Historical observations of the 13C/12C ratio of atmospheric CH4 are used to constrain the present-day methane budget using optimal estimation. Three methane emission scenarios with basis in the recent literature are evaluated against historical 13CH4 observations, considering all uncertainties together. We estimate that present-day methane emissions are composed of 64%–76% biogenic, 19%–30% fossil, and 4%–6% pyrogenic sources. It is found that, barring any changes in the isotopic signatures of sources and sink processes, satisfying the 13C/12C record requires estimates of present-day anthropogenic fuel-related emissions that are on the high end of the assumed uncertainties, even when a significant geological source is included. Extending present-day results to the time of the Last Glacial Maximum (LGM), emissions from wetlands are implied to be 40%–62% of the present-day value, the higher number being valid only for a scenario with strong (∼30 Tg/a) geological emissions and roughly 20% greater biomass burning emissions at LGM relative to the present-day.
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 Despite the important role of methane (CH4) in climate forcing, the magnitudes of its sources and sinks are still poorly quantified [Denman et al., 2007]. Anthropogenic emissions are diverse and include coal mining, natural gas use, agriculture, and waste treatment. Natural sources of CH4, especially wetlands and geological seepage [Etiope et al., 2008] are particularly uncertain, as are biomass burning [van der Werf et al., 2006], which has both natural and anthropogenic origins, and the sinks of CH4 (oxidation in soils and atmospheric chemistry). As a consequence, the future evolution of the CH4 budget, and its response to climatic changes, remain highly uncertain. CH4 observations can be used to constrain surface fluxes via inverse modeling [e.g., Mikaloff Fletcher et al., 2004a; Bruhwiler et al., 2005; Butler et al., 2005; Bousquet et al., 2006; Chen and Prinn, 2006; Bergamaschi et al., 2007; Meirink et al., 2006, 2008; Houweling et al., 1999; Hein et al., 1997]. The availability of satellite observations in particular has been shown to constrain the spatial structure of CH4 flux to much higher resolution than surface stations, thereby decreasing the dependence on a priori assumptions [Bergamaschi et al., 2009]. Nonetheless, there is a strong spatial and temporal overlap between different source types that makes the differentiation between sources, based solely on spatial and temporal footprints, difficult. Other independent constraints are therefore needed to distinguish source types from one another.
 As will be shown below, a strong determinant of the net δ13 of emissions is the amount of CH4 attributed to biomass burning and geological emissions, because these sources evidently both have high δ13 signatures. Geological seepage has recently come of interest as a strong CH4 source that has been largely neglected in CH4 budget studies [Kvenvolden and Rogers, 2005; Etiope et al., 2008]. Biomass burning is also an uncertain source, in part because fires are intermittent events with both natural and anthropogenic causes, but also because the CH4 released depends on many interacting factors (fire type, vegetation type, etc.) that are difficult to estimate even at the present-day [Bowman et al., 2009].
 Inversions of the spatial variability of present-day CH4 concentration by Bergamaschi et al. , Houweling et al.  and Bergamaschi et al.  attribute 24–34Tg/a to biomass burning and around 18Tg/a to oceanic and geological emissions. This is in contrast to Mikaloff Fletcher et al. [2004a], who deduced a biomass burning source of around 88 Tg/a from present-day measurements of δA13. Similarly, but using the historical record of δA13, Lassey et al. [2007b] found that natural sources that are enriched in 13CH4 (wildfires and/or geological seepage) may play a stronger role than generally assumed. A strong natural biomass burning and/or geological CH4 source was also suggested by Fischer et al. , based on the enhanced 13CH4 found during cold climate periods over the last 20,000 years. These results reinforce the bottom-up estimate of Etiope et al. , that geological CH4 emissions may contribute as much as 53 ± 11Tg/a.
Sowers  suggests that enhanced 13CH4 during cold climate periods over the past 10,000 years can also be explained by the reduction of emissions from boreal wetlands, the changing ratio of C3 and C4 vegetation, and changing emissions from methane clathrates in the ocean. Houweling et al.  explain the historical record of δA13 over the last millenium as the combined effect of decreasing natural emissions (during the Little Ice Age), superimposed on increasing anthropogenic emissions (first from agriculture and then from industry). Thus, fundamentally different CH4 storylines exist in the literature to explain the available observations of δA13.
