Global distribution of methane emissions: a comparative inverse analysis of observations from the TROPOMI and GOSAT satellite instruments

. We evaluate the global atmospheric methane column retrievals from the new TROPOMI satellite instrument and apply them to a global inversion of methane sources for 2019 at 2° ´ 2.5° horizontal resolution. We compare the results to an inversion using the sparser but more mature GOSAT satellite retrievals, and to a joint inversion using both TROPOMI and GOSAT. Validation of TROPOMI and GOSAT with TCCON ground-based measurements of methane columns, after 20 correcting for retrieval differences in prior vertical profiles and averaging kernels using the GEOS-Chem chemical transport model, shows global biases of -2.7 ppbv for TROPOMI and -1.0 ppbv for GOSAT, and regional biases of 6.7 ppbv for TROPOMI and 2.9 ppbv for GOSAT. Intercomparison of TROPOMI and GOSAT shows larger regional discrepancies exceeding 20 ppbv, mostly over regions with low surface albedo in the shortwave infrared where the TROPOMI retrieval may be biased. Our inversion uses an analytical solution to the Bayesian inference of methane sources, thus providing an explicit 25 characterization of error statistics and information content together with the solution. TROPOMI has ~100 times more observations than GOSAT but error correlation on the 2° ´ 2.5° scale of the inversion and large spatial inhomogeneity in the number of observations make it less useful than GOSAT for quantifying emissions at that scale. Finer-scale regional inversions would take better advantage of the TROPOMI data density. The TROPOMI and GOSAT inversions show consistent downward adjustments of global oil/gas emissions relative to a prior estimate based on national inventory reports to the United Nations 30 Framework Convention on Climate Change, but consistent increases in the south-central US and in Venezuela. Global emissions from livestock (the largest anthropogenic source) are adjusted upward by TROPOMI and GOSAT relative to the EDGAR v4.3.2 prior estimate. We find large artifacts in the TROPOMI inversion over Southeast China, where seasonal rice emissions are particularly high but in phase with extensive cloudiness, and where coal emissions may be Future advances in the TROPOMI retrieval together with finer-scale inversions and improved accounting of error correlations should 40 enable improved exploitation of TROPOMI observations to quantify and attribute methane emissions on the global scale.

