The inversions in this work use the same method as , which is described in the following. Measurements of CH₄ mole fractions (for surface inversions) or XCH₄ column-averaged mole fractions (for satellite inversions), represented by y, are related to a vector of unknown emissions x and unknown boundary conditions (the background CH₄ mole fraction) xb in the forward model, 1y=Hx+Hbxb+ϵ, where H and Hb are matrices of sensitivities of the measurements to emissions and of the measurements to boundary conditions, respectively. These are produced using the NAME atmospheric transport model (described in Sect. ). Model-measurement errors are represented by ϵ.

A prior modelled estimate of the measurements is obtained by combining the matrix sensitivities with prior CH₄ emissions and prior boundary conditions (described in Sect. ). The inversion solves for monthly scaling factors applied to the prior emissions and boundary conditions, which produce posterior emissions and boundary conditions that best reconcile the observations with uncertainties in the measurements, transport model, and priors. We employ a hierarchical Bayesian inversion framework (Eq. ), following the approach of . 2p(x,xb,θ∣y)∝p(y∣x,xb,θ)p(x∣θ)p(xb∣θ)p(θ),

where p(y∣x,xb,θ) defines the likelihood of the measurements given the emissions and boundary conditions. The prior probability density functions (PDFs) are p(x∣θ) and p(xb∣θ). The term p(θ) specifies the prior distribution for the hyperparameters that govern the model-measurement error. This hierarchical formulation of Bayes’ theorem treats the model-measurement error as an unknown variable as opposed to choosing subjective values. The hyperparameters, θ, represent both the known measurement uncertainty and the unknown model error (combined in quadrature), allowing these uncertainties to be jointly estimated with the emissions and boundary conditions during the inversion.

To avoid sampling large, non-physical negative emissions, we use a truncated normal prior PDF for both the emissions and boundary conditions. This constraint means that net uptake cannot be represented at the grid-cell level, although reductions from the prior can still capture relative decreases in emissions. The PDFs used in the inversions are described in Table .

There is no analytical solution for x and xb, we therefore use a Markov Chain Monte Carlo (MCMC) sampling approach. The emissions and boundary conditions are sampled using a No-U-Turn (NUTS) sampler , while the model-measurement error is sampled using a faster Slice sampler . In each step of MCMC sampling, the PDFs of each parameter are considered by the samplers. After which, the sample is accepted or rejected based on acceptance criteria. The probability is recomputed at each step, allowing the chain to explore the parameter space, moving towards the most likely result while occasionally considering less likely outcomes through a series of acceptances and rejections of the proposed parameter space. We use a 1500 iteration tuning phase to move away from the prior PDFs, then a 3000 iteration sampling phase, discarding the first 20 % of samples as burn-in. To ensure convergence, we apply the Gelman-Rubin diagnostic with a criterion of less than 1.05. The remaining samples, or the “trace”, provide the posterior PDF from which we calculate the mean scaling factor – used to scale the prior fluxes to the posterior fluxes – as well as the associated 95 % uncertainty intervals. For example, for total emissions from the NSA, we calculate posterior fluxes for each MCMC sample by multiplying the prior fluxes by each sampled scaling factor and then derive the mean and 95 % uncertainty intervals (in Tg yr⁻¹) from the resulting distribution.

In , an “Arctic” inversion spatial domain was used that covered the entire 360° of all latitudes greater than 36° N, selected for its ability to model the transport of particles near the poles, where particles can traverse the entire domain within the model simulation. However, this comes at a higher computational expense, making it less feasible for use with high-density TROPOMI data. This constraint is recognised as a limitation of our current study. For the current study, all of the TROPOMI inversions and sensitivity tests use a smaller “Alaska” domain (36 to 78° N and 112° E to 110° W). All surface inversions that are compared to the TROPOMI inversions also use the “Alaska” domain. Both domains are shown in Fig. . The impact of using a different domain on the surface inversion is discussed in Sect. .

In , the boundary conditions in the North Slope of Alaska inversion were fixed during the inversion, as opposed to allowing variable boundary conditions. Instead a wind direction-derived boundary condition was subtracted from the observations prior to the inversion. A sensitivity analysis conducted as part of indicated minimal difference between using fixed and variable boundary conditions. In the present study, we allow variable boundary conditions that we estimate throughout the inversions for both surface and satellite inversions to ensure consistency in the method.

We aggregate grid cells and solve for fewer unknowns in the inversion than the output grid resolution of the transport model. The grid cells are more densely concentrated in areas of interest and more sparsely distributed further from the region of interest, resulting in a total of 297 scaling factors for the emissions for each month. In the satellite inversion, we solve the boundary condition scaling factors across 4 vertical blocks – spanning 5 km intervals from 0 to 20 km – each divided into the North, East, South, and West directions, resulting in 16 scaling factors. We only use one scaling parameter in each direction (4 total) for the surface inversion.

Henceforth, the “main” inversion refers to satellite inversions that use the configuration described in this section, and any changes to this configuration are sensitivity tests (described in Sect. ).