 In this study we formulate the relationship between observed δA13 and the isotopic signatures of CH4 sources and sinks as an optimal estimation problem that takes into account the existing uncertainties in surface fluxes, chemical sinks, δ13 signatures of the sources, and isotopic fractionations of the sinks. Because this amounts to a very underconstrained problem wherein four observations are used to constrain over 20 variables, we avoid looking for a definitive most probable CH4 source/sink scenario. Instead we select three substantially different scenarios from the recent literature, adjust them relative to the observed information, and examine the resulting corrections in order to understand the remaining uncertainties in the present-day CH4 budget. The optimization is described in section 2. The baseline results are given in section 3, along with a physical interpretation and tests of the sensitivity of our results to various assumptions. Section 3 also gives a set of distinct, isotopically consistent emission scenarios and uncertainty ranges, which can be used as a priori esimates in spatially constrained inverse modeling, e.g., using satellite observations. Conclusions and points for further research are given in section 4.
Table 1 shows the net 13CH4 isotopic signature in the atmosphere, δA13, and the corresponding CH4 concentrations, measured at the South Pole in 2005 (GLOBALVIEW-CH4, NOAA Earth System Research Laboratory, Boulder, Colorado, anonymous ftp to ftp.cmdl.noaa.gov), in Antarctic ice cores for the years 1000 and 1900 [Ferretti et al., 2005] and at the Last Glacial Maximum (LGM) [Fischer et al., 2008]. The 2005 measurements represent the combined influence of present-day natural, industrial, and agricultural emissions, as well as the removal of CH4 from the atmosphere by various sinks (all of which result in 13CH4 enrichment). The year 1900 represents the early industrial (EI) era, when anthropogenic CH4 emissions were dominated by agricultural and burning sources. Here both atmospheric CH4 and δA13 were low relative to the present-day. During the early agricultural era (EA, represented by the year 1000), when anthropogenic emissions were still low relative to natural emissions, observed atmospheric CH4 is correspondingly lower but δA13 is close to the present-day level. During the drastically different climate conditions of the LGM, the average CH4 concentration was roughly half the EA level, but δA13 was higher than at any post glacial time.
Table 1. Observed Antarctic CH4 Concentration and δ13 for Four Climatic Periods
δA13 (per mille)
The present-day value is the 2005 average at the South Pole (GLOBALVIEW-CH4, NOAA Earth System Research Laboratory, Boulder, Colorado, anonymous ftp to ftp.cmdl.noaa.gov). In the second column the atmospheric CH4 burden inferred from observed concentrations by scaling the present-day burden of 4932 Tg [Denman et al., 2007] by the ratio of concentrations.
 The net δA13 results from the balance of global source and sink totals (EiSj, respectively, representing mass over time), with the sink fluxes multiplied by the fractionation factors αj specific to each process, and the emission fluxes by the isotopic ratios δi13 that characterize each source type. Assuming that both the atmospheric methane concentration and δA13 are in steady state gives the following relationship between δA13 and methane sources, sinks, and their respective δi13 signatures and sink fractionation factors αj (see Appendix A):
Equation (1) is used to optimize the global annual fluxes (in Tg/a) of ten sources (rice agriculture, domestic ruminants, waste, oil and natural gas production, coal mining and biofuel, biomass burning, wetlands, oceans, termites/wild animals, and terrestrial and marine geological seepage) and four sinks (soil consumption and reactions in the atmosphere with OH, Cl, and O(1D)), as well as their respective source signatures (δi13) or sink separation factors (εj = αj − 1). This is done by (1) formulating a priori estimates of present-day sources, sinks, and assumed uncertainties; (2) extrapolating these estimates to the past climatic periods defined above; (3) computing δA13 for each period using (1); and (4) adjusting the source and sink totals and their isotopic factors to minimize the misfit between implied and observed δA13.