3 the observations and a prior estimate . Most inverse analyses use 4-dimensional variational data assimilation (4D-Var) to solve the Bayesian problem numerically, which enables inference of emissions at any resolution but does not 75 readily provide error statistics [Meirink et al., 2008;Monteil et al., 2013;Wecht et al., 2014;Stanevich et al., 2019]. Analytical solution is possible if the CTM is linear, as is the case for methane, and has the advantage of including posterior error statistics and hence information content as part of the solution [Brasseur and Jacob, 2017]. It requires explicit construction of the Jacobian matrix of the CTM, which is computationally expensive, but this is readily done with massively parallel computing.
Once the Jacobian matrix has been constructed, it can be applied to conduct ensembles of inversions at no added cost exploring 80 the dependence of the solution on inversion parameters or observational data selection. The analytical method can be applied as a Kalman Filter by updating methane emissions sequentially [e.g., Chen and Prinn, 2006;Fraser et al., 2013;Henne et al., 2016] but optimizing all emissions together over the period of interest makes the best use of the information content from the observations [Maasakkers et al., 2019;Lu et al, 2021;. Analytically-based inversions of GOSAT satellite data have been used to pinpoint areas where the inversion results are most informed by the observations [Turner et al., 2015;85 Maasakkers et al., 2019], to diagnose the ability of the inversion to separate contributions from different source sectors  and from sources and sinks [Y. Zhang et al., 2018;Maasakkers et al., 2019], and to compare the information content from satellite and suborbital observations Baray et al., 2021].
Here we present global analytical inversions of TROPOMI and GOSAT data for 2019 at 2°´2.5° resolution to infer methane 90 sources and sinks and to attribute emissions to different sectors. This involves evaluation and intercomparison of the TROPOMI and GOSAT retrievals prior to the inversion, as any biases in the observations will propagate to the inversion results. We compare inversion results for the two instruments separately and jointly. We diagnose the information content of the inversion for each instrument and for the joint system in different regions of the world. This enables us to assess the consistency and complementarity of the two data sets. 95 2 Methane observations TROPOMI and GOSAT are in Sun-synchronous orbits with local overpass solar times of 13:30 and 13:00 respectively [Veefkind et al., 2012;Kuze et al., 2016]. We use the version 1.03 TROPOMI methane retrieval from the Netherlands Institute for Space Research [Hu et al., 2016] (http://www.tropomi.eu/data-products/methane, last accessed Aug 8, 2020) and the GOSAT methane retrieval version 9.0 of the University of Leicester obtained by the CO2 proxy method [Parker and Boesch,100 2020] (https://catalogue.ceda.ac.uk/uuid/18ef8247f52a4cb6a14013f8235cc1eb, last accessed Dec 29, 2020). We use one year data (January-December 2019) to optimize methane emissions for 2019. We only include high quality retrievals with "qa_value" ³ 0.5 for TROPOMI [S5P MPC, 2020] and "xch4_quality_flag" = 0 for GOSAT. The TROPOMI and GOSAT products are provided as column-averaged dry methane mixing ratios (XCH4) along with the prior vertical profiles used in the retrieval procedures and the averaging kernel vectors describing the altitude-dependent sensitivity of the retrievals. 105 4 The left panels of Figure 1 show the annual mean XCH4 observations from TROPOMI and GOSAT in 2019. We excluded observations poleward of 60° where (1) persistent snow cover leads to low albedo , (2) low Sun angles and extensive cloud cover make the retrieval more difficult, and (3) stratospheric CTM bias can affect the inversion [Turner et al., 2015]. The TROPOMI retrieval is successful for only 2% of scenes at 60°S -60°N, still producing 56684576 TROPOMI 115 observations, which is two orders of magnitude higher than for GOSAT (544911 observations after filtering with the quality flag). As shown in the right panel of Figure 1, TROPOMI observations are relatively sparse over persistently cloudy regions such as the wet tropics. GOSAT has relatively more success over these regions because of the use of the CO2 proxy method.
The GOSAT CH4 product also includes observations over the ocean for sunglint geometries and these are not included in the current TROPOMI product. 120 We conducted a common evaluation of the TROPOMI and GOSAT observations with ground-based Total Carbon Column Observing Network (TCCON) measurements of XCH4 [TCCON Team, 2017], using the GEOS-Chem CTM to resolve differences in prior estimates and averaging kernels between the TROPOMI, GOSAT, and TCCON retrievals [L. Zhang et al., 2010]. TCCON is a network of ground-based, sun-viewing, near Infra-Red, Fourier transform spectrometers to measure 125 greenhouse gases  and evaluate satellite retrievals [Parker et al., 2011;Houweling et al., 2014] Following L. Zhang et al. [2010], we remove the discrepancy from the use of different prior profiles in the TROPOMI, GOSAT, and TCCON retrievals by substituting a common fixed prior profile, which we take as the annual averaged TROPOMI prior profile between 30°S and 30°N. This substitution is done only for the purpose of intercomparison; it is not used subsequently 135 in the inversion. We average the individual retrievals over 2 o´2 .5 o GEOS-Chem grid cells, and apply the averaging kernels of the individual retrievals to the GEOS-Chem simulated vertical profiles to produce a model simulation of the observations. We calculate the differences Δ between satellite and TCCON measurements with reference to the GEOS-Chem CTM using: where 1 234,67898:; , 1 234,B8CD6 , and 1 234,6228> are methane column mixing ratios from TROPOMI, GOSAT, and TCCON 145 after substitution with the same prior profile. 1 234,26:_67898:; , 1 234,26:_B8CD6 , and 1 234,26:_6228> are simulated methane column mixing ratios with the appropriate averaging kernels applied. Figure 2 shows the mean differences of TROPOMI and GOSAT with TCCON for the 21 TCCON sites. There are large and correlated differences at the Izana and Zugspitze mountaintop sites where we would not expect consistency with the satellite 150 data averaged over the 2 o´2 .5 o grid. For the remaining 19 sites, the mean biases are -2.7 ppbv for TROPOMI and -1.0 ppbv for GOSAT. Of more interest for the inversion are systematic errors on regional scales (regional bias), which can be estimated by the standard deviations of ∆ (TROPOMI -TCCON) and ∆ (GOSAT -TCCON) across all TCCON sites [Buchwitz et al., 2015]. The regional bias diagnoses the reliability of the observed methane gradients for inferring methane sources in the inversion. We find regional biases of 2.9 ppbv for GOSAT and 6.7 ppbv for TROPOMI. The regional bias for GOSAT is 155 below the "breakthrough requirement" of 5 ppbv set by Buchwitz et al. [2015] as needing to be achieved for regional/global inversions of satellite observations, and the regional bias for TROPOMI is below their "threshold requirement" of 10 ppbv.
This implies that GOSAT observations are of high quality for quantifying methane sources while TROPOMI observations are still useful. The regional bias of GOSAT compared to TCCON is smaller than the value of 3.9 ppbv reported by Parker et al.
[2020a], which may reflect at least in part our accounting for differences in averaging kernels and prior vertical profiles. The 160 larger regional biases in the TROPOMI data may reflect error correlations of retrieved XCH4 and SWIR surface albedo [Hu et al., 2018;Hasekamp et al., 2019;Schneising et al., 2019].
We apply the same method for a more extensive analysis of regional differences between TROPOMI and GOSAT. Figure 3 shows the global distributions of the seasonal mean differences ∆ between the two instruments, again correcting for differences 165 in prior estimates and averaging kernels. The seasonal global mean biases for TROPOMI relative to GOSAT are consistent with the comparison to TCCON but the regional biases are larger (8.8-12.8 ppbv), and some regions show differences of magnitude comparable to the regional enhancements of Figure 1. The regional biases tend to be consistent across seasons, except for positive biases north of 40 o N in DJF that could be associated with snow cover. These biases may affect TROPOMI's constraints on the seasonal variations of methane sources. We find particularly large differences between TROPOMI and 170 GOSAT where the SWIR surface albedo is smaller than 0.1 as in Brazil, central Africa, and subarctic regions (see Figure S1).
We may therefore expect large differences between TROPOMI and GOSAT inversions for these regions.

