Verifying Methane Inventories and Trends With Atmospheric Methane Data

The 2015 Paris Climate Agreement and Global Methane Pledge formalized agreement for countries to report and reduce methane emissions to mitigate near‐term climate change. Emission inventories generated through surface activity measurements are reported annually or bi‐annually, and evaluated periodically through a “Global Stocktake.” Emissions inverted from atmospheric data support evaluation of reported inventories, but their systematic use is stifled by spatially variable biases from prior errors combined with limited sensitivity of observations to emissions (also called smoothing error), as‐well‐as poorly characterized information content. Here, we demonstrate a Bayesian, optimal estimation (OE) algorithm for evaluating a state‐of‐the‐art inventory (EDGAR v6.0) using satellite‐based emissions from 2009 to 2018. The OE algorithm quantifies the information content (uncertainty reduction, sectoral attribution, spatial resolution) of the satellite‐based emissions and disentangles the effect of smoothing error when comparing to an inventory. We find robust differences between satellite and EDGAR for total livestock, rice, and coal emissions: 14 ± 9, 12 ± 8, −11 ± 6 Tg CH4/yr respectively. EDGAR and satellite agree that livestock emissions are increasing (0.25–1.3 Tg CH4/yr/yr), primarily in the Indo‐Pakistan region, sub‐tropical Africa, and the Southern Brazilian; East Asia rice emissions are also increasing, highlighting the importance of agriculture on the atmospheric methane growth rate. In contrast, low information content for the waste and fossil emission trends confounds comparison between EDGAR and satellite; increased sampling and spatial resolution of satellite observations are therefore needed to evaluate reported changes to emissions in these sectors.

To address the USA National Academy Recommendations, we describe in this paper a Bayesian/optimal estimation (OE) framework for verifying reported methane inventories and their trends by sector (e.g., livestock, rice, coal, waste, oil, gas). As a case study we compare methane emissions and trends for 2009 to 2018 from the EDGAR v6.0 inventory (Crippa et al., 2020(Crippa et al., , 2021 to emissions inverted from satellite total column atmospheric concentration data observed by the JAXA Greenhouse Gas Observing Satellite, or GOSAT Maasakkers et al., 2019;Parker et al., 2011;Zhang et al., 2021). We use EDGAR because a gridded inventory is required to compare against emissions inverted from satellite data. We note that a country's reported inventory may simply be an inventory of facility scale emissions or alternatively an integrated total for each sector; however these can be subsequently projected to a grid using prior information about emission locations (e.g., Scarpelli et al., 2020Scarpelli et al., , 2022. A critical component needed for verifying reported inventories with atmospheric data is the characterization and mitigation of spatially varying biases resulting from errors in the prior, combined with the limited sensitivity of the atmospheric measurements to emissions, also called "smoothing error" (Rodgers, 2000). Smoothing error is typically the largest of the uncertainties in inverse problems where the role of prior assumptions is important rela tive to the information content of the data used for the inverse estimate (e.g., Figure 1, Rodgers, 2000). However, smoothing error is computationally challenging to quantify and consequently many global inverse estimates of greenhouse gas fluxes (e.g., Balsamo et al., 2021;Liu et al., 2020;Peng et al., 2022;Yin et al., 2015) rely on using different data sets and models to empirically assess the combined role of data and model uncertainties, and smoothing error. However, without this characterization of smoothing error, it is challenging to compare bottom-up (e.g., inventory) and top-down (inverted from concentration data) estimates, or intercompare top-down estimates, because it is unclear which of the error terms is causing observed differences.
Using the OE approach we estimate emissions by sector (Section 4; Cusworth et al., 2021) and its information content (uncertainty reduction, smoothing error, spatial-temporal resolution of the sectoral emissions attribution, uncertainty attribution) from the GOSAT data. We show that the variable bias from smoothing error is removed in comparison between EDGAR and these satellite based emissions using the results of the OE approach. This characterization of the information content, combined with the removal of smoothing error, is demonstrably a necessary step for providing confidence when testing the spatial distribution and posited trends of sectoral emissions in reported inventories with those informed from inverse estimates of atmospheric data.
In the next section, we next summarize the OE framework for estimating emissions from satellite concentrations, how to account for smoothing error between a comparison of satellite informed estimate and inventory, and characterization of the corresponding uncertainties. Details of the OE framework are described in the Section 4. The results section show an information content analysis of the satellite-based estimates (by sector) and its comparison to EDGAR as well as the spatial distribution of these comparisons. In particular we focus on evaluating where the satellite data has the information to robustly test the trends posited by EDGAR and its spatial distribution and where it does not. These comparisons provide new results about agricultural emissions and trends and their spatial distribution, updated interpretation to many recent results involving GOSAT data, and identifies where new observations are needed to reduce uncertainties in the global methane budget.