 We adapt three CH4 emission scenarios from the recent literature as prior emission scenarios; they are summarized in Table 2. Scenarios B07 and BQT are based on two recent inversions of satellite observations, Bergamaschi et al.  and Bousquet et al. , respectively. While these scenarios have similar nonburning natural emissions, BQT attributes about 40Tg/a less to anthropogenic biogenic emissions (rice, domestic ruminants, and waste), and correspondingly more to anthropogenic “fossil” sources (oil/gas production and coal mining/biofuel) and biomass burning (which has both natural and anthropogenic components). We additionally adapt BQT into a third scenario, GEO, by adding an additional 40Tg/a for geological seepage, thus taking into account the possible strong geological CH4 source suggested by Etiope et al. . We neglect aerobic emissions from terrestrial plants. To avoid adding a difficult-to-constrain unknown to the optimization, we refrain from separating biomass burning into natural and anthropogenic components. We assume the same prior values for the isotopic characteristics δi13 of each source. These are also listed in Table 2.
Table 2. Prior Source and Sink Magnitudes and Assumed Isotopic Signatures for Three Prior Scenarios
 In order to focus this study on sources, the same prior magnitudes and sink separation factors are assumed for three of four CH4 sinks. We assume 29Tg/a for the bacterial consumption of CH4 in dry soils [Curry, 2007], 10Tg/a for the reaction of CH4 with O(1D) [Denman et al., 2007], and 25Tg/a for the reaction of CH4 with Cl. The assumed Cl sink is an estimate based on the marine boundary layer Cl sink estimate of Allan et al. , together with the stratospheric Cl sink, which is estimated to comprise around 20–35% [Röckmann et al., 2004] of the 40Tg/a total stratospheric sink of CH4 [Denman et al., 2007] (the sensitivity of our results to this assumption is tested in section 3.2). The strongest CH4 sink, the reaction with OH, is both highly uncertain and variable [Krol et al., 1998], and is therefore estimated such that the total sink equals the total emission for each scenario, in keeping with our assumption of steady state.
 A prior uncertainty of 25% is assumed for all sources and sinks. This is sometimes less that the range between scenarios, but is done because the problem is too weakly constrained to converge to a single “best fit” scenario. We shall see that this choice of errors preserves the fundamental differences between the scenarios, allowing us to examine the physical parameters under which each scenario can be reconciled with the isotopic observations.
 For the δi13 per source, an uncertainty of 5‰ is assumed for all sources except biomass burning, where we assume an uncertainty of 8‰ in order to account for possible differences between anthropogenic and natural burning, which we examine in more detail in sections 3.3 and 3.4. For the isotopic separation factors εj of the sinks, we adopt the uncertainties estimated by Lassey et al. [2007b], assuming uncertainties of 2‰, 0.75‰ and 1.0‰ for soil consumption, the reaction with OH, and the reaction with Cl, respectively. In addition, we assume a 2‰ uncertainty for the reaction with O(1D). Thus the optimization gives more freedom to change the isotopic signature of sources, which are the focus of our study.
 The ten sources and four sinks at the present-day, plus their respective δi13 or εj, comprise the 28-element optimization vector, x. To optimize x, a given present-day source/sink configuration is first mapped to the three past times (EI, EA, and LGM) by making the following five assumptions: (1) that anthropogenic emissions (not accounting for biomass burning) were negligible at LGM, and at EI and EA were given by the nonindustrial source totals, multiplied by relative population factors aEI = 0.24 and aEA = 0.12, respectively [United Nations Population Division, 1999]; (2) that wetland emissions at LGM were proportional to the present-day wetland emission by a factor wLGM; (3) that total biomass burning emissions at past times were proportional to the present-day total burning emission by factors pEI, pEA, and pLGM; and (4) that all other (nonburning) natural emissions remained constant relative to each other in time; and (5) that the sinks changed in proportion to the total emission, but not relative to each other. As an experiment baseline, we assume wLGM = 0.62, following [Weber et al., 2010], and examine other values in section 3.5. For pyrogenic emissions, we begin with baseline values of pEI = pEA = 1.0 and pLGM = 0.8, examining other values in sections 3.3 and 3.4.
