Anthropogenic contributions to global climate change come from a range of greenhouse gases. Comparisons between them have been facilitated by defining emission equivalence relations (which we denote by ≡), usually using CO2 as a reference.

The climatic influence of greenhouse gases is commonly represented in terms of radiative forcing, F, expressed in terms of MX, the atmospheric mass of gas X, with the effect of small perturbations linearised as 1ΔF=aXΔMX, where aX is the radiative efficiency in mass units, i.e. the amount of change in radiative forcing per unit mass increase for constituent X in the atmosphere .

Equivalence relations between sources of greenhouse gases are complicated because various gases are lost from the atmosphere on a range of different timescales. This behaviour is often represented using linear response functions, where the response function, RX(t), represents the proportion of ΔSX, the perturbation in emissions of constituent X, that remains in the atmosphere after time t. Thus the mass perturbation, ΔMX, is given as a convolution integral: 2ΔMX(t)=∫0tRX(t-t′)ΔSX(t′)dt′.

The outline of this note is as follows. In Sect. 2 we show how the prescription by , which gives exact equivalence in radiative forcing between different time histories of emissions, may be elegantly expressed in terms of Laplace transforms. In Sect. 3, we adapt this representation to other metrics of emission equivalence and use it as inspiration for a new metric with a single adjustable parameter that accurately approximates equivalence in radiative forcing over timescales from decades to multiple centuries. In Sect. 4, we compare the different metrics in the time domain, and Sect. 5 discusses some of the mathematical characteristics that may bear on the political acceptance of alternative specifications of emission equivalence. We conclude in Sect. 6. Appendix A expresses the metrics in terms of frequency response, and Appendix B lists the notation. references the archived R code used in this paper.

defined an equivalence between emission histories, termed the FEI. Two emission histories are FEI equivalent if they lead to equivalent forcing at all times. In most cases, this requirement precludes point-by-point emission equivalence at all times.

Equivalent radiative forcing over all time from perturbations ΔSX and ΔSY in the emissions of gases X and Y requires 3aY∫0tRY(t-t′)ΔSY(t′)dt′=aX∫0tRX(t-t′)ΔSX(t′)dt′,for  all  t as the condition for 4ΔSY(t)≡FEIΔSX(t). Subject to the conditions of linearity, this equivalence defines the exact equality of radiative forcing. However, it is an equivalence for emission profiles and not for instantaneous values.

A special case of FEI equivalence see, for example, is when ΔSX and ΔSY both grow exponentially, with growth rate α and amplitudes cX and cY at t=0. Exponential growth has 5ΔMX(t)=∫-∞tRX(t-t′)cXexp⁡(αt′)dt′=cXexp⁡(αt)∫0∞RX(t′′)exp⁡(-αt′′)dt′′. The integral on the right is R̃X(p), the Laplace transform of RX(t), evaluated at p=α. Thus, for FEI equivalence of emissions growing exponentially at rate α one has 6cX=aYR̃Y(α)aXR̃X(α)cY. Interpreting these relations in terms of Laplace transforms can help clarify the different forms of equivalence metrics in the general case. More generally, for arbitrary emission perturbations the condition for FEI equivalence is defined by the Laplace transform of Eq. (): 7aYΔS̃Y(p)R̃Y(p)=aXΔS̃X(p)R̃X(p), giving 8ΔS̃X(p)≡FEIaYaXR̃Y(p)R̃X(p)ΔS̃Y(p)=aYaXΨ̃FEI(p)ΔS̃Y(p).

In this expression Ψ̃FEI(p)=R̃Y(p)/R̃X(p) is the Laplace transform of an integro-differential operator that acts on ΔSY(t) in the time domain. Differentiation of Eq. () shows that, for exponentially growing emissions, the asymptotic airborne fraction of a gas X is αR̃X(α) e.g., and thus the FEI curve can be defined as the ratio of asymptotic airborne fractions for growth rate p.

