For fossil fuel emissions in Europe, regional measurements from the European Methane Isotope Database (EMID; ) were combined with the global inventory from to improve the spatial accuracy of isotopic signatures. The dataset, based on the compilation by , includes isotopic data from over 13 000 locations from 347 references, though it remains largely U.S. centered. The EMID database provides additional coverage, including both literature data and new measurements from the MEMO² project for fossil fuel sources over Europe . The extended global database, combining EMID with and MEMO², consists of over 13 313 δ13C measurements from 64 countries.

The integration of these two datasets was conducted as follows:

Country-level average δ13C-CH4 signatures were computed from EMID for some European countries (e.g. UK, Germany, Poland, Romania, France), corresponding to those for which data are available in the EMID inventory.

To ensure full temporal coverage over the 1998–2022 period, extrapolation was required for certain datasets whose original time series ended prematurely, specifically for the oil and gas and wetland sub-sectors. In both cases, specific strategies were applied based on known emission trends and isotopic characteristics of source signatures.

For the oil and gas sub-sector, the dataset from ends in 2016. Post-2016 values were extrapolated only for U.S. data, based on documented trends in U.S. unconventional gas production (U.S. Energy Information Administration, EIA, https://www.eia.gov/energyexplained/us-energy-facts, last access: 14 March 2025), notably shale gas, which typically exhibits lighter δ13C-CH4 signatures than conventional sources . Basin-level variability in both gas production and isotopic composition was accounted for using data from the EIA natural gas reports (https://www.eia.gov/naturalgas/weekly/, last access: 9 March 2025) and published isotopic measurements . Outside the U.S., signatures values were held constant. We adopted a two-step method to extrapolate U.S. δ13C-CH4 values post-2016:

computing basin-weighted δ13C-CH4 values based on production volumes, using shale gas data per basin (EIA, https://www.eia.gov/dnav/ng/ng_prod_shalegas_s1_a.html, last access: 9 March 2025) and basin-specific isotopic signatures ;

For wetlands, the monthly δ13C-CH4 source signature maps from also end in 2016. To extend the dataset beyond this point, we extrapolated the 2016 seasonal cycle linearly and incorporated the long-term trend in source signatures of -0.02‰yr-1 observed over the 1984–2016 period .

To quantify the uncertainty associated with the aggregated δ13C-CH4 source signatures (σtotal), we consider two main components: (i) the propagated uncertainty from sub-sector level to sector level (σprop), i.e. the intrinsic isotopic uncertainty from sub-sector, and (ii) the aggregation uncertainties (σagg), i.e. the variability introduced when aggregating sub-sector signatures into sector-level values, using a given set of flux weights from prior inventories. Assuming that the two types of uncertainties are independent, the total uncertainty for each aggregated sector is calculated by combining these two components: 2σtotal=σprop2+σagg2

The corresponding relative uncertainty (in percentage) is then calculated as: 3Relative uncertainty (%)=σtotalδ13Csector‾×100 where σprop corresponds to the propagated uncertainty at the sector level derived from sub-sector uncertainties (‰), σagg represents the uncertainty associated with the aggregation method, estimated from sensitivity tests (‰) and δ13Csector‾ denotes the mean isotopic signature of the aggregated sector (‰).

Because the propagated uncertainty σprop is provided as a range, the resulting total uncertainty σtotal is also expressed as a corresponding range rather than a single value.

The term σprop represents the propagated uncertainty from sub-sector isotopic composition (σi) to the aggregated sector level. First, the isotopic uncertainty for each sub-sector (σi) is estimated using two complementary approaches, depending on the origin of the data:

The standard deviations of δ13C-CH4 values reported in the literature, using the dataset compiled by , which captures spatial variability and methodological dispersion;

The Monte Carlo approach provides a precise uncertainty estimate directly linked to sub-sector diversity but may underestimate broader, large-scale variability and inter-regional differences. Conversely, the literature-based standard deviation offers a more conservative benchmark that captures additional inter-study, regional, and methodological variability. The two approaches (literature-based standard deviations and Monte Carlo uncertainties) provide lower and upper estimates of the sub-sector isotopic uncertainty.

Second, the sub-sector uncertainties (σi) are propagated into the aggregated sector uncertainty component (σprop) using a flux-weighted error propagation approach: 4σprop=∑iFiFtotal2σi2 where Fi is the flux of sub-sector i (Tg yr-1), Ftotal is the total flux of the aggregated sector (Tg yr-1), and σi is the isotopic uncertainty of sub-sector i (‰).

The aggregated sector uncertainty (σprop) is reported as a range, with the lower bound corresponding to the Monte Carlo estimate and the upper bound corresponding to the literature-based estimate.

This component reflects the sensitivity of aggregated signatures to methodological choices of weighting fluxes in the construction of the aggregated sector dataset. It is evaluated through a series of sensitivity tests (see Sect. ), and computed as the standard deviation across the resulting aggregation scenarios: 5σagg=1N∑j=1N(x¯j-x¯)2 where x¯j is the mean isotopic signature derived from aggregation method j, x¯ is the overall mean isotopic signature across all tested methods, and N is the number of tested aggregation methods.

In summary, this framework provides consistent tools to estimate sectoral δ13C-CH4 uncertainties, combining sub-sector variability and aggregation effects. Beyond sectoral aggregation, isotopic uncertainties also carry a latitudinal dimension, driven notably by the distribution of C3 and C4 plants across source sectors and by stratospheric transport processes; these spatial gradients are discussed further in Sect. . Because uncertainties in isotopic signature values and their spatial allocation propagate into regional and global inversions, ultimately influencing source attribution, it is particularly important to assess them. The resulting uncertainties are directly usable in atmospheric models and inversions to inform prior error structures and guide optimization choices (see Sect. ).

The simulations were conducted using the LMDz transport model, a component of the coupled model from the Institut Pierre-Simon Laplace (IPSL-CM), developed at the Laboratoire de Météorologie Dynamique (LMD; ). The offline version was used, driven by ECMWF ERA-Interim reanalyses, as described in . LMDz operates at a horizontal resolution of 3.75°×1.875°, with 39 vertical hybrid sigma-pressure levels extending up to approximately 75 km. The model time step is 30 min. Vertical diffusion is parameterized following the local approach of , while deep convection is represented using the Kerry Emanuel scheme .

Source categories are described in Table  with the following emission fluxes: All flux datasets were selected to ensure consistency with the Global Methane Budget (GMB) framework , which provides the reference basis for methane emission assessments.