Summary of Optimal Estimation Framework
As described in the seminal work of Rodgers (2000), an estimate can be described as a function of the "true state" corresponding to the estimate (z), its prior z A , the effect of noise "n" of the measurement used to quantify the estimate δ n , and any systematic errors δ m such as from the data or atmospheric model used to invert concentrations to emissions:̂= 10.1029/2023AV000871 3 of 16 Note that we don't perfectly know the "true state," noise vector, or the systematic errors but it is useful to write the estimate in this form as it demonstrates how these terms affects the estimate, and makes the description simpler when comparing an independent measurement (or inventory) to the estimate as shown later. The A is the "averaging kernel" matrix, a function of the prior (Z A ) and posterior (̂ ) error covariances and a metric for the increase in information via the reduction in uncertainty: and also describes the sensitivity of the estimate to the true state: =̂ (Rodgers, 2000).
The first terms on the right hand side of Equation 1, z A + A(z − z A ), contains the smoothing error. To illustrate how smoothing error varies, we can assume that the "z" term in Equation 1 only describes a single parameter instead of a vector such that A is also scalar. In the limit of zero sensitivity (A = 0), the estimate ̂ will equal the prior plus the error terms, whereas if A = 1 (perfect sensitivity), the estimate will equal the true state plus the error terms. This is why the smoothing error becomes smaller as the sensitivity (or diagonal of the averaging kernel matrix) increases as shown in Figure 1. For the problem considered here the dimension of z is much larger than one (∼8,000 parameters) such that any element of ̂ depends not just on its individual true state (z) but the convolution of all other elements of the true state vector z with the averaging kernel matrix A, plus the error terms. This convolution can result in unphysical variations in elements of ̂ because of its dependency on these cross-terms, especially if the corresponding sensitivity as described by the diagonal of the averaging kernel matrix is small. Consequently, not accounting for this variable sensitivity and choice of prior in a comparison can lead to substantial biases in a comparison and corresponding mis-interpretation of the results unless smoothing error is characterized. In addition, because we use an OE framework, we can remove part of the smoothing error, that is, the effect of the choice of prior z A , on a comparison as discussed next.

Comparing an Independent Inventory to Satellite Based Emissions Estimate
To account for smoothing error when comparing an inventory to the satellite estimate we first pass the inventory (denoted as z i with uncertainty δ i , and which must be on the same gridding as the prior z A ) through the first part of Equation 1:̂= For future reference we describe this operation as "applying the observation operator." Although the use case in this paper is to evaluate a methane inventory with a satellite based emissions estimate, Equation 1 through Equation 3 apply to any inverse problem. For example, this same approach is used to compare satellite based composition measurements to independent data sets such as ozone-sondes (e.g., H. Worden et al., 2007) or up-looking Fourier Transform Spectrometers (e.g., Wunch et al., 2017) or for assimilating satellite data into a global model (e.g., Zhang et al., 2021 and refs therein).
After application of Equation 3 to the inventory, a comparison of the emissions modified by the observation operator with the satellite based emissions is given by: The effect of the a priori assumptions from z A , is now removed, mitigating the effect of smoothing error (see Section 4 for smoothing error calculation) on this comparison so that the inventory can be compared to the satellite based emissions without this large, spatially varying bias affecting the comparison. The error of the difference between satellite estimate and this adjusted inventory is then the variance of the difference: Smoothing error is typically the largest component of the error budget for emissions inverted from satellite data where the uncertainties in the prior are important relative to the information content of the data. Black symbols show the fraction of smoothing error relative to the total error for methane emissions (by sector) based on 9 years of JAXA GOSAT data on a 4 × 5 (lon/lat) grid. Red symbols show the fraction of smoothing error when the sectoral based emissions are integrated to a single number on each grid. As the sensitivity increases, the smoothing error becomes smaller. The total error budget includes smoothing error, error due to noise, and estimate of systematic errors (Methods).
Where S i , S n , and S m are the error covariances for the inventory, observation uncertainty, and systematic errors, and can be explicitly calculated or approximated (Section 4) and then used to evaluate the difference between satellite based emissions estimate and an inventory. Equation 5 shows that, while the spatially varying bias from errors in the prior z A are removed with this comparison approach, the comparison still depends on the sensitivity of the estimate via the averaging kernel matrix. Smoothing error, which includes the effect of imperfect sensitivity, is therefore mitigated but not completely removed in this comparison.

Use Case
As a demonstration of how this Bayesian/OE approach can be used to evaluate a reported inventory using satellite data, we compare sectoral emissions (and trends) from an OE based emissions estimate using total column methane data from the GOSAT satellite (Parker et al., 2011) to those from EDGAR version 6.0 (Crippa et al., 2021;Section 4). The satellite emissions are in part based on methane fluxes derived in a previous study  and then the fluxes are projected to emissions and trends by sector using an OE based sectoral partitioning algorithm (Section 4; Cusworth et al., 2021). The information content of this estimate (averaging kernel, prior and posterior covariances, and estimate of observation and systematic errors) are all provided with this sectoral based emissions estimate (Section 4) and used with the comparison to EDGAR.
We use EDGAR 6.0 as it is a gridded inventory that is different from the priors used in the satellite based estimate (Section 4) and is therefore relatively straightforward to demonstrate how the information content of the satellite based emissions estimates can test the emissions and trends posited by EDGAR 6.0. However, the approaches described here can also be used to evaluate any inventory and its trends that might be reported, for example, as part of the Global Stocktake, and then gridded so that it can be compared to the satellite estimates (e.g., Scarpelli et al., 2020Scarpelli et al., , 2022. We next describe comparisons between the satellite based estimates and EDGAR when integrated over the whole globe, the information content of this comparison, and then how this information is spatially distributed by sector.

Comparison of Integrated Sectoral Emissions and Trends Between EDGAR and Satellite
The total global emissions from EDGAR 6.0, the effect of the observation operator (Equation 3) on EDGAR, and the sectoral emissions estimates based on GOSAT data, as well as the prior used for the GOSAT based sectoral emissions estimates are shown in Figure 2a. Trends for each sector are shown in Figure 2b. The uncertainties for each of these integrated quantities account for the cross-terms between elements of the state vector (Section 4). As discussed further in Section 4, the state vector includes emissions and their trends by sector, as well as wetland fluxes for each month, and the methane chemical sink (OH). After applying the gridded observation operator to EDGAR as described in Section 4, the emissions and trends by sector are integrated for the whole globe and shown in Figures 2a and 2b. The conclusions about the integrated emissions are essentially the same as those inferred from the integrated fluxes that they are based upon and described in a previous manuscript . However, there are some differences about the trends between these previously published results and those shown in Figure 2; notably the rice, waste, and the oil and gas (O&G) trends are smaller, likely because the priors for the sectoral attribution of the emissions are different. In addition, a "relative weighting" approach is used in this previous study to scale integrated methane fluxes in each grid to emissions, and this approach does not account for the prior distribution and posterior errors of the emissions that are used in our OE based sectoral attribution approach (Section 4).

