Quantifying fossil fuel methane emissions using observations of atmospheric ethane and an uncertain emission ratio

. We present a method for estimating fossil fuel methane emissions using observations of methane and ethane, ac-counting for uncertainty in their emission ratio. The ethane:methane emission ratio is incorporated as a variable parameter in a Bayesian model, with its own prior distribution and uncertainty. We ﬁnd that using an emission ratio distribution mitigates bias from using a ﬁxed, potentially incorrect emission ratio and that uncertainty in this ratio is propagated into posterior estimates of emissions. A synthetic data test is used to show the impact of assuming an incorrect ethane:methane emission ratio and 5 demonstrate how our variable parameter model can better quantify overall uncertainty. We also use this method to estimate UK methane emissions from high-frequency observations of methane and ethane from the UK Deriving Emissions linked to Climate Change (DECC) network. Using the joint methane-ethane inverse model, we estimate annual mean UK methane emissions of approximately 0.27 (95% uncertainty interval 0.26-0.29) Tg y − 1 from fossil fuel sources and 2.06 (1.99-2.15) Tg y − 1 from non-fossil fuel sources, during

fuel source and the type of extraction or processing techniques being used. Incorrectly assuming that this ratio is fixed could introduce errors into any sector-level emissions estimates, and could alter the inference of emission trends.

Methods
In this work, a top-down hierarchical Bayesian inverse model uses observations of a secondary trace gas and its emission ratio with respect to a primary gas, to solve for emissions of the primary gas at a sectoral level. Uncertainties in the emission ratio 90 between the primary and secondary gases are statistically propagated into the emissions distributions through the hierarchical framework. The principle of this method is described below.
A forward model (Eq. 1) links observed mole fractions of a gas y to its emissions, x via a linear atmospheric chemistry and transport model H and model-measurement error . x is inferred through an 'inversion' of the forward model using Bayesian statistics. 95 y = Hx + . (1) Prior probability density functions (PDFs) must first be assigned to the parameters. To reduce the subjectivity involved when choosing these PDFs, additional 'hyper-parameters' can be included in a hierarchical Bayesian framework, which place distributions on these uncertain parameters rather than imposing them as fixed values. Ganesan et al. (2014) found that by including uncertainty in parameters (such as model-measurement error) as hyper-parameters, one could better propagate uncertainties 100 into the posterior estimate of emissions. To use these hyper-parameters in the inverse model, Bayes's theorem is extended to include the joint distributions between primary and secondary parameters θ (Ganesan et al., 2014), ρ(x|y) ∝ ρ(y|x, θ) · ρ(x|θ) · ρ(θ).
There is no analytical solution to maximise Eq. 2 so a Markov Chain Monte Carlo (MCMC) method is used produce a posterior distribution containing possible solutions for each of the parameters. This is an iterative method that randomly To solve for emissions from separate sources, the forward model is expanded to include emissions of the primary gas from two sectors A and B: Observations of the secondary gas and its emission ratio are incorporated into this model as follows. Assuming that Gas 2 is only co-emitted from sector A, with an emission ratio R relative to Gas 1, the forward model for Gas 2 is expressed as: In an application where a particle dispersion model is used to provide transport model 'footprints' (as is the case for the 120 remainder of this work) and when analysing observations of gases with long atmospheric lifetimes, atmospheric transport of both gases can be assumed to be equivalent. Therefore, the linear transport model is the same for both gases and is represented from this point onward as H.
Combining the two forward models, Eq. 3 and 4, produces a joint model where both gases inform the estimate of emissions:

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Without a framework that can consider the uncertainty in the emission ratio, R would be imposed as a fixed parameter into the sensitivity matrix at this point. In our work, the emission ratio R is treated as a variable parameter, requiring the expansion of Bayes's theorem as discussed above and shown in Eq. 6. Model-measurement uncertainty (σ y ) is also included as a hyper-parameter, again with its own prior PDF and uncertainty,

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A MCMC process is used to produce posterior distributions for both the emissions and emission ratio parameters. In this study, we used this model to estimate methane emissions from fossil fuel (FF) and non-fossil fuel (non-FF) sources, using ethane as the secondary gas. However, this model framework is highly adaptable and could be used with other tracers, for example, methane isotopologues.

