The general circulation model (GCM) LMDz is the atmospheric component of the coupled model from the Institut Pierre-Simon Laplace (IPSL-CM), developed at the Laboratoire de Météorologie Dynamique (LMD) . The version of LMDz used here is an offline version dedicated to the inversion framework created by : the precalculated meteorological fields provided by the online version of LMDz are given as input to the model, considerably reducing the computation time. The model is built at a horizontal resolution of 3.8∘ × 1.9∘ (96 grid cells in longitude and latitude), with 39 hybrid sigma-pressure levels reaching an altitude of about 75 km. The time step of the model is 30 min and the output values have a resolution of 3 h. Horizontal winds are nudged towards the ECMWF meteorological analyses (ERA-Interim) in the online version of the model. Vertical diffusion is parameterized by the local approach of , and deep convection processes are parameterized by the scheme of . The offline model LMDz is coupled with the Simplified Atmospheric Chemistry System (SACS) to represent CH4 oxidation by radicals .

We simulate atmospheric 12CH4 and 13CH4 amount fractions to retrieve both the CH4 amount fractions and the δ(13C,CH4) signal. Four clumped isotopologues (12CH4, 12CH3D, 13CH4 and 13CH3D) are simulated in one sensitivity simulation (see Sect. ) to retrieve both the δ(13C,CH4) and δ(D,CH4) compositions.

Oxidations by OH, O(1D) and Cl are included in the chemical scheme of LMDz-SACS. Time-varying 3-D fields of OH and O(1D) with daily resolution, simulated beforehand with the LMDz-INCA chemistry model , are prescribed for each oxidant species to simulate the associated chemical loss. The same meteorological data were used for generating these fields and running the simulations presented in this study.

The resulting OH field, named OH-INCA, exhibits a global mean tropospheric mass-weighted concentration of 11.1 × 105 cm-3 over 1998–2018, consistent with the previous estimates from (11.7 × 105 cm-3) and ((11.2 ± 1.3) × 105 cm-3) and well within the range derived from the Atmospheric Chemistry and Climate Model Intercomparison Project (ACCMIP) 10.3–13.4 × 105 cm-3;. It is, however, slightly larger than the very recent estimate from , obtained by constraining OH with observations of its precursors. The interhemispheric ratio is 1.14, lower than the mean value of 1.3 inferred by but more consistent with recent estimates from and an interhemispheric parity obtained from methyl-chloroform-based inversions . Global concentrations of OH-INCA increased by 4 % between 2002 and 2014.

As suggested by , the Cl concentrations derived by are prescribed here for all simulations. Their work suggests that the Cl sink accounts for only 0.8 % of the total CH4 oxidation, which is lower than other estimates used in the literature 1.8 %–5 %;.

The fractionation effect must also be represented in the modeling framework. Table  provides the fractionation coefficients applied for each loss reaction. For the OH sink, we adopted the estimate derived by . recommends using the rates but suggests increasing the uncertainty in the OH fractionation to account for the estimate (1.0054) from . As shown by , switching estimates from that of to that of has a large influence on the results, despite the fact that those authors did not optimize source signatures in their setup. As indicate that their data is of considerably higher experimental precision and reproducibility than that from previous studies, in particular , we prefer to allocate computational time to a sensitivity inversion testing a different OH field rather than testing a different OH fractionation coefficient. In addition, these estimates of fractionation coefficients come with uncertainty ranges that we could also consider in our inversions (e.g., with a Monte Carlo approach). In our case, the main limitation remains the large computational cost of one inversion (see Sect. ). In the future, we hope to be able to increase the number of sensitivity tests and account for this uncertainty. For this work, the values we adopt are the best estimates for each fractionation coefficient.

Inversions were performed using the Community Inversion Framework CIF;. This framework was designed to rationalize and bridge development efforts made by the scientific community within the same flexible, transparent and open-source system. This system was recently enhanced by to allow it to assimilate δ(13C,CH4) together with CH4 observations and to optimize both CH4 emissions and the source signatures δsource(13C,CH4) at the same time. For the purpose of this study, δ(13C,CH4) and δ(D,CH4) observations are assimilated together in the same inversion, and the system optimizes both the source signatures (δsource(13C,CH4) and δsource(D,CH4)).

The notations introduced here to describe the variational inversion method follow the convention defined by and . x is the control vector and includes all the variables optimized by the inversion system. Prior information about the control variables is included in the vector xb. Its associated errors are assumed to be unbiased and Gaussian and are described within the error covariance matrix B.

Here, the observation vector yo includes all available observations, namely the atmospheric CH4 amount fraction, δ(13C,CH4) and δ(D,CH4) data for one sensitivity test. The associated errors are also assumed to be unbiased and Gaussian and are described within the error covariance matrix R. This matrix accounts for all errors contributing to mismatches between simulated and observed values.

H is the observation operator that projects the control vector x into the observation space. This operator mainly consists of the CTM but is also followed by spatial, time and isotope-conversion operators. Following , prescribed source signatures and CH4 fluxes are first combined to generate isotope fluxes. These fluxes are then fed to the model to simulate the mixing ratios of the different isotopes over the time period considered. After the forward run, the simulated fields are interpolated to produce simulated equivalents of the observed amount fractions and isotopic compositions at specific locations and times, ensuring that a comparison between simulations and observations is possible. Adjoint versions of these forward operations are also implemented in order to perform the complementary adjoint run.

In a variational formulation of the inversion problem that allows H to be nonlinear, the cost function J is defined as 3J(x)=12x-xbTB-1x-xb+12H(x)-yoTR-1H(x)-yo.

Here, the minimum of J is reached iteratively with the descent algorithm M1QN3 , which requires several computations (40–50) of the gradient of J with respect to the control vector x: 4∇Jx=B-1x-xb+H*R-1H(x)-yo. H* denotes the adjoint operator of H.

