We aim to validate the national reporting of German CH4 emissions to the UNFCCC. A simple way to address this validation problem is the following question: By which single number should we multiply all reported German CH4 emissions based on the information from observed CH4 concentrations? We can extend this question and estimate different scaling factors for different regions and different emission sectors. In this work, we estimate scaling factors for 46 categories of CH4 fluxes for each month in 2021. The spatial definition of these flux categories is shown in Fig. . In Germany, we distinguish 11 flux categories, consisting of six regions for the agriculture sector, one flux category for land use, land use change and forestry (LULUCF) plus natural fluxes, and four regions for the sum of all remaining emissions. In summary, the state space of our inversion is defined by the flux categories and consists of only 46 numbers.

For the a priori fluxes outside Germany, we combine CAMS-REG for anthropogenic emissions with wetland emissions from the CAMS global inversion-optimized dataset , version v22r2. For Germany, we use emissions obtained from the inventory agencies, that is, the Umweltbundesamt (German Environmental Agency, Stefan Feigenspan, Theo Wernicke, and Christian Mielke, personal communication, 2024) and the Thünen Institute (Roland Fuß and John Akubia, personal communication, 2024). Moreover, we consider emissions from rivers and streams , as well as oceans .

To connect surface fluxes and observations, we need to simulate atmospheric transport. This simulation is done using the numerical weather prediction model ICON with the module for Aerosol and Reactive Trace gases (ART) at a horizontal resolution of 6.5km. Initial and lateral boundary conditions for the CH4 concentrations are taken from the CAMS global inversion-optimized dataset , version v22r2. To mitigate a possible bias in the lateral boundary conditions, we construct a smooth correction field that is added to all model predictions of the boundary contributions. This far-field correction is constructed based on observations for which the model predicts clean air with small influence of emissions from within our domain. We estimate transport uncertainties and their correlations using an ensemble of 12 members with slightly different meteorology, derived from the operational numerical weather prediction at DWD .

We use CH4 concentration observations from the European Obspack as provided on the Integrated Carbon Observation System (ICOS) carbon portal. The hourly observations are filtered by time of day and wind speed to use only observations that can be predicted well by the transport model. We use night time observations (23:00 to 05:00 local mean time) for high mountain stations and afternoon hours (11:00 to 17:00 local mean time) for all other sites, discarding observations at wind speeds below 2ms-1.

To estimate the scaling factors of the flux categories, we use a Bayesian inversion. Denoting the scaling factors as a vector s∈R46, the inversion is formulated as the optimization problem 1spost=argmins12[y-H′(s)]⊤R-1[y-H′(s)]+12(s-sprior)⊤B-1(s-sprior). Here, y denotes a vector of all observations and H′(s) is the model prediction for these observations, which includes the previously mentioned far-field correction. R is the error covariance matrix of the model–observation mismatch and B is the error covariance matrix of the a priori scaling factors sprior. Since s describes prefactors to the a priori emissions, we initially set skprior=1 for all k. In B we assume an a priori uncertainty of 2σ=0.8 (two standard deviations) for the scaling factors of most regions. This gives the inversion enough freedom to adjust the scaling factors. In large distance from Germany, the a priori uncertainty is reduced to 2σ=0.5 (see Fig. b), and for emission sectors in Germany and the Netherlands we use 2σ=1.0.

The construction of R based on the transport ensemble is discussed in detail in Part 1 . In Eq. (), R can be estimated using either a priori or a posteriori fluxes. This defines two slightly different methods that are introduced in Sect. 2.5 of Part 1 as “prior R” and “posterior R” inversion. Here, we only consider the average of the two results and the union of the two posterior uncertainty ranges.

To estimate the uncertainties of posterior fluxes conservatively, we repeat the inversion 50×2 times with each of the 50 observation sites excluded once for each of the two approximations for R. The lower and upper bounds of the resulting hundred 2σ uncertainty ranges form our posterior 95% confidence interval. This ensures that a result that is only based on a single observation site will not be considered significant.

