William M. Briggs, Statistician

Parts of this analysis were suggested by Allan MacRae, who kindly offered comments on the exposition of this article which greatly improved its readability. The article is incomplete, but I wanted to present the style of analysis, which I feel is important, as the method I use eliminates many common errors found in CO₂/Temperature studies. Any errors are, of course, entirely my own.

It is an understatement to say that there has been a lot of attention to the relationship of temperature and CO₂. Two broad hypotheses are advanced: (Hypothesis 1) As more CO₂ is added to the air, through radiative effects, the temperature later rises; and (Hypothesis 2) As temperature increases, through ocean-chemical and biological effects, CO₂ is later added to the atmosphere. The two hypotheses have, of course, different consequences which are so well known that I do not repeat them here. Before we begin, however, it is important to emphasize that both or even neither of these hypotheses might be true. More on this below.

The source of monthly temperature data is from The University of Alabama in Huntsville, which starts in January 1980. Temperature is available at different regions: global, Northern Hemisphere, etc. The monthly global CO₂ is from NOAA ERSL.

We want to examine the CO₂/temperature processes at the finest level allowed by the data, which here is monthly at the time scale, and Northern and Southern Hemisphere and the tropics at the spatial scale. The reason for doing this, and not looking at just yearly global average temperature and CO₂, is that any processes that occur at times scales less than a year, or occur only or differently in specific geographic regions, would be lost to us. In particular, it is true that the CO₂/temperature process within a year is different in the Northern and Southern hemispheres, because, of course, of the difference in timing of the seasons and changes in land mass. It is also not a priori clear that the CO₂/temperature process is the same, even at the yearly scale, across all regions. It will turn out, however, that the difference between the regional and global processes are minimal.

The question we hope to answer is, given the limitations of these data sets, with this small number of years, and ignoring the measurement error of all involved (which might be substantial), does (Hypothesis 1) increasing CO₂ now predict positive temperature change later, or does (Hypothesis 2) increasing temperatures now predict positive CO₂ change later? Again, this ignores the very real possibility that both of these hypotheses are true (e.g., there is a positive feedback).

During the course of an ordinary year, both Hypotheses 1 and 2 are true at different times, and sometimes neither is true: in the Northern Hemisphere, the temperature and CO₂ both increase until about May, after which CO₂ falls, though temperature continues to rise. In the Southern Hemisphere, temperature falls in the early months, while CO₂ rises, and so on. These well known differences are due to combinations of respiration and changes in orbital forcing.

There are, then, obvious correlations of CO₂ and temperature at different monthly lags and in different geographic regions (I use the word “correlation” in its plain English meaning and not in any statistical sense). We are not specifically interested in these correlations, which are well know and expected, and whose role in long-term climate change is minimal. The existence of these correlations present us with a dilemma, however. It might be that, for either Hypothesis 1 or 2, the time at which either CO₂ or temperature changes in response to changes in forcing is less than one year, but disentangling this climate forcing with the expected changes due to seasonality, is, while possible, difficult and would require dynamical modeling of some sort (in the language of time series, the seasonal and long-term signals are possibly confounded at time scales less than 1 year).

Therefore, instead of looking at intra-year correlations, we will instead look at inter-year correlations. This introduces a significant limitation: any real, non-seasonal, correlations less than 1 year (or at other non-integer yearly time points) will be lost and it will be possible that we are misled in our conclusions (in the language of time series, the “power” on these non-integer-year lags will be aliased onto the 1 year lag). What is gained by this approach, however, is that there is no chance of misinterpreting lags less than one year as being due to a process other than seasonality. However, the main purpose of this article is not to identify the exact dynamical and physical CO₂/temperature relationship, nor to identify the lag that best describes it; we just want to know is Hypothesis 1 or Hypothesis 2 more likely on time scales greater than 1 year?

Most of us have seen pictures like this one, which shows the monthly CO₂ for 1980-1984; also shown in the Northern Hemisphere (NH) temperature anomaly (suitably normalized to fit on the same picture).

