Technical note: On comparing greenhouse gas emission metrics

Abstract. Many metrics for comparing greenhouse gas emissions can be expressed as an instantaneous Global Warming Potential multiplied by the ratio of airborne fractions calculated in various ways. The Forcing Equivalent Index (FEI) provides a specification for equal radiative forcing at all times at the expense of generally precluding point by point equivalence over time. The FEI can be expressed in terms of asymptotic airborne fractions for exponentially growing emissions. This provides a reference against which other metrics can be compared. 5 Four other equivalence metrics are evaluated in terms of how closely they match the timescale dependence of FEI, with methane, referenced to carbon dioxide, used as an example. The 100-year Global Warming Potential over-estimates the longterm role of methane while metrics based on rates of change over-estimate the short-term contribution. A recently-proposed metric, based on differences between methane emissions 20 years apart, provides a good compromise. Analysis of the timescale dependence of metrics, expressed as Laplace transforms, leads to an alternative metric that gives closer agreement with FEI at 10 the expense of considering methane over longer time periods. The short-term behaviour, which is important when metrics are used for emissions trading, is illustrated with simple examples for the four metrics.

response function, R X (t), represents the proportion of ∆S X , the perturbation in emissions of constituent X, that remains in the atmosphere after time t. Thus the mass perturbation, ∆M X , is given as a convolution integral: The outline of this note is as follows. In Section 2 we show how the prescription by Wigley (1998), which gives exact equivalence in radiative forcing between different time histories of emissions, may be elegantly expressed in terms of Laplace transforms. In Section 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 which accurately approximates equivalence in radiative forcing over timescales 30 from decades to multiple centuries. In Section 4, we compare the different metrics in the time domain, and we conclude in Section 5. An appendix lists the notation.
2 Metrics: FEI Wigley (1998) defined an equivalence between emission histories, termed the Forcing-Equivalent Index (FEI). Two emission histories are FEI-equivalent if they lead to equivalent forcing at all times. In most cases, this requirement precludes point-by- 35 point emission equivalence at all times.
Equivalent radiative forcing over all time from perturbations ∆S X and ∆S Y in the emissions of gases X and Y requires: as the condition for Subject to the conditions of linearity, this equivalence defines exact equality of radiative forcing. However it is an equivalence for emission profiles and not for instantaneous values.
A special case of FEI-equivalence (e.g. Enting, 2018) is when ∆S X and ∆S Y both grow exponentially, with growth rate α and amplitudes c X and c Y at t = 0. Exponential growth has The integral on the right isR X (p), the Laplace transform of R X (t), evaluated at p = α. Interpreting these relations in terms of Laplace transforms can help clarify the different forms of equivalence metrics in the general case.
As a Laplace transform, the condition for FEI-equivalence is defined by the transform of (3): In this expressionR Y (p)/R X (p) is the Laplace transform of an integro-differential operator that, in the time domain, acts on Differentiation of (5) shows that, for exponentially growing emissions, the asymptotic airborne fraction of a gas X is αR X (α) (e.g. Enting, 1990) and so the FEI curve can be defined as the ratio of asymptotic airborne fractions.
The plot in Figure 1 describes the specific case of methane, CH 4 , referenced to carbon dioxide, CO 2 . The solid line, denoted 55 FEI, can be interpreted in several different, but mathematically equivalent, ways: it gives the ratio that leads to FEI-equivalence in the special case of exponentially growing emissions; it is the ratio of asymptotic airborne fractions for exponential growth, shown as a function of growth rate; it is the Laplace transform of an operator that acts on methane emission functions to produce FEI-equivalent CO 2 emissions.

Comparison of metrics
The examples given here compare four different metrics, again for the case of CH 4 referenced to CO 2 , benchmarking them against FEI. In these calculations, the response used for CO 2 is the multi-model mean from (Joos et al., 2013, Table 5) and the response of CH 4 described by a 12.4 year perturbation lifetime. In each case, these represent the response to small perturbations about current conditions, reflecting our interest in the use of metrics for trade-offs, reporting and target-setting.