The reference inversion (INV_REF) assimilates CH4 and δ(13C,CH4) observations over 1998–2018. CH4 emissions and δ(13C,CH4) source signatures for five categories of emissions – biofuel and biomass burning (BB), wetlands (WET), fossil fuels and geological sources (FFG), agriculture and waste (AGW), and other natural sources (NAT) – are optimized. The initial conditions for CH4 and δ(13C,CH4) are also optimized (see Sect. ).

For prior CH4 emissions, we adopt the bottom-up estimates compiled for the inversions performed as part of the Global Methane Budget and described in detail in . In short, anthropogenic (including biofuel) and fire emissions are based on EDGARv4.3.2 (Emissions Database for Global Atmospheric Research version 4.3.2; ) and GFED4s (Global Fire Emissions Database version 4s; ), respectively. Statistics from British Petroleum (BP) and the Food and Agriculture Organization of the United Nations (FAO) have been used to extend EDGARv4.3.2 (which ends in 2012) until 2017. The natural source emissions are based on averaged literature values: for wetlands, for termites, and and for geological (onshore) sources and oceanic sources that include geological (offshore) and hydrate sources. Prior emissions for 2018 are set equal to the emissions in 2017. Globally averaged emissions over 1998–2018 are listed in Table .

BB emissions are the combination of biomass-burning emissions from GFED4s and biofuel-burning emissions from EDGARv4.3.2. FFG emissions are the combination of oil, gas, coal, industrial and transport emissions from EDGARv4.3.2 and geological (onshore) source emissions from , whose global emissions were scaled down to 15.0 Tga-1 in the protocol of . AGW emissions are the combination of enteric fermentation, rice agriculture, manure management and waste emissions from EDGARv4.3.2. NAT emissions are the combination of termite and oceanic emissions, i.e., emissions from all natural sources apart from wetlands and geological sources.

Emissions are optimized at the grid-cell scale (one scaling factor per grid cell). For each category, diagonal elements of the matrix B are filled with the variances set to 100 % of the square of the maximum of the prior emissions over the cell and its eight neighbors during each month. Spatial error correlations (off-diagonal elements) are prescribed using an e-folding correlation length of 500 km on land and 1000 km over the oceans, without any correlation between land and ocean grid points. No temporal error correlations are prescribed.

δsource(13C,CH4) values for each emission category are also optimized and therefore included in the control vector. Prior information is built using the references given in Table . When regional information could be found, regional source signature values were prescribed for 11 continental regions (see Fig. , lower-right panel) for each subcategory. Isotopic signatures for subcategories are flux-weighted averaged to create signatures for categories, which results in signatures that are grid-cell dependent, although signatures for subcategories are set constant within a continental region. Signatures for wetlands are the only ones prescribed at the grid-cell scale, following . We optimize the source signatures at the regional scale rather than at the grid-cell scale to avoid substantial posterior differences between two adjacent grid cells. At present, there would not be enough data to corroborate, explain, or reject such differences. We therefore apply only one scaling factor per continental region and per category.

Livestock source signatures have likely been decreasing over time since the 1990s due to changes in the C3 / C4 diet within the major livestock producing countries . Also, FFG regional source signatures can vary over time due to variations in the contributions from different sectors (coal, oil and gas) to the emissions of a specific region . For AGW and FFG source signatures, we therefore optimize one scaling factor per year for each continental region. As for the other emission categories, only one scaling factor for the entire period and for each continental region is optimized. Error covariances are prescribed following the same methods as those applied to CH4 emissions.

As for δsource(D,CH4), we adopted global values suggested by and in agreement with the intervals given by . One exception is for WET sources, for which the boreal (-360 ‰) and tropical (-320 ‰) regions are differentiated. All values are summarized in Table . For each category and each continental region, only one scaling factor is optimized for the entire period.

δsource(13C,CH4) uncertainty values that are used to fill the diagonal elements of the matrix B are summarized in Table . These values have been chosen by compiling data from several studies . The observed variability over the whole globe (standard deviation σ or minimum–maximum range) presented in these studies for each category is compiled and applied here as an uncertainty (1σ), thus adopting the same value for all regions. Note that for BB sources, indicates a global standard deviation of about 20 %. However, this value is not weighted by the proportion of C3 versus C4 vegetation. Therefore, we inflated this uncertainty to 30 % to account for the uncertainty in the type of vegetation. δsource(D,CH4) uncertainty values were derived from the minimum–maximum ranges suggested by . We could have also used the standard deviation provided by . However, as the amount of data for AGW, BB, WET and NAT source signatures is very low compared to the δsource(13C,CH4) values, we prefer to use larger uncertainties and examine whether the assimilation of δ(D,CH4) observations modifies the results. For future studies, additional δ(D,CH4) data to derive realistic regional estimates, especially for non-fossil sources, would be invaluable.

We acknowledge the fact that our methods are not perfect and that the prescribed regional uncertainties might be too large compared to the regional observed uncertainties that are currently estimated, especially for δsource(13C,CH4). As our inversion system is used for the first time over a time period exceeding 10 years, it is difficult to predict the influence of the setup on the results. As the time required to run an inversion is very high at the moment (see Sect. ), we prefer to assess the behavior of this system in response to a simple (and probably slightly loose) setup and to estimate whether such uncertainties are small enough to help better constrain the CH4 budget. Additionally, source signature data representativeness is generally poor, owing in part to the small number of samples and the lack of data for several regions, particularly for non-fossil sources. It might therefore be challenging to derive a robust uncertainty using a data-driven approach for each region of the world. However, there is definitely room for improvement, and future work will build on the present work to improve this methodology and assess and prescribe better uncertainties.

Note that random uncertainties are only one side of the coin when it comes to uncertainties. Systematic uncertainties must also be investigated when using isotopic constraints. In particular, derived source signature maps for wetland sources that carry systematic uncertainties. Typically, inverse modelers address such uncertainties by conducting numerous inversions using parameters designed to account for these systematic errors. The high computational cost associated with our system prevents us from running a large number of inversions. Nevertheless, only one scaling factor is applied for each region and each category. Consequently, there is a strong regional correlation between random errors. To some degree, this approach enables the detection and correction of regional systematic errors by our system. It is important to note, however, that this correction does not rely on any existing sensitivity analysis e.g.,, but rather uses solely the information provided by atmospheric isotopic observations.

