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.

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 overestimates the long-term role of methane, while metrics based on rates of change overestimate 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 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.

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

The climatic influence of greenhouse gases is commonly
represented in terms of radiative forcing,

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,

The outline of this note is as follows.
In Sect. 2 we show how the prescription by

Equivalent radiative forcing over all time from perturbations

A special case of FEI equivalence

In this expression

The plot in Fig.

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

it gives the ratio that leads to FEI equivalence in the special case of exponentially growing emissions;

it is the Laplace transform of an operator

Ratio of airborne fractions for CH

The examples given here compare four different metrics,
again for the case of CH

Because of the commutative and associative properties of such transformations,
a transformation of the CH

In these calculations, the response used for CO

The calculations were developed for methane emissions from
active biological sources.
For fossil methane, an additional CO

The GWP with time horizon

For CH

However, this definition of equivalence has long been known
to be poor

For

Several studies

Subsequently, the search for an improved metric, termed GWP

A recent proposal for an improved GWP

When the response functions are expressed as sums
of exponentially decaying functions of time as is done here, the Laplace transforms become a sum
of partial fractions of the form

As shown in Fig.

This gives the following equivalence:

In the time domain, Eq. (

This expresses the CO

For specific calculations it may be more appropriate
to represent this metric as

Equation (

The equivalence relation (

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 CH

Figure

The results in Fig.

Figure

The nature of the FEI relation precludes close matches
in forcing from instantaneous relations between CH

The aim of our analysis has been to provide a better understanding
GWP vs. GWP

Past studies cited above suggest that an equivalence metric should
capture the context of emissions at the time.
The analysis by

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 CO

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;

if the metric for 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;

the metric needs to be “backward looking” and avoid giving present credit or debit on the basis of promises of future targets;

the backwards view should not extend too far, as the relevant actors can change over time, even in the cases of nations or multi-national groups, such as the EU, which has in the past set collective targets;

metrics defined in terms of derivatives need to
be supplemented with a specification of how this
is determined in practice,
e.g. as a difference by

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 CO

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

GWP treats this ratio as a constant for all timescales, effectively defining

Metrics relating CO

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.

The Laplace transform provides a natural formalism for analysing causal initial value systems. However, Fourier transforms and Fourier analyses have wide familiarity and can be used to describe our results.

For a periodic variation with exponentially
increasing amplitude, Eq. (5) generalises to

For

Section 3 noted that metric transformations defined by

A frequency domain interpretation can be
obtained by putting

The phases of the complex numbers in the relations above
capture the phase shifts for the various frequencies.
For the present we show only the resulting amplitudes, given by the moduli, (

Figure A1
sets

Frequency response for the various cases
of

Laplace transforms are denoted by the tilde notation with

Equivalence relations are denoted by

The R code used to perform the calculations and generate the figures is archived in FigShare at

No data sets were used in this article.

IE and NC worked on the mathematical analysis, the computer code, and the writing and checking of the manuscript.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge the contribution of Alan Lauder in bringing the issue of CH

This paper was edited by Tim Butler and reviewed by William Collins and one anonymous referee.