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The calculations were developed for methane emissions from active biological sources. For fossil methane, an additional CO 2 contribution from the oxidation of CH 4 should be included.

Global Warming Potential
The Global Warming Potential (GWP) with time horizon H defines an equivalence (denoted ≡ GWP ) for component Y given where for gas Y Although (9) is usually written without the H −1 factors, in the form above the numerator and denominator correspond to the airborne fractions of Y and CO 2 , averaged over the time horizon H, and multiplied by the factor a Y /a CO2 which corresponds 75 to GWP 0 , the H → 0 limit of GWP H . This factor can be called the instantaneous GWP.
GWP 100 , the GWP with the time horizon H = 100 years, has become the standard for greenhouse gas equivalence in international agreements.
For CH 4 , the equivalence is where all use of GWP in what follows will specifically refer to CH 4 . Relation (10) corresponds to using However, this definition of equivalence has long been known to be poor (e.g. Reilly et al., 1999), especially for emission profiles approaching stabilisation of concentrations.
For H > 100 the approximation is quite close, suggesting that the appropriate time horizon should match the e-folding time of emissions (Enting, 2018).

Derivative
Several studies (Smith et al., 2012;Lauder et al., 2013) suggested that for short-lived gases such as CH 4 , changes in emissions in the short-lived gases should be related to one-off CO 2 emissions. This suggests a metric of the form: or (as a Laplace transform): Subsequently, the search for an improved metric, termed GWP*, has been the subject of extensive studies undertaken by Allen and co-workers: (Allen et al., 2016Jenkins et al., 2018;Cain et al., 2019;Collins et al., 2019;Lynch et al., 2020).

Difference
A recent proposal for an improved GWP* (Cain et al., 2019) proposes the equivalence: The Laplace transform, as shown in Figure 1, is derived using the generic result that a time-shift by T corresponds to multiplying the Laplace transform by exp(−pT ), giving:

Reduced model 105
When, as is done here, the response functions are expressed as a sum of exponentially decaying functions of time, the Laplace transform becomes a sum of partial fractions of the form α/(p + β) so that the combination is a ratio of polynomials in p. Thus the FEI ratio will also be a ratio of polynomials which can in turn be re-expressed as a sum of partial fractions, giving an exact, but complicated, form for the FEI relation. Studies in a number of fields such as electronic engineering (e.g. Feldman and Freund, 1995) have noted that such expressions can often be usefully approximated by lower order expressions. For emission 110 equivalence, it is only practical to use very low order approximations for such a reduced model.
As shown in Figure 1, a close fit to FEI can be obtained with the reduced model (RM) given bỹ with b = 0.035.
This gives an equivalence: In the time domain, (18) becomes: where ∆Ṡ CH4 denotes the rate of change in the perturbation to CH 4 emissions.
This expresses the CO 2 -equivalent of CH 4 as a weighted average of the CH 4 emission growth rate. Consequently, the metric 120 retains the property that constant emissions of CH 4 are treated as equivalent to zero CO 2 emissions as in 'derivative' metrics (Smith et al., 2012;Lauder et al., 2013). The parameter b can be chosen to match other metrics. The value b = 0.035 is chosen so that for emissions with 1% per annum growth rate the RM metric closely matches the 100-year GWP.
For specific calculations it may be more appropriate to represent this metric as 125 Relation (20) is derived from (19) using integration by parts (or equivalently by putting p/(p + b) = 1 − b/(p + b)). It has the advantage that it is expressed in terms of emissions rather than their rates of change.
Equation 20 defines the reduced model equivalence as a difference between present emissions and a weighted average of past emissions. When considered in terms of frequency f (by setting p = 2πf × √ −1) this avoids the frequency aliasing that occurs with the 'difference' metric for periods of 20 years or integer fractions thereof (see supplementary information).

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The equivalence relation (18) can also be re-written as This defines an equivalence between the rate of change of CH 4 emissions and a combination of rate of change of CO 2 emissions (as in GWP) and current CO 2 emissions (as in the derivative-based equivalences suggested by Smith et al. (2012) and Lauder et al. (2013)).