Figure 1a compares the δA13 implied within the above framework by each scenario for the four time periods. The uncertainty of the implied δA13 is estimated via a 20,000-member ensemble of perturbations generated around each scenario, using Gaussian distributions with standard deviations given by the assumed uncertainties. It can be seen that, given the above assumptions, none of the emission scenarios fit all four periods equally well, with scenarios BQT and GEO overestimating δA13 for all times except LGM. Scenario B07, which is the most biogenically dominated, in general fits the observed δA13 best, but does not really capture the variation in δA13 across PD, EI, and EA. Given our assumed wetland and pyrogenic LGM emission factors, scenario BQT reproduces the observed high δA13 at LGM best.
 To see how each scenario changes when constrained to fit these observations, we compute the optimal state xa for each scenario using the Bayesian update equation:
 (x) represents the mapping of the elements of x to the observations of δA13, using (1) and the backward extrapolation described above. The difference between the observations of δA13 (y), and the result implied by a prior estimate of the state [(xb)] is weighted by an optimal weight matrix (K), which represents the prior uncertainty modulated by the total prior and observational uncertainty, and added to the background estimate. Because K includes the nonlinear operation ℋ, it is approximated using a 20,000-member ensemble of normally distributed random perturbations with standard deviations given by our prior assumed uncertainties. Having computed K, each ensemble member is adjusted via (2); the ensemble mean analysis is then taken as the optimized state.
 We define y to contain the δA13 observations for PD, EI, and EA, as well as the assumed present-day total emission, E. Since the only distinguishing factors between EA and LGM emissions in our framework are the wetland and pyrogenic emission reduction factors, the LGM observation only constrains these factors and does not give additional information about the other variables. We therefore perform the optimization without the LGM observation, and treat the relationship between assumed wLGM and pLGM and the implied δA13 in section 3.5.
 In section 3 we perform two main optimizations, not counting sensitivity tests: first, the scenarios are optimized such that their prior source and sink total is preserved. This shows how an enhancement of one source/sink requires a reduction in another in order to balance both the total emission and the net implied δA13. Then, all scenarios are optimized to a common total of 583Tg/a, which corresponds to the observed global average concentration for 2005 (Table 1), assuming no net growth of CH4 in 2005 and a present-day lifetime of 8.45 years [Denman et al., 2007] This yields three comparable scenarios for discussion.
Figure 1b shows the four values of δA13 that result for each scenario following optimization. The optimization manages to fit all scenarios to the observed δA13 at PD, EI, and EA, but worsens the fit to the LGM observation, which was not assimilated. This implies that our choice of factors pLGM = 0.8 and wLGM = 0.62 is not consistent with the adjusted scenarios for the LGM (to be discussed in section 3.5). Figure 2 compares the prior and optimized present-day source totals per scenario. For the sake of simplicity, and to filter out small differences in sources that are isotopically similar, the sources have been aggregated into five categories: natural and anthropogenic biogenic (NBIO, ABIO), natural and anthropogenic fossil (NFOSS and AFOSS) and pyrogenic (PYRO). It is clear that even though all scenarios are reconciled with the observations, their fundamental differences are retained.
 A common adjustment across all scenarios is an enhancement of anthropogenic fossil emissions (coal mining, biofuel, and oil/gas production) and biogenic emissions (rice, waste, and domestic ruminants), with concomitant reductions in natural biogenic (wetlands, termites, wild animals, and oceans) and pyrogenic emissions. This result can be explained by considering that the adjustment toward observations requires that the reduction of δA13 at EI must be stronger than for PD and EA. The fit to the EI observation is in part achieved by decreasing geological (NFOSS) and burning emissions (PYRO), but this adjustment affects the δA13 implied for PD, EI, and EA equally. As a consequence, AFOSS is increased in all scenarios while NBIO is lowered, which increases δA13 at PD relative to EI and EA. Increasing ABIO also raises δA13 at EA relative to EI, since fewer anthropogenic biogenic sources (which have low δ13) are assumed to exist at EA. Thus we have the overall result that to simultaneously satisfy both the present-day global δA13 and that of past times, assuming no change in biomass burning emissions since the early agricultural era, and no changes in the relative magnitudes of sinks or the isotopic signatures of sources and sinks over time, requires fossil fuel (or biofuel)-related emissions to be near the upper end of the uncertainty range.
