Carbon dioxide emissions and atmospheric concentration

By how much must carbon dioxide emissions be reduced to have a significant effect on atmospheric
CO2 concentration?  Much attention is directed to reducing emissions, but even if successful, how much will this achieve in mitigating temperature rises? These are important "end game" questions for policy makers. For policy to be effective, it is important that the linkages between these objectives are well understood.
On consideration of the implications, as presented here, a basic analysis of the available global data can be instructive.

Prior questions are whether increased atmospheric CO2 causes temperature rise and whether fossil fuel emissions are the major cause of increased CO2 concentrations. Both of these questions may be answered in the affirmative beyond reasonable doubt. The relationship between temperature and CO2 is too strong to discount. See for example the evidence presented in  A simple econometrician's guide to global warming, by  John L Perkins.

Similarly, the effect of fossil fuel emissions on atmospheric CO2 concentration can be established by a relatively simple statistical analysis. An example is that of Gary W Harding,   How Much of Atmospheric Carbon Dioxide Accumulation Is Anthropogenic?  In this analysis, Harding compared atmospheric CO2 concentration (in parts per million) with cumulative fossil fuel emissions (measured in gigatonnes, i.e. billion tonnes, of carbon). He then found the line of best fit between the two, obtaining a very high R2 value as a least squares regression statistic. Thus he concluded that anthropogenic emissions therefore explain a very substantial explanation of the atmospheric CO2 increase.  How reliable is that, and how useful is it in policy analysis?

Using data, for example from the Earth Policy Institute, it is a relatively simple but instructive task to pursue and extend these investigations and to consider their implications in the light of policy objectives.

First, Harding's results are replicated here, using similar data from 1871 to 2008. The equation, with estimated coefficients, is as follows:

          Cp =293.806  + .273098  (sum) Ec         R2 =  .996       (Equation 1)
                   (0.18)      (0.0014)                        (standard errors in parentheses)

where Cp is atmospheric CO2 in ppm and  (sum) Ec is cumulative CO2 emissions from fossil fuels. A graph of the actual and fitted values from this equation is shown in Figure 1. The strikingly good fit of the equation certainly seems convincing evidence that atmospheric CO2 accumulation is anthropogenic. The parameter estimates seem well identified. The coefficient of .273 means that CO2 concentration is increased by .273 ppm for every gigatonne of carbon emitted.


Figure 1. Atmospheric CO2 (ppm) vs cumulative emissions (gt of carbon)

Note that, as with many scientific publications, carbon dioxide is measured here in terms of  gigatonnes of carbon, not CO2.  The results, plotted against time, are shown in Figure 2.


Figure 2. Atmospheric CO2 (ppm) - cumulative emissions equation

If the relationship is estimated in different units, with both variables in the same units, the same basic result is obtained, but the emissions coefficient can be more usefully  interpreted. The result is:

 C = .215653E+07 + .546196 Ect         R2 =  .996        (Equation 2)
         (1332.8)           (0.0029)            (standard errors in parentheses)

where C is atmospheric CO2 in million tonnes of CO2 and  Ect is cumulative CO2 emissions from fossil fuels, also measured in million tonnes of CO2 (a tonne of atmospheric carbon is equivalent to 3.67 tonnes of CO2). In this case the coefficient .546 can be given the interpretation that CO2 increases represent 54.6%  of emissions, the remaining 45.4% being absorbed by the planet.

However a criticism of all these results can be mounted because both concentration and cumulative emissions contain strong time trends so that the estimated relationship contains a spurious correlation. A better estimate may be obtained by comparing changes in concentration with annual emissions, i.e. by taking first differences in the variables. If this is done, with the latter equation, the results are as follows:

         Ct - Ct-1 = 698.557 + .484044 Et           R2 =  .935       (Equation 3)
                          (139.4)       (0.011)               (standard errors in parentheses)

where Ct - Ct-1 is change in atmospheric CO2 in million tones of CO2 and  Et is annual CO2 emissions from fossil fuels. A graph of these results is shown in Figure 3. In this case a slope coefficient of .48 is obtained for the relative magnitude of additions and emissions. That is, in this first-difference formulation, atmospheric increases are slightly less than half of emissions whereas in the levels formulation they were slightly more than half. The explanation for this difference is that the levels formulation gives relatively more weight to the later period observations, when the atmospheric increase is proportionally greater. It implies that emissions are increasingly being retained in the atmosphere.