Anthropogenic sources. Emissions from agriculture, waste, fossil fuel exploitation, and biofuel combustion are taken from the EDGARv8 inventory . This dataset provides detailed activity data and region-specific emission factors, enabling realistic representation of spatial and temporal variability across sectors and regions.

Initial conditions for CH4 and δ13C-CH4 for the year 1998 were derived from an inversion covering the period 1988–2020 . A spin-up period from 1998 to 2016 was used to allow the model to adjust to realistic atmospheric gradients, particularly for δ13C-CH4 . Only the period 2016–2020 is analyzed to ensure that the model has reached a stable state and to enable direct comparison between simulations, which differ only from the perturbed parameters (see Sect. ).

Methane oxidation is simulated using the generic chemical module of the offline LMDz model . The chemical scheme includes reactions with hydroxyl radicals (OH), excited atomic oxygen (O(1D)), and chlorine radicals (Cl). Oxidant concentrations are prescribed from precomputed daily 3-D fields summarized in Table .

Hydroxyl radicals (OH). OH is the primary sink for atmospheric methane. Monthly mean OH fields are taken from LMDz-INCA simulations , driven by consistent meteorological forcing with the transport model.

Isotopic fractionation from CH4 oxidation is expressed by the kinetic isotope effects (KIEs), which quantify the difference in reaction rates between 12CH4 and 13CH4. The KIE values used for each oxidant are summarized in Table . A sensitivity analysis is conducted on the OH-KIE value, given its significant impact on the atmospheric δ13CH4 signal (see Sect. ).

Soil uptake is a significant sink for atmospheric methane, and accounts for approximately 35 TgCH4yr-1 in the global methane budget . Moreover, the soil sink is known to induce isotopic fractionation, which must be accounted for in forward simulations of the isotopic signal . In this study, rather than using a fixed negative flux, we represent soil uptake as a first-order deposition process, where the deposition velocity vdep is computed from prescribed soil fluxes and modeled surface CH4 concentrations. This approach allows the sink to respond to atmospheric gradients and enables a consistent application of isotopic fractionation. Representing soil uptake via a deposition velocity ensures that the isotopic fractionation is applied dynamically to both 12CH4 and 13CH4, in proportion to their atmospheric concentrations and the kinetic isotope effect.

The monthly deposition velocity is calculated as: 6vdep=FsoilCCH4 where Fsoil is the prior soil uptake flux from the GMB dataset , and CCH4 is the modeled surface CH4 mole fraction from inversion .

To account for isotopic fractionation during methane uptake by soils, a kinetic isotope effect (KIE) is applied. The deposition velocity for ¹³CH4 is computed as: 7vdep,13CH4=vdep,12CH4KIEδ13C with a fractionation factor KIEδ13C=1.020, based on the experimental results of .

The methodology presented in this section provides a consistent framework for simulating atmospheric CH4 mole fractions and δ13C-CH4 signals using the CIF coupled to the LMDz transport model.

Figure  presents the aggregated global maps of δ13C-CH4 source signatures averaged over the period 1998–2022. The spatial variability in these maps reflects both the diversity of isotopic signatures across sub-sectors (see Sect. ) and the spatial distribution of methane fluxes within each grid cell. The fluxes used for aggregation are detailed in Table , which also reports the mean, minimum, and maximum δ13C-CH4 source signatures. Additional maps showing individual sub-sectors are provided in Fig. S1 in the Supplement. The key spatial features for δ13C-CH4 of each sector are summarized below:

FFG. More enriched signatures (close to -40 ‰) are observed in regions with oil and gas exploitation (e.g. Middle East, USA), while more depleted values (below -50 ‰) occur in coal-dominated, oil sands, or geological seepage areas (e.g. Canada). These patterns reflect the isotopic diversity of fossil sub-sources, as documented in previous studies (Figs. S1 and S2 in the Supplement; ).

The spatial patterns described above directly influence the simulated atmospheric δ13C-CH4 signal, which may affect the results of inverse modeling. Furthermore, uncertainties in sub-sector isotopic values and their spatial allocation propagate into regional and global inversions, ultimately influencing source attribution. A detailed quantification of these uncertainties is therefore essential to assess their potential impact and guide the design of robust inversion frameworks (Sect. ).

The maps also highlight regions where different source types co-occur, potentially complicating the separation of natural and anthropogenic emissions. For example, tropical regions such as Southeast Asia, parts of Africa, and northern South America host both WET and AGW emissions in close proximity. The spatial co-location and similarity of isotopic signatures between these sources can produce blended atmospheric signals, making it challenging to disentangle their respective contributions in top-down inversions. These overlaps underscore the importance of high-resolution spatial and temporal information in both methane fluxes and isotopic signatures to improve source attribution in complex emission environments.

In summary, the updated maps reveal three robust spatial features: a strong north–south gradient in BB and WET signatures driven by C3/C4 vegetation distribution, a contrast between depleted livestock and enriched waste signatures within AGW, and a marked enrichment over major oil and gas basins for FFG. Regions of source overlap, particularly in the tropics, emerge as priority zones where high-resolution isotopic information is most needed for source attribution.

Figure  presents the monthly δ13C-CH4 source signature time series for the methane sectors considered in this study over the period 1998–2022. The time series are based on flux-weighted averages. In the following, we describe the temporal evolution and variability of each sector's isotopic signature and discuss the underlying drivers. Most aggregated source sectors show limited temporal variability in their δ13C-CH4 signature over the 1998–2022 period, with typical long-term changes below 1 ‰. The isotopic signature of FFG sources exhibits a slight enrichment over time, increasing from approximately -44.6 ‰ in the early 2000s to around -43.9 ‰ after 2020. Several factors can explain this increase. First, the rapid expansion of shale gas production, particularly in North America after 2010, introduced additional emissions with relatively enriched isotopic signatures compared to conventional natural gas . In the US, shale gas accounted for 48 % of total dry natural gas production in 2013 and reached 82 % by 2023 . Second, the relative contributions of fossil fuel subsectors have evolved over the past two decades. According to , based on the synthesis of multiple emission inventories, including EDGARv6.0 and v7.0 , GAINSv4.0 , CEDS , the share of coal-related emissions, typically more depleted in δ13C-CH4, increased from approximately 21 % of total fossil fuel emissions in the early 2000s to about 25 % in the 2010s. Meanwhile, oil and gas emissions, associated with less depleted signatures, also grew but were partially counterbalanced by this increasing coal contribution. This sectoral shift has likely moderated the enrichment of the global signature driven by shale gas. Third, geological sources are considered stable over time, both in terms of flux and isotopic signature, providing a background that buffers the sector's temporal variability. Overall, these combined effects explain the modest but persistent enrichment observed in the FFG sector's isotopic signature over the past two decades. No significant seasonal variation is detected for FFG sector, which is consistent with the absence of seasonal modulation in the underlying inventories. However, real-world temporal variability may not be fully captured by inventories. Geopolitical events (e.g. changes in gas trade flows) and evolving gas processing or distribution practices could alter the isotopic composition of emissions over time. While isotopic signatures are generally stable , shifts in the relative contribution of end-member sources or changes in gas composition could induce detectable trends . Such factors remain difficult to assess without dedicated observational constraints.

