The atmospheric mass balance of 12CH4 can be written as 1ddt12C=∑s12Fs-∑l12kl×12C×[l]=∑s12Fs-12C∑lLl, where Fs denotes the surface flux from source category s, and l denotes species contributing to chemical loss, namely Cl, OH and O(1D). We classify all sources into three isotopically distinct source categories, namely pyrogenic, fossil, and microbial CH4. While the upland soil sink of methane is included in the sources in conventional methane modeling, for reasons described in Sect.  we have modeled it as a fourth loss mechanism. The combination 12kl×[l] can be denoted as a loss rate or inverse lifetime Ll due to species l. For 13CH4, we can write a corresponding equation, 2ddt13C=∑s13Fs-13C∑lαlLl, where αl=13kl/12kl. Using the definition of δ and denoting the atmospheric isotope ratio (signature of source s) as δa (δs), respectively, we can substitute 13C=12Crstd(δa+1) and 13Fs=12Fsrstd(δs+1) and get 312Cddtδa=∑sδs-δa12Fs-12C(δa+1)∑lϵlLl, where ϵl=αl-1 and rstd=0.0112372 is a predefined standard ratio

There is no single unique value of rstd in the literature. Currently, rstd=0.011180 is used by most measurement laboratories, while values of 0.011117 and 0.011125 have also been reported recently. However, the true value of rstd impacts neither our formulation nor our results, as long as a single value is used consistently.

We use the TM5-4DVAR inversion framework , which has been used to estimate surface fluxes of CO, CO2, and CH4 in single-tracer inversions, as well as source-specific CO2 fluxes in multi-tracer inversions . At the heart of the framework is the TM5 offline tracer transport model and its adjoint, driven by ECMWF ERA-Interim reanalysis winds and run globally at 3∘ × 2∘ with 25 vertical layers defined by sigma–pressure hybrid coordinates. Two tracers are simulated in TM5: total methane or C of Eq. () and the artificial tracer Cδa′ of Eq. (). Measurements of CH4 are directly compared to modeled values of C, while measurements of δ13C are first converted to δa′ and then to Cδa′ by multiplying with values of CH4 mole fractions measured in the same air samples.

TM5-4DVAR minimizes a cost function J as a function of surface fluxes x (the set of all Fs of Sect. ) by balancing fits to atmospheric observations y with deviations from the prior fluxes x0, 12J(x)=12(Hx-y)TR-1(Hx-y)+12(x-x0)B-1(x-x0), where H is the transport, chemistry, and observation operator connecting surface fluxes with atmospheric measurements, and R and B are the error covariances of Hx-y and prior fluxes, respectively. Our formulation of R contains both the analytical measurement uncertainty and a model representativeness error proportional to local tracer gradients . For each source type (pyrogenic, fossil, and microbial), the diagonal elements of B per time step and lateral grid cell are proportional to the prior flux, or f×x0. Off-diagonal elements of B are constructed assuming an exponential decay of the prior error correlation in space and time with source-specific scales λ and τ, respectively. The values of f, λ, and τ for the different source types are given in Table . While there is no unique way of specifying these parameters, our choices yield reasonable 1σ prior uncertainties in global total microbial, fossil, and pyrogenic emissions of ∼25 Tg yr-1 (∼7 %), ∼30 Tg yr-1 (∼17 %), and ∼2 Tg yr-1 (∼6 %), respectively. The 1σ uncertainty in the annual global total CH4 emission comes to ∼40 Tg yr-1 (∼7 %) with these choices. The cost function J of Eq. () is minimized over 50 iterations by a conjugate gradient minimizer utilizing the Lanczos algorithm .

In TM5-4DVAR, we calculate J(x) of Eq. () with TM5 and ∂J/∂x with its adjoint. A traditional variational estimation would require us to run the forward and adjoint models multiple times over the entire period over which we want to estimate fluxes. However, these model runs require a significant amount of time, and iterations must be performed in succession. For example, at our 3∘ × 2∘ resolution, TM5 simulates a decade in 8 h. So to perform an inversion over 2 decades with 50 iterations (one iteration is one forward and one adjoint model run), it would take 8×2×2×50=1600 h or 67 d just for the model runs, not counting time spent in the computing queue. This is impractical given the need to do tests required of any new inversion system. Therefore, we split up our target period into several inversions that were run in parallel as shown in Fig. a. A single forward run from 1984 to 2017 produced initial C and Cδa′ fields for all inversions. This forward run was identical to scenario C_WL+ of and matched the long-term atmospheric CH4 and δ13C trends over that period. Six 5-year inversions were run simultaneously with 2 years of overlap (red bars) between inversions, starting in 1997, 2000, 2003, 2006, 2009, and 2012. After all six inversions finished, the fluxes from the middle 3-year period of each inversion (blue bars) were considered for analysis. For simulating prior and posterior mole fractions, fluxes from the non-overlapping periods (1997–2001, 2001–2004, 2004–2007, … 2013–2017) were stitched together and a single forward run was done with those fluxes.

