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The change in temperature of a system is proportional to the net inflow of heat to that system. In the case of the Earth's surface the inflow is due to the influx of shortwave radiation from the Sun. The outflow is due to the longwave thermal radiation which is proportional to the fourth power of the absolute temperature of the surface. The level of greenhouse gases such as carbon dioxide (CO2) affects the proportion of the thermal radiation retained.
Let S be the sunspot number. This is used as a proxy for the intensity of solar radiation at the top of Earth's atmosphere. If T is the absolute temperature of the Earth's surface then its outgoing radiation is proportional to σT^{4}, where σ is the StefanBoltzmann constant. The proportion of this thermal radiation which is not retained depends upon the concentration of greenhouses gases in the atmosphere, one of which is CO2. For the statistical analysis below it is presumed that the proportion not retained is a linear function of the concentration of CO2, p_{CO2}.
The estimating equation is then
where ΔT is the JanuarytoJanuary chanage in temperature. It is important to use the change of temperature over an interval rather than the change in the annual average. When a variable is, as temperature is, the cumulative sum of disturbances the process of averaging introduces statistical artifacts that interfer with the statistical analysis. For more on this topic see stochastic structure. The temperature in σT^{4} is however the annual average.
All three of the coefficients, a_{1}, b_{0} and b_{1}, should be positive if the hypothesis that an increase in the level of CO2 contributes to an increase in global temperature.
The data on CO2 concentration are derived from air samples collected at Mauna Loa Observatory, Hawaii. The source is C.D. Keeling, T.P. Whorf, and the Carbon Dioxide Research Group at the Scripps Institution of Oceanography, University of California at La Jolla, May 2005. This data covers only from 1959 to 2004 so this is the interval of analysis.
The temperature data were constructed from the data set available from NASA. The temperature data goes back to 1880. It is worthwhile to look at some data scatter diagrams to get acquainted with the statistical characteristics of the data. First consider the times series for the JanuarytoJanuary changes in temperatures.
What the diagram shows is that there were some extreme cases in the early years of the series that had more to do with the accuracy of the data than global climate. This is not a concern for the statistical work below because the analysis only covers the period for which there are data on CO2 concentrations.
The theory suggests that there should be an inverse correlation between temperature change and the level of temperature, or more precisely the fourth power of the absolute temperature.
First consider the scatter diagram for temperature change versus temperature.
There is a satifying downward slope to the data plot. When thermal radiation σT^{4} is used as the independent variable there is also a downward slope as seen below.
But what is clear is that the outliers, the extreme cases, will dominate the statistical results. This raises a note of caution in interpreting the statistical results.
 
Year  Temp Change 
Abs. Temp  pCO2  σ*T^{4}  σ*T^{4}*pCO2  Sunspots 
1959  0.0  287.22  0.000316  385.8707  0.121935  159.0 
1960  0.03  287.22  0.000317  385.8707  0.122286  112.3 
1961  0.02  287.25  0.000318  386.0319  0.122615  53.9 
1962  0.10  287.23  0.000318  385.9244  0.122901  37.6 
1963  0.05  287.13  0.000319  385.3873  0.122946  27.9 
1964  0.04  287.08  0.000320  385.1189  0.123053  10.2 
1965  0.22  287.12  0.000320  385.3336  0.123341  15.1 
1966  0.18  286.90  0.000321  384.1539  0.123444  47.0 
1967  0.13  287.08  0.000322  385.1189  0.124058  93.7 
1968  0.01  286.95  0.000323  384.4218  0.124211  105.9 
1969  0.27  286.96  0.000325  384.4754  0.124801  105.5 
1970  0.12  287.23  0.000326  385.9244  0.125676  104.5 
1971  0.31  287.11  0.000326  385.2799  0.125725  66.6 
1972  0.58  286.80  0.000328  383.6186  0.125643  68.9 
1973  0.30  287.38  0.000330  386.7312  0.12747  38.0 
1974  0.13  287.08  0.000330  385.1189  0.127201  34.5 
1975  0.14  287.21  0.000331  385.8169  0.127767  15.5 
1976  0.16  287.07  0.000332  385.0652  0.127911  12.6 
1977  0.02  287.23  0.000334  385.9244  0.128852  27.5 
1978  0.07  287.21  0.000336  385.8169  0.129449  92.5 
1979  0.16  287.28  0.000337  386.1932  0.130105  155.4 
1980  0.28  287.44  0.000339  387.0543  0.131084  154.6 
1981  0.49  287.72  0.000340  388.5646  0.132093  140.5 
1982  0.36  287.23  0.000341  385.9244  0.131635  115.9 
1983  0.23  287.59  0.000343  387.8628  0.13294  66.6 
1984  0.03  287.36  0.000344  386.6236  0.133169  45.9 
1985  0.04  287.39  0.000346  386.7850  0.133773  17.9 
1986  0.07  287.43  0.000347  387.0004  0.134343  13.4 
1987  0.17  287.50  0.000349  387.3776  0.135191  29.2 
1988  0.44  287.67  0.000351  388.2946  0.136462  100.2 
1989  0.37  287.23  0.000353  385.9244  0.136208  157.6 
1990  0.01  287.60  0.000354  387.9168  0.137396  142.6 
1991  0.01  287.59  0.000356  387.8628  0.137932  145.7 
1992  0.15  287.60  0.000356  387.9168  0.138238  94.3 
1993  0.05  287.45  0.000357  387.1081  0.138236  54.6 
1994  0.12  287.50  0.000359  387.3776  0.139014  29.9 
1995  0.10  287.62  0.000361  388.02470  0.140038  17.5 
1996  0.02  287.52  0.000363  387.4854  0.140494  8.6 
1997  0.18  287.54  0.000364  387.5932  0.141022  21.5 
1998  0.02  287.72  0.000367  388.5646  0.142440  64.3 
1999  0.41  287.70  0.000368  388.4566  0.143069  93.3 
2000  0.37  287.29  0.000369  386.2470  0.142707  119.6 
2001  0.33  287.66  0.000371  388.2406  0.144049  111.0 
2002  0.05  287.99  0.000373  390.0252  0.145507  104.0 
2003  0.17  287.94  0.000376  389.7544  0.146396  63.7 
2004  0.25  287.77  0.000377  388.8348  0.146758  40.4 
The regression results are:
The numbers shown in parentheses below the regression coefficients are the magnitudes of their tratios; i.e. the coefficients divided by the standard deviation of the regression coefficient. All but the coefficient for Sunspot number are significantly different from zero at the 95 percent level of confidence and they are of the right sign.
Shown below is a comparison of the observed temperature change and the temperature change predicted by the regression equation.
The observations are shown in black and the estimations from the regression equation are shown in fuchsia.
Another way of viewing the comparison is in the scatter diagram below of the actual and regression predicted temperature changes.
Although the tratios for the variables included in the regression equation are significant they only explain 52.2 percent of the variation in the yeartoyear temperature change. Also the effect of the CO2 in the equation includes the effects of all variables influencing temperature change which are correlated with the general trend on CO2 concentration but are not in the equation. These would include the effects of anthropogenic water vapor and anthropogenic cloudiness.
Below is a comparison of the January global temperatures with the temperatures derived by accumulating the regression estimates of the temperature changes. Again the actual (observed) temperatures are shown in black and the ones based upon the regression estimates are shown in fuchsia.
(To be continued.)
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