Information Content Analysis: How Applying the Observation Operator to EDGAR Informs Interpretation of the Satellite/EDGAR Comparison
Applying the observation operator to an independent inventory accounts for the smoothing error, and when compared to the satellite-based estimate, removes the spatially variable bias due to errors in the choice of prior (Equation 4). As shown by Equation 5, the EDGAR inventory (modified with observation operator) and satellite based estimates therefore agree if the orange line overlaps (1-sigma) with the GOSAT (red) estimate. However, agreement between estimates does not necessarily mean the comparison is informative because the satellite-based estimate and modified EDGAR can agree if the sensitivity is zero such that both simply represent the choice of prior. For this reason we also discuss the information content as quantified by the DOFS (degrees of freedom for signal) parameter as discussed in a subsequent paragraph.
Smoothing error includes not just the error in the prior for the quantity of interest (e.g., livestock emissions) but also its dependency of other elements of the state vector that affect this estimate. Figure 2c shows the averaging kernel matrix for the integrated quantities shown in Figures 2a and 2b. The row of the averaging kernel matrix describes the sensitivity of the diagonal to the true distribution of the other elements of the state vector (recall that =̂ ). The column shows how perturbing that diagonal element of the state vector affects all other state vector elements (Bowman et al., 2006;Rodgers, 2000). Figure 2c also includes the state vector elements representing wetlands and the chemical sink (WL and OH) that are jointly estimated  with the anthropogenic emissions. For example, the total livestock emissions depend substantively on the total waste and rice emissions and the chemical sink, or as shown in Equation 1 the difference between the "true value" of these quantities and their priors (or error in the priors). While the total livestock emissions estimate also depends on the trends in livestock, waste, O&G, and fires and geological emissions (LT, WT, OGT, and FGT), in practice these cross-terms should not contribute much to the livestock emissions estimate because their values are much smaller as seen by comparing Figure 2b with Figure 2a. Applying the observation operator to the inventory removes the effect of these cross-terms on the comparison with the satellite estimate as shown in Equation 4.

Information Content Analysis: Description of DOFS
Above the x-axis in Figures 2a and 2b is a parameter called "DOFS" or degrees of freedom for signal which describes the number of independent pieces of information for an estimate (Rodgers, 2000). DOFS is calculated from the trace of the sub-matrix of the averaging kernel corresponding to each sector. A value of 0 means no sensitivity to the underlying emissions whereas increasing DOFS means increasing sensitivity. Although each sector generally has a DOFS larger than one, these DOFS are distributed over many geographical regions (see Figures S1-S10 in Supporting Information S1). Nonetheless, this is a useful metric for assessing how much information is available for an observing system for each sector, especially if the DOFS are very small, indicating little sensitivity of the estimate to the true distribution. For example, as discussed previously, the modified inventory and satellite-based estimate can agree if the sensitivity (i.e., DOFS) is zero such that both simply reflect the prior.
The DOFS in Figures 2a and 2b are not for the global total emissions (and trends) but instead reflects the information in their spatial distribution as it is based on the gridded averaging kernel matrix. For example, we expect the observing system described here (based on GOSAT data) to provide the most information about the spatial distribution of the livestock sector and the least for the coal sector based on the DOFS metric. For comparison, the diagonal value for the averaging kernel shown in Figure 2c is the DOFS for the integrated totals shown in Figures 2a and 2b. Comparison of the DOFS in Figure 2a with the diagonal value in Figure 2c shows how cross-correlations between elements of the state vector affect the information in the estimate. For example, once the livestock emissions for the whole globe are integrated to a single value representative of the whole globe, it actually has a lower DOFS (diagonal of averaging kernel in Figure 2c) than that for the coal or the O&G sectors because there is stronger dependency of the integrated livestock emissions estimate on other elements of the state vector. We use the DOFS metric, along with the calculated uncertainties to support interpretation of the estimates (and comparison to EDGAR) for the spatial distribution of emissions and trends in the next sections.