The prior fluxes and their δ13C source signatures for the different categories of methane emissions are described in detail as scenario C_WL+ in . Briefly, the prior fluxes are based on bottom-up emission estimates with adjustments to match global atmospheric CH4 increases and to satisfy the global mass balance of δ13C over 1984–2017. For biomass burning or pyrogenic emissions, we use the Global Fire Emission Database (GFED) 4.1s for 1997–2016 and estimates from the Reanalysis of Tropospheric chemical composition (RETRO) project before 1997 . Other anthropogenic emissions are based on the EDGAR 4.3.2 inventory . We use natural fossil emissions reported by . Emission estimates from wild animals and termites are adopted from and , respectively. Wetland emissions and upland soil consumption of methane are estimated by a process-based model , after which the soil sink is modeled as a first-order loss process as explained in Sect. .

The δ13C source signatures used in our study are mainly spatially resolved maps based on the Global δ13C Source Signature Inventory 2020 for coal, oil and gas (ONG), biomass and biofuel burning, ruminant and wild animal sources , spatial maps for geological seeps , and wetland sources . Globally averaged values are used for waste, landfills, termites, rice, and other energy and industry, given insufficient data to develop spatial distributions for their δ13C signatures .

The sum of the bottom-up methane emission estimates described above is not consistent with top-down estimates of global total emissions based on observed atmospheric CH4 growth and estimated loss, which requires a 46 Tg yr-1 increase in the annual global emission in 2016 compared to the 1999–2006 quasi-stable period. In addition, the δ13C mass balance requires 167 Tg yr-1 emissions from fossil sources (including natural geological seeps) to be consistent with modeled sinks and the δ13C source signatures described above. Therefore, we (i) scale the ONG emissions from EDGAR 4.3.2 uniformly using annual scaling factors to reach a total of 167 Tg yr-1 from all fossil sources, (ii) impose a linear trend on wetland emissions to achieve an increase of 46 Tg yr-1 in total 2016 emissions compared to 1999–2006, and (iii) adjust emissions from agricultural and waste sectors to match the year-on-year global CH4 growth rate derived from marine boundary layer observations . This ensures that our global CH4 and 13CH4 budgets approximate the long-term trends in atmospheric CH4 and δ13C over 1984–2017.

Atmospheric methane has four loss mechanisms: atmospheric oxidation by OH and Cl throughout the atmosphere, destruction by O(1D) in the stratosphere, and surface uptake by upland soils. In all our inversions, these sinks are prescribed and not optimized. Monthly climatological CH4 loss rates in the stratosphere due to OH, Cl, and O(1D) were constructed from a run of the ECHAM5/MESSy1 chemistry transport model . Loss due to tropospheric Cl is simulated using a recent model-derived estimate of tropospheric Cl . For tropospheric OH, we use the monthly OH climatology of after scaling by 0.9 to match the declining atmospheric abundance of methyl chloroform in the early 2000s .

In most CH4 inversions, upland soil sinks are folded into the net wetland flux. However, the soil sink fractionates strongly between 12CH4 and 13CH4 , and therefore we keep it separate from wetland fluxes. We model the soil sink as a first-order reaction at the surface, in which the loss rates of 12CH4 and 13CH4 in the lowest model layer are d12C/dt=-kssFss12C and d13C/dt=-αsskssFss13C, respectively. Fss is the prior soil sink map from the Terrestrial Ecosystem Model (TEM), and kss is an arbitrary constant tuned to a value such that in a forward run with prior fluxes, the global total soil sink matches the prior total.

The fractionation between 12C and 13C for each of the loss reactions is modeled as k12/k13=1/α=CeD/T , where T is the air temperature in Kelvin. The soil sink fractionation is cast in a similar form for convenience. Coefficients C and D that we used are tabulated in Table .

To maximize the spatiotemporal coverage of in situ CH4 and δ13C data, we have developed a new database by harmonizing measurements from NOAA/INSTAAR with those from 30 other laboratories around the world . All CH4 data have been quality-checked and converted to a common CH4 scale, namely the World Meteorological Organization (WMO) X2004A scale maintained at NOAA's Global Monitoring Laboratory . For data not on the WMO X2004A scale, we applied lab-specific scale multipliers estimated based on (i) comparisons of measurements of common air samples during the WMO/IAEA Round Robin Comparison Experiment and (ii) comparisons of co-located atmospheric measurements made by NOAA and other laboratories. We constructed the uncertainty in the assimilated CH4 measurements from a combination of (i) measurement repeatability of a single sample (hereafter called the single measurement precision), (ii) lab-specific long-term reproducibility based on analyzer type and sampling frequency reported in the literature, and (iii) each lab's realization of the calibration scale. If a scale conversion was needed to bring measurements onto the WMO X2004A scale, the mole fraction uncertainty due to the scale multiplier uncertainty was added in quadrature. The final uncertainties are typically less than 9 ppb for all CH4 measurements.

