The mole fraction and isotopic composition of CH₄ are modelled using a two-box model with the boxes representing the Northern Hemisphere (NH) and Southern Hemisphere (SH) (Fig. 1). Hemisphere-specific source fluxes are compiled from six bottom-up inventories and aggregated into five source categories: agriculture, wetlands, pyrogenic, fossil fuels, and waste (Table S1 and Fig. S1 in the Supplement). Please note, “fossil fuel” and “thermogenic” emissions are used interchangeably to refer to CH₄ from fuel exploration, oil refineries, chemical processes, and power generation as compiled from EDGAR (Table S1). Pyrogenic emissions are treated as a separate category. CH₄ sinks are parameterised by sink-specific lifetimes for removal by troposphere, stratosphere, and deposition to soils, respectively. We set the inter-hemispheric exchange time to τ = 0.75 year, determined from SF₆ inversion and τ-sensitivity tests (see Fig. S5). The intra-hemispheric latitudinal gradients in CH₄, δ13C-CH₄, and δD-CH₄ are corrected for using measurement stations at different latitudes provided by the NOAA network (Fig. S4). This correction accounts for the spatial variability within each hemisphere that cannot be explicitly resolved in the two-box framework, ensuring that model outputs are comparable to observations from individual atmospheric stations. Given the 2-box setup, global-scale source totals and growth trends are expected to be robust, whereas hemispheric attribution is more sensitive to structural assumptions, including interhemispheric exchange time and prior uncertainty weighting (see Sect. 4.4).

Our atmospheric dataset comprises time series data for 6 CH₄ tracers: mole fraction (χ(CH₄); NH and SH: 1983–2024), carbon isotopic composition (δ13C-CH₄; NH: 1994–2024; SH: 1988–2024), and hydrogen isotopic composition (δD-CH₄; NH: 1992–2024; SH: 1988–2024) of CH₄, each of them being a merged dataset from stations at high northern and southern latitudes (Dasgupta et al., 2025b). We extrapolate the time series back to 1980 and forward to 2025 (Fig. S2). A 13-year spin-up is applied to ensure that inversion results are independent of initial conditions, and a 3-year spin-down allows smooth convergence back to priors (Fig. S3). Therefore, the effective “analysis period” ranges from 1994 to 2022.

Source isotopic signatures for the 5 categories are averages from the global isotope database (Sherwood et al., 2017; Menoud et al., 2022) and weighted by emission rates (Table S2). CH₄ sinks include tropospheric OH / Cl, a combined stratospheric sink including OH / O(1D)/ Cl, and soil uptake. Each sink is assigned a kinetic isotope effect (KIE) based on Cantrell et al. (1990), Saueressig et al. (1995, 1996, 2001), Röckmann et al. (2011), and Fujita et al. (2020). We use their reported fractionation factors to set effective KIEs for the model's three sink categories: tropospheric (KIE ¹³C ≈ 1.0066; D ≈ 1.317), stratospheric (KIE ¹³C ≈ 1.0144; D ≈ 1.133), and soil (KIE ¹³C ≈ 1.0201; D ≈ 1.0825) sinks.

Sensitivity tests were performed where prior flux uncertainties (±10 %–50 %) and observational error bounds (χ(CH₄): 1 ppb ¹³CH₄: 0.1–0.01 ppb; CH₃D: 0.005–0.001 ppb) were systematically varied. Reducing the prior error below 30 % increased interannual variability in posterior fluxes but degraded isotopic fits, as the inversion over-relied on the priors (Lan et al., 2021). Lowering ¹³CH₄ observational error to 0.001 ppb produced near-perfect δ13C-CH₄ fits but worsened δD-CH₄ agreement, whereas balancing errors at 1, 0.01, and 0.001 ppb yielded improved simultaneous fits. These findings highlight the trade-off between prior confidence and observational weighing in multivariate inversions and are further explored in “error-scaled” inversion scenarios (Sect. 3.3).