Livestock Emissions and Changes
As shown in Figure 2a, the largest amount of information (DOFS) from this observing system for an emissions sector is for the spatial distribution of the livestock emissions. Based on the comparison between the modified EDGAR and satellite estimate (orange and red lines respectively), we conclude that EDGAR livestock emissions are 15 ± 9 Tg CH 4 /yr too small. Most of this discrepancy is due to underestimates of livestock emissions in East Africa and Brazil, along the arc of deforestation as shown by comparing the spatial distribution of the satellite-based estimates to EDGAR ( Figure S1 in Supporting Information S1). Figure 2b shows that the trend in EDGAR 6.0 integrated livestock emissions, both before and after applying the observation operator, is consistent (within calculated uncertainties) with the satellite estimates. Figure 3 also demonstrates that the spatial distribution of this trend is broadly consistent between the EDGAR and satellite estimates, with both showing increases in emissions in Brazil, West and East sub-tropical Africa, and the Indo-Pakistan region. However, within these regions there can be substantial differences between the EDGAR and satellite-based trend estimates. Figure 3c shows one approach for using the information content from the OE based characterization for interpreting this comparison. This figure shows the ratio of the (absolute magnitude) difference from Figures 3a and 3b to the calculated uncertainty (Equation 5). Furthermore, only differences where DOFS that are larger than 0.05 ( Figure S2 in Supporting Information S1) are shown to ensure the corresponding observation has some sensitivity to the underlying trends (recall that =̂ ; Rodgers, 2000). A value of one in Figure 2c means the difference between EDGAR and satellite is within 1-σ uncertainty.
Using this information content analysis, these comparisons provide increased confidence that observed increases in livestock emissions are larger than expected (greater than 1-σ uncertainty) along the Brazilian arc of deforestation where there is substantial conversion of forest to pasture (e.g., Morton et al., 2006;Xu et al., 2021). Furthermore, we have confidence that livestock emissions are increasing in sub-tropical West and East Africa as both EDGAR and satellite agree, and there is sufficient information that the agreement is not a reflection of the prior or other elements of the state vector. On the other hand, both positive and negative trends of livestock emissions are observed in the Indo-Pakistan region, which is somewhat different than EDGAR which shows only positive trends. These livestock changes, combined with slightly decreasing wetland emissions in India , are consistent with a previous study showing nearly constant total emissions from India (Ganesan et al., 2017) over this same time period, although a subsequent paper has shown decreases in emissions in this region using the same data (Wang et al., 2021) that is inconsistent with these results.
Because the EDGAR estimate for the livestock emission trend, after the observation operator is applied, agrees well with the satellite estimate, the original EDGAR value (∼1 Tg CH 4 /yr/yr) is entirely plausible. Therefore, our final estimate for the change in livestock emissions is the low end of the satellite value to the high-end of the original EDGAR value (0.25-1.3 Tg CH 4 /yr/yr). Systematic errors in the data (e.g., errors related to albedo) or model (e.g., model transport and chemistry) could also potentially affect the trend in the satellite result if these vary from year to year (e.g., Mcnorton et al., 2020); for example, albedo can vary with different snow/surface water conditions or model transport can vary with, for example, El Nino. However, as these effects are included in the satellite and emission estimate it is more likely that these types of errors impart some year-to-year variability as opposed to longer-term temporal increases or decreases such that we do not expect these systematic errors to affect our conclusions about trends.

Oil and Gas Sector Emissions and Changes
As a contrast to the livestock sector, where there is significant information content in the comparison, we next discuss the information content of this GOSAT based observing system for evaluating the distribution of emissions and trends of the O&G sector. Note that we combine oil and gas together as they can be challenging to distinguish because of spatial overlap and frequent commonality in upstream practices (e.g., Alvarez et al., 2018;Scarpelli et al., 2022). Figure 2a shows the substantial role of the prior uncertainties and limited sensitivity of the satellite observations on the O&G emissions estimate. For example, the EDGAR 6.0 inventory sets total O&G emissions at around ∼70 Tg CH 4 /yr. After applying the observation operator to EDGAR, this value shifts downwards to ∼48 Tg CH 4 /yr and is consistent (within uncertainty) with the satellite observation (orange line) and its prior (black line). Because EDGAR modified by the observation operator agrees with the satellite estimate, we conclude that the satellite observations cannot falsify the higher global total posited by EDGAR (blue). On the other hand, as seen in Figures S3 and S4 in Supporting Information S1, this observing system can resolve total gas emissions over high emitting areas such as the Permian Basin in Texas (Cusworth et al., 2019Zhang et al., 2020) and the Turkmenistan facilities , suggesting that most of this discrepancy is due to limited sensitivity of the observing system in lower emitting regions that in aggregate can make up a large portion of the O&G emissions. Figure 4 demonstrates that this observing system has less information about trends in O&G emissions. EDGAR expects large changes in these emissions across the N. Hemisphere (Figure 4a). However, these changes in EDGAR are near zero for most of the globe after applying the observation operator because the information content is small (DOFS = 2.5 for the spatial distribution for the whole globe), such that the observation operator places the EDGAR trends near the (zero) prior. As this result is consistent with the satellite data, except in regions surrounding Turkmenistan, all that can be said from this comparison is that this observation system cannot robustly verify posited trends in O&G emissions. These results demonstrate the importance of combining sensitivity analysis (e.g., DOFS) with the uncertainty calculation to determine if an atmospheric concentration informed estimate can robustly test or verify reported emissions and their trends.

Rice, Waste, and Coal Comparisons
We next summarize what we believe are the more interesting findings with respect to the comparisons between EDGAR and satellite for the rice, waste, and coal sectors, and their trends. The supplemental contains the spatial distribution (at 5 × 4 lon/lat gridding) for the EDGAR emissions and trends for each sector, EDGAR after the observing operator is applied, the satellite based estimate, the DOFS, and the equivalent of Figure 3c for each of the emissions and corresponding trend. We refer the reader to these Figures S1-S10 in Supporting Information S1 for the subsequent discussion.

Rice (Figures S5 and S6 in Supporting Information S1)
The spatial distribution for rice emissions is consistent (within the calculated uncertainty) between satellite and EDGAR (with observation operator applied) for almost all grid cells and within 1.5 σ uncertainty (12 ± 8 Tg CH 4 / yr) for the global total. The diagonal of the averaging kernel matrix (Figure 2c) indicates substantial sensitivity (∼0.9 DOFS) for the integrated total emissions with only partial sensitivity to waste and livestock, confirming a robust estimate for total rice emissions. However, there are substantial differences between EDGAR and satellite for the rice trends. For example, EDGAR posits small increases in emissions from rice farming in Africa, Indonesia, India and East Asia, but a decline in South-East Asia ( Figure S6 in Supporting Information S1). However, after applying the observation operator to EDGAR, the small decline in rice emissions in SE Asia becomes a slight positive increase; this adjusted EDGAR is still ∼2.5 smaller (relative to the uncertainty) of the satellite data, indicating that an increase in rice emissions in this region is likely. On the other hand, while the observed emissions from rice in China are larger than EDGAR (after adjusting with the observation operator), the difference is within the uncertainty; hence we conclude that rice emissions in China are likely increasing. Combined with the results on livestock emissions, these comparisons highlight both the importance of agriculture in affecting the growth rate of atmospheric methane and also the need for accounting for smoothing error when interpreting differences between top-down and bottom-up estimates.