We used δ13C data from the Institute for Arctic and Alpine Research (INSTAAR) as well as other isotope laboratories making precise measurements of atmospheric methane with isotope ratio mass spectrometers. The INSTAAR δ13C data were measured in a subset of air samples collected from NOAA's Global Greenhouse Gas Reference Network (GGGRN). Because different labs have independent ties to primary reference materials which do not agree, we calculated offsets to bring the δ13C data onto the INSTAAR realization of the Vienna Pee Dee Belemnite (VPDB) scale . These offsets were based on measurements of cylinders, flasks filled from cylinders, or co-located sample data, and are all described in . When there was not a direct comparison, e.g., between INSTAAR and TU or INSTAAR and NIPR, we used comparisons between each of these labs and the Institute for Marine and Atmospheric research Utrecht (IMAU). Each comparison had an uncertainty associated with it, and these were combined in quadrature to account for uncertainty in the offset correction. The total uncertainty in assimilated δ13C measurements was typically less than 0.15 ‰. The final database of assimilated CH4 and δ13C measurements is available at 10.15138/64w0-0g71 .

With the following exceptions, we assimilate all the observations from this database including marine boundary layer sites, surface and tower sites over continents , and vertical profiles from routine aircraft measurements . Intermittent aircraft profiles such as from the HIPPO and ATom campaigns are not assimilated. CH4 data from flasks taken aboard routine flights between Japan and Australia as part of the CONTRAIL program have been assimilated . A subset of the CONTRAIL flasks was also analyzed for δ13C , which were not assimilated. For continental tower sites with multiple intake heights, only data from the highest intake are considered in inversions to minimize local influence. For sites with continuous CH4 analyzers, the CH4 data are averaged hourly, and only hourly averages between 11:00 and 16:00 local solar time are assimilated; these are the times when planetary boundary layer heights are likely to be best represented by transport models. For continuous CH4 analyzers on mountain tops, we only assimilate hourly averages between 00:00 and 05:00 local solar time to avoid possible up-slope contamination. Site-specific statistical filtering based on a nonparametric curve fitting routine is further applied, with the exception of vertical profiles, to remove large outliers with potential local or other contamination. The number of CH4 and δ13C measurements assimilated each year is summarized in Table , and their locations are plotted in Fig. .

The uncertainty of surface emission estimates is a combination of random and systematic uncertainties. Random uncertainties are associated with components of the inversion system whose errors are assumed to be zero on average. In the formulation of the cost function (), the prior flux x0 is assumed to have a probability density function (PDF) centered on the true flux with variance around the truth given by the prior covariance matrix B. Similarly, the model–observation mismatch Hx–y is assumed to have a PDF centered around the mismatch between the true atmospheric mole fraction and true fluxes propagated through an unbiased transport model, with variance around this mean given by R. The random uncertainty in the optimal estimate is given by B^, where 13B^-1=∂2J∂x2=HTR-1H+B-1. Variational inversion systems such as TM5-4DVAR can construct a low-rank approximation of B^ during the optimization. However, for large state vectors the B^ thus constructed is an overestimation of the true posterior uncertainty . We therefore construct an estimate of B^ by performing an ensemble of 100 independent inversions for each of the 5-year inversions of Fig. a, with prior fluxes and observations perturbed according to the covariances specified by B and R, respectively. With 100 ensemble members, our estimate of B^ is expected to be within 10 % of the exact analytical solution for B^ . Furthermore, our ensemble of inversions allows us to compute any posterior covariance and correlation between estimated fluxes, such as between large regions or different CH4 source types.

Systematic uncertainties are associated with aspects of the inversion system that are assumed to be fixed and perfectly known in principle but might in fact be biased in practice. In our inversion system, such aspects include, but are not limited to, atmospheric transport and chemistry, isotope source signatures, and the wetland inundation maps used to construct the prior wetland emissions. Because the posterior covariance estimate does not include systematic errors, we explore the impact of such errors by performing inversions with different realizations of potentially biased inputs in the following sensitivity tests.

Both the CH4-only and the CH4 + δ13C inversions fit the atmospheric CH4 data, while only the latter is consistent with atmospheric δ13C data. This is demonstrated both at surface sites from which data were assimilated and data from aircraft campaigns that were withheld for validation. Figure  shows that both inversions fit the observed CH4 time series at three NOAA baseline observatories. However, despite starting from realistic atmospheric CH4 and δ13C fields, the CH4-only inversion moves progressively farther from observed δ13C with time at those same locations, as demonstrated in Fig. . Only the CH4 + δ13C inversion fits both atmospheric CH4 and δ13C data. This is also demonstrated in Fig. , which compares modeled δ13C to δ13C measured by the HIPPO and ATom aircraft campaigns, and from regular flights between Japan and Oceania as part of the CONTRAIL program. ATom and HIPPO campaigns sampled primarily background air over the oceans at multiple latitudes and altitudes, and neither CH4 nor δ13C data from those campaigns were assimilated. CONTRAIL primarily sampled the marine background at multiple altitudes as well, except for a small number of samples taken during takeoff and touchdown in Japan. CH4 flask samples from CONTRAIL were assimilated in both inversions, but their δ13C measurements were not assimilated. The CH4-only inversion compares far less favorably to the δ13C measurements than the joint inversion. Therefore, it is reasonable to conclude that our CH4-only inversion, and very likely most traditional CH4-only inversions, does not yield a CH4 emission distribution consistent with atmospheric δ13C observations. We therefore expect our CH4 + δ13C inversion to provide more accurate emission estimates and source partitioning than our CH4-only inversion.