Synthetic data experiment 135
To investigate the influence of the ethane:methane emission ratio on posterior estimates of methane emissions, we carried out model runs as described above, using synthetic data generated from a known emissions field and a known emission ratio.
These tests used a two-sector model of identical UK 'fossil fuel' (FF) and 'non-fossil fuel' (non-FF) fluxes, with the same magnitude and spatial distribution of emissions. Total UK methane emissions from the UK National Atmospheric Emissions Inventory (NAEI) (https://naei.beis.gov.uk/) were used to represent emissions from both sectors. This test simulates a scenario 140 when fluxes from both sectors are inseparable by spatial differences alone. For these synthetic data tests we did not consider background levels of methane (i.e., the contribution to the total mole fraction from emissions outside the UK) and only tested the ability of the inversion to return the regional (UK) emissions field.
The a priori ethane:methane emission ratio, R, was assumed to be uniform across the whole domain, with a value of 0.075. This is the approximate mean ethane:methane emission ratio from natural gas sources in Europe, (Table 1). We assumed no 145 ethane emissions from the non-FF sector.
We created four-hourly synthetic methane observations at four UK tall tower sites and one coastal site in the UK Deriving Emissions linked to Climate Change (DECC) network (Stanley et al., 2018;Stavert et al., 2019)  Three sets of inversions were run: 1. Joint methane-ethane inversions where the emission ratio was fixed at values ranging from 0.5-1.5 times the true value.
This test simulates studies that hard-wire emission ratios at potentially incorrect values, without considering their uncertainty.

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2. Joint methane-ethane inversions where the emission ratio is a variable parameter with its own PDF representing the range of uncertainty in the emission ratio. The emission ratio prior PDF was given a uniform distribution ranging from 0.5-1.5 times the true value. This simulates the situation where uncertainty in the emission ratio is built into the framework. In all tests, the inversion solved for emissions as a scaling of the a priori emissions field, using basis functions to coarsen the native grid resolution of the transport model onto a 7×7 grid over the UK, with the rest of the European domain split into four larger regions. Ethane:methane emission ratios were solved for at the same resolution as methane emissions. Gaussian distributions were used for emissions parameters in these synthetic data tests. As the true emissions field is known here and 170 to represent a real-world situation where the prior mean may not necessarily be the true value, we used emission PDFs with Molar ratio (median and range) Source type Reference 0.045 (0-2.76) Global conventional oil and gas composition (Sherwood et al., 2017) 0.038 (0.001-1.0) European raw gas composition (Visschedijk et al., 2018)  a priori means equal to 125% and 75% of their true values for the FF and non-FF sectors, respectively, to simulate slightly incorrect a priori emissions fields (i.e. correct total emissions but incorrect partitioning). Both sectors were given a standard deviation of 50% of their true values. Model-measurement uncertainty was fixed at 10% of the mean pseudo-observation value for both methane and ethane.

UK methane emissions case study
We used the methane-only and joint methane-ethane inverse models to estimate monthly UK methane emissions from 2015 to 2019. We also tested the impacts of a fixed emission ratio on posterior flux estimates and investigated the propagation of uncertainties through the model when applying an uncertainty to this emission ratio.

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Methane observations were used from the five current UK DECC network sites as discussed in Sect. 2.1. Mole fraction observations of methane were made using Cavity Ringdown Spectroscopy (CRDS) instruments Picarro G2301 and G2401, calibrated using daily standard measurements, and are reported on the WMO-X2004A scale (Stanley et al., 2018;Stavert et al., 2019).
Ethane observations were made at two DECC sites, MHD and TAC, using a Medusa gas chromatography-mass spectrometry (GCMS) instrument (Prinn et al., 2018). Calibration of ethane observations is currently based on the provisional SIO-p 185 (Scripps Institution of Oceanography) scale. Frequent comparisons between Advanced Global Atmospheric Gases Experiment (AGAGE) ethane measurements (for example those made at the MHD site) and those reported by the National Oceanic and Atmospheric Administration (NOAA) at the same site, but using an independent calibration scale, show no significant long-term bias. A complete description of the ethane calibration employed here is given in Mühle et al. (2007).
Observations from the highest inlet at each tall tower site were used to reduce the impact of local fluxes and to increase the Observations of both gases were filtered to remove points when local emissions are likely to bias results, using similar methods to as described in Lunt et al. (2021). Measurements made at times when the tower inlet was sampling air from above 195 the planetary boundary layer were removed. Measurements were also removed when more than 10% of the area-integrated sensitivity at the site was from the 25 grid cells surrounding the site (i.e. local sources). Remaining observations were averaged into four-hourly periods. On average 40 (range 18 -69) % of observations were filtered each month.
The NAME model was used to produce transport 'footprints' for all observation sites. See Appendix A for more detail on how NAME was run and for an example footprint for the network of sites. As methane's lifetime of around a decade is long 200 compared to the timescale of transport within the regional domain (on the order of days), we assumed that atmospheric loss is negligible and that only transport influences the relationship between surface emissions and atmospheric concentrations.
Ethane has a shorter lifetime than methane (from approximately 2 months in summer to 6 months in winter (Helmig et al., 2016)). However, we found atmospheric loss of ethane on a 2-month timescale to have a negligible effect on the footprints over the UK and therefore we used the same transport footprints for both gases.