Coal (Figures S6 and S7 in Supporting Information S1)
As with rice emissions, the spatial distribution and the integrated emissions for coal are mostly in agreement between EDGAR (with observation operator applied) and observed. Significant sensitivity to total coal emissions ( Figure 2c) and minimal dependency on cross-terms in the state vector supports the robustness of this result. However, as shown in Figure S8 in Supporting Information S1, a negative trend in the EDGAR N. China coal emissions is replaced with a slight positive trend once the observation operator is applied, likely because of the role of the waste trend on the trends for coal as shown by the averaging kernel matrix in Figure 2c. These large differences between the original EDGAR values and the EDGAR with observation operator, highlight how correlations in the observing system can alter the interpretation, or add additional context, to a conclusion. For example, previous results also based on GOSAT data indicate a large increase in coal emissions in N. China, despite stated decreasing inventories Sheng et al., 2021), consistent with the results shown here. However, the characterization provided by the OE approach suggest that at least part of this observed increase result from spatial correlations in the inversion as demonstrated by Figure 2c . Consequently, a more robust conclusion is that the posited EDGAR decline in coal is too steep.

Waste (Figures S9 and S10 in Supporting Information S1)
These comparisons also demonstrate that waste emissions and their trends (Figures S9 and S10 in Supporting Information S1) are not well tested with this observing system as the information content is low (DOFS ∼0.4 for total emissions as seen in Figure 2c). The row of the averaging kernel matrix also shows strong dependency of waste emissions on emissions from livestock, rice, and coal. These results indicate that the waste estimate strongly depends on the choice of prior and explains why these results, in which the prior depends on both wastewater and landfills, are different from those in Worden et al. (2022) where the prior only depends on landfills; essentially the integrated estimate for waste is primarily dependent on the prior because the diagonal of the averaging kernel matrix is less than 0.5. Similarly, trends in waste emissions are not well resolved because of low information content. Higher resolution estimates are therefore needed to resolve this sector's emissions and trends from other sectors (e.g., Maasakkers et al., 2022).

Summary and Future Directions
The Global Methane Pledge and recent Conference of Parties stresses the increasing need for evaluating a countries reported methane emissions and changes with atmospheric observations to support methane emission reduction efforts, for the purpose of mitigating near term climate change. In turn, the recent USA National Academy report emphasizes the need for a framework and corresponding use-cases that includes atmospheric measurements for providing robust information about GHG emissions and their changes (National Academy, 2022). There are also nascent efforts to develop these information systems, that are based on satellite atmospheric concentration data such, as the European Copernicus project (COCO2, 2022), as well as substantive discussion on this subject across the science communities in previous and current conferences (e.g., WMO, 2023). As demonstrated by the use case in this manuscript, OE provides a framework and characterization approach for robustly comparing satellite based emission estimates to an independent inventory. The benefits of OE include (a) evaluation of the information content of the satellite top-down estimates which in turn allows us to determine where the comparison has information and where it does not, (b) improved confidence in comparisons between the satellite and inventory emissions as one of the largest uncertainties in the comparisons, smoothing error, is accounted for in the comparison and mitigated, (c) additional context in interpreting the spatial and temporal distribution of emissions via evaluation of the averaging kernel and posterior covariances, (d) a theoretical framework for attributing different sources of uncertainty, and (e) identifying where additional measurements are needed to resolve emissions and their changes.
For our use case, we compare emissions and trends from 2009 to 2019 from an observation system based on the satellite GOSAT data, the GEOS-Chem model, and choice of constraints , with emissions and trends from the EDGAR v6.0 methane inventory. We find that this observing system has the most information about the spatial distribution of livestock emissions and trends such that we are confident that EDGAR livestock emissions are underestimated but that posited increases in livestock emissions by EDGAR are consistent with satellite observed increases (0.25-1.3 Tg CH 4 /yr/yr), primarily in the Indo-Pakistan region, Africa, as well as the Brazilian arc of deforestation where there has been substantial conversion of forest for agricultural use in the last two decades (Xu et al., 2021). A smaller contribution from rice emissions in China and SE Asian also likely contributes to increasing methane; combined these results highlight the importance of food production on the global methane budget (e.g., Crippa et al., 2021) and its growth rate.
Because we calculate the information content of these estimates we can determine where the comparisons are meaningful (total uncertainties are reduced) and where they are not. Regions where robust differences relative to the uncertainty (e.g., Figure 3 right panel) occur could be targets for evaluating bottom-up models. In contrast, large regional changes in waste as well as O&G emissions posited by the EDGAR inventory cannot be tested with this satellite data set because of low information content of the observing system for these emissions. These regions with low information content are also potential targets for a more focused sub-orbital campaign, assuming upcoming missions (as subsequently discussed) cannot easily resolve their emissions.
As shown with this study and that in Worden et al. (2022), the GOSAT data have limited utility as an observing system that can be used for emissions verification; increased spatial-temporal resolution is therefore needed to evaluate a countries reported fossil emissions (e.g., Scarpelli et al., 2022), their Nationally Determined Contribution to the UNFCCC, and temporal changes. The information content (sensitivity and uncertainties) and corresponding spatial-temporal resolution of an observing system is largely driven by observation density and prior error covariance. The TROPOMI satellite instrument is now providing ∼100× higher observation density than the GOSAT instrument used in this work, although there are still some artifact issues affecting its use in global inversions (e.g., Qu et al., 2021). As these artifact issues get resolved (Lorente et al., 2021) and new instruments are launched , we can expect the information content on methane fluxes from satellite observations to increase considerably. Multiple observations have found that high emitters at small scales (10 s of meters) can make outsized contributions to regional (100's of km) methane fluxes (e.g., Cusworth et al., 2022;Frankenberg et al., 2016;Lavaux et al., 2022;Maasakkers et al., 2022;Varon et al., 2019). Combining observations that map methane enhancements at sub-kilometer scale (e.g., GHGSat, Carbon Mapper, Sentinel-2, EMIT) over intensely emitting facilities, with the global mapping available from other satellite instruments will therefore be critical for enabling climate action. The OE framework described here also provides the underlying theory for a greenhouse gas information system that can integrate these data sets with their vastly different spatial scales and use them to test and update reported inventories in a fully Bayesian manner.

