Methane emissions can be estimated using two main approaches: bottom-up and top-down modelling. Bottom-up methods model the physical and chemical processes of methane emission to create estimates of sector-level emissions, which can be distributed in space and time at a range of resolutions. However, methane emissions inventories have been shown to be inaccurate in some cases when compared to observations, which could lead to an incorrect representation of methane sources. For example, the spatial distribution of methane emissions from oil and gas sources in the Emissions Database for Global Atmospheric Research v.4.2 (EDGAR) was shown to be too heavily weighted towards locations where these fuels were distributed and used, rather than areas of fossil fuel extraction and production . Recent updates to this inventory (v.5.0 and 6.0) now include more detailed temporal and spatial profiles.

Top-down estimation of emissions uses observations of methane concentrations in the atmosphere and a chemical transport model to infer fluxes, often through Bayesian methods. These observations can be directly sampled from ambient air or remotely sensed. An estimate of emissions from a bottom-up model is typically used as prior information to inform the top-down inverse model during the inference of a posterior emissions distribution and to partition emissions to their source based on their location.

Measurements of additional trace gases can be used with a top-down approach to partition emissions, when these trace gases are co-emitted with methane from a particular source at a characteristic ratio. For example, carbon monoxide (CO) is co-emitted with methane during incomplete combustion , so it could be used to quantify emissions from biomass burning. Ethane (C2H6) is emitted by fossil fuel production and use and has no significant emissions from biogenic sources , so it can be used to quantify fossil fuel methane emissions. Methane isotopologue observations (e.g. 13CH4) can be utilised to apportion emissions in a similar method, by considering the ratio of isotopologues emitted from each source type . Studies have shown that when incorporating emission ratios or observations of additional gases into emissions quantification frameworks, the uncertainty in emission estimates of the primary gas can be reduced significantly when compared to a single gas model . However, these approaches always require a thorough understanding of the associated emission ratios, as inaccuracies in these values could introduce large posterior errors as discussed in or lead to emissions being incorrectly partitioned .

Previous studies have used ethane observations and emission ratios in a range of methods for the source partitioning of methane emissions. Typically, the enhancement in aircraft mole fraction observations of methane and ethane is compared to a bottom-up estimate of an ethane:methane ratio to assign a proportion of total regional methane emissions to a fossil fuel source e.g.. Similar methods have also been used more locally over cities or individual gas fields, where comparisons between literature and observed emission ratios have been used for source attribution of methane enhancements seen in individual plumes observed by ground-based vehicle-mounted instruments .

Ethane observations have also been incorporated more directly into joint inverse models, where emissions are optimised simultaneously to create emission profiles characteristic of each source type . Ethane and methane aircraft observations have also been optimised in a joint model to estimate surface methane fluxes, by comparing the observed ethane:methane emission ratios to a bottom-up estimate of the ratio .

Most of these previous works using observations of methane and ethane have used a fixed estimate of the ethane:methane emission ratio as a basis for the apportionment of methane emissions. Whilst some have considered trends in the ratio , most studies assume that this ratio is constant, which is unlikely to be true in most situations as the ratio can vary with location and over time, depending on the type of fossil 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.

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 emission 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/, last access: 9 February 2022) were used to represent emissions from both sectors. This test simulates a scenario 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 ). We assumed no ethane emissions from the non-FF sector.

We created 4-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 at Mace Head (MHD), Tacolneston (TAC), Bilsdale (BSD), Ridge Hill (RGL) and Heathfield (HFD) (see Fig.  for locations) by combining the synthetic emissions fields with atmospheric transport footprints made using the Met Office's Lagrangian Numerical Atmospheric-dispersion Modelling Environment (NAME) . See Appendix  for details on how NAME was run and for an example transport footprint. Synthetic ethane observations were created by combining FF ethane emissions (generated as FF methane emissions times the known uniform emission ratio of 0.075) with the transport model footprints. To mirror the DECC network, methane observations were created for all five sites, but ethane observations were only created for two sites, MHD and TAC. For both gases, Gaussian noise with a standard deviation equal to 10 % of each measurement was added to simulate instrument noise and model error.

Three sets of inversions were run.

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 hardwire emission ratios at potentially incorrect values, without considering their uncertainty.

In all tests, the inversion solved for emissions as a scaling of the a priori emission field, on a coarser grid than the native resolution of the transport model. The inversion estimated parameters for 49 regions over the UK, with the rest of the European domain split into four larger regions (see Fig.  for a representation of the inversion domain). The 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 emission field is known here and to represent a real-world situation where the prior mean may not necessarily be the true value, we used emission PDFs with 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 emission 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.

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.

Results from synthetic data tests, showing the impacts of a fixed and variable emission ratio, are summarised in Fig. . 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 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 which is 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 methane observations, the estimate for non-FF fluxes is therefore skewed in the opposite direction, with posterior mean fluxes smaller than their true value. The 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 (Fig.  rightmost points in both Fig. a and b) show that the potential errors introduced by assuming an incorrect ratio can be mitigated by including R as a variable parameter. In this case, posterior fluxes from both sectors converge closer to the true sector-level emissions, with a reduced posterior uncertainty compared to the methane-only model output but with larger uncertainty than if fixing the emission ratio. True emissions are not replicated exactly as there is some small dependence on the emissions prior. FF emissions are constrained by both methane and ethane observations, so most of the uncertainty in R is therefore carried forward into the estimates of non-FF fluxes.

