Methane and the Warming Blame Game

1. Introduction

Assessments of the climate impact of national emissions has long been recognised as an important input to climate policy [1]. Many studies have allocated countries contributions to climate change, focussing on historical warming, likely future warming, climate damages, or climate extremes [2,3,4,5,6,7,8,9,10,11]. Foremost among the motivations for this effort is the foundational UNFCCC principle of “common but differentiated responsibilites” for climate change [12]. Historical responsibility may inform policymakers considerations of equitable mitigation effort, climate justice, climate finance, obligations under loss and damage, or possible future liability and compensation claims [6,13,14,15]. The present study concerns causal attribution for warming only. Evaluating responsibility requires additional normative judgement  [6,8,16].

In 1997, Brazil proposed that national mitigation targets should be linked directly to historical warming responsibility [1]. Apart from political opposition, the Brazilian proposal faced methodological challenges [2,17,18]. As noted by den Elzen et al [2] “calculation of regional responsibility is not straightforward, because the climate system is not linear”. An obvious source of non-linearity is convexity in forcing-concentration relationships of the major greenhouse gases, carbon dioxide (CO₂), methane (CH₄) and Nitrous Oxide (N₂O) [19,20]. This difficulty was greatly reduced with the identification of Transient Climate Response to Cumulative Emissions (TCRE) as a key warming metric for CO₂ [21,22]. This meant that CO₂-induced warming could be allocated simply based on a country’s cumulative CO₂ emissions [4]. However, this simplifciation is not available for CH₄ and N₂O or short-lived climate forcers (SLCFs). Matthews et al [4] allocated CH₄-induced global warming according to the country’s share of cumulative CH₄ emissions. More recently, this ad hoc treatment of CH₄ was replaced by a warming metric-based approach using GWP* [23,24] to account for atmospheric lifetimes [7]. Linearised metrics such as GWP* have limitations when applied over long time periods, however [25].

Most allocation studies use simple climate modelling, isolating a country’s warming impact using a marginal or “leave-one-out” (LOO) method [3,5,8,10,26]. Simple Climate Models (SCMs) capture the effect of short CH₄ lifetimes and convexity in a consistent way. Recently, Li et al [8], applied an SCM and LOO with a range of equity principles to show that highly developed countries had already exceeded their “fair share” of a $1.5$ °C warming budgets before 1990. While intuitively reasonable, the marginal method may underestimate a country’s warming impact [18]. Thus, despite great progress, some of the long-standing methodological difficulties of warming attribution highlighted in the wake of the Brazilian Proposal remain unresolved.

The first and most important step in resolving this problem is to recognise that warming allocation is not a purely physical science problem. The best that can be achieved is a method of attribution that reasonable parties would agree to i.e., where no party is obviously disadvantaged. In other words, warming allocation should be treated as the outcome of a co-operative game. A formal solution to this problem is provided by warming Shapley values in Section 2 [27]. Warming allocations for country groupings are computed in Section 3. Implications for policymakers in countries with high shares of CH₄ in their emissions profiles are discussed further in Section 4.

2. Data and Methods

Allocating shares of warming to individual countries involves a number of methodological perspectives [5]. Paris agreement temperature ceilings are expressed relative the early industrial period 1851-1900 [28] and more recent allocation studies use this baseline along with territorial emissions accounting [7,8]. An SCM and marginal “leave one out” (LOO) warming allocation method is also normally used [2,3,5,6,8,10,26]. There is less consensus which climate drivers should be included in the analysis (Section 2.3)

2.1. Cooperative Games

Warming allocation requires a counterfactual approach of some kind. The warming impact of country i, $gsa{t}_i$, could be computed in an SCM by leaving out anthropogenic emissions from all other countries except i. This “leave-one-in” (LOI) approximation overestimates warming due to CH₄ and N₂O. For example, if i is a small country, atmospheric concentrations of greenhouse gases calculated in the unconstrained SCM remain close to their early-industrial values. Convexity in forcing-concentration relationships means the forcing effect of the country’s emissions are overstated leading to an overestimate of warming.

