**The man-made carbon emissions have caused the recent increase in atmospheric CO2 concentration.**

*This post sums up in english the posts dated 10 June, 9 May and 28 January 2013.*

This blog post refutes the article The phase relation between atmospheric carbon dioxide and global temperature [1] recently published by Ole Humlum, Kjell Stordahl and Jan-Erik Solheim (Humlum et al). They claim that there is no correlation between the man-made (anthropogenic) carbon emissions and atmospheric CO2 concentration. They more than suggest that the CO2 increase is caused by rising global temperatures. Their cause-effect explanation is opposite to the consensus among climate scientists, which is that the increased concentration of CO2 is man-made and that it has caused the increase in global temperature.

Mark Richardson [2] and Troy Masters / Rasmus Benestad [3] have published Comments to Humlum et al's article in which they show that the article contains serious errors. This blog post will do the same.

**Carbon emissions**

Figure 1: Man-made fossil fuel carbon emissions (CDIAC) and Mauna Loa atmospheric CO2 (ESRL) |

The blue line in Figure 1 shows the accumulated carbon emissions due to our burning of fossil fuels. These emissions are hereafter referred to simply as the carbon emissions. The data are from the Carbon Dioxide Information Analysis Center (CDIAC) [4]. CDIAC has estimated the annual carbon emissions since 1751 and the monthly emissions since 1950. In Figure 1 the emissions are summed up to show the total accumulated emissions. GtC means billion tonnes of carbon.

The red line in Figure 1 shows the Mauna Loa CO2 measurements. ppm means parts per million. The Mauna Loa measurements from March 1958, and the Global CO2 measurements from January 1980, are available at the website of Earth System Research Laboratory (ESRL) at NOAA [5].

Figure 1 shows a very strong correlation between the carbon emissions and the atmospheric CO2. In the first part of the interval the carbon emissions grow more slowly than in the last part of the interval, and so does the atmospheric CO2. I.e. Figure 1 also shows that there is a positive correlation between the rates of change of carbon emissions and of atmospheric CO2. It is therefore very surprising that Humlum et al do not find a positive correlation between these rates of change. The reason is that they make a very serious mistake in their analysis, which Mark Richardson points out in his published comment.

First, some math. Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in an independent input x. This rate of change is called the first derivative of y with respect to x. The derivative of the first derivative is the rate of change of the first derivative, and it is referred to as the second derivative. Example: Distance is an accumulated value. Speed is rate of change of distance, i.e. the first derivative of distance. Acceleration is rate of change of velocity, i.e. the second derivative of distance.

Humlum et al try in vain to find a positive correlation between the rates of change of monthly carbon emissions and of atmospheric CO2. The unit of the monthly emissions is GtC/year, showing that they are the derivatives of the emissions. Differentiation once more therefore provides the second derivative of the emissions. With other words, Humlum et al compare the

**second**derivatives of the carbon emissions with the

**first**derivatives of the atmospheric CO2. There is no reason to expect a positive correlation between these values.

Appendix B does the calculations that Humlum et al probably intended to do. The calculations show that there is a strong and positive correlation between the rates of change of carbon emissions and of atmospheric CO2.

**Global temperature**

Humlum et al more than suggest that the increase in global temperature since 1980 has caused the increase in atmospheric CO2. They calculate the correlation between the rates of change of atmospheric CO2 and of global temperature. But the effects of long-term trends disappear with their methodology. The derivative of a steadily rising curve is a constant, and constants do not contribute in correlation analyzes. Humlum et al's analysis shows that short-term variations in atmospheric CO2 occur about 10 months after short-term variations in global temperature. This is well known and generally accepted among climate scientists, but it states nothing about the relationship between stable long-term trends. This is explained in the published comments that I referred to in the beginning of the post, and in blog posts written by the authors of the comments here, here and here. I will now demonstrate the same with a simple thought experiment.