Table 3 shows the optimized source/sink totals, their optimized isotopic signatures, and the totals and estimated uncertainties of each, when the optimization is repeated while constraining the total source and total sink in all three scenarios to 583Tg/a. This upscaling of all three scenarios was found to preserve the optimized δA13 values and the relative magnitudes of the individual sources. Comparing Table 3 to Table 2 shows that the range of prior estimates has largely been reduced for emissions from oil/gas production and waste. The fraction of natural emissions, not counting natural biomass burning, has been lowered overall, and constrained from 36–41% to 33–35%. Even though the scenarios fit the observations equally well, substantial differences between them remain.
Table 3. Optimized Global Totals for Sources, Sinks, and Their Corresponding Isotopic Factors, Optimized to a Total Source and Sink of 583 Tg/a
B07 δ13 (‰)
Optimized total source (E) and sink (S) and their respective isotopic factors, as well as the natural source fraction N/E (N excluding biomass burning).
 In particular, optimized scenario GEO is still distinguished by a strong geological source (though now with a small reduction in anthropogenic fossil sources relative to BQT), and BQT by a strong biomass burning source. Comparison of the optimized δ13 signatures of the sources reveals that the differences in the scenarios are made possible by compensating adjustments in source isotopic signatures, for which we assumed prior standard deviations of 5‰. For example, in BQT and GEO, where fossil and pyrogenic emissions are stronger, the δ13 of domestic ruminants is decreased by 6‰ (i.e., it is assumed to be more depleted in 13CH4). The largest change in the prior isotopic signature is for oil and gas in scenario B07, from the typically observed ratio for fossil fuels of −40‰ to −31‰. It can also be seen that the optimized total sink fractionations are within one standard deviation of each other, and within the estimate of −7.7 ± 1.4‰ made by Lassey et al. [2007b], with scenario B07 requiring a slightly stronger total sink fractionation to offset the slightly higher total δ13 of emission.
3.2. Sensitivity to the Present-Day Cl Sink
 Our choice of the 25Tg/a Cl sink is motivated by Lassey et al. [2007b], who infer this sink from the observed seasonal cycle of δA13 in the Marine Boundary Later (MBL), while the inversion studies on which we based our prior scenarios [Bergamaschi et al., 2007; Bousquet et al., 2006] neglected Cl in the MBL as a significant sink of methane. Since the reaction with Cl is highly fractionating, a significant Cl sink implies greater offset between δA13 and the net δ13 of emission. To test the dependence of our results to this observational constraint, we have performed an alternative experiment where the Cl sink is forced to 2Tg/a, which is even less than the approximately 8Tg/a stratospheric sink [Röckmann et al., 2004]. The resulting optimized scenarios (not shown) are within a few Tg/a of the original experiment; in all cases, the effect is less than the posterior uncertainty estimate. The strongest change is a ∼ 5Tg/a increase in AFOSS, with concomitant reductions in PYRO (−1Tg/a) and ABIO (−4Tg/a), while the natural fossil emissions in the GEO scenario are hardly affected. The impact of neglecting the Cl sink in an inversion of δA13 is therefore to increase estimated anthropogenic fossil emissions (rather than geological emissions) though in terms of global quantities the effect seems to be small.
3.3. Sensitivity to Assumed Isotopic Signature of Biomass Burning Emissions
 We likewise test our choice of −18.5‰ as the isotopic signature of biomass burning, which rests on the assumption that present-day burning is predominantly anthropogenic, which implies that burned vegetation consists substantially of C4 plants [Hao and Ward, 1993; Lassey et al., 2007b]. Since this neglects the fact that biomass burning in the past was more strongly natural and thus more likely to be dominated by C3 plants, which would give it a more negative δ13, we test our assumption by decreasing the δ13 of biomass burning to −25‰. The effect on optimized emissions is very small, the strongest adjustment being a ∼ 2Tg/a reduction in optimized pyrogenic emissions in scenario B07.