Inversion method
We assemble the 2019 TROPOMI and GOSAT observations of XCH4 into an observation vector , and use the observations to optimize a state vector consisting of methane sources and sinks. We use the GEOS-Chem global CTM version 12. is essentially linear except for a small nonlinearity from the optimization of OH concentrations [Maasakkers et al., 2019]. Prior estimates G for methane sources are compiled from bottom-up inventories. We solve the Bayesian problem analytically to obtain both the posterior solution H and its error covariance matrix 1 . We conduct inversions using TROPOMI and GOSAT observations separately and together (joint inversion), and also conduct additional inversions to examine the sensitivity of 185 results to different parameters.

GEOS-Chem simulations and prior estimates
GEOS-Chem is driven by Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2) meteorological fields from the NASA Global Modeling and Assimilation Office (GMAO). The original methane simulation is described by Wecht et al. [2014]. Previous GEOS-Chem-based inversions at 4°´5° horizontal resolution had excessive 190 stratospheric methane poleward of 60 o in winter-spring due to the inability to reproduce the polar vortex dynamical barrier, and this needed to be corrected in the inversion [Turner et al., 2015;. The polar vortex dynamics are much better captured at 2°´2.5° resolution [Stanevich et al., 2019;, and we do not use satellite data poleward of 60 o in our inversion anyway. There is therefore no need for stratospheric bias correction.
195 Table 1 summarizes the prior estimates of the sources and sinks of methane, and Figure 4 shows the spatial distribution of the sources. The emissions from oil, gas, and coal exploitation are from the 2016 Global Fuel Exploitation Inventory (GFEI) version 1.0 [Scarpelli et al., 2020], which spatially allocates national emissions reported to the United Nations Framework inventory [Bloom et al., 2017], which has good performance in reproducing the observed wetland methane seasonal cycle for most regions [Parker et al., 2020b] Fung et al. [1991], and geological seepage from Etiope et al. [2019] scaled to the global magnitude of 2 Tg a -1 from Hmiel et al. [2020]. The total methane sources in the prior estimate add up to 542 Tg a -1 , which is smaller than the bottom-up inventory estimate of 594-881 Tg a -1 from the Global 210 Methane Budget 2020 [Saunois et al., 2020]. The difference is mainly caused by the higher estimates of emissions from freshwater (117 -212 Tg a -1 ), seeps (18 -65 Tg a -1 ), oil and gas (72 -97 Tg a -1 ), and coal (29 -61 Tg a -1 ) in Saunois et al. 7 [2020]. The freshwater source in our prior estimate is included in the wetland sector as represented by WetCHARTs [Bloom et al., 2017].

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The main sink of methane is oxidation by the hydroxyl radical (OH) in the troposphere [Ehhalt and Heidt, 1973], with a corresponding lifetime of 11.2 ± 1.3 years as constrained by the methylchloroform proxy [Prather et al., 2012]. Our prior estimate for the loss of methane from reaction with tropospheric OH is calculated using archived 3-D climatological monthly fields of OH concentrations from a GEOS-Chem full-chemistry simulation [Wecht et al., 2014], yielding a methane lifetime of 10.5 years due to oxidation by tropospheric OH. Additional minor losses include oxidation by tropospheric Cl atoms 225 computed using archived Cl concentration fields from Wang et al. [2019], stratospheric oxidation computed with archived 2-D monthly loss frequencies from the NASA Global Modeling Initiative model [Murray et al., 2012], and soil uptake of methane specified following Murguia-Flores et al. [2018].