Figure 2: Global temperature (GISTEMP) and global atmospheric CO2 (ESRL) |

The blue solid line in Figure 2 shows the global temperature (GISTEMP, [6]). The red solid line shows the measured global atmospheric CO2 concentration [5]. The red dashed line is explained in the next paragraph; now we can ignore it. I repeat the same calculations as Humlum et al have done with the data shown in the solid lines, and I get the same results as they do. There

**is**a positive correlation between the rates of change of global temperature and of atmospheric CO2, and the correlation is strongest when CO2 lags about 10 months after temperature. The short-term changes in temperature are followed by short-term changes in CO2. Humlum et al more than suggest that this also applies to long-term trends. However, their analysis does**not**provide any evidence for that.
In the thought experiment I detrend the CO2 curve so that its long-term trend becomes zero. The short-term variations are retained. The dotted red line in Figure 2 shows the detrended CO2. I repeat the correlation calculations described in the previous paragraph, but now with the detrended CO2 data instead of the measured CO2 data. The correlation proves to be exactly the same ! This clearly shows that the positive correlation is valid for short-term trends and not for long-term trends, because in the thought experiment the temperature has a rising long-term trend but the atmospheric CO2 has not.

Appendix A describes these calculations the thought experiment in more detail.

**Calculations that Humlum et al did not do**

Humlum et al calculated the rates of change of temperature and of atmospheric CO2. Based on these calculations they could have calculated how sensitive changes in CO2 are for changes in temperature, but they did not do that. I have done these sensitivity calculations, see details in appendix C. They show that the rise in temperature can explain only slightly more than 1 ppm of the increase in CO2 between 1980 and 2011. The rest of the increase, about 53 ppm, must be due to something else. The next paragraph shows that "something else" may be the man-made carbon emissions.

Figure 1 shows the carbon emissions and the atmospheric CO2. There is a strong correlation between the rates of change of carbon emissions and of atmospheric CO2. These results can be used to calculate how sensitive changes in CO2 are for changes in carbon emissions. I have done these sensitivity calculations, see details in appendix C. They show that the carbon emissions between 1959 and 2011 can explain as much as 68 ppm of the total increase of 71 ppm in atmospheric CO2 in this period.

**Conclusion**

Statistical analyzes of the same data as Humlum et al used show that man-made carbon emissions can explain the increase in atmospheric CO2. The rise in temperature can not. Mark Richardson shows in a blog post that the increase in atmospheric CO2 really is due to man-made carbon emissions. His reasoning is based on the conservation of mass. We know that only about half of the man-made carbon emissions remains in the atmosphere. The other half cannot just disappear out of a closed carbon cycle. It therefore must have been taken up by the nature, i.e. the land and the oceans must have been a net carbon sink. The same nature can not simultaneously have been a net source of carbon to the atmosphere. Humlum et al ignore this simple fact. My statistical analyzes fully support the conclusions of Mark Richardson.

The appendixes that follow contain plots and mathematics that document the analyzes which are the basis for this blog post.

**Appendix A Correlation between global temperature and atmospheric CO2**

The solid lines in Figure 2 show the global temperature and the atmospheric CO2 concentration since 1980. Those data are the basis for the calculations in this appendix.

Figure 3: DIFF12 values of global temperature (GISTEMP) and global atmospheric CO2 concentration (ERSL) |

The DIFF12 operator smooths the seasonal variations and calculates the derivatives, as explained in appendix D. Figure 3 shows the DIFF12 values, i.e. the rates of change, of global temperature and of atmospheric CO2 concentration.

The mean of the CO2 DIFF12 values is 1.67 ppm/year. Linear regression analysis shows that the long-term rising trend of atmospheric CO2 is 1.66 ppm/year. The values are not exactly equal because the calculations apply different formulas.

Humlum et al show similar DIFF12 curves in their figure 3 in [1].

Figure 3 indicates a strong correlation between the rates of change of temperature and of CO2. The CO2 curve lags some months behind the temperature curve. The correlation is expressed numerically as a correlation coefficient, see details in Appendix E. I have calculated the correlation coefficient with many different lags of the CO2 curve, both forward and backward in time, with the same results as Humlum et al present in [1]. Figure 4 shows the correlation coefficient as a function of the CO2 lag.