The top row of Fig.  shows the global total annual emissions from two inversions, a CH4-only or “traditional” methane inversion without δ13C data, and a joint CH4 + δ13C inversion developed in this work. The shaded regions in Fig.  denote 2σ random errors derived from 100-member Monte Carlo ensembles of inversions described in Sect. . Annual averages of the emissions and random errors are summarized in Table . Since many methane studies calculate emissions between and outside the ±30∘ latitude band, in Table  we report those numbers as well for ease of comparison. The global total emission from all categories is unaffected by the addition of δ13C data, since δ13C does not place any additional constraint on the total CH4 emission. However, the partitioning between microbial and fossil sources is changed significantly with the addition of δ13C data. Based on comparison to atmospheric data as noted in Sect. , we expect the source partitioning from our CH4 + δ13C inversion to be more accurate compared to our traditional CH4-only inversion.

Figure  also shows the total and source-disaggregated CH4 emissions from our CH4 and CH4 + δ13C inversions over three latitude bands; the tropics are bounded between 23.5∘ S and 23.5∘ N. Relative to the prior, tropical (extratropical) total emissions are adjusted upward (downward) by both inversions, and there is little sensitivity of the tropical versus extratropical partitioning to the assimilation of δ13C data. In the northern extratropics, the partitioning of CH4 emissions between the different source types does not change significantly with the addition of δ13C data. However, in the tropics the inversion with δ13C data shows significantly higher fossil (and lower microbial) emissions than the inversion without δ13C data. Fossil CH4 emissions in the southern extratropics are significantly different for most years in the presence of δ13C data, but similarly significant differences do not exist for the other source types. Finally, our estimate of pyrogenic emissions does not change significantly in Fig.  in the presence and absence of δ13C.

As explained in Sect. , we estimate possible biases in our flux estimates by running the inversion with different choices of non-optimized input. The spread in annual emissions due to alternate specifications of atmospheric chemistry (tropospheric chlorine of Sect.  and OH fractionation of Sect. ) is shown in Fig. . Analogous spreads due to different specifications of δ13C source signatures (Sect. ), wetland inundation maps (Sect. ), and initial atmospheric δ13C fields (Sect. ) are shown in Fig. . Note that the y-axis ranges in Figs.  and  are different. The average spread in annual emissions from different latitude bands and source types is summarized in Table  for each sensitivity test. The average of the annual posterior uncertainties as depicted in Fig.  is also provided in Table  as “MC-derived (2σ)” for reference, with the caveat that 2σ uncertainties are not directly comparable to the range across a few inversions.

Most of the sensitivity tests have little impact on the global total CH4 emission, and the spread in the total CH4 emission from different latitude bands is generally smaller than the posterior uncertainty of our base inversion. However, by far the largest source of error in partitioning the total emission into fossil and microbial sources comes from our representation of atmospheric chemistry, namely the distribution of tropospheric chlorine and the kinetic isotope effect of CH4 destruction by OH. Unless the uncertainty in these two factors can be reduced, our ability to use δ13C measurements to partition different source types will be seriously hampered. The uncertainty arising from our limited knowledge of δ13C source signatures, to the extent represented by the different signature maps used, is lower than the uncertainty due to atmospheric chemistry. Lastly, the uncertainty due to an incorrect specification of the initial atmospheric δ13C field is minimal, in line with our expectation that an inversion will rapidly correct for it by adjusting emissions during its spin-up period. However, we note here that the “incorrect” initial fields we constructed for the last test still satisfied the global δ13C mass balance by construction. The sensitivity to an incorrect initial condition will likely be higher if the initial field does not satisfy global δ13C mass balance.

As discussed earlier (Fig.  and discussion in Sect. ), the atmospheric methane burden has been steadily growing since 2007 after a period of quasi-stability during 1999–2006. We use our CH4 + δ13C inversion to ask whether the addition of δ13C data can provide information on the sources of the additional methane. Figure  suggests that the trend in CH4 emissions comes largely from microbial emissions in a CH4 + δ13C inversion. However, it is possible that this attribution to microbial emissions comes from our prior – which had a trend in the microbial emissions and a temporally flat fossil contribution – instead of the atmospheric data. To assess the robustness of our inferred microbial and fossil emission trends, we perform a second set of inversions with the following modifications.

We construct climatological prior fluxes and source signatures by averaging our prior emissions and signatures from 2000 to 2006. Neither the resulting priors nor the source signatures have any time trend.