Analytical inversion 230
We apply Bayesian inference to optimize a state vector consisting of (1) annual mean non-wetland methane emissions for land-containing 2 o´2 .5 o grid cells (4020 state vector elements), (2) monthly wetland methane emissions for the 14 subcontinental regions of Figure 4 (168 elements), and (3) annual hemispheric tropospheric OH concentrations (2 elements).
Trade-off is needed between spatial and temporal resolution in the state vector to avoid smoothing error in the inversion [Wecht et al, 2014] and for computational tractability. For non-wetland emissions we use high spatial resolution but only optimize the 235 annual mean values because seasonality is relatively small and predictable. For wetland emissions, we cannot assume that the prior seasonality is correct [Maasakkers et al., 2019] and instead optimize monthly emissions at coarse spatial resolution. This setup is the same as in Lu et al. [2021] and Y. Zhang et al. [2021] except for the higher horizontal resolution applied to nonwetland emissions. Together we have 4190 state vector elements, which requires a total of 4190 perturbed GEOS-Chem simulations and a base simulation to construct the full Jacobian matrix. This is readily done on a high-performance computing 240 platform as an embarrassingly parallel workload. Initial conditions on January 1, 2019 are obtained from a standard GEOS-Chem simulation using the prior emission estimates and a 10-year spin-up, and are scaled by a globally uniform factor of 0.97 in order to match the global mean column mixing ratio retrieved from TROPOMI between Jan 1 and Jan 10, 2019. This initialization efficiently reduces the normalized mean square error (NMSE) between GEOS-Chem and TROPOMI observations on January 1, 2019 from 0.37 to 0.02 and is used for both TROPOMI and GOSAT inversions. 245 The posterior estimate as defined by Bayesian inference assuming Gaussian error statistics is obtained by minimizing the scalar cost function J(x): where K is the Jacobian matrix describing the sensitivity of the observations to the state vector as simulated by GEOS-Chem, is the prior error covariance matrix, is the observational error covariance matrix assumed to be diagonal, and is a regularization parameter that accounts for the effect of unresolved correlation in the observational error. 255 is constructed by assuming 50% prior error standard deviation for all non-wetland emissions on the 2°´2.5° grid and 10% prior error standard deviation for hemispheric annual mean OH concentrations, with no error correlations. Prior error variances and covariances for monthly wetland emissions in the 14 subcontinental regions are calculated using the WetCHARTs model ensemble [Bloom et al., 2017] following Y. Zhang et al. [2021]. 260 Observational error variances (diagonal elements of ) are calculated using the residual error method [Heald et al., 2004] as the variance of the residual difference between observations and the GEOS-Chem prior simulation on the 2°´2.5° grid after subtracting the mean difference. This method sums up errors from instrument retrieval, representation, and GEOS-Chem transport. We find a global annual mean error of 13 ppbv for TROPOMI and 14 ppbv for GOSAT. For cases where the 265 calculated error is smaller than the instrument precision reported in the satellite retrieval, we use the latter instead (annual means of 9 ppbv for GOSAT and 2 ppbv for TROPOMI).
So is specified as diagonal but there is in fact some observational error covariance if only from the GEOS-Chem transport. For TROPOMI in particular, there may be many individual observations in a single GEOS-Chem grid cell for a given day, and the 270 corresponding GEOS-Chem transport errors would be perfectly correlated. Although one could average all TROPOMI observations within a 2 o´2 .5 o grid cell before ingesting them in the inversion, this would lose the averaging kernel specificity for each observation. We therefore use a regularization parameter [Hansen et al., 1999;Y. Zhang et al., 2018Y. Zhang et al., , 2020Maasakkers et al., 2019;Lu et al., 2021] to account for the off-diagonal structure missing in . Based on the corner of the Lcurve [Hansen et al., 1999] and the expected chi-square distribution of the cost function  (see Figure S2), we 275 choose = 0.002 for TROPOMI observations and = 0.5 for GOSAT observations. The regularization parameter of 0.5 for GOSAT is larger than the values of 0.05 -0.1 in Maasakkers et al. [2019], Y. Zhang et al. [2021], and Lu et al. [2021], because they used 4 o´5o resolution and several years of observations. The smaller value of for TROPOMI is due to its large number of collocated observations on the 2°´2.5° model grid. Shen et al., [2021] conducted a regional inversion of TROPOMI data using GEOS-Chem at 0.25°´0.3125° resolution and found that = 0.25 provided the best fit to the L-curve, reflecting the 280 much smaller number of collocated observations on the 0.25°´0.3125° grid.
We further balance the prior terms in the cost function by weighing the wetland emission term by the number of elements in the state vector (4020/14). This step ensures that changes in non-wetland and wetland emissions are equally expensive from a Deleted: for cost-function perspective [Maasakkers et al., 2019]. Similarly scaling the hemispheric OH terms in the cost function by the 290 number of elements in the state vector (4020/2) would lead to excessively small posterior adjustments. We therefore choose weighting factors of the OH terms (400 for TROPOMI, 450 for GOSAT) that lead to a standard deviation of 5% in the posterior OH adjustments.
There is some arbitrariness in the selection of regularization parameters and prior weighting factors in the inversion. In 295 addition to the base inversion as described above, we examined the sensitivity to the choice of with sensitivity inversions using (1) = 0.02 and (2) = 0.5 for TROPOMI, and (1) = 0.02 and (2) = 0.002 for GOSAT. We further examined the sensitivity to the choice of weighting factors with sensitivity inversions using (3) no weighting factors, (4) a weighting factor of 1 for wetland terms, and (5) a weighting factor of 2010 for the OH terms (i.e., the ratio of the number of state vector elements for non-wetland and OH terms). In this manner we performed 6 inversions using TROPOMI observations only (base 300 inversion + 5 sensitivity inversions), 6 inversions using GOSAT observations only, and 6 ´ 6 = 36 inversions using the joint TROPOMI and GOSAT observations.
The best posterior estimate obtained by minimization of the cost function J(x) is given by [Rodgers, 2000]: with posterior error covariance matrix 1 : The averaging kernel matrix defines the sensitivity of the solution to the true state:

315
where is the identity matrix. The trace of represents the number of independent pieces of information on the state vector that is gained from the observations, and is called the degrees of freedom for signal (DOFS) [Rodgers, 2000]. Note that A here is different from the retrieval averaging kernel vectors in Section 2, which described the sensitivity of methane satellite retrievals to the vertical distribution of methane.

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The posterior solution can also be presented in reduced dimensionality.
and its posterior error covariance and averaging kernel matrices are given by where * = ( ) N is the Moore-Penrose inverse [Calisesi et al., 2005].

Results and discussion
Our discussion focuses principally on results from the base inversions of the TROPOMI-only, GOSAT-only, and joint 340 TROPOMI+GOSAT observations, and uses ranges from the inversion ensemble as a more conservative estimate of posterior errors than the posterior error covariance matrix 1 . In this analysis we exclude ensemble members with unreasonable emission adjustments (e.g., negative emissions aggregated at regional scales) and OH adjustments larger than 40% (see Table S1, S2). The TROPOMI inversion has 155 DOFS, meaning that it contains 155 independent pieces of information on the distribution of methane emissions and OH concentrations. The GOSAT inversion has 238 DOFS, more than TROPOMI despite having much fewer observations. This reflects the large error correlation between individual TROPOMI observations on the 2 o´2 .5 o 355 grid of the inversion, as expressed by the difference between the regularization parameters for GOSAT observations (γ = 0.5) and TROPOMI observations (γ = 0.002). GOSAT with precise individual observations spaced by 250 km is particularly well adapted to an inversion on a 2 o´2 .5 o grid. TROPOMI would be far more valuable in a regional inversion at higher spatial resolution , although the regional biases discussed in Section 2 would still be a concern.