The scaling factors are estimated separately for each month in 2021 by using only observations from the selected month. The results for different months are thus independent. But since the posterior uncertainty estimates include systematic uncertainties, we assume that uncertainties from different months are correlated.

Figure  presents an overview of (a) the a priori CH4 fluxes accumulated over the year 2021, (c) the resulting scaling factors averaged over 2021, and the respective uncertainties (b, d). The a posteriori scaling factors (Fig. c) show the correction to the a priori emissions obtained in the inversion. A considerable increase in emissions is found for Germany and the Benelux. Lower emissions compared to the a priori are predicted for Scandinavia (see discussion in Sect. ). The scaling factors should be considered jointly with their uncertainties. The comparison of Fig. b and d shows a substantial uncertainty reduction for Germany and most of the surrounding countries, for which we chose a high a priori uncertainty.

For a more detailed comparison of a priori and a posteriori emissions and uncertainties, we consider selected national emission estimates in Fig. . Reliable inversion results are expected for countries or regions with sufficient observation coverage, strong emission signals, representation in the respective flux categories, and only moderate issues due to complex topography. These criteria are met for Germany, the Netherlands and the United Kingdom plus Ireland as grouped in Fig. . For Germany (first entry in Fig. ), the total posterior CH4 emissions (red bar) are (32±19)% higher than the anthropogenic emissions including LULUCF reported to the UNFCCC in 2024 (light blue bar). The direct comparison to the reporting neglects the unreported natural fluxes, but for Germany these are expected to be small because all relevant soil emissions are included in the LULUCF sector. The inversion significantly increases emission estimates from the agriculture sector while the combined other sectors remain nearly unchanged. Note, however, that the uncertainty in the sector attribution is large (horizontal lines, see further discussion in Sects.  and ).

For the Netherlands, we also find significantly higher emissions than in the inventory. Compared to Germany, the attribution to sectors has an even larger uncertainty, associated with fewer observations that could distinguish the sectors. Nevertheless, the total emissions from the Netherlands are comparably well constrained by the observations. For the United Kingdom and Ireland – which we combine to obtain more accurate results – the inversion yields a strong uncertainty reduction while hardly changing the total emissions, indicating a good agreement of observations and national inventory.

In most countries, the observations do not cover the whole country, or the inversion results rely on few observations. In Fig.  (gray-shaded part) we provide emission estimates also for countries or regions affected by this issue, though these have a large posterior uncertainty. Another issue arises from the definition of the flux categories, which do not necessarily follow country borders (see Fig. ). In France, Belgium, and Switzerland, the inversion scales flux categories that overlap multiple countries

Technically, the issue also affects Italy because Corsica is combined with parts of Italy in one flux category. But the a priori emissions from Corsica are so low that the effect on the national emission estimate is negligible.

Although the national emission estimates are given for the full year, a closer examination of the seasonal cycle provides additional insights. Figure  shows the monthly emission rates for the countries considered in Fig. . While the seasonal cycle is strikingly different depending on the region, we find some recurrent features. For Germany, Poland, the Netherlands, and Austria plus Czechia (panel a in Fig. ), the posterior emission rates have their minimum in May. A local minimum between April and June is also found for northern France and Belgium plus Luxembourg, see panel (b). In most countries, this minimum is followed by a local maximum in July or August, which is most prominent in the Netherlands and Austria plus Czechia (panel a).

The differences between the regions become larger in autumn and winter. In September, posterior emission rates reach their maximum in Germany and Italy, and their minimum in (northern) France. France and Belgium plus Luxembourg have their highest emission rates in winter, when Switzerland and Spain plus Portugal have their minimum. For some regions – most notably Italy and the United Kingdom plus Ireland – no clear pattern is found in the seasonal cycle for 2021 (panel c in Fig. ).