 Figure 3.  Atmospheric CO2 increases vs annual emissions (million tonnes CO2)

The difference equation can be rearranged to dynamically recompute the level of  atmospheric CO2 as follows:

        Ct  =  Ct-1  + 698.557 + .484044 Et         (Equation 4)

The results of this procedure, plotted against time, are shown in Figure 4. They do not differ greatly from the originally estimated results as shown in Figure 2.


Figure 4. Cumulative CO2 increases - emissions equation

Despite the good fit between actual and predicted values for these equations, there is a further problem. There would be serious doubt as to whether they would be a reliable guide for use in forecasting. This is because there is no inherent reason why atmospheric CO2 should increase in a fixed relationship to annual emissions. We already have evidence that they do not. Consideration of a model of absorption is a more logical way to proceed. The planetary absorption of carbon dioxide is not directly related to emissions, but principally to the stock of atmospheric CO2 and the planet's ability to absorb it. A proper model would consider both emissions and absorption and determine changes in atmospheric concentration as the balance between the two.

The inflow and outflow of carbon dioxide from the atmosphere can be thought of as a "carbon budget". An estimate of the global carbon budget has been provided by the Global Carbon Project for the years 1959-2008. The data are provided in accordance with the following identity:

        Ct - Ct-1  =  Eft  +  Elt  - Aot  -  Alt   -   Rt       (Equation 5)

 where Ct - Ct-1 is change in atmospheric CO2, Eft is annual CO2 emissions from fossil fuels consumption, Elt is annual emissions from land use changes, Aot is absorption by ocean sinks, Alt is absorption by land sinks, and Rt is the residual. A graph of the data in Equation 5 is shown in Figure 5.


Figure 5.      Carbon budget (gigatonnes of carbon)

All variables are subject to measurement error, and as can be seen from the graph, some are quite volatile, particularly land absorption. This volatility is reflected in the atmospheric estimates. The residual is the difference between atmospheric changes and sources minus sinks. Since the atmospheric estimates and fossil fuel emissions are subject to the least measurement error, it seems reasonable to simplify the analysis by using these to form an estimate of net absorption. The equation then becomes:

        Ct - Ct-1  =  Eft  +  - Ant        (Equation 6)

where "net absorption", Ant, represents (Aot  +  Alt   +   Rt  -  Elt), and may be calculated as Eft - (Ct - Ct-1). As can be seen from Figure 5, land use emissions and ocean absorption approximately cancel out, so that net absorption is comprised mainly of land absorption and the residual. The resulting data for implied net absorption are shown in Figure 6, together with the other variables of Equation 6.


Figure 6.      Carbon budget, with net absorption (gigatonnes of carbon)

The graph shows a clear upward trend in absorption, with absorption, over this time period, being generally slightly less than half of emissions, and atmospheric changes slightly more than half. A further advantage of this formulation, in terms of net absorption, is that for emissions and concentration, much longer time series are available. See for example the
Earth Policy Institute. A graph of this data. in units of million tonnes of CO2,  is shown in Figure 7.

Figure 7.  CO2 changes,  fuel emissions and net absorption (million tonnes of CO2)

The graph shows clearly the imbalance that has developed in modern times. A century ago, emissions and absorption were in approximate balance, and atmospheric CO2 changed little. As emissions have increased, so has absorption, but by only about half the amount of emissions.

It seems reasonable to assume that, other things being equal, the amount of absorption may be proportional to atmospheric CO2. However absorption, and natural emissions, which are included in net absorption, may also be temperature dependent. Hence the relationship may not be stable for use in forecasting. From the carbon budget data, it does seem that absorption from land sinks may not be keeping pace with atmospheric changes in recent decades. In view of these considerations, two equations for absorption have been estimated: one linear and one quadratic.

The linear equation is as follows:

    At  =  -51953.9 +  .024060 Ct           R2 =  .923       (Equation 7)
              (1387.63)    (0.00059)           (standard errors in parentheses)

where At is (net) absorption of CO2 in million tonnes and Ct is atmospheric CO2, also in million tonnes. The coefficient of 0.02 indicates that approximately 2 percent of atmospheric CO2 is absorbed each year.