To infer the initial CH4 and δ(13C,CH4) conditions for 1998, we run an inversion between 1988 and 1998 using the same prior emissions and isotopic signatures as that of INV_REF. We assimilate CH4 measurements from the NOAA GML network (56 stations) and δ(13C,CH4) measurements retrieved at five stations across the globe by the University of Washington (UW) between 1988 and 1996 , which we offset by 0.1 ‰ to account for measurement differences between INSTAAR and UW .

We also run a forward simulation from 1998–2010 to obtain a good spatial distribution of the δ(D,CH4) field and then apply a global offset to match the observed mean δ(D,CH4) value between 2005 and 2010. As we acknowledge that both methods are not perfect, considering the equilibration times of these isotopic compositions , we also prescribe large uncertainties in these initial conditions: 10 % for CH4, 3 % for δ(13C,CH4) and 20 % for δ(D,CH4). To optimize the initial conditions, the globe is regularly discretized using latitudinal and longitudinal bands. A step of 30∘ is applied to generate the bands, resulting in 6 × 12 = 72 regions. One scaling factor is optimized for each of these regions.

The reference inversion was first introduced in Sect. , and its setup is detailed in previous sections. Three sensitivity tests were conducted to investigate the influence of the setup of our system on posterior estimates when assimilating isotopic observations:

INV_CH4 is an inversion that only assimilates CH4 observations, not δ(13C,CH4) nor δ(D,CH4) observations, and thus only optimizes CH4 emissions for the five categories.

We also investigate the influence of the OH interannual variability (IAV) on our results. The OH IAV in the troposphere is usually derived from inversions using CH3CCl3 constraints or using global atmospheric chemistry–climate models . CH3CCl3 inversion-based studies suggest a decrease in post-2005 OH after a peak in 2000–2002. By contrast, the chemistry modeling studies derive a post-2005 stabilization after a quasi-continuous increase between 1990 and 2005, consistent with the OH IAV estimated by LMDz-INCA (see Fig. ). We therefore perform two more sensitivity tests:

INV_TURNER is designed to investigate the influence of the IAV on our results. We apply the IAV suggested by to the OH-INCA field. The associated OH field is named OH-TURNER, and its global concentrations decrease by 7 % between 2002 and 2014.

All the sensitivity tests are summarized in Table .

suggested that the results of the inversion should be discarded for the period up to 2–3 years after the beginning and the period 2–3 years before the end of the assimilation window. Although the term “spin-up” is not quite appropriate for the beginning of the window because the reason for discarding those results is slightly different, the outcome is still similar. The spin-up time, for an inversion, typically refers to a period at the beginning of the inversion when the errors in the assumed initial concentration field might influence the posterior fluxes. If the adopted spin-up period is too short, the inversion may fit the data by compensating for errors in the initial conditions with artificial emission adjustments. If the initial concentrations are also optimized, which is the case here, this effect can be reduced . However, source signatures are optimized, and, for a certain period after the start of the inversion, it is easier for the system to optimize the initial δ(13C,CH4) fields than the source signatures to fit the δ(13C,CH4) data. found that the optimized source signatures slowly move away from the prior value over time. After 2–3 years, the posterior value finally reaches a new and rather stable state. In other words, as the influence of the initial conditions on the isotopic composition decreases, the system prefers to optimize the source signatures, hence slowly reaching the posterior value. The equilibration time for δ(13C,CH4) and δ(D,CH4) is larger than the CH4 equilibration time , and therefore the signatures are more affected than the fluxes. For the end of the inversion, the reason for discarding the results is a lack of constraints, resulting in a slow return to the prior value. In this case, the use of the term “spin-down” is correct.

In addition, the strong 1997–1998 El Niño event leads to fire emission anomalies of about 20 Tga-1 according to GFED4s data and studies . Similarly, the following 1999–2000 La Niña event produced a wetland emission anomaly that persisted until 2002 . Finally, OH concentrations were also likely affected by this El Niño–Southern Oscillation (ENSO) phase . We therefore only analyze the results of the 2002–2014 period to limit the consequences of these effects. This period of time is large enough to explain the variations that caused the post-2007 CH4 renewed growth and the associated δ(13C,CH4) shift to more negative values.

The same convergence criterion was used for all inversions in order to ensure consistency between the results. The minimization process was stopped when at least 35 iterations (forward + adjoint runs) had been performed and the gradient norm ratio had fallen below 1 % of its initial value for four successive iterations.

A similar number of iterations (approx. 40) were necessary for all sensitivity tests. About 260 CPU hours were necessary to run a single iteration on LSCE (Laboratoire des Sciences du Climat et de l'Environnement) computational clusters consisting of Intel^® Xeon^® Gold 5317 central processing units (CPUs) with a frequency of 3.00 GHz. For this work, eight CPUs were run in parallel, resulting in a runtime of 32.5 h for a single iteration. More CPUs could not increase the overall performance because of some I/O (input/output) limitations of our offline model. With only one tracer to simulate, INV_CH4 therefore required about 2 months to reach the convergence criterion. Because the runtime is proportional to the number of simulated tracers, double this runtime was needed for the other inversions (two tracers), except for INV_DD, which required four times this runtime (four tracers).

While the number of CPU hours needed for these complex inversions remains reasonable, the overall runtime is excessive. It is therefore an important limitation of our system. Further developments of parallelization methods are being implemented to enable a significant reduction of the computational cost e.g.,. This method consists of breaking down the full assimilation window into multiple sub-windows and running smaller inversions in parallel for each sub-window. If source signatures remain constant, we expect the results to closely resemble those of a longer-term window inversion. Conversely, if source signatures are optimized, the influence of the initial conditions on the atmospheric isotopic composition might persist over a time that is larger than the length of the sub-window. In this case, source signatures might remain unchanged and the results could be impacted. Therefore, it is crucial to rigorously validate this parallelization method before interpreting its outcomes.