We use high frequency in situ atmospheric mole fraction measurements of CH₄ from the National Oceanic and Atmospheric Administration (NOAA) Barrow station (BRW, 71.3° N, 156.6° W) . We filter the BRW observations for low wind speeds and those coming from the direction of Utigiavik, a nearby town see. In this work, we also include a separate inversion using additional measurements from two measurement sites managed by Environment and Climate Change Canada (ECCC), Inuvik (INU, 68.3° N, 133.5° W) and Estevan Point (ESP, 49.3° N, 126.5° W) , and the Finnish Meteorological Institute (FMI) site Tiksi (TIK, 71.6° N, 128.9° E) . These sites were chosen because they are all within the study domain and have measurements within the study period (2018–2020). The locations of the sites are shown in Fig. . The measurements from these sites are filtered using a local influence filter, which excludes observations whose NAME footprints indicate they are dominated by very local surface influence, following the approach described in . The impact of using multiple sites on the surface inversion is discussed in Sect. .

We use two satellite retrieval products of column-averaged dry air mole fractions of CH₄ (XCH₄): the University of Bremen's WFM-DOAS v1.8 and the SRON operational product v2.04.00 , which we term TROPOMI-WFMD and TROPOMI-OPER, respectively. For both products, the measurement error is taken from the 1σ retrieval uncertainty.

All data have low-quality observations removed using the filters and quality flags recommended by each product, and are re-gridded to 0.234° (latitude) × 0.352° (longitude) spatial resolution (the resolution of the atmospheric transport model, see Sect. ), using a conservative interpolation method, and averaged to an hourly temporal resolution. For both TROPOMI retrievals, we removed all retrievals over the ocean due to a lack of validation data over the ocean globally. We reduced the density of the TROPOMI observations to make it computationally feasible to run our atmospheric transport model. We prioritised the data from our main region of interest (the NSA) and retained fewer observations from areas outside the NSA but within the inversion domain, which helped to constrain boundary conditions. This resampling involved dividing the domain into regions with varying data retention rates and a random sampler. After regridding, we retained all data over the NSA, 50 % from the rest of the state of Alaska, and 5 % from the remaining areas. TROPOMI-WFMD observations and observational density after regridding but before and after resampling can be seen for an example month (June 2019) over the Alaska domain in Fig. . The effects of these data retention strategies on boundary conditions were evaluated using a sensitivity test described in Sect. . After resampling, across the whole domain ∼510000 retrievals remain for TROPOMI-OPER and ∼580000 for TROPOMI-WFMD.

Due to high solar zenith angles and low sunlight, there are no observations for either TROPOMI-OPER or TROPOMI-WFMD from late October to early March. However, temporal coverage of TROPOMI-WFMD extends further into March and October than TROPOMI-OPER. As a result, TROPOMI-WFMD may have more potential to constrain emissions during the cold season. The number of retrievals over the NSA for TROPOMI-OPER and TROPOMI-WFMD (after regridding and resampling) is shown for all years in Fig. S1 in the Supplement.

The time range for our analysis is from May 2018 to October 2020. TROPOMI's fully functioning operating capacity began in late April 2018, so we do not include data from before this date.

To provide the sensitivity of XCH₄ observations to emissions from the surface we used the Met Office's Numerical Atmospheric-dispersion Modelling Environment (NAME) , a Lagrangian particle dispersion model (LPDM). We run NAME in backward mode, releasing model particles for each TROPOMI observation, which are tracked backward in time over a 30 d period. We record the sensitivity of the surface emissions (0–40 m above ground level) and the boundary conditions to the measurements. The NAME output has a resolution of 0.234° (latitude) by 0.352° (longitude). We use the “Alaska” domain shown in Fig. .

Meteorological inputs to NAME are provided by the Met Office's Unified Model , at a spatial resolution of 0.141° (latitude) × 0.094° (longitude) and 3 h temporal resolution. For each satellite product, we release particles uniformly across different heights defined by the levels within each retrieval. We ran NAME with a maximum altitude of 20 km, which corresponds to maximum level of 11 of 12 for TROPOMI-OPER and 19 of 20 for TROPOMI-WFMD. For the top levels that are above the maximum level of NAME, the CH₄ concentration was assumed to be equal to the corresponding value in the a priori profile from the retrieval. The average footprint sensitivity for TROPOMI (after resampling) are shown in Fig.  and for the BRW-only inversion and the inversion using multiple surface sites in Fig. S2.

To compare modelled mole fractions to satellite observations, we utilise the methodology derived in and described in . We convert our modelled CH₄ for each retrieval into a column mole fraction (XCH4model) for each time index, t, as follows, 3XCH4model|t=∑1npiAi⋅CH4,imodel+(1-Ai)⋅CH4,iprior, where pi is the pressure weight at each retrieval level i from the surface to the top of the atmosphere, Ai is the averaging kernel, CH4,imodel is the modelled CH₄ mole fraction and CH4,iprior is the a priori profile. TROPOMI-WFMD retrievals provide all the variables needed for Eq. (), but for TROPOMI-OPER we calculate the pressure weights as pi=Vair,i∑inVair,i, where Vair is the dry air sub-column.