Model parameters and a priori PDFs
The a priori estimate of UK methane emissions from each sector was taken from the UK Greenhouse Gas (UKGHG, Levy, 2020) model of spatially and temporally disaggregated emissions, which is based on national, annual totals from the UK National Atmospheric Emissions Inventory (NAEI). Emissions from the sectors 'energy production', 'offshore', 'industrial and domestic combustion', 'industrial processes', 'road transport' and 'other transport' were summed to form an a priori  (Table 1). Results from synthetic data tests, showing the impacts of a fixed and variable emission ratio, are summarised in Fig. 1.
Because there is no spatial distinction between sources in the prior and because the total posterior emissions are the true total, the methane-only (one gas) model returns the prior mean emissions for each sector. This lack of sectoral information from the 245 prior is also expressed in the relatively large posterior uncertainties for both sectors. In the joint methane-ethane inversion, there is more information available for the model to constrain emissions from each source. However, when the emission ratio R is fixed in the inversion at an incorrect value, the sectoral partitioning of emissions is also incorrect but is derived with high confidence. If the emission ratio is fixed at a value 50% lower than its 'true' value, posterior mean FF fluxes are estimated to be over 80% larger than their true value. As total emissions are constrained by the 250 methane observations, the estimate for non-FF fluxes is therefore skewed in the opposite direction, with posterior mean fluxes smaller than their true value. 95% confidence intervals on FF emissions in this test are too small to be visible on this scale, due to the high level of constraint from the fixed emission ratio. This synthetic data test highlights how errors could be introduced when using a fixed ethane:methane ratio that does not reflect the true uncertainty in the parameter.
Results from the joint methane-ethane model that considers the uncertainty in the emission ratio ( The joint methane-ethane inversion finds that emissions from FF sources contribute on average 15% less to total methane emissions than in the methane-only inversion. This is balanced by a proportional increase in non-FF emissions. The impact on posterior uncertainty varies across the period, with an average 15% reduction in the size of the posterior FF flux 95% 270 uncertainty interval, which increases up to 35% for some months. Our results show declining emissions over the time period, which is largely driven by emissions from non-FF sources. Annual mean posterior flux estimates are given in Table 2. These results are consistent with total emissions derived in previous inverse modelling studies using the same data (Western et al., 2020;Lunt et al., 2021).  Year FF CH4 (Tg y −1 ) Non-FF CH4 (Tg y −1 ) Total CH4 (     our monthly posterior mean emission ratios are shown in Fig. 5 (a,b). Both independently measured ratios are approximately consistent with the emission ratios estimated in this work. However, observations from Flight C191 are located far from the DECC observation network so our estimates of emission ratios over the North Sea have likely to have larger uncertainties than those closer to the towers.
As most independent observations of ethane:methane ratios over the UK have only been taken over short time periods, this 320 limits the scope of comparison available with our monthly model estimates of these ratios. Partitioning of the domain into coarse basis functions could also impact the comparison, as ratios are likely to be heterogeneous within a basis function region.
As we are focused on average emissions over the month, this should not significantly affect our results, but could limit the ability for further validation of our posterior emission ratios.
3.4 Impact of a fixed ethane:methane emission ratio on UK methane fluxes As in the synthetic data test, we tested the impact of using a fixed ethane:methane ratio on one month of posterior UK sectoral methane fluxes (Fig. 6). We ran the model for one month (April 2019), but used a range of spatially uniform emission ratios (R). As in the synthetic data test results, posterior fluxes are strongly influenced by a fixed emission ratio. For example, by assuming a fixed ratio scaling factor of 0.5 (which equates to an emission ratio of approximately 0.04, similar to literature values for natural gas fossil fuel methane sources) the estimate of mean posterior FF flux is approximately 60% higher than 330 when using a fixed ratio of 0.075 (approximately the mean emission ratio from a range of studies e.g. Table 1). As the rightmost points in both Fig. 6 (a,b)  hour, back in time for up to 30 days, and quantified their interaction with the surface and their exit locations/times from the study domain. These hourly footprints were then averaged into four-hourly footprints, to match the averaging of the observations.
Meteorological data from the UK Met Office's Unified Model (Walters et al., 2014) and a nested UK-specific 1.5 km horizontal resolution meteorological product were used to drive NAME at a one-hourly temporal resolution over the UK and at threehourly resolution over the rest of the domain. The output was stored at 0.23 • × 0.35 • spatial resolution over a domain spanning 410 -97.9 • to 39.7 • E longitude, 10.7 • to 79.3 • N latitude. This process was carried out for each observation made at each site, to build up a field of emissions sensitivity for the whole domain.  2021) showed that consistency is expected between these two methodologies.