The prior hemisphere-specific emissions for 1980–2025 (Sect. 2.3) are optimised using a Bayesian inversion system, as described in Sect. S7 in the Supplement. The inversion algorithm modifies monthly source fluxes and sink losses so that 6 modelled tracers evolve toward the observations while still remaining constrained by priors. For all results presented in Sect. 3.2–3.3, we use a fixed prior lifetime (Prather et al., 2012; Myhre et al., 2014), but we also test time-varying CH₄ lifetime scenarios derived from CAMS TM5 inversions (Fig. S6). The inversion is performed in two sequential steps. First, we perform a CH₄-only inversion to estimate the emissions and the aggregate sink that match the observed hemispheric CH₄ mole fraction, providing a consistent starting state for the isotopologue ratio calculations. Second, we run an iterative Gauss–Newton inversion jointly constrained by CH₄, δ13C-CH₄ and δD-CH₄ to refine the total budget and its allocation across source categories (see Sect. S7). The two-box model is adjusted to run with different combinations of the three tracers by selecting which isotopologues to include in the second inversion step and their associated observational uncertainties (Figs. 2, 3).

To optimise model inputs, including source isotopic signatures, sink KIEs, lifetimes, and observational-error estimates, we perform over 13 million individual inversions with perturbed prior parameter values (Table S3). Each run generates posterior source fluxes for which the 6 modelled tracers, i.e. χ(CH₄)_(NH, SH), δ13C-CH_{4 (NH, SH)}, and δD-CH_4(NH, SH) are then compared to the observations via a tracer-weighted RMSE (root mean square error; see Sect. S8). We only retain scenarios where the mean normalised RMSE between modelled and observed tracers is less than 0.1 (Fig. S7) and identify the most frequently occurring prior values among these “successful” runs (Figs. S8, S9). These modal values constitute our DPT-optimised parameter set, and the aggregated posterior results represent the DPT ensemble run (Fig. 4).

The threshold of mean normalised RMSE < 0.1 is empirically motivated: as shown in Fig. S7b, over 92 % of all posterior-observation pairs across the six tracers fall below this value, confirming that retained runs reproduce the observations within their combined measurement and model uncertainty. In absolute terms, this corresponds to residuals of ∼ 0.05 ‰–0.07 ‰ for δ13C-CH₄ and ∼ 0.2 ‰ for δD-CH₄. Sensitivity tests with stricter or looser thresholds produce near-identical ensemble medians for global source trends.

By selecting the parameter combinations that minimise the mean normalised RMSE for all six tracers, we obtained robust, data-driven refinements to the input parameters. As shown in Table 1 and Fig. S8, most source signatures remained close to their priors. For δ13C-CH₄, DPT-optimisation resulted in minor revisions to the wetland and agricultural source signatures in both hemispheres. A relatively larger adjustment was made to agricultural emission δ13C-CH₄ in SH and to a lesser extent to agricultural and wetlands δD-CH₄. Both biogenic source signatures shifted toward the heavier end of the DPT-tested range, suggesting that the δD-CH₄ data can be better matched by more enriched source signatures.

The same DPT framework was applied to optimise the tropospheric KIEs (Table 2; Fig. S9). Tropospheric KIEs for OH + Cl oxidation were slightly revised from literature values (¹³C 1.0066 and D 1.317; Saunois et al., 2020) to 1.0068 (¹³C) and 1.313 (D). These optimised KIEs yield the best simultaneous fit to the trends in χ(CH₄)_(NH, SH), δ13C-CH_4(NH, SH) and δD-CH_4(NH, SH).

These fine-tuned (within physically realistic bounds) source fingerprints and sink kinetics improve the model's ability to reproduce observed isotopic trajectories.

Figures 2 and 3 compare the modelled and observed isotopic time series, along with their corresponding emission scenarios, highlighting the performance of the three inversion setups: dual-isotope (CH₄ + δ13C-CH₄ + δD), carbon-only (CH₄ + δ13C), and hydrogen-only (CH₄ + δD).