We must adapt Eq. () such that we can calculate XCH4model in the form of the forward model, Eq. (), defined in Sect. . Our forward modelled mole fraction at each level, i, (CH4,imodel from Eq. ()) is due to emissions from the surface within our model domain, and incoming mole fractions from the boundaries, 4CH4,imodel|t=hi⋅q+hb,i⋅b, where hi is our sensitivity to the surface derived from NAME, q is our emissions vector, hb,i is our sensitivity to the boundary conditions and b is our boundary conditions vector.

As the inversion requires modelled mole fractions that are only dependent on NAME sensitivities, we construct a perturbed modelled column mole fraction, XCH4,pertmodel|t. This represents the averaging-kernel-weighted NAME contribution over the column, from the surface to the maximum level. We obtain XCH4,pertmodel|t by subtracting all the retrieval-provided prior contributions, as follows, CH4,iprior, Ai and pi are provided from the retrieval product, so the second term of Eq. () can be subtracted from XCH4model|t. Additionally, because NAME is only run up to 20km altitude, we assume the prior profile above this level and we can also subtract this known contribution from XCH4model|t. 5XCH4,pertmodel|t=XCHmodel4|t-∑1maxlevpi(1-Ai)⋅CH4,iprior-∑maxlev+1npi⋅CH4,iprior, where maxlev is the maximum level of each retrieval for which the NAME model is run, as described above. A full derivation for Eq. () is found in Sect. S1 in the Supplement. Substituting Eq. () into Eq. () gives 6XCH4,pertmodel|t=∑1maxlevpi⋅Ai⋅(hi⋅q+hb,i⋅b). Over all times, t, Eq. () can be written in the form, 7XCH4,pertmodel=Hq+Hbb. Here H and Hb are matrices of rows t, where each column is the sum of pi⋅Ai⋅hi and pi⋅Ai⋅hb,i, respectively. Equation () is now consistent with Eq. (). In practice, because q and b are not dependent on the level, they are stacked into the vector x such that, 8XCH4,pertmodel=Hx. Where each corresponding row of H is a weighted sum over all levels, allowing satellite observations to be used in the same inversion framework as surface measurements.

A priori vertical boundary conditions are derived from the CAMS global inversion-optimised greenhouse gas fluxes and concentrations v19r1 product . This product uses surface observations in an inversion to produce global atmospheric concentrations. As the product runs only until 2019, we assume that prior boundary conditions for 2020 are the same as in 2019 (discussed in Sect. ).

Our a priori emissions field comprises emissions from three major sources over the Arctic: natural emissions from tundra, anthropogenic activities, and fires. For the main inversion, we utilise the “Late-season Zona” prior, detailed in and summarised in the following. For emissions from tundra, we employ a seasonal profile derived from the findings in , complemented by spatial data from the July values of the fractional wetland satellite product SWAMPs (Surface Water Microwave Product Series) . This approach is designed to allow emissions from areas of less inundated or frozen ground, as documented in . Additionally, we incorporate anthropogenic emissions data from the Emissions Database for Global Atmospheric Research (EDGAR) v6 and fire emissions data from the Global Fire Emissions Database (GFED) v4 . The a priori emissions are shown in Fig. S3. The majority of emissions over the NSA are from the tundra sector, with significant emissions from the anthropogenic sector over the Prudhoe Bay oil fields on the Northeast coast. Finally, there are small emissions from fires in the South of the defined region. We do not include prior emissions from two other important sources in the NHLs: ocean and freshwater (lakes and rivers) emissions. In , ocean emissions from were included as a sensitivity test, and their inclusion had no significant impact on the posterior emissions. For freshwater emissions, there is substantial spatial overlap with tundra emissions over the NSA, which can lead to double counting if both sources are included separately . Including freshwater emissions in this region in the priors may therefore result in unrealistically high total prior emissions. The posterior tundra emissions can be interpreted as encompassing both tundra and freshwater methane sources.

To apportion the posterior emissions estimated in each grid cell into one of the three major source sectors, the fraction of each source in the prior emissions field was used. The distributions of each of these sectors are largely separated spatially as shown in Fig. S4.

Sensitivity tests were conducted to assess the robustness of the derived emissions to changes in the inversion setup. These tests included changing the prior used in the inversion to the “Uniform” prior from , which uniformly distributes natural emissions from wetlands and tundra across all months, using the same spatial map as the “Late-season Zona” prior but setting the emission values to the July level for every month. We tested two different grid cell aggregation formats of 104 and 421 cells (shown in Fig. S5). An additional test was carried out using a single scaling parameter for each boundary direction, such that only one block for each of the North, East, South, and West directions is solved for, in contrast to the main inversion which uses four blocks in each direction. Finally, to assess the impact of the spatial coverage of observations across the study domain, particularly on the boundary conditions, we removed TROPOMI observations from outside the state of Alaska. A summary of each sensitivity tests is shown in Table .

We additionally carried out an inversion using measurements from both TROPOMI and BRW. Combined TROPOMI and surface measurement inversions are referred to as TROPOMI-OPER+BRW or TROPOMI-WFMD+BRW throughout the text. We include an extra offset parameter, with a uniform PDF with a range of 0–10 ppb applied to the BRW measurements, as a method to account for systematic differences between the TROPOMI and BRW observations and in modelling each dataset by the NAME model.