We used the joint methane–ethane inverse model to create posterior estimates of the UK's monthly FF and non-FF methane emissions for 2015 to 2019. The methane-only model was run for the same period for comparison. Figure  gives the monthly posterior flux from the UK for the FF and non-FF sectors. Because total methane emissions are constrained by the methane observations, posterior total emissions from the methane-only and joint methane–ethane inversions are equal. The differences in results are shown in the partitioning of emissions from the two sectors.

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's 95 % 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 . These results are consistent with total emissions derived in previous inverse modelling studies using the same data .

A comparison between observed methane and ethane mole fractions and posterior modelled mole fractions from the joint methane–ethane model for two example months (April-May 2019) is given in Fig. . Percentage differences between observed and modelled mole fractions across the time series are given as histograms for each site. Baseline mole fractions from all methane sites (dashed lines in Fig. ) are consistent with those from the background site (MHD). Figure  shows a scatter plot comparing observed and modelled posterior mole fractions from the joint methane–ethane model for the full time series from 2015–2019. There is generally a good fit to observations, but the model does not always fit to the largest methane peaks from TAC, RGL and HFD, the three sites closest to areas of high emissions. Comparisons between observations and an a priori estimate of mole fractions made by combining the a priori map of fluxes with the transport model are also given in Fig. . Overall, there is an improved fit to both methane and ethane observations in the posterior estimate of mole fractions produced by the inverse model.

Model uncertainty for methane mole fractions converged at similar values for all sites across the time period, with a mean value of 7.75 ppb overall (with 75 % of all mean model error values between 5 and 10 ppb). Due to the high peaks and troughs in ethane observations, the model consistently attempted to converge the ethane total model uncertainty at the upper bound of its prior uncertainty range.

The a priori emissions and average spatial distribution of posterior emission scaling factors for 2019 are shown in Fig. . The ethane observations only indirectly constrain the much larger non-FF emissions, so there is little difference in the spatial distribution of non-FF emissions between the methane-only and joint methane–ethane inversions. However, the joint methane–ethane inversion suggests a different distribution of FF emissions, where emissions are scaled down in most locations, apart from a few regions with large positive scaling which often correlate with heavily populated areas (e.g. the West Midlands, London and south coast of England).

Across the whole period of study, posterior mean ethane:methane emission ratios varied across the domain between 0.009 and 0.2 (we discuss the implications of these posterior ratio values in Sect. ). For example, in July 2019, approximately 20 % of R values converged with clear Gaussian posterior distributions, suggesting a strongly correlated relationship between the two gases that the model was able to use to inform the posterior distribution of both the methane flux and ethane:methane ratio in that area. Posterior emission ratio PDFs in the remaining areas of the domain were more uniform, indicating a weak constraint from the observations in those areas. Mean posterior emission ratios for two different periods are shown in Fig. . Some regions, for example, central south England, London and the West Midlands, often converge with high emission ratios close to the upper bound of the a priori PDF. Grid cells where the posterior uncertainty in R (defined as posterior 95th percentile divided by posterior mean) is less than 50 % of the prior uncertainty (prior 95th percentile divided by prior mean) are highlighted in Fig. , showing areas where the observations were most able to constrain R.

We compare posterior ethane:methane emission ratios from the hierarchical inverse model to independent calculations of this ratio, made during a range of mobile observation studies. These independent datasets are sparse, and thus only a limited validation can be performed.

Ethane and methane airborne observations were made during flight C191 of the Facility for Airborne Atmospheric Measurements (FAAM) campaign over North Sea oil and gas fields on 29 July 2019, as part of the Methane Observations and Yearly Assessments (MOYA) project. A mean average ethane:methane emission ratio of 0.088 (range 0.04–0.18) was calculated from four different plume observations using two different methods, Gaussian plume fitting and linear regression. See Appendix  for information on how these ratios were calculated. In addition, the Royal Holloway mobile laboratory sampled air around potential shale gas production sites for baseline monitoring . They observed ethane:methane ratios of approximately 0.06 from local gas leaks, on 27 February 2018. Comparisons between these individual plume estimates and our monthly posterior mean emission ratios are shown in Fig. a and 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 are 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 limits the scope of comparison available with our monthly model estimates of these ratios. Partitioning of the domain into coarse grid cells could also impact the comparison, as ratios are likely to be heterogeneous within each of these regions. 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.

As in the synthetic data test, we tested the impact of using a fixed ethane:methane ratio on 1 month of posterior UK sectoral methane fluxes (Fig. ). We ran the model for 1 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 when using a fixed ratio of 0.075 (approximately the mean emission ratio from a range of studies, e.g. Table ). As the rightmost points in both Fig. a and b show, the joint methane–ethane inversion with a variable emission ratio samples the uncertainty in the emission ratio and propagates this into the posterior flux estimates. Uncertainties in the posterior flux estimates are therefore higher for both sectors than when using a fixed emission ratio but capture the overall uncertainty in the system more accurately.