The plot in Fig.  describes the specific case of methane, CH4, referenced to carbon dioxide, CO2. The solid line, denoted as FEI, can be interpreted in several different but mathematically equivalent ways:

it is the ratio of asymptotic airborne fractions for exponential growth, shown as a function of growth rate;

The examples given here compare four different metrics, again for the case of CH4 referenced to CO2, benchmarking them against FEI. A general linear, time-invariant equivalence relation can be defined by 9aCO2ΔS̃CO2-eq(p)=aCH4Ψ̃(p)ΔS̃CH4(p). In the time domain, such a metric can be regarded as a process that extracts, from the history of CH4 emissions, an index or statistic that gives CO2 equivalence. Such a metric can be assessed in radiative forcing terms by the accuracy of the approximation 10aCO2R̃CO2(p)ΔS̃CO2-eq(p)=aCH4R̃CO2(p)Ψ̃(p)ΔS̃CH4(p)≈aCH4R̃CH4(p)ΔS̃CH4(p). If the global temperature response to a change in radiative forcing is linearised using a response function U(t), as is done, for example, by , then equivalence in temperature perturbations can be analysed in terms of 11Ũ(p)aCO2R̃CO2(p)ΔS̃CO2-eq(p)=Ũ(p)aCH4R̃CO2(p)Ψ̃(p)ΔS̃CH4(p)≈Ũ(p)aCH4R̃CH4(p)ΔS̃CH4(p). In both Eqs. () and (), removing the common factors reduces the comparison to one of considering the accuracy of the approximation 12R̃CO2(p)Ψ̃(p)≈R̃CH4(p). As noted, “if CO2 equivalence is based on radiative forcing and calculated accurately for non-CO2 gases, then the temperature and sea-level implications of the [Kyoto] Protocol may be calculated from the CO2-alone case”.

Because of the commutative and associative properties of such transformations, a transformation of the CH4 source to give an equivalent CO2 source can be described in terms of how well the metric transformation, acting on the CO2 impulse response, reproduces the impulse response for CH4. The application of this relation in the frequency domain (i.e. p=2πif) is described in Appendix A.

In these calculations, the response used for CO2 is the multi-model mean from their Table 5 and the response of CH4 described by a 12.4-year perturbation lifetime . In each case, these represent the response to small perturbations from of current conditions, reflecting our interest in the use of metrics for trade-offs, reporting and target-setting. The values for aCO2 and aCH4 are also taken from , and in the latter case follow the IPCC convention of including indirect effects. These factors only appear in the relative scaling of the axes in the two parts of Fig. .

The calculations were developed for methane emissions from active biological sources. For fossil methane, an additional CO2 contribution from the oxidation of CH4, corresponding to a global warming potential (GWP) of 1, should be included.

Many previous studies of metrics have concentrated on global-scale calculations over the long term. As discussed above, this has led to the development of metrics based on rates of change. However, as discussed in Sect. 5 below, for emissions trading on shorter timescales, political acceptance is likely to favour metrics that also have equivalent influences in the short term. The short-term behaviour can be analysed by taking a notional CH4 emission profile and calculating the resulting CH4 concentrations. This is then compared to the CO2 concentrations that result from the notionally equivalent CO2 emissions.

Figure a shows a CH4 source perturbation with a rapid increase from zero to a fixed emission rate, and the CO2-equivalent emissions as determined by the various equivalence metrics. Figure b shows the CH4 concentration resulting from the methane emissions and the CO2 concentration resulting from the various CO2-equivalent emissions. In Fig. a and b, the relative scaling of the axes is given by aCH4/aCO2 so that forcing can be compared directly. In this scaling, the direct effect of CH4 has been scaled to include indirect effects, from tropospheric ozone and stratospheric water vapour, using values taken from . Note that the indirect effects are not included in the corresponding graphs given by .

The results in Fig. b clearly show the failings of the 100-year GWP for defining emission equivalence for constant sources. The forcing from GWP-equivalent CO2 (long dashes) initially lags well behind the actual forcing from CH4 but in the long term it continues to increase indefinitely long after the forcing from on-going CH4 emissions has stabilised. Compared to this behaviour, the derivative metric based on rates of change of CH4 emissions is a great improvement (chain curve). However, the CO2-equivalent forcing initially exceeds the actual forcing from CH4 and in the long-term drops below the CH4 forcing. The difference metric from (short dashes) provides a CO2-equivalent forcing that follows the actual CH4 forcing more closely with only a slight shortfall in the longer term. After t=150 the forcing from equivalence defined by the metric (short dashes) starts to increase, due to the contribution that corresponds to 0.25 times GWP when SCH4(t)≈SCH4(t-20).