The seasonal cycle in the inversion results may be partially influenced by the observation coverage because many stations lack data covering the whole year. To avoid this effect, we repeated the inversion using only stations which provide data for at least 20 days of each month. The seasonal cycle in these results does not change significantly, see Fig. . We further note that there is a seasonal cycle in the observations , which is captured well by the far field in the model though (see Fig. ). This “far field” is defined as CH₄ transported into our domain from the lateral boundaries. A possible bias in the lateral boundary conditions could influence the seasonal cycle in the estimated fluxes. Moreover, the different meteorology in summer and winter – especially influencing the planetary boundary layer and vertical mixing – can lead to a seasonal bias in our transport model . This highlights the need for careful interpretation of the seasonal cycle, as meteorological differences could introduce biases that mask true emission patterns. Another potential contribution to the seasonal cycle could arise from neglecting the OH sink of CH4 in our limited domain .

A straightforward validation of the inversion results is possible using independent validation stations. Having excluded each station once in separate inversion runs, we can use every station as an independent validation site in the respective inversion run. Figure  shows histograms of the root mean square error (RMSE) statistics obtained from the model–data mismatch before and after the inversion. The validation stations agree on average significantly better with observations when using a posteriori emissions compared to the a priori. A comparison of the same histograms for the different methods of estimating uncertainties introduced in Part 1 shows no significant differences (see Fig. ).

In this section, we complement the uncertainty estimates of our inversion results by separate measures for the sensitivity of the posterior to true emissions. The potential for detecting emissions from different sources can be identified using the posterior error covariance matrix Bpost. However, the real error reduction is also influenced by the far-field correction and the filtering of observations as detailed in Part 1 . These aspects are not fully captured in Bpost. We therefore use experiments with a “synthetic”, i.e., defined truth and pseudo-observations to test the full inversion system.

Firstly, we find that our top-down CH4 emission estimates are significantly higher than reported for Germany. Secondly, we identify the agriculture sector and possibly LULUCF and natural fluxes as the likely main source of this discrepancy. Thirdly, we recall from Part 1 that the transport error simulated in the meteorological ensemble leads to an uncertainty of 2% on the total German CH4 emissions.

Our Eulerian approach with sectoral segregation differs from other studies on CH4 inversions for single countries, e.g., for Switzerland and for the United Kingdom that use Lagrangian transport models. The latter both qualitatively attribute deviations from the inventory reporting to the agriculture sector by comparing the spatial and/or temporal patterns in the posterior fluxes to sectoral a priori fluxes. A similar strategy for sectoral segregation based on a known spatial distribution of fluxes is followed by and analyzed by . For deriving sector estimates, some inversions assume a spatial correlation of gridded emissions within each sector . Based on the same assumption, and construct ensembles of perturbed a priori fluxes to distinguish natural and anthropogenic fluxes utilizing the CarbonTracker Data Assimilation Shell . Notably, avoid the lateral boundary problem by simulating transport globally with nested zoom in Europe to estimate Finnish CH4 emissions on a coarse resolution of 1° × 1°. In the present work, we take the next step by validating sectoral emissions reported to UNFCCC and analyzing possible false attributions, making use of a significantly higher model resolution.

Our results are qualitatively in line with the discrepancy of top-down estimates and UNFCCC reporting for Germany and the Benelux found in different regional inversions for the years 2018 and earlier . Furthermore, it appears as a robust feature in our results that emissions from the UK plus Ireland agree well with reported emissions, in line with for the year 2018. For the French emissions, our inversion shows a tendency towards slightly higher emissions similar to , whereas other inversions suggest significantly higher emissions .

Although we simulate emissions and transport in a large domain, we can only provide reliable emission estimates for selected countries (compare Fig. ). Regions without notable uncertainty reduction and regions with known modeling difficulties do not benefit from our model setup. In Scandinavia, we find strong wetland emissions with insufficiently modeled fine-scale spatial and temporal variability. Combined with only small signals from non-LULUCF anthropogenic emissions, this leads to a low signal-to-noise ratio, which prevents conclusive results for Scandinavia. Furthermore, the synthesis inversion may be prone to underestimating large localized sources due to transport errors – an issue we address in Part 1 .