The CH4 observations assimilated in a CH4-only inversion constrain the total CH4 emission, and any source disaggregation relies on spatiotemporal separation of emissions as encoded in the prior emissions and their uncertainties. Since the two largest CH4 source types (microbial and fossil) have different δ13C source signatures, assimilating δ13C observations should provide additional information to separate the two sources compared to a CH4-only inversion. We can evaluate this additional information by looking at the posterior correlation between microbial and fossil emissions both globally and regionally. Posterior correlations between global annual microbial and fossil CH4 emissions, calculated from our 100-member ensemble of independent inversions as described in Sect. , are shown in Fig. . Error bars on the correlations shown in Fig.  represent the 95th percentile range of 20 000 evaluations of the correlation by randomly sampling the 100-member inversion ensemble with replacement .

For all the years shown in Fig. , a CH4-only inversion results in a strong negative correlation between global microbial and fossil emissions, consistent with the idea that atmospheric CH4 measurements constrain the total CH4 budget much better than individual source types. The addition of δ13C data reduces this negative correlation, implying that δ13C provides additional information to disentangle different CH4 source types. The degree of disentanglement, represented by the reduction in the negative correlation, is determined by the δ13C measurement coverage in a particular year and atmospheric transport connecting the emissions to those measurements. The correlation reduction is limited in our inversions by the relative sparsity of δ13C measurements; even in the most recent 2012–2017 inversion period, only 2.8 % of CH4 measurements have corresponding δ13C measurements, overwhelmingly in locations far removed from significant CH4 emissions (Fig. ). Having more δ13C measurements in general, and specifically closer to emissive regions, should allow further disentangling of the different CH4 source types.

Over smaller regions, only the northern extratropics and Asia show significant decorrelation between annual fossil and microbial emissions (Fig. b and c) with the addition of δ13C data. While several other regions show similar reductions, the reductions are typically not significant compared to the 95th percentile error bars. The significant decorrelation seen for northern extratropical and Asian emissions may be because most δ13C measurements are in the northern extratropics and downwind of Asia in the Pacific. To see similar significant decorrelation over other regions we will likely need increased δ13C coverage closer to those regions. Although Fig.  shows some δ13C measurements over North America and Europe, the majority of those measurements are from the background air sampling sites Niwot Ridge and Jungfraujoch, respectively, and therefore do not contribute to significant decorrelation of fossil and microbial emissions from those continents.

First, Figs.  and  show that our inversion assimilating only CH4 does not yield a CH4 emission distribution consistent with atmospheric δ13C. This is very likely true of CH4 inversions in general, since they have no constraints forcing them to match atmospheric δ13C gradients and trends. Starting from a prior emission distribution consistent with atmospheric δ13C trends does not ensure that the posterior emission estimates will remain consistent. Our CH4-only inversion started from a prior that reproduced the global mean atmospheric δ13C trend scenario C_WL+ of, yet the posterior deviated from it as shown in Figs.  and . We conclude that the only way to guarantee a posterior emission distribution consistent with both atmospheric CH4 and δ13C data is to assimilate them simultaneously.

Second, given an atmospheric sink scenario, our current observational coverage allows us to estimate the global total CH4 emission with a 2σ random uncertainty of ∼3.8 Tg yr-1, which is less than 1 % of the total emission. Microbial, fossil, and pyrogenic emission uncertainties are around 3.5 %, 8 %, and 10 %, respectively, at the global scale. Given these posterior uncertainties, there are significant differences between inversions with and without δ13C data in the apportionment of the total CH4 emission between microbial and fossil sources both globally and in the tropics (Fig.  and Table ). In both regions, the inclusion of δ13C data in an inversion results in a significantly higher proportion of fossil emissions compared to microbial emissions, which we consider realistic since it matches both atmospheric CH4 and δ13C data (Figs.  and ). Pyrogenic emissions are relatively insensitive to the inclusion of δ13C data.

Third, we tested the sensitivity of our results to several factors that can lead to biases or systematic errors, as detailed in Sect. . This included different maps of the δ13C isotopic source signatures, static and dynamic maps of the wetland inundation extent, different initial δ13C fields, different fractionation factors for the CH4+OH oxidation mechanism, and different fields of tropospheric Cl. The last two factors had by far the largest impacts on the large-scale apportionment between microbial and fossil emissions, even though their impact on the total CH4 budget was nil or negligible. With the OH fractionation of , the global microbial emission increases to 414 Tg yr-1 and the fossil emission drops to 131 Tg yr-1. With the lower estimate of tropospheric Cl from , the global microbial emission decreases to 345 Tg yr-1, while the fossil emission increases to 199 Tg yr-1. Since some CH4 inversions in the literature do not simulate a tropospheric Cl sink of CH4, we tested the impact of this limiting case as well. In the absence of a tropospheric Cl sink, the global microbial emission drops further to 331 Tg yr-1 and the fossil emission increases to 213 Tg yr-1. Most of these shifts in the global partitioning are accompanied by shifts in the latitudinal partitioning. All of these are significant revisions to the partitioning of Table , suggesting that the ability of atmospheric δ13C measurements to partition the total CH4 emission into different source types, at least over large regions, is limited by our knowledge of these two critical chemical processes. The uncertainty in our knowledge of δ13C source signatures, long considered a limitation on the use of δ13C data, is almost never a leading driver of uncertainty in Table , although it is usually more significant than either inundation extent or the initial δ13C field. Finally, our tests suggest that the impact of an incorrect initial δ13C field can be ameliorated by a relatively short spin-up of 1 year in an inversion, in contrast to a multi-decadal spin-up necessary for a forward model run.