Figure 4: Correlation coefficient between DIFF12 of global temperature (GISTEMP) and of global CO2 (ESRL) as a function of monthly lag of CO2. |

The correlation peak is 0.42 when CO2 lags 11 months behind temperature. This lag agrees well with a visual inspection of Figure 3.

Humlum et al calculate the correlation peak to be 0.43 when CO2 lags 9.5 months behind temperature, as shown in their figure 4a in [1]. Their correlation curve is practically equal to the curve in Figure 4. The similarity indicates that both theirs and mine calculations are correct. But Humlum et al interpret the results wrongly.

When we assess the correlation between the curves in Figure 3, we consider the shapes of the curves, and not the mean value which they vary around. This is confirmed by the formulas for the correlation coefficient [8]. This encouraged to the thought experiment described in connection with the red dashed line in Figure 2.

The DIFF12 values of the global temperature and of the detrended CO2 are equal to the ones shown in Figure 3, with one exception. The values of the detrended CO2 vary around zero, and not around 1.67 ppm/year as the DIFF12 values of the measured CO2 do. But the mean value does not contribute to the correlation coefficient, and therefore the correlation curve in the thought experiment is exactly equal to the one shown in Figure 4. This clearly shows that the positive correlation is valid for short-term trends and not for long-term trends, because in the thought experiment the temperature has a rising long-term trend but the atmospheric CO2 has not.

The global CO2 measurements start in January 1980. Humlum et al use these measurements, and for comparison reasons I do the same in this appendix. The Mauna Loa CO2 measurements start in March 1958. Calculations using the longer Mauna Loa time series provide approximately the same results as the calculations using the shorter global time series.

Calculations using other global temperature series than GISTEMP also provide approximately the same results as the calculations using the GISTEMP temperatures.

Humlum et al calculate the correlation peak to be 0.43 when CO2 lags 9.5 months behind temperature, as shown in their figure 4a in [1]. Their correlation curve is practically equal to the curve in Figure 4. The similarity indicates that both theirs and mine calculations are correct. But Humlum et al interpret the results wrongly.

**The thought experiment**When we assess the correlation between the curves in Figure 3, we consider the shapes of the curves, and not the mean value which they vary around. This is confirmed by the formulas for the correlation coefficient [8]. This encouraged to the thought experiment described in connection with the red dashed line in Figure 2.

The DIFF12 values of the global temperature and of the detrended CO2 are equal to the ones shown in Figure 3, with one exception. The values of the detrended CO2 vary around zero, and not around 1.67 ppm/year as the DIFF12 values of the measured CO2 do. But the mean value does not contribute to the correlation coefficient, and therefore the correlation curve in the thought experiment is exactly equal to the one shown in Figure 4. This clearly shows that the positive correlation is valid for short-term trends and not for long-term trends, because in the thought experiment the temperature has a rising long-term trend but the atmospheric CO2 has not.

**Longer time series with CO2 measurements**The global CO2 measurements start in January 1980. Humlum et al use these measurements, and for comparison reasons I do the same in this appendix. The Mauna Loa CO2 measurements start in March 1958. Calculations using the longer Mauna Loa time series provide approximately the same results as the calculations using the shorter global time series.

Calculations using other global temperature series than GISTEMP also provide approximately the same results as the calculations using the GISTEMP temperatures.

**Appendix B Correlation between man-made carbon emissions and atmospheric CO2**

Figure 1 shows the carbon emissions and the atmospheric CO2 concentration since 1959. Those data are the basis for the calculations in this appendix.

Figure 5: DIFF12 values of the man-made carbon emissions (CDIAC) and the Mauna Loa atmospheric CO2 (ESRL) |

The DIFF12 values of the carbon emissions increase from approximately 2½ GtC/year around 1960 to approximately 8 GtC/year in the recent years. The DIFF12 values of atmospheric CO2 also increase in the same time interval. The correlation between the DIFF12 values of the emissions and of the atmospheric CO2 is strong, and it is almost not sensible to lags in the CO2.