The AGW signature shows a slight enrichment from about -60.4 to -59.9 ‰. Since the isotopic values of the subcategories (e.g. livestock, wastewater, landfill, rice, and agricultural waste) are held constant, this change reflects a redistribution of fluxes within the AGW sector. Specifically, the relative contribution of livestock and rice cultivation, which together accounted for approximately 69 % of AGW emissions in 2000–2009, decreased to around 68 % in 2010–2019. At the same time, waste-related sources (i.e. landfills, wastewater, and agricultural waste) increased from about 31 % to 32 % of AGW emissions over the same period. This subtle but consistent shift towards more enriched waste sources likely drives the observed isotopic enrichment, particularly in rapidly developing regions with strong urban waste emissions . Moreover, the AGW sector exhibits a seasonal cycle in its isotopic signature, primarily driven by rice cultivation, which is the only sub-sector with pronounced seasonal variability in emissions. The slight reduction in the relative contribution of rice, from about 19 % to 16 % of total AGW emissions between 1998–2022, leads to a corresponding decrease in the seasonal amplitude of the δ13C-CH4 signature, reflecting the reduced influence of this highly seasonal sub-sector compared to other more temporally stable AGW sources. In addition to changes in source distribution, long-term shifts in agricultural practices may slightly influence isotopic source signatures. For instance, changes in livestock diet, such as a varying balance between C3 and C4 feeds, and large-scale changes in atmospheric δ13C-CO2 may both affect CH4 isotopic composition. showed that the progressive depletion of atmospheric δ13C-CO2 since the 1960s, driven by fossil fuel combustion, has led to a corresponding decline in δ13C of both C3 and C4 plants. This trend propagates through the food chain, ultimately affecting the δ13C-CH4 signature from ruminants. Although such effects likely occur, they are not yet explicitly represented in our dataset due to a lack of systematic isotopic observations that track temporal changes in agricultural feedstocks or cultivation practices.

The WET isotopic signature remains largely stable over the period 1998–2022, with only a minor change from approximately -58.6 to -58.7 ‰, consistent with the trend reported by . More importantly, wetlands exhibit a strong seasonal cycle in δ13C-CH4 signatures, with more depleted values during the summer months. This seasonal variation is driven by the dominance of methane emissions from boreal wetlands during summer, which are isotopically lighter than emissions from tropical wetlands .

The BB isotopic signature remains stable around -24.3 ‰ throughout the study period, consistent with the identical δ13C-CH4 signatures prescribed for biomass burning and biofuel burning subcategories . However, the sector exhibits a pronounced seasonal cycle, with less depleted values during the boreal winter. This pattern reflects the higher contribution of biomass burning emissions from the Southern Hemisphere during this period, which are characterized by relatively enriched δ13C-CH4 signatures (see Fig. ).

The NAT isotopic signature remains constant throughout the period, as expected. Values for both subcategories (oceanic and termites) are derived using climatological fluxes and fixed δ13C-CH4 values, leading to negligible variability over time.

In summary, four sectors (FFG, AGW, BB, WET) show modest but identifiable temporal signals over 1998–2022: a slight FFG enrichment driven by shale gas expansion partially offset by rising coal contributions, a slight AGW enrichment from increasing waste shares, a stable BB mean with strong seasonal cycles, and a stable WET mean with strong boreal-summer-driven seasonality. NAT remains constant by construction. These temporal patterns are small (typically <1 ‰) but systematic, and should be preserved in inversion priors.

Table  presents the uncertainty ranges associated with the δ13C-CH4 source signatures for each major sector and their respective sub-sectors. The uncertainty in aggregated δ13C-CH4 source signatures arises from two primary components: (1) the variability across sub-sectors, expressed first as the range of sub-sector uncertainties (σi), and then as a flux-weighted propagated uncertainty to the sector level (σprop), and (2) the aggregation uncertainty (σagg), reflecting the sensitivity of sectoral signatures to the prior flux distribution used for aggregation (see methodology in Sect. ). In the following, we first analyse the uncertainty intrinsic to each sectors, then examine the additional uncertainty introduced by aggregation at the sector level, and finally discuss the total propagated uncertainties and their implications for methane source attribution.

The propagated uncertainty at the aggregated sector level, denoted σprop, captures the combined effect of isotopic variability among sub-sectors within each source category. It is computed using a flux-weighted combination of individual sub-sector uncertainties (σi), as described in Sect. . These sub-sector uncertainties reflect both the intrinsic heterogeneity of emission processes and the spread of values reported in the literature. The resulting σi and σprop values for each sector are summarized in Table . Below, we describe the dominant contributors to σprop, ordered from highest to lowest sectoral uncertainty.

WET exhibits the largest propagated uncertainty (σprop=0.4–8.2 ‰), directly inherited from the wide range of sub-sector values (σi=0.4‰–8.2 ‰). The high variability reflects the influence of multiple environmental factors, including substrate type, methanogenic pathways (acetate fermentation vs. CO2 reduction), and the δ13C content of the organic matter . Tropical wetlands in particular remain under-sampled, despite representing a dominant fraction of global wetland CH4 emissions .

Beyond sectoral aggregation, the propagated uncertainties also exhibit a latitudinal structure. The distribution of C3 and C4 plants drives systematic gradients in isotopic signatures across the WET, BB, and AGW sectors: tropical regions, dominated by C4 vegetation, tend to produce more enriched δ13C-CH4 signatures, while boreal and temperate regions, where C3 plants prevail, yield more depleted values . This latitudinal gradient contributes a spatially structured component to isotopic uncertainty that is not fully captured by global or regional averages.