Fourth, atmospheric δ13C data strongly suggest that the rise in microbial emissions is the primary driver of the post-2007 growth in atmospheric CH4. While a CH4-only inversion starting from priors without a time trend attributes ∼70 % of the growth to fossil emissions, the addition of δ13C data shifts that to microbial emissions being responsible for ∼85 % of the growth. Since the latter inversion is consistent with atmospheric δ13C data while the former is not (Fig. ), we consider a majority microbial contribution to the post-2007 growth to be more realistic. A disaggregation of the growth by latitude bands suggests that a significant majority of the increase in tropical methane emissions is due to microbial and not fossil emissions. Moreover, although some of the sensitivity tests of Sect.  lead to different partitioning between microbial and fossil emissions, they all suggest a steeper trend in microbial compared to fossil emissions in Figs.  and .

Fifth, the ability of δ13C data to disentangle different CH4 source types can be quantified by the reduction in the posterior correlation between emissions from those sources owing to the addition of δ13C data compared to a CH4-only inversion. Considering the two largest source types of methane (microbial and fossil), we see significant reductions in their posterior correlation over the globe as well as the northern extratropics and Asia. The degree of decorrelation, however, is limited, and we do not see significant decorrelation over other regions. We hypothesize that this is not a limitation of our understanding of δ13C but rather of its limited observational coverage. Even in the most recent years less than 3 % of assimilated CH4 measurements were accompanied by δ13C measurements, almost exclusively from background sites. It is very likely that an increase in the observational coverage of δ13C, preferably close to source regions, will improve the capability of δ13C measurements to distinguish between different CH4 source types.

Sixth, while it is difficult to compare our emission budget directly with GCP due to different partitioning schemes, we note that our fossil fuel emissions for both the 2000–2009 and 2008–2016 periods are higher than the GCP top-down and bottom-up emissions. However, our estimate of the change in fossil fuel emissions between the two periods is significantly lower than the GCP estimates. Concurrently, our estimate of the change in microbial emissions over the same time is significantly higher than the GCP top-down estimate. Both of these discrepancies are driven by atmospheric δ13C data, since our CH4-only inversion provides changes that are consistent with GCP estimates. We therefore conclude that the microbial and fossil emission change estimates in the GCP budget are consistent with atmospheric CH4 data but not with δ13C data. Finally, our pyrogenic emission estimates are consistent with or close to the GCP estimates for both periods.

The Global Carbon Project (GCP) periodically publishes top-down and bottom-up budgets of methane emissions from a suite of models. However, a meaningful comparison between our emissions and the 2020 GCP budget is not straightforward. The GCP bottom-up (BU) budget for 2008–2017, with 737 Tg yr-1 emissions and 625 Tg yr-1 sinks, significantly overestimates the atmospheric growth rate. This is primarily due to an overestimate of both microbial (159 Tg yr-1 freshwater sources) and fossil (45 Tg yr-1 geologic sources) methane in the GCP budget, making a direct comparison with our microbial and fossil estimates meaningless. The GCP top-down (TD) estimates do not provide a fossil–microbial split of “other natural” emissions, also making a direct comparison with our estimates difficult. However, it is possible to calculate emissions for certain GCP categories from our inversions for some limited comparisons. For the period 2000–2009, we compare directly by computing the 2000–2009 average from our inversions shown in Fig. . However, since our inversions stop in 2016 and Table 3 of reports the 2008–2017 average, we first calculate the 2008–2016 mean from the GCP budget using their reported 2008–2017 mean and their 2017 emissions. For computing the GCP means quoted here, we use the spreadsheet at 10.18160/GCP-CH4-2019 in , which may result in small differences from the numbers quoted in Table 3 of due to rounding.

Our estimate of fossil fuel CH4 emissions is significantly higher than GCP estimates, consistent with earlier work by on δ13C-based source apportionment. Subtracting 35 Tg yr-1 of geologic emissions from our fossil emission estimates, we arrive at 137±2 Tg yr-1 of fossil fuel emissions during both the 2000–2009 and 2008–2016 periods, which is significantly higher than the GCP BU (TD) estimates of 111 and 127 Tg yr-1 (99 and 109 Tg yr-1), respectively. Recent measurements of 14CH4 in ice cores suggest that geologic methane emissions are much lower than we have assumed, with an upper 95th percentile confidence limit of 5.4 Tg yr-1 and mean of 1.6 Tg yr-1 . If true, this would push our fossil fuel emission estimates even higher to 170±2 Tg yr-1, indicating a significant underestimate in the GCP budget.