The quadratic equation is

   At  =  -120729  +  .081122 Ct  +  -.117436E-07  Ct2      R2 =  .930        (Equation 8)
             (18587.2)    (.015393)        (.316572E-08)          (standard errors in parentheses)

where in this case,  Ct2 is atmospheric CO2 squared. The estimation results of these equations are shown in Figure 8. While both these equations may provide similar results over the historical period, it may be expected that CO2EmAbConc.htmthey would behave quite differently in forecasting. 


Figure 8.  Absorption of CO2 vs atmospheric CO2

When plotted against time, both equations show a similar degree of precision, as shown in Figure 9, where atmospheric CO2 is calculated in ppm for comparison with Figure 2.


Figure 9. Atmospheric CO2 (ppm) - absorption equations

Having derived these alternative parameter estimates over the historical period, a further task remains: This is to provide some assumption about the future growth in emissions and to see how this translates into future concentration and absorption. Currently, global CO2 emissions are increasing at about 2 percent per annum (a large part of which is increased emissions from China). Some projections of emissions have emissions stabilising by 2030. See for example
Energy Resource Depletion and Carbon Emissions: Global Projections to 2050 by John L Perkins. A schematised equation to replicate this would have the 2 percent rate of increase decrease by about 0.1 percentage point per year.  Hence a forecasting equation for emissions may be provided as

    Et  =  (1 + 0.02(1-.001 t)) Et-1        (Equation 9)

Using this, the absorption equations above, and the identity

    Ct = Ct-1 +Et  - At                        (Equation 10)

emissions, absorption and atmospheric CO2 can all be computed simultaneously and projected into the future. This has been done to 2050, using alternatively the linear and quadratic absorption equation. The results are shown in Figure 10.


Figure 10. Projected CO2 emissions and absorption (million tonnes)

Not surprisingly, the linear and quadratic absorption equations differ markedly as atmospheric CO2 increases. This highlights the range of uncertainty that may obtain regarding future absorption. The best case is that absorption remains on a linear trajectory with respect to concentration. The quadratic equation represents perhaps a worst case scenario, where absorption peaks and then starts to decline after 2020.  In the former case, emissions and absorption get back into balance, so that atmospheric CO2 stabilises. In the latter, concentration continues to increase even though emissions decrease. The effect of these projections on atmospheric concentration, in ppm, is shown in Figure 11.


Figure 11. Projected atmospheric concentration (ppm)

For comparison, a projection of concentration based on the emissions only equation, as in Figure 4 (converted to ppm) is also shown in Figure 11. As can be seen, to achieve stability in atmospheric concentration by 2050 requires both that emissions are reduced and that absorption remains in a linear relationship with concentration. Given limitation in fossil fuel resources, the rate of increase in emissions will inevitably decline. Whether absorption can be maintained is a crucial question. There is no intrinsic reason why absorption should follow a quadratic path, as this is only a mathematical construct. However it is doubtful that the linear trajectory can be maintained, so the eventual outcome may be somewhere between the two. Exactly where will be a critical question in the latter part of this century.

What does this mean for temperature? As calculated previously, a one-degree rise in temperature may be expected with a 100 ppm rise in CO2 concentration. This relationship can be used to project temperature changes based on the computed concentration scenarios. The results are shown in Figure 12.


Figure 12. Projected temperature change (degrees C)

The different trends in concentration are reflected in trends in temperature. The fact that temperature increases are CO2-related ought to be universally accepted. Those who deny the relationship have no plausible alternative explanation of why temperatures have risen in conjunction with atmospheric CO2 over recent decades, in direct relation to emissions.

The objective of stabilising temperature must be considered in relation to both emissions and absorption. Emissions will be reduced by depletion of fossil fuel resources and by higher energy resource prices, no matter what emissions mitigation policies are adopted, or not adopted. Land use policies that are conducive to maintaining absorption are a critical ingredient of a solution. The only other alternative will be some other form of geo-engineering.

John L Perkins is a Senior Economist at the National Institute of Economic and Industry Research, Melbourne, Australia.