Figure b shows that the CO2-equivalence derived from the reduced model (dotted curve) follows the actual CH4 forcing particularly closely, as would be expected given the close agreement when the relations are expressed as Laplace transforms as shown in Fig. 1.

The nature of the FEI relation precludes close matches in forcing from instantaneous relations between CH4 and CO2 emissions. The difference and reduced model metrics relate CO2 equivalents to the past history of CH4 emissions. For a specific case, suggested an approximate equivalence to step changes in methane emissions balanced by an ongoing future CO2 uptake from growing trees.

The aim of our analysis has been to provide a better understanding GWP vs. GWP* and similar metrics. Any comprehensive analysis of what might be politically feasible needs to be done by others with greater expertise in such areas. However, there are various aspects of our analysis that bear on the practical applicability and political acceptability of various metrics and the trade-offs that need to be balanced in political choices.

Past studies cited above suggest that an equivalence metric should capture the context of emissions at the time. The analysis by (see also Eq.  above) notes that GWPH is close to FEI equivalence for growth in emissions with an e-folding time of H. Thus, a 100-year GWP was a plausible approximation at the time that it was introduced. For very large H, the GWP of short-lived gases goes to zero as 1/H, suggesting that a derivative of growth rates should define the metric for such long timescales. In contrast, for short-term trading and target setting, a metric that captures the short-term context is desirable in order to avoid distortions that would hinder political acceptability.

An important goal of defining emissions equivalence is to allow for emissions of different greenhouse gases to be substituted for each other so that a given target expressed in terms of radiative forcing (or equivalently in terms of CO2 concentration equivalence) can be achieved for the lowest economic cost. If, as is the case for GWP100, the metric overestimates the extent to which CO2-equivalent emission reductions contribute to radiative forcing, then methane reductions based on such equivalence will fall short of the CO2 concentration-equivalent target. Conversely, for short timescales where GWP100 underestimates the forcing reduction of CO2-equivalent methane reductions, short-term targets based on such equivalence will overestimate the extent of requisite methane emission reductions as in the example given by .

In considering how our analysis feeds into such considerations, we make the following notes:

the metric should capture both the long-term context needed for stabilisation and the more immediate context in which both trading and international agreements are conducted;

Finally, we note that our analysis is illustrative, using specific numbers primarily from the 5th IPCC assessment. The forthcoming 6th IPCC assessment may well make minor changes to specific numbers, such as the effective lifetime and the CO2 response function as well as things such as the inclusion of feedbacks, forcing efficiencies and indirect effects cf..

Our analysis has used the concept of FEI equivalence to analyse various definitions of greenhouse gas emission equivalence in terms of how closely equivalent emissions at a time t lead to equal radiative forcing at future times. The approach is applied to the consideration of CH4 emissions in terms of various definitions of their CO2-equivalent emissions. In the special case of exponentially growing emissions, FEI equivalence can be achieved when the emissions are scaled by the instantaneous (0 time horizon) GWP multiplied by the ratio of the asymptotic airborne fractions. This ratio depends on the e-folding growth rate. Various emission metrics can be compared in terms of how well they match this ratio at the range of relevant timescales. This analysis is equivalent to considering Laplace transforms of the impulse response functions of the respective gases.

GWP treats this ratio as a constant for all timescales, effectively defining GWPH as the instantaneous GWP multiplied by the ratio of average airborne fractions over the time horizon, H. For CH4, referenced to CO2, this means that GWPH overestimates the CH4 contribution for growth rates less than 1/H and underestimates the CH4 contribution to radiative forcing at faster growth rates.

Metrics relating CO2 equivalence to rates of change of CH4 emissions or emissions of other short-lived gases are treating the ratio of airborne fractions as proportional to the e-folding rate. This can provide a good representation of long-term behaviour relevant for stabilisation, but overestimates the role of CH4 on the shorter timescales relevant for emission trading. A range of metrics that better match FEI over a wide range of timescales from decades to millennia can be constructed. These include the metric proposed by , which uses the change in CH4 emissions over a 20-year interval, and a reduced model approximation to FEI equivalence. In each of these cases the better match is achieved at the expense of comparisons involving longer time periods.

The political acceptability of metrics other than the GWP will involve various trade-offs between accuracy and practicality. The type of analyses presented here can help analyse such trade-offs without reference to specific scenarios of changes in greenhouse gas emissions.