Figure 3 reveals marked differences in sectoral emission trajectories across the three isotopic inversion setups over the analysis period. Wetlands (Fig. 3a) exhibit relatively stable emissions from 1994 to 2007 across all three inversions, followed by a sustained increase through 2022. Mean emissions are 217.8 ± 13.9 Tg CH₄ (dual-isotope), 216.8 ± 13.3 Tg CH₄ (carbon-only), and 220.7 ± 13.7 Tg CH₄ (hydrogen-only), with a total growth of +35.3, +32.6, and +35.8 Tg across the analysis period, respectively (Table S4). Agricultural emissions (Fig. 3b) exhibit gradual increases throughout the analysis period, with mean values of 128.1 ± 7.5 Tg CH₄ (dual-isotope), 128.0 ± 7.8 Tg CH₄ (carbon-only), and 131.5 ± 7.6 Tg CH₄ (hydrogen-only), corresponding to growth of +14.8, +16.9, and +14.8 Tg. Pyrogenic emissions (Fig. 3c) display high interannual variability but a general declining trend, decreasing from ∼ 45 Tg CH₄ in the early 1990s to ∼ 32 Tg CH₄ by 2022. Mean emissions are 39.7 ± 7.6 Tg CH₄ (dual-isotope), 39.7 ± 7.9 Tg CH₄ (carbon-only), and 38.4 ± 7.4 Tg CH₄ (hydrogen-only), with reductions of -11.6, -12.1, and -11.3 Tg. Fossil fuel emissions (Fig. 3d) show the most pronounced divergence between inversion scenarios with and without δ13C-CH₄. The carbon-only inversion (110.1 ± 10.7 Tg CH₄, +28.7 Tg growth) displays steady increases throughout the period. The dual-isotope (108.4 ± 7.9 Tg CH₄, +20.7 Tg) and hydrogen-only (108.6 ± 7.4 Tg CH₄, +19.2 Tg) inversions show elevated emissions in the early 1990s, a decrease through the late 1990s, and renewed growth from 2000 onward. Waste emissions (Fig. 3e) increase steadily with excellent agreement in temporal variation across all inversions: 80.1 ± 7.5 Tg CH₄ (dual-isotope, +23.4 Tg), 79.9 ± 7.3 Tg CH₄ (carbon-only, +22.8 Tg), and 79.1 ± 7.3 Tg CH₄ (hydrogen-only, +23.1 Tg). Atmospheric lifetime (Fig. 3f) remains consistent at 9.1 ± 0.1 years across all inversions, with higher temporal variations at the hemispheric scale than at the global scale.

The dual-isotope inversion achieves the best overall fit for both δ13C-CH₄ and δD-CH₄ (Fig. S10). For δ13C, it shows a mean residual of -0.05 ‰ and a median of -0.06 ‰, slightly better than the carbon-only run (mean -0.07 ‰, median -0.06 ‰). By contrast, the hydrogen-only inversion has larger residuals (mean -0.29 ‰, median -0.30 ‰), indicating a poorer fit. For δD-CH₄, the dual-isotope inversion again performs best (mean -0.18 ‰, median -0.09 ‰), followed by the hydrogen-only setup (mean -0.55 ‰, median -0.47 ‰). The primary constraint from δD-CH₄ operates on the early-period fossil baseline rather than on tracer fit in the later period; by ∼ 2005, all inversions converge on a similar trajectory consistent with both isotopic records. The carbon-only inversion, lacking δD-CH₄ data, has the largest residuals (mean -1.74 ‰, median -1.18 ‰), confirming that δ13C-CH₄ alone cannot capture deuterium dynamics. These results demonstrate that the dual-isotope approach is the only setup capable of accurately matching both isotopic records, whereas the single-isotope inversions necessarily compromise the fit for the unconstrained tracer. Following this, we run 2 additional scenarios with both isotopes, i.e. DPT ensemble and error-scaled inversions, to fully gauge the implications of a dual isotopic constraint on the model.

We compare prior estimates with five inversion scenarios over 1994–2022 to assess how isotopic constraints re-estimate the methane budget (Table S4, Figs. 3–4).

Prior: bottom-up inventories with fixed 9-year lifetimes (Fig. 3; black).