These propagated uncertainties from sub-sectors highlight the need for more systematic and representative measurements, particularly in under-sampled regions (e.g. tropics), and sectors (e.g. coal mines). The scarcity of isotopic measurements in tropical regions is a critical limitation: despite hosting some of the largest CH4 sources globally, including tropical wetlands, rice paddies, and livestock systems, these regions remain under-represented in isotopic databases . Targeted field campaigns and isotopic monitoring networks could help reduce this uncertainty. Nonetheless, because of the inherent diversity and variability of methane formation processes, some degree of irreducible uncertainty must be accounted for and formally propagated in inversion frameworks (see Sect. ).

Aggregation is primarily required for computational efficiency in inversion frameworks (Sect. ). But, this necessary simplification introduces methodological uncertainty that must be explicitly quantified and propagated (Sect. ). The aggregation uncertainty, denoted σagg, represents the error introduced when multiple sub-sectors are combined into a single aggregated sector based on their respective emission fluxes. The corresponding values for each sector are reported in Table . Although this component is generally smaller and more stable than σprop, it increases for sectors that involve several formation pathways and isotopic fractionation, such as FFG (1.0 ‰) and AGW (1.7 ‰), where heterogeneous sub-sector compositions lead to significant propagation effects. These values illustrate that choices made in emission inventories, not only in flux magnitudes but also in source definitions and partitioning, can significantly shape sector-level isotopic signatures.

Notably, discrepancies among major methane inventories, including EDGARv8 , GAINSv4 , CEDSv2021 , and GFEIv2 , contribute significantly to this aggregation uncertainty. For instance, for the FFG sector, the aggregation uncertainty is around 2.2 % at the global scale, reflecting heterogeneity across inventories. Variability in emission factors and inventory methodologies, as described in , contribute strongly to this uncertainty. For example, oil and gas system emissions estimates vary considerably due to differences in emission factors and methodological assumptions across countries and inventories. Similarly, for the AGW sector, the aggregation uncertainty (2.8 %) reflects differences in how fluxes are allocated among agriculture and waste-related sub-sectors across inventories. For example, manure emissions can be allocated either to the agriculture or to the waste category, which can shift the aggregated isotopic signature. On the contrary, for WET, no aggregation uncertainty is reported because this sector relies on a single wetland flux dataset for weighting, precluding a cross-inventory sensitivity assessment. We note, however, that the latitudinal contrast between tropical (more enriched, ∼-50 ‰) and boreal (more depleted, ∼-70 ‰) wetlands contributes substantially to the propagated uncertainty σprop already reported for this sector (0.4 ‰–8.2 ‰). For NAT, since termite and ocean sources do not overlap geographically, the aggregation uncertainty is null.

The total uncertainty, σtotal, represents the combined effect of sub-sector variability and aggregation uncertainty. The full range of uncertainty values per sector (σtotal) is reported in Table . This range corresponds to the spread of possible standard deviations (i.e. the uncertainty spread) rather than to a range of actual δ13C-CH4 signature values. This total uncertainty varies across sectors. For example, the BB and WET sectors exhibit wide uncertainty ranges, from 0.5 ‰ to 6.0 ‰ and from 0.4 ‰ to 8.2 ‰, respectively, primarily driven by sub-sector uncertainties. For the BB sector, the total uncertainty is largely driven by the sub-sector variability of biofuel burning. For WET, the large uncertainty is mainly due to the inherent spatial and seasonal variability of wetland emissions. The FFG and AGW sectors show total uncertainties of up to 5.3 ‰ and 3.3 ‰, respectively, which is consistent with their source diversity and complex sub-sector structures. NAT sources display lower average uncertainties overall.

The columns σMenoud (standard deviation within the dataset) and Uncertainty‾Menoud (mean uncertainty) in Table  offer a comparison with extensive literature compilations . Our total uncertainties are comparable to or smaller than these literature-based estimates, suggesting that the applied aggregation methodology provides a structured and quantitative framework for uncertainty propagation that complements broader bibliographic syntheses. However, the sub-sector uncertainties (σprop) used in our calculations are derived from the same dataset compiled by . Therefore, σprop and σtotal are not fully independent from the comparison values in σMenoud and Uncertainty‾Menoud. The comparison is nonetheless informative as it illustrates how the literature-based source variability propagates through our aggregation scheme.

These sectoral uncertainties are subsequently propagated in our sensitivity simulations (see Sect. ). They serve as a quantitative basis for defining the plausible variability ranges of source signatures, allowing us to assess their influence on modeled atmospheric δ13C-CH4 distributions. A complementary qualitative assessment of systematic biases inherited from the underlying observational databases (e.g. uneven geographic coverage, methodological heterogeneity, limited temporal representativeness) is provided in Sect. . In atmospheric inversion frameworks, the total sectoral uncertainty (σtotal) can also be used to inform the specification of the prior error covariance matrix (B matrix) (see Sect. ).

Overall, the sectors rank by total uncertainty as WET (0.4 ‰–8.2 ‰) > BB (0.5 ‰–6.0 ‰) > NAT (2.0 ‰–5.4 ‰) > FFG (1.5 ‰–5.3 ‰) > AGW (1.8 ‰–3.3 ‰). Sub-sector variability dominates over aggregation effects in all sectors, and the largest reducible uncertainties lie in tropical wetlands and biomass burning regions where measurements remain sparse. These sectoral uncertainties are directly used in the sensitivity simulations (Sect. ) and inform B matrix specification in inversion frameworks (Sect. ).

For the FFG sector, our weighted mean signature is -44.2‰, which is generally consistent with but slightly more enriched compared to and whose values extend to more depleted ranges. Coal sources show greater differences, with our estimate around -43.7‰ compared to -50.7‰ in . This discrepancy can be explained by differences in data selection criteria and spatial weighting. Our estimate is derived using flux-weighted averaging that emphasizes high-emitting coal regions, such as China and India, where emissions tend to be less depleted than the global average e.g.. In contrast, the value reported by is an arithmetic mean of a broad compilation of site-level measurements, including more depleted coal emissions from regions with lower production or different geological contexts. The oil and gas isotopic signature is close to previous studies, with minor variation likely due to regional refinements. Geological emissions are fixed at -46.6‰, in line with the geochemical value from .