Another limitation comes from the challenges for the regional flux inversion caused by biases in the lateral boundary conditions. The uncertainty in lateral boundary concentrations motivates the far-field correction that is discussed in Part 1 . We expect that the far-field correction leads to more robust estimates for well-observed emissions, but it may also cause a bias towards the prior and towards lower emission estimates.

In our highly resolved transport simulation, every flux category is numerically expensive. Aiming to validate reported German emissions, we could reduce the state space of the inversion to only 46 scaling factors with monthly time resolution. This substantially limits the spatial and temporal variations that can be represented in the inversion. This approach is justified if the a priori fluxes already provide a realistic spatial distribution of all major CH4 sources within each flux category. While this may be the case in Germany and neighboring countries, the constant scaling factors for large flux categories in more distant regions may be oversimplified and could lead to less accurate results in these regions. Moreover, adjusting only a few degrees of freedom may not be sufficient to obtain realistic flux estimates in regions with limited or highly uncertain information on a priori fluxes, such as Scandinavia.

When constructing the state space, we unevenly distributed the 46 degrees of freedom on our model domain – using 11 degrees of freedom for Germany and only four for mainland France plus Belgium and Luxembourg. But the choice of flux categories affects the results and can lead to biases depending on the location of the observations . In our application, this effect is small because of the good observation coverage in Germany. This is checked in Part 1 using sensitivity tests.

We exploit the sectoral discrimination of emission in a well-observed region as a key feature of our inversion method. This relies heavily on an accurate spatial distribution and completeness of the a priori fluxes, which appears to be sufficient for the major emitting sectors in Germany. Furthermore, the sector discrimination relies on resolving comparably small spatial scales, which poses a challenge to the transport modeling. A general problem in sector attribution is that sectors with large absolute uncertainty – such as agriculture – may be falsely blamed for any change in total emissions when the observations do not clearly distinguish the sectors (see Appendix ). By quantifying this effect in the averaging kernels (see Fig. ), we confirmed that in Germany agriculture can be distinguished from other anthropogenic emissions excluding LULUCF. Small sectors like natural plus LULUCF fluxes could not be reliably distinguished from large sectors such as agriculture, and we therefore combined smaller sectors like waste and public power into the larger category “non-agr.”.

We chose the synthesis inversion for the first application of our modular inversion system, but designed this framework to be expandable to other inversion methods. For instance, most of the steps in the inversion can be applied with only minor adjustments when replacing the flux categories by an ensemble of randomly perturbed surface fluxes, similar to , or by grid cell clusters as used by . Such applications with a larger state space are limited by the computational effort of the transport simulation, which is much higher than the computational effort of the inversion itself. Similar to the inversion method, the far-field correction can be replaced by a different strategy for mitigating a boundary bias. For example, one could construct the far field based on an ensemble of boundary concentrations.

Further possibilities of extension involve other observation types, including satellite data. Our Eulerian system allows in principle the handling of large observation datasets without prohibitive computational effort, albeit changes in the construction and handling of R may be required when reaching ≳105 observations per time window. This potential is leveraged by many inversion systems that use Eulerian transport simulations e.g.,. The increasing availability of satellite data is especially interesting for constraining concentrations and emissions in regions with few or no ground-based observations, such as near the boundaries of our domain, which is an aspect to be addressed in future studies.

We identified potentials and risks in separating sectors based on the spatially highly resolved distribution of fluxes. Extending this by temporal profiles for a priori fluxes offers a yet untapped potential for future improvement of our system. Moreover, our inversion could benefit from an a priori emission ensemble reflecting the uncertainty in the spatial and temporal distribution of the fluxes. It remains to be explored whether improvements in distinguishing sectors can be achieved in our system using co-tracers such as ethane for fossil CH4 emissions or by distinguishing carbon isotopes .