The “true” warming of country i, $\delta GSA{T}_i$ can be identified with the warming Shapley value, often used in economics to solve resource allocation problems of precisely this type [27]. $\delta GSA{T}_i$ is an appropriately weighted sum of marginal contributions over all possible country coalitions S:

$$\delta \delta GSA{T}_i=\sum _{S\subseteq {\mathcal{N}}_i}{w}_S\left(gsa{t}_{S\cup i}-gsa{t}_S\right),$$

(1)

${\mathcal{N}}_i$ is the set of all a countries excluding i and the sum is over all unordered subsets of ${\mathcal{N}}_i$. Here $gsa{t}_S$ is the warming contribution of the emissions from coalition S computed in a climate model. The weights $w_S$ are ${\displaystyle \frac{N_S!\left(N-N_S-1\right)!}{N!}}$ where $N_S$ is the number of countries in coalition S and N is the total number of countries. The weights satisfy ${\sum }_S{w}_S=1$. The sum includes the null coalition of no countries.

Unlike the LOO "leave-one-out" or LOI methods, Equation 1 has the completeness property that ${\sum }_{i=1}^N\delta GSA{T}_i=gsa{t}_{\mathcal{N}}$ i.e., global warming calculated in the SCM as warming from the “grand coalition” $\mathcal{N}$, is equal to the sum of the contributions from each country. Warming Shapley values therefore represent the reasonable causal attribution of the total observed warming impact $GSAT$ to each country without any missing or excess warming. Convexity of forcing-concentration relationships suggests that warming Shapley values $\delta GSA{T}_i$ lie in the interval $({\mathrm{LOO}}_i,{\mathrm{LOI}}_i)$. This idea is explored further in Appendix C.

2.2. UNFCCC Groupings

Computationally exact evaluations of Equation 1 require $\sim 2^N$ model evaluations which is not practical for countries ($N\approx 200$). In reality, climate negotiations tend involve country groupings not individual countries. There are about 20 UNFCCC negotiating groups, reducing the problem to a few million coalitions.

UNFCCC groupings differ greatly in their historical responsibility for climate change. For example, the Umbrella Group, including the USA and other historically large industrial emitters, accounts more about one third of current warming. At the other end of the scale, Small Island Developing States (SIDS) is an influential group of 36 small countries whose combined impact on warming is of order 1 $\mathrm{m}$°C. A complication is that some countries are members of more than one UNFCCC grouping. Here, countries are assigned to the smallest group of which they are a member. For example, Brazil is a member of the 4-member BASIC and ABU (Argentina-Brazil-Uruguay). As the latter is smaller, Brazil is assigned to ABU rather than BASIC, which then consists of China, India and South Africa only. 37 countries not obviously aligned with any UNFCCC grouping are assigned to Non-Group Members. Emissions in this grouping are dominated by Turkey and Taiwan. International aviation and shipping is assigned its own group. In some instances, smaller groups are coalesced into Non-Group Members to reduce computational burden. The specific groupings used for this study are provided in the Zenodo data repository for this paper [].

2.3. Emissions Datset

This study uses country-level emissions data from the Community Emissions Data System (CEDS) [29]. This dataset covers the major greenhouse gases (Fossil and Industrial (FFI) CO₂ sources, CH₄ and N₂O) and air pollutants including SO₂. Pre-1970 CH₄ and N₂O emissions absent from CEDS were imputed using a global estimate scaled by the country’s share in 1970 [30]. CEDS does not cover F-gas emissions that account for about 1% of global warming. This omission has no material effect on the conclusions of this study. Land-use change (LUC) emissions are sometimes excluded in attribution studies due to “scientific and normative” issues [8]. Only FFI-CO₂ is included in the results of Section 3 but land-use change (LUC) emissions are included in Appendix B using the dataset from Jones et al [7]. Emissions uncertainties at country-level may be considerable, particularly for non-CO₂ gases [3,31].

2.4. SCM and Model Ensemble

The process-based SCM Hector v3.2  [32] is a suitable choice for this study because of its speed (C++ implementation), flexibility and elegant R interface. 256 Hector model configurations (ensemble) were generated consistent with observed 2003-2022 warming 1.03 ± $0.08$ °C [28]. This was done by screening a large parameter space of normally and lognormally ($ECS$,$Q_{10}$) distributed model parameters consistent with this temperature distribution [33]. Medians and mean absolute deviations (MAD) of the resulting model ensemble are shown in Table 1. The screening process induces correlations between Hector parameters, e.g., a significant positive correlation between equilibrium climate sensitivity $ECS$ and aerosol forcing parameter $AEROSCALE$.