Here, satellite inversions are compared to a surface inversion using in situ measurements from BRW. We ran satellite and surface-measurement-based inversions over the same Alaska domain to compare them to the surface-measurement-based study in , which used the more spatially extensive Arctic domain.

Figure A shows that, in the NSA, the impact of the size of the inversion domains on estimated tundra emissions varies year to year. While emissions for many months are consistent within uncertainties, inversions using the Arctic domain result in higher tundra emissions during the summer and late seasons of 2018, and the late seasons of 2019 and 2020. This suggests that domain size does impact the estimated tundra emissions in this region, potentially due to atmospheric circulation around the poles; however, the general findings using each domain are aligned. We observe significantly higher tundra emissions in our posterior results compared to our prior, including consistent high late-season emissions, a key finding in . Additionally, the average seasonal emission profiles for both domains remain the same within uncertainties, as shown in Fig. B. While we are satisfied with the similarities between the emissions derived for the NSA using either the Alaska and Arctic domains, further investigation is necessary to fully understand the implications of model domain on emission estimates from the Arctic.

A surface-based inversion was conducted using CH₄ measurements from BRW and three additional sites – INU, TIK, and ESP – within the Alaska domain, providing additional constraints outside of the NSA region on derived emissions. For 2020, the “all-surface” derived emissions agrees well with BRW-only (this study) derived emissions, as shown in Fig. A. However, for the late season months of 2018 and most of 2019, the surface-measurement-derived emissions are larger and closer to the BRW-only (Ward et al., 2024) derived emissions, which use the larger domain. The seasonal emission profile from the surface-measurement-based inversion, shown in Fig. B, remains consistent with the BRW-only (this study) inversion profile.

For the remainder of this study the BRW-only (this study) inversion using the Alaska domain is referred to as the BRW inversion.

To assess the robustness of the satellite products to varying inversion inputs and prior choices, we carried out four sensitivity tests, as detailed in Table , and observed the impact on the derived tundra and anthropogenic emissions. The results of each sensitivity test are described in the following sections, with the results in Sect.  referred to as the “Main” inversion. The monthly total emissions for all tests are shown in Fig. . We consider the monthly emissions estimate unchanged if the 95 % uncertainty intervals of the derived emissions from the sensitivity test and the main inversion overlap. March–September 2019 and 2020 total budgets are shown in Table S2.

To validate the TROPOMI-derived emissions, we forward modelled the TROPOMI posterior emissions to generate modelled mole fractions at BRW, as BRW observations were not assimilated in these inversions. The a priori emissions and boundary conditions, x and xb, from the forward model in Eq. () were scaled by the mean posterior emissions scaling factors and posterior boundary condition scaling factors, derived in the TROPOMI inversions. We compared these scaled forward model estimates to the prior forward model outputs.

Our results indicate that the TROPOMI scaled forward model provides a poorer fit to the BRW observations than the unscaled prior forward model, as shown for 2019 in Fig. . For the TROPOMI scaled forward model, the baseline is consistently negatively offset for TROPOMI-WFMD and negatively offset in the summer months and positively offset in the early season months for TROPOMI-OPER.

Additionally, forward models incorporating two of the sensitivity tests from Sect.  are shown in Fig. S14 for August 2019: one vertical block boundary conditions, and Alaska observations only. The one vertical block test increases the baseline closer to the observations for TROPOMI-WFMD. However, for TROPOMI-OPER the baseline decreases further from the observations. Limiting the measurement dataset to observations over Alaska increased the baseline for TROPOMI-OPER. For TROPOMI-WFMD, limiting measurements to Alaska only produced an anomalous trough in the baseline around 15 August 2019, which requires further investigation.

We conducted combined TROPOMI and surface measurement inversions, incorporating an additional offset parameter to our hierarchical model to account for any potential monthly biases between modelled satellite and surface mole fractions due to calibration differences, as detailed in Sect. .

For August 2019, the mean inferred offset was 37 ppb between BRW and TROPOMI-WFMD, and 6 ppb between BRW and TROPOMI-OPER. These values closely match the differences between the TROPOMI-scaled forward model and BRW observations shown in Fig. S14a for the same month, indicating that the offset parameter effectively corrects for this monthly bias.

For the TROPOMI-OPER+BRW inversion, we observed the same derived tundra emissions to the TROPOMI-OPER inversion within 95 % uncertainty intervals for most months. However, in September 2019, the TROPOMI-OPER+BRW derived tundra emissions increased compared to the BRW inversion and TROPOMI-OPER inversion outside of the 95 % uncertainty intervals (Fig. A). The TROPOMI-WFMD+BRW derived tundra emissions were greater than TROPOMI-WFMD derived tundra emissions, and closer to BRW derived tundra emissions for the summer and late season months of 2018, the late season months of 2019 and August 2020 (Fig. B). Figure S6 shows the average seasonal difference between the posterior and prior emissions from the tundra region for the TROPOMI-OPER+BRW. which are spatially similar to those from the TROPOMI-OPER inversion. For the TROPOMI-WFMD+BRW inversion, emissions from the tundra region are larger than the prior, whereas emissions from the TROPOMI-WFMD inversion are smaller across the tundra region.