You can immediately see the intra-year CO₂ “sawtooth”. This sawtooth makes it difficult to find a functional relationship of CO₂ and temperature. I do not want to model this sawtooth, because I worry that whatever model I pick will be inadequate, and I do not immediately know how to carry the uncertainty I have in the model through to the final conclusion about our Hypotheses. I also do not want to smooth the sawtooth, or perform any other mathematical operation on the observed CO₂ values within a year, because that tends to inflate measures of association.

Instead, let’s look at CO₂ in a different way:

This is yearly CO₂ measured within each month: each of the 12 months has its own curve through time. It doesn’t really matter which is which, though the two lowest curves are from the winter months (for those in the NH). What’s going on is still obvious: CO₂ is increasing year by year and the rate at which it is doing so is roughly constant regardless of which month we examine.

Looking at the data this way show that the sawtooth has effectively been eliminated, as long as we examine year-to-year changes within each month through time.

Suppose we were only interested in Decembers and in no other months. Let us plot the actual December temperature from 1980 to 2006 on the x-axis and on the y-axis plot the increase in CO₂ for the years 1981 to 2007. Shown in the thumbnail below is this plot: with black dots for the Southern Hemisphere (SH), red dots for the NH, and green dots for the tropics (redoing the analyses with global or sea surface temperatures instead of separating hemispheres produces nearly indistinguishable results). For example, in one year, the NH temperature anomaly was -0.6: this was followed in the next year by an increase of about 1.5 ppm of CO₂ (this is the left-most plot on the figure).


The solid lines estimate the relationship between temperature and the change in CO₂ (the dCO2/dt on the graph). These are loess lines and estimate the relationship between the two variables. If the loess lines were perfectly straight (and pointed in any direction), we would say the two measures are linearly correlated. The lines aren’t that straight, so the data does not appear to be that well correlated, linearly or otherwise.

Click on the figure (do this!) to see the same plot for each of the 12 months (right click on it and open it in a new window so you can follow the discussion). Notice anything? Generally, when temperature increases this year CO₂ tends to increase in the following year. Hypothesis 2 is more likely to be true given this picture.

The loess lines are not always straight, which means that a straight-line model, i.e. ordinary correlation, is not always the best model. For example, in Januaries, until the temperatures anomalies get to 0 or above, temperature and change in CO₂ have almost no relationship; after this point, the relationship becomes positive, i.e., increasing temperatures leads to increases in the change of CO₂. The strength of the relationship also depends on the month: the first six months of the year show a strong signal, but the later six show a weakening in the relationship, regardless of where in the world we are.

Coincidence? Now plot the actual December CO₂ from 1980 to 2006 on the x-axis and on the y-axis plot the change (increase or decrease) in temperature for the years 1981 to 2007. For example, in one year, the NH CO₂ was 340 ppm: this was followed in the next year by a temperature decrease of about -0.5 degrees (this is the bottom left-most plot on the figure). No real signal here:


Again, click on the figure (do this!) to see all twelve months. There does not appear to be any relationship in any month between CO₂ and change in temperature, which weakens our belief in Hypothesis 1.

It may be that it takes two years for a change in CO₂ or temperature to force a change in the other. Click here for the two-year lag between temperature and change in CO₂; and here for the two-year lag between CO₂ and change in temperature. No signals are apparent in either scenario.

As mentioned above, what we did not check are all the other possibilities: CO₂ might lead or lag temperature by 9.27, or 18.4 months, for example; or, what is more likely, the two variables might describe a non-linear dynamic relationship with each other. All I am confident of saying is, conditional on this data and its limitations etc., that Hypothesis 2 is more probable than Hypothesis 1, but I won’t say how much more probable.

It is also true that, over this period of time and using this data, CO₂ always increased. The cause of this increase sometimes was related to temperature increases (rising temperatures led to more CO₂ being released) and sometimes not. We cannot say, using only this data, why else CO₂ increased, although we know from other sources that CO₂ obviously increased because of human-cased activities.