The iterative ensemble Kalman filter (IEnKF) in a deterministic framework was introduced in Sakov et al. (2012) to extend the ensemble Kalman filter (EnKF) and improve its performance in mildly up to strongly nonlinear cases. However, the IEnKF assumes that the model is perfect. This assumption simplified the update of the system at a time different from the observation time, which made it natural to apply the IEnKF for smoothing. In this study, we generalise the IEnKF to the case of imperfect model with additive model error.The new method called IEnKF-Q conducts a Gauss-Newton minimisation in ensemble space. It combines the propagated analysed ensemble anomalies from the previous cycle and model noise ensemble anomalies into a single ensemble of anomalies, and by doing so takes an algebraic form similar to that of the IEnKF. The performance of the IEnKF-Q is tested in a number of experiments with the Lorenz-96 model, which show that the method consistently outperforms both the EnKF and the IEnKF naively modified to accommodate additive model noise.

Inverse modelling of the emissions of atmospheric species and pollutants has significantly progressed over the past 15 years. However, in spite of seemingly reliable estimates, the retrievals are rarely accompanied by an objective estimate of their uncertainty, except when Gaussian statistics are assumed for the errors, which is often an unrealistic assumption. Here, we assess rigorous techniques meant to compute this uncertainty in the context of the inverse modelling of the time emission rates – the so‐called source term – of a point‐wise atmospheric tracer. Log‐normal statistics are used for the positive source term prior and possibly the observation errors; this precludes simple Gaussian statistics‐based solutions. Firstly, through the so‐called empirical Bayesian approach, parameters of the error statistics – the hyperparameters – are first estimated by maximizing their likelihood via an expectation–maximization algorithm. This enables a robust estimation of a source term. Then, the uncertainties attached to the retrieved source rates and total emission are estimated using four Monte Carlo techniques: (i) an importance sampling based on a Laplace proposal, (ii) a naive randomize‐then‐optimize (RTO) sampling approach, (iii) an unbiased RTO sampling approach, and (iv) a basic Markov chain Monte Carlo (MCMC) simulation. Secondly, these methods are compared to a more thorough hierarchical Bayesian approach, using an MCMC based on a transdimensional representation of the source term to reduce the computational cost. Those methods, and improvements thereof, are applied to the estimation of the atmospheric caesium‐137 source terms from the Chernobyl nuclear power plant accident in April and May 1986 and Fukushima Daiichi nuclear power plant accident in March 2011. This study provides the first consistent and rigorous quantification of the uncertainty of these best estimates.

Abstract. Reliable quantification of the sources and sinks of greenhouse gases, together with trends and uncertainties, is essential to monitoring the progress in mitigating anthropogenic emissions under the Paris Agreement. This study provides a consolidated synthesis of CH4 and N2O emissions with consistently derived state-of-the-art bottom-up (BU) and top-down (TD) data sources for the European Union and UK (EU27 + UK). We integrate recent emission inventory data, ecosystem process-based model results and inverse modeling estimates over the period 1990–2017. BU and TD products are compared with European national greenhouse gas inventories (NGHGIs) reported to the UN climate convention UNFCCC secretariat in 2019. For uncertainties, we used for NGHGIs the standard deviation obtained by varying parameters of inventory calculations, reported by the member states (MSs) following the recommendations of the IPCC Guidelines. For atmospheric inversion models (TD) or other inventory datasets (BU), we defined uncertainties from the spread between different model estimates or model-specific uncertainties when reported. In comparing NGHGIs with other approaches, a key source of bias is the activities included, e.g., anthropogenic versus anthropogenic plus natural fluxes. In inversions, the separation between anthropogenic and natural emissions is sensitive to the geospatial prior distribution of emissions. Over the 2011–2015 period, which is the common denominator of data availability between all sources, the anthropogenic BU approaches are directly comparable, reporting mean emissions of 20.8 Tg CH4 yr−1 (EDGAR v5.0) and 19.0 Tg CH4 yr−1 (GAINS), consistent with the NGHGI estimates of 18.9 ± 1.7 Tg CH4 yr−1. The estimates of TD total inversions give higher emission estimates, as they also include natural emissions. Over the same period regional TD inversions with higher-resolution atmospheric transport models give a mean emission of 28.8 Tg CH4 yr−1. Coarser-resolution global TD inversions are consistent with regional TD inversions, for global inversions with GOSAT satellite data (23.3 Tg CH4 yr−1) and surface network (24.4 Tg CH4 yr−1). The magnitude of natural peatland emissions from the JSBACH–HIMMELI model, natural rivers and lakes emissions, and geological sources together account for the gap between NGHGIs and inversions and account for 5.2 Tg CH4 yr−1. For N2O emissions, over the 2011–2015 period, both BU approaches (EDGAR v5.0 and GAINS) give a mean value of anthropogenic emissions of 0.8 and 0.9 Tg N2O yr−1, respectively, agreeing with the NGHGI data (0.9 ± 0.6 Tg N2O yr−1). Over the same period, the average of the three total TD global and regional inversions was 1.3 ± 0.4 and 1.3 ± 0.1 Tg N2O yr−1, respectively. The TD and BU comparison method defined in this study can be operationalized for future yearly updates for the calculation of CH4 and N2O budgets both at the EU+UK scale and at the national scale. The referenced datasets related to figures are visualized at https://doi.org/10.5281/zenodo.4590875 (Petrescu et al., 2020b).