Chapter 3. Is the current correlation between atmospheric carbon dioxide concentration and global warming significant?

We have established a suspicion (50% to 80% probability) that the current rate of global warming may be exceptional. If it is exceptional, we seek to establish the cause. It is widely believed that increased atmospheric carbon dioxide concentrations may be the cause of global warming. We cannot look to geological records to illuminate the current period of global warming. In studying the Vostok data, we established that, up recent times, there has been equilibrium between atmospheric carbon dioxide concentrations and the concentrations in the oceans. However, this relationship has now been broken. Only about 1/3 of the increased atmospheric carbon dioxide concentration results from global warming. The remaining increase results from mankind’s activities in releasing carbon dioxide to the atmosphere. Although a significant proportion of the released carbon dioxide has dissolved in the oceans, it will be many hundreds of years before equilibrium is achieved. In this chapter, we look at recent global warming statistics to investigate whether they can be used to establish a cause/effect relationship between carbon dioxide and climate.

Climate studies cannot use formal statistical experimental design. For example, it is not possible to randomize the sequence in which conditions (such as atmospheric carbon dioxide concentrations) arise. Furthermore, successive observations are correlated. For example, both the temperature this year, and the carbon dioxide concentration this year depend, in part, on the temperature and concentration last year. In laboratory practice, such experiments would be randomized in some way. Failure to randomize the sequence of experiments is well known to introduce spurious correlations, or to conceal real effects. The problem with climate change (and release of carbon dioxide) is that we have a large, uncontrolled experiment. Living in a large uncontrolled experiment is potentially hazardous. (It would be unlikely to be approved by a medical ethics committee). From the statistical analysis viewpoint, we can deceive ourselves at the amount of data that we have. We have millions of day-by-day and hour-by-hour weather measurements. Ice cores give us thousands of climate measurements, extending over hundreds of thousands of years. However, it is uncertain how many independent measurements these data furnish. (We cannot simply count the number of data points to provide us with the statistical degrees of freedom). Climate measurements are correlated sequentially in time and spatially in distance. Thus, it is impossible to undertake formal statistical studies on the correlation between variables. We can undertake the correlations, but we are uncertain as to their statistical significance.

In this chapter, we propose a way of circumventing the statistical problem of studying the correlation between carbon dioxide concentration and global mean temperature. We have already shown (in considering the Vostok data), that geological correlations are not relevant in studying the current situation. The only conclusion that we can draw from the Vostok correlation is that about 1/3 of the measured increase in carbon dioxide concentration is caused by increased temperature. The remaining 2/3 results from mankind’s activities. Hence, we need a new approach in considering the unprecedented high carbon dioxide concentrations over recent decades. The methodology is as follows. We select a climate change model that clearly has no physical basis. We then compare the accuracy of its predictions with the accuracy of the predictions of a model relating atmospheric carbon dioxide concentration to mean global temperature. Since we are using the same data set for the two models, we can discriminate between the models using statistical tools. Unfortunately, we cannot quantify the relative probability that any particular model is correct. It is impossible to do so because we do not know the number of statistical degrees of freedom in the time series. We use data covering the 39 years from 1958 to 1996. In1958, Keeling and Whorf commenced methodical carbon dioxide measurements in Hawaii. The data set to hand when this study started extends to 1996. Serial correlation will certainly reduce the effective number of statistical degrees of freedom below 39, but we do not know by how much. However, independent of the number of degrees of freedom in the data, the following criteria hold:

1. If model A and model B have the same number of adjustable parameters, the model that predicts the outcomes with lowest variance (i.e. lowest mean error) is more likely to be correct.

2. If model A and model B show the same variance (i.e. same mean error), the model with the fewest adjustable parameters is more likely to be correct.

If we knew the number of data degrees of freedom, the relative probabilities could be calculated. However, these are not known. Hence, we can rank the models, but not quantitatively estimate their relative probabilities of being correct.

We compare five models:

1. The global mean temperature remains constant throughout the period examined.

2. The global mean temperature increases linearly with time and depends on nothing else.

3. The global mean temperature depends linearly on atmospheric carbon dioxide concentration.

4. The global mean temperature depends linearly on prices in the UK. (Thus, it depends on the cumulative price inflation in the UK).

5. The global mean temperature depends linearly on the annual number of divorces in the UK.

The models were chosen for the following reasons.

1. Model (1) was chosen because it has only one adjustable parameter; namely the single constant temperature. If none of the models give a significantly better fit to the data, global warming needs no further investigation.

2. Model (2) was chosen because it is the simplest 2-parameter model. One parameter is the temperature at a particular time. The other parameter is the temperature increment from year to year. It is the simplest model corresponding to a global warming effect that depends on nothing other some natural cycle or external event.

3. Model (3) was chosen because it directly tests the hypothesis that carbon dioxide causes global warming. For the small temperature changes observed, and the small changes in carbon dioxide concentration, any conceivable model can be approximated by a linear model. (Simply take the first terms in a Taylor expansion of any chosen model). Of course, a well-founded model will extrapolate more reliably than a linear model. This model also has two parameters. The data is published in “Trends ’93: A Compendium of Data on Global Change”, (authors Keeling C D and Whorf T P) p16, edited by Boden T A, Kaiser D P, Sepanski R J, and Stoss F W, ORNL/CDIAC-65, Oak Ridge Laboratory, Oak Ridge, Tennessee, USA (1994). It is reproduced in several publications, and is regularly updated, on line, by the authors. There is missing data for a few months in 1958 and 1964. We have made good the missing points using standard statistical techniques for such a time series.