Humlum et al show their DIFF12 curves of man-made monthly emissions and atmospheric CO2 in their figure 11 in [1]. But their curve for the man-made emissions is totally different from the blue line in Figure 5 because they show the second derivatives of the emissions. As expected they do not find a positive correlation between those second derivatives and the first derivatives of the atmospheric CO2.

Humlum et al use the global CO2 measurements starting in January 1980. Due to their error with the second derivatives they do not provide results with which the results in this appendix can be compared. Therefore, to get longer time series, this appendix uses the Mauna Loa CO2 measurements starting in March 1958.

**Appendix C Atmospheric CO2 sensitivity calculations**

Regression analysis between DIFF12 values, i.e. between rates of change, can be used to calculate how sensitive one variable is to changes in another variable. This appendix does such sensitivity calculations for atmospheric CO2, both with respect for sensitivity to changes in global temperature and to changes in man-made carbon emissions. As we will see, these calculations reveal that most of the increase in CO2 can be explained by the increase in man-made carbon emissions, and only a tiny part of the increase in CO2 can be explained by the increase in global temperature.

**Sensitivity to global temperature**

Figure 6: Scatter plot between the DIFF12 values in Figure 3, i.e. between the global temperatures (GISTEMP) and the global CO2 (ERSL) |

One of the physical explanations of the positive sensitivity is that the solubility of CO2 in the oceans decreases with increasing water temperature, and the oceans therefore outgas some CO2 to the atmosphere when they warm. The sensitivity 2.71 ppm/°C is estimated with the GISTEMP temperatures, which is combined land and sea surface temperatures. When GISTEMP is replaced with the sea surface temperatures HadSST3 [7], the sensitivity is 3.07 ppm/°C, i.e practically the same value.

The dotted line if Figure 6 shows that there is strong autocorrelation in the data. Therefore the calculated trend, i.e. the sensitivity, is not statistically significant. David C Frank et al [9] published an article in Nature in 2010 in which they estimated the sensitivity based on proxy measurements from pre-industrial time. They estimated many values with the median equal to 7.7 ppm/°C. Our estimates around 3 ppm/°C are for short time intervals before equilibrium is reached, and they therefore match fairly well with the estimates in the Nature article.

The global temperature has increased with approximately 0.5 °C between 1980 and 2011. Our sensitivity of 3 ppm/°C can explain approximately 1.5 ppm of the increase in atmospheric CO2. The median sensitivity in the article in Nature can explain a little less than 4 ppm of the increase. The total CO2 increase is 53 ppm, so only a small part of it can be explained with the increase in global temperature.

Now follows a more detailed analysis based on the trend line (sensitivity) shown in Figure 6.

The regression analysis provides both a slope and a crossing with the vertical y-axis, as shown in equation 1.

(1) dCO2 [ppm/year] = 1.65 [ppm/year] + 2.71[ppm/°C] * dT[°C/year]

Equation 1 is valid for the interval 1980 til 2011, i.e. for 32 years. The global temperature increased with 0.5 °C during these years, so dT equals to 0.0156 °C/year.

ΔCO2 in equation 2 is the increase in atmospheric CO2 during these 32 years. Equation 2 is derived by multiplying the yearly temperature increase into the last part of equation 1, and thereafter multiplying both parts with the number of years.

(2) ΔCO2 [ppm] = 1.65*32 [ppm] + 2.71*0.0156*32 [ppm]

= 52.8 [ppm] + 1.35 [ppm]

The last part 1.35 ppm is caused by the increase in temperature. The first part 52.8 ppm must be caused by something else. The section that follows shows that it probably is caused by the man-made carbon emissions.

**Sensitivity to man-made carbon emissions**

Figure 7: Scatter plot between the DIFF12 values in Figure 5, i.e. between the man-made carbon emissions (CDIAC) and the Mauna Loa atmospheric CO2 (ESRL) |

Figure 7 is similar to Figure 6, but now the horizontal axis is the DIFF12 values of the man-made carbon emissions. The vertical axis is still the DIFF12 values of global temperature. The dots are the DIFF12 values, the dotted blue line connects the values consecutive in time, and the solid line is the regression line. The slope of the regression line is 0.25 ppm/GtC, i.e. the atmospheric CO2 increases with 0.25 ppm for each GtC humans emit into it.