Background on Satellite Based Emissions of Fluxes and Attribution of Fluxes to Emissions by Sector: (Flux Inversion)
The sectoral emissions and trends inverted from the satellite data are derived in a two-step process. The first step is described in previously published results and comprises a global inversion using total atmospheric column methane data from the Japanese GOSAT (Greenhouse gases Observing SATellite) instrument (Parker et al., 2011) and the GEOS-Chem model . The GOSAT data are from the University of Leicester version 9 CO 2 proxy retrieval . The GEOS-Chem simulation shows latitude-and season-dependent biases in stratospheric methane concentrations (Saad et al., 2016;Stanevich et al., 2020), which are corrected before inversion using independent stratospheric observations by the ACE-FTS instrument (Waymark et al., 2014). The state vector for this inversion includes ( Maasakkers et al. (2016) and global oil, gas, and coal emissions by Scarpelli et al. (2020). The prior wetland emissions are from WetCHARTS v1.0 extended ensemble mean (Bloom et al., 2017).
Individual GOSAT data are assimilated into the GEOS-Chem model, which allows for the results to capture the seasonal behavior of the emissions; for example, part of the objective with the Zhang et al. (2021) results is to evaluate the seasonal processes controlling wetland fluxes . The posterior concentrations from the inversion have been verified with independent observations from the global TCCON (Total Carbon Column Observing Networks) and NOAA (National Oceanic and Atmospheric Administration) networks and can well reproduce the interannual variability of the surface methane growth rates observed at surface sites. We refer the reader to the paper by Zhang et al. (2021) for a full description and evaluation of this inversion. Note that the priors used to generate the fluxes are not completely the same as the priors used to project the fluxes to emissions by sector (Equations 5-7 in subsequent text); these different choices are in part due to improved understanding of the problem with each iteration of the GOSAT based analysis and in part due to ease of use of some of the product files, but in no way breaks the Bayesian assumptions used with the methodology described here.

Summary of Emissions Attribution
The second step in the sectoral attribution is a linear estimate of emissions by sector (and their trends) based on an OE attribution approach (see subsequent text on sectoral attribution as well as Cusworth et al., 2021;Worden et al., 2022) that projects the integrated anthropogenic fluxes and trends to emissions by sector and trends at the same 4 × 5 gridding. The emissions state vector includes livestock, waste (landfills and wastewater), rice, coal, O&G, and fires and geological. The state vector also includes the trends for these emissions as well as the wetlands and OH state vector from the flux inversion in the first step. This OE based projection from integrated anthropogenic fluxes to emissions by sector accounts for the prior distribution and uncertainties in the emissions, and includes the effects of the jointly estimated wetland fluxes and methane sink. Furthermore, because the posterior covariance is generated from this step (Section 4, subsequent section), we can calculate an averaging kernel matrix which is then used to create the "observation operator" for the emissions and trends by sector as shown by Equation 2 in the main text.

EDGAR 6.0 Inventory and Uncertainties
The inventory we choose as a use-case for evaluation is the EDGAR v6.0 inventory (https://edgar.jrc.ec.europa.eu/dataset_ghg60). We choose an EDGAR inventory as they are designed for use in atmospheric studies and hence are relatively straightforward to compare with satellite based emissions estimates. The inventory is first projected to the 4 × 5 degrees (lat/lon) gridding used with the satellite based emissions estimates. Critical towards comparing a gridded inventory with inversely estimated emissions is a realistic a priori error covariance that describes their uncertainties and any correlations between adjacent emissions. Because EDGAR does not provide an error covariance for its emissions, we use the prior covariances that were previously generated for use by the satellite based emissions estimate by sector (Worden et al., 2022; Section 4) as a convenience to simplify the demonstration of the information content calculations. However, covariances that are better informed by bottom-up emissions modeling and measurements would greatly improve quantification of the information content in satellite/inventory comparisons as this comparison would better reflect the reduction of error and where the satellite data can provide the most useful evaluation.