Figure 4 compares prior emissions with two refined dual-isotope inversions: the DPT-ensemble and error-scaled scenarios. Wetlands (Fig. 4a) show temporal patterns similar to Fig. 3, with relatively stable emissions until ∼ 007, followed by sustained increases. The DPT-ensemble estimates mean emissions of 222.6 ± 13.8 Tg CH₄ with growth of +33.5 Tg, while the error-scaled inversion yields 220.4 ± 10.3 Tg CH₄ with growth of +31.5 Tg, both lower than the prior growth (+30.1 Tg). Agricultural emissions (Fig. 4b) increase in both inversions. The DPT-ensemble (132.0 ± 7.2 Tg CH₄, +12.0 Tg growth) and error-scaled (129.3 ± 7.8 Tg CH₄, +14.7 Tg growth) scenarios both estimate lower growth than the prior (+22.3 Tg). Pyrogenic emissions (Fig. 4c) decline from ∼ 43 Tg CH₄ in 1994 to ∼ 31 Tg CH₄ by 2022, with high interannual variability. Mean emissions are 38.2 ± 7.0 Tg CH₄ (DPT-ensemble, -10.5 Tg) and 38.9 ± 7.5 Tg CH₄ (error-scaled, -11.4 Tg), both showing stronger reductions than the prior (-8.0 Tg). Fossil fuel emissions (Fig. 4d) display temporal patterns distinct from the prior, with elevated early-period emissions, a decrease through the late 1990s, and renewed growth from 2000 onward. The DPT-ensemble (106.5 ± 7.2 Tg CH₄, +18.2 Tg) and error-scaled (110.9 ± 8.4 Tg CH₄, +21.1 Tg) inversions both estimate lower growth in fossil fuel emissions than the prior (+44.0 Tg). Figure 4d also includes GAINS inventory estimates (green line) for comparison. The GAINS model estimates emissions bottom-up, i.e., quantifications of human activities contributing to emissions are multiplied by an emission factor representing the average emissions per unit of activity (Höglund-Isaksson et al., 2020). Waste emissions (Fig. 4e) increase steadily: 79.1 ± 7.6 Tg CH₄ (DPT-ensemble, +23.5 Tg) and 77.7 ± 7.3 Tg CH₄ (error-scaled, +22.4 Tg), both lower than prior growth (+29.1 Tg). Atmospheric lifetime (Fig. 4f) exhibits minimal variation, with higher temporal variations at the hemispheric scale compared to the global scale.

Figure 5 summarises the revisions to emission growth (top panel) and lifetime (bottom panel) across all inversion scenarios relative to the prior. The single-isotope inversions (carbon-only and hydrogen-only) show moderate adjustments across most sectors. In contrast, the dual-isotope, DPT-ensemble, and error-scaled inversions – all of which assimilate both δ13C-CH₄ and δD-CH₄ – consistently re-estimate wetland growth toward the SH (red shading indicates prior underestimation, with revisions of +8 to +10 Tg) while reducing NH fossil fuel growth (blue shading indicates prior overestimation, with revisions of -15 to -26 Tg). Agricultural and waste revisions are more modest across all scenarios (±5 Tg), while pyrogenic reductions are consistent (-3 to -4 Tg additional decline beyond prior). The lifetime heatmap (bottom panel) reveals hemispheric divergence: SH lifetime increases by +0.3 years across all isotope-constrained inversions, while NH lifetime decreases by -0.2 to -0.5 years depending on the scenario. Global mean lifetime remains nearly constant at 9.1 years, with deviations of ±0.1 years or less.

Prior inventories suggest steady fossil-fuel growth of +44.0 Tg between 1994 and 2022. However, isotope-constrained inversions reveal that most fossil fuel growth occurred between 2000 and 2012, with minimal additional increases thereafter. The carbon-only inversion estimates net thermogenic + pyrogenic growth at +16.6 Tg, while hydrogen-only inversions yield +7.9 Tg, and the dual-isotope solution at +9.1 Tg. Runs that include δD-CH₄ (hydrogen-only, dual-isotope, DPT-ensemble) consistently yield smaller thermogenic and pyrogenic growth, with the DPT-ensemble producing only +7.7 Tg of net thermogenic + pyrogenic growth, compared to the prior estimate of +36.0 Tg (which includes the -8.0 Tg pyrogenic reduction).

The inferred total isotopic enrichment associated with methane removal is substantially larger for deuterium (313 ‰) than for carbon (6.8 ‰). In the case of ¹³C, the sink fractionation is much smaller than the differences of isotopic source signatures between the different source categories, which span 40 ‰, whereas the ²H sink fractionation is twice as wide as the range between the average source signatures of different categories of about 150 ‰ (Table 1). Thus, δD-CH₄ provides strong sensitivity to oxidation kinetics. Among the source categories, δD is particularly sensitive to fossil CH₄ emissions, which are the only source with δD > -200 ‰. This sensitivity is particularly evident in the timing of the posterior emissions with and without δD. δD-constrained inversions assign roughly 5–10 Tg yr⁻¹ more fossil emissions to the early 1990s and about 5–8 Tg yr⁻¹ less to the 2010s than the carbon-only inversion, i.e. a higher early baseline and reduced recent growth. We note that the δD-inclusive inversions require higher fossil emissions, particularly in the late 1980s–early 1990s, but this overlaps with the model spin-up period (pre-1994) and should therefore be treated cautiously. Still, the consistent post-1994 behaviour across hydrogen-only, dual-isotope and DPT runs supports the robustness of the reduced fossil-growth result.