The 256-member model ensemble is provided in the Zenodo data repository for this paper [34]. An R package implementation of Section 2.1 and Section 2.4 with all necessary data to run the allocation is available from the GitHub site at xxxxxxxxxxxx.

3. Results

1850-2022 warming allocations were computed for fourteen UNFCCC negotiating groups plus International Aviation and Shipping. Uncertainty in warming Shapley values is found by separate evaluation of Equation 1 for each of the 256 Hector model configurations. It was verified that $gsa{t}_{\mathcal{N}}$ equals the sum of warming Shapley values for each configuration. The results are shown in Table 2.

The North American dominated Umbrella Group has the largest allocation ( 280 $\mathrm{m}$°C), followed by EU27 ( 120 $\mathrm{m}$°C) and BASIC ( 110 $\mathrm{m}$°C). Uncertainty in $\delta GSAT$ is highest for groupings with significant SO₂ emissions such as BASIC and OTHER, reflecting aerosol forcing uncertainty in Table 1. International Aviation and Shipping is likely to have a small net cooling allocation (-10 ± 10 m°C), a consequence of large historical SO₂ emissions from maritime fuels coupled with thermal inertia of the climate system. The effect of LUC-CO₂ estimates n these results are included in Table A1, Appendix A. In that case BASIC overtakes EU27 as the second largest contributor to global warming (UG 340 $\mathrm{m}$°C, BASIC 140 $\mathrm{m}$°C, EU27 120 $\mathrm{m}$°C).

Table 2 also shows LOO allocation values. $\delta GSAT\lesssim LOO$ in most cases as anticipated. $LOO>\delta GSAT$ can also arise as a consequence of a strong aerosol masking effect. The sum of LOO values deviates from $GSAT$ by -3.2±1.4% (median value ± MAD uncertainty). This number is sensitive to aerosol masking and increases to -3.9±1.1% when model configurations are restricted to below-median values of $AEROSCALE$ as discussed in Table A1, Appendix A .

LOO underestimates the warming contributions of large historical emitters such as the Umbrella Group (-3.3±0.6%) and the EU27 (-2.7±1.1%). BASIC shows a smaller deviation (-0.5±3.3) but with high uncertainty due to aerosol masking. Restricting to below-median values of aerosol forcing, LOO deviations increase slightly for Umbrella Group (-3.4±0.5%) and EU27 (-3.1±0.7%) and by a greater amount for BASIC (-2.2±1.9%). CH₄ accounts for ≈ 18% of UG warming, suggesting that LOO underestimates CH₄-induced warming by ≈ 19%. The accuracy of LOO for major historical industrial emitters is largely explained by their low shares of CH₄ emissions relative to CO₂. However, groups such as ABU or ALBA show larger discrepancies.

3.1. Methane

Warming allocations for individual countries with significant historical CH₄ emissions are of particular interest. These can be found by separating the countries from their respective UNFCCC groups. Here, warming Shapley values were evaluated with a grand coalition consisting of 13 UNFCCC groups, IAS, New Zealand, Uruguay and Ireland. These countries were selected because they have high shares of CH₄-induced warming since 1850, estimated to be 78% (URY), 71% (NZL) and 38% (IRL) when LUC emissions are excluded. Table 3 shows their warming allocations, along with three country groups from Table 2 with high shares of CH₄-induced warming, 77% (ALBA), 53% (AS), 69% (ABU).

Table 3 shows discrepancies between LOO and warming Shapley values in the range -8% to -14%. These numbers are insensitive to aerosol masking as can be seen from Table A2, Appendix A. The LOO deviations are consistent with an underestimate of CH₄-induced warming by 23% (Ireland), 20% (New Zealand) and 18% (Uruguay). The precise value is likely to depend on relative antiquity or recency of the country’s CH₄ emissions.

Many of the countries in Table 3 have large LUC-CO₂ emissions post-1850 due to deforestation. Including these emissions has a large effect on the warming allocations as seen in Table A4, Appendix B. The Brazil group warming allocation increases from 39 $\mathrm{m}$°C to 92 $\mathrm{m}$°C. New Zealand’s allocation increases from $2.3$ $\mathrm{m}$°C to $3.7$ $\mathrm{m}$°C. LOO is now more accurate because CH₄-induced warming is a proportionately smaller, and none of the deviations exceed -10% in Table A4.