Before analyzing the optimized emissions and source signatures, we verify the quality of the model's fit to the constraining observations and evaluate it against independent data. A good fit shows that the system is operational over long time periods and that posterior emissions and source signatures are consistent with the observed state of the atmosphere.

The observed globally averaged CH4 amount fraction as well as the observed globally averaged δ(13C,CH4) isotopic composition at the surface are well captured by all the posterior simulations (see Fig. b, d). INV_REF shows a Pearson's moment correlation coefficient r of 0.994 for CH4 (RMSE is 2.8 nmolmol-1) and 0.936 for δ(13C,CH4) (RMSE is 0.04 ‰). The posterior simulation therefore captures the observations much better than the prior simulation does (RMSEs of 71.2 nmolmol-1 and 1.44 ‰, respectively). The inversion that best captures the δ(13C,CH4) isotopic composition is INV_DD, with a RMSE of 0.02 ‰. This shows that assimilating δ(D,CH4) observations slightly increases the agreement with isotopic observations without performing additional iterations.

The 2002–2007 δ(13C,CH4) stabilization is well reproduced by the model in INV_REF, with a mean RMSE of 0.02 ‰ over the period. However, the post-2007 trend is not as consistent (0.05 ‰), mainly due to an overestimation of the decreasing rate (0.03 ‰a-1 against 0.02 ‰a-1). The simulated δ(13C,CH4) seasonal cycle amplitude is also slightly smaller than the observed one (0.12 ‰ against 0.14 ‰, respectively), although the two signals are well phased. Note that our results are, however, still within the prescribed observation uncertainty range.

For the sake of completeness, we also provide a comparison between δ(D,CH4) observations and prior and posterior simulations from INV_DD in Appendix B (Fig. ). After the inversion, simulations capture the observed δ(D,CH4) data much better, reducing the RMSE from 9.3 ‰ to 1.2 ‰. Although the δ(D,CH4) data are much more limited than the δ(13C,CH4) data, linear regressions indicate a small negative trend (-0.23 ± 0.12 ‰a-1) between 2005 and 2009. Additionally, the trend is positive between 2005 and 2007 (+0.86 ±  0.42 ‰a-1) and negative between 2007 and 2009 (-0.47 ± 0.25 ‰a-1). This shows that δ(D,CH4) observations might also carry some information about the renewed growth of CH4 post-2007.

The model–observation agreement varies across the stations for both CH4 and δ(13C,CH4) (see Fig. a and c). CH4 at marine boundary layer (MBL) stations (i.e., where site samples consist mainly of well-mixed MBL air) is very well reproduced by the model (mean RMSE of 17.7 nmolmol-1 and mean bias of -1.87 nmolmol-1). The model has more difficulties in simulating amount fractions at several polluted stations, such as Lac La Biche, Canada (54.95∘ N, 112.45∘ W), Shangdianzi, People's Republic of China (40.65∘ N, 117.12∘ E), Anmyeon-do, Republic of Korea (36.54∘ N, 126.33∘ E), or the Southern Great Plains, United States (36.62∘ N, 97.48∘ W), presumably owing to transport errors, representation errors and/or inaccurate estimates of CH4 prior fluxes around these stations. The δ(13C,CH4) at MBL stations is generally correctly simulated, with RMSEs of 0.2 ‰–0.3 ‰, comparable to the prescribed uncertainties. However, posterior simulations slightly overestimate δ(13C,CH4) in Northern America and underestimate it in Central America and temperate North America. This suggests an over- and underestimation of flux-weighted source signatures in these regions, respectively. This is further investigated in Sect. .

For comparison, we performed one forward simulation with posterior fluxes obtained with INV_REF to compare our simulated X‾(CH4) to independent (i.e., not assimilated) satellite observations in 2010 and evaluate the optimized atmosphere (see Fig. ). The posterior mean bias is -13.0 nmolmol-1, indicating that the GOSAT observations are higher overall than our optimized X‾(CH4), even after the inversion. Further analysis reveals that the mean bias (-17.5 nmolmol-1) in the tropics (30∘ S–30∘ N) is larger in absolute value than the bias in the northern mid-latitudes (30–60∘ N; -5.7 nmolmol-1) or in the northern high latitudes (60–90∘ N; +3.0 nmolmol-1). also reported observing a negative X‾(CH4) mean bias (-6.55 nmolmol-1) when using the TM5 model and posterior estimates deduced from a surface-based inversion. Similar to ours, the TM5 biases were mostly located in the tropics and northern mid-latitudes. reported that model errors when simulating stratospheric CH4 amount fractions could contribute to the X‾(CH4) bias. However, they did not find a strong improvement for LMDz and TM5 when replacing model simulations with MIPAS (Michelson Interferometer for Passive Atmospheric Sounding) stratospheric CH4. While this suggests that the biases in the GOSAT simulation presented here are probably not caused by stratospheric discrepancies, it is important to note that the same authors conclude that current satellite measurements of stratospheric CH4 may lack the precision necessary to eliminate these biases.

If we assume that the biases are solely the result of emission discrepancies, our findings indicate that the estimated tropical posterior emissions from our inversions might still be underestimated. Although posterior biases are lower at the surface stations in the tropics (the mean value is -5.3 nmolmol-1), the number of stations is limited in this area, especially in South America and Central Africa. , using configurations similar to ours but no isotopic constraints, found that differences between emissions from GOSAT-based and surface-based inversions mainly occurred in the tropical regions.

The in situ-only and the GOSAT-only inversions performed by provided 113 and 212 independent pieces of information, respectively, highlighting that additional constraints can be gained using satellite data. As tropical fluxes likely had a significant influence on the renewed increase of CH4 around 2007, it would be interesting to assimilate satellite observations to increase the constraints in the tropics. Furthermore, jointly assimilating satellite observations and isotopic observations might be valuable because tropical emissions largely dominate the total global release of CH4. Consequently, a change in tropical emissions might influence the global flux-weighted source signature and impact the results of an inversion performed with isotopic constraints. Our system is capable of performing such a joint assimilation; however, since we are analyzing the influence of adding isotope constraints here, we prefer to assimilate only surface data as a first step.