The anthropogenic tundra emissions are shown in Fig. S15. In May 2018, TROPOMI-OPER had exceptionally high emissions. However, in the TROPOMI-OPER+BRW the emissions are substantially lower and align with the prior emissions. Otherwise, all other months remain consistent with the TROPOMI-OPER only anthropogenic emissions, apart from June 2020 where the emissions decrease from the prior. For the TROPOMI-WFMD+BRW inversion, the derived anthropogenic emissions do not change from the TROPOMI-WFMD only inversion apart from May 2018, March 2020 and August 2020.

The root mean square error (RMSE) between the modelled and measured CH₄ mole fractions is used to assess the fit of the model to the observations. Table  shows the RMSEs for TROPOMI observations, and Table  for BRW observations, for each inversion in this study. Both prior (before inference) and posterior (after inference) RMSEs are reported. A lowering of the RMSE from the prior to the posterior for the same observations indicates a better fit, though it does not account for potential overfitting. Across all inversions, the RMSE decreases from prior to posterior for both TROPOMI and BRW, indicating improved agreement with the observations. TROPOMI-WFMD has consistently lower posterior RMSEs than TROPOMI-OPER, but also consistently higher prior RMSEs.

The RMSE ranges quantify the variability among the inversion experiments. For TROPOMI-OPER, the prior RMSE spans 23.7 to 33.7 ppb (a 10 ppb spread), which is larger than the corresponding TROPOMI-WFMD prior spread of 7.1 ppb (26.1 to 33.2 ppb). This indicates that TROPOMI-OPER is more sensitive to prior assumptions and inversion set-up. After assimilation, however, both products converge to similarly narrow posterior ranges. The OPER spread contracts to 2 ppb (16.1 to 18.1 ppb), an 80 % reduction, and TROPOMI-WFMD to 2.6 ppb (14.5 to 17.1 ppb), a 63 % reduction.

Changes in RMSE across the sensitivity tests reflect differences in the number of adjustable parameters and observations. For instance, the Alaska-only inversion has fewer observations, resulting in lower RMSEs. The 421 grid cells aggregation results in lower RMSEs than the main inversion due to increased spatial flexibility, allowing more grid cells to be optimised. In contrast, the single vertical block test reduces the degrees of freedom for adjusting boundary conditions, leading to a higher RMSE.

For BRW, the BRW-only Alaska domain inversion has the lowest posterior RMSE. The larger Arctic domain inversion, and the multiple sites and combined TROPOMI inversion, increases the RMSE.

In our inversion method, the TROPOMI model-measurement error is a quadrature combination of the measurement error taken from the retrieval and an estimated model error (see Sect. ). Figures S16–S18 show the monthly average measurement error, inferred model error (mean of all MCMC samples), and combined model-measurement error for BRW, TROPOMI-OPER, and TROPOMI-WFMD, respectively.

Over 2018–2020, the retrieval-based measurement error is much lower for TROPOMI-OPER (3 ppb; 1–6 ppb) than for TROPOMI-WFMD (15 ppb; 10—19 ppb); the reported values are the mean error over the full study period, with the numbers in parentheses representing the mean lower and upper bounds of the monthly 95 % uncertainty intervals. The inferred model error is higher for TROPOMI-OPER (16 ppb; 13–21 ppb) than for TROPOMI-WFMD (5 ppb; 2–10 ppb). These differences lead to similar combined model-measurement errors for both products: 17 ppb (14–20 ppb) for TROPOMI-OPER and 16 ppb (11–21 ppb) for TROPOMI-WFMD. The combined model-measurement error for both TROPOMI products varies seasonally, with higher uncertainties during the early and late seasons compared to summer.

For the BRW-only inversions, the average model-measurement error is 13 ppb (5–25 ppb). When BRW is combined with TROPOMI, the model-measurement error increases to 18 ppb (9–32 ppb) in the TROPOMI-WFMD+BRW inversion and to 15 ppb (7–28 ppb) in the TROPOMI-OPER+BRW inversion. Across all BRW inversions, the model error component dominates the model-measurement error, and seasonal variation is less pronounced than using TROPOMI.

TROPOMI-derived emissions can be compared with emissions derived from surface inversions to assess whether different inversion approaches yield consistent results. Here, TROPOMI-derived emissions are considered consistent with surface inversion if they align with the BRW-derived emissions within 95 % uncertainty intervals. However, as discussed in Sect. 2, the surface inversions are themselves sensitive to some aspects of the inversion set-up and inputs. Despite the limitations, in situ measurement-based inversions are a well-established method for emissions quantification in the Arctic and are valuable for comparison with TROPOMI inversions.

