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?   Read rest...

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