Global emissions are estimated by INV_REF at 590 Tga-1 when averaged over the 2002–2014 period, which is larger than prior estimates by 28 Tga-1 (see Fig. ). This change mainly arises from increases in Asia (+15 Tga-1), Central and South America (+6 Tga-1), and Africa (+3 Tga-1). About 50 % and 25 % of the increase in Asia is due to the AGW and FFG categories, respectively, suggesting that the prior estimates in EDGARv4.3.2 for these regions are underestimates. However, global emissions estimated by inverse modeling are strongly dependent on the chemical loss prescribed in the CTM. OH is responsible for most of this loss and therefore the prescribed OH field greatly influences the results of the inversion. In particular, showed that a 1 × 105 cm-3 increase in prescribed OH concentrations leads to an increase in CH4 global posterior emissions of 40 Tga-1. Here, we are more interested in the trends in various CH4 emission categories and their contributions to the total emissions. Therefore, we have only used a single OH field with several trends. The influences of the trends on the dedicated inversions are further discussed in Sect. .

The posterior global distribution of emissions across the individual categories is only slightly different from the prior one (see Fig. ). Relative posterior–prior emission differences averaged over 2002–2014 are larger for WET (+7 %) than for AGW (+4 %), FFG (+6 %) or BB (+5 %). The increase in WET emissions is mainly located in the Amazon basin (43 %) and is responsible for a small shift in the WET contribution to the total emissions (from 32.1 % to 32.6 %), which is offset by a similar reduction in the AGW contribution (from 37.9 % to 37.4 %). The tropics (90∘ S–30∘ N), the northern mid-latitudes (30–60∘ N) and the high latitudes contribute about 60 %, 35 % and 5 % to the global emissions, respectively. Apart from a small reduction in the contribution from high latitudes, the latitudinal distributions of emissions are not modified.

Global and regional source isotopic signatures are calculated using a flux-weighted average to account for the global and regional source mixture, respectively. Therefore, they may be modified by the system due to a source mixture change and/or a source signature change in a specific region.

The inversion system shifts the global source signature considerably upward from -54.5 ‰ to -52.5 ‰ (see Fig. ). The global signature is highly constrained by the fractionation coefficients and the concentrations of radicals (OH, Cl and O1D) prescribed in the CTM. Our posterior global source signature is higher (less negative) compared to other estimates , mainly because we chose to prescribe Cl concentrations and an OH fractionation that are at the low ends of the existing ranges. Each additional percent of oxidation caused by the prescribed Cl sink would lead to a drop in the global source signature of about 0.5 ‰ . Using other recent estimates of Cl tropospheric concentrations (see Sect. ), our posterior global source signature would range between -54.5 ‰ and -52.5 ‰. In addition, the application of the fractionation value derived by instead of that derived by would likely shift the global signature downward by another 1.5 ‰.

AGW, BB and FFG source signatures are shifted upward by 0.7 ‰, 0.7 ‰ and 1.3 ‰, respectively. Most notably, the posterior global flux-weighted WET signature is considerably higher (-56.6 ‰) than the prior estimate (-60.8 ‰) (see Fig. ) with regard to recent estimates . This global shift mainly arises from upward regional source signature shifts in the tropics (+3.2 ‰) and in the northern mid-latitudes (+7.1 ‰), which together contribute 97 % of the posterior–prior global WET isotopic signature difference. The remaining contribution is due to an increase in the contribution from tropical WET emissions. Our posterior global estimate of the WET source signature strongly disagrees with the recent estimates. In Appendix B, Fig.  compares our prior and posterior signatures to observations from Supplement Data 1 provided by . Overall, prior estimates show a better agreement with observations than posterior estimates. This poor agreement suggests that prescribed uncertainties might be too large (at least for WET) and supports the idea that the system yields an important adjustment of the WET signature in order to keep fossil/microbial flux partitioning unchanged. Notably, the isotopic signature in Canada is shifted upward from -70.0 ‰ (prior) to -59.4 ‰ (posterior), whereas that of Russia is shifted downward from -68.7 ‰ to -73.8 ‰. Although this demonstrates that the system is capable of applying offsets with different signs across different regions, it also appears to be unphysical, as the processes driving the source signatures in these high-latitude regions are similar . Nevertheless, the number of observations of WET source signatures is small and local uncertainties are considerable, especially in the tropics, where emissions from WET are the largest and where our system applies the most impactful adjustment. It is therefore difficult to invalidate the posterior adjustment. Further investigation, including a better assessment and prescription of the random and systematic uncertainties, is needed.

Figures  and in Appendix B show the full temporal variations for the prior and posterior source signatures. For FF, high variations indicate a change in activities associated with fossil fuel extraction, e.g., switching from one location with a specific signature to another, transitioning from one fuel type (oil, gas, coal) to another, or a combination of both. For example, the substantial shift that occurred around 2009 in the United States was caused by a large increase in emissions from the extraction of natural gas. As we chose not to prescribe temporal error correlations between different years, the system is free to optimize each year independently to better fit δ(13C,CH4) observations. For certain continental regions, such as Africa, temperate Asia, or South Asia, the interannual variability of the source signature adjustments is large and rather unrealistic, especially when compared to the emission adjustments in the same regions (see Fig. ). It is unlikely that these changes occurred without detectable changes in emissions in the same areas, especially for temperate Asia, which exhibits larger emissions from FF than in the United States. These results suggest the need to prescribe yearly temporal error correlations to dampen this artificial interannual variability. However, the example from the United States also indicates that large changes can occur, and it is reasonable to assume, considering the lack of isotopic data, that the prior data might not contain any information about potential substantial changes. Therefore, while implementing stronger temporal correlations could be a way to mitigate unrealistic interannual variability for this category, it diminishes the likelihood of detecting such changes that remain undetected by the prior data. Nevertheless, it might be sufficient to reduce the prescribed uncertainties in the source signatures in order to balance out the pressure applied by the system on the emissions and the source signatures. Overall, the same reasoning applies to AGW, although there is no evidence from the prior data that AGW source signatures can change as rapidly as those of FF. Due to the scarcity of existing data on the temporal variability of source signatures, designing a data-driven methodology to estimate potential temporal correlations, especially at the regional scale, remains highly challenging. Investigating the correlations that the system creates between the uncertainties associated with source signatures and fluxes could offer a promising avenue for extending the analysis. Due to the high computational cost of an inversion performed with our inversion system, it is impossible to derive robust posterior uncertainties. This impossibility is a major drawback, and additional studies with this system cannot be performed in the future without tackling this issue.