Optimal Estimation Description
The OE framework, used for comparing the satellite based sectoral emissions and trend to an independent methane inventory, follows the approach in Rodgers seminal work "Inverse methods for atmospheric sounding: theory and practice" (Rodgers, 2000), and used extensively in remote sensing of methane and other atmospheric trace gases (e.g., Bowman et al., 2006;Parker et al., 2011;Worden et al., 2004). Here we provide a more in-depth derivation of the basic OE equations than discussed previously in the main text.
The estimate for each emission type and its trends can be written as a function of the true state for these parameters (z), its prior z A , the noise "n" of the measurement used to quantify the fluxes, and any systematic errors δ m : Note that this is the same as Equation 1 in the main text except that we replace δ N with Gn. The posterior error covariance for this estimate, or Hessian, ̂ is the weighted sum of the prior uncertainties or covariance Z A , and the information from the observations ( − ) : The "Jacobian" K in Equation 7 describes the sensitivity of the integrated methane flux (or trend) to the observed methane concentrations in each state vector element (e.g., the state vector denoted in Figure 2c but on the spatio-temporal grid described in the main text). The gain matrix G is a matrix containing partial derivatives that relate parameters in the observation state vector (or "n" using the nomenclature in Equation 2) to parameters in the estimate state vector (or "z" using the nomenclature in Equation 2) and is defined for this problem as =̂( ) − (Bowman et al., 2006). The mapping matrix M describes the mapping relationship between the integrated flux (x) at each grid cell to the sectoral emissions that make up the flux in that grid (x = Mz, see subsequent Section 4 section on sectoral attribution). The information about the fluxes from the observations described by the Jacobians K in Equation 2 is mapped to each sector (e.g., coal, waste, gas via the mapping matrix, and then weighted further by the prior error covariance of that sector (Z A ) as discussed in the next section. We note that the Jacobians in Equation 2 can be further mapped to the Jacobians in the optimal estimate step used to relate observed satellite spectral radiances to methane concentrations (Parker et al., 2011), so that the Hessian matrix shown in Equation 2 preserves the information from the original satellite observations.

Quantifying Uncertainties
The Hessian matrix, or posterior error covariance in Equation 7, can also be written as the sum of the smoothing and observation error (Bowman et al., 2006;Rodgers, 2000;Worden et al., 2004): Z A describes the prior error covariance for the elements in the state vector (see subsequent Section 4 section).
The first term on the right is the characterization of the smoothing error and the second term is the measurement error, or the effect of observation noise on the estimate.
The uncertainties we report for an estimate are derived from the posterior covariances. For example, if we report an individual element of the covariance (e.g., total error for a sectoral emission for a certain grid point as shown in Supporting Information S1), then the uncertainty is simply the corresponding square root of the diagonal from the covariance. If on the other hand we report an integrated quantity (e.g., sum of all livestock emissions), then the corresponding uncertainty must be projected from the total error covariance (Worden et al., 2022) on the 4 × 5 grid to the integrated quantity. For example, the total uncertainties for the satellite based uncertainties shown in Figures 2a and 2b are derived by first quantifying the posterior error covariance for the integrated quantities of the state vector described by the row in Figure 2c: Where the rows of the matrix is composed of vectors the size of the row of ̂ and that have a value of 1 corresponding to the indices of the sector and 0 elsewhere. The error covariance ̂ is the sum of the Hessian plus our estimate of the model error covariance. The indices for h are the same indices shown in Figure 2c that correspond to each (integrated) elements of the state vector: L, W, R, … , WL, OH (livestock, waste, rice, … , wetlands, OH). The uncertainty for each integrated sector "i" is then the square root of the diagonal value of the covariance ̂ corresponding to the sector. To calculate the averaging kernel matrix shown in Figure 2c, we apply the same operation to the Hessian ̂ and to Z A and then recalculate the averaging kernel matrix using Equation 2 from the main text.

Sectoral Attribution
The derivation for how emissions by sector are quantified from an integrated flux is described in Cusworth et al. (2021). The approach accounts for the prior uncertainties in the emissions that make up the integrated flux as well as the sensitivity of the satellite estimate. For example, if the sensitivity of the satellite based estimate is zero (as determined from the averaging kernel matrix), then we would expect the posterior emissions estimates (by sector) to primarily reflect the prior, with some modification from the effect of cross-terms in the estimate as shown in Equation 3. As the sensitivity increases, the projected emissions depend on the relative distribution of the prior uncertainties. For example, if a grid box has two approximately equal emissions and one of the emissions has an uncertainty much larger than the other, than most of the satellite based correction to the flux is projected to the emission with the larger uncertainty. The OE framework summarized next accounts for these differences when attributing the satellite based integrated fluxes to specific emissions.
The basic equations describe a linear mapping relationship between integrated anthropogenic emissions "x" and the emissions by sector "z" within a grid that compose this flux. The linear mapping we use is a simple summation of emissions within each grid: The solution for projecting fluxes back to emissions takes the form: where the (̂ ) is the posterior emissions vector with error covariance (̂ ) and I is the identity matrix. Here S A and ̂ are the prior and posterior error covariances covariance for the fluxes corresponding to the state vector x as described in Zhang et al. (2021).
Equation 11 is similar to the "prior" swapping approach used extensively for remote sensing (e.g., Rodgers & Connor, 2003) in which one (e.g., x A ) prior can be swapped with another (e.g., Mz A ) if the Averaging kernel matrix is provided ( except that after the priors are swapped (the effect of the brackets in the above equation) it is then projected from "flux space" to "emissions space" (the effect of ̂̂−1 ).
As shown previously, the posterior error covariance matrix ̂ is calculated explicitly given M, S A , ̂ , and prior emissions error covariance matrix Z A : Note that in our previous work (Worden et al., 2022), the covariances had to be split into regions because the size of the state vector resulted in a matrix (Equation 7) that was too large to invert. We found, using simulations with smaller matrices, that this approach changed the structure of the averaging kernel matrix and imparting a small error to the estimate. However, this estimate was within the calculated uncertainty of that using the full averaging kernel matrix. To remove this difference, we now perform the matrix inversion using a state vector representative of the whole globe and not sub-regions.