4 Comparisons in the time domain
Many previous studies of metrics have concentrated on global-scale calculations over the long term. When metrics are used for emissions trading, the behaviour at shorter timescales becomes important. This can be analysed by taking a notional CH 4 emission profile and calculating the resulting CH 4 concentrations. This is then compared to the CO 2 concentrations that result from the notionally equivalent CO 2 emissions.
140 Figure 2 shows a CH 4 source perturbation with a rapid increase from zero to a fixed emission rate, and the CO 2 -equivalent emissions as determined by the various equivalence metrics. Figure 3 shows the CH 4 concentration resulting from the methane emission and the CO 2 concentration resulting from the various CO 2 -equivalent emissions. In Figures 2 and 3, the relative scaling of the axes is given by a CH4 /a CO2 so that forcing can be compared directly.
The results clearly show the failings of the 100-year GWP for defining emission equivalence in this type of context. The 145 forcing from GWP-equivalent CO 2 initially lags well behind the actual forcing from CH 4 but in the long term it continues to increase indefinitely long after the forcing from on-going CH 4 emissions has stabilised. Compared to this behaviour, the 'derivative' metric based on rates of change of CH 4 emissions is a great improvement. However, the CO 2 -equivalent forcing initially exceeds the actual forcing from CH 4 and in the long-term drops below the CH 4 forcing. The difference metric from Cain et al. (2019) provides a CO 2 -equivalent forcing that follows the actual CH 4 forcing more closely with only a slight 150 shortfall in the longer term. The increase after several centuries reflects a contribution to the metric that corresponds to 0.25 times the 100-year GWP.
The CO 2 -equivalence derived from the reduced model follows the actual CH 4 forcing particularly closely as would be expected given the close agreement when the relations are expressed as Laplace transforms.
The nature of the FEI relation precludes close matches in forcing from instantaneous relations between CH 4 and CO 2 155 emissions. The 'difference' and 'reduced model' metrics relate CO 2 equivalents to the past history of CH 4 emissions. For a specific case, Lauder et al. (2013) suggested an approximate equivalence to changes in methane emissions balanced by an ongoing future CO 2 uptake from growing trees.
We briefly note that there are trade-offs between different metrics that are difficult to balance. The goal of defining emissions equivalence is to allow for emissions of different greenhouse gases to be substituted for each other, so that a given radiative 160 forcing target can be achieved for the least economic cost. If the metric of emissions equivalence is too complex, as it is for FEI, then it may be difficult or impossible for an effective trading scheme to be implemented. If the metric is inaccurate at the relevant timescales, as is the case for GWP100, then the 'least cost' emissions pathway may overshoot the radiative forcing target, especially as stabilisation in radiative forcing is approached.

Concluding summary
165 FEI-equivalence is defined by equivalent radiative forcing at all times. Applying this to different gases constrains emissions over all time.
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 170 this ratio at the range of relevant timescales.
GWP treats this ratio as a constant, defining GWP H as the instantaneous GWP multiplied by the ratio of average airborne fractions over the time horizon, H. For CH 4 , referenced to CO 2 , this means that GWP over-estimates the CH 4 contribution for growth rates less than 1/H and under-estimates the CH 4 contribution from shorter timescales.
Metrics relating CO 2 -equivalence to rates of change of CH 4 emissions are treating the ratio of airborne fractions as pro-175 portional to the e-folding rate. This can provide a good representation of long-term behaviour relevant for stabilisation, but M X (t) Atmospheric content of constituent X. Perturbation is ∆M X (t).
p Argument of Laplace transform. Equivalent to e-folding rate when comparing exponentially growing emissions.
R X (t) Atmospheric response function for constituent X.
S X (t) Anthropogenic emission of constituent X. Perturbation is ∆S X (t). resulting from the equivalent CO2 sources, as shown in Figure 2. The relative scaling of the axes is aCH4/aCO2 so that the radiative forcing can be compared directly.