We now present trend results comparing the time periods before (2002–2007) and after (2007–2014) the renewed increase. Posterior global emissions show a net increase of 24.0 Tga-1 (see Fig. ). This occurred in most of the regions aside from Europe (-1.8 Tga-1), Canada (-0.7 Tga-1) and Oceania (-0.2 Tga-1). China, South Asia (mainly India), temperate Asia, Southeast Asia and Africa accounted for 40 %, 18 %, 18 %, 10 % and 9 % of the associated positive increase (+26.7 Tga-1), respectively. These results are consistent with prior information that estimated a rise of 27.5 Tga-1. The large contribution from China to the global increase since 2002 agrees well with the regional estimate (40 %) from . We also estimate that Central and South America did not contribute to the renewed growth, in contrast with , who suggest a large contribution from Brazil (11.5 %) to the increase in global emissions. In this region, we find that small increases in AGW emissions (+1.4 Tga-1) and FFG emissions (+0.7 Tga-1) are offset by decreases in WET emissions (-0.8 Tga-1) and BB emissions (-0.8 Tga-1).

Global AGW emissions increased by 14.2 Tga-1 between 2002–2007 and 2007–2014. Europe is the only region where these emissions substantially decreased (-1.7 Tga-1). 80 % of the net AGW increase occurred in Asia, and the rest occurred in Africa and South America. FFG emissions increased by 14.9 Tga-1, with 50 % of this increase occurring in China. These emissions notably decreased in Africa (-0.7 Tga-1), Europe (-0.2 Tga-1) and Canada (-0.1 Tga-1). By contrast, WET emissions decreased by 2.4 Tga-1, with 71 % of the net decrease located in Central and South America (33 %), Canada (25 %), and Africa (13 %). BB emissions also decreased by 2.7 Tga-1, mainly due to decreases in Southeast Asia (55 % of the net decrease) and South America (24 %). Note that the analysis period does not include the 2015 El Niño event.

Our results therefore suggest that the renewed CH4 growth post-2007 (until 2014) was equally and mainly driven by increases in global AGW (49 %) and FFG (51 %) emissions. The decreases in global WET and BB emissions as well as the increase in global OH concentrations partially balanced this renewed growth. These findings are in partial agreement with recent studies , although only explained the renewed growth with equal contributions from the AGW and FFG categories. The small decrease in BB emissions is consistent with other estimates , but the decrease in WET emissions does not agree with recent findings that suggest either a constant trend or a positive trend .

However, posterior global WET emissions show negative anomalies between 1998 and 1999 and positive anomalies between 1999 and 2004. These anomalies are mainly located in South America, where about 30 % of WET emissions originate. suggested that the 1998–2000 ENSO (El Niño–Southern Oscillation) caused negative anomalies in WET emissions between 1998 and 2000 because of El Niño and subsequent positive anomalies between 2000 and 2002 because of La Niña. The fact that positive anomalies persist until 2004 rather than 2002 in our posterior emissions cannot be easily explained. Also, the positive anomalies last 4–5 years in total, which is not consistent with the 2–3 years inferred by . As AGW emissions are also large in South America, the inversion system might be wrongly attributing large but decreasing emissions between 2002 and 2004 to WET emissions rather than AGW emissions. If the period 2002–2004, which exhibits large positive anomalies, is discarded, we find a small increase of 0.3 Tga-1 in global WET emissions between 2004–2007 and 2007–2014, which is more consistent with the studies mentioned before.

The posterior global flux-weighted δsource(13C,CH4) decreased from -52.1 ‰ in 2002 to -53.1 ‰ in 2010 and then experienced an upturn to -52.5 ‰ in 2012–2014. Notably, between 2007 and 2010, the decline in the source signature was rapid (-0.2 ‰a-1), propagating into the atmosphere and contributing to the similar trend that appears in the observed globally averaged δ(13C,CH4). The subsequent increase after 2012 led to a stabilization of the associated atmospheric signal.

Between 2002–2007 and 2007–2014, all regional isotopic signatures were shifted downward (about -0.5 ‰ over the globe), except in China (+0.6 ‰). This is mainly explained by an increase in emissions from 13C-depleted sources in most of the regions but also by a decrease in emissions from 13C-enriched sources and a decrease in AGW and FFG source signatures.

Additionally, we use a simple mathematical framework to attribute the shift in the global flux-weighted signature to the different emission categories. Here, δ‾ denotes the global flux-weighted δsource(13C,CH4). A first-order estimate of δ‾ is given by 5δ‾=∑i=1Nδi×fiF, where fi denotes the global CH4 emissions from a specific category, N is the number of emission categories (five here), F=∑i=1Nfi is the total CH4 emissions, and ri=fiF is the contribution from each category to the total emissions. A small variation in the global flux-weighted source signature, dδ‾, can therefore be calculated using the derivatives dδi and dfi: 6dδ‾=∑i=1N∂δ‾∂δi×dδi+∑i=1N∂δ‾∂fi×dfi, with 7∂δ‾∂δi=fiF∂δ‾∂fi=δi×1-riF-∑j=1j≠iNδj×fjF2.