The main inversion in Sect.  shows that the TROPOMI-OPER tundra emissions are consistent with BRW-derived emissions for multiple months and exhibit similar spatial emission patterns. In contrast, TROPOMI-WFMD exhibits different spatial patterns and shows less consistency with BRW-derived emissions. While TROPOMI-OPER demonstrates greater consistency during the early season and summer, its coverage ends in September, missing the potentially high-emitting late-season period . In the combined inversions the TROPOMI-WFMD product shows improved agreement with BRW derived tundra emissions when combined with the BRW data, as shown in Sect. . For TROPOMI-OPER, combining the BRW data results in minimal changes to derived tundra emissions, though spatial emissions patterns more closely resemble those from the BRW only inversion. For both TROPOMI products, forward modelling (Sect. ) indicates that combined inversions require an offset to account for bias between satellite and surface data. This is effectively handled using an offset parameter in Sect. .

Average NAME footprint sensitivities (Fig. A) could potentially be used as an indicator of how strongly posterior emissions are constrained, with weaker footprints expected to produce fluxes closer to the prior. However, our results show that this relationship does not hold consistently for individual years. For example, in 2019 the TROPOMI-WFMD footprints are lower than those of TROPOMI-OPER, yet the TROPOMI-WFMD tundra emissions are least constrained to the prior in the year (Fig. ). In contrast, in 2018 and 2020 the TROPOMI-WFMD tundra emissions remain closer to the prior, even though the footprint magnitudes between the two products are more similar. However, the most pronounced spatial difference in sensitivity is that TROPOMI-OPER exhibits higher footprint values over southern NSA regions, which may allow emissions from these areas to be more strongly constrained. One explanation for the difference in the footprints between the two TROPOMI products is the number of observations remaining after the retrieval and QA filtering. The footprint sensitivities alone cannot explain the inversion result, which also depends on many factors including the retrieved column values, uncertainties, potential biases in the data, and the treatment of different vertical levels in NAME (as discussed in Sect. ).

For the anthropogenic sector, the main inversion (Sect. ) indicates that TROPOMI-WFMD derived emissions are more consistent with BRW derived emissions than TROPOMI-OPER, reversing the pattern observed for tundra emissions. Anthropogenic emissions in the NSA are dominated by fugitive releases at the Prudhoe Bay oil fields, where methane is emitted as discrete point sources rather than the widespread, diffuse emissions from tundra. The BRW station is situated approximately 300 km north-west from Prudhoe Bay, and measurements are less sensitive to emissions from this source compared to the tundra south of the station. In contrast, TROPOMI directly samples over Prudhoe Bay, offering stronger constraints when clear-sky observations are available. This is confirmed by the combined inversions, where adding BRW data to the TROPOMI-only inversions caused minimal changes to anthropogenic emissions, indicating TROPOMI provides the dominant constraint. However, despite more coverage over Prudhoe Bay oil fields, TROPOMI-WFMD and TROPOMI-OPER derived different anthropogenic emissions many months, indicating retrieval-driven differences.

The intra-annual variability of anthropogenic emissions remains uncertain. The prior (EDGAR v6) assumes no monthly variation, yet all inversions in this study show some degree of monthly variability, suggesting dynamics not captured by the inventory. identified a winter peak in oil and gas methane emissions in the contiguous U.S., likely linked to higher extraction and transmission rates during the heating season. State-level activity data from the U.S. Energy Information Administration show similar trends in Alaska, including a notable spike in May 2018 – consistent with high emissions inferred by TROPOMI-OPER . TROPOMI-WFMD aligns most closely with overall seasonal activity patterns, while TROPOMI-OPER and BRW show elevated emissions during June–August, although not consistently in the same months. To further study these differences between the TROPOMI-OPER and BRW, we examined the BRW mole fractions and corresponding NAME footprints where the posterior emissions diverge. In August 2019 and 2020, where TROPOMI-OPER infers low anthropogenic emissions and BRW infers higher emissions, the posterior BRW mole fractions reproduce the observations well. However, in June 2019 and 2020, when TROPOMI-OPER infers higher anthropogenic emissions, the posterior modelled observations from BRW show a poorer fit. In some cases, this can be explained by NAME footprints exhibiting limited sensitivity over land, however, there are periods where the footprint patterns do not explain the mismatch, indicating additional factors may contribute. These diagnostics suggest that TROPOMI-OPER maybe be more sensitive to anthropogenic emissions than BRW in during some summer months. reported a small positive bias in summer and a negative bias in autumn for TROPOMI-OPER relative to TCCON at other northern high-latitude stations, which could contribute to the lower anthropogenic emissions inferred by TROPOMI-OPER in August relative to June. However, without independent validation data over the North Slope of Alaska, it is not possible to determine whether the observed differences reflect genuine emission variability or retrieval artefacts.

Finally, updates in EDGAR 2024 suggest a ∼ 60 % reduction in emissions totals over Alaska compared to EDGAR v6. If more accurate, it implies that the prior total used here may overstate actual emissions total, while the inferred seasonal variability remains unclear.

