification of the autoregressive model.
Revised on 28 March 2008 to include data through January 2008.

Eternal vigilance is the price of liberty.
— Wendell Philips, 1852.
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Figure 5 in this issue is astonishing! I hope that it can be publicized outside of TQE since beyond "An Inconvenient Truth" and others, it is the most dramatic portrayal I have seen, demonstrating the influence of fossil fuels. Fine work indeed.

— Roger Conant, Pelham MA, Mount Toby Friends meeting. [3 Dec 07]

I appreciate your enthusiasm, but Figure 5 is really not quite as impressive as it looks.

The key to understanding how the predicted line echoes the observed line so closely lies in the caption, which states that the model is an AR(1). Having used such models my entire professional career, I stupidly forgot that they are not everyone's bread and butter. I will try to explain here, and then I will go back to the original and insert a few explanatory sentences.

When trying to construct an appropriate model for a dynamic process, it is always a good idea to build into the model the idea that external influences (e.g. solar activity and CO2 emissions) cause the state variable (temperature of the earth) to change somewhat from its previous state. Typically, the state variable has a certain inertia — it only gradually moves away from the previous state, and towards a new equilibrium. Therefore, it behooves the modeler to include an inertial term in the prediction equation. We call such models "autoregressive". The terminology AR(1) is intended to communicate the idea that the model is autoregressive with exactly one such inertial term.

There are many physical and statistical reasons why this is a good thing to do, but it does have one disadvantage: the graph showing predictions and observations will necessarily appear to show a very good fit to the data; the more inertia the better the apparent fit. In effect, the goodness of the prediction based on external variables is being confounded with (i.e. lumped together with) the strength of the inertia. There are ways of disentangling these two distinct influences for those who are not proficient enough do it by eyeball, but I don't normally make such distinctions in my technical papers. In this case I should have, clearly.

To summarize, the model is good but not brilliant. Some of the tight parallelism between the red and blue lines is due to the inertial term, not the all-important external variables. — Loren


Publisher: Russ Nelson, St. Lawrence Valley (NY) Friends Meeting.

Editor: Loren Cobb, Boulder (CO) Friends Meeting.

Editorial Board

  • Chuck Fager, Director, Quaker House, Fayetteville, NC.
  • Virginia Flagg, San Diego (CA) Friends Meeting.
  • Valerie Ireland, Boulder (CO) Friends Meeting.
  • Jack Powelson, Boulder (CO) Meeting of Friends.
  • Norval Reece, Newtown (PA) Friends Meeting.
  • William G. Rhoads, Germantown (PA) Monthly Meeting.
  • J.D. von Pischke, a Friend from Reston, VA.
  • John Spears, Princeton (NJ) Friends Meeting.
  • Geoffrey Williams, Attender at New York Fifteenth Street Meeting.

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Copyright © 2007 by Loren Cobb. All rights reserved.
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