Using this simplified linear relationship, we find that the 0.34 ‰ decrease in δ‾ (see Fig. ) between 2002–2007 and 2007–2014 was due to

a decrease in the global AGW source signature (resulting in a shift in δ‾ of -0.22 ‰),

This decrease is partially offset by

a large increase in FFG emissions (resulting in a shift in δ‾ of +0.24 ‰),

The reference inversion assimilates both CH4 and δ(13C,CH4). To quantify the impact of assimilating δ(13C,CH4) data, INV_CH4 assimilates CH4 observations only and does not simulate the isotopic composition. Differences between INV_REF and INV_CH4 therefore provide insight into the influence of the isotopic constraint. Notably, these two inversions show similar results for CH4 emissions. As both inversions are constrained by the same global sink, global emissions estimated by INV_CH4 are only 0.3 Tga-1 lower on average over the 2002–2014 period. Tropical emissions are increased compared to INV_REF, and the contribution from tropical emissions to the total emissions is shifted from 59.8 % to 60.3 % (+2.5 Tga-1), mainly due to an increase in WET tropical emissions (+1.6 Tga-1), which is offset by decreases in the northern mid-latitudes (-1.2 Tga-1) and high latitudes (-0.2 Tga-1). WET emissions increase by 1.1 Tga-1 in INV_CH4 between 2002–2007 and 2007–2014 (Table ) instead of decreasing in INV_REF. The increases in AGW, FFG and WET emissions contribute 50 %, 47 % and 3 % of the post-2007 renewed growth, respectively, and are therefore slightly different from the results of INV_REF. To summarize, adding the isotopic constraint (INV_REF as compared to INV_CH4) slightly decreases the contribution from tropical emissions to global emissions, removes a very small contribution from WET emissions to the post-2007 renewed growth, and slightly changes the contributions from emission increases to the post-2007 renewed growth.

Differences between INV_REF and INV_CH4 are small, presumably as a result of the large prior uncertainties in source signatures, which allows the inverse system to adjust the atmospheric isotopic compositions at a low cost by changing signatures rather than emissions. To test this hypothesis, we run INV_LOCKED assuming a perfect knowledge of isotopic signatures (no uncertainties in the prior source isotopic signatures). Although the global total emissions obtained with INV_REF and INV_LOCKED are very similar, the individual contributions from each emission category are modified (Fig. c). On average over the 2002–2014 period, FFG emissions are increased by 24 % compared to INV_REF, mainly due to large relative increases in China and the Middle East (+30 % to 50 %). Global FFG emissions amount to 153 Tga-1, making them more consistent with the large revisions (150–200 Tga-1) derived by with recent isotopic data. WET emissions located in boreal regions and in South America are decreased by around 30 %, whereas WET emissions from Central Africa are slightly increased, leading to a decrease in global WET emissions of 14 %. Finally, BB emissions are increased by 41 % and AGW emissions are slightly decreased by 7 %, with globally uniform changes (Fig. a and b). Furthermore, INV_LOCKED explains the renewed growth through contributions from enhanced FFG (46 %), AGW (37 %) and WET (17 %) emissions between 2002–2007 and 2007–2014 (Table ). WET emissions therefore actively participate in the post-2007 CH4 growth in INV_LOCKED, as opposed to INV_REF. The FFG, AGW and WET emission increases are, however, offset by a large decrease in BB emissions, nearly three times larger than in the other inversions. Also, emission IAVs are increased for all categories. In particular, the BB emission peaks in 2006 and 2009 are much higher in INV_LOCKED than in INV_REF relative to the mean over the period. Such variations are probably too large to be realistic when compared to prior data and other inversion studies e.g.,. However, they provide an upper bound for emission trends as constrained by isotopic values. In many inversion studies e.g.,, isotopic signatures are fixed, and our results suggest that this may lead to significant errors in CH4 emission trends. This stresses the importance of finding the right balance between overconstrained signatures, as in INV_LOCKED, and likely underconstrained signatures, as in INV_REF. At present, either isotopic constraints are too loose to yield critical information about sectorial and regional CH4 emissions or our estimates of the associated uncertainties are overestimated in our methodology, which is also a possibility that we will address in future studies.

Finally, assimilating δ(D,CH4) observations and optimizing δsource(D,CH4) source signatures in INV_DD has a very small influence on our posterior emission estimates, as indicated in Table . The most significant difference observed is a small positive shift of +0.5 ‰ in the BB posterior source signature compared to INV_REF. Consequently, with our setups, assimilating δ(D,CH4) does not appear to provide any substantial additional constraint on the CH4 budget estimate. Several factors may contribute to this result: (1) the existing network provides comparatively few δ(D,CH4) observations in comparison to δ(13C,CH4) observations; (2) δ(D,CH4) observations only span the period from 2005–2010 and so the full run cannot be fully constrained by this data; and (3) the constraints may be too weak due to an overestimation of the prescribed uncertainties in δsource(D,CH4) source signatures. As including δ(D,CH4) in the inversion doubles the computational cost compared to a setup like INV_REF, we recommend that δ(D,CH4) should not be assimilated in our system until either the computational cost can be reduced, more observations become available or lower uncertainties are established. However, a hypothetical network of δ(D,CH4) measurements obtained at a reasonable frequency and spanning a longer period of time could efficiently complement δ(13C,CH4) observations and provide a wealth of information . More specifically, reactions with OH, O1D and Cl have fractionation coefficients that depend on the isotope. Therefore, incorporating δ(D,CH4) constraints might help to disentangle the effects of the associated sinks and provide additional insights into the global sink and its mixture. However, optimizing the sinks introduces additional degrees of freedom and complexifies the inverse problem. With the current system and at such a high resolution for the optimized variables, we recommend against the simultaneous optimization of both the source signatures and the sinks. However, a coarser resolution for the optimized variables, or at least for the sink, might be able to accommodate a simultaneous optimization.

Last but not least, we have tested the impacts of OH trends on our results. INV_FLATOH and INV_REF show very similar results (Table ). The main difference is that INV_FLATOH infers a smaller increase in total emissions (+18 Tga-1) between 2002–2007 and 2007–2014 than INV_REF does (+24 Tga-1). As a smaller sink is prescribed in INV_FLATOH compared to INV_REF, the increase in total emissions required to fit the observations of CH4 amount fractions is also smaller. The contributions from the emission categories to the total emissions are barely affected (± 0.2 %). The contributions from AGW and FFG emissions to the increase in the total emissions between the two periods are exactly the same as in INV_REF. In addition, as the inter-hemispheric OH ratio is not modified, the contributions from the tropics, mid-latitudes and high latitudes are identical for each emission category. Overall, the differences between INV_REF and INV_FLAT are negligible and do not affect the conclusions deduced from the INV_REF results.