Assuming that methane emissions from geological seeps do not change significantly over decadal timescales, we estimate a change of 0±2 Tg yr-1 in fossil fuel CH4 emissions from 2000–2009 to 2008–2016. This is markedly different from the BU (TD) GCP budget, which estimates an increase of 16 Tg yr-1 (10 Tg yr-1) between the two periods. While our baseline estimates for the two periods may be influenced by systematic biases (Sect. ), the change between the two periods is relatively robust. With the alternate specification of tropospheric chlorine and alternate fractionation due to the OH oxidation , which are the two biggest sources of bias in source apportionment by δ13C, the change in our fossil fuel emission estimate between the two periods is 1 and -1.7 Tg yr-1, respectively, which is well within our uncertainty estimate of 2 Tg yr-1 and significantly lower than both the GCP BU and GCP TD estimates. The GCP BU (TD) estimate of an increase of 16 Tg yr-1 (10 Tg yr-1) between the two periods is closer to our estimate of 8.4±5.6 Tg yr-1 from a CH4-only inversion, which is not consistent with δ13C data.

For reasons mentioned above, we cannot directly compare our microbial emission estimates to GCP emission estimates. However, if we assume that methane from termites, wild animals, and oceans does not change over decadal timescales, we can compare the change in the GCP TD estimate of wetlands, agriculture, and waste from 2000–2009 to 2008–2016 with the change in microbial emissions in our inversion estimates. The GCP TD budget estimates a change of 12.6 Tg yr-1 between those two periods compared to our estimate of 26±2 Tg yr-1 from a joint δ13C and CH4 inversion and of 18±6 Tg yr-1 from a CH4-only inversion. Thus, the change in microbial emissions in the GCP TD budget is at the lower end of but consistent with our estimate from a CH4-only emission, while it is not consistent with our budget after incorporating δ13C data. We cannot perform a similar analysis with the GCP BU budget because freshwater emissions cannot be assumed to be static over decadal timescales.

Finally, our pyrogenic emission estimates for both the 2000–2009 and 2008–2016 periods are 30±0.6 Tg yr-1, with a change of 0.3±0.5 Tg yr-1. These are close to the GCP BU (TD) estimates of 31 and 30 Tg yr-1 (29 and 31 Tg yr-1), respectively. Neither the GCP budgets nor our inversion show significant changes in pyrogenic methane emissions between the two periods.

derive sector-specific changes in methane emissions from a two-box model by first creating an ensemble of emission scenarios consistent with known emissions of methane and their uncertainties, then comparing the resultant hemispheric mean CH4 and δ13C in the atmosphere with measurements. Based on the ensemble members that agree with hemispheric mean CH4 and δ13C time series, they come up with sector-specific emission changes. We note here that according to Figs. 1b and 2a of , none of their choices fit the atmospheric δ13C data very well. We hypothesize that this is due to both the biases in their bottom-up emission estimates, which could not be sufficiently corrected due to tight assumed uncertainties in them, and the inability of a box model to interpret spatial gradients of CH4 and δ13C. Specifically, Fig. 1b of shows that none of the ensemble members can reproduce the recent downward trend in δ13C. Therefore, their estimated fractional contribution of microbial (fossil) emissions to the recent growth in CH4 is likely to be too low (high).

Table  summarizes the changes in methane emissions from the “quasi-stable” period of 2000–2006 to two later periods during the renewed growth, 2007–2012 and 2013–2017. For both the periods, we estimate significantly smaller contributions from fossil emissions compared to . While the microbial contributions look similar in magnitude, they constitute a smaller fraction of the total change for . Lastly, while estimate a downward trend in pyrogenic emissions, we do not estimate a trend significantly different from zero. To summarize, we estimate a larger (smaller) influence of increasing microbial (fossil) emissions behind the recent CH4 growth, which is likely to be more accurate because it better reproduces the observed δ13C trend in the atmosphere (compare Fig.  here with Fig. 1b of ). We cannot separate the anthropogenic from natural microbial contributions in our framework, and we note that such a separation by relies on their modeled wetland fluxes and not on atmospheric CH4 or δ13C data.

describe an alternative variational inversion framework using the LMDz-SACS model to assimilate CH4 and δ13C measurements. We find it heartening that others have decided to tackle this complicated problem. Since they reserve decadal dual-tracer inversions for future work, we will compare their technique with ours to highlight the similarities and differences. The biggest difference lies in the decision of to optimize δ13C source signatures compared to our choice of keeping them fixed for a specific inversion. While δ13C source signatures are uncertain for many methane sources, we explain our reasons for not optimizing them in Sect. . Instead, we explore the impact of source signature uncertainty with different constructions of the source signature map as detailed in Sect. . In the end, at least for large geographical regions, the uncertainty from source signatures did not prove to be a leading uncertainty (Table ). The second major difference between the two inversion frameworks lies in the construction of the prior CH4 fluxes. While use a prior that approximately matches the atmospheric CH4 growth rate, we construct our priors to match both the CH4 growth rate and the δ13C trend over 2 decades. We suspect this and the linearity of our formulation due to not optimizing source signatures to be the reasons why our inversion required a shorter spin-up time compared to .