4. Model (4) was chosen because it represents a null base model. There is no conceivable way that global warming can be controlled by UK inflation. Hence, if a 2-parameter model does not give a better fit (lower variance) than this model, it has no claim to be a valid explanation of global warming. The data is available on the Safalra website (www.safalra.com). We have compounded the quoted annual inflation rates (given as percentages), rather than employ their cumulative (2-figure precision) values directly. We used this data set because it is easily available; otherwise, it was a random choice.

5. Model (5) was chosen for the same reason as model (4), just in case model (4) turned out to be a lucky (or unlucky) choice. The data is available on the UK Government Statistics web site (www.statistics.gov.uk).

The models predictions are illustrated in Figure 11. The fits achieved are summarised in Table 5.-

Figure 11. Comparison of temperature correlations relating temperature to time, to carbon dioxide concentration, to UK inflation and to UK divorce rate.

Table 5. Correlation of observed recent global temperature rise.

We see that the additional parameter that allows mean global temperature to change over time does not make a dramatic improvement over the assumption that we are just seeing random temperature changes. However, if we extended the fit over the longer time frame of the previous chapter, it would be clear that the mean error of the constant temperature assumption would increase. Despite its requirement for fewer parameters, we are reasonably confident that the temperature is changing.

There is no dramatic difference between the 2-parameter models. However, superficially, it looks as if UK inflation is a better predictor of global warming than is atmospheric carbon dioxide concentration. The explanation that temperatures are just rising with time is also almost as plausible (statistically) as the explanation that the rise is driven by atmospheric carbon dioxide.

For those unswervingly committed to the Carbon Hypothesis, the explanation for the above findings is obvious. Carbon dioxide is causing the observed temperature rise. Its concentration is increasing reasonably uniformly over time. Hence, any other parameter that increases reasonably uniformly with time will coincidentally also give a good fit with the observed temperature rise. However, this argument is two-edged. If global temperatures are being driven to rise roughly uniformly by some other factor, any factor that increases steadily with time will also correlate with the rising temperature. Carbon dioxide concentration is one such factor. Hence, statistically any correlation between atmospheric carbon dioxide concentration and temperature may be purely coincidence.

We can conclude that, from statistics alone, there is no reason to believe that the correlation between global warming and increased carbon dioxide concentrations is other than coincidence. It follows that this coincidence cannot form part of the science supporting the Carbon Hypothesis.

Annexe to Chapter 3. How can we establish a statistically significant model that relates global warming to carbon dioxide releases?

The above analysis is, in itself, inconclusive. However, it gives clear guidance on the steps that must be taken in order to establish that carbon dioxide causes global warming. There must be a mechanistic model that clearly links atmospheric carbon dioxide concentration with global warming. It is insufficient to note that, in the laboratory, carbon dioxide absorbs infrared radiation. We must show that, in the atmosphere, it absorbs sufficient energy to account for the observed global warming. The analysis in this chapter also shows that any mechanistic model must have the following characteristics:

1. It must have no more than two adjustable parameters.

2. If it has two adjustable parameters, it must fit the observed temperature data with a mean error of significantly less than 0.12 degrees C. If it fits the observed temperature data with a mean error of 0.12 degrees C (or more), it must have not more than one adjustable parameter.

It is worth noting what is meant by an “adjustable parameter”. In the above analysis, we have adjusted one parameter such that, at a particular condition, temperature is correctly predicted. (In the actual analysis, we happen to have chosen to ensure that the mean temperature is correctly predicted at the mean value of the factor that might drive temperature. In the examples, the potential “driving factors” were time, CO₂ concentration, inflation, and divorces per annum). In cases (2) to (5), we have adjusted a second parameter to give the observed temperature increment per year, per ppm of carbon dioxide etc. Thus, in both cases, we have used actual climate (temperature) measurements to adjust our simple models. Any model that allows parameters to be adjusted based on climate measurements of any sort consumes adjustable parameters. Parameters independently measured in the laboratory (for example, the emissivity of soil) are, for this purpose, not adjustable parameters. We pay attention to these constraints in the next chapter. The Appendix to this report clearly shows that carbon dioxide concentration does not determine global temperature. For example, consider that we select an atmospheric carbon dioxide concentration of 250 ppm. The corresponding atmospheric temperatures vary over a range from 6.9C below present to almost 0.5C above present. This temperature range is a characteristic of a global climate that is always changing. Modellers have to fix one parameter to force their models to match the current temperature. Thus, one parameter is adjusted. Similarly, the Appendix shows that, at a given temperature and carbon dioxide concentration, the temperature can rise or fall. Thus, modellers have to fix a second parameter to force their model to predict the current rate of temperature rise. Thus, these modellers have already consumed their two adjustable parameters. Some workers produce models that, based on atmospheric carbon dioxide concentration alone, predict that the current temperature should be much higher. They then introduce just enough shielding from aerosol and smoke releases to reduce the temperature to the observed value. Incorporating this shielding introduces a further adjustable model parameter.

Most current climate models are global circulation models. These models incorporate the major wind patterns, and many incorporate the geography of the world, with local variations of climate. Many of these models have hundreds of parameters; indeed some models have thousands of parameters. The models are mostly too complex to ascertain exactly which parameters have been adjusted to ensure agreement with current weather or climate data. It is impossible for any referee of a published paper to check the model (referees are rarely furnished with the source code). All that a referee can do is to check that the assumptions seem to be reasonable, and that the results seem to be consistent with the assumptions and with other work. It would be prohibitively expensive to do more. On this basis, we believe that global circulation models are unlikely to be of value in determining whether carbon dioxide causes global warming. The models have value in determining the implications of global warming (the likely incidence of flooding, of drought and other manifestations of climate change). However, we need a different diagnostic tool to give a simple model with only one or two adjustable parameters. We attempt to introduce such a model in Chapter 4.

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