3.4. Temporal Variation in Pyrogenic Emissions
 It was found in section 3.1 that, in order to satisfy both the EI δA13 minimum and the PD observation, optimization increases anthropogenic fossil emissions relative to the prior. This happens in part because pyrogenic emissions, which are isotopically heavy and were estimated above to be unchanged between EI/EA and PD, constitute a much larger part of EI/EA emissions, thus requiring a strong adjustment in the other emission types, which is in turn compensated for by large increases in anthropogenic fossil emissions at the present-day.
 This result is therefore related to our assumed pyrogenic emission factors pEI and pEA, and is examined more closely in Figure 3, which shows how the optimized present-day AFOSS changes as we change the value assumed for pEI and pEA from 0.5 to 1.5. Superimposed in Figure 3 are the results of two alternative optimizations, one (dashed lines) where the isotopic signature of biomass burning is lowered to −25‰ (reflecting a possible reduction in anthropogenic biomass burning), and one (dot-dashed lines) where the isotopic signature of wetland emissions is increased to −52‰, which considers the extreme possibility that wetland emissions are dominated by inundated trees instead of grasses [Rice et al., 2010]. The baseline value (pEI = pEA = 1.0) is indicated by a vertical line. As predicted, estimated AFOSS increases with pEI/pEA. Assuming the high extreme value of δ13 for wetlands results in an optimized AFOSS that is about 3Tg/a lower, while assuming the higher value for biomass burning raises the optimized AFOSS by at most 1Tg/a.
 Charcoal records [Power et al., 2008] suggest that pyrogenic CH4 emissions were slightly higher at EI and EA than PD (presumably due to more aggressive agricultural burning), though the actual CH4 emissions for these eras depend strongly on vegetation and fire type and are therefore far less clear. If EI/EA biomass burning emissions differed from the present-day by 20%, the impact on estimated present-day anthropogenic fossil emissions is within about 5Tg/a, while the range between scenarios is much larger. The BQT prior total for AFOSS (122Tg/a) results from inverse modeling of present-day measurements of both concentration and δA13, and is thus substantially backed up by observation. It is indicated by a horizontal solid line in Figure 3. It is not possible to achieve this total in the optimization of our three scenarios without a change in EI/EA pyrogenic emissions that exceeds 50%.
3.5. Implication for LGM Emissions
 We now turn our attention to the observed δA13 at LGM. Clearly, our assumption that wLGM = 0.62 (62% wetland emissions relative to PD) and pLGM = 0.8 (80% pyrogenic emissions relative to PD) is no longer consistent with this observation given the optimized scenarios (Figure 1). If pLGM is increased, the implied LGM emissions will be isotopically heavier, which means that the wLGM required to satisfy the observation will increase. In other words, if there are substantial natural fossil emissions, the required wetland emission reduction will be less. Figure 4 shows the change in implied δA13 at LGM as a function of wLGM, for unchanged LGM burning emissions (pLGM = 1), and a 20% reduction and enhancement of emissions at LGM (pLGM = 0.8 and 1.2, respectively). The plot is shown for scenarios B07 (Figure 4a) and GEO (Figure 4b), omitting BQT because it shows values in between those of B07 and GEO. The observed LGM δA13 is indicated by a horizontal black line.
 Charcoal records indicate LGM fire activity that is lower than the present-day [Power et al., 2008]. Assuming that pLGM = 0.8 we find that emissions from wetlands had to be 30% of the present-day value in scenario B07, and about 42% in scenario GEO, in order for the implied δA13 to meet the observed value. For comparison, the finding of Weber et al. , that LGM wetland emissions were by 58–65% of the PD emission, is indicated in Figure 4 by a dashed line at 0.62. Similarly, the result of Kaplan , that wetland emissions at LGM were reduced to 76% of the PD value, is indicated by a dot-dashed line. For the biogenically dominated scenario B07, simultaneously fitting the observed δA13 and either estimate of wetland emissions reduction requires increases in biomass burning that are likely to be unrealistic (pLGM > 1.2). However, with the strong geological emissions of scenario GEO, the Weber et al.  value can be reconciled with the observations for a 20% enhancement of pyrogenic emissions.