Information content from the inversions 345
Deleted: data sets Y. Zhang et al. [2021] previously reported an inversion of 2010-2018 GOSAT data using GEOS-Chem at 4 o´5o resolution.
That inversion achieved 179 DOFS, compared to 238 DOFS in our inversion for just one year of GOSAT data at 2 o´2 .5 o resolution. The higher DOFS in our case reflects the higher dimension of our emission state vector (2 o´2 .5 o versus 4 o´5o grid 365 cells), combined with higher weight per observation (γ = 0.5 versus 0.05) because of lower error correlation on the 2 o´2 .5 o scale. As pointed out above, the GOSAT data are particularly well suited to a 2 o´2 .5 o resolution for the inversion. The finer 2 o´2 .5 o resolution allows for improved sectoral and national attribution of inversion results as will be done in Section 4.3.
In the joint inversion, TROPOMI observations add additional DOFS to the GOSAT posterior at 0°-30° N (mainly over India 370 and the Middle East, Figure 5 and Figure S3), where TROPOMI has more observations than in the rest of the world ( Figure   1). TROPOMI has lower averaging kernel sensitivities at 30°-60° N and 0°-60° S, and the information content over these two regions mostly comes from GOSAT. This could reflect the limitation of using a single global regularization parameter γ for the TROPOMI observations, because the observations should have more weight (larger γ) when they are less dense. Improving this aspect of the inversion is a target for future work. 375 The 155 DOFS for TROPOMI are partitioned as 151 for non-wetland emissions, 3 for wetlands, and 1 for OH. The 238 DOFS for GOSAT are partitioned as 232 for non-wetland emissions, 5 for wetlands, and 1 for OH. The wetland emissions are largely unchanged in both inversions because of error weighting in the cost function that penalizes departure from the prior estimate.
Without this error weighting, the TROPOMI inversion would yield unrealistic wetland emissions and seasonalities (case 3 in 380 Table S1). The problem may reflect systematic biases in the TROPOMI retrieval due to the low SWIR surface albedo over wetland surfaces (e.g., Brazil and central Africa, see Figure S4, and boreal wetlands in Canada and Russia), combined with seasonal imbalance in observations (cloudiness for tropical wetlands, sun angle and snow for boreal wetlands) and seasonal biases at high northern latitudes (Figure 3). The GOSAT-only inversion without error weighting for wetlands shows no such problems, but we still apply error weighting in that base inversion for comparison to TROPOMI. Improvement in TROPOMI 385 retrievals over wetlands is clearly needed. In the meantime, our further discussion of results in Section 4.3 will focus on the non-wetland emissions.
The posterior/prior ratio of global OH concentrations is 0.96 for both the TROPOMI and GOSAT inversions and 0.91 for the joint inversion. Methane lifetimes against oxidation by tropospheric OH range from 10.7 to 11.0 years in the ensemble of 390 TROPOMI inversions excluding case 3 (Table S1) and from 10.7 to 11.1 years in the GOSAT inversions (Table S2). These corrections improve agreement with the observationally constrained methane lifetime of 11.2 ± 1.3 years [Prather et al., 2012].
The north/south interhemispheric OH ratio (NH/SH) is 1.03 in the prior estimate, 0.93 in the TROPOMI inversion, 1.15 in the GOSAT inversion, and 1.03 in the joint inversion, suggesting that the observations do not usefully constrain this ratio. Patra 400 et al. [2014] estimated a ratio of 0.97 ± 0.12 from methyl chloroform observations. unbiased state compared to TROPOMI and a -1.7 ppbv global bias relative to GOSAT (Section 2). It underestimates 2019 GOSAT observations everywhere by an average of 14.6 ppbv ( Figure 6), implying the need to increase methane sources and/or decrease OH concentrations. It also underestimates TROPOMI over most of the world but overestimates in some regions (notably the subarctic) that may reflect TROPOMI retrieval biases as discussed in Section 2.

410
Both TROPOMI and GOSAT inversions reduce the negative differences between simulations and observations. The improvement can be measured by the value of the cost function J(x) in Equation 3, which decreases by 35% for the TROPOMI inversion and 54% for the GOSAT inversion. GOSAT observations are still underestimated by an average of 5.3 ppbv in the GOSAT inversion because the information from the observations is not sufficient to fully correct the bias in the prior estimate.
Cross-evaluation of the posterior simulation with the independent data set (TROPOMI or GOSAT) also shows improvement. 415 The fit to the GOSAT data is improved everywhere even with the TROPOMI inversion. TROPOMI shows problematic regions where the inversion overcorrects the prior bias. This will be discussed further in Section 4.3.