Sources of Uncertainty When Comparing Satellite Based Emissions Estimate and Trend to Inventory
As discussed in the main text, the different sources of error when comparing the satellite based emissions to an independent inventory are described by the variance of the difference between the satellite estimate and inventory modified by the observation operator: Where S i is the covariance for the inventory uncertainties δ i , S n is the observation error projected to emissions, and S m is the model error. Note that the observation error, S n , either is directly calculated using a chain of partial derivatives that relates noise to concentration data to fluxes to emissions, or it can be calculated by subtracting the "smoothing" or resolution error from the Hessian Z (Rodgers, 2000): We quantify the final term in Equation 13 (model error) using the "residual error" approach discussed in Zhang et al. (2021) which compares the difference of single observations and model values relative to yearly means of the observations minus the model. This yields a total observational error variance from which the contribution from the observing instrument can be subtracted to yield the model transport error variance. Application to satellite methane column observations indicates model transport error standard deviations of typically 5-10 ppb . However, for the purpose of this study we conservatively assume the model error is the same as the observation error S n , which is based on concentration errors that are ∼14 ppb.
If only a partial inventory is provided, such as a countries reported emission inventory, then the errors still need to account for the spatially varying sensitivity of the estimate as described by the averaging kernel. In this case there is a "cross-state" term (Worden et al., 2004;H. Worden et al., 2007), in which the emissions and its prior uncertainties in one region affect the estimated emissions in another. Equation 5 then becomes: Here the indices "ii" represents the submatrix of A and the covariances S corresponding to the reported countries emissions (e.g.,), and the index "j" represents the emissions for all other parts of the inventory.

Description of Priors and Prior Covariances Used With Sectoral Attribution
Sectoral attribution for emissions based on atmospheric concentration data requires use of a priori emissions and prior covariances. For the OE approach described here, this means a realistic representation of z A (priors) and Z A (prior covariances). As discussed previously, the priors used for the sectoral emissions attribution overlap but are not completely the same as those used to generate the satellite based fluxes (see earlier text) The priors and prior covariances and how they are generated and used for the sectoral emissions are described in Worden et al. (2022) and the reader is directed to this paper (Section 2.3 Generation of Prior Emissions, Covariances, and Uncertainties) for details. In summary, the a priori emissions are based on a combination of emissions inventories. Livestock emissions are based on the results of a 2013 NASA Carbon Monitoring System study (Wolf et al., 2017). Rice, landfill, solid waste, and waste water emissions are based on EDGAR 5.0 (https://edgar.jrc. ec.europa.eu/dataset_ghg50, Crippa et al., 2019). A key difference between the prior emissions in this study versus the Worden et al. (2022) study is that the EDGAR 5.0 waste water emissions are now explicitly included with the landfill emissions, increasing the prior amount for the total. Fossil emissions are based on reported emissions to the UNFCCC as part of the global stock-take (Scarpelli et al., 2020. Wetland emissions are based on an ensemble of wetland, semi-empirical process models (Bloom et al., 2017). Fires are based on emissions from the Global Fire Emissions Database version 4 (Giglio et al., 2013) and combined with natural seeps that are based on a 2019 inventory (Etiope et al., 2018). The covariances for OH are taken from Zhang et al. (2021).
Constructing a realistic a priori covariance is challenging as the matrix must be invertible (Equation 3) while realistically describing the prior uncertainties and their correlations (e.g., Super et al., 2020). This challenge is amplified because the uncertainties and how they might be correlated are generally not well known (Janssens-Maenhout et al., 2019). Ideally, a covariance is constructed from an ensemble of different emissions estimates at the desired 10.1029/2023AV000871 14 of 16 scale. Because these are not yet available for the different sectors used in this research, we use a downscaling approach instead as described in our previous work (Worden et al., 2022) and which is similar to that described in Super et al. (2020). The approach is to first sub-divide the globe into eight regions which are primarily associated as an "Annex 1" or "non-Annex 1." Our primary constraint in calculating a prior covariance is that projecting it from its original gridding (in this previous work it is at 1 × 1 degree) down to a single value representative of the region, the total uncertainty should be 15% for a region primarily composed of Annex 1 countries while the value is 30% for regions primarily associated with "non-Annex 1" countries (Janssens-Maenhout et al., 2019). Mechanistically this means that if we project, for example, the covariance for coal emissions for an Annex 1 region down to a single number than it will have an uncertainty very close to 15%. To achieve these result, the uncertainty at any grid cell must be quite large (∼80% at 1° resolution) with substantial correlation with nearby emissions (∼0.8 correlation with emissions within 400 km). These numbers are not unrealistic based on regional studies of uncertainties in California and Texas (Maasakkers et al., 2016) such that we adopt this general approach throughout the globe. Note that it is straightforward to use alternative priors and prior covariances in the sectoral attribution (Equations 5-7) if a researcher can provide them and has good rationale for their internal covariance structure. Future results will likely use covariances that depend on multiple emission inventories and their correlations with the downscaling approach described here.
As discussed in Worden et al. (2022), the priors and prior covariances are initially calculated on a 1 × 1 degree grid box. For the study in this manuscript the gridding is 4 × 5 degrees (latitude/longitude). Consequently, we project the covariances at 1 × 1 to 4 × 5 (lat/lon) through simple interpolation. Our approach for quantifying the prior and covariance for the trends is to (a) set the prior for the trend to be zero at every grid box and (b) to set the covariance for the trend equal to 5% of the corresponding variance for the emissions at every grid, that is, the covariance for the trend is simply the covariance for the emissions multiplied by 0.05 2 . The prior covariances for each region are then mapped to the whole globe at 4 × 5 degrees. However, only elements of the globe that have non-zero emissions are used to keep the state vector small such that the matrix described by Equation 3 is robustly invertible (i.e., ̂̂−1 = ). Mechanistically, this means using only those emissions that are larger than 1% of the largest emission value on the globe. With this threshold we reduce the size of the emissions part of the state vector by ∼50% but retain ∼98% of the total emissions such that the revised total is within 8 Tg CH 4 /yr of the ∼350 Tg CH 4 /yr total. We then redistribute this small remaining residual to the retained grid boxes so that the total emissions on this smaller subset is the same as the original.