In the AGW sector, livestock methane isotopic values (-65.8‰) are more depleted than reported by (-63.0‰) and (-63.6‰). This difference likely stems from our use of the spatially explicit source signature maps from , which account for regional differences in C3/C4 feed composition using global maps of biomass C3/C4 ratios . Flux-weighted averaging based on these maps emphasizes regions dominated by C3 vegetation, such as temperate zones, resulting in more depleted signatures. Our estimate is also consistent with the -64.9‰ value reported for 2012 by . Waste-related emissions, including landfill, wastewater, and agricultural waste, are updated following , resulting in relatively values: -50.9‰ for wastewater, -56.2‰ for landfill, and -54.9‰ for agricultural waste. Our landfill and wastewater values align with but differ by 2 ‰–4 ‰ from and . This increase is attributed to changes in waste management practices, notably increased biogas production, which tends to leak methane with relatively higher δ13C . For agricultural waste specifically, the inter-dataset spread (-52 to -57.8 ‰) exceeds the sub-sector uncertainty σi, pointing to genuine methodological heterogeneity in the definition of this sub-sector (crop residues vs. manure vs. composting). Rice emissions, also from , are set at -59.9‰, slightly more enriched than the -63.0‰ used in due to differences in the amount of compiled literature.

For BB, our weighted mean of -24.3‰ from matches closely the value from , while being slightly more depleted than in . WET signatures, derived largely from , have a weighted mean of -58.6‰, slightly enriched compared to (-63.3‰) and consistent with . The ∼4.7 ‰ offset with reflects the over-representation of boreal measurements in their literature compilation vs. our flux-weighted average, which emphasizes tropical wetlands. Recent tropical airborne measurement programs report wetland source signatures of -59.0±1.3 ‰ in Bolivian Amazonia and -59.3±2.0 ‰ for a Zambia–Bolivia composite , consistent with the tropical wetland values used in our maps from . The slightly more enriched values produced by our maps over African papyrus-dominated wetlands likely reflect that these recent airborne measurements have not yet been assimilated into process-based wetland isotope models, contributing to the systematic biases on tropical wetlands discussed below.

Natural sources (NAT) retain previous estimates, with termites at -63.4‰ and oceans at -42.0‰, reflecting values from and . The weighted mean of -51.9‰ accounts for the relative contribution of these sources. For termites, we adopted a value of -63.4‰, consistent with and . The depleted signature of -76.1‰ reported by was not used here due to concerns about potential outliers. This choice explains the discrepancy with the more depleted mean of , who included the value in their compilation.

Beyond these sub-sector-specific differences, the inter-dataset inconsistencies are of the same order as the sub-sector uncertainty σi and point to systematic biases inherited from the underlying observational databases, which σi does not capture. At the sub-sector level, σi reflects the statistical dispersion within the sampled measurements , not the representativeness of that sample relative to the true global distribution of emission sources. We identify three main systematic biases:

Uneven geographic coverage: the compilation from underlying is dominated by North American data, EMID improves European coverage but Africa, South America, and large parts of Asia remain under-sampled; tropical wetlands are represented through fewer campaigns than boreal/temperate ones, and recent tropical measurement programs have not yet been integrated;

The propagation of these biases to the aggregated maps depends on the flux-weighting scheme: biases affecting small-flux sub-sectors (termites, oceans) have limited impact, while those affecting high-emitting sub-sectors (livestock, oil and gas, tropical wetlands) propagate more directly to the modeled atmospheric δ13C-CH4 signal, consistent with the sensitivity hierarchy reported in Table .

Generally, the updated δ13C-CH4 source signature maps show clear sectoral and regional patterns, with depleted signatures for wetlands and agriculture, and enriched signatures for fossil fuel and biomass burning emissions. Temporal variations are limited over the study period, except in sectors where emissions were known to vary in time. Uncertainty analysis highlights significant variability, particularly for the agriculture and waste sector. The updated maps are broadly consistent with recent datasets. In the next section, we assess how the uncertainties over inputs propagate to the modeled atmospheric δ13C-CH4 signal through sensitivity simulations.

We note that the term “aggregation” is used in two distinct methodological senses in this section. The first refers to the choice of flux inventory used to weight sub-sector isotopic signatures into sector-level values, which generates the aggregation uncertainty σagg reported in Table  (AGGREG_EDGAR, AGGREG_GFEI, AGGREG_CEDS and AGGREG_GAINS simulations). The second refers to the sectoral granularity of the atmospheric model itself, i.e. the number of source categories effectively transported (5 by default vs. 14 in the NO_AGGREG simulation). Both are tested below but answer complementary questions: the former probes the sensitivity of the aggregated signature values, while the latter probes the sensitivity of the modeled atmospheric signal to the level of sub-sector resolution.

Figure  summarizes the sensitivity of flux aggregation to different prior datasets. Panel (a) shows the variability of flux estimates between inventories, highlighting where emission fluxes are most uncertain. Since this test focuses specifically on uncertainties arising from the flux-weighted aggregation of isotopic signatures, only anthropogenic sectors (FFG, AGW and BB) are included (see Sect. ). Panel (b) presents the resulting variability in the aggregated δ13C-CH4 source signatures by sector. Panel (c) shows the sensitivity of simulated atmospheric δ13C-CH4 signals to these aggregated isotopic changes at surface level. It is important to note that panel (c) displays a single sensitivity map representing the combined impact of flux-weighted isotopic signatures variations across all anthropogenic sectors on isotopic signal simulated at the surface. Because only isotopic signatures are perturbed while the underlying emission fluxes remain unchanged, this test does not directly affect CH4 mole fractions. The only noticeable effect on CH4 mole fractions occurs in the “NO_AGGREG” setup, in which the number of aggregated source categories was increased from 5 to 14 to evaluate the impact of the trade-offs between computational efficiency and isotopic detail (see Sect. ), resulting in a very small change in simulated CH4 mole fractions (about 5.6 ppb on average; Table ). Figure S3 in the Supplement shows the same information as in Fig.  but expressed in terms of relative standard deviation (RSD). Together, these results illustrate how inventory discrepancies propagate through to atmospheric simulations of δ13C-CH4 signal.