Global distribution
Our posterior/prior ratios for the 2019 GOSAT inversion in Figure 5 show large upward adjustments of non-wetland emissions 420 in the south-central US, Venezuela, and the Middle East, consistent in magnitude with the previous inversion of 2010-2018 GOSAT data by Y. Zhang et al. [2021], who used the same prior estimate. These two inversions also have consistent magnitude of downward adjustments in the western US, Europe, Russia, and North China Plain. We find larger upward adjustments than Y. Zhang et al. [2021] in India, East Africa, and Brazil, which they identified as regions with rapidly increasing emissions over , where we assume that the partitioning between sectors in a given grid cell to be correct in the prior inventory and that the posterior/prior ratio applies equally to all sectors in the grid cell. This assumption is due to the lack of additional information 435 (e.g., isotopic fractionation [Ghosh et al., 2015;G. Zhang et al., 2016;Zazzeri et al., 2017]) to separate different sources. Our restricted adjustment of wetland emissions due to increased weight in the cost function means that errors in wetland emissions could be projected to non-wetland sectors. For example, for the TROPOMI-only inversion, global posterior non-wetland emissions are 361 Tg a -1 in the base inversion and 389 Tg a -1 in the sensitivity inversion without increased weight for wetland emissions (Table S1). For the GOSAT inversion the effect is much less, 399 versus 404 Tg a -1 (Table S2). 440 Table 1 compiles our sectoral attributions of inversion results. Of particular interest is the oil/gas sector, for which the global prior estimate (66 Tg a -1 ) is based on 2016 UNFCCC national inventory reports. We find global decreases in the joint TROPOMI+GOSAT inversion to 56 Tg a -1 , largely driven by decreases in Russia. This is consistent with the correction in Russian oil/gas emissions reported to the UNFCCC, from 27 Tg a -1 in the communication used by the GFEI to 16 Tg a -1 in the 445 latest communication [UNFCCC, 2020]. Livestock emissions (the single largest anthropogenic methane source) are adjusted upward by the joint inversion from 116 Tg a -1 in the EDGAR v4.3.2 prior estimate to 139 Tg a -1 . In China, both GOSAT and TROPOMI inversions adjust non-wetland methane emissions downward in the North China Plain ( Figure 5). This has been a long-standing result of inversions of satellite data using EDGAR v4.1 and v4.2 as prior estimate [Monteil et al., 2013;Thompson et al., 2015;Alexe et al., 2015;Turner et al., 2015] and has been attributed to an overestimate 455 of emissions from the coal sector which dominates total EDGAR emissions in the region. More recent inversions using the UNFCCC-based GFEI as prior estimate have found the same result , but GFEI takes its spatial allocation of coal emissions from EDGAR v4.3.2. A more detailed bottom-up analysis by Sheng et al. [2019] finds most of the Chinese coal emissions to be in South China, in contrast to EDGAR which places them in the North China Plain.

Major source regions
Our TROPOMI inversion over Southeast China shows spatially inconsistent results with the GOSAT inversion ( Figure 5) and 460 overcorrects the fit to observations (Figure 6), which may be due to aliasing between coal and rice emissions. Rice cultivation is the dominant source of methane in Southeast China in our prior estimate, but the emissions have large seasonality and peak in summer when cloudiness is pervasive and TROPOMI observations are few, as shown in Figure 8. GOSAT is less affected by cloudiness (Figure 8), on account of its use of the CO2 proxy retrieval method. We therefore exclude posterior estimates Deleted: This follows the same methodology previously used by from TROPOMI and the joint inversions from Figure 7. Because China accounts for a large fraction of global rice [Chen et al., 2013] and coal emissions [Cheng et al., 2011;Miller et al., 2019], we also exclude these entries from Table 1. At national scale, the GOSAT inversion adjusts anthropogenic methane emissions downward from 67 Tg a -1 to 56 Tg a -1 in China, very close to the value of 55 Tg a -1 in the latest report by China to the UNFCCC in 2014. 475 All three inversions adjust methane emissions upwards in India. The results from the base inversion are at the higher end of the range from the inversion ensemble, but the 41-57 Tg a -1 range of national emissions spanned by the ensemble is still much higher than previous inversions of GOSAT and in-situ data including 33 Tg a -1 for 2010-2018 by Y. Zhang et al. [2021] and 22 Tg a -1 for 2010-2015 by Ganesan et al. [2017]. This may reflect the rapid increase of Indian emissions over the 2010-2018 480 period previously identified by Y. Zhang et al. [2021] and attributed principally to livestock.
In Brazil, the large upward adjustments from 19 Tg a -1 (prior) to 35 Tg a -1 (GOSAT) and 32 Tg a -1 (joint) in the posterior estimates are consistent with previous top-down estimates [Maasakkers et al., 2019;. TROPOMI shows adjustments in the opposite direction, likely reflecting observational bias associated with low SWIR surface albedo (Figure 3) 485 and limited number of observations. The joint inversion is dominated by results from GOSAT on account of the much higher averaging kernel sensitivities for the inversion ( Figure 5).
All inversions adjust emissions downwards in Europe (prior: 37 Tg a -1 , TROPOMI: 26 Tg a -1 , GOSAT: 33 Tg a -1 , joint: 29 Tg a -1 ), consistent with the previous downward adjustments in the 2010-2018 mean in the GOSAT inversion and the small negative 490 trend in methane emissions [Y. Zhang et al., 2021]. The largest reductions of emissions in Europe are from coal and oil/gas emissions.
In CONUS, the large upward adjustment in the south-central region reflects the well-known underestimate of oil/gas emissions by the US EPA inventory in that region Smith et al., 2017;Peischl et al., 2018;Alvarez et al., 2018;495 Maasakkers et al., 2021;Gorchov Negron et al., 2020;Lyon et al., 2021]. The posterior estimates from both TROPOMI and GOSAT adjust national methane emissions slightly downwards from 30 Tg a -1 to 26 Tg a -1 (TROPOMI) and 29 Tg a -1 (GOSAT) over CONUS, close to the posterior estimates of 31 Tg a -1 from the 0.5 o´0 .625 o inversion over 2010-2015 . The joint inversion adjusts emissions upwards to 40 Tg a -1 due to the larger averaging kernel sensitivity over the south-central US, where emissions have large upward adjustments. 500