The physical explanations of the positive sensitivity is obvious. Each ppm of CO2 in the atmosphere contains 2 GtC. Without a net natural carbon sink the sensitivity should therefore have been 0.5 ppm/GtC.

The dotted line in Figure 7 shows that there is strong autocorrelation in the data. Therefore the calculated trend, i.e. the sensitivity, is not statistically significant.

The man-made carbon emissions have increased with approximately 270 GtC between 1959 and 2011. The sensitivity 0.25 ppm/GtC can explain approximately 68 ppm of the increase in atmospheric CO2. But let us do the math more thoroughly, just as we did for the sensitivity to the rise in temperature.

The linear regression analysis provides both a slope and a crossing with the vertical y axis, as shown in equation 3.

(3) dCO2 [ppm/year] = 0.06 [ppm/year] + 0.25[ppm/GtC] * dC[GtC/year]

Equation 3 is valid for the interval between 1959 and 2011, i.e for 53 years. The man-made carbon emissions have increased with approximately 270 GtC in these 53 years, so dC equals to 5.1 GtC/year.

ΔCO2 in equation 4 is the increase in atmospheric CO2 during these 53 years. Equation 4 is derived by multiplying the yearly emission increase into the last part of equation 3, and thereafter multiplying both parts with the number of years.

(4) ΔCO2 [ppm] = 0.06*53 [ppm] + 0.25*5.1*53 [ppm]

= 3.2 [ppm] + 67.6 [ppm]

The last part in equation 4 dominates, and it is caused by the increase in the man-made carbon emissions. The first part 3.2 ppm must be caused by something else. The regression analysis of the DIFF12 values in Figure 7 does not tell what “something else” is. But the regression analysis of the DIFF12 values in Figure 6 substantiates that “something else” probably is the increase in global temperatures.

The results from the regression analyzes of the scatter plots in Figure 6 and Figure 7 fit well with each other.

**Appendix D The DIFF12 operator**

For each month the DIFF12 operator calculates the difference between the average of the last 12 months and the average of the 12 preceding months. In this way the operator both smooths the seasonal variations and calculates the derivative.

Example: The atmospheric CO2 concentration has seasonal variations, as seen in Figure 2. The DIFF12 values are shown with the red line in Figure 3. The seasonal variations are removed, but the other short-term variations are retained. The value in June 2001 is the average of the values from July 2000 till June 2001 minus the average of the values from July 1999 till June 2000. The unit of the CO2 concentration is ppm, and the unit of the DIFF12 values is therefore ppm/year.

**Appendix E Correlation coefficients**

There are different formulas to calculate correlation coefficients. Humlum et al do not disclose the formula they are using. Pearson's formula [8] is the most common one, and it is reasonable to assume that this one is used when nothing else is stated. But the Pearson's formula provides higher correlation coefficients than the ones presented by Humlum et al in [1].

The Spearman and Kendall rank correlation coefficients are alternatives to Pearson. Kendal tau-b [8] provides the same correlation coefficients as the ones presented by Humlum et al. Kendall tau-b is not sensitive to outliers in the data, and it is often used in statistical hypothesis testing. It is therefore appropriate to use Kendall's tau-b formula in our context. I use Kendal tau-b in this blog post both for this reason and because it provides the same coefficients as the ones presented by Humlum et al.

The conclusions in both Humlum et al's article and in this blog post are not affected by the choice of the formula for the correlation coefficient.

**References**

Ole Humlum, Kjell Stordahl and Jan-Erik Solheim (Humlum et al).

Mark Richardson

Troy Masters / Rasmus Benestad

Carbon Dioxide Information Analysis Center (CDIAC)

Earth System Research Laboratory (ESRL) at National Oceanic & Atmospheric Administration (NOAA)

National Aeronautics and Space Administration, Goddard Institute for Space Studies

**7**. Global Sea Surface HadSST3

Hadley Centre

Wikipedia

David C Frank et al.

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