The largest flux uncertainties (Fig. S3a in the Supplement) are observed in the Fossil Fuels and Geological (FFG) and Agriculture and Waste (AGW) sectors (65 % and 41 % respectively). These uncertainties stem from differences between inventories, which exhibit regional discrepancies. For example, in Central Asia (Turkmenistan, Afghanistan, Uzbekistan), there are significant differences in fossil fuel emission estimates. In Turkmenistan, GAINS estimates 1259 ktCH4yr-1, while EDGARv8 and CEDS report 1343 and 1351 ktCH4yr-1 respectively and GFEI 888 ktCH4yr-1. Although totals appear similar, the spatial allocation and sectoral breakdown differ markedly between inventories. This is partly because inventories rely on national reports submitted to the UNFCCC, ensuring consistency at the country level but not necessarily in spatial detail or sub-sector attribution. Additionally, satellite-based studies (e.g. ) have identified emission events in regions such as Turkmenistan, associated with fossil fuel infrastructure. However, such episodic or localized emissions are generally not included in bottom-up inventories, which may underestimate the true emission rates. In the AGW sector, a high RSD is observed in Botswana. This is linked to the aggregation structure in GAINS, where large African regions are aggregated, whereas other inventories provide country-level estimates.

The impact of these flux uncertainties on δ13C-CH4 source signatures (Fig. b) is especially pronounced in the AGW sector, which shows the highest isotopic sensitivity (1.7 ‰), particularly in the Middle East and parts of Africa. This is related to the relative contributions of sub-sectors with distinct isotopic signatures, such as livestock (more depleted) and waste (less depleted), as defined in the inventories. For instance, in Ethiopia (2016–2020), livestock accounts for 79 % of AGW emissions in CEDS and GAINS, but only 67 % in EDGARv8, which is reflected in the region's isotopic signature variability. Regional variations in livestock diets, driven by the local balance between C3 and C4 forage plants, further contribute to this spread . For instance, in Ethiopia, the balance between C3 and C4 forage grasses varies with altitude, leading to distinct isotopic signatures within the same country . AGW is disaggregated in our framework into five sub-sectors (livestock, rice, landfills, wastewater, and agricultural waste; Table ), each with its own isotopic signature and flux distribution. Livestock and rice together account for ∼62 % of AGW emissions and carry the most depleted signatures (-65.8 ‰ and -59.9 ‰), while waste-related sub-sectors (-50.9 to -56.2 ‰) make up the remaining 38 %. This contrast is the main driver of the propagated uncertainty σprop reported for AGW in Table . To assess whether finer sectoral granularity, i.e. the number of source categories effectively resolved in the atmospheric model, would meaningfully change the modeled atmospheric signal, the NO_AGGREG simulation increases the number of source categories from 5 to 14 (Table ). This is conceptually distinct from the aggregation uncertainty σagg defined in Sect. , which quantifies the sensitivity of the aggregated signature to the choice of flux inventory used for weighting (Fig. ). The resulting differences are small: +0.061 ‰ globally for δ13C-CH4 (Fig. S4 in the Supplement). Localized differences of up to ±0.5 ‰ in δ13C-CH4 occur in regions where sub-sector composition contrasts strongly with the global average, notably in South Asia and the Middle East. These results indicate that increasing sectoral granularity beyond five sub-sectors does not substantially alter the modeled atmospheric isotopic signal at the global scale. Reducing AGW-related uncertainty would therefore benefit more from better-constraining the signature values of the existing sub-sectors and from refining inventory-level flux partitioning between livestock and waste, particularly in rapidly developing regions, than from increasing the number of source categories.

In contrast, the FFG sector shows lower isotopic sensitivity despite higher flux uncertainties (2.2 %). This is explained by the more homogeneous isotopic signatures of coal, oil and gas and geological, and the smaller relative isotopic differences among sub-sectors. The uncertainties are highest in North America, they are mainly associated with the difference in relative contribution of oil and gas vs. coal in the datasets. The variability in Chinese coal emissions is also well documented, with CEDS based on EDGARv4.2 previously overestimating emissions compared to recent regional inventories . These results highlight the key role of emission partitioning within aggregated categories in shaping the final isotopic source signature, particularly in the AGW sector. These uncertainty values are used to define the σagg parameter in the uncertainty analysis (see Sect. ).

The sensitivity of simulated atmospheric δ13C-CH4 signals to aggregation choices remains globally low, with a mean RSD of 0.32 % (Fig. S3c in the Supplement) and a mean SD of approximately 0.06 ‰ (Fig. c). This suggests that, despite regional discrepancies, aggregation uncertainties have a limited impact on large-scale atmospheric isotopic patterns. Moreover, the values shown in Fig. S3c represent relative sensitivities computed over the entire model domain, which tends to dilute localized sensitivity hotspots. When compared to the RSD of observed δ13C-CH4 values at surface stations e.g., the simulated sensitivities is smaller. This indicates that real-world atmospheric variability exceeds the response induced by inventory-driven aggregation uncertainties, and further support the limited impact of this specific error source on the modelisation of the atmospheric isotopic signal at observational sites (more details in Sect. ).

In summary, two distinct aggregation-related effects have been tested in this section. First, the choice of flux inventory used to weight sub-sector signatures (the aggregation uncertainty σagg) introduces sizeable variability in the aggregated signature values themselves, notably for AGW and FFG, but propagates only weakly to the modeled atmospheric δ13C-CH4 signal. Second, increasing the sectoral granularity from 5 to 14 source categories (NO_AGGREG test) modifies the modeled signal by only +0.06 ‰ globally. Both aggregation-related choices therefore have a limited impact on atmospheric simulations, supporting the transferability of the updated maps across inversion systems and inventories.

Figure  summarizes the sensitivity of atmospheric simulations to uncertainties in methane flux estimates from key source sectors: wetlands, freshwaters, and the total anthropogenic emissions (i.e. the sum of all anthropogenic sources). Unlike the previous section, where only isotopic signatures were perturbed through flux-weighted aggregation, here the underlying emission fluxes themselves are varied in the model simulations. Panel (a) shows the SD of CH4 emissions across inventories, and highlight where flux uncertainties are the greatest. For a detailed breakdown of anthropogenic subsectors (e.g. fossil fuels, waste, agriculture), refer to Figs. S6 and S7 in the Supplement, which show their individual contributions. Panels (c) and (e) display the impact of uncertainties in wetlands, freshwaters, and anthropogenic fluxes on modeled CH4 mole fractions and δ13C-CH4 signals, respectively, at surface level. Figure S5 in the Supplement shows the same information in terms of RSD. Theses figures show how uncertainties in sectoral emissions propagate into atmospheric simulations.