The downward revision of the fossil increase suggests an underestimation of fossil-fuel emissions in the used inventory in the 1990s and early 2000s, followed by a modest overestimation in the late 2010s (Fig. 4). Although fossil-fuel production continued to rise over this period (IEA, 2023), our posterior emissions stayed comparatively flat. This may result from several non-exclusive and overlapping factors – including declining emission intensity per unit production, the adoption of mitigation measures, changes in extraction and fuel-mix practices, or errors and biases in bottom-up reporting – rather than any single cause. This interpretation is supported by independent inventory estimates from the GAINS model, which also reports lower emission intensities in recent decades (Höglund-Isaksson et al., 2020) and by inversion studies that do not find large fossil-driven growth and in some cases point to increased biogenic contributions (Thompson et al., 2018; Schaefer et al., 2016). Taken together, these lines of evidence support our conclusion that the fossil-fuel growth is likely overestimated in the used bottom-up inventory.

Traditional inversion setups that fix thermogenic isotopic signatures may underestimate regional shifts toward lower-emission production systems, such as modern shale-gas operations in the USA between 2000 and 2012 (Uveges et al., 2025). The DPT framework explores fossil δD within ±5 ‰ of literature values (Table S3), encompassing both conventional gas and heavier shale gas signatures (Uveges et al., 2025; Riddell-Young et al., 2025). Even at the heavier end of this range, the atmospheric δD constraint requires substantial biogenic increases to explain the post-2006 growth, because the isotopic separation between fossil and biogenic methane in δD space (∼ 110–150 ‰) is large relative to the explored fossil δD range. It is also worth noting that if countries overreport fossil-fuel emissions in their national inventories, which serve as the foundation for bottom-up estimates, the resulting prior estimates will overestimate these emissions relative to atmospheric inversion results. Recent evaluation of China's carbon emissions, for example, found that coal-related CO₂ output was overestimated due to incorrect assumptions about fuel quality, leading to a downward correction of ∼ 14% for 2013 emissions and a cumulative reduction of ∼ 10.6 Gt CO₂ over 2000–2013 (Liu et al., 2015; Hu et al., 2025). Similar reporting uncertainties may affect CH₄ inventories.

The decline in pyrogenic emissions (-8.0 Tg in priors) aligns well with satellite observations of shrinking burn areas and improved fire management (van der Werf et al., 2017), and inversions consistently confirm this downward trajectory. These results imply that fossil fuels have played a steady but secondary role in post-2006 CH₄ growth, while biomass burning has declined more consistently than bottom-up inventories suggested. The reduction in net thermogenic growth from +36.0 to +7.7 Tg may indicate that certain emission drivers are potentially more influential in recent atmospheric trends than previously thought.

Bottom-up priors suggest that wetlands, agriculture, and waste collectively contributed approximately +81.5 Tg to global biogenic CH₄ growth from 1994 to 2022. However, all five inversion setups revise this estimate downward to between +68.6 and +73.7 Tg, with the DPT-ensemble yielding +69.0 Tg (Table S4). All five are cost-minimised solutions to the same observational constraints under different prior assumptions, and all converge on a ∼ 15% reduction in biogenic growth compared to prior estimates, while still making it the primary driver of post-2006 CH₄ growth.

Across our inversion suite, the posterior global mean lifetime is 9.1 ± 0.1 years, i.e., +0.1 year relative to the fixed prior (9.0 years). Examining the trend over the analysis period, the inverted lifetime reveals a slight net shortening of ∼ 0.1 years. Hemispherically, the two-box posteriors consistently indicate an NH shortening of ∼ 0.2 ± 0.1 years and an SH lengthening of ∼ +0.3 ± 0.1 years.