4. Discussion

Methane is a short-lived-climate-forcer (SLCF) with lifetime ≈ 12 years in the current atmosphere [24,35]. It is also a major greenhouse gas, with a convex (square-root) relationship between radiative forcing and atmospheric concentration [19,20]. Recent scientific and climate policy [8,23,24,25,26] work emphasises CH₄-induced warming, rather than CO₂-equivalents, to reflect the distinct climate physics of this gas. For CO₂, cumulative emissions and transient warming impact are interchangeable because they are simply related through TCRE. However, an SCM [32,36] is needed to evaluate the warming impact of CH₄ and other non-CO₂ drivers particularly over long time periods. SCMs relate radiative forcing to subsequent warming with fast components and slow components associated with multiple equilibration timescales [37].

The results of Section 3 illustrate the 1850-2022 warming allocation problem using Hector v3.2 [32] and an extended CEDS dataset [29]. LOO is calculated as the reduction in global warming when a country’s emissions are omitted. It an extensively used approximate allocation method [2,3,5,8,9,10,26]. However, Table 2,Table 3, Table A5 and Table A6 illustrate the fact that LOO values sum to less than global warming calculated in the grand coalition of all countries. This is undesirable because it means that 23 $\mathrm{m}$°C of the warming in Table 2 is unallocated, for example. Li et al [8] used a simple re-scaling i.e., if 3% of total warming is missing then all LOO country allocations are increased by 3%. The missing warming is equivalent to ≈ 50 $\mathrm{G}$$\mathrm{t}$ CO₂, far larger than the cumulative emissions of most countries.

Methane’s radiative efficiency was 55% higher in 1850 compared to today because atmospheric concentration of was only ≈ 42% of its current value [20]. The means that the warming impact of a CH₄-emitting country is lowered by the emissions of all other countries. A “leave-one-in” (LOI) allocation method removes this effect by neglecting the contribution of all other countries to increased concentration. In some respects, LOI is an equally plausible allocation method to LOO. However, the sum of LOI allocations is greater than calculated global warming. Therefore, neither LOO nor LOI can be regarded as a satisfactory solution of the warming allocation problem. This is discussed further in Appendix C.

Section 2.1 provides a formal solution to the causal attribution problem in terms of warming Shapley values. The sum of attributions now equals the global value with no unallocated warming. LOO and Shapley values for UNFCCC groupings are compared in Table 2,Table 3, Table A5 and Table A6. The results show that deviations are not uniform but are larger in countries with a higher share of CH₄ emissions. The numerical results suggest that LOO is accurate for CO₂-induced warming but underestimates CH₄-induced warming by ≈ 20%. For example, about 18% or the Umbrella Group’s warming is caused by CH₄. This implies LOO underestimates warming by -3.6% in good agreement with Table 2. The success of LOO is largely explained by the fact that CO₂ is the dominant source of warming, particularly when LUC-CO₂ is included, and that linear TCRE relationship is accurate over the full historical period. Larger deviations are expected when CH₄ emissions dominate but these never exceed ≈ 20%.

Methane present climate policy-makers with an acute set of challenges. The findings of Section 3 place countries such as Ireland or New Zealand somewhat further into carbon debt, while Brazil and Uruguay are closer to exhausting their warming budget than previously thought [8]. Furthermore, national climate policy frameworks, such as those based on carbon budgets, often implicitly aim to define an “acceptable” national share of global warming. This study suggests that warming Shapley values are an appropriate tool for such national assessments. The implications of this approach in future mitigation scenarios, particularly extended to 2100, require investigation. A robust allocation method is likely to benefit future policy.

Two points should be emphasised. Firstly, this study considered the global warming causal attribution problem relative to the early industrial period. It has not considered responsibility which may involve further normative judgements[16]. Only in a consequentialist or strict liability approach are these two concepts equivalent [14]. Secondly, this paper has no implications for the warming impact of methane at global level.

In conclusion, despite the apparent failure of the 1997 Brazilian Proposal [1], attribution of warming impact is likely to remain an important driver of future climate policy. Warming Shapley values to resolve the CH₄-induced warming allocation problem with no unallocated or excess warming.