Methane flux uncertainties are highest for anthropogenic sources (mean RSD of 129 %), followed by wetlands (96 %) and freshwaters (22 %) (Fig. S5a in the Supplement). These differences arise from inventory discrepancies, the inherent complexity of methane emission processes (e.g. large spatio-temporal variability, dependence on environmental conditions and management practices), and the uneven availability of observational data across regions and sectors, which limits the capacity to constrain emissions. Among anthropogenic subsectors, fossil fuels exhibit the largest uncertainty (RSD 83.8 %, Fig. S6), driven by inconsistent national reporting, the use of variable emission factors, and the presence of poorly constrained super-emitters . Waste emissions follow (RSD 51.9 %), and include landfills, agricultural waste, and wastewater. The limited spread across inventories for landfills is mostly due to the use of similar Tier 1 methods and data sources, not better emission constraints. Substantial uncertainties persist due to variations in emissions arising from different climate conditions, landfill management practices, and the inherent temporal and geographical variability of landfill emissions . Wastewater emissions remain particularly uncertain due to variability in treatment processes and limited measurements . Biofuel burning shows a high RSD (55.7 %) but low absolute impact due to its smaller flux. Agriculture (RSD 42.2 %) contributes significantly in absolute terms due to its large emissions. Regarding natural sources, wetland emission uncertainties stem from multiple factors: inconsistent wetland extent maps , uncertainties in methane production and oxidation modeling , and the influence of environmental drivers such as temperature and water table depth . The dominant source of long-term uncertainty is wetland areal extent , while seasonal variability is primarily driven by meteorology . Tropical wetlands remain particularly uncertain due to sparse data coverage despite their importance for global feedbacks . Freshwater emissions are also uncertain due to poorly mapped inland waters, complex seasonal dynamics, and diverse emission pathways (e.g. diffusion, ebullition, plant-mediated transport) . However, as only one freshwater dataset was available the sensitivity shown here reflects the introduction of freshwater emissions into the simulation (ON/OFF comparison) rather than a quantified uncertainty across multiple estimates.

Panel (c) of Fig.  shows how flux uncertainties propagate into modeled CH4 mole fractions. The strongest sensitivity is linked to freshwater emissions, with an average variability of 68 ppb (4.4 %). This reflects a significant contribution to total methane emissions (+53 TgCH4yr-1) and strong regional impacts, particularly near the Caspian Sea, where freshwater sources dominate and where no wetland or oceanic fluxes were present in the reference simulation, thus amplifying the local sensitivity (Fig. S2). The Caspian Sea is a large endorheic saline lake whose emissions are treated as freshwater in our framework given the absence of a dedicated dataset; its brackish nature and proximity to major oil and gas infrastructure introduce additional uncertainty in this region. Anthropogenic fluxes contribute to a variability of 30 ppb (2.4 %), especially in major fossil fuel production regions such as Siberia and industrialized areas like Eastern China. Wetland fluxes result in a variability of 25 ppb (1.7 %), concentrated in high-emission regions such as the Amazon, Southeast Asia, and the Congo Basin.

Similarly, modeled δ13C-CH4 signals (panel e) show a spatial pattern that mirrors CH4 mole fraction sensitivity. Wetland and freshwater flux uncertainties both lead to an average isotopic variability of 0.2 ‰ (0.9 %), while anthropogenic fluxes cause 0.3 ‰ (1.6 %). These findings emphasize that sectoral flux uncertainties substantially influence regional isotopic signal, particularly in areas with high methane emissions.

Observed SDs from surface monitoring sites (shown as colored circles) are also displayed in panels (c) and (e) for comparison. In several cases, modeled isotopic sensitivities exceed the observed SD, especially for freshwater, induced CH4 variability, highlighting their relevance for inversion performance (see Sect. ). As illustrated in panels (c) and (e), the maps convey two layers of information: regional hotspots where large uncertainties may hamper local flux attribution (e.g. Caspian region, Chinese industrial basins, Congo wetlands, etc.), and background sensitivity over well-mixed or remote areas, which are critical for constraining hemispheric to global budgets. This distinction is further discussed in Sect. .

In summary, freshwater fluxes dominate the CH4 mole fraction sensitivity, followed by anthropogenic and wetland fluxes. For δ13C-CH4, anthropogenic fluxes induce the largest variability, with wetland and freshwater contributions of similar order. Sectoral flux uncertainties produce both regional hotspots (Caspian, Congo, Chinese basins) and broader background sensitivities, both relevant for inversion design.

Figure  also illustrates the uncertainties related to atmospheric chemistry parameters, focusing mainly on the kinetic isotope effect (KIE) of methane oxidation by tropospheric hydroxyl radicals (OH), which are the primary oxidant of methane. Panel (b) presents the SD of the total tropospheric OH column (pressure levels below 250 hPa), highlighting regions with the largest OH uncertainties. Panel (d) shows how these uncertainties affect modeled CH4 mole fractions at surface level. Panel (f) displays how uncertainty over KIE impacts atmospheric δ13C-CH4 signals. OH uncertainty does not affect δ13C-CH4 signal, and OH-KIE does not impact CH4 mole fractions. Figure S4 shows the same information in terms of RSD.

Uncertainties in OH fields (Fig. b and Fig. S5b in the Supplement) are substantial e.g.. SD are particularly high over tropical continental regions such as Amazonia, South Asia, and the African savannas, where OH concentrations peak due to intense photochemistry and high humidity and where strong inter-model differences coexist . These regions also display marked longitudinal contrasts in tropospheric OH, driven by zonal asymmetries in convection, lightning NOx, and biomass burning, which are not consistently captured across global chemistry models . Recent studies further suggest that tropospheric oxidative capacity is itself evolving over time: infer an increasing global OH abundance from radiocarbon monoxide (14CO) observations, with implications for the interpretation of recent CH4 trends . Satellite-based observations have also been explored as a means to better characterize tropospheric OH distributions and reduce these uncertainties .