We note that in an inversion system, such modest adjustments could reflect true changes in sink strength, but they may also compensate for source re-estimations. Therefore, we compare the results to independent evidence on OH changes: studies suggest that the overall OH loss rate is generally “well buffered”. The assigned uncertainty was approximately 10 %–15 % uncertainty during the 2000s (Saunois et al., 2020; Nisbet et al., 2019), which means that a 9 % increase in tropical OH (Anderson et al., 2021; Stavert et al., 2021; Stevenson et al., 2020) is near the detection limit of global-scale budget estimates, similar to our findings. If OH (and/or Cl) have indeed increased modestly, a small but real shortening of lifetime would be expected. Our inversion's global lifetime change (±0.1 year) aligns well between these perspectives: large enough to reflect plausible sink variability, yet small enough to be consistent with a largely stable OH background.

The hemispheric divergence, with NH lifetime declining (-0.2 ± 0.1 years) and SH lifetime increasing (+0.3 ± 0.1 years), qualitatively aligns with independent evidence of stronger oxidant increases in the NH (Anderson et al., 2021) and the southward re-estimation of wetland emissions. Starting from our two-box model structure, the SH box exhibits a baseline longer lifetime (∼ 9.3 years vs. 9.0 years NH) even in the prior, reflecting fundamental differences in oxidant regimes between hemispheres. We acknowledge that 3-D models are better suited for spatial attribution; our two-box framework limits robust hemispheric conclusions. We therefore focus on the more reliable global estimate: a slight lifetime decrease of ∼ 0.1 years. While modest, this adjustment improves the match to observed δ13C-CH₄ and δD-CH₄ trends, particularly by preventing overcorrection of isotopic signals when rebalancing source contributions.

The pronounced interannual variability in posterior wetland emissions reflects compensatory adjustments within an underdetermined inversion rather than model instability. Wetlands carry the largest absolute prior uncertainty (∼ 75 Tg yr⁻¹ at 30 %), so year-to-year adjustments needed to balance atmospheric growth and lifetime variability are most strongly expressed in this category. Agriculture and waste, occupying intermediate isotopic space, are similarly susceptible. For this reason, sectoral growth is assessed using 3-year endpoint averages (Table S4) and 5-year moving averages (Fig. 4), which are robust across all five inversion scenarios. Our sensitivity experiments reveal that varying prior emission uncertainties from 30 % amplifies interannual (year-to-year) variability in posterior emissions. When prior uncertainties tighten below 30 %, absolute uncertainties scale with source size (e.g., wetlands: ∼ 75 → 50 Tg; pyrogenic: ∼ 12 → 8 Tg). To balance isotopic ratios annually, the inversion preferentially adjusts the largest sources because the same fixed uncertainty allows much larger absolute variability, leading to compensatory oscillations in posterior emissions and lifetime that can degrade isotopic fits. Only a minor fraction of this variability, if any, may be caused by teleconnections such as ENSO as seen in pyrogenic and wetland posterior emissions corresponding to El Niño and La Niña years, respectively. Similarly, relative observational error weighting between the 3 tracers critically influences inversion behaviour. Optimal performance occurs when relative uncertainties match isotopologue abundance ratios (1 ppb CH₄, 0.01 ppb ¹³CH₄, 0.001 ppb CH₃D).

All inversions presented thus far have used fixed uncertainties across sources: 30 % for all emission categories and 10 % for the tropospheric lifetime. To test whether the results are sensitive to this choice, we implemented an error-scaled inversion that adjusts prior uncertainty weights based on the relative flux magnitude of each source. Specifically, larger sources are assigned proportionally smaller absolute uncertainties, while the tropospheric lifetime uncertainty is tightened from 10 % to 9 % based on a sensitivity analysis that shows this configuration optimally balances source and sink adjustments. This error-scaled configuration yields more moderate hemispheric re-estimation than the unweighted dual-isotope inversion while maintaining improved fits to isotopic observations. For example, SH wetland growth is +19.1 Tg (DPT) versus +13.1 Tg (error-scaled), while NH wetland growth is +14.4 Tg (DPT) versus +18.5 Tg (error-scaled). This demonstrates that hemispheric attributions are sensitive to prior weighting, although the global totals remain robust (+69.0 Tg vs. +68.6 Tg biogenic growth).