There are also a few differences in implementation between the two frameworks. Most notably, estimate the posterior uncertainty as the spread between different inversion configurations, correctly stating that an evaluation of the posterior covariance matrix would require significantly more computing resources. We evaluate that posterior covariance matrix for both CH4 + δ13C and CH4-only inversions and present both types of uncertainty, namely the systematic uncertainty as the spread between multiple inversion configurations and the random (Bayesian) uncertainty as the spread of an ensemble of 100 independent inversions. The configurations we explore for the systematic uncertainty are also different from and include alternate specifications of the Cl oxidant and the isotopic discrimination of the CH4 + OH reaction. We find the latter two to be the most significant drivers of uncertainty for partitioning CH4 emissions using δ13C data.

While our implementation of the inversion is different from , our goals are very similar. We look forward to long-term inversions of CH4 and δ13C data using LMDz-SACS so that we may compare and contrast with our results presented here and figure out how best to use isotopic measurements to solve the atmospheric methane puzzle.

In addition to the publications compared in detail above, some others have used combinations of CH4 and δ13C observations to infer sector-specific sources of methane and the causes behind the recent increase, either in 2D or 3D atmospheric models. In terms of attributing the recent growth of CH4, the most significant difference between our work and either of those studies is that our inferred change in biomass burning emissions is smaller. For example, our GFED 4.1s prior has a reduction of 1.35 Tg yr-1 in pyrogenic emissions from 2003–2006 to 2007–2015 . This is changed to a reduction of 0.81 Tg yr-1 by our CH4 + δ13C inversion (Fig. ) and to a reduction of 0.19±0.72 Tg yr-1 when we start from a prior without trends (Fig. ). This suggests that the estimate of reduced pyrogenic emissions is driven primarily by its presence in the prior and not by δ13C data, which are sparse near regions with large fire emissions. To contrast, infer a reduction of 2.9 Tg yr-1 in pyrogenic emissions between the same periods, allowing for a larger increase in fossil emissions to balance CH4 and δ13C trends. Finally, we note that even though the INV_FIXED inversion of with climatological priors allows for variations in OH and is therefore not directly comparable to our analogous inversion, their INV_FIXED estimates no reduction in pyrogenic emissions Table 6 of. This supports our hypothesis that a reduction in pyrogenic emissions is driven primarily by the prior and not by atmospheric δ13C data.

In agreement with our estimates, also attribute the majority of the growth between 2007 and 2014 to microbial and not fossil sources. They infer a 3±2 Tg yr-1 reduction in pyrogenic emissions during that period compared to a reduction of 0.58 Tg yr-1 inferred by our CH4+δ13C inversion (0.50±1.66 Tg yr-1 reduction if we start with a climatological prior). We note that our prior pyrogenic emission for 2014 is 3.45 Tg yr-1 higher than 2007, and a CH4-only inversion estimates an increase of 1.65 Tg yr-1 between the two years (Fig. ); therefore, even the small reduction we estimate must be driven by δ13C data. However, since this reduction is smaller than the 3±2 Tg yr-1 estimated by , our fossil emission increase is also smaller to balance the δ13C trend. Finally, do not estimate a significant role of the chemical sink behind the recent CH4 growth, consistent with our previous work .

While we feel confident in the CH4 emission estimates reported here, there are several areas which we plan to explore and improve in future work.

While our effort to quantify source-specific methane emissions from atmospheric δ13C measurements has yielded several significant results, it has also uncovered areas for further progress. First, Fig.  shows a lack of δ13C data over the tropics and close to source regions. This is due to the historical choice of preferentially measuring δ13C at background sites and the general lack of tropical samples. More δ13C measurements in historically under-sampled regions would let us better separate tropical emissions and attribute continental emissions to specific sources. Second, while the soil sink has a small impact on CH4-only studies, its impact on δ13C studies is large owing to the strong fractionation. Recent studies have suggested that the soil sink may be underestimated in current biogeochemical models and may have trends due to a changing climate , which are both aspects that need better characterization. Third, our knowledge of the OH fractionation of 13CH4 and the distribution of tropospheric Cl is a source of significant uncertainty in partitioning methane sources using δ13C data (Sect. ). There is a wide range of estimates for OH fractionation in the literature and an analogously wide range of distributions of tropospheric Cl . The uncertainty in both of these needs to be reduced in order to use δ13C data more effectively. Finally, while we have ruled out a sink-only explanation for the recent growth of methane , a small contribution from sink variations is possible. Since direct measurements of OH are few and far between and atmospheric levels of MCF are too low for quantifying recent variations in OH, we need alternate ways of separating source and sink influences. One possibility is to use measurements of δD in methane to constrain the sink . We plan to explore the potential of δD measurements in future work.