To assess the robustness of TROPOMI-based inversions, sensitivity tests varying the inversion setup were conducted (Sect. ). Derived emissions are considered robust if they do not change when the inversion set up changes, meaning that the results are stable and reliable across different modelling configurations. The stability would indicate that the inversion method and TROPOMI data can consistently produce accurate emissions estimates despite variations in the setup, thereby enhancing the confidence in the derived emissions. TROPOMI derived tundra emissions showed significant changes with alterations to the prior seasonal cycle. Both the early season (March–May) and late season (September–October) periods were especially affected when the prior emissions were given a uniform seasonal profile with unrealistically high emissions during the cold season (see Table ). This highlights the importance of using a well-informed prior, especially in the cold season, consistent with findings by . However, the BRW inversion was much less sensitive to the uniform prior and still produced plausible posterior emissions despite the unrealistic emissions in the cold season. Overall, the strong dependence of TROPOMI-derived tundra emissions on the prior suggests a lack of robustness, a concern that could extend to other high-latitude Arctic regions. Tests involving grid cell aggregations, boundary condition vertical blocks, and Alaska observations showed smaller or insignificant changes in derived tundra emissions. However, the grid cell aggregations sensitivity test showed spatial differences, with TROPOMI-OPER showing more spatial variation than TROPOMI-WFMD. The RMSE spread across the sensitivity tests shows that TROPOMI-OPER has higher prior sensitivity than TROPOMI-WFMD, but after the inversion, both products converge to similarly narrow posterior RMSE ranges, indicating that the inversion reduces experiment-to-experiment variability. For anthropogenic emissions, all sensitivity tests resulted in changes in TROPOMI-derived emissions for very few months. Unlike for tundra emissions, no sensitivity test directly altered the prior anthropogenic emissions, meaning their sensitivity to prior inputs was not assessed, and this should be included in future work.

An important consideration is how observations from outside the NSA influence derived emissions within it. For the surface inversions, excluding sites outside Alaska (i.e. using only BRW on the same domain) resulted in lower tundra emissions compared to the inversion using more sites. In contrast, for the TROPOMI inversions, restricting observations to Alaska alone, rather than the whole inversion domain, led to increased tundra emissions in some summer months, accompanied by a change in the optimised boundary conditions relative to the prior boundary conditions. Although these setups are not directly comparable, these contrasting results suggest that information from outside the region of interest plays different roles in the surface and satellite inversions.

Data providers have corrected for bias in both TROPOMI products used in this study to improve accuracy at high latitudes, and the retrievals have been validated using XCH₄ measurements from TCCON sites . However, the geographic locations of TCCON, COCCON and AirCore sites relative to the NSA pose significant limitations for bias correction. The nearest TCCON site, Eureka in Nunavut, Canada, is about 2000 km away and the nearest AirCore measurements are in Europe. The closest ground-based column XCH₄ measurements come from a COCCON station in Fairbanks, Alaska, located approximately 300 km south of the NSA, and COCCON stations are not currently used for the correction or validation of either of the TROPOMI products used in this study. As notes, anomalies have been detected in the TROPOMI-OPER product that are not near to TCCON stations, which is likely to affect the TROPOMI-WFMD product. Consequently, despite the presence of a few ground-based measurement stations in the Arctic, potential improvements or anomalies over the NSA, as well as other high-latitude regions such as the East Siberian Lowlands and the Taymyr Peninsula , could be overlooked.

Compared to previous estimates of emissions in the Arctic in August 2018 which typically has high emissions , a pronounced trough is observed in TROPOMI-OPER tundra emissions over the NSA during August 2018. August is a month of low data coverage over the NSA (Fig. S1), and this is likely due to persistent high cloud cover. Both TROPOMI-OPER and TROPOMI-WFMD implement cloud filtering using data from the Visible Infrared Imaging Radiometer Suite (VIIRS) onboard the Suomi-NPP satellite , to ensure high-quality retrievals under clear-sky conditions. During the 2018–2020 study period, August 2018 experienced the highest cloud coverage, with clouds present during 88 % of daylight hours (Fig. S19). The sparse data in August 2018 likely results in the inversion adhering closely to the prior in this month, producing lower emissions estimates than in surrounding months due to insufficient observational constraint. This underlines the need to continue and expand in situ measurements in important but persistently cloudy methane emitting regions. Future satellite missions, such as the Franco-German Methane Remote Sensing LIDAR Mission (MERLIN) , planned to launch in 2028, will potentially have higher data density in cloudy scenes, as the high resolution LIDAR beam will reach the surface through gaps in optically thick clouds.

Despite lower reported measurement errors and prior RMSE, TROPOMI-OPER consistently results in higher posterior RMSE and larger inferred model error compared to TROPOMI-WFMD. This discrepancy might indicate that TROPOMI-OPER measurement errors are underestimated, leading the inversion to assign more of the model-measurement error to model errors over measurement errors. The greater spatial and temporal variability observed in TROPOMI-OPER scaling factors and monthly emissions estimates suggests it contains finer-scale variations that are more challenging for the inversion to resolve. In contrast, TROPOMI-WFMDs smoother spatial patterns and higher retrieval uncertainties may provide the inversion with more flexibility to fit the observations, resulting in lower posterior RMSE and smaller model error estimates.