Runs that include δD-CH₄ (hydrogen-only, dual-isotope, DPT-ensemble) consistently yield smaller thermogenic and pyrogenic growth than the carbon-only inversion (+7.9–9.1 Tg vs. +16.6 Tg). δD-CH₄ is particularly responsive to the enriched deuterium signature of fossil CH₄ (also pyrogenic), and how fast that CH₄ is oxidised by OH (Stell et al., 2021). The physical basis for this distinction lies in the magnitude of KIEs and the ranges of source signatures. The tropospheric KIE for deuterium (1.313) corresponds to a fractionation of 313 ‰, compared to 6.8 ‰ for carbon (KIE 1.0068), meaning the deuterium KIE deviation is approximately 46 times larger. This amplification is further enhanced by the large separation between biogenic and thermogenic source signatures in δD space (Table 1): biogenic sources range from -299 ‰ to -339 ‰, while fossil fuels cluster near -191 ‰ to -192 ‰, a separation of ∼ 110 ‰–150 ‰. For comparison, the δ13C separation is only ∼ 20 ‰ (biogenic: -58 ‰ to -64 ‰; fossil: -44 ‰). This results in higher fossil emissions at the beginning of our analysis period, and thus a smaller growth over the following decades, compared to the prior emissions.

The emission trends derived from our δD–CH₄–constrained inversions closely align with those of other recent isotope studies, which collectively suggest that the post-2006 CH₄ growth is primarily microbial in origin, rather than fossil. Riddell-Young et al. (2025) found that both δ13C- and δD-based mass balances attribute over 70 Tg yr⁻¹ of the 2006–2023 CH₄ increase to microbial sources, with little to no fossil trends after 2013. Our δD-inclusive inversions similarly produce only a very small net thermogenic and pyrogenic growth (∼ 8–10 Tg) and significantly larger biogenic growth (∼ 70 Tg). In our 2-box modelling setup, this is associated with a redistribution of emissions toward SH wetlands, reinforcing the notion that microbial sources dominate recent CH₄ increases. Fujita et al. (2025) likewise obtained a near-flat fossil emission trajectory after the early 2000s, closely matching the magnitude and trend of our posterior fossil emissions (Fig. 4d). Moreover, their inferred increase in OH (Fig. 6g) is consistent with the slight CH₄ lifetime decrease NH in our inversions (Table S4), indicating that modest sink strengthening, rather than rising fossil emissions, helps reconcile the post-2006 CH₄ growth. Chandra et al. (2024) highlighted major inventory biases in GAINS and EDGAR fossil sectors; our δD inversions support this interpretation but indicate that correcting early-period fossil baselines, rather than invoking strong post-2000 growth, better reconciles observed isotope trends. Together, these comparisons confirm that δD-CH₄ constraints yield smaller trends in fossil contributions than earlier inventories or single-isotope studies, providing independent support for a microbial-dominated CH₄ rise and underscoring the value of δD-CH₄ for refining emission histories.

Our two-box framework offers a conceptual and computationally efficient method for combining CH₄, δ13C, and δD observations, with the option to perform DPT experiments using millions of individual inversion runs. On the other hand, the two-box model approach has obvious limitations that impact spatial attribution and limit the attribution of the inferred emission adjustments. This approach, by design, collapses latitudinal structure into two hemispheric reservoirs, which removes regional signals, does not resolve mid-latitude or tropical gradients, and importantly, does not include intrahemispheric transport. The conclusions drawn at the global scale, i.e., the dominance of biogenic growth and the reduced fossil-fuel trend, are consistent across all inversion scenarios and are insensitive to lifetime partitioning between hemispheres. Hemispheric-scale results, including the NH/SH wetland split and hemispheric lifetime trends, are more sensitive to model structure and should be interpreted with caution. One question is then how the hemispheric averages are constructed. In our work, we used high-latitude station records because they provide reliable and consistent temporal trends, including robust inter-laboratory offset estimation (Dasgupta et al., 2025a). This requires corrections for the latitudinal gradient to infer hemispheric averages, an approach that carries error when gradients change over time. Alternatively, the inter-laboratory offsets could also be applied to lower-latitude time series, where direct inter-station comparisons are more problematic because of stronger regional influences from nearby sources. However, if regional variability is included anyway, using explicit 3D models that incorporate spatial variability and atmospheric transport is likely the more insightful approach. Spatially resolved inversions may then provide more detailed insight into the origin of the derived emission adjustments compared to the prior. Based on our results, including δD, it can indeed provide additional constraints compared to δ13C alone. Thus, it is also important to expand δD observations, which are still limited compared to δ13C.