Conclusions
We used one year (2019) of atmospheric methane column observations from the new TROPOMI satellite instrument in a global inverse analysis of methane sources at 2 o´2 .5 o resolution, and compared results to the same analysis using the more mature but sparser GOSAT instrument as well as the combination of the two instruments. By analytical solution to the inverse Deleted: ed problem, we were able to quantitatively compare the information content from the two satellite datasets. This includes 520 averaging kernel sensitivities and degrees of freedom for signal (DOFS) that quantify the number of independent pieces of information on the distribution of methane emissions.
We began by validating the global observations from TROPOMI and GOSAT by common reference to the ground-based TCCON methane column measurements, using the GEOS-Chem CTM to correct for the effects of different prior estimates 525 and averaging kernels in the retrievals from each instrument. Results show that TROPOMI and GOSAT are globally biased by -2.7 ppbv and -1.0 ppbv respectively. Their regional biases relative to TCCON are 7 ppbv and 3 ppbv, respectively, sufficiently small for inverse analyses of methane emissions on regional to global scales. Intercomparison between TROPOMI and GOSAT shows larger regional differences exceeding 20 ppbv, generally in places where the SWIR surface albedo is low and TROPOMI retrievals would be subject to biases [Lorente et al., 2021]. GOSAT is less sensitive to albedo-driven biases 530 because of its CO2 proxy retrieval method, as compared to the full-physics retrieval in TROPOMI.
We find that the GOSAT inversion has a global DOFS of 232 for non-wetland methane emissions on the 2´2.5 o grid, larger than the TROPOMI inversion (DOFS of 151) despite the TROPOMI data being much denser. This is because individual TROPOMI observations have large error correlations on the 2°´2.5° grid of the inversion, whereas the GOSAT observations 535 with their 250 km separation are ideally suited for our 2°´2.5° inversion scale. Finer-scale inversions, as done for regional studies, would be far more effective at exploiting the information from TROPOMI. A better representation of error correlation, accounting for the relative sparsity of TROPOMI data in cloudy regions, would also increase the value of TROPOMI data in global inversions. Combining the TROPOMI and GOSAT data in a joint inversion increases the DOFS to 244, with most of the added information from TROPOMI in the 0°-30° N latitudinal band including India and the Middle East. 540 The TROPOMI and GOSAT inversions for 2019 show consistent upward adjustments of anthropogenic methane emissions over Venezuela (oil/gas) and the south-central US (oil/gas), and downward adjustments over Europe (oil/gas and coal), Russia (oil/gas), and the North China Plain (coal). These adjustments are relative to the official national inventory reports to the UNFCCC in 2016 and used as prior estimates in our inversion. The TROPOMI and GOSAT inversions also show consistent 545 upward adjustments over East Africa where livestock emissions are large. Global livestock emissions increase from 116 Tg a -1 in the EDGAR v4.3.2 prior estimate to 139 Tg a -1 in the joint GOSAT+TROPOMI inversion. Some regions show large inconsistencies between TROPOMI and GOSAT inversions, and we find that these generally reflect TROPOMI regional biases in low-albedo regions. The strict cloudiness filter used in TROPOMI observations is also problematic in methane source regions such as wetlands and rice agriculture that have extensive and sometimes seasonal cloud cover. 550 Our results demonstrate the potential of applying TROPOMI observations to constrain methane emissions on a global scale through inverse analyses, but also stress the need for caution. The methane retrieval from TROPOMI is still in an early stage, Deleted: as top-down constraints on methane emissions Deleted: that and the current operational product appears to have systematic biases in low-albedo regions. Future generations of the retrieval may address these data quality flaws [Lorente et al., 2021]. Improved accounting of model transport error correlations is also needed to fully exploit the inversion of TROPOMI observations on a global scale. In the meantime, GOSAT provides a highquality record of methane observations going back to 2010 and we have shown that 1 year of GOSAT observations can usefully inform emissions on a 2 o´2 .5 o grid. GOSAT will be increasingly useful in the future to attribute methane trends and to validate 560 future generations of the TROPOMI retrieval.

895
MB is the global mean bias of TROPOMI relative to GOSAT and RB is the regional bias as defined by the standard deviation of ∆ on the 4 o ×5 o grid.