On the contrary, INV_TURNER infers a decline of 1.6 % in global emissions between 2002–2007 and 2007–2014 (Table ), mainly driven by a large decrease in WET emissions (-15 Tga-1) and a slightly larger decrease in BB emissions (-4 Tga-1) than in INV_REF. With this prescribed OH IAV, the renewed CH4 growth post-2007 is therefore entirely caused by a large decline in the global OH sink between the two periods. Changes in AGW and FFG emissions in INV_TURNER are still positive but are 2–3 times smaller than in INV_REF. Using all the information provided by the sensitivity tests inferring a net increase in emissions (i.e., without INV_TURNER), this increase is principally attributed to fossil sources (50 ± 3 %) and agriculture and waste sources (47 ± 5 %). Nevertheless, there is substantial variation in the results between configurations that optimize source signatures and those that do not.

The decline in the OH sink between the two periods affects the δ(13C,CH4) atmospheric signal in two opposite ways:

If the OH sink is the only sink, a decline in OH concentrations has no effect on δ(13C,CH4) in the long term (several decades) because the mean fractionation is not affected. However, in the short term (a decade), as OH concentrations decrease, 12CH4 and 13CH4 atmospheric lifetimes increase. Due to the fractionation effect, there is a time lag between increases in 12CH4 and 13CH4 amount fractions. 12CH4 accumulates faster than 13CH4, leading to a decrease in δ(13C,CH4).

– referred to as “BA22” in this subsection – quantify the global CH4 budget and investigate the post-2007 renewed growth, using the TM5-4DVAR inversion framework to assimilate both CH4 and δ(13C,CH4) measurements. In our opinion, their work is strongly relevant and tackles this complex topic with an appropriate and robust methodology. As our goals are similar, we compare our systems and methodologies here.

First, it is worth mentioning that our system is capable of assimilating δ(D,CH4)) observational data and optimizing the associated source signatures. Although this feature has a small influence on our results in this work, we believe that its relevance will grow as more δ(D,CH4)) data become available and the associated uncertainties decrease.

As already stated in their paper, the main difference between our systems is the optimization of source signatures. BA22 prefer to investigate the influence of the source signature uncertainties with different sensitivity tests that adopt various source signature maps. This choice relies on the fact that they can run a large number of inversions at low cost using a parallel configuration. It is a good strategy to assess the influence of systematic errors in source signatures. In our work, we did not investigate this influence, and we decided to optimize source signatures in order to consistently account for random errors in source signatures and emissions at the same time. As both interact and impact the atmospheric composition in very complex ways, it seemed important to us to perform at least one inversion combining all the uncertainties.

We agree with BA22 that the second major difference between the two studies lies in the construction of the prior CH4 fluxes. However, they suggest that we constructed a prior that approximately matches the atmospheric CH4 growth rate in . In and in the present study, we derived our prior fluxes and source signatures solely on the basis of bottom-up estimates and literature data. For CH4 emissions, the fact that prior simulations match the atmospheric CH4 growth rate shows that bottom-up estimates are roughly consistent with atmospheric data, even before the inversion process.

We prefer not to adjust prior fluxes to match observational δ(13C,CH4) data because, in this case, we assume that bottom-up estimates suffer from strong systematic uncertainties, which is difficult to demonstrate. When adjusting prior CH4 fluxes, it is also assumed that CH4 emissions derived by bottom-up estimates are more likely to be wrong than source signatures estimates. We believe that the opposite is more plausible because observational data on source signatures are very scarce at present. Therefore, we prefer to start from robust and validated data and let the inversion system combine them with the assimilated atmospheric observations and the random uncertainties. BA22 start with a flat prior, and the posterior results deviate significantly from the prior. As the prior data do not seem to have a strong influence on the posterior results, adjusting prior fluxes prior to the inversion should have no effect on the results. However, this is yet to be confirmed in our case, i.e., with a nonlinear observation operator.

BA22 are able to calculate posterior uncertainties using a large ensemble of inversions. It is a precious feature that we do not possess at present. This is made possible by the relatively low computational cost of their configuration (an adjusted prior and a linear formulation) but also by the fact that they divide the full assimilation window into shorter sub-windows (5 years) that are run in parallel. A 1-year overlap with previous and next sub-windows is applied. It would be interesting to compare the posterior results obtained with this parallelized configuration to an inversion with a complete assimilation window. As the relaxation time for isotopic composition in the atmosphere in response to a perturbation is much larger decades; than that for CH4 itself, we are concerned that using such short time periods might affect posterior results, especially if the observation operator is nonlinear. Modifying the prior data to fit the observed isotopic composition, as in BA22, might be a prerequisite for the success of this method.

It is clear that these setup differences propagate to posterior results. Using the additional δ(13C,CH4) data, BA22 find that fossil CH4 emissions and microbial emissions contributed about 15 % and 85 %, respectively, of the post-2007 CH4 growth. As presented in the previous sections, our results are completely different. Most notably, they find a contribution of 30 % from fossil emissions to the total emissions on average over 1999–2016. While our reference inversion finds a much smaller number (21 %), our inversion with fixed source signatures (INV_LOCKED) gives a closer value (26 %). The small source partitioning discrepancy between INV_LOCKED and BA22's inversion results might also be caused by a difference in prescribed isotopic fractionation, as suggested by the sensitivity analysis of .

We cannot fully explain why BA22's conclusions about the causes of the renewed CH4 growth post-2007 differ so substantially from our own. However, it appears that BA22 also use a robust methodology to study the global CH4 budget and the renewed growth. Despite significant differences, we find good complementarity between our approaches and hope to learn from each other in order to improve our systems and reconcile our results.