global warming temperature data

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 CO₂concentration 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:

        C^p =293.806 + .273098 (sum) E^c         R² = .996       (Equation 1)
                   (0.18)      (0.0014)                        (standard errors in parentheses)

where C^p is atmospheric CO₂ in ppm and (sum) E^c 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.

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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.

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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 E^ct R² = .996 (Equation 2)
(1332.8) (0.0029) (standard errors in parentheses)

where C is atmospheric CO₂ in million tonnes of CO₂ and E^ct 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:

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

where C_t - C_t-1 is change in atmospheric CO₂ in million tones of CO₂ and E_t 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.

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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:

C_t = C_{t-1 +}698.557 + .484044 E_t (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.

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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:

C_t - C_t-1 = E^f_t + E^l_t - A^o_t - A^l_t - R_t (Equation 5)

where C_t - C_t-1 is change in atmospheric CO₂, E^f_t is annual CO₂ emissions from fossil fuels consumption, E^l_t is annual emissions from land use changes, A^o_tis absorption by ocean sinks, A^l_tis absorption by land sinks, and R_tis the residual. A graph of the data in Equation 5 is shown in Figure 5.

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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:

C_t - C_t-1 = E^f_t + - Aⁿ_t (Equation 6)

where "net absorption", Aⁿ_t, represents (A^o_t + A^l_t + R_t - E^l_t), and may be calculated as E^f_t - (C_t - C_t-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.

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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.

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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:

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

where A_t is (net) absorption of CO₂ in million tonnes and C_t 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

A_t = -120729 + .081122 C_t + -.117436E-07 C_t² R² = .930 (Equation 8)
(18587.2) (.015393) (.316572E-08) (standard errors in parentheses)

where in this case, C_t² 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.

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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

    E_t = (1 + 0.02(1-.001 t)) E_t-1       (Equation 9)
_{Using this, the absorption equations above, and the identity}    C_t = C_t-1 +E_t - A_t                       (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.

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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.

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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.

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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.