The sensitivity of simulated CH4 mole fractions to OH variability (Fig. d) is relatively uniform globally, with an average SD of 49.6 ppb (RSD of 3.6 %). This confirms that the oxidative sink is a dominant factor controlling methane concentrations and that its uncertainties propagate broadly rather than being confined to specific regions . While OH-related uncertainties propagate globally and uniformly, their impact near source regions appears limited (Fig. d). This is particularly relevant given ongoing concerns about CH4–OH interactions in polluted areas, where local nonlinearities may arise due to complex dependencies of OH concentrations on emissions of NOx, CO, and volatile organic compounds (VOCs) . In theory, elevated CH4 concentrations could partially saturate the OH sink, especially in regions with high levels of co-emitted VOCs and NOx that alter oxidative capacity. However, this expected nonlinearity is not strongly expressed in the CH4 sensitivity maps. Because OH concentrations are prescribed and do not respond to CH4 levels in our configuration, the oxidative capacity is higher than it would be under interactive chemistry. This conservative setup further dampens any potential CH4–OH saturation effects, explaining the relatively uniform CH4 sensitivity patterns shown in Fig. d. Moreover, the SD induced by OH variability exceeds the observed SD of CH4 mole fractions at most monitoring sites, indicating that OH-related uncertainties alone can introduce model variability greater than observational noise.

Regarding the kinetic isotope effect, Figs. f and S5f in the Supplement show that uncertainties in the OH-KIE induce a geographically homogeneous SD of 0.4 ‰ (RSD of 2.2 %) in the atmospheric δ13C-CH4 signal, exceeding the observed SD at surface stations . This sensitivity is driven by only two published experimental determinations of the 12C/13C KIE for CH4+OH: (1.0054±0.0009 at 296 K) and (1.0039±0.0004 at 296 K). No measurements exist below 278 K, although theoretical calculations suggest the KIE may increase at lower temperatures . At steady state, showed that the 0.0015 difference between the two values yields a ∼1.3 ‰ shift in atmospheric δ13C-CH4, illustrating the high leverage of this parameter. In practice, previous inversions have adopted either value, in e.g. , and in e.g. . This choice is absorbed by the posterior source mixture. Accordingly, we recommend treating the full Cantrell–Saueressig range in inversion frameworks (see Sect. ).

Beyond OH, two additional sinks contribute to the δ13C-CH4 budget and deserve explicit discussion: oxidation by chlorine (Cl) and soil uptake. Both are included in our forward simulations (Sect. ) but were not perturbed in dedicated sensitivity experiments.

The Cl sink accounts for a small fraction of total CH4 oxidation. Recent estimates converge on a tropospheric contribution of ∼1%–3 % of the total chemical sink , with the latest Global Methane Budget reporting a climatological tropospheric Cl sink of 6[1–13] TgCH4yr-1 , substantially smaller and better constrained than earlier estimates . Despite this small magnitude, the Cl reaction carries an exceptionally large kinetic isotope effect (KIE ≈1.066 at 298 K; ), more than an order of magnitude larger than that of OH, so even modest uncertainties in Cl concentrations translate into substantial shifts in modeled δ13C-CH4 (see Table ). further identified the combined uncertainty in fractionation (OH-KIE and Cl contribution) as the single most important factor limiting isotope-based source partitioning at the global scale . quantified this influence within the same CIF-LMDz-SACS framework used here, and reported a near-linear sensitivity of +11.7 TgCH4yr-1 and -1.0 ‰ in the globally averaged source signature per 1000 molec.m-3 increase in mean tropospheric Cl, with stratospheric Cl alone contributing a ∼0.30 ‰ surface enrichment via stratosphere–troposphere exchange and modifying the δ13C-CH4 seasonal cycle amplitude by up to 10 %–20 % depending on latitude. Because our configuration adopts the Cl field from , consistent with the most recent tropospheric chlorine chemistry, the Cl-related uncertainty in our simulations is bounded by the ranges quantified in , which are of the same order of magnitude as the OH-KIE sensitivity reported in Table . A dedicated Cl sensitivity experiment was therefore not repeated here to avoid duplicating a recent and comprehensive analysis with the same model.

Soil uptake contributes ∼31[17–39] TgCH4yr-1 to the global CH4 budget , or about 5 % of the total CH4 sink, with a moderate KIE of ∼1.020 , intermediate between OH (1.0039) and Cl (1.066) (see Table ). In our framework, it is implemented as a first-order deposition process with isotope-dependent deposition velocities (Sect. ). Given its small relative contribution to the total sink, its moderate KIE, and the well-constrained global magnitude reported in recent budgets, soil uptake uncertainties are expected to have a substantially smaller impact on the modeled δ13C-CH4 signal than the OH-KIE (Table ), and were therefore not perturbed in the Monte Carlo ensemble.

In summary, OH-KIE is the dominant chemistry-related driver of δ13C-CH4 variability, while OH fields dominate CH4 mole fraction sensitivity. The Cl sink also exerts a strong leverage on δ13C-CH4 through its large KIE; its impact within the same CIF-LMDz-SACS framework has been comprehensively quantified by and is therefore not duplicated here.

Figures  and S8 in the Supplement present the sensitivity of the simulated atmospheric δ13C-CH4 signal to uncertainties in source-specific isotopic signatures, based on the Monte Carlo simulations (see Sect. ). Panel (a) displays the SD of the prescribed δ13C-CH4 source signatures used as input to the simulations. The highest signature RSDs are associated with the BB sector, with an RSD of 36 % and a SD of 7.8 ‰, followed by FFG sector (RSD: 13.0 %, SD: 6.1 ‰), NAT sector (RSD: 12.0 %, SD: 5.3 ‰), WET sector (RSD: 9.2 %, SD: 5.2 ‰), and AGW sector (RSD: 7.3 %, SD: 4.6 ‰). These RSD values are consistent with the sector-specific uncertainty ranges summarized in Table S3. The spatial pattern of these uncertainties reflects the regional sampling domains used in the Monte Carlo parameterization (see Sect. ).

Panel (b) shows the resulting variability in modeled atmospheric δ13C-CH4 at surface level. These simulations only perturb the isotopic composition of the emissions while keeping total CH4 fluxes fixed, as a result, there is no corresponding effect on CH4 mole fractions. Despite having a comparatively lower SD in its source signature, the AGW sector emerges as the dominant driver of atmospheric δ13C-CH4 uncertainty, with a mean sensitivity of 0.32 ‰ (1.74 %). This impact is particularly pronounced in regions with high AGW emissions, such as India, where EDGARv8 estimates an annual mean emission of 26 TgCH4 per year over the period 2016–2020. Other sectors contribute less significantly to overall isotopic variability but still have regionally relevant effects: BB contributes 0.16 ‰ (0.84 %), FFG 0.04 ‰ (0.43 %), NAT 0.07 ‰ (0.38 %), and WET 0.02 ‰ (0.10 %).