3.6. Implication for Changing CH4 Lifetime
 Our optimizations do not include as a constraint the observed atmospheric CH4 concentrations for the past time periods (Table 1), and thereby avoid making assumptions about changes in CH4 lifetime between the four climate periods. However, if for example our implied past time sources and sinks are small compared to the observed burden at each time (along with the assumption of a steady state), then we are required to assume either an overall enhancement of natural emissions at the past times (EI or EA) and/or a change in the atmospheric lifetime of CH4. To see this, one can relate the total emission for periods T = EI and EA to the corresponding approximate burden, divided by lifetime:
Here τ represents the present-day CH4 lifetime, and A′, P and N represent the nonindustrial-anthropogenic, pyrogenic, and natural emission totals, respectively, for present-day; these are scaled by the factors aT and pT (assumed a priori), and nT and sT, which represent the ratio between past time and present-day natural emissions (nT), and lifetime (sT). BT represents the atmospheric burden at each time period, and can be approximated by scaling the present-day burden of 4932 Tg by the ratio of present-day and past time Antarctic/South Pole concentrations in Table 1. This neglects the difference between global mean and Antarctic CH4 resulting from the latitudinal gradient, which is estimated as 1–4% for LGM [Chappellaz et al., 1997; Brook et al., 2000] and 2–4% for EA and EI [Dällenbach et al., 2000; Brook et al., 2000], and thus translates to at most about 5% difference in the estimated burden. Having assumed values for aT it is possible to use equation (3) to infer either nT or sT, while assuming the other is unity, as a function of pT.
 We find that the maximum implied changes for both periods, if pEI and pEA are varied from 0.5 to 1.5, is a 3% reduction of either the CH4 lifetime or natural CH4 emissions at EI or EA, which is marginal. A somewhat reduced EA/EI lifetime is certainly plausible [e.g., Crutzen and Brühl, 1993].
4. Summary and Conclusions
 We have presented an optimal estimation of the present-day methane budget using the observed δA13 isotopic record, taking into account the uncertainties in sources, sinks, and δi13 and εj factors. It was found that it is possible to fit the observations by slightly adjusting three present-day CH4 emission scenarios, without changing the fundamental properties that distinguish each scenario. This yields three distinct ways of partitioning the present-day CH4 emission budget, all of which satisfy the δA13 record: scenario B07 is dominated by biogenic sources, especially anthropogenic ones (rice and waste), while BQT and GEO feature more prominent contributions from biomass burning (33–38Tg/a), and GEO an additional strong contribution (36 ± 10Tg/a) from geological seepage. The 14Tg/a range of biomass burning emissions covered by these scenarios is within the interannual variability estimated for biomass burning emissions [e.g., Butler et al., 2005]. Comparing the scenarios to the approximate CH4 burden, all three imply only marginal changes in CH4 lifetime or nonpyrogenic natural emissions over the last millenium.
 In order to differentiate the δA13 implied for the present-day from the early industrial and early agricultural eras, we find that present-day anthropogenic emissions related to fossil fuels (coal mining and biofuel, oil and gas production) need to be on the high end of the assumed uncertainty range, especially if biomass burning in the past was high relative to the present-day. This, along with relatively lower present-day natural biogenic emissions than were assumed a priori, was a common result in the optimization of all scenarios; even in the presence of strong geological emissions, which contribute to the total fossil fraction of emission. The optimization of scenario GEO shows that, while the geological source helps to explain the high δA13 at LGM, geologic emissions do not readily explain the difference in δA13 between, e.g., the early industrial period and present-day. For scenario B07, the shift from biogenic to anthropogenic fossil emissions is primarily pronounced as an increase of estimated oil/gas emission to 72Tg/a, in line with scenarios BQT and GEO. The posterior estimate for anthropogenic emissions in B07 is in agreement with the recently published Emission Database for Global Atmospheric Research (EDGAR) version 4 emissions [EDGAR, 2010].
