The literature on global warming presumes that greenhouse effect of water vapor does not have to be considered explicitly because the level of water vapor will be a function of temperature and the level of temperature will be determined by the concentration of the other greenhouse gases, notably CO₂. If the level of water vapor in the atmosphere were only a function of temperature then this might be valid but the relationship is more in the nature of a stochastic relationship; i.e.,

W = f(T) + u
 

where W is the water vapor concentration, T is the temperature and u is the effect of the the unmeasured variables affecting W. The variable u may be considered a random variable. The nature of the relationship between W and T can be seen from the following graph.

The magnitude of the greenhouse effect from water vapor is so large compared to the other greenhouse gases that the fluctuations in u may be quantitatively more important than the fluctuations in the concentrations of the other greenhouse gases.

The complete model is of the nature

T = g(Z+W)
W = f(T) + u
 

where Z is the concentration of the other greenhouse gases besides water vapor. These should be weighted by their radiative efficiencies relative to CO₂. Likewise W should be measured in terms of equivalent amount of CO₂. Climate Change 2001, the report of the Intergovernmental Panel on Climate Change (IGCC), does not give the radiative efficiency of water vapor or of that CO₂ so this is not a feasible computation at this time, but in principle it could be done. A study by Raymond Pierrehumbert seems to indicate that the radiative efficiency of water vapor relative to that of CO₂ is 1.5.

The incremental changes about an average would be approximately

ΔT = g'(Z+W)[ΔZ + f'(T)ΔT + u]
and solving for ΔT
ΔT = [1/(1 − g'(Z+W)f'(T)]g'(Z+W))[ΔZ + u]
 

The factor 1/(1 − g'(Z+W)f'(T)] is approximately two, according to the IGCC 2001. This means that g'(Z+W)f'(T) is approximately one half.

The correlation coefficient between ΔT and ΔZ would be the same as that between ΔT and u. The relative importance of variation in ΔZ and variation in u on the variation in ΔT depends upon the relative variances of ΔZ and u.

Data on ΔZ is not available at this time but the changes in the level of CO₂ is a good proxy for it since CO₂ constitutes the dominant share of the non-watervapor greenhouse gases and the time patterns for methane and nitrous oxide, the other major greenhouse gases, is quite similar to that of CO₂.

A regression of the global temperature on the CO₂ level (Mauna Loa data) for the years 1959 to 2004 gives a coefficient of determination (R²) of 0.775, meaning that 77.5 percent of the variation in global temperature over that period is explained by the variation in the CO₂ level over that period. This means that the variance of u is about 29 percent of the variance of ΔZ.

The full display of the information from the regression is:

ΔT = 8.96 + 1.18ΔZ
                 (0.096)
R² = 0.7747
 

An R² of 0.775 corresponds to a correlation of 0.88 between ΔT and ΔZ. This seems to be a reasonably high level, but it also means that 22.5 percent of the variation is not attributable to the variation in CO₂. Furthermore, a significant amount of the correlation stems from the fact that both variables have a trend. If there is a causal relationship between the two variables then the common trend is important, but suppose there is no causal relationship. The existence of a trend in any two variables would result in a possibly spurious correlation between. A method for checking to see how much of the correlation stems from the trends is to determine the correlation between the deviations of the variables from their trends. This done by regressing both variables on time and computing the deviations from the trend lines. When this is carried out the coefficient of determination is is 0.2967, which corresponds to a correlation coefficient of 0.54. This correlation could be more impressive than the 0.88 correlation coefficient between the two undetrended variables. If there were no causal relationship between the two variables the expected value of the correlation between the two detrended variables would zero. The question is whether the value of 0.54 is statistically significantly different from zero and this is a matter of what is the standard deviation of correlation coefficients computed between two random variables. It appears that the standard deviation of sample correlation coefficients is equal to 1/√(N-3) where is N is the sample size. For the above regression the sample size is 46 so the standard deviation for the sample correlation coefficient is 0.1525. The expected value of the correlation between two independent and unrelated random variables is zero so the sample value of 0.54 is 3.5 standard deviation units away from zero and this is signficant at the 99 percent level of confidence.

Another way to confirm that the statistical relationship between global temperature and the level of CO₂ is not just a matter of the two variable having a trend is to regress global temperature on CO₂ levels and a linear trend. If the correlation between temperature and CO₂ were due solely to their having trends the t-ratio, the ratio of the coefficient for CO₂ levels to the standard deviation of that coefficient, would not be statistically different from zero.

ΔT = 11.124 + 3.626ΔZ −3.419TREND
              (0.851)      (1.184)
R² = 0.8113
 

The coefficient of determination for the regression of temperature on CO₂ levels and a linear trend is 0.811. The t-ratio for the CO₂ coefficient 4.23, a value highly significantly different from zero at the 99 percent level of confidence.

The time path of CO₂ levels is nonlinear so a further test of the relationship would be to include a quadratic trend as well as a linear trend in the regression analysis. This regression yields a coefficient of determination of 0.841. The t-ratio for the coefficient for CO₂ levels is 4.17, again a value highly significantly different from zero at the 99 percent level of confidence. Thus the statistical relationship between global temperature and CO₂ levels passed all the tests of significance with flying colors.

These results are very reasonable. There is a solid statistical relationship between the global temperature and the level of CO₂ that is not just upon common trends. However the vatiation in CO₂, along with all of the variables which are correlated with CO₂ levels including the water vapor feedback, account for only 77.5 percent of the variation in global temperatures. This means that other factors, including aerosol levels but also variations in water vapor that are not driven by temperature changes, account for 22.5 percent of the variation in global temperature.

Since the factor 1/(1 − g'(Z+W)f'(T)] is approximately two and the coefficient for ΔZ is 1.18 this means that g'(W+Z) equals 0.59. But g'(Z+W)f'(T) has to be 0.5 for the factor to be two. Thus f'(T) must be 0.5/0.59 or 0.85. What this means is that if the global temperature increases by 1°C then the concentration of water vapor measured in terms of equivalent CO₂ has to increase by 0.85 ppm. Since water vapor has a radiative efficiency 1.5 times that of CO₂ the increase in actual concentration of water vapor is 0.565 ppm per degree increase in temperature.

(To be continued.)