The cold season in the Arctic tundra has been shown to be a larger contributor to annual CH₄ emissions than previously thought . Compared to previous satellite products, the increased data coverage of TROPOMI during this period, capturing high-resolution measurements at the start of the late season and end of the early season, potentially enables improved monitoring of this understudied season. This is especially valuable given the practical challenges of deploying additional in situ measurement sites due to harsh cold season conditions and the remoteness of Arctic tundra. The main inversions in this study found that after imposing prior tundra emissions with higher late season emissions derived from , the posterior tundra emissions in September–October increased further from the prior, aligning with these previous studies that find unexpectedly large emissions from the cold season.

However, despite additional data coverage, the tundra emissions derived from the TROPOMI inversions demonstrated a lack of robustness in the early and late seasons. This lack of robustness may stem from several factors, including issues with XCH₄ retrievals during these periods due to higher solar zenith angles, albedo effects from snow cover, and seasonal stratospheric influences, such as the polar vortex. These challenges are reflected in the model errors derived in the inversions. In the TROPOMI inversions, the model errors are consistently higher during the early and late seasons compared to the growing season (Figs. S17 and S18). In contrast, BRW model errors show year-to-year variability when model errors are highest (Fig. S16) – not consistently in the cold season. This indicates that there may be a systematic difference in how well the transport model captures boundary-layer versus upper-atmosphere dynamics.

conducted inversions using previous versions of the TROPOMI-WFMD (v1.2) and TROPOMI-OPER (v01.02.02 and v01.03.01) products for 2018 to assess the utility of TROPOMI products at high latitudes. Unlike our study, which concentrates on the NSA, their research divides the analysis into larger geographical areas circumpolar north of 45° N. Their inversion uses the CTE-CH₄ inverse model, which employs an ensemble Kalman filter and an Eulerian atmospheric transport model, which significantly differs from that used here.

recently applied a different LPDM, FLEXPART, to estimate methane emissions from the West Siberian Lowlands – another key northern high-latitude source region. To reduce computational costs, they developed a retrieval averaging routine that aggregates satellite observations onto a coarser spatial grid, based on the standard deviation of the original observations. This contrasts with our observation filtering approach, which prioritised measurements over our specific region of interest. However, their inversion domain is considerably smaller than ours, meaning that a larger proportion of available observations are relevant to their domain by default. The inversion system used by , FLEXINVERT, is a variational inversion method that uses a conjugate gradient minimisation with the FLEXPART LPDM.

A direct comparison of the derived emissions derived here and those from and is not possible due to differing geographical focusses. However, similar to , we find that TROPOMI-derived tundra (or natural in ) emissions are lower than those from in situ site-based inversions. Notably, our results show larger deviations from the prior than both and , who report posterior emissions more closely aligned with their prior estimates. This discrepancy may reflect differences in the accuracy of the a priori emissions, or the relative strength of prior constraints in the inversion frameworks, which can limit how much the posterior can deviate from the prior.

The treatment of model-measurement errors differs between our inversion, and , yet they all converge on similar total model-measurement uncertainties, with imposing the highest minimum error. fix their transport model error at 15 ppb and impose a 5 ppb minimum retrieval error (total ≥20 ppb). prescribe combined model-measurement errors of 14–20 ppb based on retrieval errors and an estimate of background uncertainty. In contrast, our hierarchical inversion treats the model-measurement error as a hyperparameter that is inferred to reduce subjective assumptions and values fall in the 11–21 ppb range (Sect. ).

In all three studies, it is assumed that the model-measurement errors are uncorrelated. For future TROPOMI inversions, recommended increasing model-measurement uncertainty or accounting for model-measurement error correlation in both space and time. The HBMCMC method in this work assumes that uncertainties are uncorrelated both spatially and temporally. In reality, model-measurement errors in satellite data could be highly spatially correlated. From the transport model side, systematic biases in meteorological input data, such as wind speed and direction, can affect multiple measurements in the same area. Additionally, simplified parameterisation of fine-scale processes like turbulence and convection can lead to spatially correlated errors. For satellite measurements, retrievals in close proximity could be impacted by surface albedo biases, such as from snow cover. Calibration issues, such as instrument calibration drifts and inconsistent calibration corrections, can introduce systematic biases over large areas. Data processing, such as spatial interpolation of XCH₄ values, can also introduce correlations by spreading errors from one location to adjacent areas.

As discussed previously, transport models exhibit seasonal biases and inaccuracies in CH₄ profiles at high latitudes . Most TROPOMI-based inversions use Eulerian transport models, but this work, alongside , employs an LPDM. LPDMs become linearly computationally expensive with increased satellite data due to the rapid growth in the number of particles modelled as resolution increases. In contrast, most Eulerian models use a fixed spatial emissions grid, making them more efficient for large datasets (although some Eulerian models have used nested grids ). However, they also must account for the chemical loss of CH₄ through reactions with OH, introducing uncertainties due to the spatial and temporal variability of different OH fields . Improving the computational cost of producing footprint sensitivities is the subject of ongoing work e.g.,, which could result in the use of TROPOMI data on larger spatial domains – such as full-Arctic model simulations – being feasible using LPDM backward-running simulations.