 Overall, optimization constrains the present-day natural fraction of emissions to 32–35%, not counting natural biomass burning. In contrast, the optimized scenarios differ in the fossil fraction of sources. B07 has 19% fossil sources, BQT has 25%, and GEO 30%. This range is possible within the allowed uncertainty range of the δ13 signatures of the sources; that is, scenario B07 tends to estimate these to be on the high, and scenario GEO on the low end, of the uncertainty range. From observations of atmospheric radiocarbon, Quay et al.  estimate that fossil emissions, which are devoid of radiocarbon, make up 18 ± 9% of total present-day emissions, while more recent work by Lassey et al. [2007a] argues that a fossil fraction of 30 ± 2.3% is more likely. Thus, while fossil fractions of about 20–29% can evidently be accommodated within the global δ13 budget, the results of Lassey et al. [2007a] would suggest that the BQT and GEO scenarios are more likely, though it would mean that the present-day emissions from coal mining and biofuel are about 34Tg/a stronger than estimated by B07, and also much larger than assumed in the recent EDGAR 4 inventory [EDGAR, 2010].
 We find the maximum feasible wetland emissions at LGM to be ∼ 60% of present-day wetland emissions, in agreement with Weber et al. , but this requires a ∼ 36Tg/a geological CH4 source (scenario GEO), and a ∼ 20% enhancement in LGM biomass burning emissions relative to the present-day. For the biogenically dominated scenario B07, and without any LGM enhancement of biomass burning, estimated wetland emissions are about 40% the present-day value.
 The interpretation of the LGM observations in section 3.5 neglects possible relative changes in CH4 sinks at LGM. For example, both Kaplan  and Schaefer and Whiticar  estimate that the soil sink was reduced at the LGM relative to PD. Schaefer and Whiticar  estimate that this would result in a ∼ 1.7‰ weaker total sink fractionation, which would require a corresponding increase in the δ13 of emission. In our framework this translates to roughly a 0.15 reduction of wLGM for a given pLGM, i.e., an additional 15% reduction of LGM wetland emissions relative to the present-day, which, given reasonable estimates of biomass burning at LGM, would increase the disagreement between our results and the wetland emission reductions estimated by Kaplan  and Weber et al. . A more exact number would require an in-depth analysis of LGM atmospheric chemistry, and is beyond the scope of this study. Observations of δD may offer further constraints on the past and present time CH4 budgets; here we point to the recent work of Sowers  and Mischler et al. . As mentioned above, in order to avoid convoluting the results with too many unknown factors, this study has not in detail considered changes in the δ13 signatures of the individual sources over the last 25,000 years, which could have arisen from various climatic effects, with several complex possibilities [Whiticar and Schaefer, 2009]. We have also not explicitly accounted for possible isotopic differences in emissions from boreal peatlands (which are dominated by C3 plants, Still et al. ) and tropical wetlands, though the experiment of section 3.4 suggests that even strong changes in the δ13 of wetland emissions would have a small effect on the outcome of this study. The optimal estimation approach outlined here could be extended to take these factors into account, though such an approach would also increase the degrees of freedom of an already underconstrained problem, which further increases the necessity for more observations and better prior knowledge of the relevant sources and sinks (as well as their isotopic signatures) at each point in time.
 It remains to be seen how the scenarios resulting from this analysis compare to observations with spatial and temporal structure, in particular from satellites. Inversions of satellite measurements by Bergamaschi et al. , Meirink et al.  and Bergamaschi et al.  have suggested that an emission budget similar to scenario B07 is likely, though these studies did not test alternative a priori scenarios. Accounting for geological emissions in an inversion of high-resolution spatial observations requires an a priori inventory of the global distribution of geological seepage, which to our knowledge has not yet been published.
 The relationship between source/sink fluxes and isotopic signatures at some time, and the observed δA13 at that time, is derived by writing out the source/sink mass balance for 13CH4. Defining the atmospheric ratio RA = 13CH4/12CH4 and the concentration of total CH4 as CA, this balance is given by
Assuming steady state of both CA and RA, we have
Substituting in the definition of δ13 yields the following expression:
 Our present-day state is taken to be the year 2005, which follows nearly a decade of constant atmospheric CH4 concentrations. For the three past times examined in this paper, changes in the global CH4 concentrations are assumed slow given the decadal lifetime of CH4.
 The authors gratefully acknowledge helpful discussions with Nanne Weber, as well as the comments of two anonymous reviewers, in the preparation of this manuscript. This work is part of the EU Project HYMN, which is funded by the EU 6th Framework Programme (GOCE) under contract 037048.